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Register a new userExploring High Multiplicity Amplitudes: The QM Analogue of the Spontaneously Broken Case
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Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step towards this problem we consider the quantum mechanical equivalent of $1 to n$ scattering amplitudes in a spontaneously broken $phi^4$-theory by extending our previous results on the quartic oscillator with a single minimum to transitions $langle n lvert hat{x} rvert 0 rangle$ in the symmetric double-well potential with quartic coupling $lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $langle n lvert hat{x} rvert 0 rangle sim exp left( F (lambda n) / lambda right)$ in the limit of large $n$ and $lambda n$ fixed. We apply the methods of exact perturbation theory put forward by Serone et al. to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas.
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Calculations of $1to N$ amplitudes in scalar field theories at very high multiplicities exhibit an extremely rapid growth with the number $N$ of final state particles. This either indicates an end of perturbative behaviour, or possibly even a breakdown of the theory itself. It has recently been proposed that in the Standard Model this could even lead to a solution of the hierarchy problem in the form of a Higgsplosion. To shed light on this question we consider the quantum mechanical analogue of the scattering amplitude for $N$ particle production in $phi^4$ scalar quantum field theory, which corresponds to transitions $langle N lvert hat{x} rvert 0 rangle$ in the anharmonic oscillator with quartic coupling $lambda$. We use recursion relations to calculate the $langle N lvert hat{x} rvert 0 rangle$ amplitudes to high order in perturbation theory. Using this we provide evidence that the amplitude can be written as $langle N lvert hat{x} rvert 0 rangle sim exp(F(lambda N)/lambda)$ in the limit of large $N$ and $lambda N$ fixed. We go beyond the leading order and provide a systematic expansion in powers of $1/N$. We then resum the perturbative results and investigate the behaviour of the amplitude in the region where tree-level perturbation theory violates unitarity constraints. The resummed amplitudes are in line with unitarity as well as stronger constraints derived by Bachas. We generalize our result to arbitrary states and powers of local operators $langle N lvert hat{x}^q rvert M rangle$ and confirm that, to exponential accuracy, amplitudes in the large $N$ limit are independent of the explicit form of the local operator, i.e. in our case $q$.
We perform a comprehensive study of on-shell recursion relations for Born amplitudes in spontaneously broken gauge theories and identify the minimal shifts required to construct amplitudes with a given particle content and spin quantum numbers. We show that two-line or three-line shifts are sufficient to construct all amplitudes with five or more particles, apart from amplitudes involving longitudinal vector bosons or scalars, which may require at most five-line shifts. As an application, we revisit selection rules for multi-boson amplitudes using on-shell recursion and little-group transformations.
We explore the possibility that scale symmetry is a quantum symmetry that is broken only spontaneously and apply this idea to the Standard Model (SM). We compute the quantum corrections to the potential of the higgs field ($phi$) in the classically scale invariant version of the SM ($m_phi=0$ at tree level) extended by the dilaton ($sigma$). The tree-level potential of $phi$ and $sigma$, dictated by scale invariance, may contain non-polynomial effective operators, e.g. $phi^6/sigma^2$, $phi^8/sigma^4$, $phi^{10}/sigma^6$, etc. The one-loop scalar potential is scale invariant, since the loop calculations manifestly preserve the scale symmetry, with the DR subtraction scale $mu$ generated spontaneously by the dilaton vev $musimlanglesigmarangle$. The Callan-Symanzik equation of the potential is verified in the presence of the gauge, Yukawa and the non-polynomial operators. The couplings of the non-polynomial operators have non-zero beta functions that we can actually compute from the quantum potential. At the quantum level the higgs mass is protected by spontaneously broken scale symmetry, even though the theory is non-renormalizable. We compare the one-loop potential to its counterpart computed in the traditional DR scheme that breaks scale symmetry explicitly ($mu=$constant) in the presence at the tree level of the non-polynomial operators.
The rational parts of 5-gluon one-loop amplitudes are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We found complete agreement with the previously well-known results of Bern, Dixon and Kosower obtained by using the string theory method. Intermediate results for some combinations of Feynman diagrams are presented in order to show the efficiency of the method and the local cancellation between different contributions.
The rational parts of 6-gluon one-loop amplitudes with scalars circulating in the loop are computed by using the newly developed method for computing the rational parts directly from Feynman integrals. We present the analytic results for the two MHV helicity configurations: $(1^-2^+3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^+6^+)$, and the two NMHV helicity configurations: $(1^-2^-3^+4^-5^+6^+)$ and $(1^-2^+3^-4^+5^-6^+)$. Combined with the previously computed results for the cut-constructible part, our results are the last missing pieces for the complete partial helicity amplitudes of the 6-gluon one-loop QCD amplitude.
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