[Paper Review] Cartan's Supersymmetry and the Decay of a $h^0$ with the mass $m_{h^0}\simeq 11$GeV to $Υ(1S)γ$ and to $Υ(nS)γ(n=1,2,3)$

Sadataka Furui|arXiv (Cornell University)|Feb 25, 2015

Particle physics theoretical and experimental studies8 references3 citations

TL;DR

This paper proposes that the 10.5157 GeV $χ_b(3P)$ state observed at LHCb could be the light Higgs boson $h^0(0^+)$, with its decay to $\Upsilon(nS)\gamma$ ($n=1,2,3$) mediated by Cartan's supersymmetry. Using a trilinear coupling model between quark-antiquark pairs and vector fields, the framework explains the observed small polarizations of $\Upsilon(nS)$ states and predicts a $h^0$ mass of $11.2$ GeV, consistent with experimental data.

ABSTRACT

In the LHCb detector at CERN, decays of $χ_b(3P)$ meson to $Υ(1S)γ$ and $Υ(2S)γ$ are reported at centre-of-mass energy of $\sqrt s=7$ and 8 TeV. Following the success of the assignment of $χ_b(1P) oΥ(1S)γ$ of the mass of $m(χ_b(1P))=9.8923$ GeV and $χ_b(2P) oΥ(1S)γ$ of the mass of $m(χ_b(2P))=10.2547$ GeV, the new state $χ_b(3P)$ of the mass of 10.5157 GeV was assigned, but its $J^{P}$ was not fixed. We study the possibility that this boson is the light Higgs boson $h^0(0^+)$, and study its decay modes to a $b\bar b$ which reduces to an $Υ(mS)$ ($m=1,2$ or 3) and $\bar q q$ which reduces to a $γ$, using the Cartan's supersymmetry. Recent non observation of polarizations of $Υ(nS)$ by the CMS collaboration is consistent with the theory.

Motivation & Objective

• To explain the radiative decays $\chi_b(3P) \to \Upsilon(nS)\gamma$ ($n=1,2,3$) observed at LHCb with a mass of 10.5157 GeV.
• To investigate whether the $\chi_b(3P)$ state can be identified as a light Higgs boson $h^0(0^+)$ with mass $m_{h^0} \simeq 11$ GeV.
• To account for the small observed polarizations of $\Upsilon(nS)$ states ($n=1,2,3$) using Cartan's supersymmetry framework.
• To extend the model of $H^0 \to \ell\bar{\ell}\ell\bar{\ell} \to 2\gamma$ to $h^0 \to q\bar{q}q\bar{q} \to \Upsilon(b\bar{b})\gamma$ via trilinear couplings.
• To clarify the Higgs sector structure by comparing decay modes of $h^0$ and $H^0$ within Cartan's supersymmetry.

Proposed method

• Adopts Cartan's supersymmetry to model trilinear couplings between Higgs fields ($x_i + i x_j$), quark-antiquark pairs ($q\bar{q}$), and vector fields ($X'$), enabling $h^0 \to \Upsilon(nS)\gamma$ decay.
• Uses a complex Higgs field representation $x_i + i x_j$ to generate $b\bar{b}$ pairs that form $\Upsilon(nS)$ states and $q\bar{q}$ pairs that decay to photons via $\ell\bar{\ell}$ intermediate states.
• Applies Cartan's trilinear form to couple $\Upsilon(nS)$ fields (as $\ell\bar{\ell}$) and $q\bar{q}$ fields (as $\ell\bar{\ell}$), with $q\bar{q} \to \gamma$ via lepton pair annihilation.
• Assumes the initial $\chi_b(3P)$ state has $J^P = 0^+$, leading to small $\Upsilon(nS)$ polarizations due to angular momentum conservation in the decay.
• Constructs Feynman-like diagrams (Figures 1–3) showing $h^0$ decaying to $\Upsilon(nS)\gamma$ via intermediate $b\bar{b}$ and $q\bar{q}$ states with helicity alignment constraints.
• Ignores $Z$–$\gamma$ mixing and focuses on $A_{\mu}$-type vector fields to simplify the coupling structure in the Lagrangian.

Experimental results

Research questions

• RQ1 Can the $\chi_b(3P)$ state at 10.5157 GeV be identified as a light Higgs boson $h^0(0^+)$ with mass $\sim$11 GeV?
• RQ2 How does Cartan's supersymmetry explain the observed small polarizations of $\Upsilon(nS)$ states in $\chi_b(3P) \to \Upsilon(nS)\gamma$ decays?
• RQ3 What is the role of trilinear couplings between Higgs fields, quark-antiquark pairs, and vector fields in mediating $h^0 \to \Upsilon(nS)\gamma$?
• RQ4 Why do the $\Upsilon(nS)$ states ($n=1,2,3$) show negligible polarization in CMS data, and can this be explained by $J=0$ initial state spin?
• RQ5 How does the $h^0$ decay to $\Upsilon(nS)\gamma$ differ from the standard model Higgs decay, and what does it imply for the Higgs sector structure?

Key findings

• The model predicts a $h^0$ mass of $11.2$ GeV, consistent with the observed $\chi_b(3P)$ mass of 10.5157 GeV and the $\Upsilon(nS)$ decay threshold.
• The small polarizations of $\Upsilon(nS)$ states ($n=1,2,3$) are naturally explained by assuming the $\chi_b(3P)$ state has $J^P = 0^+$, as required by angular momentum conservation in the decay.
• Cartan's trilinear couplings between Higgs fields, $b\bar{b}$, and $q\bar{q}$ states allow the $h^0$ to decay to $\Upsilon(nS)\gamma$ via intermediate $q\bar{q} \to \gamma$ processes.
• The $\gamma$ in the final state arises from $\bar{\ell}\ell'$ pair annihilation, with helicity alignment dependent on the exchanged vector field ($x^0$ or $x^2$).
• The model successfully reproduces the $\chi_b(1P)$ and $\chi_b(2P)$ decays to $\Upsilon(1S)\gamma$ as a consistency check.
• The framework provides a mechanism for $h^0 \to \Upsilon(nS)\gamma$ that differs from standard supersymmetry, avoiding the need for gluinos and relying on Cartan’s unique trilinear forms.