| 19 | | The |
| 20 | | {{{ |
| 21 | | #!latex |
| 22 | | $SU(3)_1 \times SU(3)_2 \to SU(3)_C$ |
| 23 | | }}} |
| 24 | | breaking induced by the expectation value of the |
| 25 | | {{{ |
| 26 | | #!latex |
| 27 | | ({$\bf 3,\bar{ 3}$}) |
| 28 | | }}} |
| 29 | | scalar field Phi generates color-octet and color-singlet scalars. The most general renormalizable potential for Phi is: |
| 30 | | {{{ |
| 31 | | #!latex |
| 32 | | $V(\Phi)=-m^2_{\Phi}\text{Tr}(\Phi\Phi^\dagger) -\mu (\text{det }\Phi+\text{H.c.})+\frac{\xi}{2}\left[ \text{Tr}(\Phi\Phi^\dagger) \right]^2+\frac{k}{2}\text{Tr}(\Phi\Phi^\dagger\Phi\Phi^\dagger) \ ,$ |
| 33 | | }}} |
| 34 | | where |
| 35 | | {{{ |
| 36 | | #!latex |
| 37 | | $\text{det } \Phi = \frac{1}{6}\epsilon^{ijk}\epsilon^{i'j'k'}\Phi_{ii'}\Phi_{jj'}\Phi_{kk'} \ ,$ |
| 38 | | }}} |
| 39 | | and where, without loss of generality, one can choose mu > 0. Assuming |
| 40 | | {{{ |
| 41 | | #!latex |
| 42 | | $m^2_\Phi >0$, |
| 43 | | }}} |
| 44 | | Phi acquires a (positive) diagonal expectation value: |
| 45 | | {{{ |
| 46 | | #!latex |
| 47 | | $\langle \Phi \rangle = u \cdot \mathcal{I} \,.$ |
| 48 | | }}} |
| 49 | | The Phi expansion around the vacuum gives: |
| 50 | | {{{ |
| 51 | | #!latex |
| 52 | | $\Phi=u+\frac{1}{\sqrt{6}}\left(\phi_R+i\phi_I\right)+\left(G^a_H+iG^a_G\right)T^a \ ,$ |
| 53 | | }}} |
| 54 | | where |
| 55 | | {{{ |
| 56 | | #!latex |
| 57 | | $\phi_R$, $\phi_I$ |
| 58 | | }}} |
| 59 | | are singlets under SU(3)_C Additionally, |
| 60 | | {{{ |
| 61 | | #!latex |
| 62 | | $G^a_G$, $a=1,\dots,8$, |
| 63 | | }}} |
| 64 | | are the Nambu-Goldstone bosons associated with the color-symmetry breaking, and |
| 65 | | {{{ |
| 66 | | #!latex |
| 67 | | $G^a_H$ |
| 68 | | }}} |
| 69 | | are color octets. |
| 70 | | |
| 71 | | GH can be produced in pairs through its interactions with gluons: |
| 72 | | {{{ |
| 73 | | #!latex |
| 74 | | $\frac{g^2_s}{2}f^{abc}f^{ade}G^b_{\mu}G^{\mu d}G^c_H G^e_H +g_s f^{abc} G^a_{\mu} G^b_H \partial^{\mu} G^c_H \ ,$ |
| 75 | | }}} |
| 76 | | or it can be produced singly via gluon-gluon fusion. This occurs at one-loop order through the cubic interaction |
| 77 | | {{{ |
| 78 | | #!latex |
| 79 | | $\frac{\mu}{6} d_{abc} G^a_H G^b_H G^c_H \,,$ |
| 80 | | }}} |
| 81 | | which arises from the |
| 82 | | {{{ |
| 83 | | #!latex |
| 84 | | $\mu(\det\Phi+\text{H.c.})$ |
| 85 | | }}} |
| 86 | | term in the potential; where |
| 87 | | {{{ |
| 88 | | #!latex |
| 89 | | $d_{abc}$ |
| 90 | | }}} |
| 91 | | is the SU(3) totally symmetric tensor. The single production of GH can be described by the effective coupling |
| | 21 | The color octet (GH) is a neutral heavy hadronic resonance. The Feynman diagram for the production of GH and its decay to a single top quark and a charm quark is shown below. |
| | 22 | [[Image(Coloron.png)]] |
| | 23 | The single production of GH can be described by the effective coupling |
