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 |