Triviality

Notes on Renormalization (Draft)

Abstract

Renormalization of the effective potential is considered using autonomous and perturbative conventions. The primary focus of attention is the triviality of $\lambda \left[\Phi^{2}\right]_{4}^{2}$. Several examples are considered: 1. An interacting scalar field, 2. Scalar quantum electrodynamics, and 3. Scalar Yang-Mills theory. Effects of finite temperature, background spacetime curvature, and multiple loops are also considered. The Halpern-Huang approach and long-range effects are included for completeness.

Interacting scalar field

Let

\[S\left[\Phi,\Psi\right] = S_{\Phi}\left[\Phi\right] + S_{\Psi}\left[\Psi\right] + S_{I}\left[\Phi,\Psi\right] = \int d^{4}x \, \mathcal{L}_{\Phi \Psi}\]

where

\[S_{\Phi}\left[\Phi\right] = \int d^{4}x \left(\frac{1}{2} (\partial_{\mu}\Phi_{o})^{2} + \frac{1}{2}m^{2}_{\Phi o} \Phi_{o}^{2} - \frac{1}{4} \lambda^{\Phi}_{o} \Phi_{o}^{4}\right),\] \[S_{\Psi}\left[\Phi\right] = \int d^{4}x \left(\frac{1}{2} (\partial_{\mu}\Psi_{o})^{2} + \frac{1}{2}m^{2}_{\Psi o} \Psi_{o}^{2} - \frac{1}{4} \lambda^{\Psi}_{o} \Psi_{o}^{4}\right),\]

and

\[S_{I}\left[\Phi,\Psi\right] = \frac{g_{o}}{4}\int d^{4}x \, \Phi_{o}^{2} \Psi_{o}^{2}.\]

Equivalently,

\[\mathcal{L}_{\Phi \Psi} = \frac{1}{2} (\partial_{\mu}\Phi_{o})^{2} + \frac{1}{2} (\partial_{\mu}\Psi_{o})^{2} - U(\Phi,\Psi),\]

where

\[U(\Phi,\Psi) = -\frac{1}{2}m^{2}_{\Phi o} \Phi_{o}^{2} + \frac{1}{4} \lambda^{\Phi}_{o} \Phi_{o}^{4} - \frac{1}{4}g_{o} \Phi_{o}^{2} \Psi_{o}^{2} - \frac{1}{2}m^{2}_{\Psi o} \Psi_{o}^{2} + \frac{1}{4}\lambda^{\Psi}_{o}\Psi_{o}^{4}\]

is the classical potential.

If $m_{\Phi o} = m_{\Psi o} = m_{o}$ and $\lambda^{\Phi}{o} = \lambda^{\Psi}{o}= - g_{o} /2 = \lambda_{o}$ then

\[S\left[\Phi,\Psi\right] = S\left[\phi^{i}\right] = \int d^{4}x \left(\frac{1}{2} (\partial_{\mu}\phi^{i}_{o})^{2} + \frac{1}{2} m^{2}_{o} (\phi^{i}_{o})^{2} - \frac{\lambda_{o}}{4}\left[(\phi^{i}_{o})^{2} \right]^{2}\right)\]

where $\phi^{i}{o} = \left(\Phi{o},\Psi_{o}\right)$, $(\phi^{i}{o})^{2} = \Phi{o}^{2} + \Psi_{o}^{2}$, and $\left[(\phi^{i}{o})^{2} \right]^{2} = \Phi{o}^{4} + 2 \Phi_{o}^{2} \Psi_{o}^{2} + \Psi_{o}^{4}$.

This is the $O\left(2\right)$ symmetric linear sigma model.

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To one loop in perturbation theory, the effective action is

\[\Gamma\left[\Phi,\Psi\right] = S\left[\Phi,\Psi\right] + \frac{i}{2} \log \det \left[ \begin{array}{cc} \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Phi} & \frac{ \delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Psi \delta \Phi} \\ \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Psi} & \frac{ \delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Psi \delta \Psi} \end{array} \right] - i (\mathrm{connected}\ \mathrm{diagrams}) + \int d^4x \, \delta \mathcal{L}_{\Phi \Psi},\]

where

\[\frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Phi} = \partial^{2} + m^{2}_{\Phi o} - 3 \lambda^{\Phi}_{o} \Phi_{o}^{2} + \frac{1}{2} g_{o} \Psi_{o}^{2} = \partial^{2} + M^{2}_{\Phi o},\] \[\frac{ \delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Psi \delta \Psi} = \partial^{2} + m^{2}_{\Psi o} - 3 \lambda^{\Psi}_{o} \Psi_{o}^{2} + \frac{1}{2} g_{o} \Phi_{o}^{2} = \partial^{2} + M^{2}_{\Psi o},\]

and

\[\frac{ \delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Psi} = g_{o} \Phi_{o} \Psi_{o}.\]

More specifically,

\[\begin{aligned} \det \left[ \begin{array}{cc} \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Phi} & \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Psi \delta \Phi} \\ \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Phi \delta \Psi} & \frac{\delta^{2} \mathcal{L}_{\Phi\Psi}}{\delta \Psi \delta \Psi} \end{array} \right] &= \left(\partial^{2} + M^{2}_{\Phi o}\right)\left( \partial^{2} + M^{2}_{\Psi o}\right) - g^{2}_{o} \Phi_{o}^{2} \Psi_{o}^{2} \\ &= \det \left[\left(\partial^{2} + M^{2}_{X}\right) \left( \partial^{2} + M^{2}_{Y}\right)\right] \\ &= \det \left[\partial^{2} + M^{2}_{X}\right] \det \left[\partial^{2} + M^{2}_{Y}\right] \end{aligned}\]

where $M_X$, $M_Y$ solve

\[M^{2}_{X} + M^{2}_{Y} = M^{2}_{\Phi o} + M^{2}_{\Psi o},\]

and

\[M^{2}_{X} M^{2}_{Y} = M^{2}_{\Phi o} M^{2}_{\Psi o} - g_{o}^{2} \Phi_{o}^{2} \Psi_{o}^{2}.\]

For $X,Y$ unknown, $X + Y = A + B$ and $XY = AB - C^{2}$ is solved by

\[X = \frac{A + B}{2} \pm \frac{A - B}{2} \sqrt{1 + \left[\frac{2C}{A - B}\right]^2}, \quad Y = \frac{AB - C^2}{X}.\]

If $C = \mathcal{O}(g)$, then $X = A + \mathcal{O}(g^2)$ and $Y = B + \mathcal{O}(g^2)$.

The effective potential is defined for constant $\phi^{i}_{c}$ background fields

\[U_{eff}(\phi^{i}_{c}) = -\frac{1}{(VT)} \Gamma\left[\phi^{i}_{c}\right].\]

Hereafter, all fields appearing are constant background fields. Using dimensional regularization, let

\[I_{-1}(m) \equiv \int \frac{d^{4}k}{(2 \pi)^{4}} \frac{1}{(k^{2} + m^{2})^{2}} = \frac{(4 \pi)^{\epsilon/2}}{4 \pi^{2}} \Gamma(1+\epsilon/2)\frac{1}{\epsilon m^{\epsilon}},\] \[I_{0}(m) \equiv \int \frac{d^{4}k}{(2 \pi)^{4}} \frac{1}{k^{2} + m^{2}},\] \[\begin{aligned} I_{1}(m) &\equiv \int \frac{d^{4}k_{E}}{(2 \pi)^{4}} \log (k^{2}_{E} + m^{2}) \\ &= - \frac{1}{8} m^{4} I_{-1}(M) + \frac{m^{4}}{64 \pi^{2}}\left(\log\left[\frac{m^{2}}{M^{2}}\right]-\frac{3}{2}\right) \\ &= a_{1} m^{4} \mathscr{I} + a_{2} m^{4} \log\left[\frac{m^{2}}{M^{2}}\right] - a_{3}, \end{aligned}\]

with

\[a_{1} = -\frac{1}{8}, \quad a_{2} = \frac{1}{4(4\pi)^{2}}, \quad a_{3} = \frac{3}{8(4\pi)^{2}}, \quad \mathscr{I} = I_{-1}(M).\]

The functional determinant is then

\[\begin{aligned} \frac{1}{(VT)} \log \det (\partial^{2} + m^2) &= \frac{1}{(VT)} \mathrm{Tr} \log (\partial^{2} + m^2) \\ &= \frac{1}{(VT)} \sum_{k} \log (-k^2 + m^2) \\ &= \int \frac{d^{4}k_{E}}{(2 \pi)^{4}} \log (k_{E}^{2} + m^{2}) \\ &= I_{1}(m). \end{aligned}\]

In this notation, the bare one-loop effective potential is

\[\begin{aligned} U_{eff}(\Phi_{o}, \Psi_{o}) &= -\frac{1}{2}m^{2}_{\Phi o} \Phi_{o}^{2} + \frac{\lambda^{\Phi}_{o}}{4} \Phi_{o}^{4} - \frac{g_{o}}{4} \Phi_{o}^{2} \Psi_{o}^{2} - \frac{1}{2}m^{2}_{\Psi o} \Psi_{o}^{2} + \frac{\lambda^{\Psi}_{o}}{4} \Psi_{o}^{4} \\ &\quad + (M^{2}_{\Phi o})^{2} \left[ a_{1} \mathscr{I} + a_{2} \log\left(\frac{M^{2}_{\Phi o}}{M^{2}}\right) - a_{3} \right] \\ &\quad + (M^{2}_{\Psi o})^{2} \left[ a_{1} \mathscr{I} + a_{2} \log\left(\frac{M^{2}_{\Psi o}}{M^{2}}\right) - a_{3} \right] \end{aligned}\]

where

\[(M^{2}_{\Phi o})^{2} = m^{4}_{\Phi o} + 2 m^{2}_{\Phi o} \left( \frac{1}{2} g_{o} \Psi_{o}^{2} - 3 \lambda^{\Phi}_{o} \Phi_{o}^{2} \right) + \left( \frac{1}{2} g_{o} \Psi_{o}^{2} - 3 \lambda^{\Phi}_{o} \Phi_{o}^{2} \right)^{2},\]

and

\[(M^{2}_{\Psi o})^{2} = m^{4}_{\Psi o} + 2 m^{2}_{\Psi o} \left( \frac{1}{2} g_{o} \Phi_{o}^{2} - 3 \lambda^{\Psi}_{o} \Psi_{o}^{2} \right) + \left( \frac{1}{2} g_{o} \Phi_{o}^{2} - 3 \lambda^{\Psi}_{o} \Psi_{o}^{2} \right)^{2}.\]

The renormalization equations are

$\Phi^2$ term:

\[\left[-\frac{1}{2} m^{2}_{\Phi o} - 6 m^{2}_{\Phi o} a_{1} \lambda^{\Phi}_{o} \mathscr{I} + m^{2}_{\Psi o} a_{1} g_{o} \mathscr{I}\right] \Phi_{o}^{2} = -\frac{1}{2} m^{2}_{\Phi} \varphi^{2},\]

$\Phi^4$ term:

\[\left[ \frac{\lambda^{\Phi}_{o}}{4} + 9 a_{1} (\lambda^{\Phi}_{o})^{2} \mathscr{I} + a_{1} \frac{g_{o}^{2}}{4} \mathscr{I} \right] \Phi_{o}^{4} = \frac{1}{4} \lambda^{\Phi} \varphi^{4},\]

$\Phi^2 \Psi^2$ term:

\[\left[ \frac{g_{o}}{4} + 3 a_{1} g_{o} \lambda^{\Psi} \mathscr{I} + 3 a_{1} g_{o} \lambda^{\Phi}_{o} \mathscr{I} \right] \Phi_{o}^{2} \Psi_{o}^{2} = \frac{1}{4} g \varphi^{2} \Psi^{2},\]

$\Psi^2$ term:

\[\left[ -\frac{1}{2} m^{2}_{\Psi o} - 6 m^{2}_{\Psi o} a_{1} \lambda^{\Psi}_{o} \mathscr{I} + m^{2}_{\Phi o} a_{1} g_{o} \mathscr{I} \right] \Psi_{o}^{2} = -\frac{1}{2} m^{2}_{\Psi} \Psi^{2},\]

$\Psi^4$ term:

\[\left[ \frac{\lambda^{\Psi}_{o}}{4} + 9 a_{1} (\lambda^{\Psi}_{o})^{2} \mathscr{I} + a_{1} \frac{g_{o}^{2}}{4} \mathscr{I} \right] \Psi_{o}^{4} = \frac{1}{4} \lambda^{\Psi} \Psi^{4},\]

Also,

\[M^{2}_{\Phi o} = m^{2}_{\Phi o} - 3 \lambda^{\Phi}_{o} \Phi_{o}^{2} + \frac{1}{2} g_{o} \Psi_{o}^{2} = m^{2}_{\Phi} - 3 \lambda^{\Phi} \varphi^{2} + \frac{1}{2} g \Psi^{2},\]

and

\[M^{2}_{\Psi o} = m^{2}_{\Psi o} - 3 \lambda^{\Psi}_{o} \Psi_{o}^{2} + \frac{1}{2} g_{o} \Phi_{o}^{2} = m^{2}_{\Psi} - 3 \lambda^{\Psi} \Psi^{2} + \frac{1}{2} g \varphi^{2}.\]

Let $g$ be the expansion parameter:

\[\begin{aligned} Z_{\Phi} &= c^{Z_{\Phi}}_{o} + c^{Z_{\Phi}}_{1} g + c^{Z_{\Phi}}_{2} g^{2} + \ldots, \\ \lambda^{\Phi}_{o} &= c^{\Phi}_{o} + c^{\Phi}_{1} g + c^{\Phi}_{2} g^{2} + \ldots, \\ m^{2}_{\Phi o} &= c^{m^{2}_{\Phi}}_{o} + c^{m^{2}_{\Phi}}_{1} g + c^{m^{2}_{\Phi}}_{2} g^{2} + \ldots, \\ g_{o} &= c^{g}_{1} g + c^{g}_{2} g^{2} + \ldots, \\ Z_{\Psi} &= c^{Z_{\Psi}}_{o} + c^{Z_{\Psi}}_{1} g + c^{Z_{\Psi}}_{2} g^{2} + \ldots, \\ \lambda^{\Psi}_{o} &= c^{\Psi}_{o} + c^{\Psi}_{1} g + c^{\Psi}_{2} g^{2} + \ldots, \\ m^{2}_{\Psi o} &= c^{m^{2}_{\Psi}}_{o} + c^{m^{2}_{\Psi}}_{1} g + c^{m^{2}_{\Psi}}_{2} g^{2} + \ldots. \end{aligned}\]

The $\Phi_{o}$ field is renormalized in the autonomous fashion. From the $\Phi_{o}^{4}$ equation:

\[\begin{aligned} \lambda^{\Phi}_{o} - \frac{9}{2} (\lambda^{\Phi}_{o})^{2} \mathscr{I} - \frac{g_{o}^{2}}{8} \mathscr{I} &= c^{\Phi}_{o} + c^{\Phi}_{1} g + c^{\Phi}_{2} g^{2} - \frac{9}{2} (c^{\Phi}_{o} + c^{\Phi}_{1} g)^{2} \mathscr{I} - \frac{1}{8} (c^{g}_{1})^{2} \mathscr{I} g^{2} \\ &= \frac{\lambda^{\Phi}}{(Z^{\Phi}_{o})^{2}} \\ &= \frac{\lambda^{\Phi}}{(c^{Z_{\Phi}}_{o})^{2}} \left(1 - \frac{c^{Z_{\Phi}}_{1}}{c^{Z_{\Phi}}_{o}} g + (c^{Z_{\Phi}}_{2})^{-1} g^{2} + \ldots\right)^{2}, \end{aligned}\]

with

\[(c^{Z_{\Phi}}_{2})^{-1} = -\frac{c^{Z_{\Phi}}_{2}}{c^{Z_{\Phi}}_{o}} + \left( \frac{c^{Z_{\Phi}}_{1}}{c^{Z_{\Phi}}_{o}} \right)^{2}.\]

Three regimes are considered:

• $\lambda^{\Phi} = \mathcal{O}(1)$:
  $c^{\Phi}{o} = \frac{2}{9 \mathscr{I}}, \quad c^{\Phi}{1} = \text{stuff}, \quad c^{\Phi}{2} = \frac{\mathscr{I}}{8}(c^{g}{1})^{2} + \text{stuff}$

• $\lambda^{\Phi} = \mathcal{O}(g)$:
  $c^{\Phi}{o} = \frac{2}{9 \mathscr{I}}, \quad c^{\Phi}{1} = \frac{\lambda^{\Phi}}{g (c^{Z_{\Phi}}{o})^{2}}, \quad c^{\Phi}{2} = \left[ \frac{1}{8} (c^{g}{1})^{2} + \frac{9}{2} (c^{\Phi}{1})^{2} \right] \mathscr{I}$

• $\lambda^{\Phi} = \mathcal{O}(g^{2})$:
  $c^{\Phi}{o} = \frac{2}{9 \mathscr{I}}, \quad c^{\Phi}{1} = 0, \quad c^{\Phi}{2} = \frac{1}{8} (c^{g}{1})^{2} \mathscr{I} + \frac{\lambda^{\Phi}}{(g c^{Z_{\Phi}}_{o})^{2}}$

The $\Psi$ field is renormalized with perturbation theory. From the $\Psi_{o}^{4}$ equation there are two regimes:

• $\lambda^{\Psi} = \mathcal{O}(g)$:

\[c^{\Psi}_{o} = 0, \quad c^{\Psi}_{1} = \frac{\lambda^{\Psi}}{g (c^{Z_{\Psi}}_{o})^{2}}, \quad c^{\Psi}_{2} = \left[ \frac{1}{8} (c^{g}_{1})^{2} + \frac{9}{2} (c^{\Psi}_{1})^{2} \right] \mathscr{I}\]

• $\lambda^{\Psi} = \mathcal{O}(g^{2})$:

\[c^{\Psi}_{o} = 0, \quad c^{\Psi}_{1} = 0, \quad c^{\Psi}_{2} = \frac{1}{8} (c^{g}_{1})^{2} \mathscr{I} + \frac{\lambda^{\Psi}}{(g c^{Z_{\Psi}}_{o})^{2}}\]

From the $\Phi_{o}^{2}\Psi^{2}$ equation:

\[g_{o}\left[1 - \frac{3}{2} (\lambda^{\Phi}_{o} \mathscr{I} + \lambda^{\Psi}_{o} \mathscr{I})\right] = \frac{g}{Z_{\Phi} Z_{\Psi}},\]

and expanding both sides:

\[\begin{aligned} c^{g}_{1} + c^{g}_{2} g + c^{g}_{3} g^{2} + \dots &= \frac{1}{Z_{\Phi} Z_{\Psi}} \left(1 + \frac{3}{2} (\lambda^{\Phi}_{o} + \lambda^{\Psi}_{o}) \mathscr{I} \right) \\ &= \frac{1}{c^{Z_{\Phi}}_{o} c^{Z_{\Psi}}_{o}} \left(1 - \frac{c^{Z_{\Phi}}_{1}}{c^{Z_{\Phi}}_{o}} g + (c^{Z_{\Phi}}_{2})^{-1} g^{2} + \ldots \right) \\ &\quad \times \left(1 - \frac{c^{Z_{\Psi}}_{1}}{c^{Z_{\Psi}}_{o}} g + (c^{Z_{\Psi}}_{2})^{-1} g^{2} + \ldots \right) \left(1 + \frac{3}{2} (\lambda^{\Phi}_{o} + \lambda^{\Psi}_{o}) \mathscr{I} \right) \end{aligned}\]

where

\[\begin{aligned} c^{g}_{1} &= \frac{1}{c^{Z_{\Phi}}_{o} c^{Z_{\Psi}}_{o}} \left(1 + \frac{3}{2} (c^{\Phi}_{o} + c^{\Psi}_{o}) \mathscr{I} \right) = \frac{4}{3 c^{Z_{\Phi}}_{o} c^{Z_{\Psi}}_{o}}, \\ c^{g}_{2} &= \frac{1}{c^{Z_{\Phi}}_{o} c^{Z_{\Psi}}_{o}} \left(\frac{c^{Z_{\Phi}}_{1}}{c^{Z_{\Phi}}_{o}} + \frac{c^{Z_{\Psi}}_{1}}{c^{Z_{\Psi}}_{o}} + \frac{3}{2}(c^{\Phi}_{1} + c^{\Psi}_{1}) \mathscr{I} \right), \\ c^{g}_{3} &= \frac{1}{c^{Z_{\Phi}}_{o} c^{Z_{\Psi}}_{o}} \left((c^{Z_{\Phi}}_{2})^{-1} + (c^{Z_{\Psi}}_{2})^{-1} + \frac{3}{2}(c^{\Phi}_{2} + c^{\Psi}_{2}) \mathscr{I} \right). \end{aligned}\]

Also,

\[(c^{Z_{\Psi}}_{2})^{-1} = -\frac{c^{Z_{\Psi}}_{2}}{c^{Z_{\Psi}}_{o}} + \left(\frac{c^{Z_{\Psi}}_{1}}{c^{Z_{\Psi}}_{o}}\right)^{2}.\]

To be continued …

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