Spontaneous symmetry breaking without vacuum energy: the effective potential

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The effective Lagrangian of a classical scalar field corresponding to the vacuum expectatuion value of a quantum scalar field which displays spontaneous symmetry breaking without vacuum energy is given by

\mathcal{L} = \sqrt{-g} \left( \xi \Phi R - \omega g^{\mu\nu} \frac{\partial_\mu \Phi \partial_\nu \Phi}{\Phi} - \upsilon g^{\mu\nu} \frac{\partial_\mu \Phi \partial_\nu \chi}{\chi} - \eta \Phi g^{\mu\nu} \frac{\partial_\mu \chi \partial_\nu \chi}{\chi^2} - \mu \Phi \chi^2 \right) + \sum (-1)^n X_n X_{n+1},

\mathcal{L}_i = \sqrt{-g} \left( \xi' \Phi^2 \chi^2 R - \omega' \chi^2 g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi - \upsilon' \Phi \chi g^{\mu\nu} \partial_\mu \Phi \partial_\nu \chi - \eta' \Phi^2 g^{\mu\nu} \partial_\mu \chi \partial_\nu \chi - \mu' \Phi^2 \chi^4 \right),

where the parameters are functions of the dimensionless scalar \hbar \chi^2 / \Phi which maintains the invariance under the global scale transformation

g_{\mu\nu} \rightarrow \Omega^2 g_{\mu\nu}, ~ \Phi \rightarrow \Omega^{-2} \Phi, ~ \chi \rightarrow \Omega^{-1} \chi, ~ X_{n + 1} \rightarrow X_{n + 1},

but explicitly breaks the length-scale covariance. The scale transformation corresponds to conversions of the units of length, time and reciprocal mass in the same proportion, under which the numerical values of c and \hbar do not change. The dimensions of \Phi and \chi are respectively \text{M} \text{T}^{-1} and \text{L}^{-1}. The inertial coupling K = \sqrt{-g} X_1 has the dimensions \text{L}^2 \text{M}^{-1} \text{T} and \hbar K has the dimensions \text{L}^4. The scalar K \Phi^2 has the dimensions of action and is invariant under scale transformations. Its value in empty space determines the absolute scale of mass. The Lagrangian is also invariant under the transformation \chi \to - \chi.

The potential density is defined by

\mathcal{V} = \frac{V(x, y)}{X_1} = \frac{\mu(x) y + \mu'(x) y^2}{X_1},

with x = \hbar \chi^2 / \Phi and y = K \Phi \chi^2. Its contributions to the stress-energy tensor and the equations of \Phi and \chi are respectively

- \frac{g_{\mu\nu}}{X_1 \sqrt{-g}} y \frac{\partial V}{\partial y},

- \frac{1}{X_1 \Phi} \left( - x \frac{\partial V}{\partial x} + y \frac{\partial V}{\partial y}\right),

- \frac{2}{X_1 \chi} \left( x \frac{\partial V}{\partial x} + y \frac{\partial V}{\partial y} \right).

Spontaneous symmetry breaking requires that V(x, y) be minimum at x = x_0 > 0, y = y_0 > 0. In empty space, the equations of \Phi and \chi are equivalent to

\hbar \frac{\chi_0^2}{\Phi_0} = x_0,~~ K_0 \Phi_0 \chi_0^2 = y_0 = - \frac{\mu(x_0)}{2\mu'(x_0)} = - \frac{\frac{\mathrm{d} \mu}{\mathrm{d} x} (x_0)}{\frac{\mathrm{d} \mu'}{\mathrm{d} x} (x_0)}

and the absolute scale of mass is determined by

\frac{\hbar}{K_0 \Phi_0^2} = \frac{x_0}{y_0},

with

\frac{\mathrm{d} \mu^2 / \mu'}{\mathrm{d} x}(x_0) = 0.

For an otherwise general function V(x, y), the contribution to the cosmological constant from spontaneous symmetry breaking is null in empty space. As

y \frac{\partial V}{\partial y} = \mu(x) y + 2 \mu'(x) y^2 = V(x, y) - X_0 X_1^2 + K^2 (\xi' \Phi^2 \chi^2 R - \omega' \chi^2 g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi - \upsilon' \Phi \chi g^{\mu\nu} \partial_\mu \Phi \partial_\nu \chi - \eta' \Phi^2 g^{\mu\nu} \partial_\mu \chi \partial_\nu \chi ),

the potential in the stress-energy tensor is proportional to [ V(x, y) - X_0 X_1^2 ] / K. In empty space, R = 0 and X_0 X_1^2 = - \mu'(x_0) y_0^2 = V(x_0, y_0). In states that have the same X_0 X_1^2 as the empty space, say canonical states, the inertial coupling is determined by

K^2 \frac{\mathcal{L}_i}{\sqrt{-g}} = V(x_0, y_0).

The empty space is the canobical state with minimum energy.

Defining f(x) = - \mu(x) / 2\mu'(x) and L(x) = \mu(x)^2 / 4\mu'(x) so that \mu(x) = -2L(x) / f(x) and \mu'(x) = L(x) / f(x)^2, V(x, y) takes the form

V(x, y) = \left(- \frac{2y}{f(x)} + \frac{y^2}{f(x)^2} \right) L(x).

Its minimum is given by V(x_0, y_0) = - L(x_0), x_0 is the maximum point of L(x) and y_0 = f(x_0). Due to \mu' \Phi^2 \chi^4 in K, the linear terms in the equations of \Phi and \chi in canonical states are proportional to

\frac{x_0^2}{K_0 \Phi_0} \frac{\mathrm{d}^2 V}{\mathrm{d} x^2}(x_0) \delta x,

with \delta x = -\hbar \frac{\chi_0^2}{\Phi_0^2} \delta \Phi + 2\hbar \frac{\chi_0}{\Phi_0} \delta \chi, \Phi = \Phi_0 + \delta \Phi and \chi = \chi_0 + \delta \chi. The correspondence between x and the expectation value of a scalar field associated with a quantum effective action, \varphi, is set by

x = \frac{x_0}{\varphi_0^2} \varphi^2,

L(x) = - \hbar K_0 V_{\text{eff}}(\varphi),

where \varphi_0 is the vacuum expectation value of the scalar field in the absence of an external current and V_{\text{eff}}(\varphi) is the effective potential. As V_{\text{eff}}(\varphi) is minimum at \varphi = \varphi_0, L(x) is maximum at x = x_0 as required. If \varphi is canonically normalized, its mas is given by

\left( \frac{mc}{\hbar} \right)^2 = \frac{\mathrm{d}^2 V_{\text{eff}}}{\mathrm{d} \varphi^2}(\varphi_0) = - \frac{4 x_0^2}{\hbar K \varphi_0^2} \frac{\mathrm{d}^2 L}{\mathrm{d} x^2}(x_0),

\varphi_0^2 = \frac{2 \Phi_0}{\hbar} \frac{4[\omega(x_0) + y_0 \omega'(x_0)][\eta(x_0) + y_0 \eta'(x_0)] - [\upsilon(x_0) + y_0 \upsilon'(x_0)]^2}{4[\omega(x_0) + y_0 \omega'(x_0)] + \eta(x_0) + y_0 \eta'(x_0) + 2[\upsilon(x_0) + y_0 \upsilon'(x_0)]}.

The dimensionless scalar field defined by

\hbar \frac{\chi_0^2}{\Phi_0^2}\delta\Phi + \frac{\eta(x_0) + y_0 \eta'(x_0) + \upsilon(x_0) + y_0 \upsilon'(x_0)}{4[\omega(x_0) + y_0 \omega'(x_0)] + \upsilon(x_0) + y_0 \upsilon'(x_0)} 2\hbar \frac{\chi_0}{\Phi_0}\delta\chi

is massless. The Planck mass nomalization of \Phi and \chi is set by - \mu(x_0) = \xi(x_0) + y_0 \xi'(x_0) = 1/8 and \Phi_0 = c^3/ 2\pi G so that

x_0 = - \left( \frac{8\pi G\hbar}{c^3} \right)^2 V_{\text{eff}}(\varphi_0),

with \mu(x_0) < 0 and V_{\text{eff}}(\varphi_0) < 0.

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Spontaneous symmetry breaking without vacuum energy

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here.

Abstract.

The contribution to the cosmological constant from spontaneous breaking of generic internal symmetries of classical fields is cancelled by constraining the part of the Lagrangian that is invariant under lenght–scale transformations. The constraint determines the dependence of the inertial coupling on the metric and the matter fields. For low energy density, this dependence is suppressed by quartic interaction terms of scalar fields in empty space. The absolute scale of the scalar field mass parameters and the canonical normalization of the matter fields are set by the inertial coupling in empty space. The mechanism uses an infinite sequence of auxiliary fields in a special series whose sum is defined by a linear and additive summation method. It is a loophole in the theorem that spontaneous breaking of the scale symmetry without vacuum energy requires fine–tuning. By introducing a Jordan–Brans–Dicke field, the Lagrangian is scale invariant in all spacetime dimensions. The correspondence between a classical scalar field and the vacuum expectation value of a quantum scalar field is defined by an effective Lagrangian.

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A scale invariant effective Lagrangian of a classical scalar field which displays spontaneous symmetry breaking without vacuum energy is given by

\mathcal{L} = \mathcal{L}_g + \sum (-1)^n X_n X_{n+1},

\mathcal{L}_g is the gravitational coupling sector of the Lagrangian,

\mathcal{L}_g = \sqrt{-g}\left(\frac{\xi}{2}R\Phi - \frac{\omega}{2}g^{\mu\nu}\frac{\partial_\mu\Phi\partial_\nu\Phi}{\Phi} - \frac{\eta}{2} g^{\mu\nu} \frac{\partial_\mu\chi\partial_\nu\chi}{\chi^2}\Phi + \frac{\mu^2}{2} \frac{\chi^2}{\Phi}\right),

X_0 = \sqrt{-g}\mathcal{L}_i,

\mathcal{L}_i is the inertial coupling sector of the Lagrangian,

\mathcal{L}_i = \sqrt{-g}\left(\frac{\xi'}{2}R\chi^2 -\frac{\omega'}{2}g^{\mu\nu}\frac{\partial_\mu\Phi\partial_\nu\Phi}{\Phi^2}\chi^2 - \frac{\eta'}{2} g^{\mu\nu} \partial_\mu\chi\partial_\nu\chi - \frac{\lambda}{4} \frac{\chi^4}{\Phi^2}\right),

the parameters are dimensionless functions of K\chi^2/\Phi and \log(K\chi^2/\Phi), K\chi^2/\Phi is a positive dimensionless scalar and K = \sqrt{-g}X_1 is the inertial coupling, which corresponds to the reciprocal of the square of the unit of mass. K\Phi has the dimensions of square length. The auxiliary fields X_{n+1} are independent of g_{\mu\nu}, \Phi and \chi.

The contributions to the stress–energy tensor from the extra factor \sqrt{-g} in X_0 and K cancel the effective cosmological constant.

Due to the extra factor \sqrt{-g} in X_0, all odd auxiliary fields are multiplied by the absolute value of the Jacobian determinant and all even auxiliary fields are divided by the square of the Jacobian determinant under coordinate transformations.

In D dimensions, the Lagrangian is invariant under a scale transformation

g_{\mu\nu}\to\Omega^2 g_{\mu\nu}, \Phi\to\Omega^{-D+2}\Phi, \chi\to\Omega^{-D+1}\chi, X_{n+1}\to X_{n+1},

the scaling factor \Omega is a positive constant. The scale transformation corresponds to conversions of the units of length, time and reciprocal mass in the same proportion, by which the numerical values of c and \hbar do not change.

The scalars K\chi^2 / \Phi and (K\Phi)^{\frac{D-2}{2}}\Phi are invariant. (K\Phi)^{\frac{D - 2}{2}}\Phi has the dimensions of action and its value in empty space determines the absolute scale of mass.

The sum of the special series \sum(-1)^nX_nX_{n+1} is defined by whichever linear and additive summation method that sums the Grandi series 1-1+1-1+\cdots to a real number \sum(-1)^n, for example,

\sum(-1)^nX_nX_{n+1}=X_0X_1+\sum_{k=1}^\infty X_{2k}(-X_{2k-1}+X_{2k+1}),~ \sum(-1)^n=1

or

\sum(-1)^nX_nX_{n+1} = X_0X_1 - X_1X_2 + \sum_{k=1}^\infty X_{2k+1}(X_{2k} - X_{2k+2}),~\sum(-1)^n=0.

The action is stationary with respect to variations in any finite set of auxiliary fields.

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On Hermes and the theft of Apollon’s cattle, picturehymn and book.

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