New Physics?

From TechWiki

The next level..?

1 The 5th Force
- 1.1 The Setup
- 1.2 The Solution
2 The Non-Perturbative Higgs Mechanism

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The 5th Force

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The Setup

Look up the equations of general relativity.

Adding a scalar quantity $Λ$ , called the cosmological constant, to the field equations:

$G^{\mu \nu} + \Lambda g^{\mu \nu}= - \frac{8 \pi G_N}{c^4} T^{\mu \nu}$ .

Einstein called this the biggest blunder of his life...

A modern day interpretation:

$G^{\mu \nu} = - \frac{8 \pi G_N}{c^4} \left( T^{\mu \nu} + \frac{c^4 \Lambda}{8 \pi G_N} g^{\mu \nu} \right) =: - \frac{8 \pi G_N}{c^4} \left( T^{\mu \nu} + T_{vac}^{\mu \nu} \right) .$

This leads to the idea, that the vacuum has an intrinsic energy content:

$T^{\mu \nu}_{vac} = \left( \begin{matrix} c^2 \rho_\Lambda &0&0&0\\ 0 & - c^2 \rho_\Lambda &0&0 \\ 0&0& - c^2 \rho_\Lambda &0 \\ 0&0&0& - c^2 \rho_\Lambda \end{matrix} \right),$

where

$\rho_\Lambda := \frac{c^2 \Lambda}{8 \pi G_N},$

is the vacuum energy density.

The Poisson equation for the Newtonian gravitational potential $φ$ incorporating the vacuum density is:

$\triangle \phi = 4 \pi G_N \left(2 \rho_{r,m} - 2 \rho_\Lambda \ \right).$

$ρ r, m$ is the usual radiation-matter density.

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The Solution

A simple model can illustrate the physical effects of this equation.

$\vec F (\vec x) = m_0 \vec g (\vec x),$

is the force exerted onto a test particle with mass $m 0$ at the point $\vec x$ in a gravitational field. This defines the field vector $\vec g$ . For the gravitational potential $φ$ and $\vec g$ the following relations hold:

$\vec g = - \nabla \phi,$

$\phi = - \int \vec g \cdot d\vec s$ .

It can be derived that

$\nabla \cdot \vec g = -\triangle \phi = -4 \pi G_N \rho$ .

The general solutions are

$\phi (\vec x) = -G_N \int \rho (\vec x^\prime) \frac{1}{|\vec x - \vec x^\prime|} d^3x^\prime ,$

$\vec g (\vec x) = -G_N \int \rho (\vec x^\prime) \frac{\vec x - \vec x^\prime}{|\vec x - \vec x^\prime|^3} d^3x^\prime$ .

Substituting $\rho \to 2\rho_m - 2 \rho_\Lambda$ , employing a spherical symmetrical mass distribution $\rho_m = \rho_m (r= |\vec x|)$ and integrating over a spherical volume with radius $r$ yields

$\vec g = \vec g_{rel} - \vec g_{\Lambda},$

where

$\vec g_{rel}(\vec x) = 2 \vec g_{Newton}(\vec x),$

and a new term

For $\vec F = F \vec x / r$ the magnitudes of the forces are thus

F r e l = 2 F N e w t o n

and

 $F Λ = c 2 m 0 Λ r$ .

This concludes that relativistic matter generates a gravitational attraction twice as strong as in the Newtonian limit. However, it is also obvious that $Λ$ is the source of an additional long-range force $F_\Lambda \propto r$ which acts in the opposite direction of the gravitational force. This repulsive force is sometimes referred to as 'anti-gravity' or the 'fifth force'.

The consequence of it is that galaxies placed twice as far apart as two others will feel the effect of double the repulsive force. So the greater the volume of the universe due to its expansion the faster the expansion will become, i.e. the expansion is accelerated.

This has been recently observer: [1], [2]

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The Non-Perturbative Higgs Mechanism

Note that this a tentative idea I came up with in 2001, and probably gets a high score on the Crackpot index ;-)

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The Idea

Take the math of the Higgs mechanism but don't interpret it within a perturbative quantum field theory framework. This means that there is

no need for a physical Higgs boson,

which, by the way, I bet won't be found at the LHC. The interpretation of what the Higgs mechanism then is, should perhaps be best framed along the lines of The Mathematical Universe, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, and perhaps A New Kind of Science.

To quote (arXiv:hep-th/0304245v2)
It is important to stress that there is a logical difference between the Higgs mechanism and the existence of the Higgs particle: the latter provides an elegant and simple implementation of the former. [...] Nevertheless, let us emphasize right away that all efforts to implement the Higgs mechanism of the standard model without a physical Higgs boson have failed so far.

Look up some equations of the Higgs mechanism.

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The Recipe

Parameterization of $φ$ . The four degrees of freedom stay unchanged, $4 φ$ .
Taking unitary gauge. Three degrees of freedom vanish but reappear as polarizational degrees of freedom as the result of the Higgs mechanism, $12=4_\phi+2_{W^1}+2_{W^2}+2_{W^3}+2_{B} \to 12=1_\phi +3_{W^+}+3_{W^-}+3_{Z}+2_{A}$ .
Taking the vacuum expectation value, i.e., the spontaneous symmetry breaking, $\phi_1^0 \to v$ , i. e. $1_\phi \to 0_v$ .
Substitution of $\langle \phi \rangle$ into the Lagrangian.

Note that this procedure never used the perturbative ansatz

$h(x) := \phi^0_1 (x) - v$

which yields a physical Higgs field $h$ . Note for the vacuum expectation values: $\langle \phi^0_1 \rangle = v$ and $\langle h \rangle = 0$

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The Results

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No Physical Higgs Boson

There is no Higgs particle in this framework. It will be interesting to see if the as yet unobserved Higgs boson will be detected at CERN's LHC anytime in 2007 or not...

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The Same Mass Terms

The idea of the Higgs mechanism was to give a process for the generation of boson and fermion mass terms from the invariant Lagrangian of the Standard Model describing massless gauge bosons interacting with massless fermions.

The non-perturbative interpretation of the Higgs mechanism generates the same mass terms from $\mathcal L^{kin}_H$ :

m γ = 0

$m_{W^\pm} =\frac{1}{2} v g ,$

$m_{Z} = \frac{1}{2} v \sqrt{g^2 + g^{\prime2}},$

$m_l = \frac{v \lambda_{Y,l} }{\sqrt2},$

$m_q = \frac{v \lambda_{Y,q}}{\sqrt 2} .$

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Additional Terms

In the standard version of the Higgs mechanism, all other terms originate from $v + h$ in $\mathcal V_H$ :

- λ H v 2 h 2

$\frac{1}{2} \mu^2 h^2$ ,

μ 2 v h

- λ H v 3 h

- λ H v h 3

$-\frac{1}{4}\lambda_H h^4$ ,

$\frac{1}{2} \mu^2 v^2$ ,

$-\frac{1}{4}\lambda_H v^4$ .

The first term is identified as the Higgs particle's mass

$m_h^2 = \lambda_H v^2 h^2$ .

The problem is that $λ H$ , giving the characteristics of the scalar potential, cannot be derived from the theory itself. All the other terms are ignored due to being 'unphysical'.

In the non-perturbative interpretation, the only additional terms are the last two terms seen above:

$\frac{1}{2} \mu^2 v^2$ ,

$-\frac{1}{4}\lambda_H v^4$ .

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Very Interesting...

The two higher order terms in the non-perturbative Higgs mechanism can be combined:

This term can be interpreted as an effective energy density

$\rho_H := \frac{\mu^4}{4 \lambda_H}$

Recalling these equations, this implies an effective cosmological constant (in natural units) of the form

In essence, the (non-perturbative) Higgs mechanism proposed here, i.e., the Higgs mechanism without a physical Higgs particle, is a simplification of the conventional view, resulting in mass terms for fermions and boson and an intrinsic energy density for the vacuum!

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References

A newsgroup summary
The paper
J.F. Gunion, H.E. Haber, G. Kane, S. Dawson; "The Higgs Hunter's Guide"; Addison-Wesley; 1990 (and most textbooks on gauge theory or quantum field theory)

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Categories: Physics

New Physics?

From TechWiki

Contents

The 5th Force

The Setup

The Solution

The Non-Perturbative Higgs Mechanism

The Idea

The Recipe

The Results

No Physical Higgs Boson

The Same Mass Terms

Additional Terms

Very Interesting...

References

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