Problems in Classical Electroweak Theory

The classical electroweak theory forms the core of the standard model of particle physics. The great success of both the electroweak theory and the standard model include e.g. the prediction of the intermediate vector bosons {W^\pm, Z} and the Higgs boson. In spite of its success, there are a number of issues and difficulties for the classical electroweak theory, which we will address in this blog post.

In the next blog post, we shall introduce the PID electroweak theory, resolving all these difficulties. In particular, the PID approach provides a first principle approach for introducing the Higgs field.

1. Classical electroweak theory

In essence, the electroweak theory is the generalization of the Fermi theory, and provides a useful computational tool for transition probability and amplitudes. It is a {U(1) \times SU(2)} gauge theory incorporating the Higgs field, and its main ingredients include

  • It involves three {SU(2)} gauge potentials, {W^1_\mu, W^2_\mu, W^3_\mu}, and and one {U(1)} potential {B_\mu};
  • The Higgs scalar doublet {\phi=(\phi^+, \phi^0)} was introduced into the Yang-Mills Lagrangian action in order to derive proper mass generation mechanism for the intermediate bosons.
  • With the gauge potentials, the following combinations are introduced to represent the intermediate vector bosons {W^\pm_\mu}, {Z_\mu} and the electromagnetic potential {A_\mu}, respectively:

    \displaystyle W^\pm_\mu =\frac{1}{\sqrt2} ( W^1_\mu \pm i W^2_\mu), \ \ \ \ \ (1)

    \displaystyle Z_{\mu}=\cos\theta_wW^3_{\mu}+\sin\theta_wB_{\mu}, \ \ \ \ \ (2)

    \displaystyle A_{\mu}=-\sin\theta_wW^3_{\mu}+\cos\theta_wB_{\mu}, \ \ \ \ \ (3)

2. Lack of weak force formulas

This problem is that all weak interaction theories have to face, and it is also that all existing theories cannot solve.

In fact, the classical electroweak theory, there are four gauge field potentials:

\displaystyle W^1_{\mu},\ W^2_{\mu},\ W^3_{\mu},\ B_{\mu},

and we don’t know which of these potentials plays the role of weak interaction potential.

3. Violation of Principle of Representation Invariance (PRI)

We have discovered a basic principle, called the principle of representation invariance (PRI), for the {SU(N)} gauge theory, which describes an {interacting} {N} particle system; see the previous post for details about PRI.

Elements in {SU(N)} are expressed as { \Omega =e^{i\theta^a\tau_a}}, where {\{\tau_1, \cdots ,\tau_{N^2-1}\}} is a basis of the set of traceless Hermitian matrices, and plays the role of a coordinate system in this representation. Consequently, an {SU(N)} gauge theory should be invariant under the following global transformation of the representation bases:

\displaystyle \tilde{\tau}_a=x^b_a\tau_b, \ \ \ \ \ (4)

where { X=(x^b_a) } is a a nondegenerate complex matrix. We call such invariance of the {SU(N)} gauge theory the principle of representation invariance (PRI).

PRI is a logic requirement for any gauge theory, and has profound physical consequences. In particular, by PRI, any linear combination of gauge potentials from two different gauge groups are prohibited.

In the classical electroweak theory, a key ingredient is the linear combinations of {W^3_{\mu}} and {B_{\mu}}. By PRI,

\displaystyle W^3_{\mu}\ \text{ is\ the\ third\ component\ of a}\ SU(2)\ \text{ tensor } \{W^a_\mu\},

\displaystyle B_{\mu}\ \text{ is\ the}\ U(1)\ \text{ gauge\ field}.

Hence, for the combinations of two different types of tensors:

\displaystyle Z_{\mu}=\cos\theta_wW^3_{\mu}+\sin\theta_wB_{\mu},

\displaystyle A_{\mu}=-\sin\theta_wW^3_{\mu}+\cos\theta_wB_{\mu},

used in the classical electroweak theory and the standard model of particle physics, violate PRI.

4. Decoupling obstacle

The classical electroweak theory has a difficulty for decoupling the electromagnetic and the weak interactions. In reality, electromagnetism and weak interaction often are independent to each other. Hence, as a unified theory for both interactions, one should be able to decouple the model to study individual interactions. However, the classical electroweak theory manifests a radical decoupling obstacle.

For example, if there is no weak interaction involved, then

\displaystyle W^{\pm}_{\mu}=0,\ \ \ \ Z_{\mu}=0, \ \ \ \ \ (5)

hold true. In this case, the theory should return to the {U(1)} gauge invariant Maxwell equations. But we see that

\displaystyle A_{\mu}=\cos\theta_wB_{\mu}-\sin\theta_wW^3_{\mu},

where {B_{\mu}} is a {U(1)} gauge field, and {W^3_{\mu}} is a component of {SU(2)} gauge field. Therefore, {A_{\mu}} is not independent of {SU(2)} gauge transformation. In particular, the condition (5) means

\displaystyle W^1_{\mu}=0,\ \ \ \ W^2_{\mu}=0,\ \ \ \ W^3_{\mu}=-\tan \theta_wB_{\mu}. \ \ \ \ \ (6)

Now we take the transformation (4) for the generators of {SU(2)}, {W^a_{\mu}} becomes

\displaystyle \left(\tilde{W}^1_{\mu}, \tilde{W}^2_{\mu}, \tilde{W}^3_{\mu} \right)=\left( y^1_3W^3_{\mu}, y^2_3W^3_{\mu}, y^3_3W^3_{\mu} \right),\ \ \ \ (y^b_a)^T=(x^b_a)^{-1}.

It implies that under a transformation (4), a nonzero weak interaction can be generated from a zero weak interaction field of (5)-(6):

\displaystyle \tilde{W}^{\pm}_{\mu}\neq 0,\ \ \ \ \tilde{Z}_{\mu}\neq 0\ \ \ \ \text{ as}\ y^a_3\neq 0\ (1\leq a\leq 3),

and the nonzero electromagnetic field {A_{\mu}\neq 0} will become zero:

\displaystyle \tilde{A}_{\mu}=0\ \ \ \ \text{ as}\ \ \ \ y^3_3=\cot \theta_w.

Obviously, it is not reality.

5. Artificial Higgs mechanism

In the classical electroweak action, the Higgs sector {\mathcal{L}_H} is not based on a first principle, and is artificially added into the action.

6. Presence of a massless and charged boson {\phi^+}

In the WS theory, the Higgs scalar doublet {\phi=(\phi^+, \phi^0)} contains a massless boson {\phi^+} with positive electric charge. Obviously there are no such particles in reality. In particular, the particle {\phi^+} is formally suppressed in the classical electroweak theory by transforming it to zero. However, from a field theoretical point of view, this particle field still represents a particle. This is one of major flaws for the electroweak theory and for the standard model.

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