The Higgs Hunter's Guide

John F. Gunion

• Verlag: CRC Press
• Jahr: 2018
• ISBN:
  9780429976070
• Auflage: 1
• Thema: Physical Sciences
• Thema: Physics

The Higgs Hunter's Guide is a definitive and comprehensive guide to the physics of Higgs bosons. In particular, it discusses the extended Higgs sectors required by those recent theoretical approaches that go beyond the Standard Model, including supersymmetry and superstring-inspired models.

Chapter 1

Introduction and Preliminaries

Although the Higgs mechanism [1] was used to introduce mass into the Standard Model [2,3] two decades ago, experimental sensitivity to a Standard Model Higgs boson remains extremely limited. Masses below about 2m_μ can be excluded by a combination of low energy experimental data on nuclear transitions and rare decays of K mesons. Recent results in K and B decays probably rule out masses from 2m_μ to 2m_τ. Upsilon decays are potentially sensitive to masses above 2m_μ, up to about 5 GeV, but uncertainties regarding the exact magnitude of the expected decay rate to Higgs prevent firm conclusions at this time; although no Higgs bosons have been observed in such decays. Certainly, it will be 1990 (at the earliest) before experiments begin to probe the mass region above 5 GeV, where one might most naively expect to find the Standard Model Higgs boson.

As is often emphasized [4, 5, 6], the Higgs sector of the Standard Model is not understood from a fundamental point of view, although it performs technically in an entirely satisfactory way as an effective low energy theory, without conflict or contradiction. The physics that underlies electroweak symmetry breaking is simply not certain, although theorists may have guessed how it functions.

1.1 The Higgs Mechanism

Let us review the Higgs mechanism, to recall how the Higgs boson arises as the direct physical manifestation of the origin of mass in the Standard Model.^* The Standard Model is a gauge theory. The SU(2) × U(1) gauge invariance of the theory requires masses of the gauge bosons to be zero, since the presence of a mass term for the gauge bosons violates gauge invariance (M²A_μA^μ is not invariant if A_μ → A_μ – ∂_μχ where χ is a function of position in space-time, so M² must be zero). The Higgs mechanism circumvents this constraint by beginning with a gauge invariant theory having massless gauge bosons, and ending with a spectrum having massive gauge bosons, after algebraic transformations on the Lagrangian. The physics leading to a gauge boson mass and a physical Higgs boson is contained in the simple Abelian case, which we now review.

Assume there exists a complex scalar boson ϕ and a massless gauge boson A^μ. Assume the Lagrangian of the theory has the form

$\mathcal{L}={\left(D_{\mu }\varphi \right)}^{*}\left(D^{\mu }\varphi \right)+{\mu }^2{\varphi }^{*}\varphi -\lambda {\left({\varphi }^{*}\varphi \right)}^2-\frac{1}{4}{F}^{\mu v}{F}_{\mu v}.$

(1.1)

The parameters are constrained by λ > 0 (so that the potential is bounded from below), and μ² > 0. F^μν is the antisymmetric tensor of the gauge boson field, F^μν = ∂^μA^ν − ∂^νA^μ. Invariance of the theory under a local gauge transformation,

$\begin{array}{l}\varphi \to \varphi \text{'}=e^{ig\chi \left(x\right)}\varphi \\ A^{\mu }\to A{\text{'}}^{\mu }=A^{\mu }-{\partial }^{\mu }\chi \left(x\right),\end{array}$

(1.2)

is guaranteed if in the Lagrangian we use the covariant derivative D^μ = ∂^μ + igA^μ, in place of the ordinary partial derivative ∂^μ.

The potential for the scalar field has its minimum value at $\varphi =v/\sqrt{2}=\sqrt{{\mu }^2/2\lambda }$. It is appropriate to expand ϕ near its minimum to find the spectrum of the theory, so write

$\varphi =\left[v+h\left(x\right)\right]/\sqrt{2}$

(1.3)

where where h(x) is a real field. Substituting this into $\mathcal{L}$, we have explicitly

$\begin{array}{l}\mathcal{L}=\frac{1}{2}\left[\left({\partial }_{\mu }-ig{A}_{\mu }\right)\left(v+h\right)\left({\partial }^{\mu }-ig{A}^{\mu }\right)\left(v+h\right)\right]\\ +\frac{1}{2}{\mu }^2{\left(v+h\right)}^2-\frac{1}{4}\lambda {\left(v+h\right)}^4-\frac{1}{4}{F}^{\mu v}{F}_{\mu v}.\end{array}$

(1.4)

This contains several important terms. There is a term $\left(g^2{v}^2/2\right)A_{\mu }{A}^{\mu }$ that should be interpreted as a mass term for the gauge boson. There is a term $-\lambda v^2{h}^2$ that is a mass term for the scalar boson. There are interaction terms h³, h⁴, hAA, and h²AA, with related strengths. The theory with a complex scalar boson and a massless gauge boson has been reinterpreted as a theory with a real scalar boson and a massive gauge boson, because the scalar potential had its minimum at a value of ϕ that was non-zero. This way of giving mass to the gauge boson is called the Higgs mechanism [1].

Four things should be emphasized for our purposes. First there is a real boson, h, that should occur as a physical boson—the Higgs boson. Second, its mass depends on λ and on v. The gauge boson mass determines v, but λ is a parameter characteristic of the scalar potential and no one has ever found a way to calculate or determine λ without finding experimental information about the Higgs spectrum itself. Therefore the mass of the Higgs boson is unknown. Third, the interaction terms (plus those that occur when fermions are given mass) determine the production mechanisms and decays of the Higgs boson (i.e., how it couples to particles that are accelerated or ...