Scalar chromodynamics

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (July 2022) (Learn how and when to remove this message) An editor has performed a search and found that sufficient sources exist to establish the subject's notability. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Scalar chromodynamics" – news · newspapers · books · scholar · JSTOR (July 2022) (Learn how and when to remove this message) (Learn how and when to remove this message)

In quantum field theory, scalar chromodynamics, also known as scalar quantum chromodynamics or scalar QCD, is a gauge theory consisting of a gauge field coupled to a scalar field. This theory is used experimentally to model the Higgs sector of the Standard Model.

It arises from a coupling of a scalar field to gauge fields. Scalar fields are used to model certain particles in particle physics; the most important example is the Higgs boson. Gauge fields are used to model forces in particle physics: they are force carriers. When applied to the Higgs sector, these are the gauge fields appearing in electroweak theory, described by Glashow–Weinberg–Salam theory.

This article discusses the theory on flat spacetime ${\displaystyle \mathbb {R} ^{1,3}}$, commonly known as Minkowski space.

The model consists of a complex vector valued scalar field ${\displaystyle \phi }$ minimally coupled to a gauge field ${\displaystyle A_{\mu }}$.

The gauge group of the theory is a Lie group ${\displaystyle G}$. Commonly, this is ${\displaystyle {\text{SU}}(N)}$ for some ${\displaystyle N}$, though many details hold even when we don't concretely fix ${\displaystyle G}$.

The scalar field can be treated as a function ${\displaystyle \phi :\mathbb {R} ^{1,3}\rightarrow V}$, where ${\displaystyle (V,\rho ,G)}$ is the data of a representation of ${\displaystyle G}$. Then ${\displaystyle V}$ is a vector space. The 'scalar' refers to how ${\displaystyle \phi }$ transforms (trivially) under the action of the Lorentz group, despite ${\displaystyle \phi }$ being vector valued. For concreteness, the representation is often chosen to be the fundamental representation. For ${\displaystyle {\text{SU}}(N)}$, this fundamental representation is ${\displaystyle \mathbb {C} ^{N}}$. Another common representation is the adjoint representation. In this representation, varying the Lagrangian below to find the equations of motion gives the Yang–Mills–Higgs equation.

Each component of the gauge field is a function ${\displaystyle A_{\mu }:\mathbb {R} ^{1,3}\rightarrow {\mathfrak {g}}}$ where ${\displaystyle {\mathfrak {g}}}$ is the Lie algebra of ${\displaystyle G}$ from the Lie group–Lie algebra correspondence. From a geometric point of view, ${\displaystyle A_{\mu }}$ are the components of a principal connection under a global choice of trivialization (which can be made due to the theory being on flat spacetime).

The Lagrangian density arises from minimally coupling the Klein–Gordon Lagrangian (with a potential) to the Yang–Mills Lagrangian.^[1]: 102 Here the scalar field ${\displaystyle \phi }$ is in the fundamental representation of ${\displaystyle {\text{SU}}(N)}$:

where

• ${\displaystyle F_{\mu \nu }}$ is the gauge field strength, defined as ${\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }+ig[A_{\mu },A_{\nu }]}$. In geometry this is the curvature form.
• ${\displaystyle D_{\mu }\phi }$ is the covariant derivative of ${\displaystyle \phi }$, defined as ${\displaystyle D_{\mu }\phi =\partial _{\mu }\phi -ig\rho (A_{\mu })\phi .}$
• ${\displaystyle g}$ is the coupling constant.
• ${\displaystyle V(\phi )}$ is the potential.
• ${\displaystyle {\text{tr}}}$ is an invariant bilinear form on ${\displaystyle {\mathfrak {g}}}$, such as the Killing form. It is a typical abuse of notation to label this ${\displaystyle {\text{tr}}}$ as the form often arises as the trace in some representation of ${\displaystyle {\mathfrak {g}}}$.

This straightforwardly generalizes to an arbitrary gauge group ${\displaystyle G}$, where ${\displaystyle \phi }$ takes values in an arbitrary representation ${\displaystyle \rho }$ equipped with an invariant inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$, by replacing ${\displaystyle (D_{\mu }\phi )^{\dagger }D^{\mu }\phi \mapsto \langle D_{\mu }\phi ,D^{\mu }\phi \rangle }$.

The model is invariant under gauge transformations, which at the group level is a function ${\displaystyle U:\mathbb {R} ^{1,3}\rightarrow G}$, and at the algebra level is a function ${\displaystyle \alpha :\mathbb {R} ^{1,3}\rightarrow {\mathfrak {g}}}$.

At the group level, the transformations of fields is^[2]

${\displaystyle \phi (x)\mapsto U(x)\phi (x)}$

${\displaystyle A_{\mu }(x)\mapsto UA_{\mu }U^{-1}-{\frac {i}{g}}(\partial _{\mu }U)U^{-1}.}$

From the geometric viewpoint, ${\displaystyle U(x)}$ is a global change of trivialization. This is why it is a misnomer to call gauge symmetry a symmetry: it is really a redundancy in the description of the system.

The theory admits a generalization to a curved spacetime ${\displaystyle M}$, but this requires more subtle definitions for many objects appearing in the theory. For example, the scalar field must be viewed as a section of an associated vector bundle with fibre ${\displaystyle V}$. This is still true on flat spacetime, but the flatness of the base space allows the section to be viewed as a function ${\displaystyle M\rightarrow V}$, which is conceptually simpler.

If the potential is minimized at a non-zero value of ${\displaystyle \phi }$, this model exhibits the Higgs mechanism. In fact the Higgs boson of the Standard Model is modeled by this theory with the choice ${\displaystyle G={\text{SU}}(2)}$; the Higgs boson is also coupled to electromagnetism.

By concretely choosing a potential ${\displaystyle V}$, some familiar theories can be recovered.

Taking ${\displaystyle V(\phi )=M^{2}\phi ^{\dagger }\phi }$ gives Yang–Mills minimally coupled to a Klein–Gordon field with mass ${\displaystyle M}$.

Taking ${\displaystyle V(\phi )=\lambda (\phi ^{\dagger }\phi )^{2}-\mu _{H}^{2}\phi ^{\dagger }\phi }$ gives the potential for the Higgs boson in the Standard Model.

• Scalar electrodynamics
• Quantum chromodynamics