Wilson Loop Expectations for Non-abelian Finite Gauge Fields Coupled to a Higgs Boson at Low and High Disorder

Abstract

We consider computations of Wilson loop expectations to leading order at large $\beta$ in the case where a non-abelian finite gauge field interacts with a Higgs boson. By identifying the main order contributions from minimal vortices, we can express the Wilson loop expectations via an explicit Poisson random variable. This paper treats multiple cases of interests, including the Higgs boson at low and high disorder, and finds efficient polymer expansion like computations for each of these regimes.

Appendices

Details on Vertex Decompositions

In this section, we will describe assorted facts on Higgs boson vertex configurations that would be useful for our decomposition theorem.

The following lemma details the behaviors of the Higgs boson charges on the complement of the support of a configuration.

Lemma A.1

Let P be the support of a configuration $\mathcal {C}$ of Higgs boson and gauge fields. As previously, we use V(P) and E(P) to denote the set of vertices and edges that are associated with P. Let the complement of V(P) in $V_N$ be divided into separate connected components $V(P)^c = CC_1 \cup CC_2 \cup \cdots \cup CC_N$. Then the following statements hold,

• There is a single charge $c_i$ such that each vertex $v_i$ in $CC_i$ is assigned the same Higgs boson charge $c_i$, e.g. $\phi _{v_i} = c_i, \forall v_i \in CC_i$.

• If v is a vertex in V(P) that is connected by an edge in $E_N$ to a vertex $v_i \in CC_i$, then $\phi _v =c_i$, where $c_i$ is the common color of each vertex in $CC_i$.

Proof

We start with the proof of the first item.

If $CC_i$ is not monocharged, then there are two vertices $v_i,w_i$ in $CC_i$ that are not the same color as well as a path $p_i$ consisting entirely of vertices in $CC_i$ connecting $v_i$ and $w_i$. At least one of the edges, $e_i$, on this path p must have opposite charges on its neighboring vertices. This would imply that $e_i$ would be in $\text {supp}(\mathcal {C})$; furthermore, it would imply that the vertices of this edge would belong in V(P) rather than the complement. This is a contradiction. This proves the first item.

Now, we prove the second item. If $v_i$ is a vertex in $CC_i$ and w is a vertex in V(P) adjacent to v that does not have the same charge $c_i$, then the edge $e=(v_i,w)$ w has the same charge as v would be in $\text {supp}(\mathcal {C})$. Thus, the vertex v must be in V(P). This is a contradiction. $\square$

The behavior of the Higgs boson configurations is much like the Ising model. One can imagine regions of monocharged components containing regions of other monocharged components recursively. It is important to understand the relationship between these monocharged components and the support of the configuration. The following lemma details these relationships.

Lemma A.2

Let $(\sigma ,\phi )$ be a configuration $\mathcal {C}$ of Higgs boson and gauge fields. Let V be a some connected monocharged set(i.e., all the vertices are assigned the same Higgs boson charge) in $V_N$ and define the set MC(V) as follows,

$$\begin{aligned} MC(V):= \{w \in V_N: \exists \text { path } p(v \rightarrow w) \text { s.t. } \forall \text { vertices } a \in p, \phi _a = \phi _w \}. \end{aligned}$$

(A.1)

Namely, MC(V) is the monocharged cluster of vertices in $V_N(V)$ that can be connected to V with vertices of the same Higgs boson charge as V.

We define EC(MC(V)), the ‘exterior connection’, as the set of edges connecting vertices in MC(V) to its complement in $V_N$.

$$\begin{aligned} EC(MC(V)):=\{e=(v,w) \in E_N: v \in MC(V), w \in {MC(V)}^c\}. \end{aligned}$$

(A.2)

We finally define PB(MC(V)) , the ‘plaquette boundary’, to be the set of plaquettes in $P_N$ that have one of the edges in EC(MC(V)) as a boundary edge. Namely,

$$\begin{aligned} PB(MC(V)):=\{ p \in P_N: \exists e \in EC(MC(V)) \text{ s.t. } e \in \delta p \}. \end{aligned}$$

(A.3)

Then, PB(MC(V)) is a subset of $\text {supp}(\mathcal {C})$. As a shorthand, we can call PB(MC(V)) the boundary of MC(V).

Now assume further that there is no vertex in MC(V) that is a boundary vertex of $\delta \Lambda _N$. We can make the following statements on the decomposition of PB(MC(V)) into connected components.

There is a unique connected component, which we will call the external boundary of MC(V); we will denote this by EB(MC(V)). It satisfies the following properties.

• EB(MC(V)) is connected.

• Embed the subset MC(V) into the full lattice ${\mathbb {Z}}^d$ and split ${MC(V)}^c$ in ${\mathbb {Z}}^d$ into its connected components ${MC(V)}^c = CC_1 \cup CC_2 \cup \cdots \cup CC_N$. Let $CC_1$ be the unique non-compact component, so it extends to $\infty$. Define $EC(CC_1)$ to be the set of edges connecting $EC(CC_1)$ to its complement and $PB(CC_1)$ to be the set of plaquettes having at least one edge from $EC(CC_1)$ on its boundary. Then, $EB(MC(V)) = PB(CC_1)$.

Remark A.1

The statement that PB(MC(V)), as we have defined it above, is in $\text {supp}(\mathcal {C})$ is by definition. The second part of the above lemma is to formally state the intuition that a set that is monocharged but connected and compact is separated from the outside vertices that do not share its charge by a single connected boundary. All other boundaries of the set MC(V) are internal and separate it from the islands of charge that are internal to MC(V).

Notation

• G and H are used for group names, while g and h are used for group elements.

• $\sigma$ is used to represent the gauge field configuration while $\sigma ^g$ represents field configurations with a gauge fixing with respect to a tree. item $\rho$ denotes a representation of a group.

• $\phi$ is used to denote Higgs field configurations.

• $\eta$ is used to denote an auxiliary field resulting from gauging.

• $W_{\gamma }$ is used to denote the Wilson Loop Action.

• E is the notation used for generic sets of edges, with e for an individual edge. P is used as the notation for generic sets of plaquettes with p for a given plaquette, and V is notation for generic sets of vertices, with v used for individual vertices.

• ${\mathcal {V}}$ is used for vortices while ${\mathcal {K}}$ is used for knots.

• ${\mathcal {H}}$ and $\mathfrak {H}$ will be used to denote Hamiltonian values.

• $f^u$ and $\mathfrak {f}^u$ is shorthand for our Higgs boson action in the Hamiltonian.

• Z and ${\mathcal {Z}}$ are used for the partition functions, the former with respect to an event, and the latter used for plaquette sets in the support.

• There are many quantities that have analogous versions in Sects. 2, Sect. 4, and Sect. 5. ${\tilde{\cdot }}$ is used in Sect. 4, while ${\hat{\cdot }}$ is used in Sect. 5. The vanilla version is used in 2. Some examples of notation that behave this way are the constants $\alpha _{\beta ,\kappa }$, $A_{\beta ,\kappa }$, ${\mathfrak {c}}$ and the functions $\Phi$.

• $\Phi$ denotes our polymer counting function.

• $\Omega$ denotes events under consideration.

• $\alpha _{\beta ,\kappa }$ appears as a counting bound for our polymer counting function.

• $A_{\beta ,\kappa }$ is the constant that appears as the basis of the exponent in the main order term of our Wilson loop expectation.

• ${\mathfrak {c}}$ appears intermediately as a ratio of the probability of ‘bad’ events to ‘good; events.