Three QFT Models with Scalars | M.E. Irizarry-Gelpí

Sun 02 April 2017
Physics
#scalars

In this post I would like to consider three quantum field theories with scalar fields only that serve as toy models for certain supersymmetric gauge theories. Each model has two kinds of scalar fields: $N_{f}$ complex flavor fields and $N_{c}$ real color fields. The complex flavor fields will be denoted by $\Phi$ and labeled with an integer. Each complex flavor field has a distinct mass and coupling to the real color fields.

The A-Model

The interactions in the A-model are cubic. There is only one real color field ($N_{c} = 1$). For each complex flavor field, the interaction term is

$$ \Phi^{*} \Phi A. $$

If you look at an elastic flavor two-body process of the form

$$ \Phi_{1} + \Phi_{2} \longrightarrow \Phi_{1} + \Phi_{2}, $$

you see tree-level two-body interactions. These tree-level two-body interactions appear in all elastic processes, and can be used to build ladder diagrams and their generalizations. The four-point ladder diagrams for this model can have any non-negative integer number of loops.

The simplest tree-level process is the elastic four-point process mentioned above, with an exchange of $A$ quanta.

This model is important in six-dimensional spacetime (related to $\mathcal{N} = (2, 0)$ superconformal field theory).

The AB-Model

The interactions in the AB-model are quartic. There are two real color field ($N_{c} = 2$). For each complex flavor field, the interaction term is

$$ \Phi^{*} \Phi AB. $$

If you look at an elastic flavor two-body process of the form

$$ \Phi_{1} + \Phi_{2} \longrightarrow \Phi_{1} + \Phi_{2}, $$

you see one-loop-level two-body interactions. These one-loop-level two-body interactions appear in all elastic processes, and can be used to build "ladder" diagrams and their generalizations. When connected in series, the one-loop two-body interactions will lead to four-point ladder diagrams with only odd number of loops (e.g. 1, 3, 5, ...).

Besides one-loop two-body interactions, you can have one-loop many-body interactions with an even number of bodies. These many-body interactions are constructed with three-body interactions where the $A$ line connects to a different vertex from the $B$ line. For the two-body case, the loop in one-loop is a digon (i.e. two sides) so you cannot have a three-body interaction. The next case is four-body where the loop is a square (i.e. four sides). In this case you have four vertices, and each vertex is connected to two other vertices.

The simplest tree-level process is the elastic six-point process

$$ \Phi_{1} + \Phi_{2} + B \longrightarrow \Phi_{1} + \Phi_{2} + B $$

This corresponds to an exchange of $A$ quanta.

This model is important in four-dimensional spacetime (related to $\mathcal{N} = 4$ super Yang-Mills theory).

The ABCD-Model

The interactions in the ABCD-model are sextic. There are four real color field ($N_{c} = 4$). For each complex flavor field, the interaction term is

$$ \Phi^{*} \Phi ABCD. $$

If you look at an elastic flavor two-body process of the form

$$ \Phi_{1} + \Phi_{2} \longrightarrow \Phi_{1} + \Phi_{2}, $$

you see three-loop-level two-body interactions. These three-loop-level two-body interactions appear in all elastic processes, and can be used to build "ladder" diagrams and their generalizations. When connected in series, the three-loop two-body interactions will lead to four-point ladder diagrams with only odd number of loops (e.g. 3, 7, 11, ... or $L = 4n+3$).

Besides three-loop two-body interactions, you can have many-loop many-body interactions with an even number of bodies. These can be constructed with two-loop three-body interactions where each vertex has two digons coming out of it, or with four-body interactions where four lines come out of a vertex and connect to four distinct vertices.

There are six possible digons ($AB$, $AC$, $AD$, $BC$, $BD$, $CD$) and thus three possible three-body vertices (with $AB$ and $CD$; with $AC$ and $BD$; with $AD$ and $BC$). These three body vertices can be used to form polygons with digons as sides. A $2n$-body interaction constructed with such three-body vertices correspond to $L = 2n + 1$ loops, so only odd number of loops. For example, for the four-body interaction you get five loops.

The four-body interactions can be used to construct antiprisms. The simplest antiprism has two digons and four triangles giving a six-loop four-body interaction. This is different from the five-loop four-body interaction made with three-body interactions. An antiprism with $2n$ vertices gives a $2n$-body interaction with $2n+2$ faces and thus $L = 2n+2$ loops, so only even number of loops. Constructing the six-body interaction requires an antiprism with six vertices (an octahedron) and eight faces.

The simplest tree-level process is the elastic ten-point process

$$ \Phi_{1} + \Phi_{2} + B + C + D \longrightarrow \Phi_{1} + \Phi_{2} + B + C + D $$

This corresponds to an exchange of $A$ quanta.

This model is important in three-dimensional spacetime (related to $\mathcal{N} = 6$ ABJ(M) theory and $\mathcal{N} = 8$ BL theory).