compute the analytic structure of the Feynman Diagrams of a model - Maple Programming Help

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Physics[FeynmanDiagrams] - compute the analytic structure of the Feynman Diagrams of a model

 Calling Sequence FeynmanDiagrams(InteractionLagrangian, vertices = ..., numberoflegs = ..., numberofloops = ..., externallegs = ..., excludepropagators = ..., crossedpropagators = ..., includetadpoles = ..., fields = ...)

Parameters

 InteractionLagrangian - the interaction terms of a Lagrangian depending on quantum fields vertices = ... - (optional) the same as points = ..., list of points at which the Lagrangian is evaluated; these are the vertices of the corresponding Feynman diagrams; default value is 1 vertex numberoflegs = ... - (optional) the number of external legs of the Feynman diagrams, these legs are represented in the output as _NP(...) normal ordered products; default value is all possible cases numberofloops = ... - (optional) integer or set of integers; the number of loops of the Feynman diagrams: each line joining two vertices of the loop is a propagator, represented in the output as a list of two fields externallegs = ... - (optional) the same as normalproducts = ..., the right-hand side is a function _NP(...), or a set of them, where each _NP(...) contains fields or field names as arguments, which are the normal ordered products of unpaired fields expected in the output; to these unpaired fields are associated the external legs of the related Feynman diagrams excludepropagators = ... - (optional) a list of two fields representing a propagator, or a set of these lists, indicating that terms involving these propagators are to be excluded from the output; the right-hand side can also be one of the keywords free or cross crossedpropagators = ... - (optional) it can be true or false (default) to indicate that crossed propagators (related to crossed terms in the quadratic part of the Lagrangian) are not zero and so must be taken into account includetadpoles = ... - (optional) it can be the false (default) or true to include or discard terms containing tadpoles, that is, lines that start and end in the same vertex. fields = ... - (optional) either a function or its name, or a set or list of them; indicates the fields of the InteractionLagrangian;

Description

 • A scattering matrix $S$ relates the initial and final states of an interacting system. In an N-dimensional spacetime with coordinates $X$, S can be written as:

$S=T\left({ⅇ}^{I\left(∫L\left(X\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆX\right)}\right)$

 where $I$ is the imaginary unit, $L$ is the interaction Lagrangian, written in terms of quantum fields depending on the spacetime coordinates $X$. The $T$ symbol means time-ordered. For the terminology used in this page, see chapter IV of Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982. This exponential can be expanded as a sum of terms.

$S=1+{S}_{1}+{S}_{2}+{S}_{3}+\mathrm{...}$

 Each ${S}_{n}$ contains integrals of time-ordered products of the interaction Lagrangians evaluated at different points (vertices of the Feynman Diagram). For example, the second order term is of the form:

${S}_{2}=∫∫T\left(L\left(X\right),L\left(Y\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆX\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆY$

 where $X$ and $Y$ represent different points in spacetime.
 • Using Wick's theorems, the FeynmanDiagrams command computes the expanded form of these time-ordered products of interaction Lagrangians in the integrands of each ${S}_{n}$.
 • The expansion of each ${S}_{n}$consists of a sum of terms, each of which is a product of certain pairings of the operator field functions entering $L$ with the normal product of the remaining unpaired operators of free fields. The pairings between fields are the propagators, the Green functions of the associated free fields.
 • To each of the terms of the expansion of ${S}_{n}$ corresponds a Feynman diagram with $n$ vertices (the number of interaction Lagrangians in ${S}_{n}$). Within the diagram, each of the pairings of fields has a corresponding internal line, and each field entering the normal products of the unpaired field operators has a corresponding external leg. The output of the FeynmanDiagrams command is thus a sum of terms that has a one-to-one correspondence to the sequence of Feynman diagrams for the given interaction Lagrangian.
 NOTE: the current implementation of FeynmanDiagrams can expand up to ${S}_{3}$ (products of up to three interaction Lagrangians), and therefore can produce the analytic representation of Feynman diagrams with no more than three vertices.

Examples

In the Maple Standard GUI, to have the output of this help page use textbook notation, open it as a worksheet (see icon in the toolbar) and execute the input lines below after setting Physics[Setup](mathematicalnotation = true);

Load the package and set three coordinate systems to work in.

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$
 $\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$ (1)
 > $\mathrm{Coordinates}\left(X,Y,Z\right)$
 ${\mathrm{Default differentiation variables for d_, D_ and dAlembertian are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right)\right\}$
 ${\mathrm{Systems of spacetime Coordinates are:}}\left\{{X}{=}\left({\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right){,}{Y}{=}\left({\mathrm{y1}}{,}{\mathrm{y2}}{,}{\mathrm{y3}}{,}{\mathrm{y4}}\right){,}{Z}{=}\left({\mathrm{z1}}{,}{\mathrm{z2}}{,}{\mathrm{z3}}{,}{\mathrm{z4}}\right)\right\}$
 $\left\{{X}{,}{Y}{,}{Z}\right\}$ (2)
 > $L≔\mathrm{λ}{\mathrm{φ}\left(X\right)}^{4}+\mathrm{σ}{\mathrm{η}\left(X\right)}^{3}$
 ${L}{≔}{\mathrm{\lambda }}{}{{\mathrm{\phi }}{}\left({X}\right)}^{{4}}{+}{\mathrm{\sigma }}{}{{\mathrm{\eta }}{}\left({X}\right)}^{{3}}$ (3)

The expressions entering ${S}_{1}$ (only one vertex and so one evaluation point), representing the connected Feynman graphs for this interaction Lagrangian and discarding terms with tadpoles, is:

 > $\mathrm{FeynmanDiagrams}\left(L\right)$
 ${\mathrm{\lambda }}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{4}}:{+}{\mathrm{\sigma }}{}:{{\mathrm{\eta }}{}\left({X}\right)}^{{3}}:$ (4)

To include terms with tadpoles use the option includetadpoles = true or just includetadpoles

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{includetadpoles}\right)$
 ${6}{}{\mathrm{\lambda }}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{2}}:{}\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({X}\right)\right]{+}{\mathrm{\lambda }}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{4}}:{+}{\mathrm{\sigma }}{}:{{\mathrm{\eta }}{}\left({X}\right)}^{{3}}:$ (5)

You can filter these result to obtain only the tree-level terms (see the Options subsection of the Description).

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{numberofloops}=0,\mathrm{includetadpoles}\right)$
 ${\mathrm{\lambda }}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{4}}:{+}{\mathrm{\sigma }}{}:{{\mathrm{\eta }}{}\left({X}\right)}^{{3}}:$ (6)

After using FeynmanDiagrams once, the fields $\mathrm{phi}$ and $\mathrm{eta}$ get automatically set as quantum operators, so you do not need to set them as such again. Alternatively, you can always set them as quantumoperators (before or after calling FeynmanDiagrams) using the Setup command.

 > $\mathrm{Setup}\left(\mathrm{quantumop}=\left\{\mathrm{φ},\mathrm{η}\right\}\right)$
 ${\mathrm{* Partial match of \text{'}quantumop\text{'} against keyword \text{'}quantumoperators\text{'}}}$
 $\left[{\mathrm{quantumoperators}}{=}\left\{{\mathrm{\eta }}{,}{\mathrm{\phi }}\right\}\right]$ (7)

The structure of ${S}_{2}$ (that is, the expanded form of the time-ordered product of interaction Lagrangians entering the integrand of the second term in the expansion of the scattering matrix) is:

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right]\right)$
 ${16}{}{{\mathrm{\lambda }}}^{{2}}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{3}}{}{{\mathrm{\phi }}{}\left({Y}\right)}^{{3}}:{}\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]{+}{96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}{+}{72}{}{{\mathrm{\lambda }}}^{{2}}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{2}}{}{{\mathrm{\phi }}{}\left({Y}\right)}^{{2}}:{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{2}}{+}{18}{}{{\mathrm{\sigma }}}^{{2}}{}:{\mathrm{\eta }}{}\left({X}\right){}{\mathrm{\eta }}{}\left({Y}\right):{}{\left[{\mathrm{\eta }}{}\left({X}\right){,}{\mathrm{\eta }}{}\left({Y}\right)\right]}^{{2}}{+}{9}{}{{\mathrm{\sigma }}}^{{2}}{}:{{\mathrm{\eta }}{}\left({X}\right)}^{{2}}{}{{\mathrm{\eta }}{}\left({Y}\right)}^{{2}}:{}\left[{\mathrm{\eta }}{}\left({X}\right){,}{\mathrm{\eta }}{}\left({Y}\right)\right]$ (8)

In the result above, the tree-level terms involving propagators are associated with connected, one-particle-reducible Feynman graphs, and can be obtained by specifying $\mathrm{numberofloops}=0$.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberofloops}=0\right)$
 ${16}{}{{\mathrm{\lambda }}}^{{2}}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{3}}{}{{\mathrm{\phi }}{}\left({Y}\right)}^{{3}}:{}\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]{+}{9}{}{{\mathrm{\sigma }}}^{{2}}{}:{{\mathrm{\eta }}{}\left({X}\right)}^{{2}}{}{{\mathrm{\eta }}{}\left({Y}\right)}^{{2}}:{}\left[{\mathrm{\eta }}{}\left({X}\right){,}{\mathrm{\eta }}{}\left({Y}\right)\right]$ (9)

NOTE: the current implementation of FeynmanDiagrams can expand products of up to three interaction Lagrangians, so a maximum of three spacetime points are allowed, and therefore produces the analytic representation of Feynman diagrams with no more than three vertices.

Compute only the terms corresponding to Feynman diagrams with two external legs.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberoflegs}=2\right)$
 ${96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}{+}{18}{}{{\mathrm{\sigma }}}^{{2}}{}:{\mathrm{\eta }}{}\left({X}\right){}{\mathrm{\eta }}{}\left({Y}\right):{}{\left[{\mathrm{\eta }}{}\left({X}\right){,}{\mathrm{\eta }}{}\left({Y}\right)\right]}^{{2}}$ (10)

Compute only the terms corresponding to Feynman diagrams with two external legs corresponding to the field phi, regardless of the vertex to which they are attached (it could be X or Y).

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{normalproducts}=\mathrm{_NP}\left(\mathrm{φ},\mathrm{φ}\right)\right)$
 ${96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}$ (11)

The same computation but including tadpoles

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{normalproducts}=\mathrm{_NP}\left(\mathrm{φ},\mathrm{φ}\right),\mathrm{includetadpoles}\right)$
 ${72}{}{{\mathrm{\lambda }}}^{{2}}{}:{{\mathrm{\phi }}{}\left({X}\right)}^{{2}}:{}{\left[{\mathrm{\phi }}{}\left({Y}\right){,}{\mathrm{\phi }}{}\left({X}\right)\right]}^{{2}}{}\left[{\mathrm{\phi }}{}\left({Y}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]{+}{72}{}{{\mathrm{\lambda }}}^{{2}}{}:{{\mathrm{\phi }}{}\left({Y}\right)}^{{2}}:{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{2}}{}\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({X}\right)\right]{+}{96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}$ (12)

Compute only the terms with two external legs corresponding to the normal products $\mathrm{eta}\left(\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right),\mathrm{eta}\left(\mathrm{y1},\mathrm{y2},\mathrm{y3},\mathrm{y4}\right)$.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{normalproducts}=\mathrm{_NP}\left(\mathrm{φ}\left(X\right),\mathrm{φ}\left(Y\right)\right)\right)$
 ${96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}$ (13)

Compute the terms leading to Feynman graphs with two loops.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberofloops}=2\right)$
 ${96}{}{{\mathrm{\lambda }}}^{{2}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right):{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{3}}$ (14)

In ${S}_{2}$, among the terms resulting from expanding the time-ordered product of two interaction Lagrangians (Feynman diagrams with two vertices), there are no terms involving three loops.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberofloops}=3\right)$
 ${0}$ (15)

Compute the terms entering the integrand of ${S}_{3}$, having four external legs (normal ordered products involving four fields); in this example, these are of the form $\left[\mathrm{phi},\mathrm{eta}\right]$, where each field is applied at X, Y, or Z.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right],\left[Z\right]\right],\mathrm{numberoflegs}=4\right)$
 ${10368}{}{{\mathrm{\lambda }}}^{{3}}{}:{\mathrm{\phi }}{}\left({X}\right){}{\mathrm{\phi }}{}\left({Y}\right){}{{\mathrm{\phi }}{}\left({Z}\right)}^{{2}}:{}{\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Y}\right)\right]}^{{2}}{}\left[{\mathrm{\phi }}{}\left({X}\right){,}{\mathrm{\phi }}{}\left({Z}\right)\right]{}\left[{\mathrm{\phi }}{}\left({Y}\right){,}{\mathrm{\phi }}{}\left({Z}\right)\right]$ (16)

Following is an example of an interaction Lagrangian with a spinor-electromagnetic vertex. First, define the vector field ${A}_{\mathrm{mu}}$.

 > $\mathrm{Define}\left({A}_{\mathrm{μ}}\right)$
 ${\mathrm{Defined objects with tensor properties}}$
 $\left\{{{A}}_{{\mathrm{\mu }}}{,}{{\mathrm{\gamma }}}_{{\mathrm{\mu }}}{,}{{\mathrm{\sigma }}}_{{\mathrm{\mu }}}{,}{{X}}_{{\mathrm{\mu }}}{,}{{Y}}_{{\mathrm{\mu }}}{,}{{Z}}_{{\mathrm{\mu }}}{,}{{\partial }}_{{\mathrm{\mu }}}{,}{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\delta }}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{{\mathrm{\epsilon }}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right\}$ (17)

Check the type of letter used to represent spinor indices (or, change this setting according to your preference using Setup) and set $A$ as a quantum operator.

 > $\mathrm{Setup}\left(\mathrm{spinorindices}=\mathrm{lowercaselatin_is},\mathrm{anticommutativeprefix}=Q,\mathrm{op}=A\right)$
 ${\mathrm{* Partial match of \text{'}op\text{'} against keyword \text{'}quantumoperators\text{'}}}$
 $\left[{\mathrm{anticommutativeprefix}}{=}\left\{{Q}{,}{\mathrm{_λ}}\right\}{,}{\mathrm{quantumoperators}}{=}\left\{{A}{,}{\mathrm{\eta }}{,}{\mathrm{\phi }}\right\}{,}{\mathrm{spinorindices}}{=}{\mathrm{lowercaselatin_is}}\right]$ (18)

So, to represent spinor indices, use lowercase Latin letters, and the spinor fields will be represented by any name prefixed by $Q$, and the matrix indices related to the matrix product involving Dirac matrices are written explicitly in the interaction Lagrangian.

 > $L≔\mathrm{α}\stackrel{&conjugate0;}{Q[j]\left(X\right)}{{\mathrm{Dgamma}}_{\mathrm{μ}}}_{j,k}Q[k]\left(X\right)A[\mathrm{μ}]\left(X\right)$
 ${L}{≔}{\mathrm{\alpha }}{}\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{}{{Q}}_{{k}}{}\left({X}\right){}{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{j}{,}{k}}$ (19)

Compute the expanded form of the integrand entering ${S}_{2}$, only the terms with 4 external legs, so all the corresponding graphs have no loops.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberoflegs}=4\right)$
 $:\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{}{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){}{{Q}}_{{l}}{}\left({Y}\right){}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right):{}\left[{{Q}}_{{k}}{}\left({X}\right){,}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}{+}:{{Q}}_{{k}}{}\left({X}\right){}{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}{}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right):{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{,}{{Q}}_{{l}}{}\left({Y}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}{+}:{{Q}}_{{k}}{}\left({X}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{}{{Q}}_{{l}}{}\left({Y}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}:{}\left[{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){,}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}$ (20)

The terms entering the integrand of ${S}_{2}$ and that only have 2 external legs, so all the corresponding graphs have one loop.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right]\right],\mathrm{numberoflegs}=2\right)$
 $:{{Q}}_{{k}}{}\left({X}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}:{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{,}{{Q}}_{{l}}{}\left({Y}\right)\right]{}\left[{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){,}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}{+}:\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{}{{Q}}_{{l}}{}\left({Y}\right):{}\left[{{Q}}_{{k}}{}\left({X}\right){,}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}\right]{}\left[{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){,}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}{+}:{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right):{}\left[{{Q}}_{{k}}{}\left({X}\right){,}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}\right]{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{,}{{Q}}_{{l}}{}\left({Y}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{2}}$ (21)

Compute the expanded form of the integrand entering ${S}_{3}$ for this model.

 > $\mathrm{FeynmanDiagrams}\left(L,\mathrm{vertices}=\left[\left[X\right],\left[Y\right],\left[Z\right]\right]\right)$
 $:{{Q}}_{{k}}{}\left({X}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}{}{{A}}_{{\mathrm{\gamma }}}{}\left({Z}\right):{}\left[{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){,}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right)\right]{}\left[{{Q}}_{{l}}{}\left({Y}\right){,}\stackrel{{&conjugate0;}}{{{Q}}_{{m}}{}\left({Z}\right)}\right]{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{,}{{Q}}_{{n}}{}\left({Z}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\delta }}}\right)}_{{\mathrm{~m}}{,}{\mathrm{~n}}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{3}}{-}:{{Q}}_{{k}}{}\left({X}\right){}{{A}}_{{\mathrm{\alpha }}}{}\left({Y}\right){}\stackrel{{&conjugate0;}}{{{Q}}_{{m}}{}\left({Z}\right)}:{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{,}{{Q}}_{{l}}{}\left({Y}\right)\right]{}\left[\stackrel{{&conjugate0;}}{{{Q}}_{{i}}{}\left({Y}\right)}{,}{{Q}}_{{n}}{}\left({Z}\right)\right]{}\left[{{A}}_{{\mathrm{\mu }}}{}\left({X}\right){,}{{A}}_{{\mathrm{\gamma }}}{}\left({Z}\right)\right]{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\delta }}}\right)}_{{\mathrm{~m}}{,}{\mathrm{~n}}}{}{\left({{\mathrm{\gamma }}}_{{\mathrm{\beta }}}\right)}_{{\mathrm{~i}}{,}{\mathrm{~l}}}{}{\left({{\mathrm{\gamma }}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\right)}_{{\mathrm{~j}}{,}{\mathrm{~k}}}{}{{\mathrm{\alpha }}}^{{3}}{-}:\stackrel{{&conjugate0;}}{{{Q}}_{{j}}{}\left({X}\right)}{}{{Q}}_{{l}}{}\left({Y}\right){}{{A}}_{{\mathrm{\gamma }}}{}\left({Z}\right):$