order = m : $m$ indicates the truncation order of the expansion $S=1+S_1+S_2+\mathrm{...}$ of the scattering matrix $S$ in coordinates representation, or of the corresponding matrix element $⟨f|S|i⟩$ in momentum representation. If not indicated, in coordinates representation the default value of order is 3, and in momentum representation it is automatically determined from the indicated incomingparticles, outgoingparticles and numberofloops.

incomingparticles = ... : when given, the output is the matrix element $⟨f|S|i⟩$ in momentum representation; incomingparticles is a list of field names representing the incoming particles $\left|i\right.⟩$, and all of outgoingparticles and numberofloops should also be given.

outgoingparticles = ... : (required when incomingparticles are indicated), a list of field names representing the outgoing particles $\left.⟨f\right|$, in $⟨f|S|i⟩$.

numberofloops = n : (required when incomingparticles are indicated, otherwise optional) the right-hand side is an integer, or a set of integers. This option indicates that the returned expansion $S=1+S_1+S_2+\mathrm{...}$ in coordinates representation, or the matrix element $⟨f|S|i⟩$ in momentum representation, should only include terms whose corresponding Feynman diagrams contain $n$ loops, or the numbers of loops indicated in the set of integers. When working in coordinates representation and numberofloops is not indicated, all the terms containing up to 3 loops are returned.

diagrams : synonyms: graphs, plot, produces, in addition, the display of the Feynman diagrams (drawings) corresponding to each term of the algebraic expression being returned. In coordinates representation, the correspondence between the terms in the output and diagrams is one-to-one. In momentum representation, only the diagrams representing different topologies are displayed.

output = ... : (related to momentum representation), if given, the right-hand side is probabilitydensity, so that given s incomingparticles and r outgoingparticles, the output is the average number of scattered particles per unit time and volume, having momenta within the infinitesimal volumes ${\mathrm{dp}}_1,\mathrm{...},{\mathrm{dp}}_r$. That transition probability density is constructed using $⟨f|S|i⟩$ and the formula (13) of Sec.21 of ref.[1]

reduciblegraphs = ... : (optional) it can be true or false (default), to include or discard from the expansion of $S$, or from the matrix element $⟨f|S|i⟩$, the terms corresponding to graphs with loops that are 1-particle reducible.

includetadpoles = ... : (optional) it can be true or false (default), to include or discard from the expansion of $S$, or from the matrix element $⟨f|S|i⟩$, the terms containing tadpoles, that is, lines that start and end in the same vertex.

numberofvertices = n : $n$ indicates that the expected output is just $S_n$, the $n^{\mathrm{th}}$ term of the expansion of $S$ in coordinates representation, or just $⟨f|S_n|i⟩$ in momentum representation. The Feynman diagrams of $S_n$ are the ones that contain $n$ vertices.

numberofexternallegs = ... : synonym numberoflegs, related to coordinates representation, the right-hand side is a positive integer, or a set of them, indicating the number of external legs of the Feynman diagrams corresponding to each term of the algebraic expression returned. Default value is all possible configurations of external legs.

normalproducts = ... : related to coordinates representation, the right-hand side is a function _NP(...), or a set of them, where each _NP(...) contains field names as arguments. These are the normal ordered products of unpaired fields entering $S=1+S_1+S_2+\mathrm{...}$. To these unpaired fields are associated the external legs of the related Feynman diagrams. This option is useful to specify one process (identified by the number and kind of external legs in _NP(...)) among all the possible configurations of external legs returned by default.

excludepropagators = ... : 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.

numberofverticesinloops = n : $n$ indicates that the expected output, either $S=1+S_1+S_2+\mathrm{...}$ in coordinates representation, or $⟨f|S|i⟩$, in momentum representation, should contain only terms where the loops contain n vertices (e.g. 3 = triangle, 4 = square, etc.)

putindicesinsquareofmomentum = ... : (optional), related to momentum representation, it can be true or false (default), to put spacetime indices in all the occurrences of squares of momentum within Feynman integrals. By default these indices are put only when the matrix element $⟨f|S|i⟩$ being returned contains no integrals.

gauge = ... : the right-hand side can be Feynman, Landau, unitary or arbitrary, respectively resulting in the values 1, 0, $\mathrm{\infty }$ and unset for the FeynmanDiagrams:-xi gauge parameter.

usepropagators = ... : the right-hand side can be true (default) or false; when set to false FeynmanDiagrams returns using an abstract representation in terms of FeynmanDiagrams:-Propagator, instead of a specific propagator whose form depends on the the spin of the underlying field.

externalnormalization = ... : the right-hand side can be true (default) or false. When true, the normalization factor related to the external legs follows the conventions of [1]; when false, these normalization factors are all set equal to 1.

simplifytensorindices = ... : (optional), related to momentum representation, it can be true or false (default), to perform a simplification of the tensor indices appearing in the returned $⟨f|S|i⟩$.

factortreelevel = ... : (optional), related to momentum representation, it can be true or false (default), to factor the tree-level terms appearing in the returned $⟨f|S|i⟩$.

labels = ... : related to coordinates representation, the right-hand side is a list of lists with labels (Capital letters), say $\left[\left[X\right]\,\left[Y\right]\,\mathrm{...}\right]$, representing the spacetime points used as dummy integration variables in each $S_n$. These labels also represent the points at which the interaction Lagrangians are evaluated when computing their time-ordered- products entering those integrands. Default value is the dependency of the fields of the interaction Lagrangian, as well as any other system of coordinates defined using the Coordinates command.

crossedpropagators = ... : related to coordinates representation, 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. This option only works up to order = 3

quiet : (optional) indicates that the computation should proceed without displaying messages on the screen, for instance regarding the matching of optional keywords

A scattering matrix $S$ relates the initial and final states, $\left|i\right.⟩$ and $\left|f\right.⟩$, of an interacting system. In an 4-dimensional spacetime with coordinates $X$, $S$ can be written as:

where $i$ is the imaginary unit and $L$ 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 for instance chapter IV, "The Scattering Matrix", in ref.[1]. This exponential can be expanded as

where

and $T\left(L\left(X_1\right)\,\mathrm{...}\,L\left(X_n\right)\right)$ is the time-ordered product of n interaction Lagrangians evaluated at different points. Note the factor $\frac{i^n}{n!}$ included in the definition of $S_n$ used in this page.

with(Physics):

Setup(mathematicalnotation = true, coordinates = [X, Y, Z], quantumoperators = phi);

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\,Z=\left(\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\,Y\,Z\right\}\,\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{quantumoperators}=\left\{\mathrm{\phi }\right\}\right]$

L := lambda*phi(X)^4;

$L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^4$

S = FeynmanDiagrams(L);

$S=1+\mathrm{I}\int \mathrm{\lambda }:{\mathrm{\phi }\left(X\right)}^4:\ⅆX^4+\frac{{\mathrm{I}}^2}{2!}\int \int 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\ⅆX^4\ⅆY^4+\frac{{\mathrm{I}}^3}{3!}\int \int \int 1728{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2{\mathrm{\phi }\left(Y\right)}^2{\mathrm{\phi }\left(Z\right)}^2:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]+10368{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2\mathrm{\phi }\left(Y\right)\mathrm{\phi }\left(Z\right):\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^2+2592{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2{\mathrm{\phi }\left(Y\right)}^2:{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]}^2{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^2+10368{\mathrm{\lambda }}^3:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]}^2{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^2+3456{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^3+576{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3{\mathrm{\phi }\left(Z\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]\ⅆX^4\ⅆY^4\ⅆZ^4$

S__treelevel = FeynmanDiagrams(L, numberofloops = 0, diagrams);

$\mathrm{S\_\_treelevel}=1+\mathrm{I}\int \mathrm{\lambda }:{\mathrm{\phi }\left(X\right)}^4:\ⅆX^4+\frac{{\mathrm{I}}^2}{2!}\int \int 16{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\ⅆX^4\ⅆY^4+\frac{{\mathrm{I}}^3}{3!}\int \int \int 576{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3{\mathrm{\phi }\left(Z\right)}^2:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]\ⅆX^4\ⅆY^4\ⅆZ^4$

FeynmanDiagrams(L, incomingparticles = [phi, phi], outgoingparticles = [phi, phi], numberofloops = 0, diagrams);

$\frac{\frac{3\mathrm{I}}{2}\mathrm{\lambda }\mathrm{\delta }\left({\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}-{\mathrm{P\_\_3}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}-{\mathrm{P\_\_4}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}{{\mathrm{\pi }}^2\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}\mathrm{E\_\_3}\mathrm{E\_\_4}}}$

FeynmanDiagrams(L, numberofexternallegs = 2, diagrams);

$\frac{{\mathrm{I}}^2}{2!}\int \int 96{\mathrm{\lambda }}^2:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^3\ⅆX^4\ⅆY^4+\frac{{\mathrm{I}}^3}{3!}\int \int \int 3456{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^3+10368{\mathrm{\lambda }}^3:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]}^2{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^2\ⅆX^4\ⅆY^4\ⅆZ^4$

FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 2, diagrams, output = probabilitydensity);

$\frac{\left(\prod _{i=1}^1{n}_i\right){\left|F\right|}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(\prod _{f=1}^1{{\overrightarrow{\mathrm{dP}}}^{}}_f^3\right)}{2\mathrm{\pi }}\mathbf{where}F=\int \int \frac{\frac{3\mathrm{I}}{8}{\mathrm{\lambda }}^2}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}+\mathrm{p\_\_2}+\mathrm{p\_\_3}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4\ⅆ{\mathrm{p\_\_3}}^4$

F = FeynmanDiagrams(L, incoming = [phi], outgoing = [phi], numberofloops = 2, putindices);

$F=\int \int \frac{\frac{3\mathrm{I}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}+{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right)}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left(\left({\mathrm{P\_\_1}}_{\mathrm{\mu }}+{\mathrm{p\_\_2}}_{\mathrm{\mu }}+{\mathrm{p\_\_3}}_{\mathrm{\mu }}\right)\left({\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+{\mathrm{p\_\_2}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+{\mathrm{p\_\_3}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}_{\mathrm{\nu }}{\mathrm{p\_\_2}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}_{\mathrm{\alpha }}{\mathrm{p\_\_3}}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4\ⅆ{\mathrm{p\_\_3}}^4$

FeynmanDiagrams(L, numberofloops = 3);

$\frac{{\mathrm{I}}^3}{3!}\int \int \int 3456{\mathrm{\lambda }}^3:{\mathrm{\phi }\left(X\right)}^2:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^3+10368{\mathrm{\lambda }}^3:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Z\right)\right]}^2{\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Z\right)\right]}^2\ⅆX^4\ⅆY^4\ⅆZ^4$

FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 3);

$\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{11}\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{p\_\_3}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\left(-\mathrm{P\_\_1}+\mathrm{P\_\_2}-\mathrm{p\_\_3}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_3}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+2\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{11}\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(-\mathrm{P\_\_2}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\left(-\mathrm{P\_\_1}+\mathrm{P\_\_2}-\mathrm{p\_\_3}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_3}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{11}\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(-\mathrm{P\_\_1}-\mathrm{p\_\_2}+\mathrm{P\_\_2}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\left(-\mathrm{P\_\_2}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4$

S[2] = FeynmanDiagrams(L, numberofvertices = 2, includetadpoles);

$S_2=\frac{{\mathrm{I}}^2}{2!}\int \int 16{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]+144{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^2:{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^2\left[\mathrm{\phi }\left(Y\right)\,\mathrm{\phi }\left(Y\right)\right]+96{\mathrm{\lambda }}^2:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^3+72{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^2{\mathrm{\phi }\left(Y\right)}^2:{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^2\ⅆX^4\ⅆY^4$

Setup(quantumoperators = eta);

$\left[\mathrm{quantumoperators}=\left\{\mathrm{\eta }\,\mathrm{\phi }\right\}\right]$

L__2 := lambda*phi(X)^4 + sigma*eta(X)^3;

$\mathrm{L\_\_2}≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^4+\mathrm{\sigma }{\mathrm{\eta }\left(X\right)}^3$

FeynmanDiagrams(L__2, normalproducts = _NP(phi, eta));

$0$

S__treelevel = FeynmanDiagrams(L__2, order = 2, numberofloops = 0);

$\mathrm{S\_\_treelevel}=1+\mathrm{I}\int \mathrm{\sigma }:{\mathrm{\eta }\left(X\right)}^3:+\mathrm{\lambda }:{\mathrm{\phi }\left(X\right)}^4:\ⅆX^4+\frac{{\mathrm{I}}^2}{2!}\int \int 16{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]+9{\mathrm{\sigma }}^2:{\mathrm{\eta }\left(X\right)}^2{\mathrm{\eta }\left(Y\right)}^2:\left[\mathrm{\eta }\left(X\right)\,\mathrm{\eta }\left(Y\right)\right]\ⅆX^4\ⅆY^4$

FeynmanDiagrams(L__2, order = 2, numberofloops = 1);

$\frac{{\mathrm{I}}^2}{2!}\int \int 18{\mathrm{\sigma }}^2:\mathrm{\eta }\left(X\right)\mathrm{\eta }\left(Y\right):{\left[\mathrm{\eta }\left(X\right)\,\mathrm{\eta }\left(Y\right)\right]}^2+72{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^2{\mathrm{\phi }\left(Y\right)}^2:{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^2\ⅆX^4\ⅆY^4$

FeynmanDiagrams(L__2, order = 2, numberofloops = 2);

$\frac{{\mathrm{I}}^2}{2!}\int \int 96{\mathrm{\lambda }}^2:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^3\ⅆX^4\ⅆY^4$