DoFun 3.0: Functional equations in Mathematica

Markus Q. Huber

Institut fur Theoretische Physik, Justus-Liebig–Universitat Giessen, Heinrich-Buff-Ring 16, 35392 Giessen,

Germany

Anton K. Cyrol, Jan M. Pawlowski

Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg,Germany

Abstract

We present version 3.0 of the Mathematica package DoFun for the derivation of functional equa-tions. In this version, the derivation of equations for correlation functions of composite operatorswas added. In the update, the general workflow was slightly modified taking into account expe-rience with the previous version. In addition, various tools were included to improve the usageexperience and the code was partially restructure for easier maintenance.

Keywords: Dyson-Schwinger equations, functional renormalization group equations, correlationfunctions, quantum field theory, composite operators

PROGRAM SUMMARY

Program Title: DoFun

Version number: 3.0.0

Licensing provisions: GPLv3

Programming language: Mathematica, developed in version 11.3

Operating system: all on which Mathematica is available (Windows, Unix, MacOS)

PACS: 11.10.-z,03.70.+k,11.15.Tk

Nature of problem: Derivation of functional renormalization group equations, Dyson-Schwinger equations

and equations for correlations functions of composite operators in symbolic form which can be translated

into algebraic form.

Solution method: Implementation of algorithms for the derivations of these equations and tools to trans-

form the symbolic to the algebraic form.

Unusual features: The results can be plotted as Feynman diagrams in Mathematica. The output is compat-

ible with the syntax of many other programs and is therefore suitable for further (algebraic) computations.

1. Introduction

Computer algebra systems are an integral part of particle physics and physics in general. Manyspecialized tools exist and supplement generic programs like Mathematica [1]. Especially in pertur-bative calculations in high-energy physics they are indispensable, see, e.g., [2–4]. In recent years,non-perturbative functional methods, see [5–17] for reviews, have also reached a point where thehelp of computer algebra systems is helpful or even mandatory. To assist in these cases, a rangeof dedicated tools was developed [18–22].

Here, we present a continuation of that work with a new and extended version 3 of the programDoFun (Derivation Of FUNctional equations) [18, 20]. Its purpose is the derivation of Dyson-Schwinger equations (DSEs), functional renormalization group equations (RGEs) and - added inthis version - correlation functions of composite operators. The output can be arranged in sucha way that it is compatible with other programs to perform traces, like FormTracer [22], FORM

Email addresses: markus.huber@physik.jlug.de (Markus Q. Huber),j.pawlowski@thphys.uni-heidelberg.de (Jan M. Pawlowski)

Preprint submitted to CPC August 9, 2019

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[23–27], HEPMath [28] or FeynCalc [29–31], and perform numeric calculations by exporting it tonumeric code [21].

In the past, DoFun was helpful in several projects using large systems of equations which areextremely tedious or even impossible to derive manually. But even for manually manageable casesa computer-assisted derivation is very useful. Examples for the usage of DoFun include Yang-Millstheory and QCD in various gauges in vacuum, e.g., [32–52] and beyond, e.g., [53–56], effectivemodels for QCD, e.g., [56–63], asymptotic gravity [64] and other models [65]. DoFun is often usedin combination with other programs like Form [24–27], FormTracer [22], CrasyDSE [21] or xPert[66].

Since publication of version 2 it became apparent that some aspects of DoFun should be im-proved to optimize the workflow and in particular to fully incorporate some cases which were notincluded in the original version. A main consequence of this a change in the handling of fields.Originally the natures of fields were guessed from the input. While this makes many use casessimple, it leads to problems for other cases, e.g., complex scalar fields. Thus, fields have to bedefined now explicitly which avoids ambiguous situations.

A totally new feature is the derivation of correlation functions for composite operators. Itrelies on a simple identity, but the calculations can be quite cumbersome as typically many fieldsare involved which lead to many loops. We also added a few new useful tools, for example, theidentification of 1PI diagrams or the extraction of diagrams of a certain type by name. Thegraphical representation was also modified using now Graph[] instead of GraphPlot[] which isslightly more versatile.

Finally, we moved the code to a public git repository (https://github.com/markusqh/DoFun)to make use of a modern development infrastructure and provide a platform for bug reporting.

In the following we first explain how to install DoFun and get access to the documentation inSec. 2. In Sec. 3 we give a short overview of the derivation of functional equations. Sec. 4 containssome additional details. We summarize in Sec. 5. The appendices contain various summaries ofnew functions and usage changes from version 2 to 3. For the readers familiar with DoFun 2,changes they have to consider are listed there as well. Readers who want to get started right awayshould read the installation instructions and can then continue with the provided documentationthat includes examples.

2. Installation and usage

DoFun was developed in Mathematica 11.3 and tested in Mathematica 12. Earlier versionswere not tested but will most likely work. DoFun can be installed with the installation script fromthe git repository by evaluating

In[1]:= Import["https://raw.githubusercontent.com/markusqh/\

DoFun/master/DoFun/DoFunInstaller.m"]

Alternatively, one can download it from https://github.com/markusqh/DoFun/releases andcopy it to the Mathematica applications folder.1

The documentation of DoFun is available in Mathematica’s Documentation Center : Add-onsand Packages → DoFun. Direct access to the documentation of a function is also possible via??function or pressing F1 when the cursor is inside the function name.

3. Derivation of functional equations

In this section, the derivation of Dyson-Schwinger and flow equations is described. More detailscan be found in the articles on DoDSE [67] and DoFun 2 [20], but we reproduce the main stepsas a quick reference. We also discuss differences in the implementation in DoFun 3. In Section 3.4the equations for composite operator correlation functions are discussed and exemplified using theenergy-momentum tensor.

1On a typical Linux system this would be ˜/.Mathematica/Applications.

2

3.1. Basic definitions

A basic quantity for all derivations is the effective action Γ[Φ] which depends on the collectivefield Φ. An index encodes the field type, the position or momentum argument and all indicesassociated with an internal symmetry group. Repeated indices are summed and integrated over ifnot noted otherwise. The effective action is defined via a Legendre transformation:

Γ[Φ] := supJ

(−W [J ] + JiΦi). (1)

The Ji’s are the sources for the fields Φi. The generating functional W [J ] is related to the bareaction S[φ] as follows:

Z[J ] =

∫D[φ]e−S+φjJj =: eW [J]. (2)

The fields φ that appear here are the quantum fields which are related to the average fields Φ by

Φi ≡ 〈φi〉J =δW

δJi= Z[J ]−1

∫D[φ]φie

−S+φjJj . (3)

Setting the sources J to zero leads to the physical expectation values of the fields φ: Φphys :=〈φi〉J=0.

For the effective action a vertex expansion around the physical ground state is employed:

Γ[Φ] =∞∑n=0

1

N i1...in

∑i1...in

Γi1...in(Φi1 − Φi1,phys) . . . (Φin − Φin,phys). (4)

The N i1...in are symmetry factors. The physical n-points functions Γi1...in are obtained by deriva-tives of the effective action and setting the sources to zero:2

Γij :=ΓijJ=0 =δ2Γ[Φ]

δΦiδΦj

∣∣∣∣∣Φ=Φphys

, (5a)

Γi1...in :=Γi1...inJ=0 =δnΓ[Φ]

δΦi1 . . . δΦin

∣∣∣∣∣Φ=Φphys

. (5b)

The (field-dependent) propagators are the inverse of the two-point functions:

DijJ :=

δW [J ]

δJiδJj=

[(δ2Γ

δΦ2

)−1]ij

, (6)

Again, the physical propagators are obtained for J = 0:

Dij = DijJ=0. (7)

In the following we will need the derivatives of propagators, fields and vertices with respect tofields. With the relations

δ

δΦiΓj1...jnJ =

δΓ

δΦiδΦj1 . . . δΦjn. (8)

and

δ

δΦiDjkJ = −εjmi

[(δ2Γ

δΦ2

)−1]jm(

δ3Γ

δΦiδΦmδΦn

)[(δ2Γ

δΦ2

)−1]nk

. (9)

2In contradistinction to DoFun 2 we do not include a minus sign in the definition of the vertices by default. SeeAppendix A.3 for details on signs and how to enable the previous behavior for compatibility.

3

we arrive at the simple and complete set of derivative rules

δ

δΦiΦj = δij , (10a)

δ

δΦiΓj1...jnJ = Γij1...jnJ , (10b)

δ

δΦiDjkJ = −εjmi Djm

J ΓimnJ DnkJ . (10c)

Note that lowering and raising indices is trivial within the convention introduced here. If Grass-mann fields are involved, the function εjmi takes care of corresponding signs due to their anti-commutative nature. It is defined as

εjk...i =

{1 i bosonic field

(−1)# Grassmann fields in jk... i fermionic field(11)

Note that we use only left-derivatives.3

3.2. Derivation of Dyson-Schwinger equations

The master equation is derived from the integral of a total derivative,

0 =

∫D[φ]

δ

δφie−S+φjJj =

∫D[φ]

(− δSδφi

+ Ji

)e−S+φjJj =

− δSδφ′i

∣∣∣∣∣φ′i=δ/δJi

+ Ji

Z[J ] . (12)

Plugging in Eq. (2), we can switch to the generating functional of connected correlation functions,W [J ],

− δSδφi

∣∣∣∣∣φi=

δW [J]δJi

+ δδJi

+ Ji = 0 , (13)

where

e−W [J]

(δ

δJi

)eW [J] =

δW [J ]

δJi+

δ

δJi, (14)

was used. Performing a Legendre transformation we obtain the master equation for 1PI functions:

δΓ

δΦi=

δS

δφi

∣∣∣∣∣φi=Φi+D

ijJ δ/δΦj

. (15)

By applying further derivatives and setting the sources to zero at the end, DSEs for any n-pointfunction can be obtained. For more details we refer to Refs. [17, 20, 67] and for a short descriptionof a graphical derivation to Ref. [18].

The implementation of the derivation in DoFun is as follows:

• Perform the first derivative.

• Replace the fields according to Eq. (15).

• Perform additional derivatives.

• Set the sources to zero and get the phyical propagators and vertices.

• Get signs from ordering the fermions in a canonical way and the ε-functions.

• Identify equal diagrams.

3In DoFun 2 left- and right-derivatives were used.

4

Note that equal diagrams could be identified earlier leading to fewer intermediate expressions, butthis way turns out to be more flexibel. If systems get too large, though, one could lower the numberof terms by taking this into account.

3.3. Derivation of functional renormalization group equations

We will follow here the standard derivation of the flow equation for the so-called effectiveaverage action given in Ref. [68]. For flow equations we introduce a momentum scale k in the bareaction S[φ] via a regulator term. It serves to integrate out quantum fluctuations in a controlledway:

∆Sk[φ] =1

2φiR

ijk φj . (16)

All functionals depend now on k. In the limit k → 0, the full effective action is recovered. Theeffective average action, defined by a modified Legendre transformation,

Γk[Φ] = −Wk[J ] + JiΦi −1

2ΦiR

ijk Φj , (17)

is used instead of the standard effective action Γ[Φ]. This leads to k-dependent correlation functionsΓi1...ink . The master equation, which describes the dependence of the effective average action onthe scale k, is the Wetterich equation, [68],

∂kΓk[Φ] =1

2

[1

Γ(2)k [Φ] +Rk

]ji∂kR

ijk

=1

2Tr

1

Γ(2)k [Φ] +Rk

∂kRk

=1

2TrDk,J∂kRk , (18)

where Γ(2)k [Φ] is the second derivative of the effective average action. The supertrace Tr includes

a minus sign for Grassmann fields. Dk,J is the field-dependent propagator including the regulatorterm. Equations for n-point functions are obtained by applying n derivatives to Eq. (18) using thedifferentiation rules from Eq. (10).

The implementation of the derivation in DoFun is as follows:

• Instead of Eq. (18), the following expression is used to minimize the number of diagramsduring the derivation (the index J is suppressed here):

∂tΓk[Φ] =1

2Tr ∂t ln

(Γ

(2)k [Φ] +Rk

), (19)

where t = ln(k/Λ) with Λ being a UV cutoff scale. The derivative ∂t only acts on theregulator Rk.

• The starting expression is

1

2εila ∂tD

ilΓalj . (20)

The indices i and j are not closed. This will be done at the end when also the derivative ∂tis applied.

• Further derivatives are applied with the rules of Eq. (10).

• The sources are set to 0 to obtain physical propagators and vertices.

• The trace is closed by setting i = j. If the corresponding propagator belongs to Grassmannfields, a minus sign is added.

5

• The expressions are reorganized in the canonical way, viz., bosons left of anti-Grassmannfields left of Grassmann fields. Within these three groups external indices are left of internalones and the latter are organized by the vertex they connect to. This leads to signs fromanti-commuting fields and is also required to be able to recognize equal diagrams which arethen summed up.

• The derivative ∂t is applied.

3.4. Correlation functions of composite operators

Any full correlation function can be expressed in terms of dressed propagators and vertices as[8]

〈F (φ)〉 = F

(Φi +Dij

J

δ

δΦj

). (21)

For correlation functions of composite operators O(φ), this leads to

〈O(φ(x))O(φ(y))〉 = O

(Φi(x) +Dij

J

δ

δΦj

)O

(Φi(y) +Dij

J

δ

δΦj

), (22)

where the x- and y-dependence is indicated partially. The derivatives to be performed here aresimilar to the case of DSEs and we can use the corresponding functions. For n fields in theexpectation value, up to n− 2 loops can appear.

From a technical point of view it is convenient to write the composite operator as a generaln-point function contracted by an auxiliary function we denote as C. C behaves like a vertex andallows using many functions of DoFun in a straightforward way. To illustrate this, consider the op-erator Oij(x) = φai (x)φaj (x). The corresponding two-point function can be written as (integrationand summation over repeated indices is implied)

〈Oij(x)Okl(y)〉 = Ci′a,j′bi,j (x;x1, x2)Ck

′c,l′dk,l (y;x3, x4)〈φai′(x1)φbj′(x2)φck′(x3)φdl′(x4)〉 (23)

with

Ci′a,j′bi,j (x;x1, x2) = δi

′iδj′jδabδ(x1 − x)δ(x2 − x). (24)

As a specific example we take the correlation function of the spatial, traceless part of theenergy-momentum tensor of Yang-Mills theory:

πij(x) = F aµi(x)F a,µj (x)− 1

3δijF

aµk(x)F a,µk(x). (25)

The correlation function we want to calculate is

Gππ(x, y) = 〈πij(x)πij(y)〉, (26)

which, for example, gives access to the shear viscosity via the Kubo relation [69]. We symbolicallywrite the energy-momentum tensor as

πij = π(2)ij + π

(3)ij + π

(4)ij , (27)

where the numbers in parentheses indicates the number of gluon fields. Gππ(x, y) can then be split

into parts G(k,l)ππ (x, y) corresponding to pairs of π

(k)ij :

Gππ(x, y) =

4∑k,l=1

〈π(k)ij π

(l)ij 〉. (28)

The minimal number of loops appearing in G(k,l)ππ (x, y) is b(k + l − 1)/2c, and the maximal number

is k + l − 2, viz., Gππ(x, y) has up to six loops.4 We restrict ourselves to two loops here, as

4bc is the floor function.

6

expressions become too long otherwise, but the procedure is the same for the dropped expressions.Thus, we only take into account

Gππ(x, y) = π(2,2)ij + π

(2,3)ij + π

(3,2)ij + π

(3,3)ij + π

(2,4)ij + π

(4,2)ij . (29)

The steps up to here amount to the following in DoFun. The composite operator is representedby a field, so we must define it together with the gluon field:

In[2]:= setFields[{A,FF}]

As action we define the symbolic Yang-Mills action without ghosts which do not contribute here:

In[3]:= action={{A, A}, {A, A, A}, {A, A, A, A}};

For G(x, y) we define two auxiliary functions:

In[4]:= F[j_] := Module[{j1, j2, j3, j4, j5, j6, j7, j8, j9},

{op[CO[{FF, j}, {A, j1}, {A, j2}], {A, j1}, {A, j2}]/2!,

op[CO[{FF, j}, {A, j3}, {A, j4}, {A, j5}], {A, j3}, {A, j4},

{A, j5}]/3!,

op[CO[{FF, j}, {A, j6}, {A, j7}, {A, j8}, {A, j9}], {A, j6},

{A, j7}, {A, j8}, {A, j9}]/4!}]

pi[i_, j_, k_, l_] := op[F[i][[k - 1]], F[j][[l - 1]]]

The second quantity is the combination of the parts with k and l gluon legs. G(x, y) can then bewritten as

In[5]:= G1 = pi[i, j, 2, 2] + pi[i, j, 2, 3] + pi[i, j, 3, 2] +

pi[i, j, 3, 3] + pi[i, j, 2, 4] + pi[i, j, 4, 2];

The next steps are the replacement of the fields and setting the sources to zero:

In[6]:= G2 = replaceFieldsCO@(G1);

G3 = setSourcesZero[G2, action, {}];

We select now only diagrams up to two loops:

In[7]:= G4 = Select[G3, getLoopNumber[#] <= 2 &];

This result can also be obtained directly with the function doCO:

In[8]:= G4 = doCO[action, G1, getLoopNumber[#] <= 2 &];

The last argument used is optional and selects here the terms with one and two loops.G4 contains connected and disconnected diagrams. Since the relevant quantity is actually the

expectation value of the commutator of the composite operator, they will finally vanish and wedrop them here. Of the originally 72 diagrams, many of which are identical, though, now 63 remain:

In[9]:= GConn = getConnected[G4];

We sum up those graphs:

In[10]:= GConnId =

identifyGraphs[GConn, {{FF, i}, {FF, j}}];

Using the plot styles

In[11]:= fieldRules = {{A, Red}, {FF, Thick, Orange}};

we can plot the result:

In[12]:= DSEPlot[GConnId, fieldRules]

7

ij -1

=

+1

2

ij+

1

4

ij

+1

6

ij+

1

4

ij

+1

4

ij

+1

4

i j+

1

2

ij

+1

4

i j

+1

2

ij

+1

4

i j+

1

4

i j +1

2

i j

Figure 1: The correlation function Eq. (26) up to two loops. Red, continuous lines are gluons, the triangle representsthe composite operator πij with the external leg plotted blue and dotted.

ij -1

=

+1

2

ij+

1

4

ij

+1

6

ij+

1

4

ij

+1

4

i j+

1

2

ij

+1

2

ij

+1

2

i j

Figure 2: The correlation function Eq. (26) up to two loops with some diagrams discarded that vanish due to colorcontractions.

It is depicted in Fig. 1. We see three types of diagrams which are not 1PI. They all vanish, becauseeither the combination of a three-gluon vertex and a two-gluon-leg part of πij yields zero due tothe color structure [70] or because the combination of a gluon propagator and the three-gluon-legpart of πij yields zero.

We thus continue only with the 1PI part:

In[13]:= G = get1PI[GConnId];

A graphical representation is depicted in Fig. 2. For the algebraic expression, we would needto define the gluon propagator, the three- and four-gluon vertices and the components of theenergy-momentum tensor. Since the final expressions are not very elucidating, we only show thecommands to transform the symbolic to the algebraic expressions. First, we define which indicesthe fields have:

In[14]:= defineFieldsSpecific[{A[mom, col, lor],

FF[mom, lors, lors]}]

This assigns the gluon field A a momentum, a color and a Lorentz index and the energy-momentumtensor a momentum and two spatial Lorentz indices.5 The algebraic expression is obtained with

In[15]:= getAE[G, {{FF, i, p, k, l}, {FF, j, -p, m, n}}]

The arguments in the lists assign, for example, the external field FF with the index i the momentump and the spatial Lorentz indices k and l. Since we have not defined the propagators and verticesthey are only formally given. If we had defined them, we could now continue to perform the traces.

Finally, we show some three-loop diagrams as an example of higher contributions to Eq. (26)in Fig. 3.

4. Some details

This section collects some more detailed information about certain aspects of DoFun.

4.1. Structure of DoFun

The main part of DoFun is contained in three package files:

5The names of the indices are arbitrary and do not have any special meaning within DoFun.

8

Figure 3: Examples for three-loop contributions to G(x, y).

• DoDSERGE.m: This package contains the functions for deriving DSEs, flow equations andcomposite operator correlation functions.

• DoFR.m: This packages provides tools to derive Feynman rules from a given action. Changesin version 3 are minor and it should work with old programs without problems. Only signsfor fermions should be checked because all derivatives are left-derivatives now.

• DoAE.m: This package contains functions to transform symbolic to algebraic expressions.The functionality is basically identical to verson 2 with the exception that composite oper-ators were added. It should work with old programs without problems.

Additional files/directories are the following:

• Kernel/init.m is called for loading DoFun and automatically checks for updates.

• DoFunInstall.m can be called for installing DoFun on a computer. It copies all required filesto the appropriate places.

• Documentation contains the documentation which can be accessed in the DocumentationCenter via Add-ons and Packages → DoFun.

4.2. Fields

All fields are assigned a specific type out of the following list: boson, fermion, antiFermion,complex and antiComplex. The field type is obtained with fieldType[]. To assign thesetypes, the function setFields[] is used. It also takes care of setting various other properties offields, e.g., if they are commuting or anti-commuting, which can be checked with cFieldQ[] andgrassmannQ[]. Fields know about their anti-fields which can be determined with antiField[].

4.3. Treatment of anti-commuting fields

Signs from the anti-commuting nature of fermions are taken into account by the function definedin Eq. (11). It is called sf[] and automatically simplifies in many cases. It is inserted wheneverneeded and only at the end, when all single fields of the superfield are written out, the signs aredetermined with the function getSigns[].

The notation for vertices uses the following convention for the places of indices:

Γψψψψijkl =δ4Γ

δψiδψjδψkδψl(30)

This entails that fields in the derivation functions are put in inverse order, because they are appliedfrom left to right. For example, the DSE corresponding to Eq. (30) is derived by

9

In[16]:= doDSE[action, {{psi, l}, {psi,

k}, {psibar, j}, {psibar, i}}]

5. Summary

Calculations in quantum field theory can easily become too tedious to perform them manually.Automatizing them not only alleviates the calculations but is in some cases mandatory. Thesoftware DoFun we presented here is a tool for functional calculations which can profit from suchan automatization. It can derive various functional equations from a given action and yields resultsin a form suitable for further manipulations by other programs. The present version 3 simplifiessome aspects, introduces new tools and adds composite operators to the tool box.

6. Acknowledgments

We thank Jens Braun, Tobias Denz, Marc Leonhardt, Mario Mitter, Coralie Schneider, NilsStrodthoff and Nicolas Wink for useful discussions and input on how to improve DoFun. Fundingby the FWF (Austrian Science Fund) under Contract No. P27380-N27 is gratefully acknowledged.The work is also supported by EMMI, the BMBF grant 05P12VHCTG, and is part of and sup-ported by the DFG Collaborative Research Centre SFB 1225 (ISOQUANT) as well as by the DFGunder Germany’s Excellence Strategy EXC- 2181/1 - 390900948 (the Heidelberg Excellence ClusterSTRUCTURES).

Appendix A. Usage changes from DoFun 2

This section lists differences between DoFun 2 and 3 and is intended for users already familiarwith the former. New users can skip this section.

Appendix A.1. Quick fact sheet on how to update to DoFun 3

Task Description

Field definitions Fields need to be explicitly declared before any calculations withsetFields[].

Sign conventions Vertices are defined now as the positive derivative of the effectiveaction which leads to additional minus sign in diagrams. The oldbehavior can be restored by setting $signConvention=1.

Derivatives All derivatives are now left-derivatives. This may affect signs offermionic vertices.

Algebraic expressions The order of the resulting field arguments of propagators and ver-tices might be different now. When converting the symbolic intoalgebraic expressions, the definitions of propagators and verticesmight thus need to be adapted.

Appendix A.2. Types of fields and setFields[]

Fields have an explicit type now. They can be (real) bosons, complex fields or fermions. Tohave a clear connection between fields and their anti-fields, the latter have the types anti-complexfield and anti-fermionic, respectively. The field type can be obtained with fieldType[]. Beforedoing any calculations, the field type has to be declared with setFields[]. It also sets otherproperties like if they are commuting (cFieldQ[]) or anti-commuting (grassmannQ[]).

Appendix A.3. Sign conventions

• The definition of vertices changed compared to DoFun 2. They are defined now as

Γi1...in :=Γi1...inJ=0 =δnΓ[Φ]

δΦi1 . . . δΦin

∣∣∣∣∣Φ=Φphys

. (A.1)

This is determined by the default value of $signConvention=-1. It can be reset to 1 torecover the old behavior.

• For Grassmann-valued fields only left-derivatives are used. This can lead to different signscompared to using left- and right-derivatives.

10

Appendix B. New functions

Various new functions were added. The ones accessible for the user are listed below:

• setFields[]: Defines properties of fields, see Sec. Appendix A.2.

• doCO[]: Derive equation for the correlation function of a composite operator.

• getVertexNumbers[], getDiagramType[], extractDiagramType[] and groupDiagrams[]:Tools to classify and extract diagrams. Known diagram types are stored in diagramTypes,which can be extended by definitions of the user. For example, all diagrams of triangle typecan be extracted from the expression exp by

extractDiagramType[exp, "triangle"]

Currently the following diagrams are defined: oneLoop, tadpole, sunset, squint, triangle3,swordfish3, box, triangle4, swordfish4, fivePoint4

• sortCanonical[]: Puts fields in a canonical order as required for diagram identification.Replaces the function orderFermions[].

• getSigns[], sf[]: Signs from permuting fields are determined with getSigns[] using theinformation stored in the auxiliary function sf[].

• getConnected[], getDisconnected[],connectedQ[], disconnectedQ[]: Extract (dis)connecteddiagrams.

• getNon1PI[], get1PI[], onePIQ[]: Extract 1PI/non-1PI diagrams.

Appendix C. Limitations and disclaimer

DoFun was carefully tested. Nevertheless we cannot guarantee that the program is free offlaws. However, we encourage everybody who finds a bug to report it on GitHub via https:

//github.com/markusqh/DoFun/issues.At the moment of publication of this article, we know of the following limitations:

• Identification of diagrams only works up to two-loop. It should be noted that graph identi-fication is a non-trivial problem in general. We decided not to put too much effort into this,because in relevant cases the identifications can still be done manually. In practice, we thinkthe only relevant case are equations of composite operators. Identification can also fail whenmixed propagators appear.

• Mathematica 12 introduced some changes in plotting graphs which either introduced a bugor removed a certain feature on purpose. Thus, at the moment it may cause problems to plotdiagrams with more than two propagators connecting the same vertices. A warning messageis printed if this happens and the styles of the propagators may be incorrect.

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