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\begin_body

\begin_layout Title
Loop amplitude calculations
\begin_inset Newline newline
\end_inset

on a computer
\end_layout

\begin_layout Subtitle
Do-It-Yourself guide
\end_layout

\begin_layout Author
\begin_inset CommandInset href
LatexCommand href
name "Vitaly Magerya"
target "vitaly.magerya@tx97.net"
type "mailto:"
literal "false"

\end_inset


\end_layout

\begin_layout Institute
Institut für Theoretische Physik,
\begin_inset Newline newline
\end_inset

Karlsruher Institut für Technologie
\begin_inset Newline newline
\end_inset

(ITP KIT)
\end_layout

\begin_layout Date
Hamburg, CAPP 2021
\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\V}[1]{\mathbf{#1}}
{\mathbf{#1}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\M}[1]{\mathbb{#1}}
{\mathbb{#1}}
\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\d}{\mathrm{d}}
{\mathrm{d}}
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status collapsed

\begin_layout Plain Layout
Goals
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Goal of the lecture: take a 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
loop amplitude
\end_layout

\end_inset

, calculate it 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
on a computer
\end_layout

\end_inset

: analytically (and then numerically), from start to finish, with our own
 code.
\begin_inset Formula 
\[
\fig{eetohadrons}
\]

\end_inset


\end_layout

\begin_layout Frame
Target quantity: the 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
total cross-section of 
\begin_inset Formula $e^{+}e^{-}$
\end_inset

 annihilation to hadrons
\end_layout

\end_inset

,
\begin_inset Formula 
\[
\sigma(e^{+}e^{-}\to\text{hadrons})=\sigma(e^{+}e^{-}\to\text{partons});
\]

\end_inset

we shall calculate 
\begin_inset Formula $\mathcal{O}(\alpha_{s}^{2})$
\end_inset

 corrections to it (and then 
\begin_inset Formula $\mathcal{O}(\alpha_{s}^{3})$
\end_inset

 too).
\end_layout

\begin_layout Frame
Model: QCD with 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $N_{f}$
\end_inset

 massless quarks
\end_layout

\end_inset

, and 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $N_{t}$
\end_inset

 massive quarks
\end_layout

\end_inset

 of mass 
\begin_inset Formula $m$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Motivation: the 
\begin_inset Formula $R$
\end_inset

 ratio
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[trim=5mm 0mm 0mm 67mm,clip,height=60mm]{fig/rpp2020-sigma_R_ee.pd
f}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Formula $R\equiv\sigma_{\mathrm{tot}}(e^{+}e^{-}\to\text{hadrons})/\sigma_{\mathrm{tot,leading}}(e^{+}e^{-}\to\gamma^{*}\to\text{muons})$
\end_inset

, from 
\begin_inset CommandInset href
LatexCommand href
name "Ezhela,~Lugovsky,~Zenin~'03"
target "https://arxiv.org/abs/hep-ph/0312114"
literal "true"

\end_inset

, available at 
\begin_inset CommandInset href
LatexCommand href
name "pdg.lbl.gov/2020/hadronic-xsections"
target "https://pdg.lbl.gov/2020/hadronic-xsections/"
literal "false"

\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Calculating 
\shape italic
e
\shape default

\begin_inset script superscript

\begin_layout Plain Layout
+
\end_layout

\end_inset


\shape italic
e
\shape default

\begin_inset script superscript

\begin_layout Plain Layout
–
\end_layout

\end_inset

 annihilation to hadrons
\end_layout

\end_inset


\end_layout

\begin_layout Frame
The square matrix element of the total cross-section is
\begin_inset Formula 
\[
\left|M(e^{+}e^{-}\to\mathrm{partons})\right|^{2}=\sum_{n}\int\mathrm{d}\mathrm{PS}_{n}\left|\sum\!\sfig{eetopartons}\right|^{2}.
\]

\end_inset

Often calculating it is easier via the optical theorem:
\begin_inset Formula 
\[
\left|M\right|^{2}=-2\,\mathrm{Re}\left(\sfig{eeoptical}\right).
\]

\end_inset

Further, we can split this into the leptonic (
\begin_inset Formula $L$
\end_inset

) and the hadronic (
\begin_inset Formula $H$
\end_inset

) tensors:
\begin_inset Formula 
\[
\left|M\right|^{2}=\frac{2}{q^{4}}\,\mathrm{Re}\left(L_{\mu\nu}H^{\mu\nu}\right),\ L_{\mu\nu}\equiv\sfig{eeleptonic},\ H^{\mu\nu}\equiv\sfig{eehadronic}.
\]

\end_inset

All the loop integration and 
\begin_inset Formula $\alpha_{s}$
\end_inset

 corrections are in 
\begin_inset Formula $H^{\mu\nu}$
\end_inset

, and 
\begin_inset Formula $L_{\mu\nu}$
\end_inset

 is simple.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Tensor structures and projectors
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Because 
\begin_inset Formula $H^{\mu\nu}(q)$
\end_inset

 is a Lorentz-covariant tensor, it can only have this general structure:
\begin_inset Formula 
\[
H^{\mu\nu}(q)=g^{\mu\nu}F_{1}(q)+q^{\mu}q^{\nu}F_{2}(q).
\]

\end_inset

Because of Ward identities, 
\begin_inset Formula $q_{\mu}H^{\mu\nu}=H^{\mu\nu}q_{\nu}=0$
\end_inset

, the structure is further restricted to
\begin_inset Formula 
\[
H^{\mu\nu}(q)=F_{1}(q)\,\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}\right).
\]

\end_inset

To invert this and get 
\begin_inset Formula $F_{1}$
\end_inset

 from 
\begin_inset Formula $H^{\mu\nu}$
\end_inset

 we must construct a projector:
\begin_inset Formula 
\[
F_{1}=H^{\mu\nu}\frac{g_{\mu\nu}}{d-1}.
\]

\end_inset

Calculting scalar values like 
\begin_inset Formula $F_{1}$
\end_inset

 is simpler than tensor values like 
\begin_inset Formula $H^{\mu\nu}$
\end_inset

, and a tensor decomposition like this along with projector constuction
 is almost always needed.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Calculation plan
\end_layout

\end_inset


\end_layout

\begin_layout Frame

\end_layout

\begin_deeper
\begin_layout Enumerate
Generate Feynman diagrams for the process.
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $F_{1}=\sfig{dia0}+\sfig{dia1}+\sfig{dia8}+\dots$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

 with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 output.
\end_layout

\end_deeper
\begin_layout Enumerate
Apply the Feynman rules.
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $F_{1}=\int\mathrm{d}^{d}l_{1}\,\mathrm{Tr}\!\left(\gamma^{\mu}\slashed{k}_{1}\gamma^{\nu}\slashed{k}_{2}\right)\cdots+\dots$
\end_inset


\end_layout

\begin_layout Itemize
Custom 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 code.
\end_layout

\end_deeper
\begin_layout Enumerate
Resolve Dirac and color tensor summation, convert to the scalar integral
 families.
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $F_{1}=N_{f}C_{a}\cdots I_{123}+\dots$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 
\begin_inset Formula $\to$
\end_inset

 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset


\begin_inset Formula $\to$
\end_inset

 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Use IBP relations to reduce to smaller set of 
\begin_inset Quotes eld
\end_inset

master integrals
\begin_inset Quotes erd
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $F_{1}=\left(N_{f}C_{a}\cdots+\dots\right)I_{111}+\dots$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 
\begin_inset Formula $\to$
\end_inset

 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Kira
\end_layout

\end_inset

 
\begin_inset Formula $\to$
\end_inset

 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Evaluate the master integrals.
\end_layout

\begin_deeper
\begin_layout Itemize
Numerically: sector decomposition with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Analytically: differential equations with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Fuchsia
\end_layout

\end_inset

.
\end_layout

\end_deeper
\end_deeper
\begin_layout Frame
General idea: use 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 to glue everything together.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Diagram generation
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Feynman diagrams with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

 is a widely used program for Feynman diagram generation available at 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

http://cfif.ist.utl.pt/~paulo/qgraf.html
\end_layout

\end_inset

.
\end_layout

\begin_layout Frame
To generate diagrams with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

:
\end_layout

\begin_deeper
\begin_layout Enumerate
Create 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
qgraf.dat
\end_layout

\end_inset

, defining the process (incoming particles, outgoing particles, loop count,
 names of the momenta, vetoed diagrams), and listing the 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
model
\end_layout

\end_inset

 and 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
style files
\end_layout

\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
See 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
qgraf.dat.example
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Create a 
\begin_inset Flex Emphasize
status open

\begin_layout Plain Layout
model file
\end_layout

\end_inset

 with a list of fields and vertices.
\end_layout

\begin_deeper
\begin_layout Standard
See 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
qgraf-modfile
\end_layout

\end_inset

 for the QCD model we'll use.
\end_layout

\end_deeper
\begin_layout Enumerate
Create a 
\begin_inset Flex Emphasize
status open

\begin_layout Plain Layout
style file
\end_layout

\end_inset

 (output template) defining the output format.
\end_layout

\begin_deeper
\begin_layout Standard
See 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
qgraf-stylefile
\end_layout

\end_inset

 for the Mathematica output style definition we'll use.
\end_layout

\end_deeper
\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Frame
Also introducing: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
qgraf.sh
\end_layout

\end_inset

 and 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
mkdia.py
\end_layout

\end_inset

 programs that will make our 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

 usage simpler.
\end_layout

\begin_layout Frame
Use the 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 notebook 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
show-diagrams.nb
\end_layout

\end_inset

 to view the diagrams.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

 diagram structure
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Short summary:
\end_layout

\begin_deeper
\begin_layout Itemize
In 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

 diagrams consist of vertices and propagators (edges).
\end_layout

\begin_layout Itemize
Each vertex has a list of 
\begin_inset Quotes eld
\end_inset

fields
\begin_inset Quotes erd
\end_inset

 (legs).
\end_layout

\begin_layout Itemize
Vertex fields (legs) and their momenta are presented as incoming into the
 vertex.
\end_layout

\begin_layout Itemize
Both vertices and their legs are identified by their indices.
\end_layout

\begin_layout Itemize
Vertices and internal legs have positive indices.
\end_layout

\begin_layout Itemize
External legs have negative indices: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-1
\end_layout

\end_inset

, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-3
\end_layout

\end_inset

, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-5
\end_layout

\end_inset

, ...
 for incoming particles, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-2
\end_layout

\end_inset

, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-4
\end_layout

\end_inset

, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
-6
\end_layout

\end_inset

, ...
 for outgoing.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Frame
\begin_inset Formula 
\[
\fig{qgraf}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Feynman rules
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Notation on a computer
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Feynman rules 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
in a book
\end_layout

\end_inset

:
\begin_inset Formula 
\[
\fig{AA1l}=
\]

\end_inset


\begin_inset Formula 
\[
\int\frac{\mathrm{d}^{d}l}{\left(2\pi\right)^{d}}\,ig_{e}Q_{f_{1}}\delta_{i_{1}i_{2}}\delta_{f_{1}f_{2}}\,\mathrm{Tr}\!\left(\gamma^{\mu}\frac{\slashed{l}-\slashed{q}}{\left(q-l\right)^{2}+i0}\gamma^{\nu}\frac{\slashed{l}}{l^{2}+i0}\right)ig_{e}Q_{f_{2}}\delta_{i_{2}i_{1}}\delta_{f_{2}f_{1}}.
\]

\end_inset


\end_layout

\begin_layout Frame
\begin_inset VSpace defskip
\end_inset

Feynman rules 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
on a computer
\end_layout

\end_inset

: 
\begin_inset Formula 
\begin{align*}
 & \text{\code{In[1]\ :=\ One\ Feynman\ diagram,\ please???}}\\
\\
 & \text{\code{Syntax::sntxf:\ "One\ Feynman\ diagram"}}\\
 & \text{\code{cannot\ be\ followed\ by\ ",\ please???".}}
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 notation, I
\end_layout

\end_inset


\end_layout

\begin_layout Frame
To operate on Feynman rules in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

, we need to choose a notation.
 Any will work; we'll use the following.
\end_layout

\begin_layout Frame
Index names:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\mu_{i}$
\end_inset

, Lorentz indices, 
\begin_inset Formula $1\dots d$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
lor[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $i_{i}$
\end_inset

, fundamental color indices, 
\begin_inset Formula $1\dots N_{c}$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
fun[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $a_{i}$
\end_inset

, adjoint color indices, 
\begin_inset Formula $1\dots N_{a}=\frac{1}{2}\left(N_{c}^{2}-1\right)$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
adj[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $f_{i}$
\end_inset

, light quark flavors (up, down, etc), 
\begin_inset Formula $1\dots N_{f}$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
flv[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $t_{i}$
\end_inset

, heavy quark flavors (top, maybe bottom), 
\begin_inset Formula $1\dots N_{t}$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
flvt[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $s_{i}$
\end_inset

, Dirac (spinor) indices, 
\begin_inset Formula $1\dots4$
\end_inset

: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
spn[
\end_layout

\end_inset


\begin_inset Formula $i$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Frame
Tensors:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $f^{a_{1}a_{2}a_{3}}$
\end_inset

 , 
\begin_inset Formula $SU(N_{c})$
\end_inset

 structure constants: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
colorf[
\end_layout

\end_inset


\begin_inset Formula $a_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $a_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $a_{3}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $T_{i_{1}i_{2}}^{a}$
\end_inset

, 
\begin_inset Formula $SU(N_{c})$
\end_inset

 generators, 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
colorT[
\end_layout

\end_inset


\begin_inset Formula $a$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $i_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $i_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $\left(\gamma^{\mu}\right)_{s_{1}s_{2}}$
\end_inset

, Dirac matrices: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
gammachain[gamma[
\end_layout

\end_inset


\begin_inset Formula $\mu$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
],
\end_layout

\end_inset


\begin_inset Formula $s_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $s_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Formula $\left(\slashed{p}\right)_{s_{1}s_{2}}$
\end_inset

, Dirac slash notation: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
gammachain[slash[p],
\end_layout

\end_inset


\begin_inset Formula $s_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $s_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

;
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 notation, II
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Propagator denominators:
\end_layout

\begin_deeper
\begin_layout Itemize
Massless: 
\begin_inset Formula $\frac{1}{p^{2}+i0}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
den[p]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Massive: 
\begin_inset Formula $\frac{1}{p^{2}-m^{2}+i0}\to$
\end_inset


\begin_inset space ~
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
den[p,m2]
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
Momenta components: 
\begin_inset Formula $p^{\mu}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
momentum[p,
\end_layout

\end_inset


\begin_inset Formula $\mu$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Scalar products: 
\begin_inset Formula $p\cdot q\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
sp[p,q]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Delta functions:
\end_layout

\begin_deeper
\begin_layout Itemize
Quark flavor: 
\begin_inset Formula $\delta_{f_{1}f_{2}}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
deltaf[
\end_layout

\end_inset


\begin_inset Formula $f_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $f_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Heavy quark flavor: 
\begin_inset Formula $\delta_{t_{1}t_{2}}\to$
\end_inset


\begin_inset space ~
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
deltaft[
\end_layout

\end_inset


\begin_inset Formula $t_{1}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $t_{2}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Generic: 
\begin_inset Formula $\delta_{xy}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
delta[
\end_layout

\end_inset


\begin_inset Formula $x$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
,
\end_layout

\end_inset


\begin_inset Formula $y$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Itemize
Quark electric charges:
\end_layout

\begin_deeper
\begin_layout Itemize
Light: 
\begin_inset Formula $Q_{f_{i}}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
chargeQ[
\end_layout

\end_inset


\begin_inset Formula $f_{i}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Heavy: 
\begin_inset Formula $Q_{t_{i}}\to$
\end_inset

 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
chargeQt[
\end_layout

\end_inset


\begin_inset Formula $t_{i}$
\end_inset


\begin_inset Flex Code
status open

\begin_layout Plain Layout
]
\end_layout

\end_inset

.
\end_layout

\end_deeper
\end_deeper
\begin_layout Frame
Everything else (products, sums, powers, etc): the normal 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 expressions.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Feynman rules: propagators
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Quark propagator: 
\begin_inset Formula $i\delta_{f_{1}f_{2}}\delta_{i_{1}i_{2}}\frac{\left(\slashed{p}\right)_{s_{2}s_{1}}}{p^{2}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Heavy quark propagator: 
\begin_inset Formula $i\delta_{t_{1}t_{2}}\delta_{i_{1}i_{2}}\frac{\left(\slashed{p}+\mathds{1}m_{t}\right)_{s_{2}s_{1}}}{p^{2}-m^{2}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Gluon propagator: 
\begin_inset Formula $-i\delta_{a_{1}a_{2}}\left(\frac{g_{\mu_{1}\mu_{2}}}{p^{2}}-\left(\xi-1\right)\frac{p^{\mu_{1}}p^{\mu_{2}}}{\left(p^{2}\right)^{2}}\right)$
\end_inset

.
\end_layout

\begin_layout Itemize
Ghost propagator: 
\begin_inset Formula $i\delta_{a_{1}a_{2}}\frac{1}{p^{2}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Photon propagator: not needed, photons are external for 
\begin_inset Formula $H^{\mu\nu}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Frame
See 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
Amplitude[]
\end_layout

\end_inset

 in 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
allthecode.m
\end_layout

\end_inset

 for the implementation.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Feynman rules: vertices
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Anti-quark/quark/gluon vertex: 
\begin_inset Formula $ig_{s}\delta_{f_{1}f_{2}}T_{i_{1}i_{2}}^{a_{3}}\left(\gamma^{\mu_{3}}\right)_{s_{1}s_{2}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Anti-quark/quark/photon vertex: 
\begin_inset Formula $ig_{e}Q_{f_{1}}\delta_{f_{1}f_{2}}\delta_{i_{1}i_{2}}\left(\gamma^{\mu_{1}}\right)_{s_{1}s_{2}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Anti-ghost/ghost/gluon vertex: 
\begin_inset Formula $g_{s}f^{a_{3}a_{2}a_{1}}p_{1}^{\mu_{3}}$
\end_inset

.
\end_layout

\begin_layout Itemize
Three-gluon vertex: 
\begin_inset Formula 
\[
g_{s}f^{a_{1}a_{2}a_{3}}\left(g^{\mu_{1}\mu_{2}}\left(p_{1}-p_{2}\right)^{\mu_{3}}+\left(123\to231\right)+\left(123\to312\right)\right).
\]

\end_inset


\end_layout

\begin_layout Itemize
Four-gluon vertex:
\begin_inset Formula 
\[
-ig_{s}\left(f^{\nu\mu_{1}\mu_{2}}f^{\nu\mu_{3}\mu_{4}}\left(g^{\mu_{1}\mu_{3}}g^{\mu_{2}\mu_{4}}-g^{\mu_{1}\mu_{4}}g^{\mu_{2}\mu_{3}}\right)+\left(1342\right)+\left(1423\right)\right).
\]

\end_inset

Note the fresh summation index 
\begin_inset Formula $\nu$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Tensor summation
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Dirac tensor summation with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

 can expand traces of gamma matrices:
\begin_inset Formula 
\begin{align*}
\mathrm{Tr}\left(\gamma^{\mu}\slashed{p}\right)=\, & \left(\gamma^{\mu}\right)_{s_{1}s_{2}}\left(\slashed{p}\right)_{s_{2}s_{1}}\\
=\, & \text{\code{gammachain[gamma[lor[mu]],spn[1],spn[2]]*}}\\
 & \text{\code{*gammachain[slash[p],spn[2],spn[1]]}}\\
\to\, & \text{\code{gammachain[gamma[mu],slash[p],spn[1],spn[1]]}}\\
\to\, & \text{\code{g\_(}}i\text{\code{,mu,p)}}\\
 & \text{\code{traceN} }i;\\
\to\, & \text{\code{p(mu)}}\\
\to\, & \text{\code{momentum[p,lor[mu]]}}
\end{align*}

\end_inset

On the 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

 side:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $i$
\end_inset

 can be any arbitrary (but unique) integer;
\end_layout

\begin_layout Itemize
\begin_inset Flex Code
status open

\begin_layout Plain Layout
mu
\end_layout

\end_inset

 must be declared as an index: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
index mu = d
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Flex Code
status open

\begin_layout Plain Layout
p
\end_layout

\end_inset

 must be declared as a vector: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
vector p
\end_layout

\end_inset

;
\end_layout

\begin_layout Itemize
\begin_inset Flex Code
status open

\begin_layout Plain Layout
p
\end_layout

\end_inset

 must be a single variable, not an expression.
\end_layout

\end_deeper
\begin_layout Frame
See: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.dirac-trace.frm
\end_layout

\end_inset

.
\end_layout

\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Color tensor summation with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Color.h
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Color.h
\end_layout

\end_inset

 is a 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

 package for 
\begin_inset Formula $SU(N)$
\end_inset

 tensor summation, available at 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

https://www.nikhef.nl/~form/maindir/packages/color/
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Usage in summary:
\begin_inset Formula 
\begin{align*}
\mathrm{Tr}\left(T^{a_{1}}T^{a_{1}}\right)=\, & T_{i_{1}i_{2}}^{a_{1}}T_{i_{2}i_{1}}^{a_{1}}\\
=\, & \text{\code{colorT[ajd[1],fun[1],fun[1]]*}}\\
 & \text{\code{*colorT[ajd[2],fun[2],fun[1]]}}\\
\to\, & \text{\code{T(fun1,fun2,adj1)*T(fun2,fun1,adj1)}}\\
 & \text{\code{\#call\ docolor}}\\
\to\, & \text{\code{NA*I2R}}\\
\to\, & N_{a}\,T_{f}
\end{align*}

\end_inset

See: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.color-trace.frm
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Quark flavor summation
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Flavor-related factors are complicated by the dependence of charge on flavor
 (
\begin_inset Formula $Q_{f}$
\end_inset

).
 Three cases are relevant:
\begin_inset Formula 
\begin{align*}
\sfig{flavor.nf} & \sim N_{f},\\
\sfig{flavor.sumqf} & \sim{\textstyle \sum}Q_{f},\\
\sfig{flavor.sumqf2} & \sim{\textstyle \sum}Q_{f}^{2},
\end{align*}

\end_inset

and the same three cases for the heavy quarks (
\begin_inset Formula $N_{t}$
\end_inset

, 
\begin_inset Formula $\sum Q_{t}$
\end_inset

, 
\begin_inset Formula $\sum Q_{t}^{2}$
\end_inset

).
\end_layout

\begin_layout Frame
No library for this; just some 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

 code that:
\end_layout

\begin_deeper
\begin_layout Enumerate
Apply each 
\begin_inset Formula $\delta_{f_{1}f_{2}}$
\end_inset

 factor by renaming 
\begin_inset Formula $f_{1}$
\end_inset

 into 
\begin_inset Formula $f_{2}$
\end_inset

 (if 
\begin_inset Formula $f_{1}\ne f_{2}$
\end_inset

).
\end_layout

\begin_layout Enumerate
Recognize the possible remaining cases (
\begin_inset Formula $Q_{f}^{2}\delta_{ff}$
\end_inset

, 
\begin_inset Formula $Q_{f}\delta_{ff}$
\end_inset

, 
\begin_inset Formula $\delta_{ff}$
\end_inset

).
\end_layout

\end_deeper
\begin_layout Frame
See: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.flavor-trace.frm
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
IBP relations
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Integral family construction (for IBP)
\end_layout

\end_inset


\end_layout

\begin_layout Frame
An IBP integral family with 
\begin_inset Formula $L$
\end_inset

 loop momenta 
\begin_inset Formula $l_{i}$
\end_inset

, and 
\begin_inset Formula $E$
\end_inset

 external momenta 
\begin_inset Formula $p_{i}$
\end_inset

, is the set of integrals
\begin_inset Formula 
\[
I_{\nu_{1},\nu_{2},\dots,\nu_{N}}\equiv\int\frac{\mathrm{d}^{d}l_{1}}{\left(2\pi\right)^{d}}\dots\frac{\mathrm{d}^{d}l_{L}}{\left(2\pi\right)^{d}}\frac{1}{D_{1}^{\nu_{1}}\dots D_{N}^{\nu_{N}}},
\]

\end_inset

where 
\begin_inset Formula $N=L\left(L+1\right)/2+LE$
\end_inset

, and the denominators,
\begin_inset Formula 
\[
D_{i}=\left(c_{ij}\,l_{j}+c'_{ik}\,p_{k}\right)^{2}-m_{i}^{2}+i0,
\]

\end_inset

are linearly independent when viewed as polynomials in the scalar products
\begin_inset Formula 
\[
s\equiv\left\{ l_{i}\!\cdot\!l_{j},\ l_{i}\!\cdot\!p_{j}\right\} .
\]

\end_inset

This ensures that each 
\begin_inset Formula $s_{i}$
\end_inset

 can be uniquely expressed in terms of 
\begin_inset Formula $D_{j}$
\end_inset

:
\begin_inset Formula 
\[
D_{i}=M_{ij}\,s_{j}+C_{i}\quad\Rightarrow\quad s_{i}=\left(M^{-1}\right)_{ij}\left(D_{j}-C_{j}\right).
\]

\end_inset

To express an integral in 
\begin_inset Formula $I_{\nu\cdots}$
\end_inset

 notation: replace all 
\begin_inset Formula $s$
\end_inset

 by 
\begin_inset Formula $D$
\end_inset

, count 
\begin_inset Formula $D_{i}$
\end_inset

 powers.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Integral symmetries
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Compare these two integrals:
\begin_inset Formula 
\[
I_{1}=\int\frac{\mathrm{d}^{d}l}{\left(p_{1}-l\right)^{2}\left(p_{1}+p_{2}-l\right)^{2}l^{2}}\text{ and }I_{2}=\int\frac{\mathrm{d}^{d}l'}{\left(p_{1}+l'\right)^{2}\left(p_{2}-l'\right)^{2}l'^{2}}.
\]

\end_inset

To see that they are equal, use Feynman parameters:
\begin_inset Formula 
\[
I_{i}=\Gamma\!\left(3-\frac{d}{2}\right)\int\mathrm{d}x_{1}\mathrm{d}x_{1}\mathrm{d}x_{1}\,x_{1}x_{2}x_{3}\,\delta(1-x_{1}-x_{2}-x_{3})\,\mathcal{U}_{i}^{3-d}\mathcal{F}_{i}^{d/2-3},
\]

\end_inset


\begin_inset Formula 
\[
\mathcal{U}_{1}=x_{1}+x_{2}+x_{3},\ \mathcal{F}_{1}=\left(x_{1}+x_{2}\right)x_{3}p_{1}^{2}+\left(x_{1}+x_{3}\right)x_{2}p_{2}^{2}+2x_{2}x_{3}p_{1}\!\cdot\!p_{2},
\]

\end_inset


\begin_inset Formula 
\[
\mathcal{U}_{2}=x_{1}+x_{2}+x_{3},\ \mathcal{F}_{2}=\left(x_{2}+x_{3}\right)x_{1}p_{1}^{2}+\left(x_{1}+x_{3}\right)x_{2}p_{2}^{2}+2x_{1}x_{2}p_{1}\!\cdot\!p_{2}.
\]

\end_inset

Both expressions become identical under 
\begin_inset Formula $x_{1,2,3}\leftrightarrow x_{3,2,1}$
\end_inset

.
 In momenta space this corresponds to 
\begin_inset Formula $l'=l-p_{1}$
\end_inset

.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Pak~'11"
target "https://arxiv.org/abs/1111.0868"
literal "true"

\end_inset

;
\begin_inset space ~
\end_inset


\begin_inset CommandInset href
LatexCommand href
name "\\noun{Feynson}"
target "https://github.com/magv/feynson"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Automated implementation: 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Feynson
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/magv/feynson"
target "https://github.com/magv/feynson"
literal "false"

\end_inset

).
\end_layout

\begin_layout Frame
Example: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.feynson.symmetrize.in
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Scaleless (zero) integral detection
\end_layout

\end_inset


\end_layout

\begin_layout Frame
For efficiency, scaleless integrals should be put to zero early.
\end_layout

\begin_layout Frame
Consider the triangle family with one off-shell leg:
\begin_inset Formula 
\[
I_{a,b,c}\equiv\sfig{triangle}
\]

\end_inset

Two subsectors of this family are zero:
\begin_inset Formula 
\[
I_{0,b,c}\equiv\sfig{triangle-noa}=0,\qquad\text{and}\qquad I_{a,b,0}\equiv\sfig{triangle-noc}=0.
\]

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Sufficient criteria
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "Lee~'13"
target "https://arxiv.org/abs/1310.1145"
literal "true"

\end_inset

): a family (or a subsector) is zero if there are such 
\begin_inset Formula $x$
\end_inset

-independent 
\begin_inset Formula $k_{i}$
\end_inset

, that
\begin_inset Formula 
\[
\sum_{i}k_{i}\,x_{i}\frac{\partial}{\partial x_{i}}\left(\mathcal{F}(x)+\mathcal{U}(x)\right)=\mathcal{F}(x)+\mathcal{U}(x),
\]

\end_inset

where 
\begin_inset Formula $\mathcal{F}$
\end_inset

, 
\begin_inset Formula $\mathcal{U}$
\end_inset

, and 
\begin_inset Formula $x$
\end_inset

 give the corresponding Feynman parameterization.
\end_layout

\begin_layout Frame
Implementation: any IBP solver, also 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Feynson
\end_layout

\end_inset

.
\end_layout

\begin_layout Frame
Example: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.feynson.zero-sectors.in
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
IBP software
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Fire6
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "bitbucket.org/feynmanIntegrals/fire"
target "https://bitbucket.org/feynmanIntegrals/fire"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Smirnov,~Chukharev~'19"
target "https://arxiv.org/abs/1901.07808"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Fast and parallel
\end_layout

\end_inset

 Laporta-style IBP reduction implementation.
\end_layout

\begin_layout Itemize
Has a (terrible) 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 interface, but C++ core.
\end_layout

\begin_layout Itemize
Can use modular arithmetic to control intermediate expression swell, and
 for much greater parallelizibility (thousands of cores).
\end_layout

\begin_layout Itemize
Requires 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
LiteRed
\end_layout

\end_inset

 to discover symmetries within integral families.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Kira
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "kira.hepforge.org"
target "https://kira.hepforge.org/"
literal "false"

\end_inset

):
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Maierhöfer,~Usovitsch,~Uwer~'18"
target "http://arxiv.org/abs/1705.05610"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Fast and parallel
\end_layout

\end_inset

 Laporta-style IBP reduction implementation.
\end_layout

\begin_layout Itemize
Automatically finds symmetries within and between families.
\end_layout

\begin_layout Itemize
Optionally uses modular arithmetic methods.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Klappert,~Lange,~et~al~'20"
target "https://arxiv.org/abs/2008.06494"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Main drawback: excessive memory use (e.g.
 200GB for a 4-loop problem).
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
IBP software, contd.
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
LiteRed
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "inp.nsk.su/~lee/programs/LiteRed"
target "https://inp.nsk.su/~lee/programs/LiteRed/"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Lee~'13"
target "https://arxiv.org/abs/1310.1145"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Heuristic-driven IBP relation solution for general indices.
\end_layout

\begin_layout Itemize
Written in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

, 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
easy to use
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Not parallelizable, and 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
too slow
\end_layout

\end_inset

 for larger problems.
\end_layout

\begin_layout Itemize
Guarantees the minimal number of masters.
\end_layout

\begin_layout Itemize
Contains auxiliary functions for integral differentiation, Feynman parameterizat
ion, and dimentional recurrence construction.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Forcer
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/benruijl/forcer"
target "https://github.com/benruijl/forcer"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Ruijl,~Ueda,~Vermaseren~'17"
target "https://arxiv.org/abs/1704.06650"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Hand-crafted reduction for massless 2-point functions up to 4
\begin_inset space ~
\end_inset

loops.
\end_layout

\begin_layout Itemize
Written in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

, parallelizable.
\end_layout

\begin_layout Itemize
The 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
fastest thing for 2-point functions
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Frame
Many others: 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Reduze}"
target "https://arxiv.org/abs/1201.4330"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Air}"
target "https://arxiv.org/abs/hep-ph/0404258"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Fmft}"
target "https://arxiv.org/abs/1707.01710"
literal "true"

\end_inset

/
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Matad}"
target "https://arxiv.org/abs/hep-ph/0009029"
literal "true"

\end_inset

, private implementations, etc.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
IBP reduction with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Kira
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Frame

\end_layout

\begin_layout Frame
Usage in short:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "kira.hepforge.org"
target "https://kira.hepforge.org/"
literal "false"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Define kinematics (
\begin_inset Flex Code
status open

\begin_layout Plain Layout
config/kinematics.yaml
\end_layout

\end_inset

).
\end_layout

\begin_layout Itemize
List integral families (
\begin_inset Flex Code
status open

\begin_layout Plain Layout
config/integralfamilies.yaml
\end_layout

\end_inset

).
\end_layout

\begin_layout Itemize
Create a jobs file (e.g.
 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
jobs.yaml
\end_layout

\end_inset

), defining
\end_layout

\begin_deeper
\begin_layout Itemize
for which integrals to 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
write down
\end_layout

\end_inset

 IBP relations (
\begin_inset Formula $r$
\end_inset

 and 
\begin_inset Formula $s$
\end_inset

 bounds);
\end_layout

\begin_layout Itemize
for which integrals to 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
solve
\end_layout

\end_inset

 IBP relations (
\begin_inset Formula $r$
\end_inset

, 
\begin_inset Formula $s$
\end_inset

, and 
\begin_inset Formula $d$
\end_inset

 bounds, or a list).
\end_layout

\end_deeper
\begin_layout Itemize
Get the results as 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 (or 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

) substitution tables.
\end_layout

\end_deeper
\begin_layout Frame
Guide to the notation:
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $r$
\end_inset


\end_layout

\end_inset

 is the sum of denominator powers (positive indices);
\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $s$
\end_inset


\end_layout

\end_inset

 is the sum of numerator powers (negative indices);
\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $d$
\end_inset


\end_layout

\end_inset

 is the sum of dots (indices 
\begin_inset Formula $\ge2$
\end_inset

);
\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $t$
\end_inset


\end_layout

\end_inset

 is the number of denominators.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Formula 
\[
\begin{array}{l}
\left.\begin{array}{lcccc}
\hphantom{I_{3,3,3,-3,0}}\, & r & s & d & t\end{array}\right.\\
\left.\begin{array}{lcccc}
I_{1,2,1,0,0} & 4 & 0 & 1 & 3\\
I_{1,1,3,-1,0} & 5 & 1 & 2 & 3
\end{array}\right\} 11100_{2l}=\text{Sector\ id }7\\
\left.\begin{array}{lcccc}
I_{0,2,2,0,0} & 4 & 0 & 2 & 2\\
I_{0,1,2,-2,0} & 3 & 2 & 1 & 2
\end{array}\right\} 01100_{2l}=\text{Sector\ id }6
\end{array}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Sector decomposition
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Sector decomposition software
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/gudrunhe/secdec"
target "https://github.com/gudrunhe/secdec"
literal "false"

\end_inset

):
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Python
\end_layout

\end_inset

 core that generates C++ files.
\end_layout

\begin_layout Itemize
Can use different Monte-Carlo integration algorithms: Randomized Quasi-Monte
 Carlo
\begin_inset space ~
\end_inset

(
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Qmc}"
target "https://github.com/mppmu/qmc"
literal "true"

\end_inset

), 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Vegas
\end_layout

\end_inset

/
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Suave
\end_layout

\end_inset

/
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Divonne
\end_layout

\end_inset

/
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Cuhre
\end_layout

\end_inset


\begin_inset space ~
\end_inset

(
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Cuba}"
target "http://www.feynarts.de/cuba/"
literal "true"

\end_inset

), 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Cquad
\end_layout

\end_inset


\begin_inset space ~
\end_inset

(
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Gsl}"
target "http://www.gnu.org/software/gsl/"
literal "true"

\end_inset

).
\end_layout

\begin_layout Itemize
Optional GPU support (with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qmc
\end_layout

\end_inset

).
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Fiesta
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "bitbucket.org/feynmanIntegrals/fiesta"
target "https://bitbucket.org/feynmanIntegrals/fiesta/"
literal "false"

\end_inset

):
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 core, generates/compiles to C++ behind the scenes.
\end_layout

\begin_layout Itemize
Custom Monte-Carlo integration on both CPU and GPU.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Sector decomposition with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Frame
Usage overview (details: 
\begin_inset CommandInset href
LatexCommand href
name "secdec.readthedocs.io"
target "https://secdec.readthedocs.io/"
literal "false"

\end_inset

):
\end_layout

\begin_deeper
\begin_layout Enumerate
Setup 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset

 by installing dependencies and running 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
make
\end_layout

\end_inset

 in its release archive.
\end_layout

\begin_layout Enumerate
Use the 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Python
\end_layout

\end_inset

 module 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset

 to define the integral and generate the code for the integration library.
\end_layout

\begin_layout Enumerate
Compile the integration library.
\end_layout

\begin_layout Enumerate
Import the integration library from 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Python
\end_layout

\end_inset

 (or 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
C++
\end_layout

\end_inset

), call it to perform integration.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Differential equations
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Method of differential equations
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Consider a family of integrals depending on external momenta 
\begin_inset Formula $p_{i}$
\end_inset

 and masses 
\begin_inset Formula $m_{i}$
\end_inset

:
\begin_inset Formula 
\[
I_{\nu_{1},\nu_{2},\dots,\nu_{N}}\equiv\int\frac{\mathrm{d}^{d}l_{1}}{\left(2\pi\right)^{d}}\dots\frac{\mathrm{d}^{d}l_{L}}{\left(2\pi\right)^{d}}\frac{1}{D_{1}^{\nu_{1}}\dots D_{N}^{\nu_{N}}}=I_{\nu_{1},\nu_{2},\dots,\nu_{N}}\!\left(\left\{ p_{i}\!\cdot\!p_{j},m_{i}^{2}\right\} \right).
\]

\end_inset

With dimensional analysis one of the parameters can be scaled out, making
 remaining arguments dimensionless:
\begin_inset Formula 
\[
I\left(\left\{ p_{i}\!\cdot\!p_{j},m_{i}\right\} \right)=\left(p_{1}^{2}\right)^{\frac{d}{2}L-\sum_{i}\nu_{i}}\,I\left(\left\{ x_{i}\right\} \right),\qquad\left\{ x_{i}\right\} =\left\{ \frac{p_{i}\!\cdot\!p_{j}}{p_{1}^{2}},\frac{m_{i}^{2}}{p_{1}^{2}}\right\} .
\]

\end_inset

Idea: if 
\begin_inset Formula $\left\{ x_{i}\right\} \ne\emptyset$
\end_inset

, we can construct differential equations in 
\begin_inset Formula $x_{i}$
\end_inset

, and solve them instead of performing the loop integration directly.
\end_layout

\begin_layout Frame
(In practice: just set 
\begin_inset Formula $p_{1}^{2}=1$
\end_inset

, restore it in the end by dimensionality).
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Constructing differential equations
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Suppose IBP relations were solved, and we have a set of master integrals
 
\begin_inset Formula $I_{i}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Enumerate
Differentiate each 
\begin_inset Formula $I_{i}$
\end_inset

 by one of the parameters, 
\begin_inset Formula $x=m_{a}^{2}\text{ or }p_{a}\cdot p_{b}$
\end_inset

:
\begin_inset Formula 
\[
\partial_{m_{a}^{2}}I=\int\frac{\mathrm{d}^{d}l_{1}}{\left(2\pi\right)^{d}}\dots\frac{\mathrm{d}^{d}l_{L}}{\left(2\pi\right)^{d}}\,\partial_{m_{a}}\frac{1}{D_{1}^{\nu_{1}}\dots D_{N}^{\nu_{N}}},
\]

\end_inset


\begin_inset Formula 
\[
\partial_{p_{a}\cdot p_{b}}I=\left(G^{-1}\right)_{ia}p_{i}\!\cdot\!\left(\partial_{p_{b}}I\right),\quad\partial_{p_{a}\cdot p_{a}}I=\frac{1}{2}\left(G^{-1}\right)_{ia}p_{i}\!\cdot\!\left(\partial_{p_{a}}I\right),
\]

\end_inset

where 
\begin_inset Formula $G$
\end_inset

 is the Gram matrix: 
\begin_inset Formula $G_{ij}\equiv p_{i}\cdot p_{j}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Express the derivatives as integrals in the same family:
\begin_inset Formula 
\[
\partial_{x}I_{i}=\sum_{k}C_{k}\,I_{\nu_{1}^{(k)},\nu_{2}^{(k)},\dots,\nu_{N}^{(k)}}.
\]

\end_inset


\end_layout

\begin_layout Enumerate
Use the IBP tables to reduce those integrals back to the master integrals,
 thus obtaining a linear differential equation system for 
\begin_inset Formula $I_{i}$
\end_inset

:
\begin_inset Formula 
\begin{align*}
\partial_{x}I_{i} & \overset{\text{IBP}}{=}\sum_{j}M_{ij}\,I_{j}.
\end{align*}

\end_inset


\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Example: self-energy with one mass
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Consider a family of self-energy with one mass integrals
\begin_inset Formula 
\[
I_{a,b}\equiv\sfig{selfenergy}=\int\frac{\mathrm{d}^{d}l}{\left(2\pi\right)^{d}}\frac{1}{\left(q-l\right)^{2a}\left(l^{2}-m^{2}\right)^{b}}.
\]

\end_inset

This family has two master integrals:
\begin_inset Formula 
\begin{align*}
I_{1} & \equiv I_{0,1}=\sfig{selfenergy-01}=\int\frac{\mathrm{d}^{d}l}{\left(2\pi\right)^{d}}\frac{1}{l^{2}-m^{2}},\\
I_{2} & \equiv I_{1,1}=\sfig{selfenergy-11}=\int\frac{\mathrm{d}^{d}l}{\left(2\pi\right)^{d}}\frac{1}{\left(q-l\right)^{2}\left(l^{2}-m^{2}\right)}.
\end{align*}

\end_inset

Constructing differential equation system:
\begin_inset Formula 
\[
\partial_{m^{2}}\underbrace{\begin{pmatrix}\vphantom{\frac{2}{\left(m^{2}\right)}}I_{1}\\
\vphantom{\frac{2}{\left(m^{2}\right)}}I_{2}
\end{pmatrix}}_{\V I}=\begin{pmatrix}\vphantom{\frac{2}{\left(m^{2}\right)}}I_{0,2}\\
\vphantom{\vphantom{\frac{2}{\left(m^{2}\right)}}}I_{1,2}
\end{pmatrix}\overset{\text{IBP}}{=}\underbrace{\begin{pmatrix}\vphantom{\frac{2}{\left(m^{2}\right)}}\frac{-2+d}{2m^{2}} & 0\\
\vphantom{\frac{2}{\left(m^{2}\right)}}\frac{2-d}{2m^{2}\left(m^{2}-q^{2}\right)} & \frac{-3+d}{m^{2}-q^{2}}
\end{pmatrix}}_{\M M}\underbrace{\begin{pmatrix}\vphantom{\frac{2}{\left(m^{2}\right)}}I_{1}\\
\vphantom{\frac{2}{\left(m^{2}\right)}}I_{2}
\end{pmatrix}}_{\V I}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Solution via the epsilon-form
\end_layout

\end_inset


\end_layout

\begin_layout Frame
If we have a differential equation system for 
\begin_inset Formula $\V I$
\end_inset

,
\begin_inset Formula 
\[
\partial_{x}\V I(x,d)=\M M(x,d)\,\V I(x,d),\qquad\text{where }d=4-2\varepsilon,
\]

\end_inset

we can transform to a new basis 
\begin_inset Formula $\V J$
\end_inset

 via the transformation matrix 
\begin_inset Formula $\M T$
\end_inset

,
\begin_inset Formula 
\[
\V I(x,\varepsilon)=\M T(x,\varepsilon)\,\V J(x,\varepsilon),
\]

\end_inset

and for 
\begin_inset Formula $\V J$
\end_inset

 the same differential equation system will look like
\begin_inset Formula 
\[
\partial_{x}\V J=\M T^{-1}\left(\M M\M T-\partial_{x}\M T\right)\V J\equiv\M M'\,\V J.
\]

\end_inset

Idea: if this system is in an 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
\begin_inset Formula $\varepsilon$
\end_inset

-form
\end_layout

\end_inset

 (has the dependence on 
\begin_inset Formula $\varepsilon$
\end_inset

 factorized),
\begin_inset Formula 
\[
\partial_{x}\V J(x,\varepsilon)=\varepsilon\,\M S(x)\,\V J(x,\varepsilon),
\]

\end_inset

then the solution for 
\begin_inset Formula $\V J$
\end_inset

 as a series in 
\begin_inset Formula $\varepsilon$
\end_inset

 becomes trivial:
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Henn~'13"
target "https://arxiv.org/abs/1304.1806"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Formula 
\[
\V J(x,\varepsilon)\equiv\varepsilon^{k_{0}}\sum_{k=0}^{\infty}\varepsilon^{k}\V J^{(k)}(x),
\]

\end_inset


\begin_inset Formula 
\[
\sum\varepsilon^{k}\V J^{(k)}=\sum\varepsilon^{k+1}\M S\,\V J^{(k)}\quad\Rightarrow\quad\V J^{(k)}(x)=\int^{x}\M S(x')\,\V J^{(k-1)}(x')\mathrm{d}x'+\V C^{(k)}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Example, cont.: the epsilon-form
\end_layout

\end_inset


\end_layout

\begin_layout Frame
The 
\begin_inset Formula $\varepsilon$
\end_inset

-form can be achieved with a transformation to the following basis 
\begin_inset Formula $\V J$
\end_inset

:
\begin_inset Formula 
\[
\V I=\begin{pmatrix}\left(-1+2\varepsilon\right)m^{2} & 0\\
-1+\varepsilon & \left(-1+\varepsilon\right)\left(m^{2}-q^{2}\right)
\end{pmatrix}\V J.
\]

\end_inset

Differential equation system in 
\begin_inset Formula $\V J$
\end_inset

 then has the form
\begin_inset Formula 
\[
\partial_{m^{2}}\V J=\varepsilon\begin{pmatrix}-\frac{1}{m^{2}} & 0\\
-\frac{1}{q^{2}}\frac{1}{m^{2}}-\frac{1}{q^{2}}\frac{1}{m^{2}-q^{2}} & -\frac{2}{m^{2}-q^{2}}
\end{pmatrix}\V J.
\]

\end_inset

Accordingly, the solution is
\begin_inset Formula 
\begin{align*}
\V J^{\left(0\right)} & =\V C^{(0)},\\
\V J^{(1)} & =\V C^{(1)}+\begin{pmatrix}-C_{1}^{(0)}\int^{m^{2}}\frac{\mathrm{d}m'^{2}}{m'^{2}}\\
\dots
\end{pmatrix},\\
\V J^{(2)} & =\V C^{(2)}+\begin{pmatrix}-C_{1}^{(1)}\int^{m^{2}}\frac{\mathrm{d}m'^{2}}{m'^{2}}+C_{1}^{(0)}\int^{m^{2}}\frac{\mathrm{d}m'^{2}}{m'^{2}}\int^{m'^{2}}\frac{\mathrm{d}m''^{2}}{m''^{2}}\\
\dots
\end{pmatrix}.
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Iterated integrals
\end_layout

\end_inset


\end_layout

\begin_layout Frame
The solution to differential equation in an 
\begin_inset Formula $\varepsilon$
\end_inset

-form always come as iterated integrals of the form of 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
multiple polylogarithms
\end_layout

\end_inset

 (a.k.a Goncharov polylogarithms):
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Goncharov~'98"
target "https://arxiv.org/abs/1105.2076"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Formula 
\[
G(w_{1},w_{2},\dots,w_{n};x)\equiv\int_{0}^{x}\frac{\mathrm{d}t_{1}}{t_{1}-w_{1}}\int_{0}^{t_{1}}\frac{\mathrm{d}t_{2}}{t_{2}-w_{2}}\cdots\int_{0}^{t_{n}}\frac{\mathrm{d}t_{n}}{t_{n}-w_{n}}.
\]

\end_inset

Special case for the trailing zeros:
\begin_inset Formula 
\[
G(\underbrace{0,\dots,0}_{n};x)\equiv\frac{1}{n!}\log^{n}(x).
\]

\end_inset

Integration and differention, the simple case:
\begin_inset Formula 
\[
\int\mathrm{d}x\frac{1}{x-a}G(\vec{w};x)=G(a,\vec{w};x)+C,\quad\frac{\mathrm{d}}{\mathrm{d}x}G(w_{1},\vec{w};x)=\frac{1}{x-w_{1}}G(\vec{w};x).
\]

\end_inset

Software: 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{GiNaC}"
target "https://www.ginac.de/"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Hpl}"
target "https://www.physik.uzh.ch/data/HPL/"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{Harmpol}"
target "https://www.nikhef.nl/~form/maindir/packages/harmpol/harmpol.html"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{HyperInt}"
target "https://bitbucket.org/PanzerErik/hyperint/"
literal "true"

\end_inset

, 
\begin_inset CommandInset href
LatexCommand href
name "\\noun{PolyLogTools}"
target "https://gitlab.com/pltteam/plt"
literal "true"

\end_inset

, etc.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Example, cont.: the solution in GPLs
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Rewriting the iterated integrals in terms of 
\begin_inset Formula $G$
\end_inset

:
\begin_inset Formula 
\begin{align*}
\V J^{\left(0\right)} & =\V C^{(0)},\\
\V J^{(1)} & =\V C^{(1)}+\begin{pmatrix}-C_{1}^{(0)}G(0;m^{2})\\
\dots
\end{pmatrix},\\
\V J^{(2)} & =\V C^{(2)}+\begin{pmatrix}-C_{1}^{(1)}G(0;m^{2})+C_{1}^{(0)}G(0,0;m^{2})\\
\dots
\end{pmatrix}.
\end{align*}

\end_inset

It is trivial to generate this answer to any required order.
 Note that the answer will only have 
\begin_inset Formula $w_{i}\in\{0,q^{2}\}$
\end_inset

.
 By rescaling the weights (or setting 
\begin_inset Formula $q^{2}$
\end_inset

 to
\begin_inset space ~
\end_inset

1), this can be turned into 
\begin_inset Formula $w_{i}\in\left\{ 0,1\right\} $
\end_inset

, the 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
harmonic polylogarithms
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Multiple polylogarithms, more properties
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Differentiation in the general case
\end_layout

\end_inset

 when 
\begin_inset Formula $w_{i}$
\end_inset

 may depend on 
\begin_inset Formula $x$
\end_inset

:
\begin_inset Formula 
\[
\frac{\d}{\d x}G(w_{1},\dots,w_{n};y)=\sum_{i}G(w_{1},\dots,\xcancel{w_{i}},\dots,w_{n};y)\frac{\d}{\d x}\log\frac{w_{i}-w_{i-1}}{w_{i}-w_{i+1}},
\]

\end_inset

where 
\begin_inset Formula $w_{0}\equiv y$
\end_inset

, and 
\begin_inset Formula $w_{i+1}\equiv0$
\end_inset

.
\end_layout

\begin_layout Frame
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Integration in the general case
\end_layout

\end_inset

 when 
\begin_inset Formula $w_{i}$
\end_inset

 may depend on 
\begin_inset Formula $x$
\end_inset

 is nontrivial, one must first represent 
\begin_inset Formula $G$
\end_inset

 in a way that the integration variable only appears in the last argument.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Brown~'08"
target "https://arxiv.org/abs/0804.1660"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Automated via 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
HyperInt
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "bitbucket.org/PanzerErik/hyperint"
target "https://bitbucket.org/PanzerErik/hyperint/"
literal "false"

\end_inset

):
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Panzer~'14"
target "https://arxiv.org/abs/1403.3385"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset Formula 
\begin{align*}
 & \text{\code{\$\ maple}}\\
 & \text{\code{>\ read\ "HyperInt.mpl";}}\\
 & \text{\code{>\ fibrationBasis(Hlog(x,[y,x,y]),[x,y]);}}\\
 & \text{\code{Hlog(x,[y,y,y])-Hlog(x,[y,0,y])}}
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Multiple polylogarithms and their relatives
\end_layout

\end_inset


\end_layout

\begin_layout Frame
The integral representation of multiple polylogarithms (
\begin_inset Formula $G$
\end_inset

) is equivalent to the infinite sum representation (
\begin_inset Formula $\mathrm{Li}$
\end_inset

):
\begin_inset Formula 
\begin{align*}
\mathrm{Li}_{m_{1},\dots,m_{n}}\!\left(x_{1},\dots,x_{n}\right) & =\sum_{i_{1}>\dots>i_{n}>0}\frac{x_{1}^{i_{1}}}{i_{1}^{m_{1}}}\cdots\frac{x_{n}^{i_{n}}}{i_{n}^{m_{n}}}=\\
 & =\left(-1\right)^{n}G\bigl(\underbrace{0,\dots,0}_{m_{1}-1},\frac{1}{x_{i}},\dots,\underbrace{0,\dots,0}_{m_{2}-1},\frac{1}{x_{1}x_{2}\cdots x_{n}};1\bigr).
\end{align*}

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Note: there are conflicting conventions for the order of the indices in
 the 
\begin_inset Formula $\mathrm{Li}$
\end_inset

 summation.
 Above is the 
\begin_inset Quotes eld
\end_inset

physicist
\begin_inset Quotes erd
\end_inset

 notation (used in e.g.
 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Hpl
\end_layout

\end_inset

, 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
GiNaC
\end_layout

\end_inset

, and the MZV datamine).
\end_layout

\begin_layout Itemize
The 
\begin_inset Quotes eld
\end_inset

mathematician
\begin_inset Quotes erd
\end_inset

 notation is reverse: 
\begin_inset Formula $0<i_{1}<\dots<i_{n}$
\end_inset

; it was used by Goncharov, and is also used in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
HyperInt
\end_layout

\end_inset

.
 The order of indices in the Multiple Zeta Values is also reversed there.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Multiple polylogarithms and their relatives, II
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Logarithms
\end_layout

\end_inset

 are GPLs with a single weight: 
\begin_inset Formula 
\[
\log x=G(0;x),\qquad\log\left(\frac{a-x}{a}\right)=G(a;x).
\]

\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Nielsen’s generalized polylogarithms
\end_layout

\end_inset

 are GPLs with 
\begin_inset Formula $w_{i}\in\{0,1\}$
\end_inset

:
\begin_inset Formula 
\[
S_{n,p}(x)=\left(-1\right)^{p}G(\underbrace{0,...,0}_{n},\underbrace{1,...,1}_{p};x).
\]

\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Harmonic polylogarithms
\end_layout

\end_inset

 (HPLs) are GPLs with 
\begin_inset Formula $w_{i}\in\{0,\pm1\}$
\end_inset

:
\begin_inset Formula 
\[
H_{\dots,+m,-n,0}(x)=(-1)^{m}G(\dots,\underbrace{0,\dots,0,1}_{m},\underbrace{0,\dots,0,-1}_{n},0;x).
\]

\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Two-dimensional HPLs
\end_layout

\end_inset

 are GPLs with 
\begin_inset Formula $w_{i}\in\left\{ 0,1,1−z,−z\right\} $
\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Multiple Zeta Values
\end_layout

\end_inset

 
\begin_inset Formula $\zeta_{\vec{w}}$
\end_inset

 are just 
\begin_inset Formula $H_{\vec{w}}(1)$
\end_inset

 (in the 
\begin_inset Quotes eld
\end_inset

physicist
\begin_inset Quotes erd
\end_inset

 notation).
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Fixing the integration constants
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Differential equations only give the solution up to the integration constants.
 Finding these constants is the essential difficulty of the method.
 There are many ways.
\end_layout

\begin_deeper
\begin_layout Itemize
By evaluating the integrals in a limit where they simplify.
\end_layout

\begin_deeper
\begin_layout Itemize
Large mass limit.
 Small mass limit.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Smirnov~'02"
target "https://www.springer.com/us/book/9783540423348"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Even massless integrals can be evaluated by adding masses to them, and connectin
g the large mass limit to the massless limit via the differential equations.
\end_layout

\end_deeper
\begin_layout Itemize
Using the knowledge of the analytic properties of the integrals.
\end_layout

\begin_deeper
\begin_layout Itemize
E.g.: enforcing regularity in the kinematic limits.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Gehrmann,~Remiddi~'00"
target "https://arxiv.org/abs/hep-ph/0008287"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
From partial knowledge of the integrals values, such as a Mellin moment.
\end_layout

\begin_deeper
\begin_layout Itemize
One can integrate over the semi-inclusive integrals to obtain the fully
 inclusive ones.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Gituliar~'15"
target "https://arxiv.org/abs/1512.02045"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_deeper
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Example, cont.: integration constants
\end_layout

\end_inset


\end_layout

\begin_layout Frame
First, we consider 
\begin_inset Formula $I_{1}$
\end_inset

 (the vacuum bubble) 
\begin_inset Quotes eld
\end_inset

simple
\begin_inset Quotes erd
\end_inset

, and look it up in a book:
\begin_inset Formula 
\[
\sfig{selfenergy-01}=-\frac{i\pi^{\frac{d}{2}}}{\left(2\pi\right)^{d}}\Gamma\left(1-\frac{d}{2}\right)\left(m^{2}-i0\right)^{\frac{d}{2}-1}.
\]

\end_inset

For 
\begin_inset Formula $I_{2}$
\end_inset

: in the limit of zero mass the massive diagram becomes massless,
\begin_inset Formula 
\[
\lim_{m^{2}\to0}\sfig{selfenergy-11}=\sfig{selfenergy-11-nomass}.
\]

\end_inset

This massless integral we also know from a book:
\begin_inset Formula 
\[
\sfig{selfenergy-11-nomass}=\frac{i\pi^{\frac{d}{2}}}{\left(2\pi\right)^{d}}\frac{\Gamma^{2}\!\left(\frac{d}{2}-1\right)\Gamma\!\left(2-\frac{d}{2}\right)}{\Gamma\!\left(d-2\right)}\left(-q^{2}-i0\right)^{\frac{d}{2}-2}.
\]

\end_inset

Then, we can expand these in series' in 
\begin_inset Formula $\varepsilon$
\end_inset

, compare with the series from 
\begin_inset Formula $\varepsilon$
\end_inset

-form solution, and fix all 
\begin_inset Formula $\V C^{(k)}$
\end_inset

.
\end_layout

\begin_layout Frame
See: 
\begin_inset Flex Code
status open

\begin_layout Plain Layout
example.massive-self-energy.nb
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
The fundamental solution
\end_layout

\end_inset


\end_layout

\begin_layout Frame
A 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
fundamental solution
\end_layout

\end_inset

 to 
\begin_inset Formula $\partial_{x}\V I=\M M\,\V I$
\end_inset

 is an 
\begin_inset Formula $n$
\end_inset

 by 
\begin_inset Formula $n$
\end_inset

 matrix of independent solutions, such that any solution can be expressed
 as a linear combination of its columns.
\end_layout

\begin_layout Frame
A fundametal solution for a system in an 
\begin_inset Formula $\varepsilon$
\end_inset

-form, 
\begin_inset Formula $\partial_{x}\V J=\varepsilon\,\M S\,\V J$
\end_inset

, can be constructed as a series in 
\begin_inset Formula $\varepsilon$
\end_inset

:
\begin_inset Formula 
\[
\M W=\mathds{1}+\varepsilon\int_{x_{0}}^{x}\d x'\,\M S(x')+\varepsilon^{2}\int_{x_{0}}^{x}\d x'\,\M S(x')\int_{x_{0}}^{x'}\d x''\,\M S(x'')+\dots.
\]

\end_inset

The general solution is then just 
\begin_inset Formula $\M W$
\end_inset

 multiplied by a vector of integration constants 
\begin_inset Formula $\V C$
\end_inset

:
\begin_inset Formula 
\[
\V J(x,\varepsilon)=\M W(x,\varepsilon)\,\V C(\varepsilon),
\]

\end_inset

where the constants themselves are a series in 
\begin_inset Formula $\varepsilon$
\end_inset

,
\begin_inset Formula 
\[
\V C(\varepsilon)\equiv\varepsilon^{k_{0}}\sum_{k=0}^{\infty}\varepsilon^{k}\V C^{(k)}.
\]

\end_inset

This is an alternative way to write down a solution for 
\begin_inset Formula $\V J$
\end_inset

, with the benefit of being immediately extendable to the multivariate case.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
The multivariate case
\end_layout

\end_inset


\end_layout

\begin_layout Frame
If differential equations in multiple variables are considered, and a combined
 
\begin_inset Formula $\varepsilon$
\end_inset

-form is achieved,
\begin_inset Formula 
\[
\partial_{x_{i}}\V J(\vec{x},\varepsilon)=\varepsilon\,\M S_{i}(\vec{x})\,\V J(\vec{x},\varepsilon),
\]

\end_inset

then writing down the solution in this case can be made easy by:
\end_layout

\begin_deeper
\begin_layout Enumerate
Choosing an integration contour along the axes 
\begin_inset Formula $x_{i}$
\end_inset

 in an some order, for example from 
\begin_inset Formula $(0,0,\dots)$
\end_inset

 to 
\begin_inset Formula $(x_{1},0,\dots)$
\end_inset

, then to 
\begin_inset Formula $(x_{1},x_{2},0,\dots)$
\end_inset

, etc.
\end_layout

\begin_layout Enumerate
Writing down the fundamental solutions along each segment, 
\begin_inset Formula $\M W_{i}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Because segments are chosen such that only 
\begin_inset Formula $x_{i}$
\end_inset

 changes along each, 
\begin_inset Formula $\M W_{i}$
\end_inset

 can be calculated the same as in single-variate case.
\end_layout

\end_deeper
\end_deeper
\begin_layout Frame
The general solution for 
\begin_inset Formula $\V J$
\end_inset

 is then
\begin_inset Formula 
\[
\V J(\vec{x},\varepsilon)=\M W_{n}(x_{1},\dots,x_{n},\varepsilon)\cdots\M W_{n}(x_{1},\varepsilon)\,\V C(\varepsilon).
\]

\end_inset

Overall this result will be a sum of terms of this form:
\begin_inset Formula 
\[
G({\text{arguments depending on }x_{1},\dots,x_{n-1}};x_{n})\cdots G({\text{constants}};x_{1}).
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Lee algorithm
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Lee algorithm for finding the epsilon-form
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Overall idea: to go from a differential equation system
\begin_inset Formula 
\[
\partial_{x}\V I=\M M\,\V I,\quad\text{where }\M M(x,\varepsilon)=\sum_{i}\frac{\M A_{i}(\varepsilon)}{\left(x-x_{i}\right)^{k_{i}}},
\]

\end_inset

to an 
\begin_inset Formula $\varepsilon$
\end_inset

-form
\begin_inset Formula 
\[
\partial_{x}\V J=\varepsilon\,\M S\,\V J,\quad\text{where }\M S(x)=\sum_{i}\frac{\M S_{i}}{x-x_{i}},
\]

\end_inset

apply a series of simple basis transformation 
\begin_inset Formula $\V I=\M T\,\V J$
\end_inset

, such that each brings the system a bit closer to an 
\begin_inset Formula $\varepsilon$
\end_inset

-form.
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Lee~'14"
target "https://arxiv.org/abs/1411.0911"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Enumerate
If a higher pole is present (i.e.
 
\begin_inset Formula $k_{i}\ne1$
\end_inset

) then use a transformation that reduces the rank of 
\begin_inset Formula $\M A_{i}$
\end_inset

, eventually eliminating it.
 (
\begin_inset Quotes eld
\end_inset

Fuchsification
\begin_inset Quotes erd
\end_inset

).
\end_layout

\begin_layout Enumerate
Else, for the eigenvalues of 
\begin_inset Formula $\M A_{i}$
\end_inset

 of the form 
\begin_inset Formula $n+k\varepsilon$
\end_inset

, use a transformation that shifts 
\begin_inset Formula $n$
\end_inset

 by 
\begin_inset Formula $\pm1$
\end_inset

, eventually setting it to zero.
 (
\begin_inset Quotes eld
\end_inset

Normalization
\begin_inset Quotes erd
\end_inset

).
\end_layout

\begin_layout Enumerate
If all 
\begin_inset Formula $\M A_{i}$
\end_inset

 eigenvalues are proportional to 
\begin_inset Formula $\varepsilon$
\end_inset

, use a transformation that makes the whole 
\begin_inset Formula $\M A_{i}$
\end_inset

 proportional to 
\begin_inset Formula $\varepsilon$
\end_inset

.
 (
\begin_inset Quotes eld
\end_inset

Factorization
\begin_inset Quotes erd
\end_inset

).
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Lee algorithm, fuchsification
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Consider 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
a “balance” transformation between 
\begin_inset Formula $x_{1}$
\end_inset

 and 
\begin_inset Formula $x_{2}$
\end_inset


\end_layout

\end_inset

:
\begin_inset Formula 
\[
\M T(x,\varepsilon)=\overline{\M P}(\varepsilon)+\frac{x-x_{2}}{x-x_{1}}\M P(\varepsilon),\quad\text{with }\M P^{2}=\M P,\text{ and }\M P+\overline{\M P}=\mathds{1}.
\]

\end_inset

If the matrix 
\begin_inset Formula $\M M$
\end_inset

 has a pole at 
\begin_inset Formula $x=x_{1}$
\end_inset

 of power 
\begin_inset Formula $n>1$
\end_inset

,
\begin_inset Formula 
\[
\M M=\M A_{-n}\left(x-x_{1}\right)^{-n}+\M A_{-n+1}\left(x-x_{1}\right)^{-n+1}+\dots,
\]

\end_inset


\end_layout

\begin_layout Frame
then either 
\end_layout

\begin_deeper
\begin_layout Enumerate
there is such 
\begin_inset Formula $\M P$
\end_inset

 and 
\begin_inset Formula $x_{2}$
\end_inset

 , that the transformed 
\begin_inset Formula $\M A_{-n}$
\end_inset

 is of lower rank than 
\begin_inset Formula $\M A_{-n}$
\end_inset

 , while the leading expansion order around 
\begin_inset Formula $x=x_{2}$
\end_inset

 does increase beyound 
\begin_inset Formula $n=1$
\end_inset

; or
\end_layout

\begin_layout Enumerate
the 
\begin_inset Formula $\varepsilon$
\end_inset

-form cannot be reached by any rational transformation.
\end_layout

\end_deeper
\begin_layout Frame
Then, a series of balance transformations can decrease the rank of all 
\begin_inset Formula $\M A_{-n}$
\end_inset

 with 
\begin_inset Formula $n\ne1$
\end_inset

 to zero, transforming 
\begin_inset Formula $\M M$
\end_inset

 into the 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Fuchsian form
\end_layout

\end_inset

:
\begin_inset Formula 
\[
\M M(x,\varepsilon)=\sum\frac{\M A_{i}(\varepsilon)}{x-x_{i}}.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Lee algorithm, normalization
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Differential equations for master integrals are observed to have a special
 feature: when transformed into Fuchsian form,
\begin_inset Formula 
\[
\M M=\sum\frac{\M A_{i}}{x-x_{i}},
\]

\end_inset

the eigenvalues of 
\begin_inset Formula $\M A_{i}$
\end_inset

 often have the form of 
\begin_inset Formula $n+k\varepsilon$
\end_inset

, where 
\begin_inset Formula $n\in\mathbb{N}$
\end_inset

.
\end_layout

\begin_layout Frame
Now, a balance can be found between 
\begin_inset Formula $x_{1}$
\end_inset

 and 
\begin_inset Formula $x_{2}$
\end_inset

 that does not spoil the Fuchsian form.
 Such a balance will shift one of the eigenvalues of 
\begin_inset Formula $\M A_{1}$
\end_inset

 by 
\begin_inset Formula $+1$
\end_inset

, one of 
\begin_inset Formula $\M A_{2}$
\end_inset

 by 
\begin_inset Formula $−1$
\end_inset

.
\end_layout

\begin_layout Frame
Then, a series of such balances can transform 
\begin_inset Formula $\M M$
\end_inset

 into a 
\begin_inset Flex Emph
status open

\begin_layout Plain Layout
normalized Fuchsian form
\end_layout

\end_inset

: where all 
\begin_inset Formula $\M A_{i}$
\end_inset

 have eigenvalues proportional to 
\begin_inset Formula $\varepsilon$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Itemize
Sometimes eigenvalues of the form 
\begin_inset Formula $\frac{1}{2}+n+k\varepsilon$
\end_inset

 are encountered.
 If they are present at up to three different 
\begin_inset Formula $\M A_{i}$
\end_inset

 , then it is possible to change the integration variable from 
\begin_inset Formula $x$
\end_inset

 to such 
\begin_inset Formula $y$
\end_inset

, that the differential equation in 
\begin_inset Formula $y$
\end_inset

 has all the eigenvalues in the form 
\begin_inset Formula $n+k\varepsilon$
\end_inset

.
\end_layout

\begin_layout Itemize
Sometimes eigenvalues are not linear in 
\begin_inset Formula $\varepsilon$
\end_inset

.
 This often means the master integral basis is linearly dependent.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Lee algorithm, factorization
\end_layout

\end_inset


\end_layout

\begin_layout Frame
Finally, once all the eigenvalues of 
\begin_inset Formula $\M M$
\end_inset

 residues are proportional to 
\begin_inset Formula $\varepsilon$
\end_inset

, then either
\end_layout

\begin_deeper
\begin_layout Enumerate
the whole matrix can be made proportional to 
\begin_inset Formula $\varepsilon$
\end_inset

 by a transformation that does not depend on 
\begin_inset Formula $x$
\end_inset

; or
\end_layout

\begin_layout Enumerate
the 
\begin_inset Formula $\varepsilon$
\end_inset

-form can not be reached.
\end_layout

\end_deeper
\begin_layout Frame
Such transformation is searched for via an ansatz.
\end_layout

\begin_layout Frame
Note that on this step, and on all previous ones too, there exists multiple
 transformations that can achieve the desired form.
 As a result, the 
\begin_inset Formula $\varepsilon$
\end_inset

-form is not unique.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Lee algorithm, the multivariate case
\end_layout

\end_inset


\end_layout

\begin_layout Frame
If a system of differential equations in multiple variables is considered:
\begin_inset Formula 
\[
\partial_{x_{i}}\V I(\vec{x},\varepsilon)=\M M_{i}(\vec{x},\varepsilon)\,\V I(\vec{x},\varepsilon),
\]

\end_inset

then it is useful to have a single transformation 
\begin_inset Formula $\M T(x,\varepsilon)$
\end_inset

 that transforms all of the differential equation systems into 
\begin_inset Formula $\varepsilon$
\end_inset

-form simultaneously,
\begin_inset Formula 
\[
\partial_{x_{i}}\V J(\vec{x},\varepsilon)=\varepsilon\,\M S_{i}(\vec{x})\,\V J(\vec{x},\varepsilon),\qquad\text{with }\V I=\M T\,\V J.
\]

\end_inset

Single-variable Lee algorithm can be reused for this by:
\end_layout

\begin_deeper
\begin_layout Enumerate
Reducing 
\begin_inset Formula $\M M_{1}$
\end_inset

 to an 
\begin_inset Formula $\varepsilon$
\end_inset

-form with 
\begin_inset Formula $\M T_{1}(x_{1},x_{2},\dots,\varepsilon)$
\end_inset

.
\end_layout

\begin_layout Enumerate
Transforming all the equations with 
\begin_inset Formula $\M T_{1}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Reducing 
\begin_inset Formula $\M M_{2}$
\end_inset

 to an 
\begin_inset Formula $\varepsilon$
\end_inset

-form with a such a 
\begin_inset Formula $\M T_{2}(x_{2},\dots,\varepsilon)$
\end_inset

 that is independent of 
\begin_inset Formula $x_{1}$
\end_inset

 and 
\begin_inset Formula $\varepsilon$
\end_inset

.
\end_layout

\begin_layout Enumerate
Transforming all the equations with 
\begin_inset Formula $\M T_{2}$
\end_inset

.
 This will not spoil the 
\begin_inset Formula $\varepsilon$
\end_inset

-form in 
\begin_inset Formula $x_{1}$
\end_inset

 because 
\begin_inset Formula $\M T_{2}$
\end_inset

 does not depend on 
\begin_inset Formula $x_{1}$
\end_inset

 or 
\begin_inset Formula $\varepsilon$
\end_inset

.
\end_layout

\begin_layout Enumerate
Repeating similarly for the rest of 
\begin_inset Formula $\M M_{i}.$
\end_inset


\end_layout

\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Epsilon-form software
\end_layout

\end_inset


\end_layout

\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Fuchsia
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/magv/fuchsia.cpp"
target "https://github.com/magv/fuchsia.cpp"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Gituliar,~Magerya~'17"
target "https://arxiv.org/abs/1701.04269"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Implements the Lee algorithm.
\end_layout

\begin_layout Itemize
Initial version in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Python
\end_layout

\end_inset

/
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
SageMath
\end_layout

\end_inset

, newer version in C++/
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
GiNaC
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Differential equation systems in multiple variables are supported.
\end_layout

\begin_layout Itemize
Can certify if a system is irreducible (with exceptions).
\end_layout

\begin_layout Itemize
Can suggest variable changes that will make it reducible.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Canonica
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/christophmeyer/CANONICA"
target "https://github.com/christophmeyer/CANONICA"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Meyer~'17"
target "https://arxiv.org/abs/1701.04269"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Uses the rational ansatz method.
\end_layout

\begin_deeper
\begin_layout Standard
Constructs an ansatz for the transformation matrix up to some powers of
 the variables, then solves for the coefficients.
\end_layout

\end_deeper
\begin_layout Itemize
Written in 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Supports systems with multiple variables.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Epsilon
\end_layout

\end_inset

 (
\begin_inset CommandInset href
LatexCommand href
name "github.com/mprausa/epsilon"
target " https://github.com/mprausa/epsilon"
literal "false"

\end_inset

)
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sidenote{
\end_layout

\end_inset


\begin_inset CommandInset href
LatexCommand href
name "Prausa~'17"
target "https://arxiv.org/abs/1701.00725"
literal "true"

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Implements the Lee algorithm.
 
\end_layout

\begin_layout Itemize
Doesn't support systems with multiple variables.
\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Section
Summary
\end_layout

\begin_layout Frame
\begin_inset Argument 4
status open

\begin_layout Plain Layout
Summary
\end_layout

\end_inset


\end_layout

\begin_layout Frame
We have talked about:
\end_layout

\begin_deeper
\begin_layout Itemize
Computing loop amplitudes with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Qgraf
\end_layout

\end_inset

, 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Form
\end_layout

\end_inset

, 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Color.h
\end_layout

\end_inset

, 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Feynson
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Mathematica
\end_layout

\end_inset

 as the hot glue.
\end_layout

\begin_layout Itemize
IBP reduction with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Kira
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Numerical evaluation with 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
pySecDec
\end_layout

\end_inset

.
\end_layout

\begin_layout Itemize
Differential equations, the 
\begin_inset Formula $\varepsilon$
\end_inset

-form, 
\begin_inset Flex Noun
status open

\begin_layout Plain Layout
Fuchsia
\end_layout

\end_inset

.
\end_layout

\end_deeper
\begin_layout Frame
\begin_inset VSpace 5mm
\end_inset


\end_layout

\begin_layout Frame

\end_layout

\begin_deeper
\begin_layout Standard
\align center

\shape italic
\begin_inset Flex Only
status open

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout

2
\end_layout

\end_inset


\begin_inset Flex Emph
status open

\begin_layout Plain Layout
Thank you for your attention.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Frame

\end_layout

\end_body
\end_document
