#  Four-loop QCD MOM beta functions
from the three-loop vertices at the symmetric point

Alexander Bednyakov<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980,
Russia  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Leninskii pr., 5, Moscow 119991, Russia  Andrey Pikelner
<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia

###### Abstract

For the first time, we compute three-loop contributions to all triple vertices
in QCD at the symmetric point. The analytic results are obtained in massless
QCD with an arbitrary color group in the Landau gauge. All new loop integrals
are expressed in terms of harmonic polylogarithms at the sixth root of unity.
These corrections allow us to derive expressions for the four-loop QCD beta
function in a set of momentum-subtraction schemes.

##  I Introduction

The strong coupling constant $\alpha_{s}\equiv g_{s}^{2}/(4\pi)$ is a
fundamental parameter of QCD. It enters the predictions of many Standard Model
(SM) observables, e.g., the Higgs production cross-section Anastasiou  _et
al._ (2015) at the LHC, and the uncertainty of $\alpha_{s}$ significantly
influences the comparison between theory and experiment. In itself,
$\alpha_{s}$ is not a physical observable, its value depends on the scale
$Q^{2}$ and the utilized renormalization prescription (or scheme). The freedom
in choosing a normalization point manifests itself in “running” governed by
the renormalization group (RG) equations

$\displaystyle\frac{d\alpha_{s}(Q^{2})}{d\ln Q^{2}}=\beta(\alpha_{s}(Q^{2})).$
(1)

Given the renormalization scheme (RS), one can calculate the $\beta$-function
order by order in perturbation theory (PT). To parametrize the strength of
strong interactions, one usually uses the ${\rm\overline{MS}}$scheme and
quotes the value of $\alpha_{s}^{(5)}(M_{Z})=0.1179(10)$ Tanabashi  _et al._
(2018) defined in effective QCD with $n_{f}=5$ massless flavors and evaluated
at the $Z$-boson mass. This definition is of great convenience when studying
inclusive observables dominated by short-distance effects. For almost 20 years
the four-loop $\beta^{\rm\overline{MS}}$ van Ritbergen  _et al._ (1997);
Czakon (2005) in the ${\rm\overline{MS}}$ scheme has been utilized in state-
of-the-art QCD calculations, and only recently the five-loop result Baikov
_et al._ (2017); Luthe  _et al._ (2017); Herzog  _et al._ (2017a); Chetyrkin
_et al._ (2017) became available in the literature. The latter allows one to
significantly improve the agreement between $\alpha_{s}$ determinations (see,
e.g., Ref. Tanabashi  _et al._ (2018) for details) in precision measurements
carried out at different energy scales.

At lower energies other $\alpha_{s}$ definitions can be more convenient. For
example, in lattice QCD one can introduce the strong coupling as a particular
RG-invariant combination (so-called _invariant_ charge) of the vertex and two-
point Green functions evaluated at fixed external momenta in a fixed (usually
Landau) gauge (see, e.g., Refs. Boucaud  _et al._ (2014); Zafeiropoulos  _et
al._ (2019)). Since lattice results Aoki  _et al._ (2019) give rise to one of
the most precise determinations of $\alpha_{s}^{(5)}(M_{Z})$, it is important
to study the strong coupling and its running in the momentum-subtraction (MOM)
schemes. In these RSs the vertex functions and propagators are normalized in
such a way that at a certain kinematic point there are no corrections beyond
the tree level. As a consequence, the renormalized couplings coincide with the
corresponding invariant charges and can be directly compared to the
nonperturbative lattice results.

Figure 1: For symmetric point integrals $p_{1}^{2}=p_{2}^{2}=q^{2}$ and for
auxiliary integrals we use $p_{1}^{2}=p_{2}^{2},q^{2}=x\,p_{1}^{2}$

Contrary to the ${\rm\overline{MS}}$ scheme, in the MOM scheme one needs to
know the Green functions beyond the divergent terms. Some choices of
kinematics can make the calculation quite challenging , especially of
vertices. In this paper, we consider three-point functions that depend on
momenta $p_{1}$, $p_{2}$, and $q=p_{1}+p_{2}$ (see Fig. 1) and utilize the
symmetric normalization point (SMOM) in the Euclidean region
$p_{1}^{2}=p_{2}^{2}=q^{2}=-Q^{2}$. One-loop QCD renormalization in the SMOM
scheme in a general linear gauge has been known for quite a long time,
following the pioneering work in Ref.Celmaster and Gonsalves (1979). At the
moment, only two-loop expressions for the QCD vertices in this kinematics are
available in the literature in numerical Chetyrkin and Seidensticker (2000)
and analytical Gracey (2011a) forms, allowing one to find three-loop Chetyrkin
and Seidensticker (2000); Gracey (2011b) strong-coupling beta functions in
SMOM RS. In our work, we exploit modern Feynman-integral evaluation methods to
improve these results by one more order of PT. To simplify our calculation, we
routinely use the Landau gauge, in which the gluon propagator is transversal
and there is no need to consider the gauge-parameter renormalization.

In the context of perturbative QCD (pQCD) one can relate renormalized
couplings $a_{\rm ren}\equiv\alpha^{\rm ren}_{s}/(4\pi)$ defined in different
RSs. In what follows, we consider $a_{\rm ren}=\\{a_{\rm ggg},a_{\rm
ccg},a_{\rm qqg}\\}$, which satisfy

$\displaystyle\mu^{-2\varepsilon}a_{\rm bare}$ $\displaystyle=Z_{a_{\rm
ren}}a_{\rm ren}=\left[\frac{Z_{\rm ggg}^{2}}{Z_{\rm gg}^{3}}\right]a_{\rm
ggg}=\left[\frac{Z_{\rm ccg}^{2}}{Z_{\rm cc}^{2}Z_{\rm gg}}\right]a_{\rm
ccg}=\left[\frac{Z_{\rm qqg}^{2}}{Z_{\rm qq}^{2}Z_{\rm gg}}\right]a_{\rm
qqg},$ (2)

with $a_{\rm bare}$ being the bare coupling defined in the dimensionally
regularized theory with $d=4-2\varepsilon$. The bare coupling is divergent and
is related to the renormalized ones $a_{\rm ren}$ via divergent factors
$Z_{a_{\rm ren}}$. The latter are combinations of the renormalization
constants of the three-gluon ($Z_{\rm ggg}$), ghost-gluon ($Z_{\rm ccg}$), and
quark-gluon ($Z_{\rm qqg}$) vertices together with those of the gluon ($Z_{\rm
gg}$), ghost ($Z_{\rm cc}$) and quark ($Z_{\rm qq}$) fields.

The vertex renormalization constants denoted collectively by $Z_{V}$ are
obtained order by order in perturbation theory from the bare $\Gamma^{V}_{\rm
bare}$ vertices by making sure that the renormalized counterpart
$\Gamma^{V}_{\rm ren}$

$\displaystyle\Gamma^{V}_{\rm ren}(q_{i},a_{\rm ren})=Z_{V}(a_{\rm
ren})\Gamma^{V}_{\rm bare}(q_{i},a_{\rm bare}),\quad\mu^{-2\varepsilon}a_{\rm
bare}=Z_{a_{\rm ren}}a_{\rm ren}$ (3)

is finite for all external momenta $q_{i}$ and satisfies certain normalization
conditions.

In the ${\rm\overline{MS}}$scheme only divergent terms are subtracted, and one
can prove that $a_{{\rm\overline{MS}}}=a_{\rm ggg}^{\rm\overline{MS}}=a_{\rm
ccg}^{\rm\overline{MS}}=a_{\rm qqg}^{\rm\overline{MS}}$. On the contrary, in
the SMOM schemes one also subtracts finite terms and requires that there are
no $\mathcal{O}(a_{\rm ren})$ corrections to a particular $\Gamma^{V}_{\rm
ren}$ at the symmetric point, characterized by momentum $Q^{2}$. To avoid the
appearance of $\log(Q^{2}/\mu^{2})$ in the renormalization constants, it is
convenient to choose $Q^{2}=\mu^{2}$.

The vertex functions of our interest have color and space-time indices and one
can decompose them in terms of basis tensors with Lorentz-invariant
coefficients (form-factors). The choice of the basis is not unique and we make
use of a decomposition Gracey (2011a) taht is valid for the symmetric point.
To save space, we present here only terms Chetyrkin and Seidensticker (2000)
that are relevant for the definition of the required vertex renormalization
constants (3):

$\displaystyle\Gamma^{abc}_{\mu}(p_{1},p_{2})$
$\displaystyle=-ig_{s}f^{abc}\left(p^{\nu}_{1}g_{\nu\mu}\Gamma^{\rm
ccg}(-\mu^{2})+\ldots\right),$ (4)
$\displaystyle\Gamma^{abc}_{\mu\nu\rho}(p_{1},p_{2})$
$\displaystyle=ig_{s}f^{abc}\left(T_{\mu\nu\rho}\Gamma^{\rm
ggg}(-\mu^{2})+\ldots\right),$ (5)
$\displaystyle\Gamma^{a}_{\mu,ij}(p_{1},p_{2})$
$\displaystyle=g_{s}T^{a}_{ij}\left(\gamma_{\mu}\Gamma^{\rm
qqg}(-\mu^{2})+\ldots\right).$ (6)

Here all momenta are assumed to be outgoing, $p_{1}$ in Eq.(4) corresponds to
the antighost, and $T_{\mu\nu\rho}=g_{\mu\nu}(p_{1}-p_{2})_{\rho}+\ldots$
represents the tensor that enters into the tree-level three-gluon vertex. The
SU(N) generators $T^{a}_{ij}$ in the quark-gluon vertex (6) satisfy
$[T^{a},T^{b}]_{ij}=if^{abc}T^{c}_{ij}$ with structure constants $f^{abc}$.

The expressions for the bare form factors $\Gamma^{V}_{\rm bare}$ with
$V=\\{{\rm ggg,~{}ccg,~{}qqg}\\}$ are extracted from the tensor vertices by
means of the projectors given in Ref. Gracey (2011a). To define $a_{\rm ggg}$
in the respective SMOM scheme (MOMggg) via Eq. (2), we require that at the
symmetric point $\Gamma^{\rm ggg}_{\rm ren}(-\mu^{2})=1$, i.e. $Z_{\rm
ggg}^{-1}=\Gamma^{\rm ggg}_{\rm bare}(-\mu^{2})$. In the same way one can
relate the bare coupling $a_{\rm bare}$ to $a_{\rm ccg}$ or $a_{\rm qqg}$ by
requiring that either $\Gamma^{\rm ccg}_{\rm ren}(-\mu^{2})=1$ (MOMh), or
$\Gamma^{\rm qqg}_{\rm ren}(-\mu^{2})=1$ (MOMq). Since field renormalization
constants also enter into Eq.(2), we have to impose conditions on $Z_{\rm
gg}$, $Z_{\rm cc}$, and $Z_{\rm qq}$ in the SMOM scheme: there should be no
corrections to the corresponding tree-level propagator for the external
momentum $q^{2}=-\mu^{2}$.

From Eq. (2) one can deduce that two renormalization prescriptions for $a_{\rm
ren}$, say $a_{R}$ and $a_{{\rm\overline{MS}}}$, are related via finite
correction factors $X_{R}$:

$\displaystyle a_{R}$
$\displaystyle=\left(Z_{a_{R}}/Z_{a_{{\rm\overline{MS}}}}\right)a_{{\rm\overline{MS}}}\equiv
a_{{\rm\overline{MS}}}X_{R}=a_{{\rm\overline{MS}}}\left[1+\sum_{l}X^{(l)}_{R}a_{{\rm\overline{MS}}}^{l}\right].$
(7)

Given $X_{R}$ at $L$ loops, one can determine the $(L+1)$ MOM-scheme beta
functions from $\beta_{{\rm\overline{MS}}}$ via a relation that is valid in
the Landau gauge,

$\displaystyle\beta_{R}$
$\displaystyle\equiv\frac{da_{R}}{d\ln\mu^{2}}=\frac{\partial
a_{R}(a_{{\rm\overline{MS}}})}{\partial
a_{{\rm\overline{MS}}}}\cdot\beta_{{\rm\overline{MS}}}(a_{{\rm\overline{MS}}}),\qquad
a_{{\rm\overline{MS}}}=a_{{\rm\overline{MS}}}(a_{R}),$ (8)

where in the final step we invert Eq. (7) to express $\beta_{R}$ in terms of
$a_{R}$. The main aim of this paper is to calculate the three-loop corrections
$X^{(3)}_{R}$ to the relations (7) between $a_{\rm ggg}$, $a_{\rm ccg}$,
$a_{\rm qqg}$ and $a_{{\rm\overline{MS}}}$. As one of the applications of our
result, we use the four-loop beta function in the ${\rm\overline{MS}}$scheme
van Ritbergen  _et al._ (1997); Czakon (2005) to find the corresponding beta
functions $\beta_{\rm ggg}$, $\beta_{\rm ccg}$, and $\beta_{\rm qqg}$ in the
considered SMOM schemes.

##  II Details of the calculation

We generate Feynman diagrams with DIANA Tentyukov and Fleischer (2000), and
obtain 8, 106, and 2382 graphs for the $\Gamma^{\rm ggg}_{\rm ren}$ three-
gluon vertex at one, two, and three loops, respectively. Both ghost-gluon and
quark-gluon vertices give rise to 2, 33, and 688 diagrams at the same loop
levels. After the application of projectors Gracey (2011a) and taking fermion
and color tracesvan Ritbergen  _et al._ (1999), we are left with scalar
Feynman integrals, which we reduce to a set of two one-loop, eight two-loop,
and 51 three-loop master integrals by means of Reduze 2von Manteuffel and
Studerus (2012) and FIRE6Smirnov and Chuharev (2019).

The main challenge is calculating of the full set of three-loop three-point
integrals in SMOM kinematics, which was not available in the literature. To
compute master integrals, we rely on the _linear reducibility_ of massless
vertex-type integrals with arbitrary off-shell momenta, which is proven to
take place up to three-loop orderChavez and Duhr (2012); Panzer (2014). Due to
the latter property, for our integrals in more restricted kinematics we choose
two strategies of evaluation. First of all, we derive a new basis of finite
master integrals von Manteuffel  _et al._ (2015) and try to compute them
directly by means of the HyperInt packagePanzer (2015) in terms of generalized
polylogarithms (GPLs). Unfortunately, we are unable to calculate the most
complicated integrals and instead use another strategy based on the solution
of the system of differential equations (DEs). The method of DEs cannot be
applied directly to the calculation of single scale integrals we are
interested in, and we construct a set of auxiliary integrals with arbitrary
external $q^{2}=x\,p_{1}^{2}$ (Fig.1). After switching to a new variable $z$,
$x=2-z-1/z$, we reduce the original DE system to the so called
$\varepsilon$-formHenn (2013). Due to the presence of singularities at complex
points, we make use of the epsilon packagePrausa (2017), which is capable of
dealing with the latter. The obtained system in the $\varepsilon$-form is
easily solved order by order in the $\varepsilon$ expansion. The solution is
given by linear combinations of GPLs $G(a_{1},\dots,a_{n};z)$, where $a_{i}$
correspond to different sixth roots of unity, with a number of unknown
constants to be fixed from boundary conditions. The latter can be obtained by
matching the expansion in the limit $q^{2}\to 0$, corresponding to $z\to 1$,
with the explicit result of large-momentum asymptotic expansion. Since $z=1$
is a singular point of the DEs, naive Taylor expansion is not sufficient to
fix all of the constants and we exploit the EXP packageHarlander  _et al._
(1998); Seidensticker (1999) to generate the series in terms of massless
propagators. We compute the latter by the MINCER package Gorishnii  _et al._
(1989); Larin  _et al._ (1991) keeping exact dependence111Available at
https://www.nikhef.nl/~form/maindir/packages/mincer/mincerex.tgz. on the
space-time dimension variable.

In this way, we obtain the analytic results for auxiliary integrals depending
on $z$. Taking the limit $z\to e^{i\pi/3}$, which is regular and corresponds
to SMOM kinematics, we compute the required single-scale master integrals. The
correctness of the analytic calculation is verified numerically using
pySecDecBorowka  _et al._ (2018). Substituting the integrals in the
expressions for the three-loop $\Gamma^{V}_{\rm bare}$ and expanding in
$\varepsilon$ up to the necessary order, we see that the maximal
transcendental weight in the final result is, as expected, $2L=6$ with $L$
being the loop order. Using the basis and the reduction rules from Ref.Henn
_et al._ (2017), we are able to simplify the expressions for $\Gamma^{V}_{\rm
bare}$ significantly. As a consequence, we obtain the SMOM renormalization
constants, conversion factors (7), and the beta functions (8) in a rather
compact form.

##  III Results and conclusion

All of the necessary renormalization constants up to three-loop order were
calculated iteratively via Eq. (3). We also reproduced the well-known three-
loop ${\rm\overline{MS}}$ expressions, required to derive the relations (7)
between the SMOM couplings and $a_{{\rm\overline{MS}}}$ evaluated at the same
scale $Q^{2}=\mu^{2}$. To save space, we present the results in numerical form
with all QCD color factors substituted explicitly:

$\displaystyle X_{\rm ccg}=1$
$\displaystyle+a_{{\rm\overline{MS}}}\left(18.54827536-1.111111111\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{2}\left(641.9400677-85.55595017\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+1.234567901\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{3}\left(26810.13185-5350.674817\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+240.8472277\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-1.371742112\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right),$
(9) $\displaystyle X_{\rm qqg}=1$
$\displaystyle+a_{{\rm\overline{MS}}}\left(16.71577458-1.111111111\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{2}\left(472.1590958-83.11121681\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+1.234567901\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{3}\left(16997.21982-4340.986026\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+228.6939963\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-1.371742112\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right),$
(10) $\displaystyle X_{\rm ggg}=1$
$\displaystyle+a_{{\rm\overline{MS}}}\left(26.49248887-3.416806434\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{2}\left(960.4627178-202.0850109\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+7.687393017\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}\right)$
$\displaystyle+a_{{\rm\overline{MS}}}^{3}\left(42285.00716-12133.42891\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+902.7134506\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-14.34154686\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right).$
(11)

The analytic results for the general gauge group are available as a
Supplementary Material and have several remarkable properties. We managed to
simplify the two-loop part Gracey (2011b) and expressed it solely in terms of
powers of $\pi$, odd $\zeta$ values, and two polygamma functions
$\psi^{(1)}(1/3)$ and $\psi^{(3)}(1/3)$. It turns out that to write down the
three-loop contribution we only need to introduce three additional constants:
$\psi^{(5)}(1/3)$, and two combinations $H_{5}$ and $H_{6}$ of GPLs with
uniform transcendental weights five and six, respectively. Evaluating
$H_{5,6}$ numerically Vollinga and Weinzierl (2005) with high precision and
using the PSLQ algorithmFerguson  _et al._ (1999), we reconstructed the new
constants222A similar set of constants generated by cyclotomic polylogarithms
was considered in Ref.Ablinger  _et al._ (2011). through a more restricted
basis Kniehl  _et al._ (2017) of real parts of harmonic polylogarithms of the
argument $e^{i\pi/3}$.

As a first application of Eqs.(9),(10), and (11) we obtain a set of SMOM beta
functions via Eq.(8), thus extending the results of Ref. Gracey (2011b) to
four-loop order:

$\displaystyle\beta_{\rm ccg}$ $\displaystyle=\beta_{\rm uni}(a_{\rm
ccg})-a_{\rm
ccg}^{4}\left(2813.492952-617.6471546\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+21.50281811\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}\right)$
$\displaystyle-a_{\rm
ccg}^{5}\left(96089.34786-23459.32128\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+1735.992218\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-33.24145137\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right),$
(12) $\displaystyle\beta_{\rm qqg}$ $\displaystyle=\beta_{\rm uni}(a_{\rm
qqg})-a_{\rm
qqg}^{4}\left(1843.652731-588.6548459\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+22.58781183\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}\right)$
$\displaystyle-a_{\rm
qqg}^{5}\left(68529.68547-15466.43194\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+1093.568841\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-18.85323795\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right),$
(13) $\displaystyle\beta_{\rm ggg}$ $\displaystyle=\beta_{\rm uni}(a_{\rm
ggg})-a_{\rm
ggg}^{4}\left(1570.9844+0.56592607\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}-67.089536\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}+2.6581155\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}\right)$
$\displaystyle-a_{\rm
ggg}^{5}\left(94167.261-27452.645\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}+4152.5388\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{2}-543.68484\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{3}+20.429348\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}}^{4}\right),$
(14)

with $\beta_{\rm
uni}(a)=-a^{2}(11-2/3\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}})-a^{3}(102-38/3\,{\color[rgb]{0,0.35546875,0.58984375}n}_{{\color[rgb]{0,0.35546875,0.58984375}f}})$
being the universal two-loop scheme-independent contribution.

The obtained formulas can be used in a number of ways. For example, one can
improve the precision of matching between the lattice and pQCD results. In
addition, the possibility to switch from one RS to another in truncated PT
series for observables, or more generally RG-invariant quantities (see, e.g.
Refs. Gracey (2014); Herzog  _et al._ (2017b)), provides us with an additional
handle on theoretical uncertainties beyond simple scale variation. Moreover,
the computed integrals can also be used in SMOM calculations of more
complicated three-point Green functions with operator insertions, e.g., for
studies of the light-quark masses as in Refs. Almeida and Sturm (2010); Gracey
(2011c).

###### Acknowledgements.

We thank A. Kotikov, V. Magerya, and S. Mikhailov for fruitfull discussions.
The work of A.P. is supported by the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS.” The work of A.B. is supported by
the Grant of the Russian Federation Government, Agreement No. 14.W03.31.0026
from 15.02.2018.

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# Measuring Information Diffusion in Code Review at Spotify

Michael Dorner  Blekinge Institute of TechnologyKarlskronaSweden
0000-0001-8879-6450<EMAIL_ADDRESS>,  Daniel Mendez
0000-0003-0619-6027 Blekinge Institute of TechnologyKarlskronaSweden
fortissMunichGermany<EMAIL_ADDRESS>,  Ehsan Zabardast  Blekinge
Institute of TechnologyKarlskronaSweden 0000-0002-1729-5154
<EMAIL_ADDRESS>,  Nicole Valdez  SpotifyStockholmSweden
<EMAIL_ADDRESS>and  Marcin Floryan  SpotifyStockholmSweden
<EMAIL_ADDRESS>
(2024)

###### Abstract.

Background: As a core practice in software engineering, the nature of code
review has been frequently subject to research. Prior exploratory studies
found that code review, the discussion around a code change among humans,
forms a communication network that enables its participants to exchange and
spread information. Although popular in software engineering, there is no
confirmatory research corroborating this theory and the actual extent of
information diffusion in code review is not well understood.

Objective: In this registered report, we propose an observational study to
measure information diffusion in code review to test the theory of code review
as communication network.

Method: We approximate the information diffusion in code review through the
frequency and the similarity between (1) human participants, (2) affected
components, and (3) involved teams of linked code reviews. The measurements
approximating the information diffusion in code review serve as a foundation
for falsifying the theory of code review as communication network.

††journalyear: 2024††copyright: none††conference: ACM/IEEE International
Symposium on Empirical Software Engineering and Measurement (ESEM); October
20–25, 2024; Barcelona, Spain

##  1\. Introduction

The theory is compelling: Modern software systems are often too large, too
complex, and evolve too fast for an individual developer to oversee all parts
of the software and, thus, to understand all implications of a change.
Therefore, most software projects rely on code review to foster discussions on
changes and their impacts before they are merged into the code bases. During
those discussions, the participants exchange information and when needed and
deemed relevant, the information is passed on in subsequent code reviews.
Thereby, the information diffuses in the communication network that emerges
from code review.

This theory is based on the solid and thorough exploratory research that
identified information exchange as a key expectation towards code review
(Bacchelli2013, 2, 3, 5, 19, 6)—also beyond teams and architectural boundaries
(Bacchelli2013, 2, 3, 4)—which makes code review a communication network.

While this theory is plausible, exploratory research alone is not
sufficient—it also requires the confirmatory counterpart, which is currently
missing. Exploratory research begins with specific observations, distills
patterns in those observations, and derives theories from the observed
patterns using inductive reasoning. The nature of exploratory research leads
to limited generalizability as they are drawn from specific cases. As such, it
is more susceptible to researcher bias due to the absence of a predefined
theory. Deductive research starts with a general theory, makes predictions
(often in the form of hypotheses), and evaluates whether that prediction holds
true or not in empirical observations. In research, we need both exploratory
and confirmatory research to minimize bias and maximize the validity and
reliability of theories efficiently. Figure 1 shows the empirical research
cycle involving both exploratory (theory-generating) and confirmatory (theory-
testing) research.

TheoryPredictionObservationPatternInductive reasoningExploratory
researchDeductive reasoningConfirmatory research Figure 1. The empirical
research cycle (in analogy to (Mendez2019, 9)): While exploratory research is
theory-generating using inductive reasoning (starting with observations),
confirmatory research is theory-testing using deductive reasoning (starting
with a theory). This research is confirmatory.

In the proposed study, we aim to fill that gap: The objective is to test the
theory of code review as communication network. Instead of using classical
statistical tests for the hypothesis testing, we quantify the extent of
information diffusion in the code review system at Spotify, which may or may
not contradict the underlying theory of code review as communication network
or its universality. A single empirical code review system with no or marginal
information diffusion could not be aligned with the existing theory of code
review as communication network in general; further constraints, context, or
limitations must be considered. Therefore, we measure information diffusion in
code review at Spotify across social, organizational, and architectural
boundaries.

##  2\. Background

###  2.1. Code Review As Communication Network

The theory of code review as communication network is based on different
exploratory studies that investigated the motivations and expectations towards
code review in an industrial context (Bacchelli2013, 2, 3, 5, 19, 6). In a
synthesis of expectations and motivations towards code review reported by the
exploratory studies, (Dorner2024, 8) identified that information111We
consistently use _information_ instead of knowledge as in prior work
(Bacchelli2013, 2, 3, 5, 19, 6) throughout this study and concur, thereby,
with (Dorner2024, 8) and (Pascarella2018, 17). Although not equivalent,
information encodes knowledge since knowledge is the meaning that may be
derived from information through interpretation. This means that we may see
information as a superset of knowledge. Hence, not all information is
necessarily knowledge, but all knowledge is information. This allows us to
subsume different stances, definitions, and notions of knowledge without an
epistemological reflection upon the various definitions of knowledge.
Furthermore, we can refrain from delineating the notion of knowledge from the
notion of truth, the latter being too often an inherent connotation of
knowledge. We may well postulate that not everything communicated is true.
Opinions, expectations, misunderstandings, or best guesses are also part of
any engineering and development process and do not meet knowledge and,
consequently, truth by all definitions. exchange as the cause for all effects
expected from code review (Dorner2024, 8).

###  2.2. Code Review Networks

In contrast to other work (e.g., (Dorner2022, 7, 11, 22, 4, 21)) using social
network analysis to investigate code review, we use a code review network as
the name suggests: a network of code reviews whose nodes represent code
reviews and links indicate the references to other code reviews explicitly and
manually added by human code review participants.

This modelling approach is not novel. (Li2018, 15) and (Hirao2019, 12) used
this modelling approach to explore the links between code reviews. (Hirao2019,
12) explored the links between code reviews in six open-source projects that
use Gerrit as a code review tool (Hirao2019, 12). (Li2018, 15) extended the
modelling and investigation beyond pull requests to issues and identified
patterns among the linkings (Li2018, 15). Although the context (i.e., open-
source software development), research objective, and analyses of those
studies are not comparable, our modelling approach for the code review
network, which we will discuss in Section 3.2 in detail, is similar but
differs as we exclude all non-human linking activities (in contrast to
(Li2018, 15, 12) and use code review only (in contrast to (Li2018, 15)).

###  2.3. Measuring Information Diffusion in Code Review

Although different qualitative studies report information sharing as a key
expectation towards code review (Bacchelli2013, 2, 3, 5, 19, 6), only three
prior studies have quantified information exchange in code review.

In an _in-silico_ experiment, (Dorner2024, 8) simulated an artificial
information diffusion within large (Microsoft), mid-sized (Spotify), and small
code review systems (Trivago) modelled as communication networks (Dorner2024,
8). We measured the minimal topological and temporal distances between the
participants to quantify how far and how fast information can spread in code
review. We found evidence that the communication network emerging from code
review scales well and spreads information fast and broadly, corroborating the
findings of prior qualitative work. The reported upper bound of information
diffusion, however, describes information diffusion in code review under best-
case assumptions, which are unlikely to be achieved. While the upper bound of
information diffusion helps us already to understand the boundaries of code
review as a communication network, it still does not substitute a more
profound empirical measurement, for which we set the foundation with this
registered report.

In the first observational study, (Rigby2013, 18) extended the expertise
measure proposed by (Mockus2002, 16) (Mockus2002, 16). The study contrasts the
number of files a developer has modified with the number of files the
developer knows about (submitted files $\cup$ reviewed files) and found a
substantial increase in the number of files a developer knows about
exclusively through code review.

A second observational study (Sadowski2018, 19) reports (a) the number of
comments per change a change author receives over tenure at Google and (b) the
median number of files edited, reviewed, and both—as suggested by (Rigby2013,
18) (Rigby2013, 18). The study finds that the more senior a code change author
is, the fewer code comments he or she gets. The authors “postulate that this
decrease in commenting results from reviewers needing to ask fewer questions
as they build familiarity with the codebase and corroborates the hypothesis
that the educational aspect of code review may pay off over time.” In its
second measurement, the study reproduces the measurements of (Rigby2013, 18)
but reports it over the tenure of employees at Google. They showed that
reviewed and edited files are distinct sets to a large degree.

Although the proposed file-based network creation is a sophisticated approach
and may serve as a complement measurement in future studies, we found the
following limitations in the measurement applied in prior work:

  * •

File names may change over time, which introduces an unknown error to those
measurements.

  * •

The software-architectural or other technical aspects (e.g., programming
language, coding guidelines) of code make the measurements difficult to
compare in heterogeneous software projects.

  * •

We are unaware of empirical evidence that passive exposure to files in code
review would lead to improved developer fluency.

  * •

The explanatory power of both measurements is limited since the authors set
arbitrary boundaries: (Rigby2013, 18) excluded changes and reviews that
contain more than ten files, and (Sadowski2018, 19) limited the tenure of
developers to 18 months and aggregated the tenure of developers by three
months.

Furthermore, our code-review-based approach differs in two aspects: First,
information in code review is not only encoded in the source code but also is
also in the discussions within a code review. A file-based approach does not
reveal this type of information diffusion. Our code-review-based approach
includes information encoded in the affected files and in the related
discussions but also subsumes information on other abstraction layers of the
software system. Second, a file-based approach assumes a passive and implicit
information diffusion. That is, information is passively absorbed during
review by the developers. In contrast, the information diffusion captured by a
code-review-based approach like ours is an active information diffusion, that
is, a developer actively and explicitly links information that she or he deems
to be worth linking, which makes linking a human, explicit, and active
decision.

##  3\. Research Design

We designed this study as an observational study (Ayala2022, 1) measuring the
information diffusion in code review at Spotify. The measurement is not an end
in itself but serves as the foundation for our hypothesis test222Hypothesis
testing is often associated with statistical hypothesis testing, which is not
applicable in our case, regardless of its flavor (frequentist or Bayesian). A
statistical hypothesis test is built upon propositions in the context of a
population using data drawn from a sample. We, however, do not sample in this
study. As for any observational study, we do not aim for generalization but
aim to describe and uncover associations and patterns without regard to causal
relationships (Ayala2022, 1).: A single empirical code review system—Spotify’s
code review system, for example—with no or marginal information diffusion
could not be aligned with the existing theory of code review as a
communication network in general, and the theory as it stands would be
falsified (_reductio ad absurdum_). The theory must then be revised or
reformulated more precisely (e.g., by adding limitations, constraints, or
conditions).

In the next subsection, we state our hypotheses and discuss how we
qualitatively reject our hypotheses outside classical statistical tests.

###  3.1. Hypotheses

If code review is a communication network that enables the exchange of
information (theory $T$) as identified by different exploratory studies
(Dorner2024, 8), then information substantially spread in code review

  * •

between code review participants (hypothesis $H_{1}$) and

  * •

between code software components (hypothesis $H_{2}$) and

  * •

between teams (hypothesis $H_{3}$).

We can formulate this sentence as the propositional statement

(1) $T\implies\left(H_{1}\land H_{2}\land H_{3}\right).$

That means our theory $T$ can be falsified in its universality if one of our
hypotheses cannot withstand an empirical measurement. Instead of defining
arbitrary thresholds for rejecting our hypothesis, we propose a qualitative
rejection criterion. This implies we will reject our hypotheses based on a
comprehensive discussion of the observations of information diffusion in code
review at Spotify.

As for any observational study, the measurement model, measuring system, and
actual measurement define the quality of the study. Therefore, we present our
measurement model, the measuring system, and the actual measurement following
the definitions in the _International Vocabulary of Metrology_ (isovim, 13) in
the next subsections in detail.

###  3.2. Measurement model

A _measurement model_ is the mathematical relation among all quantities known
to be involved in a measurement. In this section, we describe the three
approaches to quantifying information diffusion in code review, which are the
foundation for the qualitative rejection of our hypotheses.

We use a code review network to model information diffusion in code review. We
define a code review network—in its verbatim meaning—as a network of code
reviews whose nodes represent code reviews and whose links indicate a
reference between code reviews, explicitly and manually added by code review
participants. We argue that the explicit and manual referencing by code review
participants is a strong indicator of actual information exchange from one
code review to another. This assumption allows us to measure information
diffusion without analyzing the specific information that was exchanged and
its context.

Mathematically, we model those code review networks as a directed graph
$G=(C,R)$ where

  * •

$C$ is a set of vertices representing code reviews and

  * •

$R$ is a set of edges which are ordered pairs of vertices representing the
references between code reviews:

$R\subseteq\left\\{(a,b)\mid(a,b)\in C^{2}\text{ and }a\neq b\right\\}$

The direction of those edges represents the reference: The directed edge
$(a,b)$ represents a code review $a$ referencing code review $b$.

Figure 2 depicts such a simple and small code review network with five code
reviews linked to each other.

$c_{1}$$c_{4}$$c_{2}$$c_{0}$$c_{3}$ Figure 2. An exemplary code review network
with code reviews as vertices and references between them as edges. Code
reviews can reference one (see $c_{1}$, which references $c_{2}$), multiple
(see $c_{2}$, which references $c_{0}$ and $c_{3}$), or no other code review
(see $c_{4}$).

The relative number of linked code reviews is the first approach to
quantifying information diffusion in code review and, therefore, the first
input for our discussion on its significance.

In a second approach, we approximate information diffusion in code review by
measuring the similarity (or dissimilarity) of code review participants,
software architecture, or organizational structure in linked code reviews: The
more dissimilar the set of participants, affected code components, or involved
teams of the linked code reviews, the broader the information spread in code
review is.

Therefore, we enhance each code review with further information for each
hypothesis:

  * •

$f_{1}:C\rightarrow\\{\text{participants}\\}$ where a code review is mapped to
its participants addressing $H_{1}$

  * •

$f_{2}:C\rightarrow\\{\text{components}\\}$ where a code review is mapped to
the affected components addressing $H_{2}$

  * •

$f_{3}:C\rightarrow\\{\text{teams}\\}$ where the code review is mapped to the
owning teams of the affected component addressing $H_{3}$

Through those enhancements, we gain insights into information diffusion into
three orthogonal dimensions: A social dimension, where information diffuses
between code review participants; a software architectural dimension, where
information diffuses software components under review; and an organizational
dimension, where information diffuses between teams. Those orthogonal
dimensions allow us to investigate information diffusion from different
angles: Information may spread between components but may never leave the team
boundaries since both components are owned by the same team.

After enhancing, we apply two different similarity measures based on the type
of enhancement to make the linked code reviews comparable along the three
dimensions:

  * •

Code review participants and teams are sets. We apply the _Jaccard index_ to
quantify the similarity between two sets. The Jaccard index (or Jaccard
similarity coefficient) for two sets $A$ and $B$ is defined by

(2) $\displaystyle J(A,B)=\frac{|A\cap B|}{|A\cup B|}=\frac{|A\cap
B|}{|A|+|B|-|A\cap B|}\;.$

  * •

For the tree-like component structure, set-based operations fall short.
Instead, we use the _graph edit distance_ , which is a measure of similarity
(or dissimilarity) between two component graphs (Gao2010, 10). The graph edit
distance finds the minimal set of edit operations (insertion, deletion,
substitution), in terms of cost, needed to transform one graph into
another.333The graph edit distance between two graphs resembles the likely
better-known string edit distance between strings: With the interpretation of
strings as connected, directed acyclic graphs of maximum degree one, classical
definitions of edit distance such as Levenshtein distance or Hamming distance
may be interpreted as graph edit distances between suitably constrained
graphs.

Mathematically, we define the graph edit distance as

(3) $\displaystyle
GED(G_{1},G_{2})=\min_{(e_{1},...,e_{k})\in\mathcal{P}(G_{1},G_{2})}\sum_{i=1}^{k}c(e_{i})$

where $\mathcal{P}(G_{1},G_{2})$ denotes the set of edit paths transforming
$G_{1}$ into (a graph isomorphic to) $G_{2}$ and $c(e)\geq 0$ is the cost of
each graph edit operation $e$.

Both similarity measures are normalized, i.e., $[0,1]$. The smaller the
similarity measures, the more dissimilar the set of participants, affected
code components, or involved teams of the linked code reviews. This allows us
to approximate information diffusion in code review by measuring the
similarity (or dissimilarity) of code review participants, software
architecture, or organizational structure in linked code reviews: The smaller
the similarity measures, the more dissimilar the set of participants, affected
code components, or involved teams of the linked code reviews. The
distribution of those similarities will indicate to what extent information
spread across the boundaries mentioned before.

In Figure 3, we exemplify how we will use the similarities measures for
discussion on falsifying the theory by three possible archetypes of cumulative
distributions of all three similarity measures and their relation to the
theory test.

$0$$0.2$$0.4$$0.6$$0.8$$1$$0.2$$0.4$$0.6$$0.8$$1$Similarity (a) A large
dissimilarity between linked code reviews along all three dimensions implies
that information diffuses beyond all three boundaries; our theory is not
falsified.

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$Similarity (b) A large
dissimilarity of code review participants and a large similarity of teams
among linked code reviews imply that information diffuses within a team rather
than between teams; our theory is falsified in its universality.

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$Similarity (c) A large
similarity among all dimensions means that information diffuses to a small
extent between participants, teams, and components; our theory is falsified.

Figure 3. Three examples of cumulative distributions of the information
diffusion measured in form of similarity of linked code reviews with respect
to participants, components, and teams: Depending on the discussions of those
results, we may or may not reject our hypotheses and, thus, falsify our
theory.

Aside from the two quantitative approaches, we plan to include also a visual
approach. The ownership of code components allows us to cluster components per
owning team, providing a more intuitive, human-comprehensive perspective.
Figure 4 uses a circular graph layout of the components grouped by the owning
teams. The components are linked via the code review network $G=(C,R)$. We
hope this visualization helps identify hot and cold spots and reveals the
first patterns of information diffusion.

Figure 4. A circular layout of the components grouped by the owning teams and
linked by code reviews. This visualization may be too cluttered for massive
information diffusion.

However, depending on the extent of information between components and teams,
the visualization may highlight the hot and cold spots of information
diffusion, but it can also be visually overwhelming in case of a massive
information diffusion.

###  3.3. Measuring system

A measuring system is the set of measuring instruments and other components
assembled and adapted to give information used to generate measured values
within specified intervals for quantities of specified kinds. As common in
software engineering, our measuring system is a data extraction and analysis
pipeline.

Since our measuring system is not trivial, involves a lesser-known GitHub API
endpoint, and requires different data sources, we describe our measuring
system in this dedicated section. Figure 5 provides a high-level overview of
our measuring instrument, which we describe in detail in the following.

GitHub Enterprise instance Get all teams Get all repos Get all pulls Get files
Get all timeline events Add pull creation to events Filter events Extract
events of type referenced REST APIREST APIREST APIREST APIREST API Identify
component Backstage Identify component owner Participants grouped by code
review teamsrepositoriespullseventseventseventscomponentsfiles $G=(C,R)$
$f_{3}:C\rightarrow\text{teams}$ $f_{1}:C\rightarrow\text{participants}$
$f_{2}:C\rightarrow\text{components}$ Measurement model (Section 3.2) Data
sources Activities Data flow Model input Legend Figure 5. An overview of our
measuring instrument. The raw data is extracted from two different sources,
Spotify’s internal GitHub Enterprise (light blue) and Backstage instance
(yellow). The results feed into the measurement model.

The first data source for our measuring instrument is the GitHub Enterprise
instance and its REST or GraphQL API. For our measurement, we follow the REST
API. In GitHub, a pull request is a code review. GitHub automatically
tracks444https://docs.github.com/en/get-started/writing-on-github/working-
with-advanced-formatting/autolinked-references-and-urls#issues-and-pull-
requests when a user references an issue and pull requests in such. Since
internally, a code review is an issue in GitHub, we can tap the GitHub REST
API endpoint for timeline events of
issues555https://docs.github.com/en/enterprise-
server@3.10/rest/issues/timeline?apiVersion=2022-11-28#list-timeline-events-
for-an-issue. The timeline events contain all events triggered by activities
in a pull request or issue, including the automated links to other pull
requests or issue. GitHub’s event endpoint /events is not suitable for
extracting the event data because this API endpoint returns only a maximum of
300 events and only for the last 90
days666https://docs.github.com/en/enterprise-
server@3.10/rest/activity/events?apiVersion=2022-11-28#about-github-events.
The outcome of the crawling is a list of all events.

Tapping the timeline events API requires the related pull requests. The GitHub
search is not suitable for including or excluding pull requests since it
limits its results to 1000 results per search, which is not enough at
Spotify’s scale. Therefore, we had to collect all pull requests from all
repositories from all teams from GitHub.

We need the pull request information for two further steps:

  * •

For each pull request, we also extract all files in a pull request to map
those files to components in later steps.

  * •

Since there is pull request creation event
available777https://docs.github.com/en/enterprise-
server@3.10/rest/overview/issue-event-types?apiVersion=2022-11-28, we add
those information from the pull endpoint.

We then filter the list of events according to the sampling frame and exclude
all events from bots.

After filtering, we extract

  * •

all events of type reference888https://docs.github.com/en/enterprise-
server@3.10/rest/overview/issue-event-types?apiVersion=2022-11-28#referenced
and its payload, the referenced pull request (code review) which results in a
code review network $G=(C,R)$, the first input of our measurement model, and

  * •

all human participants grouped by each code review which results in the
mapping of code review to its participants
$f_{1}:C\rightarrow\text{participants}$, a second input for our measurement
model.

We believe that the GitHub referencing system is a reliable source. Two
studies rely on this referencing system in GitHub (Kavaler2019, 14, 23).
However, both use the so-called -mentions that reference a user but not the
references to issues or pull requests.

The second source for our measurement model is the software architecture
description tracking all components. Spotify uses a tool called
Backstage999https://backstage.io for tracking its software architecture. For
each pull request, we extracted all files and mapped the files to components.
A software component is a self-contained, reusable piece of software that
encapsulates the internal construction and exposes its functionality through a
well-defined interface so other components can use the functionality. Software
components can take many forms, including libraries, modules, classes,
functions, or even entire microservices or applications. Components are
hierarchically structured and may contain files or recursively other
components. At Spotify, the component structure maps to the virtual folder
structure of the source code. That means software components are specific
folders that contain files.

Since the component structure evolves over time, we map the files to the
component structure at the time when the code reviews are referenced.
Therefore, we use the available historical daily snapshots of the software
architecture at Spotify.

To identify the component of the files in a pull request efficiently, we
create a file graph reflecting the paths of all changed files per code review
and a time-varying component graph reflecting the component structure for each
given day. The leaves of the intersection of both graphs represent the
components for the files changed in a pull request. Figure 6 sketches the
intersection of both graphs. This mapping code reviews to components
$f_{2}:C\rightarrow\text{components}$ is the third input for our measurement
model.

$\cap$$=$ File graph Component graph Intersection graph

FileComponentFolderComponent of changed filesLegend

Figure 6. By intersecting the file graph (representing the changed files
discussed in a code review) and the component graph (representing the
component structure), we can extract the components of the changed files by
extracting the leaves of the intersection graph efficiently.

For each identified component, we also identify its owner. Component ownership
refers to the concept of assigning responsibility and accountability for a
particular software component to an individual or an organizational unit
within an organization. Spotify uses weak code-ownership (Smite2023, 20). The
mapping code review to owner $f_{3}:C\rightarrow\text{teams}$ is the fourth
input for our measurement model.

###  3.4. Measurement

The _measurement_ is the process of experimentally obtaining values that can
be reasonably attributed to a quantity together with any other available
relevant information.

For our measurement, we use Spotify’s internal GitHub Enterprise and the
Backstage instance. It comprises all Spotify-internal code reviews and
components. We will run our measurement in 2024. Our sampling frame is one
year and includes the timeframe $[\text{2019-01-01},\text{2019-12-31}]$. This
timeframe, outside of the ongoing developments at Spotify, allows us to
publish all data in an anonymized way. However, the extent of information
diffusion we will find might require us to shorten the timeframe.

##  4\. Limitations

In general, the chain of evidence of our study depends on two main factors:
(1) the measurement model, measuring system, and actual measurement, and (2)
the thoroughness of our discussion for qualitatively rejecting the hypotheses
and, thereby, falsifying the theory of code review as communication network.

Although we will not be able to provide the complete raw data and only a
prototypical extraction pipeline for Backstage, we believe that our thorough
description of our measurement model, measuring system, and the actual
measurement at Spotify provides a solid foundation for this line of research.
Our replication package will contain the necessary yet anonymized data to
reproduce and replicate our study beyond the context of Spotify. However, as
for every data-driven study, missing, incomplete, faulty, or unreliable data
may significantly affect the validity of our study. To mitigate those risks,
we conducted a pilot study in October 2023. Although we have not encountered
such threats to validity, we cannot exclude data-related limitations.
Therefore, this section will also cover the limitations that come from
excluding or missing data once our data collection is completed.

However, we believe the two most critical limitations of our study lie in the
nature of a qualitative falsification of theories. Although traditional
statistical hypothesis tests also have their limitations and, ultimately, also
represent an implicit and qualitative discussion, we believe that a discussion
remains more prone to bias, most importantly because there are no clear
criteria to reject the hypotheses upfront. Such clear rejection and
falsification criteria are not possible and meaningful upfront for this
research; all thresholds, values, or estimates would be arbitrary. However, we
believe that a comprehensive discussion makes a potential bias explicit and
allows other researchers to conclude differently. Additionally, we will
publish our measurement system and all intermediate anonymized data to enable
other researchers to replicate our work.

Second, even if our data and a thorough discussion suggest falsifying our
theory by rejecting one of the hypotheses, our modelling approach may not
capture the (relevant) information diffusion in code review. Although we have
strong indications that the explicit referencing of code reviews is an active
and explicit information diffusion triggered by human assessment, we are not
aware of empirical evidence that supports our assumption.

Although already discussed in Section 3, we emphasize again that the findings
of the extent of information diffusion will not be generalizable. We do not
believe that this is a major limitation of our research design since our
argumentation is based on contradiction (_reductio ad absurdum_).

This section will also include a detailed discussion of limitations that
originate in incomplete or missing data when they become visible after the
data collection and analysis.

###### Acknowledgements.

We thank Spotify for supporting this research and the anonymous reviewers for
their valuable and extensive feedback. This work was supported by the KKS
Foundation through the SERT Project (Research Profile Grant 2018/010) at
Blekinge Institute of Technology.

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