point. Let us illustrate this by specifying the form of these reduced
subgraphs as follows:

(5.51)

The crossed dots of order $(n)$ denote the counterterm insertion due to
shrinking the respective $n$-loop subgraph to a point. Since we assume that to
each vertex $V_{i}$ there is associated an entering momentum $p_{i}$,
shrinking a subgraph comprised of vertices $V_{i_{1}},\dots,V_{i_{k}}$, leads
to a combination of incoming momenta $p_{i_{1}}+\dots+p_{i_{k}}$ for that
counterterm vertex, as indicated in the graphs. In choosing independent
momenta, we can make use of momentum conservation. For a reduced subgraph with
$n$ vertices, it is sufficient to specify $n-1$ incident momenta to the
vertices. They uniquely characterize the momenta of a given reduced subgraph.
What is more, all momenta of the graph can be reconstructed by linear
combinations of these independent external momenta. A specific choice is given
in Eq. (5.50). Clearly, in this way all inequalities of the sector (5.20) are
implemented and the combination of all the selected independent incoming
momenta of the $\bar{H}_{i}$ span all independent incoming momenta of the full
graph and can be used as independent variables in the result. The variable
substitution leads to the following replacement of the integration measure,
analogous to Eq. (5.43),

$\displaystyle\int_{({\cal C},\sigma)}d\alpha_{1}\ldots d\alpha_{17}$
$\displaystyle=2^{L}\int_{\begin{subarray}{c}t_{6}=0\ldots\infty\\\ t_{1\ldots
5}=0\ldots
1\end{subarray}}\prod_{i=1}^{6}dt_{i}t_{i}^{(2I_{H_{i}}-1)}\int_{0}^{1}\prod_{k}d\beta_{k}\,,$
(5.52)

where $I_{H_{i}}$ is the number of internal lines in $H_{i}$.

This example provides us with sufficient information to construct the general
result for the integral representation of a general 1PI graph $G$ in a
specific sector $({\cal C},\sigma)$. As in the example, the sector defines a
chain of subgraphs $H_{i}$ (as many as there are loops; one of them is equal
to the full graph $G$). The sector also defines a particular replacement of
all $\alpha$’s in terms of $t_{i}$ and $\beta_{k}$; for each subgraph $H_{i}$
it is also useful to define the variable $\xi_{i}$ for the product of all the
“other $t_{i}$”. All lines of $G$ are partitioned into lines of the reduced
graphs $\bar{H}_{i}$, and for each $\bar{H}_{i}$ one can choose a set of
independent incoming momenta $\underline{q}_{H_{i}}$, which in total span all
incoming momenta of the full graph. Since each line carries one mass variable
and one $u$-variable, we can also partition these variables into sets of
masses $m_{H_{i}}$ and sets of $u$’s, $u_{H_{i}}$, corresponding to the
respective $\bar{H}_{i}$.

With these variables we can rescale physical quantities, generalizing Eq.
(5.46), as

$\displaystyle\tilde{\underline{q}}_{H_{i}}$
$\displaystyle=t_{i}\xi_{i}\underline{q}_{H_{i}}\,,$ (5.53a)
$\displaystyle\tilde{m}_{H_{i}}$ $\displaystyle=t_{i}\xi_{i}m_{H_{i}}\,,$
(5.53b) $\displaystyle\tilde{u}_{H_{i}}$
$\displaystyle=(t_{i}\xi_{i})^{-1}u_{H_{i}}\,.$ (5.53c)

We allow the integral to contain a numerator expressed as a derivative with
respect to $u$-variables as in Eqs. (3.37,3.38a), but we assume that the
derivative operator $Z$ in the numerator is a product of $Z_{H_{i}}$, where
each $Z_{H_{i}}$ only depends on variables specific to $\bar{H}_{i}$. This is
always the case in actual Feynman diagrams. For simplicity we follow Ref. [4]
and assume that all $Z_{H_{i}}$ are homogeneous polynomials in the variables
$\partial/\partial u_{H_{i}}$ and $m_{H_{i}}$ of some degree
$r_{\bar{H}_{i}}$. Then we can write

$\displaystyle\tilde{Z}_{H_{i}}$
$\displaystyle=(t_{i}\xi_{i})^{r_{\bar{H}_{i}}}{Z}_{H_{i}}$ (5.54)

where $\tilde{Z}_{H_{i}}$ is the same homogeneous polynomial expressed with
$\partial/\partial\tilde{u}_{H_{i}}$ and $\tilde{m}_{H_{i}}$. Writing
$D=4-2\epsilon$ we can finally define a power-counting degree of each reduced
subgraph $\bar{H}_{i}$ and the complete (sub)graphs $H_{i}$ as

$\displaystyle\omega_{\bar{H}_{i}}$
$\displaystyle=4L_{\bar{H}_{i}}-2I_{\bar{H}_{i}}+r_{\bar{H}_{i}}\,,$ (5.55a)
$\displaystyle\omega_{H_{i}}$
$\displaystyle=\sum_{\begin{subarray}{c}H^{\prime}\subseteq H_{i}\\\
H^{\prime}\in{\cal C}\end{subarray}}\omega_{\bar{H}^{\prime}}\,.$ (5.55b)

This clearly corresponds to the superficial power-counting degree of the
original momentum integral.

With these building blocks we can formulate the general result for the
integral specified in Eqs. (3.37,3.38a). Decomposing the integral in sectors
as in Eq. (5.22),

$\displaystyle{\cal T}_{G}$ $\displaystyle=\sum_{({\cal C},\sigma)}{\cal
T}_{G,({\cal C},\sigma)}\,,$ (5.56)

and setting again $D=4-2\epsilon$, the result for each sector can be written
as

$\displaystyle{\cal T}_{G,({\cal C},\sigma)}$
$\displaystyle=c_{D}^{L}2^{L}\int_{\begin{subarray}{c}t_{L}=0\ldots\infty\\\
t_{1\ldots L-1}=0\ldots
1\end{subarray}}\prod_{i=1}^{L}\frac{dt_{i}}{t_{i}}(t_{i}\xi_{i})^{-\omega_{\bar{H}_{i}}+2\epsilon}\tilde{Z}_{H_{i}}$
$\displaystyle\qquad\times\int_{0}^{1}\prod_{k}d\beta_{k}\tilde{d}_{G}^{-D/2}e^{iW_{G}}\Bigg{|}_{u=0}\,.$
(5.57)

The properties of the appearing objects are:

  * •

All variables $t_{i}$, $\xi_{i}$, $\beta_{k}$ and the rescaled physical
variables $\tilde{\underline{q}}_{H_{i}}$, $\tilde{m}_{H_{i}}$,
$\tilde{u}_{H_{i}}$, and the power-counting degrees $\omega_{\bar{H}_{i}}$ are
defined above.

  * •

The explicit powers of $t_{i}$ correspond to the original superficial power-
counting degrees of the momentum integrals over the subdiagrams $H_{i}$. For
each $t_{i}$ integral, a factor $(t_{i}\xi_{i})^{2\epsilon}$ was split off
which may be viewed as the essence of the $D$-dimensional integration measure.

  * •

The remaining integrand $\tilde{d}_{G}^{-D/2}e^{iW_{G}}$ has no explicit
dependence on $t_{L}$ at all. It depends on $t_{L}$ only via the rescaled
physical variables. The other $t_{i}$ with $i=1\ldots L-1$ typically appear
explicitly, however.

  * •

The function $\tilde{d}_{G}$ is a rescaled Symanzik polynomial which satisfies
$\tilde{d}_{G}\geq 1$ in the integration region.

  * •

For $\varepsilon>0$ in the $+i\varepsilon$ prescription, the function
$e^{iW_{G}}$ is exponentially decreasing for large $t_{L}$.

  * •

The product $\tilde{d}_{G}^{-D/2}e^{iW_{G}}$, therefore, is analytic in
$\epsilon$ and $C^{\infty}$ in $t_{i}$, $\beta_{k}$ and the rescaled physical
variables $\tilde{\underline{q}}_{H_{i}}$, $\tilde{m}_{H_{i}}$,
$\tilde{u}_{H_{i}}$.

This statement is the starting point for the inductive proof of
renormalization in DReg given in Ref. [4], and it is a direct consequence of
Lemma 4 of that reference.

####  5.2.6 Ingredient 4: Integrand Relation between Graphs and Subgraphs

An important step in the proof is the application of subtraction operators
$T_{H}$ to a graph $G$. In order to analyze this operation, relationships
between the original graph $G$, the reduced graph $G/_{H}$ and the subgraph
$H$ are needed. These relationships are again essentially independent of
$D$-dimensional treatments. They rely on detailed analysis of the graphs
themselves, and the relationships between graphs and the
$\alpha$-parametrizations.

The required theory involves incidence matrices and graph theoretical
representations of the Symanzik polynomial ${\cal U}$, or $\tilde{d}_{G}$, and
the exponent $W_{G}$. Although the theory is very elegant and not too
difficult we do not develop it here. Hence we only list several important
statements without proof. For the proofs we refer to Ref. [4] and references
therein. Further discussions were given e.g. in Refs. [69, 51, 124]

Consider the Symanzik polynomial ${\cal U}_{G}$ for a graph $G$, and let $H$
be a subgraph of $G$. ${\cal U}$ is a homogeneous polynomial in all $\alpha$’s
of degree $L$. Consider the case where all $\alpha$’s corresponding to the
subgraph $H$ are rescaled by a factor $\rho$, while all other $\alpha$’s
remain fixed. Then, for small $\rho$ we have

$\displaystyle{\cal U}_{G}(\text{$\alpha$'s in $H$ rescaled by $\rho$})$
$\displaystyle=\underbrace{{\cal
U}_{G/_{H}}}_{\text{$\rho$-independent}}\quad\underbrace{{\cal
U}_{H}}_{\propto\rho^{L_{H}}}+{\cal O}(\rho^{L_{H}+1})\,,$ (5.58)

i.e. at the lowest non-vanishing order the Symanzik polynomial factorizes into
the two individual Symanzik polynomials for the reduced graph and the
subgraph. If $G$ and $H$ are part of an integration sector as defined above,
then variables $t_{G}$, $t_{H}$ (and possibly further $t_{i}$) and $\beta_{k}$
exist and rescaled Symanzik polynomials $\tilde{d}$ can be defined for each of
these graphs. In this case, $\tilde{d}_{G}$ for the full graph cannot depend
on $t_{G}$ but it can depend on $t_{H}$, while $\tilde{d}_{G/_{H}}$ and
$\tilde{d}_{H}$ can neither depend on $t_{G}$ nor on $t_{H}$. Their
relationship is the factorization

$\displaystyle\tilde{d}_{G}|_{t_{H}=0}$
$\displaystyle=\tilde{d}_{G/_{H}}\tilde{d}_{H}\,.$ (5.59)

A similar relationship can be established for the exponent $W_{G}$ appearing
in the general result of the integral (5.57). Defining $W_{H}$ and
$W_{G/_{H}}$ using the same variable transformations for the sector $({\cal
C},\sigma)$, suitably adapted to the subgraph and reduced graph, the
relationship is

$\displaystyle W_{G}|_{t_{H}=0}$
$\displaystyle=W_{G/_{H}}+W_{H}|_{t_{H}=0}\,,$ (5.60)

if all these quantities are expressed in terms of rescaled variables
$\tilde{\underline{q}}$, $\tilde{u}$, $\tilde{m}$. This property can be
established in an elementary way once the exponents are constructed via
incidence matrices.

For the same conditions, a further, more intricate property can also be
established and is important. It is the following property involving
derivatives,

$\displaystyle\frac{d}{dt_{H}}\tilde{d}_{G}^{-D/2}e^{iW_{G}}\Bigg{|}_{t_{H}=0}$
$\displaystyle=\xi_{H}U_{H}\left[\frac{d}{dt_{H}}\tilde{d}_{H}^{-D/2}e^{iW_{H}}\right]_{t_{H}=0}\cdot\tilde{d}_{G/_{H}}^{-D/2}e^{iW_{G/_{H}}}\,.$
(5.61)

Here $U_{H}[X]$ denotes an insertion operator which effectively inserts its
argument $X$ as a vertex into a bigger graph. To achieve this insertion, the
external momenta of the argument $X$ must become internal momenta of the
bigger graph, in this case of $G/_{H}$. Technically, $U_{H}$ acts by shifting
in its argument the momentum variables $\tilde{\underline{q}}_{H}$ by terms
involving derivatives with respect to $u$-variables for the bigger graph
$G/_{H}$.

This is a statement of pivotal importance for the full proof of the theorem
stated in Sec. 5.2.1 since it allows relating divergences of a full graph to
divergences of counterterm graphs and thus allows making manifest the
cancellation of subdivergences. It is essentially the content of Lemma 5 of
Ref. [4].

####  5.2.7 Ingredient 5: Simple Integrals and Non-Analytic Functions of $D-4$

Now we discuss several simple integrals and special functions that arise in
DReg due to the $D$-dimensionality of spacetime. They encapsulate how the
regularization acts, how divergences arise as $1/(D-4)$ poles and how
divergences cancel by adding suitable counterterms. We set again
$D=4-2\epsilon$.

First we discuss a simple type of integral, defined as

$\displaystyle f(z)$ $\displaystyle=\int_{0}^{\infty}dtt^{z-1}g(t)\,,$ (5.62)

where $z$ is a complex variable and $g(t)$ is a $C^{\infty}$ function which
either decreases exponentially for $t\to\infty$ or which involves the step
function $\theta(1-t)$ cutting off the integral at $t=1$. This simple integral
appears in the general result (5.57) but also in the 1-loop example (5.48). In
all these cases the $t$-integration involves one factor which is of the form
$t^{n-1+2\epsilon}$, where $n$ is an integer. This corresponds to the above
form for $z=n+2\epsilon$. This factor is non-analytic in $t$ around $t=0$. The
remaining $t$-dependences in Eqs. (5.57,5.48) are complicated but are
$C^{\infty}$ functions in $t$ which indeed fulfil the requirements on $g(t)$
listed above. In case of the $t_{L}$ integration, the remaining integrand
exponentially decreases, in case of all other $t_{i}$ integrations, the
integration stops at $t_{i}=1$.

The above function $f(z)$ is a generalization of the $\Gamma$-function, where
$g(t)=e^{-t}$. The $\Gamma$-function is known to have simple poles at $z=0$,
$z=-1$, $z=-2$, …. It is easy to see that the same is true for the more
general $f(z)$. Clearly, when for $\text{Re}(z)>0$ the integral defining
$f(z)$ converges and defines an analytic function. To study negative
$\text{Re}(t)$ we can add to and subtract from $g(t)$ a Taylor polynomial
$\sum g^{(k)}(0)t^{k}/k!$, where $g^{(k)}$ denotes the $k$-th derivative.
Integrating this polynomial from $0$ to $1$ we obtain

$\displaystyle f(z)$
$\displaystyle=\int_{0}^{\infty}dtt^{z-1}\left[g(t)-\theta(1-t)\sum_{k=0}^{n}\frac{g^{(k)}(0)}{k!}t^{k}\right]+\sum_{k=0}^{n}\frac{g^{(k)}(0)}{k!}\frac{1}{z+k}\,.$
(5.63)

For any non-negative integer $n$ and for $\text{Re}(z)>0$ the value and
convergence properties of the integral are not changed. However, the square
bracket behaves like $t^{n+1}$ for small $t$, hence the integral now converges
even for negative $z$, as long as $\text{Re}(z)>-n-1$. Hence this formula
represents an analytic continuation of $f(z)$ onto the entire complex $z$
plane. It makes also manifest that this analytically continued $f(z)$ has
single poles at $z=-0$, $z=-1$, $z=-2$, ….

We can rewrite the result in the form of an integration rule for the typical
$t$-integrals appearing in DReg by replacing $z=-n+2\epsilon$ with integer
non-negative $n$ and $\epsilon\approx 0$. We then have the rule

$\displaystyle\int_{0}^{\infty}dtt^{-n-1+2\epsilon}g(t)$
$\displaystyle=\frac{1}{n!}\left(\frac{d}{dt}\right)^{n}g(t)\Bigg{|}_{t=0}\frac{1}{2\epsilon}+\text{regular
expression}\,,$ (5.64)

where the form of the regular expression can be read off from Eq. (5.63). The
$t$-integrals in the general formula (5.57) are to be analytically continued
in this way. Hence this rule immediately shows that any $t$-integration can
only lead to single $1/\epsilon$-poles and not to more complicated divergences
as $\epsilon\to 0$.

Next, we consider two special simple classes of non-analytic functions of $t$.
They are defined as the two kinds of sets (for integer $K,L$)

$\displaystyle K$ $\displaystyle<L:$ $\displaystyle J_{K}^{L}$
$\displaystyle=\left\\{f(t,\epsilon)=\frac{c_{1}t^{2\epsilon}+\ldots+c_{L}t^{2L\epsilon}}{\epsilon^{K}}=\text{finite
for }\epsilon\to 0\right\\}\,,$ (5.65a) $\displaystyle K$ $\displaystyle\leq
L:$ $\displaystyle\tilde{J}_{K}^{L}$
$\displaystyle=\left\\{f(t,\epsilon)=\frac{c_{0}+c_{1}t^{2\epsilon}+\ldots+c_{L}t^{2L\epsilon}}{\epsilon^{K}}=\text{finite
for }\epsilon\to 0\right\\}\,.$ (5.65b)

In the definitions of the sets, the lower index $K$ refers to the
$\epsilon$-power in the denominator, and the upper index $L$ can be thought of
as the loop number at which the functions become of interest. The coefficients
$c_{i}$ are arbitrary except for the constraint that the defined functions are
finite for $\epsilon\to 0$.

Let us illustrate how such functions can appear by considering a 2-loop
diagram $G$ with a 1-loop subdiagram $H$. We imagine a calculation not only of
the diagrams themselves but of the entire renormalization procedure, taking
into account suitable counterterm diagrams cancelling subdivergences. In the
imagined calculations we use the general formula (5.57). If the 1-loop diagram
$H$ is computed in isolation, it involves one $t_{1}$-integral whose essential
non-analytic part is simply

$\displaystyle t_{1}^{2\epsilon}\in J_{0}^{1}\,,$ (5.66)

which is an element of the set $J_{0}^{1}$ and which may be attributed to the
$D$-dimensional measure. The result of the $t_{1}$-integration via the rule
(5.64) then leads particularly to a $1/(2\epsilon)$ pole, and a counterterm
for diagram $H$ can be defined that cancels this divergence. In the 2-loop
calculation of $G$ the 1-loop diagram $H$ appears as a subdiagram with
corresponding $t_{1}$ integration. Here the $t_{1}$ variable is accompanied by
$\xi_{1}$, which is here simply $\xi_{1}=t_{2}$. After the $t_{1}$
integration, the non-analytic factor $\xi_{1}^{2\epsilon}$ remains and
combines with the $1/(2\epsilon)$ pole. In the corresponding counterterm
diagram, where the subdiagram $H$ is replaced by the counterterm cancelling
its $1/(2\epsilon)$ pole there is no $t_{1}$ integration and no appearance of
the variable $\xi_{1}$. Therefore, after the $t_{1}$ integration and after
combining with the counterterm diagram cancelling the subdivergence, a
combined function

$\displaystyle\frac{t_{2}^{2\epsilon}-1}{2\epsilon}\in\tilde{J}_{1}^{1}$
(5.67)

appears. The finiteness of functions in the set $\tilde{J}_{1}^{1}$ reflects
the successful cancellation of the subdivergence. Proceeding with the
computation of the 2-loop diagram $G$, this function is combined with the
measure factor, such that the interesting non-analytic part of the
$t_{2}$-integrand is

$\displaystyle t_{2}^{2\epsilon}\frac{t_{2}^{2\epsilon}-1}{2\epsilon}\in
J_{1}^{2}\,.$ (5.68)

This example illustrates the general idea: Functions in $J_{K}^{L}$ are the
functions that actually appear as the non-analytic factors in the $t_{L}$
integrations at the $L$-loop level during the renormalization procedure. After
carrying out a $t_{L}$ integral and after combining with the suitable
counterterm contribution, a function in the set $\tilde{J}_{K+1}^{L}$ appears.
At the next loop level the integrand needs to be prepared by suitable
rearrangements and combined with the measure factor $t_{L+1}^{2\epsilon}$ to
produce a function of the set $J_{K+1}^{L+1}$, and so on.

For this reason it is helpful to study the properties of functions in these
sets on their own, before tackling the actual loop integrations. Some
particularly useful properties are:

$(i)$

any function $f\in J_{K}^{L}$ has the limit $f(t,0)=\text{const}\times(\ln
t)^{K}$.

$(ii)$

for a function $f\in J_{K}^{L}$, the integral
$\int_{1}^{t}\frac{dt^{\prime}}{t^{\prime}}f(t^{\prime},\epsilon)$ produces an
element of the next set $\tilde{J}_{K+1}^{L}$.

$(iii)$

the converse is also true, i.e. every element of $\tilde{J}_{K+1}^{L}$ can be
written in terms of such an integral.

$(iv)$

a function $f\in J_{K}^{L}$ where the first argument is a product can be
factorized as $f(\xi
t,\epsilon)=\sum_{j}f_{1j}(\xi,\epsilon)f_{2j}(t,\epsilon)$ where all
functions on the right-hand side are elements of $f_{nj}\in J_{K_{nj}}^{L}$
where $K_{1j}+K_{2j}=K$. This property is obviously important to prepare
higher-loop integrands such that $t$ integrals act on isolated functions
depending only on $t$, not on $\xi$.

$(v)$

there is a simple product rule $f_{K_{1}}^{L_{1}}f_{K_{2}}^{L_{2}}\in
J_{K_{1}+K_{2}}^{L_{1}+L_{2}}$ for functions $f_{K_{i}}^{L_{i}}\in
J_{K_{i}}^{L_{i}}$. This property is also important on the multiloop level in
case a multiloop diagram contains two disjoint divergent subdiagrams.

The properties can all be proved using elementary integration tricks and
l’Hopital’s rule for limits. Such properties of these functions are the
content of Lemma 2 of Ref. [4].

####  5.2.8 Sketch of Proof by Induction

All explained ingredients are important in the full proof of the central
Theorem 1 in Ref. [4] and stated in Sec. 5.2.1. Here we give a sketch of this
proof. The proof applies the $\alpha$ parametrization of integrals decomposed
into sectors as in Eq. (5.22). The renormalization procedure is then expressed
in terms of the forest formula (5.41). This formula provides the basis for an
inductive proof, where a graph $G$ and a sector are fixed and then all factors
$(1-T_{H_{i}})$ in the forest formula are successively applied in the correct
ordering. The base case of the induction is provided by the general formula
(5.57). The induction step needs to carry out the actual integration over one
$t$ variable and some $\beta$ variables corresponding to the next
$(1-T_{H_{i}})$ factor. The step uses the properties of the special functions
of $\epsilon$ defined in Sec. 5.2.7, and the relationships between the graph,
subgraph and reduced graph described in Sec. 5.2.6.

Obtaining the precise form of the induction hypothesis is highly nontrivial,
but it can be motivated using all the developed insight. It can be formulated
as follows. Consider a 1PI graph $G$ and a sector $({\cal C},\sigma)$. All
following quantities are specific to this sector but for brevity we will omit
all indices denoting this dependence. The graph has $L_{G}$ loops and the
sector contains $L_{G}$ subgraphs $H_{1},\ldots,H_{L_{G}}$. Without loss of
generality we assume the labelling such that the subgraphs are already ordered
according to their allowed appearance in the forest formula (5.41), such that
if $H_{j}\supseteq H_{i}$ then also $j\geq i$ (the ordering is not unique).
Then after evaluating $L\leq L_{G}$ factors in the forest formula we obtain
the expression (suppressing the dependence on the sector $({\cal C},\sigma)$)

$\displaystyle{\cal
R}_{X}(G)\equiv(1-T_{H_{L}})\cdot\ldots\cdot(1-T_{H_{2}})\cdot(1-T_{H_{1}})\cdot
G\,.$ (5.69)

This represents a partially renormalized graph where $L$ loops and $L$
subgraphs have already been treated in previous induction steps. Sec. 5.2.4
gave arguments that this expression should have simple divergence properties
when acted upon by further $T_{H_{i}}$ operators. Despite this, the partially
renormalized expression on its own clearly can have very complicated
analytical structure and can still have non-polynomial divergences which the
proof needs to deal with. The label $X$ denotes the set of all subgraphs which
have already been treated, and we also define $X_{0}$ as the subset of $X$
which contains only maximal subgraphs, i.e.

$\displaystyle X$ $\displaystyle=\\{H_{1},\ldots,H_{L}\\}\,,$ $\displaystyle
X_{0}$ $\displaystyle=\\{M_{1},\ldots,M_{S},H_{L}\\}\,,$ (5.70)

where it is used that $H_{L}$ itself is necessarily a maximal subgraph in $X$
and where names have been given to all other elements of $X_{0}$.

The induction hypothesis states that after evaluating all $t_{i}$ and
$\beta_{k}$ integrals corresponding to lines in the already treated graphs in
$X$, we obtain

$\displaystyle{\cal R}_{X}(G)={}$ sum of terms like $\displaystyle\int_{\geq
L+1}\prod_{M\in
X_{0}}\xi_{M}^{-\omega_{M}}\tilde{f}_{M}(\xi_{M},\epsilon)g_{G,X}\Bigg{|}_{\tilde{u}=0}\,,$
(5.71) where the integration factors for the remaining integrals are
abbreviated as $\displaystyle\int_{\geq L+1}={}$ $\displaystyle
c_{D}^{L_{G}-L}\int\prod_{i=L+1}^{L_{G}}\frac{dt_{i}}{t_{i}}(t_{i}\xi_{i})^{-\omega_{\bar{H}_{i}}+2\epsilon}\int\prod_{k\in
G/_{X_{0}}}d\beta_{k}\tilde{Z}_{H_{i}}\,.$ (5.72)

Here the integration boundaries of the $t_{i}$ and $\beta_{k}$ integrals are
as in Eq. (5.57), and the notation $k\in G/_{X_{0}}$ corresponds to all
indices $k$ corresponding to any line outside the already treated graphs in
the set $X$. In the product over the maximal subgraphs $M$ (which includes the
case $M=H_{L}$), each $M$ is equal to one particular $H_{j(M)}$ and for
simplicity we identify the indices $\xi_{M}\equiv\xi_{{j(M)}}$.

We provide the following comments on the induction hypothesis:

  * •

The “sum of terms like” refers to the expression in the integrand which really
is of the form $\sum_{a}\prod_{M}\tilde{f}_{M,a}g_{G,X,a}$. Since the proof
can be carried out for each such term we drop the index $a$ and this
summation.

  * •

The integration variables $t_{i}$ and $\beta_{k}$ and the $\tilde{u}_{k}$
variable for the already treated graphs do not exist anymore since they have
been integrated over/set to zero. Hence the only appearing $t_{i}$,
$\beta_{k}$ and $\tilde{u}_{k}$ are the ones for $i=L+1,\ldots,L_{G}$ and for
$k\in G/_{X_{0}}$.

  * •

The sets of physical variables $\tilde{\underline{q}}_{H_{i}}$,
$\tilde{m}_{H_{i}}$ and the remaining $\tilde{u}_{H_{i}}$ (for $H_{i}\notin
X$) are rescaled only by the remaining $t_{i}$’s. I.e. Eq. (5.53) applies in a
modified form where on the right-hand side $t_{i}=1\,\forall i\leq L$ and
where the $\tilde{u}_{H_{i}}$ for $i\leq L$ do not exist.

  * •

The particularly nontrivial and interesting part of the statement is the
integrand in Eq. (5.71). It displays the analytic structure of the partially
renormalized graph and the result of all the evaluated $t_{i}$ and $\beta_{k}$
integrals. The result is a product of functions $\tilde{f}_{M}$, which are
non-analytic in the remaining $t_{i}$, and the function $g_{G,X}$.

  * •

Each function $\tilde{f}_{M}$ is an element of a set $\tilde{J}_{K}^{L}$ with
$K\leq L$. These functions are thus non-analytic in the remaining $t_{i}$ but
have a finite limit for $\epsilon\to 0$, reflecting the successful subtraction
of subdivergences. The functional form of each $\tilde{f}_{M}$ is further
specific to the chain of subgraphs $X_{M}=\\{H^{\prime}\subseteq
M,H^{\prime}\in{\cal C}\\}$, and does not depend on any details of graphs or
parts of graphs outside $M$. Only the argument $\xi_{M}$ has a dependence on
$t_{i}$ variables corresponding to bigger graphs.

  * •

The function $g_{G,X}$ carries the complicated dependence on all physical
variables and all other $t_{i}$ and $\beta_{k}$ variables. $g_{G,X}$ is
$C^{\infty}$ in all these remaining integration variables and all the physical
variables $\tilde{\underline{q}}_{H_{i}}$, $\tilde{m}_{H_{i}}$ and
$\tilde{u}_{H_{i}}$ rescaled as defined above. It is analytic in $\epsilon$,
again reflecting the cancellation of subdivergences, and it has no explicit
dependence on $t_{L_{G}}$ corresponding to the full graph $G$ (except for the
product $t_{L_{G}}^{2}\varepsilon$, similar to Eq. (5.47)). Its functional
form is specific to the full graph $G$ and the treated graphs $H_{i}\in X$.

The induction base case is the one where $L=0$ and no subgraph has been
treated yet. In this case the sets $X$ and $X_{0}$ are empty and ${\cal
R}_{X}(G)$ simply refers to the unrenormalized result ${\cal T}_{G}$. The form
of the unrenormalized result is given in Eq. (5.57) and it directly confirms
the induction hypothesis (5.71) with
$g_{G,\emptyset}=2^{L_{G}}\tilde{d}^{-D/2}e^{iW_{G}}$.

For a sketch of the induction step we assume $L\geq 1$ and assume the partial
renormalization was carried out up to loop number $L-1$ and that the induction
hypothesis holds at loop number $L-1$. It is then useful to introduce a
notation for the previously treated subgraphs and previously treated maximal
subgraphs. We write

$\displaystyle X^{\prime}$ $\displaystyle=\\{H_{1},\ldots,H_{L-1}\\}$
$\displaystyle X^{\prime}_{0}$
$\displaystyle=\\{m_{1},\ldots,m_{s}\\}\cup\\{M_{1},\ldots,M_{S}\\}\,,$ (5.73)

and we keep the definitions of Eq. (5.70) such that
$X=X^{\prime}\cup\\{H_{L}\\}$ and such that the subgraphs $m_{i}$ are the
maximal subgraphs of $H_{L}$. The remaining subgraphs are $H_{L}$ as well as
$H_{i}$ with $i\geq L+1$, the lines and $\beta_{k}$ are the ones with $k\in
G/_{X^{\prime}_{0}}$ or equivalently the ones with $k\in G/_{X_{0}}$ or with
$k\in\bar{H}_{L}$. The induction hypothesis for loop number $L-1$ can
therefore be cast into the form

$\displaystyle{\cal R}_{X^{\prime}}(G)$ $\displaystyle=\text{sum of terms like
}$ $\displaystyle\int_{\geq L+1}\prod_{M\in
X_{0}\setminus\\{H_{L}\\}}\xi_{M}^{-\omega_{M}}\tilde{f}_{M}(\xi_{M},\epsilon)$
$\displaystyle\times
c_{D}\int\frac{dt_{L}}{t_{L}}(t_{L}\xi_{L})^{-\omega_{\bar{H}_{L}}+2\epsilon}\tilde{Z}_{H_{L}}\int\prod_{k\in\bar{H}_{L}}d\beta_{k}$
$\displaystyle\times\prod_{m_{i}}\xi_{m_{i}}^{-\omega_{m_{i}}}\tilde{f}_{m_{i}}(\xi_{m_{i}},\epsilon)g_{G,X^{\prime}}\Bigg{|}_{\tilde{u}=0}\,.$
(5.74)

In this way of writing the role of the graph $H_{L}$ which is to be treated
next is exhibited, while the factors in the first line contain the same
integration factors and almost the same $\tilde{f}_{M}$ factors as Eq. (5.71).
The physical variables appearing here inside the $\tilde{Z}_{H_{i}}$ and
$g_{G,X^{\prime}}$ are rescaled with all $t_{i}$ for $i\leq L$, and all
comments made for the induction hypothesis apply with suitable modifications.

In the induction step we need to assume the validity of Eq. (5.74) and carry
out the next step, construct ${\cal R}_{X}(G)$ and prove that it takes the
form (5.71) with all listed properties. The construction involves the
evaluation of all integrals in the last two lines of Eq. (5.74). It also
involves the application of the next subtraction operator $(1-T_{H_{L}})$,
which also only affects the last two lines of Eq. (5.74) in particular because
the integration factors $\int_{L+1}$ stay unchanged if the subgraph $H_{L}$ is
replaced by its counterterm.

We begin with several immediate simplifications of the factors in the last two
lines of Eq. (5.74). First we observe that all the $\xi_{m_{i}}$ in the last
line are all equal to each other, and they are equal to
$\xi_{m_{i}}=t_{L}\xi_{L}$. The reason is that the $\xi_{m_{i}}$ are the
products of $t_{j}$ for all subgraphs in ${\cal C}$ which contain $m_{i}$ and
that the $m_{i}$ are maximal subgraphs of $H_{L}$. One consequence is that the
$\omega_{\bar{H}_{L}}$\- and $\omega_{m_{i}}$-dependent terms combine simply
to $(t_{L}\xi_{L})^{\omega_{H_{L}}}$. A less trivial consequence is that all
non-analytic functions and the measure factor for $t_{L}$ can be combined as

$\displaystyle f_{H_{L}}(t_{L}\xi_{L},\epsilon)$
$\displaystyle=(t_{L}\xi_{L})^{2\epsilon}\prod_{m_{i}}\tilde{f}_{m_{i}}(t_{L}\xi_{L},\epsilon)\,,$
(5.75)

which is an element of the set $J_{K}^{L}$ for some $K<L$ thanks to the
properties of the functions discussed in Sec. 5.2.7. Second, after the
$\beta_{k}$ integrations and after applying the derivative operator
$\tilde{Z}_{H_{L}}$ and setting $\tilde{u}_{H_{L}}=0$, we obtain

$\displaystyle\bar{g}_{G,X^{\prime}}=\tilde{Z}_{H_{L}}\int\prod_{k\in\bar{H}_{L}}d\beta_{k}g_{G,X^{\prime}}\Bigg{|}_{\tilde{u}_{H_{L}}=0}\,.$
(5.76)

This function is still $C^{\infty}$ in the remaining variables and analytic in
$\epsilon$. Hence the last two lines of Eq. (5.74) can be written as

$\displaystyle
c_{D}\int\frac{dt_{L}}{t_{L}}(t_{L}\xi_{L})^{-\omega_{H_{L}}}f_{H_{L}}(t_{L}\xi_{L},\epsilon)\bar{g}_{G,X^{\prime}}\,.$
(5.77)

The more difficult part of the induction step is the evaluation of the $t_{L}$
integral and the application of the $(1-T_{H_{L}})$ subtraction operator. Two
cases need to be distinguished. The first case is when the next step is the
final step of renormalization, i.e. when $L=L_{G}$ and $H_{L}=G$. The second
case is when $L<L_{G}$ and $H_{L}$ is still a proper subgraph of the full
graph $G$.

To sketch the first case with $L=L_{G}$ and $H_{L}=G$ we note that in this
case the second line of Eq. (5.74) is just the factor 1 since there are no
remaining integrations and there are no other maximal subgraphs $M$. Likewise,
the remaining $\xi_{L}=1$, and from the induction hypothesis we know that the
variable $t_{L}=t_{L_{G}}$ does not appear explicitly in
$\bar{g}_{G,X^{\prime}}$ — this variable only enters via rescaled physical
variables $\tilde{q}_{H_{i}}$ and $\tilde{m}_{H_{i}}$ i.e. via products of
$t_{L}$ and physical momenta and masses. Plugging in the general form of the
function $f_{H_{L}}$ yields a sum of terms like

$\displaystyle\sum_{n}\frac{c_{n}t_{L}^{2n\epsilon}}{\epsilon^{K}}\bar{g}_{G,X^{\prime}}\,,$
(5.78)

which need to be integrated over $t_{L}$ with the measure $\int
dt_{L}t_{L}^{-\omega_{H_{L}}-1}$. This integral is performed via the general
rule (5.64). This rule leads to a regular expression and a singular term. The
regular expression can be shown to be analytic in $\epsilon$ and $C^{\infty}$
in all other variables. The singular term contains poles in $\epsilon$ and
takes the form

$\displaystyle\sum_{n}\frac{c_{n}}{2n\epsilon^{K+1}}\frac{1}{\omega_{H_{L}}!}\left[\left(\frac{d}{dt_{L}}\right)^{\omega_{H_{L}}}\bar{g}_{G,X^{\prime}}\right]_{t_{L}=0}\,.$
(5.79)

This singular term can be shown to have all desirable properties. The poles in
$\epsilon$ are at most of degree $1/\epsilon^{L_{G}}$. The coefficients are
polynomials in the physical variables, masses and momenta, of degree
$\omega_{H_{L}}$.292929 Here, and in Ref. [4], the factor of the dimensional
regularization scale $\mu^{2\epsilon}$ is omitted from the definition of
renormalized amplitudes. If this factor is included it is also possible to
prove that the divergent polynomial is independent of $\mu$. It is therefore
possible to define the subtraction operator $(1-T_{G})$ for this sector such
that it subtracts this polynomial divergence; the resulting finite remainder
satisfies all properties listed after the induction hypothesis (5.71). It is
further possible to define the full divergent part of the full diagram,
$T\overline{\cal R}_{G}$, as the sum of all these singular terms arising in
this way in all sectors. This object has all properties required for a
possible contribution to a counterterm Lagrangian: in position space it is
local, it has the correct power-counting degree, and its value depends only on
the graph $G$ and not on its embedding into bigger graphs.

Finally we also sketch the remaining induction step for the case $L<L_{G}$ and
$H_{L}\neq G$. Here, evaluating the $t_{L}$ integral and applying the
subtraction operator $(1-T_{H_{L}})$ to Eq. (5.77) leads to three terms: the
regular expression from the $t_{L}$ integration, the singular expression from
the $t_{L}$ integration and the counterterm contribution from $T_{H_{L}}$,
where $T_{H_{L}}$ is defined via the full renormalization of the graph $H_{L}$
in isolation. All terms need to be rearranged by using properties of the $f$
functions discussed in Sec. 5.2.7, in particular of the factorization property
of these functions. Furthermore, the singular expression of the $t_{L}$
integration has to be rearranged by using properties such as (5.61) for the
relationships between graphs, subgraphs and reduced graphs. In these ways it
is possible to show that the combination of all terms acquires the form of the
induction hypothesis (5.71) and that all announced properties are fulfilled.

In this way all properties announced in Sec. 5.2.1 are established, except for
the equivalence to BPHZ. Illustrating this point requires comparing the
structure of appearing integrals in the DReg and the BPHZ approaches. For this
we refer to the original literature [22, 4].

##  6 Renormalization and Symmetry

In the preceding section we have seen how the renormalization programme allows
to subtract the divergences from Feynman diagrams. Importantly, the
subtraction terms are polynomials in momenta constrained by power counting,
and the subtraction is equivalent to adding certain counterterms to the
Lagrangian. By choosing a certain renormalization scheme, the remaining
ambiguities of finite counterterms can be fixed and the Lagrangian
supplemented by those counterterms defines a finite $4$-dimensional theory.

In this section we consider the problem of renormalization in the presence of
symmetries, specifically gauge invariance. On the one hand, symmetries put
additional restrictions on certain quantities which allows for
simplifications. On the other hand, we also have to ask about the
compatibility of symmetries and regularization and whether they can be
restored if intermediately broken. Since regularization may in general spoil
the classical symmetry, we shall require its validity as part of the
definition of our theory. The symmetry of interest for us is gauge invariance
promoted to BRST invariance as described in Sec. 2.3. On the level of Green
functions, this symmetry is implemented by the Slavnov-Taylor identity as
described in Sec. 2.5. In a more compact notation (cf. Eq. (2.102)), it can be
written as

$\mathcal{S}(\Gamma_{\mathrm{ren}})=\int\,\mathrm{d}^{4}x\,\frac{\delta\Gamma_{\mathrm{ren}}}{\delta\phi(x)}\frac{\delta\Gamma_{\mathrm{ren}}}{\delta
K_{\phi}(x)}\stackrel{{\scriptstyle!}}{{=}}0.$ (6.1)

Here we assumed for simplicity that all symmetry transformations, i.e. both
linear and non-linear, are coupled to sources $K_{\phi}$. The Slavnov-Taylor
identity is the pivotal tool in the proof of renormalizability of quantized
Yang-Mills gauge theories, including the proof that the quantum theory
actually is physically sensible.

The first proofs of the renormalizability of non-Abelian gauge theories were
given by ’t Hooft, Lee and Zinn-Justin in Refs. [125, 126, 127, 128, 129,
130], all employing various versions of Slavnov-Taylor identities. These
proofs establish not only the finiteness and validity of the Slavnov-Taylor
identity but also the interpretation of the quantum theory with a unitary and
gauge-fixing independent S-matrix defined on a Hilbert space of quantum states
with positive norm. Later, the proofs were generalized by Becchi, Rouet, Stora
and Tyutin (BRST) to the case where nothing is known about symmetry properties
of the employed regularization scheme, establishing the approach of algebraic
renormalization [41, 42, 43, 44], see also the reviews by Piguet/Rouet and
Piguet/Sorella [119, 47]. A particularly satisfactory formulation is achieved
with the Kugo/Ojima formalism [45] where the existence of a nilpotent operator
$Q_{B}$ is derived from the Slavnov-Taylor identity. $Q_{B}$ generates BRST
transformations on the level of asymptotic states and its role on the level of
quantum states is similar to the role of the BRST operator $s$ on the
classical level, see Eqs. (2.56–2.59). It may be used to define the physical
Hilbert space as the quotient space

$\mathcal{H}_{\text{phys}}=(\mathrm{ker}\,Q_{B})/(\mathrm{im}\,Q_{B})\,.$
(6.2)

Hence two states are equivalent if they differ by a total $Q_{B}$-variation. A
single state is called physical if $Q_{B}\ket{\psi}=0$, provided it is not
some total variation, i.e. $\ket{\psi}\neq Q_{B}\ket{\chi}$ for some
$\ket{\chi}$, in which case it would be equivalent to the zero vector. The
fields act Lorentz covariantly on the whole space including unphysical states
and because of the Slavnov-Taylor identity, $Q_{B}$ commutes with the
S-matrix. Hence the physical S-matrix defined on the physical Hilbert space
maps physical states to physical states, it is Lorentz invariant, unitary and
causal. All these properties can be shown in a very elegant way [45]. We thus
see that if we make sure that the Slavnov-Taylor identity is obeyed after
renormalization, we are guaranteed a consistent quantum field theory.

Hence the logic now is the following. In section 2 we defined gauge theories
which classically satisfy the BRST symmetry. Then we established dimensional
regularization as a framework for treating such theories perturbatively in
loop orders. Now we are in a position to define our renormalized theory with
the fundamental Slavnov-Taylor identity intact and study the possible
obstructions posed by regularization. To this end we shall first discuss the
counterterm structure for manifestly preserved symmetries during
renormalization in Sec. 6.1. Then in Sec. 6.2 we give a brief overview of the
field of algebraic renormalization which is the appropriate setting in which
to discuss breaking and restoration of symmetries. Finally we discuss how the
general analysis of algebraic renormalization can be specialized to the case
of dimensional regularization in Sec. 6.3.

###  6.1 Counterterms in Symmetry-Preserving Regularization

We first recall the simple case where a symmetry is manifestly preserved at
all steps of the calculation. This is the standard case often encountered in
textbook discussions and practical calculations using DReg in QED and QCD, for
reasons described already in Sec. 4.3. There one frequently uses so called
renormalization transformations of the generic form

$\displaystyle g$ $\displaystyle\rightarrow g+\delta g$ (6.3a)
$\displaystyle\phi_{i}$ $\displaystyle\rightarrow\sqrt{Z_{ij}}\,\phi_{j},$
(6.3b)

for coupling constants $g$ and quantum fields $\phi_{i}$ with associated
parameter and field renormalization constants $\delta g$ and $\delta
Z_{ij}=Z_{ij}-\delta_{ij}$. The renormalization constants are to be understood
as power series in loop orders or equivalently in the renormalized parameters.

This procedure is applied onto the classical action $S_{0}$ and thereby
defines a bare action $S_{\mathrm{bare}}$, cf. (3.1), itself giving rise to
the counterterm action

$S_{\mathrm{ct}}=S_{\mathrm{bare}}-S_{0}.$ (6.4)

The divergent parts of these generated counterterms cancel UV divergences of
loop diagrams, and the finite parts of the counterterms can be used to fulfil
certain renormalization conditions as mentioned in Sec. 3.1.

In terms of the Slavnov-Taylor identities, the standard case is expressed by
the statement

$\mathcal{S}(\Gamma_{\mathrm{reg}})=0,$ (6.5)

which, as explained in Sec. 4.3, means that the regularized Green functions
already satisfy the Slavnov-Taylor identity. If applicable, similar equations
should hold for other identities such as the ones discussed in Sec. 2.6 (e.g.
ghost equation). This is indeed the case in QED and QCD in DReg at all orders.
The basis of this statement was explained in Sec. 4. The manifest symmetry at
the regularized level (6.5) has two implications for the structure of
renormalization. First, the possible divergences are restricted by Eq. (6.5)
which, in turn, also restricts the structure of counterterms needed to cancel
divergences. Second, possible finite counterterms are also restricted by Eq.
(6.5), together with the ultimate requirement (6.1) for the renormalized
theory. Both implications can be simultaneously evaluated as follows. Assuming
that the theory has been renormalized up to order $\mathcal{O}(\hbar^{n-1})$,
we are interested in the $\mathcal{O}(\hbar^{n})$-order counterterms
$\mathcal{L}_{\mathrm{ct}}^{n}$ and the $\mathcal{O}(\hbar^{n})$ divergences
of the regularized theory. The renormalized theory at order
$\mathcal{O}(\hbar^{n})$ can be written as

$\Gamma^{(n)}_{\text{ren}}=\Gamma^{(n)}_{\text{reg,fin}}+\Gamma^{n}_{\text{reg,div}}+S^{n}_{\text{ct}}.$
(6.6)

For further analysis it is customary to introduce the linearized Slavnov-
Taylor operator $s_{\Gamma}$, defined by expanding the Slavnov-Taylor operator
$\mathcal{S}(\Gamma)$ for both linearly and non-linearly transforming fields
$\phi$ and $\Phi$, respectively,

$\mathcal{S}(\Gamma)=\int\,\mathrm{d}^{4}x\,\frac{\delta\Gamma}{\delta
K_{i}(x)}\frac{\delta\Gamma}{\delta\Phi_{i}(x)}+\int\,\mathrm{d}^{4}x\,s\phi_{i}(x)\frac{\delta\Gamma}{\delta\phi_{i}(x)},$
(6.7)

as follows,

$\mathcal{S}(\Gamma+\zeta\mathcal{F})=\mathcal{S}(\Gamma)+\zeta
s_{\Gamma}\mathcal{F}+\mathcal{O}(\zeta^{2}),$ (6.8)

for some functional $\mathcal{F}$. Its concrete form is given by

$s_{\Gamma}=\int\,\mathrm{d}x\,\left(\frac{\delta\Gamma}{\delta
K_{i}(x)}\frac{\delta}{\delta\Phi_{i}(x)}+\frac{\delta\Gamma}{\delta\Phi_{i}(x)}\frac{\delta}{\delta
K_{i}(x)}+s\phi_{i}(x)\frac{\delta}{\delta\phi_{i}(x)}\right).$ (6.9)

Of special interest is the case of the classical action $\Gamma_{\text{cl}}$,
for which we define

$b\equiv s_{\Gamma_{\text{cl}}},$ (6.10)

as the linearized Slavnov-Taylor operator based on the classical action. In
agreement with the nilpotency of the BRST operator (2.59), the algebraic
structure of the Slavnov-Taylor operator leads to two nilpotency relations

$\displaystyle s_{\Gamma}\mathcal{S}(\Gamma)$ $\displaystyle=0,$ (6.11)
$\displaystyle s_{\Gamma}s_{\Gamma}$
$\displaystyle=0\quad\text{if}\quad\mathcal{S}({\Gamma})=0.$ (6.12)

Substituting the decomposition of Eq. (6.6) into Eqs. (6.5) and (6.1), we
first get

$b\,\Gamma^{n}_{\text{reg,div}}=0.$ (6.13)

This establishes the restriction on the possible divergences. Second, we
obtain

$b\,S^{n}_{\text{ct}}=0,$ (6.14)

both for the divergent and the finite parts. The most general solution of this
equation in terms of admissible counterterm actions yields the counterterm
structure which is sufficient to cancel the divergences and required to
establish the symmetry. The corresponding calculations were carried out in the
original references on the renormalization of Yang-Mills theories cited at the
beginning of this section; textbook discussions can be found e.g. in the
textbooks by Zinn-Justin, Weinberg and Böhm/Denner/Joos [50, 32, 34].

For most theories of interest including the SM, the outcome is the familiar
statement cited in the beginning (cf. Eq. (6.3)) that all counterterms can be
obtained by renormalization transformation of the classical action. A second
related outcome is then that any two consistent regularization/renormalization
prescriptions which both fulfil the symmetry requirement (6.1) can only differ
by a reparametrization of the form (6.3).303030 This is a stronger statement
than the one of Eq. (5.9) because a smaller number of parameters is affected.

###  6.2 Broken Symmetries and Algebraic Renormalization

Now we turn to the case of interest for e.g. chiral gauge theories in which
the symmetry is not manifestly preserved by the regularization. This case is
characterized by

$\mathcal{S}(\Gamma_{\text{reg}})\neq 0,$ (6.15)

in contrast to Eq. (6.5). Clearly, the required structure of the counterterms
is more complicated. Now, the divergences and required divergent counterterms
may be non-symmetric and not fulfil Eq. (6.13). In this case one has to
determine them by explicit calculation of the divergences of Green functions
instead of reading off their structure from a renormalization transformation
such as (6.3). In this way the theory can be rendered finite despite the
broken symmetry (6.15).

Even on the finite level, the symmetry breaking (6.15) might still persist.
Finite counterterms then have to be determined such that the fully
renormalized theory fulfils the basic requirement (6.1). In some cases it can
actually be impossible to find such counterterms; the symmetry is then said to
be broken by an anomaly. Since we consider the Slavnov-Taylor identity as part
of the definition of the theory, an anomalous breaking of the Slavnov-Taylor
identity means that the theory is inconsistent and not renormalizable. In
cases without an anomaly it is indeed possible to recover the symmetry by
appropriately chosen finite counterterms.

Even though the precise form of the symmetry breaking depends on the
regularization, it is possible to study the general case of (6.15) in a
regularization-independent way. This study is the content of algebraic
renormalization, pioneered by BRST [41, 42, 43, 44], see also the reviews [47,
119]. The main insight of the procedure is that the possible breakings are
restricted in two ways. On the one hand, they are restricted by the Slavnov-
Taylor identity itself, similar to the possible divergent structures in Eq.
(6.13). On the other hand, they are restricted by a regularization-independent
version of the quantum action principle.

Those two restrictions taken together provide a regularization-independent
analysis of the renormalization of gauge theories. In the following we shall
first sketch the quantum action principle in the BPHZ framework of
renormalization, where it was originally established and subsequently used for
algebraic analysis, as well as exhibit a connection to the regularized quantum
action principle of DReg. The central point is then to review how the
aforementioned restrictions can be used to restore the broken symmetry by
suitable counterterms provided there are no anomalies.

####  6.2.1 The Quantum Action Principle in BPHZ

As discussed in Sec. 5.1, the BPHZ approach to renormalization constituted one
of the first full discussions of all-order renormalization, rigorously
establishing the possibility to obtain finite Green functions and S-matrix
elements in agreement with basic postulates such as causality and unitarity.
In this framework Lowenstein and Lam derived various theorems now summarized
as the quantum action principle [107, 106, 109, 108, 110]. The theorems are
similar to the regularized quantum action principle in DReg discussed in Sec.
4. The difference is that the theorems discussed here are valid in strictly
$4$ dimensions, for the fully renormalized theory.

Further, this form of the quantum action principle is generally valid not only
in the BPHZ framework but in all regularization/renormalization frameworks
that are equivalent; hence it also applies to results obtained using DReg, if
the $\text{LIM}_{D\to 4}$ defined in Eq. (3.5) has been taken. The algebraic
method is based on this general formulation and its results hold for all such
equivalent frameworks. In BPHZ, finite expressions and the Gell-Mann-Low
formula are defined by an iterative operation on momentum space integrals
whereby Taylor series contributions up to some UV-subtraction degree are
subtracted from the integrands giving finite integrals by power counting.
Further, normal products, i.e. products of fields and their derivatives at the
same space-time point, may be defined as finite parts of certain Wick-ordered
insertions into the Green function. One can derive so called Zimmermann-
identities which linearly relate over-subtracted normal products, i.e. of
higher UV-degree than the canonical operator dimension, to minimally
subtracted ones. These prove a powerful tool in e.g. deriving field equations
and studying anomalies.

A first version of the quantum action principle can be used to express the
relation of some infinitesimal variation of Green functions, or equivalently
generating functionals, with the insertion of a normal product. Ref. [107]
considers differential vertex operations (DVO) which are insertions of
integrated normal ordered local field polynomials into the Gell-Mann-Low
formula corresponding to the respective Green function

$\Delta\cdot G_{i_{1},\dots,i_{n}}(x_{1},\dots,x_{n})=\langle
0|T\int\,\mathrm{d}y\,N[P(y)]\phi^{i_{1}}(x_{1})\dots\phi^{i_{n}}(x_{n})|0\rangle.$
(6.16)

Then one can connect the variation of the Green function w.r.t. some parameter
with those DVO’s, i.e. taking some infinitesimal variation as
$\mathcal{L}_{\text{int}}\rightarrow\mathcal{L}_{\text{int}}+\sum_{k}\varepsilon_{k}P_{k}(x)$,
it follows

$\frac{\partial
G^{\varepsilon}}{\partial\varepsilon^{k}}\bigg{|}_{\varepsilon=0}=i\Delta\cdot
G.$ (6.17)

This result is valid for BPHZ renormalized disconnected, connected and 1PI
Green functions, and therefore also for the corresponding generating
functionals.

It can be used to derive the renormalized QAP for a generic parameter of the
theory $\lambda$,

$\frac{\partial\Gamma}{\partial\lambda}=i\Delta_{\lambda}\cdot G,$ (6.18)

where
$\Delta_{\lambda}=\int\,\mathrm{d}x\,N[\frac{\partial\mathcal{L}}{\partial\lambda}]$.

There are several further versions of the quantum action principle with
regards to variations of parameters or (external) fields. In particular Refs.
[109, 108, 110] established a version of the action principle w.r.t variations
of dynamical fields (see e.g. Ref. [108], Eq. (5.4)). The left-hand side being
equal to zero due to conservation of some current, the resulting relation
corresponds to Eq. (2.99) for the more general case of a non-invariant
Lagrangian $\delta\mathcal{L}\neq 0$ under some symmetry transformation. It is
rigorously established in terms of the generating functional for general Green
functions renormalized in the BPHZ framework, and it can be connected to the
generating functional of 1PI Green functions via Legendre transformation.

Thus the finite BPHZ framework is a setting in which formally derived
identities among generating functionals such as the ones described in Secs.
4.1 or 2.5 can be given a sensible all-order meaning.

In addition, in any regularization/renormalization procedure in agreement with
the basic postulates there is a way to cancel divergences and to obtain finite
Green functions. These may differ from the ones obtained in BPHZ (or any other
regularization), but in view of the theorems discussed in Sec. 5.1 the
differences can only amount to local counterterms at each order.

In the following we summarize important statements of the quantum action
principle valid for any such finite Green functions defined via any consistent
regularization and subtraction of divergences. The statements can be cast in a
variety of forms, similarly to Sec. 4.1. Here we provide the formulation for
the effective action $\Gamma$, as reviewed in Ref. [47]. First, equations of
motion for the generating functionals can be written as

$\displaystyle\frac{\delta\Gamma}{\delta\phi_{i}(x)}-\Delta_{i}(x)\cdot\Gamma$
$\displaystyle=0\,.$ (6.19)

For variations with respect to parameters we have

$\displaystyle\frac{\partial\Gamma}{\partial\lambda}$
$\displaystyle=\int\mathrm{d}x\,\Delta(x)\cdot\Gamma\,.$ (6.20)

As discussed in Secs. 2.5 and 2.4, in the case of nonlinear symmetry
transformations it is useful to couple the composite operators to some
external field, say $\rho^{a}(x)$. Then one can arrive at the following
version of the quantum action principle relevant for such nonlinear symmetry
transformations,

$\frac{\delta\Gamma}{\delta\rho_{a}(x)}\frac{\delta\Gamma}{\delta\phi_{i}(x)}=\Delta^{ai}(x)\cdot\Gamma\,.$
(6.21)

In all previous equations (6.19–6.21), the quantities $\Delta$ denote
insertions of local composite field operators, whose dimensions are bounded by
power counting and whose tree-level value is fixed in terms of the classical
expression $\Gamma_{\text{cl}}$. For example, in case of Eq. (6.19),
$\Delta_{i}$ is a local composite field operator whose dimension is bounded by
$(D-d_{i})$, where $d_{i}$ denotes the power-counting dimension of the
corresponding field $\phi_{i}$, and

$\Delta_{i}=\frac{\delta\Gamma_{cl}}{\delta\phi_{i}}+\mathcal{O}(\hbar)\,.$
(6.22)

####  6.2.2 Comparing Quantum Action Principles in BPHZ and DReg

The quantum action principles discussed in the previous subsection for BPHZ
and in Sec. 4 for DReg are similar but different. Here we briefly comment on
their relationship. The BPHZ version is valid for any
regularization/renormalization procedure, including DReg. However, it is valid
for the finite theory, in DReg for the theory after taking
$\text{LIM}_{D\,\to\,4}$ as defined in Eq. (3.5). The definition of this limit
includes setting evanescent quantities (such as the $(D-4)$-dimensional metric
$\hat{g}^{\mu\nu}$) to zero. The insertions $\Delta$ appearing e.g. in Eqs.
(6.21) are always finite, $4$-dimensional normal product insertions into the
finite Green functions.

In contrast, in the DReg case, the counterpart equation (4.6) is valid for
general $D\neq 4$, including evanescent quantities. In addition, if the
identity corresponds to a symmetry such as the Slavnov-Taylor identity which
is valid at tree level and in $4$ dimensions, then the insertion $\Delta$
appearing in Eq. (4.6) is purely evanescent.

It may not be immediately obvious how this can be reconciled with the purely
$4$-dimensional case of BPHZ. This is however important as we shall be making
use of general considerations following from the algebraic framework while
working in DReg. In fact, both versions of the quantum action principle are
valid and useful. The BPHZ version is useful to establish general existence
proofs which we can rely on also within DReg, but the DReg version is useful
for explicit computations since there the explicit form of the insertion
$\Delta$ is known.

The key is provided by the Bonneau identities established in Refs. [73, 74].
These identities precisely state that the insertion of an evanescent operator
in DReg as in Eq. (4.6) may in the $\text{LIM}_{D\to 4}$ be rewritten as an
insertion of a finite, $4$-dimensional operator as in Eq. (6.21). In this way,
the BPHZ quantum action principle can also be rederived from the one in DReg.

On the technical level, the Bonneau relationship also provides the
coefficients in the expansion of evanescent operator insertions in terms of
$4$-dimensional, finite insertions. They are given by the residue of simple
$1/(D-4)$ pole of the insertion of the evanescent operator into Green
functions. The proof is essentially achieved by taking dimensionally
renormalized amplitudes $\mathcal{R}_{G}$ associated to a graph $G$ and
comparing the vertex insertions
$\hat{g}^{\mu\nu}[\mathcal{O}_{\mu\nu}\cdot\mathcal{R}_{G}]$ on the one hand
with vertex insertions with
$[\hat{g}^{\mu\nu}\mathcal{O}_{\mu\nu}]\cdot\mathcal{R}_{G}$ on the other
hand.

At the one-loop level, the Bonneau identities are not surprising since
evanescent quantities can only contribute in the $\text{LIM}_{D\to 4}$ if they
hit $1/(D-4)$ poles, which at the one-loop level have local coefficients which
may be interpreted as a $4$-dimensional local operator. However, their
validity lies in their all-order nature. We mention here that Bonneau
identities can also be used to obtain information on renormalization group
equations in the presence of symmetry breakings of the regularization, see
e.g. Refs. [131, 25, 28].

####  6.2.3 Algebraic Renormalization and Symmetry Restoration

With the quantum action principle at our disposal we can now describe the
logic of algebraic renormalization of gauge theories. The starting point are
possible breakings of the Slavnov-Taylor identity (or similar identities) as
given by Eq. (6.15) due to the regularization. The quantum action principle
provides a useful tool in restricting the structure of the breaking and in
determining whether the symmetry can be restored, i.e. whether there are
anomalies. For that we proceed inductively order by order in perturbation
theory. The goal is to determine the required finite, symmetry-restoring
counterterms $S_{\text{ct}}^{n}$ step by step for each $n$.

At lowest order, at the classical level $n=0$, the Slavnov-Taylor identity is
valid by construction. This forms the basis of the inductive procedure. Let us
then suppose the theory is renormalized completely, hence it is finite and the
Slavnov-Taylor identity is fulfilled at some order $n-1$. In addition, at the
next order $n$, the divergences are already cancelled by appropriate singular
counterterms. Hence we have

$\mathcal{S}(\Gamma_{\text{subren}}^{(n),\text{fin}})=\mathcal{O}(\hbar^{n}),$
(6.23)

where we have introduced the notation

$\Gamma_{\text{subren}}^{(n),\text{fin}}=\Gamma^{(n)}_{\text{subren}}+S^{n}_{\text{sct}},$
(6.24)

which denotes the effective action finite at order $n$ after
subrenormalization and adding the necessary divergent $n$-loop counterterms.
This quantity corresponds to the set of finite Green functions for which the
validity of the quantum action principle in BPHZ has been proven, and it can
be defined in any other regularization scheme equivalent to BPHZ.

The task is then to study the possible breakings on the RHS of Eq. (6.23) as
well as the possible structure of counterterms. As mentioned before the
breaking is restricted in two ways. First, we may employ the quantum action
principle to find,

$\mathcal{S}(\Gamma_{\text{subren}}^{(n),\text{fin}})=\hbar^{n}\Delta\cdot\Gamma_{\text{subren}}^{(n),\text{fin}}=\hbar^{n}\Delta+\mathcal{O}(\hbar^{n+1}).$
(6.25)

The important point is that $\Delta$ is a local polynomial in fields and
derivatives, also restricted by power counting. This property was announced in
Sec. 2.5, where the Slavnov-Taylor identity was formally derived from the path
integral.

Second, applying the linearized BRST operator $s_{\Gamma_{\text{cl}}}\equiv b$
to Eq. (6.25) using Eq. (6.11) and extracting the $\mathcal{O}(\hbar^{n})$
terms, we arrive at a consistency condition (also called the Wess-Zumino
consistency condition),

$b\Delta=0.$ (6.26)

Hence the possible breaking $\Delta$ is restricted very similarly (cf. Eq.
(6.13)) to the possible divergences $\Gamma_{\text{div}}$ in Sec. 6.1. Both
$\Gamma_{\text{div}}$ in Eq. (6.13) and $\Delta$ in Eq. (6.26) are local
polynomials restricted by power counting which are annihilated by $b$, but
$\Gamma_{\text{div}}$ is of ghost number $0$ whereas $\Delta$ has ghost number
$1$. Now one can make a distinction. If $\Delta$ is a $b$-exact term, i.e. if
there exists another local polynomial $\Delta^{{}^{\prime}}$ with

$\Delta=b\Delta^{{}^{\prime}},$ (6.27)

it is called a trivial element of the cohomology of the BRST operator. In this
case we can supplement the original action with a new $n$-loop order
counterterm

$S^{n}_{\text{fct}}=S^{n}_{\text{fct,non-
inv}}+S^{n}_{\text{fct,inv}}=-\Delta^{{}^{\prime}}+S^{n}_{\text{fct,inv}},$
(6.28)

where the last term reflects the freedom to add to the action any finite,
symmetric counterterm, obeying $b\,S^{n}_{\text{fct,inv}}=0$. Hence, we end up
with

$\mathcal{S}(\Gamma_{\text{subren}}^{(n),\text{fin}}+\hbar^{n}S^{n}_{\text{fct}})=\mathcal{S}(\Gamma_{\text{subren}}^{(n),\text{fin}})+b\,\hbar^{n}S^{n}_{\text{fct}}+\mathcal{O}(\hbar^{n+1})=\mathcal{O}(\hbar^{n+1}),$
(6.29)

where the last step follows from the induction hypothesis. Compatibility with
ghost and gauge fixing equation is shown in Ref. [47].

Hence, under the condition (6.27), we can find a counterterm action
$S_{\text{fct,non-inv}}^{n}$ which defines finite, non-invariant counterterms
that repair the symmetry. Furthermore, it is possible to add any number of
finite, invariant counterterms to the action as they satisfy
$b\,S_{\text{fct,inv}}=0$ and hence do not disturb the STI. These invariant
counterterms behave like the finite counterterms discussed in Sec. 6.1 and can
be used to satisfy certain renormalization conditions.

One task of the algebraic renormalization programme is therefore to determine
the most general solution of the equation $b\Delta=0$. If all possible
solutions are $b$-exact, then this constitutes a proof that the Slavnov-Taylor
identity can be established at all orders in the renormalized theory.

However, if we cannot write the breaking $\Delta$ as a $b$-exact term, the
symmetry cannot be repaired. This is an anomaly. In case of the Slavnov-Taylor
identity such an anomaly is disastrous since it destroys the interpretation of
the theory as a sensible quantum theory, see the discussion at the beginning
of the present section. Anomalies are thus non-trivial elements of the
cohomology of the $b$-operator, i.e. expressions which are annihilated by $b$
but are not $b$-exact.

The previous remarks constitute crucial insights of the BRST formalism [41,
42, 43, 44]. The analysis of whether a gauge theory is renormalizable, i.e.
whether the Slavnov-Taylor identity can be restored at each order, can be made
on a purely classical level, by finding all possible solutions of Eq. (6.26)
and checking whether they are all $b$-exact.

The actual computation can be found in the original references and in the
reviews [119, 47]. It can be sketched as follows. From the Wess-Zumino
consistency condition (6.26) and the nilpotency of the BRST operator, one can
derive a set of equations, the so called descent equations. Solving these
gives a general expression of the possible anomalies of a theory. In the
present case of interest for a generic Yang-Mills theory it can be shown that
the consistency condition simplifies to $s\Delta(G,c)=0$, see e.g. [47], with
dependence on the gauge and the ghost field only. Writing $\Delta$ as an
integrated local product and solving the descent equations, it leads to the
famous Adler-Bell-Jackiw gauge anomaly first discovered in Refs. [6, 7,
8],313131Note the different relative sign of the first term of Eq. (6.30)
compared to [47] which comes from a different sign convention in the covariant
derivative, see Eq. (2.6).

$\Delta=L\times\varepsilon_{\mu\nu\rho\sigma}\mathrm{Tr}\int\mathrm{d}^{4}x\,c_{a}\partial^{\mu}\left(-gd_{A}^{abc}\partial^{\nu}G_{b}^{\rho}G_{c}^{\sigma}+g^{2}\frac{\mathcal{D}_{A}^{abcd}}{12}G_{b}^{\nu}G_{c}^{\rho}G_{d}^{\sigma}\right),$
(6.30)

where $L$ is a coefficient that can be determined from explicit calculations
and which depends on the theory inputs. The group symbols are given by,

$d_{A}^{abc}=\mathrm{Tr}\Big{(}T^{a}_{\text{adj}}\big{\\{}T^{b}_{\text{adj}},T^{c}_{\text{adj}}\big{\\}}\Big{)},$
(6.31)

and

$\mathcal{D}_{A}^{abcd}=d_{A}^{nab}f^{ncd}+d_{A}^{nac}f^{ndb}+d_{A}^{nad}f^{nbc},$
(6.32)

where $T^{a}_{\text{adj}}$ denotes adjoint generators under which ghosts and
gauge fields transform, cf. Eq. (2.2). Expression (6.30) must vanish by
itself, i.e. cannot be absorbed by counterterms, for the theory to be
consistent. In the case of a single left-handed fermion it can be shown by a
one-loop calculation that the anomaly is proportional to

$\frac{1}{2}\,d_{A}^{abc}\,\mathrm{Tr}(T^{a}\\{T^{b},T^{c}\\}),$ (6.33)

which means that its cancellation depends on an appropriate choice of the
matter content of the theory. The famous Adler-Bardeen theorem guarantees that
if the gauge anomaly vanishes at one-loop order, it also vanishes at all
orders, cf. [47]. The expression in Eq. (6.33) cannot vanish by itself, but in
such a theory with a family of left-handed fermions, their charges may add up
to zero as is the case in the SM. For some gauge groups such as SU(2), the
above expression vanishes identically due to the vanishing of some group
symbols. Hence there can be no anomaly.

In summary we have sketched how algebraic renormalization allows identifying
the general structure of the breaking of the Slavnov-Taylor identity. It
constitutes a setting in which the restoration of the symmetry can be proven
to all orders for trivial elements of the BRST cohomology such as spurious
breakings introduced by the BMHV algebra. In the case of non-spurious
breakings, e.g. the gauge anomaly, one can derive explicit conditions for its
cancellation which a sensible theory must satisfy. Further, nonrenormalization
theorems, as in the case of the Adler-Bardeen theorem, can be shown and allow
evaluating the gauge anomaly in a simple way. The main technical tool which
serves to establish these findings is the general quantum action principle
valid in many equivalent subtraction schemes. A key advantage of the algebraic
proof is that there is no need for an invariant regularization which for e.g.
chiral gauge theories does not exist.

####  6.2.4 Outlook and Further Remarks on Anomalies and Algebraic
Renormalization

At this point we interject a brief outlook on anomalies and further
applications of the techniques of algebraic renormalization. Next to the
perturbative chiral gauge anomalies discussed above and discovered in Refs.
[6, 7, 8] there exist global chiral anomalies [132] and pertubative mixed
gauge-gravitational anomalies [133, 134, 135]. A chiral gauge model can be
renormalized only if all these chiral anomalies cancel, which may be achieved
by a proper choice of fermion representations of for the chiral model, for
example see Ref. [11] and references therein. Eq. (6.30) is necessary but not
sufficient if gravity and nonperturbative effects are taken into account.

Important theories such as the Standard Model of particle physics, are
renormalizable. In particular, the electroweak SM was completely treated in
algebraic renormalization in Ref. [57], establishing the SM as a fully all-
order consistent, renormalizable theory. Ref. [56] gave a similar proof using
the background field gauge (see footnote 3), and Ref. [59] gave a similar
proof for the supersymmetric SM. These papers complement earlier extensive
discussions of the renormalization of the electroweak SM by e.g. Refs. [136,
60], see also Ref. [137].

The validity of the Slavnov-Taylor identity and the techniques of algebraic
renormalization can also be used to establish further interesting physics
properties of quantum gauge theories such as the renormalized electroweak SM.
E.g. charge universality can be established based on both gauge choices [136,
48], see also Ref. [137] for further discussions. As another example, the
renormalization of Higgs vacuum expectation values in spontaneously broken
gauge theories can be controlled via a suitable Slavnov-Taylor identity [138,
139].

###  6.3 Algebraic Symmetry Restoration in the Context of DReg

So far in this section we have studied the role of symmetries in the process
of renormalization. If the symmetry is respected by the regularization, it
implies a great simplification for the UV counterterms. If it is not,
algebraic renormalization constitutes a general setup which allows identifying
symmetry violations and restoring the symmetry. Here we specialize the general
procedure to the case of DReg. We use the BMHV scheme with non-anticommuting
$\gamma_{5}$ in which gauge invariance may be broken.

####  6.3.1 Formulation of Symmetry and Symmetry Breaking in DReg

The ultimate symmetry requirement is the Slavnov-Taylor identity expressing
BRST invariance of the full renormalized theory, Eq. (6.1). In the context of
DReg this requirement can be formulated as

$\mathop{\text{LIM}}_{D\,\to\,4}\,(\mathcal{S}_{D}(\Gamma_{\mathrm{DRen}}))=0.$
(6.34)

As defined in Sec. 3.1, $\Gamma_{\mathrm{DRen}}$ denotes the renormalized
effective action, still in $D$ dimensions but including all counterterms
cancelling $1/\epsilon$ divergences and restoring symmetries. The limit refers
to the operation of letting $\epsilon\rightarrow 0$ as well as putting
evanescent quantites such as $\hat{g}^{\mu\nu}$ to zero.

In order to discuss the inductive procedure, we consider some order $n$ and
suppose the theory has been renormalized and all counterterms have been
constructed up to the previous order $n-1$. This provides us with

$\displaystyle\Gamma^{(n)}_{\text{subren}}\,,$ (6.35)

again using the notation of Sec. 3.1. At this point we know from Sec. 5 that
the divergences at the $n$-th order can be cancelled by adding a local
counterterm action $S_{\text{sct}}^{n}$. It may or may not be true that the
divergences follow the simple pattern described in Sec. 6.1. In general we can
always write

$\displaystyle S_{\text{sct}}^{n}$
$\displaystyle=S_{\text{sct,inv}}^{n}+S_{\text{sct,non-inv}}^{n}\,,$ (6.36)

where the first term corresponds to symmetric counterterms as described in
Sec. 6.1 and the second term corresponds to whatever other divergent
counterterms are required.

After subtracting these divergences the theory is finite at the order $n$ and
the Slavnov-Taylor identity may be written as

$\displaystyle\mathcal{S}_{D}(\Gamma_{\mathrm{subren}}^{(n)}+S_{\text{sct}}^{n})=\hbar^{n}\Delta_{D}+\mathcal{O}(\hbar^{n+1})\,,$
(6.37)

where $\Delta_{D}$ is a possible finite breaking term, still evaluated in $D$
dimensions. The subrenormalized and finite effective action introduced for the
algebraic analysis in Eq. (6.24) is now given by
$\mathop{\text{LIM}}_{D\,\to\,4}(\Gamma_{\mathrm{subren}}^{(n)}+S_{\text{sct}}^{n})$,
and the counterpart of Eq. (6.25) is given by the 4-dimensional limit

$\displaystyle\mathop{\text{LIM}}_{D\,\to\,4}\,\Delta_{D}$
$\displaystyle=\Delta_{\text{from Eq.\ (\ref{InductionStep})}}\,.$ (6.38)

This finite quantity $\Delta$ is the one constrained by algebraic
renormalization and discussed after Eq. (6.25). That is, it is a local
breaking term which satisfies the Wess-Zumino consistency conditions and which
can be cancelled by adding suitable counterterms (we assume that there is no
genuine anomaly).

The practical question is then how to obtain first the breaking term $\Delta$
and then the symmetry-restoring counterterms. There are two strategies for
this. The first, obvious option is to evaluate all Green functions appearing
on the LHS of Eq. (6.37) including their finite parts, plug them into the
Slavnov-Taylor identity and determine the potentially non-vanishing breaking.
This straightforward procedure is convenient in that it operates on ordinary
Green functions. Its drawback is that most finite parts of Green functions —
in particular parts that are non-polynomial in the momenta — will be in
agreement with the symmetry and hence drop out of Eq. (6.37), such that the
calculation can become unnecessarily complicated. Nevertheless this direct
approach has been used in the literature, e.g. in Refs. [140, 141, 142, 143,
144] on applications on chiral gauge theories and supersymmetric gauge
theories. In the subsequent section 7.4.1 we will also illustrate this
approach with a concrete example.

A second, alternative approach is provided by using the regularized quantum
action principle in DReg, described in Sec. 4. This theorem guarantees that we
can rewrite the LHS of Eq. (6.34) as

$\mathcal{S}_{D}(\Gamma_{\mathrm{DRen}})=(\widehat{\Delta}+\Delta_{\mathrm{ct}})\cdot\Gamma_{\mathrm{DRen}}.$
(6.39)

The possible breaking of the Slavnov-Taylor identity is thus rewritten as an
operator insertion of the composite operator
$\widehat{\Delta}+\Delta_{\mathrm{ct}}$, which is defined as

$\displaystyle\widehat{\Delta}$ $\displaystyle=\mathcal{S}_{D}(S_{0})\,,$
(6.40a) $\displaystyle\widehat{\Delta}+\Delta_{\text{ct}}$
$\displaystyle=\mathcal{S}_{D}(S_{0}+S_{\text{ct}})\,,$ (6.40b)

In this approach, the breaking $\Delta$ may be computed in terms of the RHS of
(6.39). The advantage lies in significantly restricting possible non-vanishing
contributions. In particular, $\widehat{\Delta}$ is evanescent; hence it can
contribute in the $\mathrm{LIM}_{D\rightarrow 4}$ only in combination with
$1/\epsilon$ singularities of Feynman diagrams.

The RHS of (6.39) can be expanded in loop orders as

$\widehat{\Delta}+\sum_{i=1}^{\infty}\hbar^{i}\left(\widehat{\Delta}\cdot\Gamma^{i}_{\mathrm{DRen}}+\sum_{k=1}^{i-1}\Delta^{k}_{\mathrm{ct}}\cdot\Gamma^{i-k}_{\mathrm{DRen}}+\Delta^{i}_{\mathrm{ct}}\right).$
(6.41)

Plugging the previous definitions into Eq. (6.34), we arrive at an equation
expressing the symmetry requirement exactly at the order $n$,

$\mathop{\text{LIM}}_{D\,\to\,4}\,\left(\widehat{\Delta}\cdot\Gamma_{\mathrm{DRen}}^{n}+\sum_{k=1}^{n-1}\Delta^{k}_{\mathrm{ct}}\cdot\Gamma^{n-k}_{\mathrm{DRen}}+\Delta^{n}_{\mathrm{ct}}\right)=0,$
(6.42)

for all $n\geq 1$. The individual terms in this equation have divergent and
finite parts, but by construction the entire expression is finite; hence the
cancellation of divergences may be used as a consistency check of practical
calculations. For the determination of symmetry-restoring counterterms, Eq.
(6.42) should be viewed as follows. At the order $n$ and after
subrenormalization and adding divergent $n$-loop counterterms, everything in
Eq. (6.42) is already known except the finite counterterms of order $n$. They
enter via $\Delta^{n}_{\mathrm{ct}}$, which in turn depends on the to-be-
determined counterterms. The following subsubsection will make the dependence
explicit. Hence Eq. (6.42) can be regarded as the optimized defining relation
for the symmetry-restoring counterterms in DReg.

We close with the remark that Eq. (6.42) does not fully determine all finite
counterterms. It only determines the required form of counterterms in order to
restore the symmetry. However, Eq. (6.42) is blind to several types of
counterterms: finite and symmetric counterterms (which often correspond to a
renormalization transformation as described in Sec. 6.1) drop out; such
counterterms can therefore still be adjusted at will e.g. to satisfy the
renormalization conditions corresponding to an on-shell or a different
desirable renormalization scheme. In addition, evanescent and finite
counterterms also drop out and may be added to optimize the counterterm
action.

####  6.3.2 Practical Restoration of the Symmetry

Here we illustrate the blueprint for the practical restoration of the
symmetry, if Eq. (6.42) is used as a basis.

We begin at the one-loop level and start from the regularized but
unrenormalized effective action $\Gamma^{(1)}$.323232We slightly simplify the
notation and use $\Gamma^{(1)}$ in the following equations of this
subsubsection to denote the unrenormalized effective action up to one-loop
order. According to the general notational scheme defined in Sec. 3.1, this
could also be called $\Gamma^{(1)}_{\text{subren}}$.  At the one-loop level
the regularized action plus counterterms as well as the symmetry breaking
induced by the counterterms at one-loop order are given by

$\displaystyle\Gamma_{\text{DRen}}^{(1)}$
$\displaystyle=\Gamma^{(1)}+S_{\text{sct}}^{1}+S_{\text{fct}}^{1}\,,$ (6.43a)
$\displaystyle\Delta_{\text{ct}}^{1}$
$\displaystyle=\mathcal{S}_{D}(S_{0}+S_{\text{ct}})^{1}=b_{D}S_{\text{sct}}^{1}+b_{D}S_{\text{fct}}^{1}\,,$
(6.43b)

where the last part of the last equation is a specific rearrangement possible
at the one-loop level, and where the linearized Slavnov-Taylor operator
$b_{D}$ is defined in analogy to $b$ in Eq. (6.10). The general equations
establishing the cancellation of divergences and symmetry restoration, (6.42),
become

$\displaystyle S^{1}_{\text{sct}}+\Gamma^{1}_{\text{div}}$
$\displaystyle=0\,,$ (6.44a)
$\displaystyle\big{(}\widehat{\Delta}\cdot\Gamma^{1}_{\text{}}+\Delta_{\text{ct}}^{1}\big{)}_{\text{div}}$
$\displaystyle=0\,,$ (6.44b)
$\displaystyle\mathop{\text{LIM}}_{D\,\to\,4}\,\big{(}\widehat{\Delta}\cdot\Gamma^{1}_{\text{}}+\Delta_{\text{ct}}^{1}\big{)}_{\text{fin}}$
$\displaystyle=0\,.$ (6.44c)

Compared to the general Eq. (6.42), terms that vanish at one-loop order were
dropped. The quantities that need to be explicitly computed here are the one-
loop divergences $\Gamma^{1}_{\text{div}}$ and the one-loop diagrams with one
insertion of the evanescent operator $\widehat{\Delta}$,
$\widehat{\Delta}\cdot\Gamma^{1}_{\text{}}$. The first of these equations then
determines the divergent one-loop counterterms $S_{\text{sct}}^{1}$, and the
second equation provides a consistency check of the divergences. In view of
Eq. (6.43b), the last line contains $b_{D}S_{\text{fct}}^{1}$ and thus
determines the symmetry-restoring one-loop counterterms.

Next we consider the two-loop order. At the two-loop level, the corresponding
equations for the effective action and the symmetry breaking of counterterms
are

$\displaystyle\Gamma_{\text{DRen}}^{(2)}$
$\displaystyle=\Gamma_{\text{subren}}^{(2)}+S_{\text{sct}}^{2}+S_{\text{fct}}^{2}\,,$
(6.45a) $\displaystyle\Delta_{\text{ct}}^{2}$
$\displaystyle=\mathcal{S}_{D}(S_{0}+S_{\text{ct}})^{2}=\mathcal{S}_{D}(S_{0}+S_{\text{ct}}^{1})^{2}+b_{D}S_{\text{sct}}^{2}+b_{D}S_{\text{fct}}^{2}\,,$
(6.45b)

where the upper index 2 corresponds to extracting the two-loop terms. The last
equation exhibits the appearance of the genuine two-loop counterterms in a way
specific to the two-loop level. The equations corresponding to finiteness and
symmetry restoration read

$\displaystyle S^{2}_{\text{sct}}+\Gamma^{2}_{\text{subren,div}}$
$\displaystyle=0\,,$ (6.46a)
$\displaystyle\big{(}\widehat{\Delta}\cdot\Gamma_{\text{\text{subren}}}^{2}+\Delta_{\text{ct}}^{1}\cdot\Gamma_{\text{}}^{1}+\Delta_{\text{ct}}^{2}\big{)}_{\text{div}}$
$\displaystyle=0\,,$ (6.46b)
$\displaystyle\mathop{\text{LIM}}_{D\,\to\,4}\,\big{(}\widehat{\Delta}\cdot\Gamma_{\text{\text{subren}}}^{2}+\Delta_{\text{ct}}^{1}\cdot\Gamma_{\text{}}^{1}+\Delta_{\text{ct}}^{2}\big{)}_{\text{fin}}$
$\displaystyle=0\,.$ (6.46c)

Here we have to calculate first the two-loop divergences to obtain the two-
loop divergent counterterms. Then we have to calculate diagrams with
insertions of $\widehat{\Delta}$ up to the two-loop level (and including one-
loop subrenormalization), as well as one-loop diagrams with insertions of
$b_{D}$-transformed one-loop counterterms. The second equation must
automatically hold and provides a check. The third equation then determines
the genuine finite two-loop symmetry-restoring counterterms
$b_{D}S_{\text{fct}}^{2}$, which appear via Eq. (6.45b) in
$\Delta_{\text{ct}}^{2}$.

In summary the recipe is as follows,

  * •

UV-renormalize the theory, previously renormalized up to order $n-1$, at order
$n$ to obtain the singular counterterms,

  * •

calculate genuine $n$-loop Green functions with one-time insertion of
$\widehat{\Delta}$ for their divergent and finite part,

  * •

calculate the $k$-loop order insertion into $(n-k)$-loop order graphs and
determine their divergent and finite contributions,

  * •

check that the divergences thus obtained sum up to zero,

  * •

collect the finite contributions and choose monomials $X$ such that $b_{D}\,X$
cancels them. This is always possible as discussed in the previous
subsections.

####  6.3.3 The Counterterm Lagrangian in the BMHV Scheme

The output of the regularization/renormalization programme is the renormalized
effective action and the required counterterm action consisting of singular
and finite counterterms. In the context of the BMHV scheme the previous
subsections show that the counterterm action can in general contain five
different kinds of terms,

$S_{\text{ct}}=S_{\text{sct,inv}}+S_{\text{sct,non-
inv}}+S_{\text{fct,inv}}+S_{\text{fct,restore}}+S_{\text{fct,evan}}.$ (6.47)

This equation is a more detailed version of the generic decomposition
explained in Sec. 3.1 into singular and finite counterterms. For both the
singular and the finite counterterms we may isolate a symmetry-invariant
piece, which has the pattern of symmetric counterterms discussed in Sec. 6.1
and typically corresponds to counterterms generated by a renormalization
transformation as

$S_{0}\stackrel{{\scriptstyle\text{ren.\ transf.\
(\ref{RenTrafo})}}}{{\longrightarrow}}S_{0}+S_{\mathrm{sct,inv}}+S_{\text{fct,inv}}.$
(6.48)

In general, the conditions of Sec. 6.1 are not met and symmetry-violating
counterterms are required. Accordingly, the next type of counterterms

$S_{\text{sct,non-inv}}$

corresponds to additional singular counterterms needed to cancel additional
$1/\epsilon$ poles of loop diagrams that cannot be cancelled by symmetry-
invariant counterterms. They may be evanescent and, starting from $2$-loop
order, also $4$-dimensional (non-evanescent). They cannot be obtained by
renormalization transformations. We note that the subtraction of evanescent
$1/\epsilon$ poles is a necessity for the consistency of higher orders (see
also Ref. [5] for a review discussing this point).

Next,

$S_{\text{fct,restore}}$

corresponds to finite counterterms needed to restore the Slavnov-Taylor
identity and thus the underlying gauge invariance. They are the central
objects of the present discussion and the outcome of the practical recipe of
Sec. 6.3.2. Determining these counterterms is one of the key tasks in the
usage of the BMHV scheme. Once those counterterms are found, the theory can be
considered to be renormalized.

As mentioned before, the symmetry-restoring counterterms are not unique.
Clearly, they may be modified by shifting around any symmetry-invariant
counterterm between $S_{\text{fct,inv}}$ and $S_{\text{fct,non-inv}}$, since
invariant terms would drop out of Eqs. (6.42,6.44c,6.46c). The overall sum of
$S_{\text{fct,inv}}+S_{\text{fct,non-inv}}$ can only be fixed by imposing a
renormalization scheme (such as e.g. the on-shell scheme), and the split into
$S_{\text{fct,inv}}$ and $S_{\text{fct,non-inv}}$ can only be fixed by picking
a convention. To illustrate this point, let us assume the counterterm
Lagrangian must contain a non-gauge invariant term
$zA^{\mu}\,\square\,A_{\mu}$ where $z$ is a coefficient and $A^{\mu}$ a gauge
field. Two different options for the counterterm Lagrangians would then be

$\displaystyle\mathcal{L}_{\text{fct,non-inv}}$
$\displaystyle=zA^{\mu}\,\square\,A_{\mu},$
$\displaystyle\mathcal{L}_{\text{fct,inv}}$ $\displaystyle=\delta
Z(A^{\mu}\,\square\,A_{\mu}+(\partial A)^{2}),$ (6.49a)
$\displaystyle\mathcal{L}_{\text{fct,non-inv}}$ $\displaystyle=-z(\partial
A)^{2},$ $\displaystyle\mathcal{L}_{\text{fct,inv}}$ $\displaystyle=(\delta
Z+z)(A^{\mu}\,\square\,A_{\mu}+(\partial A)^{2}).$ (6.49b)

The invariant counterterm here corresponds to an invariant counterterm
generated by a field renormalization from the usual gauge invariant kinetic
term $F^{\mu\nu}F_{\mu\nu}$. According to the assumption, both options restore
the symmetry, and they lead to the identical renormalized theory. The field
renormalization constant $\delta Z$ can be used to adopt a desired
renormalization condition.

Finally,

$S_{\text{fct,evan}}$

corresponds to additional counterterms that are both finite and evanescent.
Adding or changing such counterterms can change e.g. a purely 4-dimensional
counterterm $A^{\mu}\bar{\psi}\bar{\gamma}_{\mu}\psi$ to a fully
$D$-dimensional counterterm $A^{\mu}\bar{\psi}{\gamma}_{\mu}\psi$. These
counterterms vanish in the 4-dimensional limit, but they can affect
calculations at higher orders. They also drop out of Eqs. (6.42,6.44c,6.46c).
Hence one viable option is that the symmetry-restoring counterterms
$S_{\text{fct,restore}}$ are always defined by using strictly 4-dimensional
quantities only. However, this is not the only option; in concrete cases
elevating 4-dimensional terms to fully $D$-dimensional ones may simplify
expressions appearing at higher orders. At any rate, each such choice
generates a different, valid, renormalized theory. From a practical point of
view it is desirable to make a computationally simple choice.

##  7 Practical Treatment of Chiral Gauge Theories in the BMHV Scheme of DReg

In recent years, the treatment of chiral gauge theories with the non-
anticommuting $\gamma_{5}$ BMHV scheme has received increasing interest.
Applications in the SM at the multiloop level and in effective field theories
with additional operators involving chiral fermions have become more
important, see e.g. the discussions in Refs. [145, 137, 146, 147]. Accordingly
the usefulness of regularization/renormalization schemes for which ultimate
consistency is fully established is becoming more appreciated. After the
pioneering one-loop discussion of gauge theories with chiral fermions in Refs.
[131, 148], Ref. [25] extended the analysis to general chiral gauge theories
including scalar fields and Yukawa couplings to chiral fermions. Ref. [26]
pioneered the application of the BMHV scheme to chiral gauge theories at the
two-loop level with a first, abelian example. Ref. [27] extended the one-loop
analysis to the case of the background field gauge fixing and to the full
gauge–fermion sector of the electroweak SM.

In this section, we give concrete illustrations of how to treat chiral gauge
theories in the BMHV scheme with non-anticommuting $\gamma_{5}$. The
discussion is based on our results in Refs. [26, 25]. The following subsection
7.1 provides an extended overview of the procedure and a guide for the present
section.

###  7.1 Overview and Guide to the Present Section

In Sec. 2, we discussed the basic defining gauge invariance of gauge theories
and reformulated it in terms of BRST symmetry and Slavnov-Taylor and Ward
identities. In Sec. 6 we explained how these symmetry identities are elevated
to defining properties of the renormalized theory at higher orders. For the
gauge theories we study here, it is known that these defining symmetry
identities can be fulfilled in any consistent regularization/renormalization
procedure, by appropriately defining counterterms. In Sec. 3 we explained the
definition of dimensional regularization and the BMHV scheme for $\gamma_{5}$,
which in general breaks gauge invariance in the presence of chiral fermions.
In Sec. 5 we explained the proof that dimensional regularization constitutes
one of the consistent regularization/renormalization procedures.

As a result it is in principle established that the dimensional regularization
including BMHV scheme for $\gamma_{5}$ may be used for chiral gauge theories.
Further, Sec. 6.3 also provided a blueprint how to determine the required
counterterm structure in concrete calculations. In this section, we carry out
such concrete calculations and illustrate all required steps in detail.

In the Abelian chiral gauge theory defined below, we expect the validity of
simple QED-like Ward identities; the simplest one corresponds to the
transversality of the photon self-energy. It turns out that in the BMHV
scheme, the actual one-loop self-energy violates this transversality (see Eq.
(7.35)). The violation affects both the divergent and the finite part in the
BMHV scheme of dimensional regularization. The breaking, however, is a
polynomial in the momentum; hence it can be cancelled by adding a local
counterterm to the Lagrangian — in line with the general existence statement
mentioned above. After adding this counterterm the required transversality is
fulfilled. The concrete required form of the counterterm can be found in Eqs.
(7.39,7.41).

A question is then what is the most efficient way to determine such symmetry
breakings in general. Answers were given in Sec. 6.3.1 and can be illustrated
as follows. One way in principle is to explicitly evaluate all Green functions
and test the validity of all Ward and Slavnov-Taylor identities between all
Green functions. The explicit computation of the non-transverse terms in Eq.
(7.35) provides an example. Given that there are in principle infinitely many
identities between Green functions and given that the computation of Green
functions involves also complicated non-local terms that cannot contribute to
the symmetry violation, this strategy is not the most efficient.

Sec. 6.3.1 also explained a shortcut that is based on the regularized quantum
action principle of dimensional regularization discussed in Sec. 4. Staying
with the example of the photon self-energy, the terms violating the
transversality in Eq. (7.35) and then in Eq. (7.37) may be equivalently
obtained by computing one special Feynman diagram, shown in Eqs. (7.44,7.45).
This diagram involves an insertion of the operator $\widehat{\Delta}$ which
reflects the breaking of chiral gauge invariance in $D$ dimensions, and the
quantum action principle guarantees that the evaluation of this diagram
reproduces directly the breaking of the transversality of the photon self-
energy. The simplification is threefold: First and foremost, since
$\widehat{\Delta}$ is evanescent, only the ultraviolet divergent part of the
diagram can contribute — hence the evaluation is simpler (the degree of
simplification dramatically increases for more complicated Green functions and
at higher orders). Second, in the general case there are much fewer diagrams
with insertions of $\widehat{\Delta}$ than ordinary diagrams. Third, since
only divergent parts contribute it is clear that the symmetry
breaking/restoration procedure requires only the computation of power-counting
divergent diagrams with insertions of $\widehat{\Delta}$.

This more efficient but less obvious strategy based on the quantum action
principle was applied to chiral gauge theories at the one-loop level in Refs.
[25, 131, 148, 27] with and without scalar sector and to an Abelian chiral
gauge theory at the two-loop level in Ref. [26]. It was also applied to the
case of supersymmetric gauge theories in the context of dimensional reduction
at the two- and three-loop level in Refs. [102, 105].

In the largest part of the present section we focus on the simpler case of an
Abelian chiral gauge theory. We begin in Sec. 7.2 by defining the considered
model and collecting all relevant symmetry identities. Then we discuss the
subtleties in the continuation to $D$ dimensions and determine the insertion
operator $\widehat{\Delta}$. Sec. 7.3 provides a more technical overview of
the procedure to determine the symmetry-restoring counterterms than the
previous remarks. In Sec. 7.4 we then discuss the explicit computations in the
abelian model in detail. We begin with the case of the photon self-energy
mentioned above and illustrate both strategies to determine the symmetry-
restoring counterterms, then we progress to other Green functions and to the
two-loop level. Thereafter Sec. 7.5 discusses the case of non-Abelian Yang-
Mills theories and presents explicit calculations and results at the one-loop
level.

###  7.2 Definition of an Abelian Chiral Gauge Theory

Here we define a concrete Abelian chiral gauge theory which will be used in
explicit calculations. It is first defined in $4$ dimensions along with its
symmetry requirements in Sec. 7.2.1; then the definition is extended to $D$
dimensions within the framework of the BMHV $\gamma_{5}$ scheme and resulting
BRST symmetry breaking is exhibited in Sec. 7.2.2.

####  7.2.1 Chiral Electrodynamics in 4 dimensions

Following Sec. 2.6, the 4-dimensional classical Lagrangian for quantum
electrodynamics (QED) is given by

$\mathcal{L}_{\text{QED}}=i\overline{\psi}_{i}\not{D}_{ij}{\psi}_{j}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^{2}-\bar{c}\partial^{2}c+\rho^{\mu}s{A_{\mu}}+\bar{R}^{i}s{{\psi}_{i}}+R^{i}s{\overline{\psi}_{i}},$
(7.1)

with U(1) ghost and external BRST sources included.333333In contrast to Sec.
2.6, we already integrated out the Nakanishi-Lautrup field $B(x)$, i.e. we
used $B=-(\partial_{\mu}A^{\mu})/\xi$ already in the Lagrangian. The only
generator in this theory is the real and diagonal charge
$Q_{ij}=Q_{i}\delta_{ij}$, so that the covariant derivative reads

$D_{ij}^{\mu}=\partial^{\mu}\delta_{ij}+ieA^{\mu}{Q}_{ij}\,.$ (7.2)

We now define a similar, but chiral Abelian gauge theory. We separate the
fermionic content into left-handed and right-handed chirality parts,

$\psi_{R/L}=\mathbb{P}_{R/L}\psi,\quad\mathbb{P}_{R/L}=\frac{\mathbbm{1}\pm\gamma_{5}}{2},$
(7.3)

and allow only purely right-handed fermions to appear as dynamical
fields.343434This is a choice made to simplify the discussion. E.g. the U(1)Y
sector of the SM contains both left-handed and right-handed fermions with
different gauge quantum numbers. It could be treated similarly. The
4-dimensional and purely right-handed classical Lagrangian of the model then
reads

$\mathcal{L}_{\chi\text{QED}}=i\overline{\psi_{R}}_{i}\not{D}_{ij}{\psi_{R}}_{j}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^{2}-\bar{c}\partial^{2}c+\rho^{\mu}s{A_{\mu}}+\bar{R}^{i}s{{\psi_{R}}_{i}}+R^{i}s{\overline{\psi_{R}}_{i}},$
(7.4)

where the interaction, coupling only to the right-handed fermions, is defined
by the covariant derivative as

$D_{ij}^{\mu}=\partial^{\mu}\delta_{ij}+ieA^{\mu}{\mathcal{Y}_{R}}_{ij}\,.$
(7.5)

Emphasizing the similarity with the U(1)Y sector of the Standard Model we call
the generator ${\mathcal{Y}_{R}}_{ij}={\mathcal{Y}_{R}}_{i}\delta_{ij}$ the
hypercharge. It can be seen that the left-handed fermions $\psi_{L}$ are now
decoupled from the theory. In order to avoid triangle anomalies we need to
impose the following additional anomaly cancellation condition to the
hypercharge,

$\text{Tr}(\mathcal{Y}_{R}^{3})=0\,.$ (7.6)

Following Sec. 2, the non-vanishing BRST transformations for this model are

$\displaystyle s{A_{\mu}}$ $\displaystyle=\partial_{\mu}c\,,$ (7.7a)
$\displaystyle s{\psi_{i}}$
$\displaystyle=s{{\psi_{R}}_{i}}=-i\,e\,c\,{\mathcal{Y}_{R}}_{ij}{\psi_{R}}_{j}\,,$
(7.7b) $\displaystyle{}s{\overline{\psi}_{i}}$
$\displaystyle=s{\overline{\psi_{R}}_{i}}=-i\,e\,\overline{\psi_{R}}_{j}c{\mathcal{Y}_{R}}_{ji}\,$
(7.7c) $\displaystyle s{\overline{c}}$
$\displaystyle=B\equiv-\frac{1}{\xi}\partial A,$ (7.7d)

where $s$ is the nilpotent generator of the BRST transformations, which acts
as a fermionic differential operator. This 4-dimensional tree-level action

$S_{0}^{(4D)}=\int d^{4}x\,\mathcal{L}_{\chi\text{QED}}$ (7.8)

satisfies the following Slavnov-Taylor identity

$\mathcal{S}(S_{0}^{(4D)})=0\,,$ (7.9)

where the Slavnov-Taylor operator, with the field content we consider, was
already given in Eq. (2.116). At this point, we emphasize two additional
functional identities that hold in 4 dimensions and that were derived and
discussed in Sec. 2.6. The first is the ghost equation,

$\displaystyle\bigg{(}\frac{\delta}{\delta\bar{c}}+\partial_{\mu}\frac{\delta}{\delta\rho_{\mu}}\bigg{)}S_{0}^{(4D)}$
$\displaystyle=0\,.$ (7.10)

The second is the functional form of the abelian Ward identity353535Here we
keep the Nakanishi-Lautrup field $B(x)$ explicitly. However, one could
integrate it out here as well using $B=-(\partial_{\mu}A^{\mu})/\xi$.

$\displaystyle\bigg{(}\partial^{\mu}\frac{\delta}{\delta
A^{\mu}(x)}+ie{\mathcal{Y}}_{R}^{j}\sum_{\Psi}(-1)^{n_{\Psi}}\Psi(x)\frac{\delta}{\delta\Psi(x)}\bigg{)}S_{0}^{(4D)}$
$\displaystyle=-\Box B(x)\,,$ (7.11)

suitably adapted to the present theory $\chi$QED and its field content. The
summation extends over the charged fermions and their sources,
$\Psi\in\\{{\psi_{R}}_{j},\overline{\psi_{R}}_{j},R^{j},\bar{R}^{j}\\}$ and
$n_{\Psi}\in\\{0,1,0,1\\}$.

Functional relations such as the ghost equation and the local Ward identity
are part of the definition of our theory in 4 dimensions. Once we perform the
regularization and renormalization procedure, the requirement that those
identities still hold imposes important restrictions as we will soon see in
the explicit loop calculations. But first we extend the model to $D$
dimensions and examine the consequences of this extension.

####  7.2.2 Definition of Chiral Electrodynamics in DReg

We can immediately see that the extension of $\chi$QED to $D$ dimensions is
not unique due to the right-handed chiral current
$\overline{\psi_{R}}_{i}\gamma^{\mu}{\psi_{R}}_{j}$. The extension to $D$
dimensions of this term has three _inequivalent choices_ , each of them
_equally correct_ :

$\displaystyle\overline{\psi}_{i}\gamma^{\mu}\mathbb{P}_{R}\psi_{j}\,,$
$\displaystyle\overline{\psi}_{i}\mathbb{P}_{L}\gamma^{\mu}\psi_{j}\,,$
$\displaystyle\overline{\psi}_{i}\mathbb{P}_{L}\gamma^{\mu}\mathbb{P}_{R}\psi_{j}\,.$
(7.12)

They are different because
$\mathbb{P}_{L}\gamma^{\mu}\neq\gamma^{\mu}\mathbb{P}_{R}$ in $D$ dimensions.
Each of these choices leads to a valid $D$-dimensional extension of the model
that is renormalizable using dimensional regularization and the BMHV scheme
and is expected to produce the same final results in physical 4 dimensions
after the renormalization procedure is performed. However, the intermediate
calculations and the $D$-dimensional results will differ, depending on the
choice for this interaction term. The third option, which is equal to

$\overline{\psi}\mathbb{P}_{L}\gamma^{\mu}\mathbb{P}_{R}\psi=\overline{\psi}\mathbb{P}_{L}\overline{\gamma}^{\mu}\mathbb{P}_{R}\psi=\overline{\psi_{R}}\overline{\gamma}^{\mu}\psi_{R}\,,$
(7.13)

is the most symmetric one and leads to the simplest intermediate expressions.
Notice that this choice is actually the most straightforward one since it
preserves the information that right-handed fermions were present on the left
and on the right sides of the interaction term before the extension, see also
the review [15].

The second, more serious problem, is that as it stands the pure fermionic
kinetic term
$i\overline{\psi_{R}}_{i}\not{\partial}{\psi_{R}}_{i}=i\overline{\psi}_{i}\mathbb{P}_{L}\not{\partial}\mathbb{P}_{R}\psi_{i}$
projects only the purely $4$-dimensional derivative, leading to a purely
4-dimensional propagator

$\frac{i\,\mathbb{P}_{R}\,\not{p}\,\mathbb{P}_{L}}{\bar{p}^{2}},$ (7.14)

and to unregularized loop diagrams. As discussed in Sec. 3.4, the only valid
choice for the propagator in the $D$-dimensional theory in the context of
dimensional regularization is

$\frac{i\,\not{p}}{p^{2}},$ (7.15)

so we are thus led to consider the full Dirac fermion $\psi$ with both a left
and right-handed component, and use instead the fully $D$ dimensional
covariant kinetic term $i\overline{\psi}_{i}\not{\partial}{\psi}_{i}$. It can
be re-expressed in terms of projectors as follows:

$\begin{split}i\overline{\psi}_{i}\not{\partial}{\psi}_{i}&=i\overline{\psi}_{i}\overline{\not{\partial}}{\psi}_{i}+i\overline{\psi}_{i}\widehat{\not{\partial}}{\psi}_{i}\\\
&=i(\overline{\psi}_{i}\mathbb{P}_{L}\not{\partial}\mathbb{P}_{R}{\psi}_{i}+\overline{\psi}_{i}\mathbb{P}_{R}\not{\partial}\mathbb{P}_{L}{\psi}_{i})+i(\overline{\psi}_{i}\mathbb{P}_{L}\not{\partial}\mathbb{P}_{L}{\psi}_{i}+\overline{\psi}_{i}\mathbb{P}_{R}\not{\partial}\mathbb{P}_{R}{\psi}_{i})\end{split}$
(7.16)

Notice that the fictitious, sterile left-chiral field $\psi_{L}$ is
introduced, which appears only within the kinetic term and nowhere else, it
does not interact so it does not couple in particular to the gauge bosons of
the theory, and we enforce it to be invariant under gauge transformations.

Unfortunately, the choice of the $D$-dimensional propagator, crucial for loop
regularization, that led to the introduction to the left-handed component in
the kinetic term, breaks the gauge invariance of the fermionic part of the
Lagrangian, which is evident if we separate it in this way:

$\displaystyle\mathcal{L}_{\text{fermions}}$
$\displaystyle=\mathcal{L}_{\text{fermions,inv}}+\mathcal{L}_{\text{fermions,evan}}\,,$
(7.17a) $\displaystyle\mathcal{L}_{\text{fermions,inv}}$
$\displaystyle=i\overline{\psi}_{i}\overline{\not{\partial}}{\psi}_{i}-e{\mathcal{Y}_{R}}_{ij}\overline{\psi_{R}}_{i}\not{A}{\psi_{R}}_{j}\,,$
(7.17b) $\displaystyle\mathcal{L}_{\text{fermions,evan}}$
$\displaystyle=i\overline{\psi}_{i}\widehat{\not{\partial}}{\psi}_{i}\,,$
(7.17c)

where the first term contains purely 4-dimensional derivatives and gauge
fields and preserves the gauge and BRST invariance, since the fictitious left-
chiral field $\psi_{L}$ is a gauge singlet. The invariant term can also be
written as a sum of purely left-chiral and purely right-chiral terms involving
the 4-dimensional covariant derivative as

$\displaystyle\mathcal{L}_{\text{fermions,inv}}$
$\displaystyle=i\overline{\psi_{L}}_{i}\overline{\not{\partial}}{\psi_{L}}_{i}+i\overline{\psi_{R}}_{i}\overline{\not{\partial}}{\psi_{R}}_{i}-e{\mathcal{Y}_{R}}_{ij}\overline{\psi_{R}}_{i}\not{A}{\psi_{R}}_{j}$
(7.18a)
$\displaystyle=i\overline{\psi_{L}}_{i}\overline{\not{\partial}}{\psi_{L}}_{i}+i\overline{\psi_{R}}_{i}\overline{\not{D}}{\psi_{R}}_{i}\,,$
(7.18b)

where the gauge invariance is obvious. The second term in Eq. 7.17a is purely
evanescent, i.e. it vanishes in 4-dimensional limit. If we rewrite the
evanescent term as

$\displaystyle\mathcal{L}_{\text{fermions,evan}}$
$\displaystyle=i\overline{\psi_{L}}_{i}\widehat{\not{\partial}}{\psi_{R}}_{i}+i\overline{\psi_{R}}_{i}\widehat{\not{\partial}}{\psi_{L}}_{i}\,,$
(7.19)

it can be easily seen that it mixes left- and right-chiral fields with
different gauge transformation properties. This causes the breaking of gauge
and BRST invariance — the central difficulty of the BMHV scheme.363636 We
remark that the problem is not specific to the case where the left-handed
fermion is sterile. As Eq. (7.19) shows the problem generally exists if the
left-handed and right-handed fermions have different gauge quantum numbers.
Refs. [131, 27] consider this case and end up with essentially the same
breaking of BRST invariance in $D$ dimensions and the same further
consequences.

We can summarize this symmetry property and the symmetry breaking as

$\displaystyle s_{D}\mathcal{L}_{\text{fermions,inv}}$ $\displaystyle=0\,,$
(7.20a) $\displaystyle s_{D}\mathcal{L}_{\text{fermions,evan}}$
$\displaystyle\neq 0\,,$ (7.20b)

where $s_{D}$ is the obvious extension of the BRST operator (7.7) to $D$
dimensions.

Since the extension of BRST transformation of fields in $D$ dimensions is
straightforward, our $D$ dimensional action is then

$\begin{split}S_{0}=\;&\int
d^{D}x\bigg{(}i\overline{\psi}_{i}\not{\partial}{\psi}_{i}+e{\mathcal{Y}_{R}}_{ij}\overline{\psi_{R}}_{i}\not{A}{\psi_{R}}_{j}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^{2}\\\
&-\bar{c}\partial^{2}c+\rho^{\mu}(\partial_{\mu}c)+i\,e\,\bar{R}^{i}c\,{\mathcal{Y}_{R}}_{ij}{\psi_{R}}_{j}+i\,e\,\overline{\psi_{R}}_{i}c{\mathcal{Y}_{R}}_{ij}R^{j}\bigg{)}\,\\\
\equiv\;&\sum_{i}S^{i}_{\overline{\psi}\psi}+\sum_{i}\overline{S^{i}_{\overline{\psi}_{R}A\psi_{R}}}+S_{AA}+S_{\text{g-fix}}+S_{\bar{c}c}+S_{\rho
c}+S_{\bar{R}c\psi_{R}}+S_{Rc\overline{\psi_{R}}}\,,\end{split}$ (7.21)

where also useful abbreviations for the individual terms were introduced.
Similar to the fermion Lagrangian, the full $D$-dimensional action may be
written as the sum of two parts, an “invariant” and an “evanescent” part,

$\displaystyle S_{0}$ $\displaystyle=S_{0,\text{inv}}+S_{0,\text{evan}}\,,$
(7.22a) $\displaystyle S_{0,\text{evan}}$ $\displaystyle=\int
d^{D}x\,i\overline{\psi}_{i}\widehat{\not{\partial}}{\psi}_{i}\,.$ (7.22b)

The second part $S_{0,\text{evan}}$ consists solely of one single, evanescent
fermion kinetic term, the remnant of the $D$-dimensional propagator.

Now we quantify the symmetry breaking caused by the BMHV scheme, the non-
anticommuting $\gamma_{5}$ and the resulting evanescent term in the action.
Acting with the $D$-dimensional BRST operator on the $D$-dimensional tree-
level action Eq. 7.21 gives:

$s_{D}S_{0}=s_{D}S_{0,\text{inv}}+s_{D}S_{0,\text{evan}}=0+s_{D}\int
d^{D}x\,i\overline{\psi}_{i}\widehat{\not{\partial}}{\psi}_{i}\equiv\widehat{\Delta},$
(7.23)

where the non-vanishing integrated breaking term $\widehat{\Delta}$ is given
by

$\widehat{\Delta}=-\int
d^{D}xe\,{\mathcal{Y}_{R}}_{ij}\,c\,\left\\{\overline{\psi}_{i}\left(\overset{\leftarrow}{\widehat{\not{\partial}}}\mathbb{P}_{R}+\overset{\rightarrow}{\widehat{\not{\partial}}}\mathbb{P}_{L}\right)\psi_{j}\right\\}\equiv\int
d^{D}x\,\widehat{\Delta}(x).$ (7.24)

Acting with the $D$-dimensional Slavnov-Taylor operator $\mathcal{S}_{D}$ on
the tree-level action, we obtain

$\mathcal{S}_{D}(S_{0})={s}_{D}S_{0,\text{inv}}+{s}_{D}S_{0,\text{evan}}=0+\widehat{\Delta}\,,$
(7.25)

hence the Slavnov-Taylor identity in $D$ dimensions is violated by the same
BRST breaking term at tree-level.373737 The simpler linear equations (2.110)
to (2.113) specific for Abelian theories are manifestly valid also in $D$
dimensions. We will not discuss them further, but they have the consequence
that higher-order corrections, including counterterm actions, cannot depend on
the ghost/antighost and source fields. For this reason, the linearized
Slavnov-Taylor operator here reduces to BRST transformations, $b_{D}=s_{D}$.

As mentioned in the overview Sec. 7.1, this breaking term will be a crucial
tool in practical calculations. This breaking will be used as a composite
operator insertion in Feynman diagrams. It generates an interaction vertex
whose Feynman rule (with all momenta incoming and derived from the combination
$i\widehat{\Delta}$) is:

| $\begin{aligned}
&=-\frac{e}{2}\,{\mathcal{Y}_{R}}_{ij}\left((\widehat{\not{p_{1}}}+\widehat{\not{p_{2}}})+(\widehat{\not{p_{1}}}-\widehat{\not{p_{2}}})\gamma_{5}\right)_{\alpha\beta}\\\
&=-e\,{\mathcal{Y}_{R}}_{ij}\left(\widehat{\not{p_{1}}}\mathbb{P}_{R}+\widehat{\not{p_{2}}}\mathbb{P}_{L}\right)_{\alpha\beta}\,.\end{aligned}$
---|---
(7.26)

As discussed in the context of Eq. (2.90), in this way the functional
derivatives of $i\widehat{\Delta}\cdot\Gamma$ correspond to 1PI Feynman
diagrams with one insertion of the Feynman rule (7.26). An analogous Feynman
rule is derived for charge-conjugated fermions.

It is important to notice that this breaking $\widehat{\Delta}$ is evanescent,
i.e. it vanishes in the 4-dimensional limit. This results from the evanescent
original term (7.22b) and has the consequence that insertions of
$\widehat{\Delta}$ can only contribute in power-counting divergent Feynman
diagrams.

###  7.3 Symmetry Restoration Requirements

Before beginning the explicit calculations we recall and collect the required
symmetry identities and the strategy for symmetry restoration in a more
technical way than in the overview Sec. 7.1. We begin by collecting the
required symmetry identities.

Symmetry identities expressing gauge/BRST invariance are considered part of
the definition of the theory. Hence they are required to be fulfilled at all
orders, see Secs. 2.6 and 6.2 for detailed discussions.

The symmetry requirements are defined for the renormalized and finite
4-dimensional effective action of the form

$\Gamma_{\text{ren}}=S_{0}^{(4D)}+\mathcal{O}(\hbar),$ (7.27)

where we again highlight that the effective action coincides with the
classical action up to higher-order corrections, and that loop corrections are
of higher order in $\hbar$, see Eq. (2.86) and Sec. 3. The first symmetry
requirement is BRST (and underlying gauge) invariance, which is expressed as
the Slavnov-Taylor identity

$\mathcal{S}(\Gamma_{\text{ren}})=0,$ (7.28)

for the renormalized theory. Notice that in $\chi$QED the fields $c$,
$\bar{c}$ and $\rho^{\mu}$ do not have higher order corrections, so relations

$\displaystyle\frac{\delta\Gamma_{\text{ren}}}{\delta c(x)}$
$\displaystyle=\frac{\delta S_{0}^{(4D)}}{\delta c(x)}\,,$
$\displaystyle\frac{\delta\Gamma_{\text{ren}}}{\delta\bar{c}(x)}$
$\displaystyle=\frac{\delta S_{0}^{(4D)}}{\delta\bar{c}(x)}\,,$
$\displaystyle\frac{\delta\Gamma_{\text{ren}}}{\delta\rho^{\mu}(x)}$
$\displaystyle=\frac{\delta S_{0}^{(4D)}}{\delta\rho^{\mu}(x)}\,.$ (7.29)

hold trivially, since the respective derivatives of the tree-level action are
linear in the dynamical fields as described in Sec. 2.6. The fact that the
ghost does not have higher loop corrections will play a part in reducing the
number of diagrams appearing in higher orders, compared to an analogous Yang-
Mills theory. The local Ward identity

$\bigg{(}\partial^{\mu}\frac{\delta}{\delta
A^{\mu}(x)}+ie{\mathcal{Y}}_{R}^{j}\sum_{\Psi}(-1)^{n_{\Psi}}\Psi(x)\frac{\delta}{\delta\Psi(x)}\bigg{)}\Gamma_{\text{ren}}=-\Box
B(x)\,,$ (7.30)

is an automatic consequence of the Slavnov-Taylor identity as we have shown in
Sec. 2.6.

We record here the application of the Ward identity to the photon self-energy
as an example that will later be illustrated in explicit computations. If we
rewrite the Ward identity in the momentum-space representation and take a
variation with the respect to photon field, we obtain the requirement

$ip_{\nu}\frac{\delta^{2}{\widetilde{\Gamma}_{\text{ren}}}}{\delta{A_{\mu}(p)}\delta{A_{\nu}(-p)}}=0\,,$
(7.31)

which corresponds to the transversality of the photon self-energy.

All previous symmetry identities must hold after regularization and
renormalization at each loop order. If the symmetries are broken in the
intermediate regularization procedure, as is the case when we use the BMHV
scheme, they must be restored order by order in perturbation theory, by adding
suitable counterterms.

The symmetry identities are covered by the general analysis of algebraic
renormalization discussed in Sec. 6.2.3, and the theory has no gauge anomaly,
see Eq. (7.6). This guarantees that the procedure of symmetry restoration
works at all orders.

Now we recapitulate the practical strategies for the concrete determination of
symmetry-restoring counterterms, following the detailed outline given in Sec.
6.3. The application will be discussed in the subsequent subsections, where we
treat not only the chiral model $\chi$QED but also compare it with the
familiar case of ordinary QED to highlight the features of the BMHV treatment
of $\gamma_{5}$.

The first obvious difference is that ordinary QED is a vector-like gauge
theory, and DReg preserves all relevant symmetry identities manifestly at each
step: the counterpart to the tree-level breaking $\widehat{\Delta}$ in Eq.
(7.24) vanishes as already discussed in Sec. 4.3. Hence generating
counterterms by a renormalization transformation is sufficient, see the
discussion in Sec. 6.1 and Eq. (6.3).

For the case of $\chi$QED, the existence of a tree-level symmetry breaking,
$\widehat{\Delta}\neq 0$, necessitates symmetry-restoring counterterms. Hence,
generating counterterms by a renormalization transformation is not sufficient,
and the general structure is the one discussed in Sec. 6.3.3, i.e. the
combination

$S_{\text{sct,inv}}+S_{\text{sct,non-
inv}}+S_{\text{fct,inv}}+S_{\text{fct,restore}}+S_{\text{fct,evan}}\,.$ (7.32)

Sec. 6.3 presented two basic strategies to carry out the required computations
of the crucial symmetry-restoring counterterms $S_{\text{fct,restore}}$. The
first is based on the explicit computation of ordinary Green functions and
explicitly checking symmetry identities. Its essential equation is Eq. (6.37),
which requires computing

$\mathcal{S}_{D}(\Gamma_{\mathrm{subren}}^{(n)}+S_{\text{sct}}^{n})$

at each new order $n$. If this expression is non-zero, finite counterterms
have to be found and added to the action such that the symmetry breaking is
canceled.

The second strategy is based on using the regularized quantum action principle
and represented by Eq. (6.42),

$\mathop{\text{LIM}}_{D\,\to\,4}\,\left(\widehat{\Delta}\cdot\Gamma_{\mathrm{DRen}}^{n}+\sum_{k=1}^{n-1}\Delta^{k}_{\mathrm{ct}}\cdot\Gamma^{n-k}_{\mathrm{DRen}}+\Delta^{n}_{\mathrm{ct}}\right)=0.$

The computation of full Green functions and evaluating Slavnov-Taylor
identities is replaced by the computation of Green functions with insertions
of breaking operators such as $\widehat{\Delta}$. This equation is specialized
to Eqs. (6.44) and (6.46) at the one- and two-loop level.

In the following subsections we will illustrate Feynman diagrammatic
computations for both strategies. The more efficient second strategy is
illustrated also at the two-loop level. We will then see how the desired
symmetry-restoring counterterms are determined.

###  7.4 Explicit Calculations and Results in the Abelian Chiral Gauge Theory

In this section explicit calculations in the Abelian chiral gauge theory
defined above in Sec. 7.2 are performed in the BMHV scheme of DReg and all
necessary counterterms are provided up to the two-loop level. In particular,
the evaluation of the photon self-energy at the one-loop (sections 7.4.1 and
7.4.2) and the two-loop level (Sec. 7.4.3) is highlighted and the results are
compared to ordinary QED. As announced in Sec. 7.1, there are two different
ways of determining symmetry-restoring counterterms. While the method in Sec.
7.4.1 amounts to the explicit evaluation of the full photon self-energy, i.e.
a full Green function, including its finite part, Sec. 7.4.2 employs the
direct method based on the regularized quantum action principle where the
symmetry breaking is determined via special Feynman diagrams with an insertion
of the $\widehat{\Delta}$-operator, which reflects the breaking of chiral
gauge invariance. Sec. 7.4.2 then concludes by providing the full one-loop
counterterm action for chiral QED in the BMHV scheme. Similarly, in Sec. 7.4.3
the two-loop counterterms for the photon self-energy are obtained using the
latter method based on the regularized quantum action principle but are
verified by comparing with the explicit result for the full photon self-energy
including its finite part. Concluding, Sec. 7.4.4 provides the full two-loop
renormalization of chiral QED in the BMHV scheme.

####  7.4.1 One-Loop Photon Self-Energy and Symmetry-Restoring counterterms

To better understand the features of the BMHV scheme, we now focus on explicit
loop calculations. We take the photon self-energy and compare its results in
ordinary QED and chiral QED. The photon self-energy is subject to the simplest
Ward identity (7.31) — it must be transverse, to guarantee the correct
physical interpretation of the theory describing a massless spin 1 particle
with two transverse polarizations.

The photon self-energy is denoted as

$\begin{aligned} i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)\,=\,\end{aligned}$ | $\begin{aligned} .\end{aligned}$
---|---

We use the notation explained in Sec. 2.4, corresponding to the 1-particle
irreducible diagrams with external fields and momentum as indicated.383838
However, in this subsection we use a slightly simpler notation than in Sec.
3.1 for unrenormalized/subrenormalized expressions. We drop the subscript
${}_{\text{subren}}$ and simply write $\Gamma^{1}$ for the unrenormalized one-
loop effective action and $\Gamma^{2}$ for the subrenormalized two-loop
effective action. Accordingly, the following equations correspond to the
unrenormalized one-loop photon self-energy.

We begin by recalling the well-known one-loop result of ordinary QED with
massless fermions as defined in Eq. (7.1),

$\displaystyle i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{div,QED}}=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}\epsilon}\frac{4\,\text{Tr}(Q^{2})}{3}(p^{\mu}p^{\nu}-p^{2}g^{\mu\nu})\,,$
(7.33a) $\displaystyle
i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{fin,QED}}=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}}\frac{2\,\text{Tr}(Q^{2})}{3}\left[\left(\frac{10}{3}-2\ln(-p^{2})\right)({p}^{\mu}{p}^{\nu}-{p}^{2}{g}^{\mu\nu})\right]\,.$
(7.33b)

Here and in all following results we set $D=4-2\epsilon$ and suppress the
dimensional regularization scale $\bar{\mu}^{2}=\mu^{2}\,4\pi e^{-\gamma_{E}}$
in dimensionful logarithms. We see that the result is transverse and satisfies
the Ward identity (7.31) both in its divergent and finite parts. Adding the
counterterm action

$\displaystyle S_{\text{sct,QED}}^{1}=\;$
$\displaystyle\frac{-\hbar\,e^{2}}{16\pi^{2}\epsilon}\frac{4\text{Tr}(Q^{2})}{3}S_{AA}+\ldots\,,$
(7.34)

where the dots denote terms unrelated to the photon self-energy, cancels the
divergences and preserves the validity of the Ward identity. The factor
$\hbar$ was explicitly restored to highlight that the counterterm action is of
one-loop order. As is well known, this counterterm action can be generated via
a photon field renormalization transformation.

In comparison, the result for the one-loop photon self-energy diagram in
$\chi$QED with massless fermions as defined in Eq. (7.4) reads

$\displaystyle
i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{div,$\chi$QED}}=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}\epsilon}\frac{2\,\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\left[(\overline{p}^{\mu}\overline{p}^{\nu}-\overline{p}^{2}\overline{g}^{\mu\nu})-\frac{1}{2}\widehat{p}^{2}\overline{g}^{\mu\nu}\right]\,,$
(7.35a) $\displaystyle
i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{fin,$\chi$QED}}=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\Bigg{[}\left(\frac{10}{3}-2\ln(-p^{2})\right)(\overline{p}^{\mu}\overline{p}^{\nu}-\overline{p}^{2}\overline{g}^{\mu\nu})$
$\displaystyle\qquad\qquad\qquad-\bigg{(}\overline{p}^{2}+\widehat{p}^{2}\Big{(}\frac{8}{3}-\ln(-p^{2})\Big{)}\bigg{)}\overline{g}^{\mu\nu}\Bigg{]}\,.$
(7.35b)

From this illustrative example, we can extract several interesting comments.
First, and most obviously, transversality is violated by the last terms in
Eqs. (7.35). This will be our main focus. But also the transverse part shows
two differences compared to ordinary QED. Since the interaction vertex in
$\chi$QED differs from the one given in standard QED by

$\displaystyle V_{\text{QED}}$ $\displaystyle\to-ie\gamma^{\mu}Q_{ij},$
$\displaystyle V_{\text{$\chi$QED}}$ $\displaystyle\to-
ie\bar{\gamma}^{\mu}\mathbb{P}_{R}\mathcal{Y}_{R,ij},$ (7.36)

it projects the fermion loop content, so the transverse part becomes purely
$4$-dimensional, explaining the appearance of the covariants
$\overline{g}^{\mu\nu}$ and $\overline{p}^{\mu}$ in Eqs. (7.35). Further, due
to this projection only half the number of fermionic degrees of freedom appear
in the loop for the chiral case, resulting in the relative factor of 2 with
respect to ordinary QED.

Let us now focus on the breaking of transversality in the photon self-energy.
The divergent breaking term in Eq. (7.35a) is proportional to
$\widehat{p}^{2}$, i.e. it is evanescent. In contrast, the finite breaking
term in Eq. (7.35b) contains finite expressions that do not vanish in the
4-dimensional limit. The finite breaking also contains evanescent terms that
vanish in the $\mathop{\text{LIM}}_{D\to 4}$; these will be ignored in the
following.

We can exhibit the breaking explicitly by plugging the photon self-energy into
the Ward identity (7.31); we obtain

$\displaystyle
ip_{\nu}\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{div+fin,$\chi$QED}}$
$\displaystyle=\frac{ie^{2}}{16\pi^{2}}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\Bigg{[}-\frac{1}{\epsilon}\widehat{p}^{2}\overline{p}^{\mu}-\overline{p}^{2}\overline{p}^{\mu}\Bigg{]}\neq
0\,.$ (7.37)

Here we have ignored the finite, evanescent term, as announced. In line with
the derivation of the Ward identity from the Slavnov-Taylor identity via
derivatives with respect to a ghost field, see Eq. (2.117), the result is
equivalent to the violation of the Slavnov-Taylor identity

$\displaystyle[\mathcal{S}(\Gamma)]_{A^{\mu}c}^{1}$
$\displaystyle=\frac{ie^{2}}{16\pi^{2}}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\Bigg{[}-\frac{1}{\epsilon}\widehat{p}^{2}\overline{p}^{\mu}-\overline{p}^{2}\overline{p}^{\mu}\Bigg{]}\,,$
(7.38)

where the left hand side denotes functional derivatives in momentum space,
similarly to the notation of $\Gamma_{AA}$.

A decisive feature of the breaking terms is their locality — the breaking
terms in all the previous equations are polynomials of the momentum in
momentum space, and this translates into local expressions on the level of the
(effective) action. This locality is in line with the general statement
discussed in Sec. 6.2.3 which forms the basis of algebraic renormalization.
This means that a local counterterm can be defined that cancels the symmetry
breaking.

In view of the explicit results, the required counterterms for the sector of
the photon self-energy can be read off as follows. We first discuss the
divergent counterterms. The divergent counterterms can be split into an
invariant and a non-invariant part as in Eq. (7.32) as
$S_{\text{sct}}=S_{\text{sct,inv}}+S_{\text{sct,non-inv}}$ such that the one-
loop parts relevant for the photon self-energy in $\chi$QED read

$\displaystyle S_{\text{sct,inv,$\chi$QED}}^{1}$
$\displaystyle=\frac{-\hbar\,e^{2}}{16\pi^{2}\epsilon}\frac{2\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\overline{S_{AA}}+\ldots\,,$
(7.39a) $\displaystyle S_{\text{sct,non-inv,$\chi$QED}}^{1}$
$\displaystyle=\frac{-\hbar\,e^{2}}{16\pi^{2}\epsilon}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\int
d^{D}x\,\frac{1}{2}\bar{A}_{\mu}\widehat{\partial}^{2}\bar{A}^{\mu}+\ldots\,,$
(7.39b)

where the dots denote terms unrelated to the photon self-energy. As in the
case of ordinary QED, the divergences are canceled, and the invariant
counterterm can be generated via a photon field renormalization
transformation. In contrast to ordinary QED, however, the non-invariant term
is required, and it cannot be obtained from a renormalization transformation
but must be read off by hand.

Obviously, adding these counterterms does not only cancel the divergences of
the photon self-energy but it also cancels the divergences in the breaking of
the Ward/Slavnov-Taylor identities (7.37,7.38). Specifically adding the
counterterms to the action modifies the Slavnov-Taylor identity
$\mathcal{S}(\Gamma)$ to
$\mathcal{S}(\Gamma+S_{\text{sct,$\chi$QED}}^{1})=\mathcal{S}(\Gamma)+s_{D}S_{\text{sct,$\chi$QED}}^{1}+\ldots$,
where the dots denote higher-order terms and where

$\begin{split}s_{D}S_{\text{sct,$\chi$QED}}^{1}=\Delta_{\text{ct}}^{1}\big{|}_{\text{div}}&=-\frac{\hbar}{16\pi^{2}\epsilon}\frac{e^{2}\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\,\int
d^{D}x\,(\overline{\partial}_{\mu}c)\,(\widehat{\partial}^{2}\bar{A}^{\mu})\,.\end{split}$
(7.40)

In momentum-space, with incoming $A^{\mu}$ momentum $p$, this is precisely the
negative of the divergent term in Eq. (7.38). This is an automatic consequence
of the finiteness.

Now we discuss the required finite counterterms to the photon self-energy. The
explicit result (7.35b) shows that the transversality is restored by the
following finite counterterm:

$\displaystyle S^{1}_{\text{fct,$\chi$QED}}$
$\displaystyle=\frac{\hbar}{16\pi^{2}}\int
d^{4}x\frac{-e^{2}\text{Tr}(\mathcal{Y}_{R}^{2})}{6}\bar{A}_{\mu}\overline{\partial}^{2}\bar{A}^{\mu}+\ldots$
(7.41)

In momentum space this counterterm cancels the non-transverse
$\overline{p}^{2}$-term of (7.35b) (we recall that the remaining non-
transverse finite terms are evanescent and vanish in the
$\mathop{\text{LIM}}_{D\to 4}$). On the level of the Slavnov-Taylor identity,
adding the finite counterterm modifies the Slavnov-Taylor identity
$\mathcal{S}(\Gamma)$ by the term

$\displaystyle s_{D}S^{1}_{\text{fct,$\chi$QED}}$
$\displaystyle=-\frac{\hbar}{16\pi^{2}}\int
d^{D}x\,\frac{e^{2}\text{Tr}(\mathcal{Y}_{R}^{2})}{3}(\overline{\partial}_{\mu}c)(\overline{\partial}^{2}\bar{A}^{\mu})\,.$
(7.42)

In momentum space, this is the negative of the finite term in Eq. (7.38).

In total, after adding all counterterms (7.39,7.41) to the photon self-energy
and taking the $\mathop{\text{LIM}}_{D\to 4}$, the renormalized one-loop
photon self-energy is

$\displaystyle i\widetilde{\Gamma}_{AA}^{\nu\mu}(p)|^{1}_{\text{ren,
$\chi$QED}}=$
$\displaystyle\frac{ie^{2}}{16\pi^{2}}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\Bigg{[}\left(\frac{10}{3}-2\ln(-p^{2})\right)(\overline{p}^{\mu}\overline{p}^{\nu}-\overline{p}^{2}\overline{g}^{\mu\nu})\Bigg{]}\,.$
(7.43)

It is finite, defined in 4 dimensions, and it is properly transverse. One may
still add further, finite, symmetric counterterms. These can be derived from
usual field and parameter renormalization but are not our focus here.

####  7.4.2 One-Loop Photon Self-Energy — Direct Computation of Symmetry
Breaking

In the previous subsection we determined the required counterterms (7.39,7.41)
by carrying out an explicit computation of a Green function, including its
finite part, and by explicitly evaluating the breaking of the relevant
symmetry identity. We now show how the determination of the counterterms can
be performed in a simpler way. We still illustrate it for the one-loop photon
self-energy, but the advantage of that simplification will become more and
more prominent for higher orders and more complicated Green functions.

Instead of evaluating the full photon self-energy including its finite part
(7.35), the following is sufficient: First we need the divergent part of the
photon self-energy, i.e. only (7.35a). This of course determines the divergent
counterterms (7.39) unambiguously.

Second, we need the violation of the symmetry, expressed in terms of Eq.
(7.38). This violation can be obtained in a more direct way, by using the
regularized quantum action principle discussed in Sec. 4. This tells us that
the violation $\mathcal{S}(\Gamma)\neq 0$ is directly given by diagrams with
insertions of the composite operator $\widehat{\Delta}$, corresponding to the
tree-level violation of the Slavnov-Taylor identity in $D$ dimensions. For the
photon self-energy, the violation (7.38) can be obtained directly by computing
the Green function $[\widehat{\Delta}\cdot\widetilde{\Gamma}_{Ac}^{\mu}]$,
i.e. the 1-particle irreducible Green function with an insertion of
$\widehat{\Delta}$ and external $A^{\mu}$ and $c$ fields.

At one-loop order there is only one diagram.
$i[\widehat{\Delta}\cdot\widetilde{\Gamma}_{Ac}^{\mu}]^{(1)}\,=$ (7.44)
The result of this single diagram is

$\displaystyle
i[\widehat{\Delta}\cdot\widetilde{\Gamma}_{Ac}^{\mu}]^{1}_{\text{div}}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}\epsilon}\,\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\widehat{p_{1}}^{2}\overline{p_{1}}^{\mu}\,,$
(7.45a) $\displaystyle
i\,[\widehat{\Delta}\cdot\widetilde{\Gamma}_{Ac}^{\mu}]^{1}_{\text{fin}}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\overline{p}_{1}^{2}\overline{p}^{\mu}_{1}\;.$
(7.45b)

We see that the result of this diagram indeed agrees with the right-hand side
of Eq. (7.38), as it is guaranteed by the regularized quantum action
principle.

The important point is the technical simplification: the computation of this
diagram is technically easier than the computation of the finite part of the
photon self-energy since only power-counting divergent parts of the loop
integrals are relevant. We reiterate that the technical advantage is much more
dramatic at higher orders and for more complicated Green functions.

It is instructive to rewrite the result in coordinate space,

$\displaystyle[\widehat{\Delta}\cdot\Gamma]^{(1)}_{\text{div}}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}\epsilon}\,\frac{\text{Tr}(\mathcal{Y}_{R}^{2})}{3}\int