Any consistent regularization scheme induces an apparent violation of gauge
invariance in non-anomalous chiral gauge theories. This violation shows up in
amplitudes evaluated in perturbation theory and can be removed by the
inclusion of finite counterterms. In this context, renormalization is more
sophisticated than in a vector-like gauge theory. Two steps can be
distinguished in the subtraction procedure. A first one is required to remove
infinities. At a given order in perturbation theory, this can be done by
adding a set of local divergent counterterms. At this stage, the theory
delivers finite results, but the corresponding amplitudes do not preserve
gauge invariance in general. Indeed, the latter is broken by finite terms that
can be systematically deleted by adding local finite counterterms.

The two steps can be reiterated at each order of perturbation theory and can
be implemented directly at the level of the generating functional of the 1PI
Green’s functions of the theory. Starting from the regularized functional
$\Gamma^{\rm reg}[\phi]$, divergencies are canceled by the local counterterm
$\Gamma_{\rm ct}[\phi]$, such that $\Gamma[\phi]=\Gamma^{\rm
reg}[\phi]+\Gamma_{\rm ct}[\phi]$ produces finite results. By further adding
the local finite counterterm $S_{\rm ct}[\phi]$, we finally get the functional
$\Gamma_{\rm inv}[\phi]=\Gamma[\phi]+S_{\rm ct}[\phi]$ that satisfies the WI
of the theory. Of course, such a separation of the subtraction procedure into
two moves is purely conventional. What matters is the overall combination
$\Gamma_{\rm ct}[\phi]+S_{\rm ct}[\phi]$, which can be split into the sum of a
divergent term and a finite one in infinitely many ways. In practical
computation, however, the two above steps appear to be very convenient and
have been adopted in our approach.

The main result of this work is a general analytic expression of the finite
one-loop counterterm $S_{\rm ct}[\phi]$ for a renormalizable chiral gauge
theory including gauge bosons and fermions transforming in arbitrary
representations of the gauge group. A very appealing feature of this result is
that the counterterm $S_{\rm ct}[\phi]$ is determined for any possible
consistent regulator belonging to a wide class. We only require that the
chosen regularization scheme obeys the Quantum Action Principle, preserves
Lorentz invariance in four dimensions, and gauge invariance when the theory is
vector-like. The physical information is entirely encoded in the gauge
variation $L_{a}\Gamma[\phi]$.232323The dependence of $S_{\rm ct}[\phi]$ on
the subtraction procedure is specified by $L_{a}\Gamma[\phi]=L_{a}(\Gamma^{\rm
reg}[\phi]+\Gamma_{\rm ct}[\phi])$. This can be expressed as a linear
combination of local operators of dimension four, whose coefficients can be
determined by a one-loop computation for each given regularization scheme. The
counterterm $S_{\rm ct}[\phi]$ automatically follows from the knowledge of
these coefficients.

We started by quantizing the theory with the Background Field Method and by
choosing the Background Field Gauge, which guarantees the gauge invariance of
the functional $\Gamma_{\rm inv}[\phi]$ at the level of background fields. In
this respect, we differ from previous approaches, where the theory is
quantized with the help of a traditional gauge fixing that breaks the gauge
symmetry down to the rigid BRST invariance. The WI of the functional
$\Gamma_{\rm inv}[\phi]$ resulting from the Background Field Gauge are easier
to deal with compared to the non-linear Slavnov-Taylor identities consequences
of the BRST invariance: they simply read $L_{a}\Gamma_{\rm inv}[\phi]=0$.

A key ingredient of our derivation is the non-redundant parametrization of the
gauge variation $L_{a}\Gamma[\phi]$ at the one-loop order, which has been
established independently from the adopted regularization by exploiting
several properties of the theory. The Quantum Action Principle guarantees
that, order by order in perturbation theory, $L_{a}\Gamma[\phi]$ is a finite
local polynomial in the fields and their derivatives preserving the symmetries
of the regulator. Last but not least, the WZ consistency conditions greatly
reduce the number of independent coefficients needed to describe
$L_{a}\Gamma[\phi]$. Similar considerations restrict the form of the sought-
after counterterm $S_{\rm ct}[\phi]$. Its analytic expression can be fully
determined in complete generality – up to gauge-invariant contributions – from
the equality $L_{a}(\Gamma[\phi]+S_{\rm ct}[\phi])=0$.

One of the most widely used regularization in practical computation is DR and
an important part of our work has been devoted to specifying our general
results to such a scheme. Within a path-integral formalism, we have computed
the gauge variation of the whole one-loop renormalized functional
$\Gamma[\phi]$ in the BMHV scheme. The result was also reproduced in several
parts via a diagrammatic computation. The full set of one-loop finite
counterterms in DR for the class of theories under investigation has been
obtained and is compactly summarized in Eq. (4.3.4).

To exemplify our result, we have computed the one-loop finite counterterm for
the SM in the limit of vanishing Yukawa couplings, when DR and the BMHV scheme
for $\gamma_{5}$ are chosen. This can be seen as a first step toward the
automation of one-loop computations in an even more general class of theories
such as chiral gauge theories including a scalar sector, like the SM, or non-
renormalizable ones, such as the SMEFT. The need for local counterterms
restoring gauge invariance in SMEFT one-loop computations have already been
emphasized [27, 28, 29] and we are confident that our approach, suitably
generalized, can represent a useful tool in this context.

## Acknowledgements

The research of C.C. was supported by the Cluster of Excellence Precision
Physics, Fundamental Interactions, and Structure of Matter (PRISMA+ \- EXC
2118/1) within the German Excellence Strategy (project ID 39083149).

##  Appendix A General solution of the Wess-Zumino conditions

### A.1 Wess-Zumino consistency conditions in terms of the $C^{k}_{A}$

#### A.1.1 Bosonic sector

#### P-even sector

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ C^{0}_{[pc]}=0$
$\displaystyle\left.\left(C_{cbp}^{3}+f_{cbe}C^{0}_{pe}\right)\right|_{\textrm{symm.
in pc}}=0$ $\displaystyle\leavevmode\nobreak\
(C^{3}_{pbc}+f_{pbe}C^{0}_{ce})+(C^{4}_{cpb}+C^{5}_{cpb})-(C^{4}_{pcb}+C^{5}_{pcb})=0$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
2(C^{3}_{pbc}+f_{pbe}C^{0}_{ce})+(C^{2}_{cpb}+C^{2}_{pcb})-2(C^{4}_{pcb}+C^{5}_{pcb})-4C^{6}_{pcb}=0$
$\displaystyle\leavevmode\nobreak\
(C^{3}_{pbc}+f_{pbe}C^{0}_{ce})-(C^{2}_{cpb}+C^{2}_{pcb})+C^{3}_{cpb}+C^{3}_{pcb}-2(C^{4}_{pcb}+C^{5}_{pcb})=0$
$\displaystyle\leavevmode\nobreak\
(C^{3}_{pbc}+f_{pbe}C^{0}_{ce})+(C^{3}_{pcb}+f_{pce}C^{0}_{eb})-2(C^{4}_{pcb}+C^{5}_{pcb})-2C^{6}_{pcb}=0$
$\displaystyle\left.\left(C^{3}_{cbe}f_{pde}-C^{3}_{pde}f_{cbe}-C^{2}_{cbe}f_{pde}+C^{2}_{pde}f_{cbe}\right)\right|_{\textrm{symm.
in bd}}=2C^{7}_{[pc](bd)}$
$\displaystyle\left.\left(C^{2}_{cbe}f_{pde}-C^{2}_{pde}f_{cbe}\right)\right|_{\textrm{symm.
in bd}}=2C^{8}_{[pc](bd)}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
C^{2}_{cbe}f_{pde}-C^{2}_{pde}f_{cbe}+2C^{5}_{c(ed)}f_{pbe}+2C^{5}_{p(ed)}f_{cbe}+2C^{8}_{(c|db|p)}-2C^{8}_{pc(db)}=0$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
2(C^{3}_{cbe}-C^{2}_{cbe})f_{pde}-2(C^{3}_{pde}-C^{2}_{pde})f_{cbe}+2C^{4}_{c(ed)}f_{pbe}+2C^{4}_{p(ed)}f_{cbe}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-4C^{7}_{pc(db)}+2C^{8}_{(c|d|p)b}=0$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
C^{2}_{cbe}f_{pde}-C^{2}_{pde}f_{cbe}+2C^{6}_{c(ed)}f_{pbe}+2C^{6}_{p(ed)}f_{cbe}+2C^{7}_{cd(pb)}+2C^{7}_{pd(cb)}-2C^{8}_{pc(db)}=0$
(A.1) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
C^{2}_{ced}f_{pbe}+C^{2}_{cbe}f_{pde}-2C^{2}_{p(ed)}f_{cbe}+2C^{5}_{p(ed)}f_{cbe}+2C^{6}_{p(ed)}f_{cbe}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+2C^{7}_{pd(cb)}-2C^{8}_{pc(db)}+C^{8}_{pdbc}=-f_{pce}C^{2}_{ebd}$
$\displaystyle\leavevmode\nobreak\
(C^{3}_{ced}-C^{2}_{ced})f_{pbe}+(C^{3}_{cbe}-C^{2}_{cbe})f_{pde}-2(C^{3}_{p(ed)}-C^{2}_{p(ed)})f_{cbe}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+2C^{4}_{p(ed)}f_{cbe}-2C^{7}_{pc(db)}+C^{8}_{pdcb}=-(C^{3}_{ebd}-C^{2}_{ebd})f_{pce}$
$\displaystyle\left.\left(2C^{4}_{c(eb)}f_{pde}-2C^{7}_{pc(bd)}+C^{8}_{pbcd}\right)\right|_{\textrm{symm.
in bd}}=-C^{4}_{e(bd)}f_{pce}$
$\displaystyle\left.\left(2C^{5}_{c(eb)}f_{pde}+C^{8}_{pdbc}-C^{8}_{pcbd}\right)\right|_{\textrm{symm.
in bd}}=-C^{5}_{e(bd)}f_{pce}$
$\displaystyle\left.\left(2C^{6}_{c(eb)}f_{pde}+2C^{7}_{pd(bc)}-C^{8}_{pcbd}\right)\right|_{\textrm{symm.
in bd}}=-C^{6}_{e(bd)}f_{pce}$
$\displaystyle\left.\left(f_{phe}C^{7}_{ce(bd)}+f_{che}C^{7}_{pe(bd)}+f_{pbe}C^{8}_{cehd}+f_{cbe}C^{8}_{pehd}+4C^{10}_{cphbd}+4C^{10}_{pchbd}\right)\right|_{\textrm{symm.
in bd}}=0$
$\displaystyle\left.\left(f_{pde}C^{7}_{ce(bh)}+2f_{phe}C^{7}_{cd(be)}-2f_{che}C^{7}_{pd(be)}+4C^{10}_{pcdbh}+f_{che}C^{8}_{pedb}\right)\right|_{\textrm{symm.
in bh}}=-f_{pce}C^{7}_{ed(bh)}$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
2f_{che}C^{7}_{pe(bd)}+f_{pde}C^{8}_{cehb}+f_{phe}C^{8}_{cdeb}+C^{8}_{cdhe}f_{pbe}+C^{8}_{pehd}f_{cbe}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
f_{che}C^{8}_{pdeb}-f_{cbe}C^{8}_{pdhe}+8C^{10}_{pchbd}=-f_{pce}C^{8}_{edhb}$
$\displaystyle\left.\left(4f_{pha}C^{10}_{cabdf}-4f_{cha}C^{10}_{pabdf}\right)\right|_{\textrm{symm.
in hb, df, hb $\leftrightarrow$ df}}=-f_{pce}C^{10}_{ehbdf}$

#### P-odd sector

$\displaystyle\begin{aligned}
&f_{pde}C^{1}_{c(eb)}+f_{cde}C^{1}_{p(eb)}+C^{9}_{pb[cd]}+C^{9}_{cb[pd]}=0\\\
&\left.(2f_{pde}C^{1}_{ceb}+2C^{9}_{pb[cd]})\right|_{\textrm{symm. in
bd}}=-f_{pce}C^{1}_{e(db)}\\\
&\left.(4C^{11}_{c[pbdf]}+4C^{11}_{p[cbdf]}+f_{pde}C^{9}_{ce[bf]}+f_{cde}C^{9}_{pe[bf]})\right|_{\textrm{antisymm.
in bdf}}=0\\\
&\left.\left(2f_{pfe}C^{9}_{cb[ed]}-f_{cfe}C^{9}_{pe[bd]}-f_{cde}C^{9}_{pe[fb]}-2f_{cfe}C^{9}_{pb[ed]}\right)\right|_{\textrm{antisymm.
in df}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+12C^{11}_{p[cbdf]}+f_{pbe}C^{9}_{ce[fd]}=-f_{pce}C^{9}_{eb[fd]}\\\
&\left.\left(4f_{phb}C^{11}_{c[badf]}-4f_{chb}C^{11}_{p[badf]}\right)\right|_{\textrm{antisymm.
in adfh}}=-f_{pce}C^{11}_{e[afdh]}\end{aligned}$ (A.2)

#### A.1.2 Fermionic sector

$\displaystyle\begin{aligned}
&-iC^{14}_{pcX}-C^{12}_{cX}T^{p}_{X}+T^{p}_{X}C^{12}_{cX}-T^{c}_{X}C^{12}_{pX}+T^{c}_{X}C^{13}_{pX}-if_{pcb}C^{12}_{bX}=0\\\
&-iC^{14}_{pcX}-C^{13}_{cX}T^{p}_{X}+T^{p}_{X}C^{13}_{cX}+C^{13}_{pX}T^{c}_{X}-C^{12}_{pX}T^{c}_{X}-if_{pcb}C^{13}_{bX}=0\\\
&-if_{pqb}C^{14}_{cbX}+if_{cqb}C^{14}_{pbX}+C^{14}_{cqX}T^{p}_{X}-T^{p}_{X}C^{14}_{cqX}+C^{14}_{pqX}T^{c}_{X}-T^{c}_{X}C^{14}_{pqX}-if_{pcb}C^{14}_{bqX}=0\\\
&-iC^{14}_{cpX}-iC^{14}_{pcX}-C^{12}_{cX}T^{p}_{X}+T^{p}_{X}C^{13}_{cX}-C^{12}_{pX}T^{c}_{X}+T^{c}_{X}C^{13}_{pX}=0\end{aligned}$
(A.3)

### A.2 Parametrizations for the remaining $C^{k}_{A}$

$\displaystyle C^{7}_{abcd}=$ $\displaystyle\leavevmode\nobreak\
c^{7}_{LRRR}(-T^{abcd}_{LLLL}+T^{abcd}_{LRRR}+T^{abcd}_{RLLL}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LRRR}+T^{abdc}_{RLLL}-T^{abdc}_{RRRR}$
(A.4)
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LRRR}+T^{acdb}_{RLLL}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LRRR}+T^{adcb}_{RLLL}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LLRR}(-T^{abcd}_{LLLL}+T^{abcd}_{LLRR}+T^{abcd}_{RRLL}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LLRR}+T^{abdc}_{RRLL}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LRRL}+T^{acdb}_{RLLR}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LRRL}+T^{adcb}_{RLLR}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LRLR}(-T^{abcd}_{LLLL}+T^{abcd}_{LRLR}+T^{abcd}_{RLRL}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LRLR}+T^{abdc}_{RLRL}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LRLR}+T^{acdb}_{RLRL}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LRLR}+T^{adcb}_{RLRL}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LLLR}(-T^{abcd}_{LLLL}+T^{abcd}_{LLLR}+T^{abcd}_{RRRL}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LLLR}+T^{abdc}_{RRRL}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LRLL}+T^{acdb}_{RLRR}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LRLL}+T^{adcb}_{RLRR}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LRRL}(-T^{abcd}_{LLLL}+T^{abcd}_{LRRL}+T^{abcd}_{RLLR}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LRRL}+T^{abdc}_{RLLR}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LLRR}+T^{acdb}_{RRLL}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LLRR}+T^{adcb}_{RRLL}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LLRL}(-T^{abcd}_{LLLL}+T^{abcd}_{LLRL}+T^{abcd}_{RRLR}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LLRL}+T^{abdc}_{RRLR}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LLRL}+T^{acdb}_{RRLR}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LLRL}+T^{adcb}_{RRLR}-T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7}_{LRLL}(-T^{abcd}_{LLLL}+T^{abcd}_{LRLL}+T^{abcd}_{RLRR}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{LRLL}+T^{abdc}_{RLRR}-T^{abdc}_{RRRR}$
$\displaystyle\qquad\qquad-T^{acdb}_{LLLL}+T^{acdb}_{LLLR}+T^{acdb}_{RRRL}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{LLLR}+T^{adcb}_{RRRL}-T^{adcb}_{RRRR})$
$\displaystyle\leavevmode\nobreak\
c^{7\prime}_{LLRR}(-2T^{acbd}_{LLLL}+T^{acbd}_{LLRR}+T^{acbd}_{LRRL}+T^{acbd}_{RLLR}+T^{acbd}_{RRLL}-2T^{acbd}_{RRRR}-2T^{adbc}_{LLLL}$
$\displaystyle\qquad\qquad+T^{adbc}_{LLRR}+T^{adbc}_{LRRL}+T^{adbc}_{RLLR}+T^{adbc}_{RRLL}-2T^{adbc}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7\prime}_{LLLR}(-2T^{acbd}_{LLLL}+T^{acbd}_{LLLR}+T^{acbd}_{LRLL}+T^{acbd}_{RLRR}+T^{acbd}_{RRRL}-2T^{acbd}_{RRRR}-2T^{adbc}_{LLLL}$
$\displaystyle\qquad\qquad+T^{adbc}_{LLLR}+T^{adbc}_{LRLL}+T^{adbc}_{RLRR}+T^{adbc}_{RRRL}-2T^{adbc}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7\prime}_{LRRR}(-T^{acbd}_{LLLL}+T^{acbd}_{LRRR}+T^{acbd}_{RLLL}-T^{acbd}_{RRRR}-T^{adbc}_{LLLL}+T^{adbc}_{LRRR}+T^{adbc}_{RLLL}-T^{adbc}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{7\prime}_{LRLR}(-T^{acbd}_{LLLL}+T^{acbd}_{LRLR}+T^{acbd}_{RLRL}-T^{acbd}_{RRRR}-T^{adbc}_{LLLL}+T^{adbc}_{LRLR}+T^{adbc}_{RLRL}-T^{adbc}_{RRRR})+$
$\displaystyle
c^{7\prime}_{LLRL}(-T^{acbd}_{LLLL}+T^{acbd}_{LLRL}+T^{acbd}_{RRLR}-T^{acbd}_{RRRR}-T^{adbc}_{LLLL}+T^{adbc}_{LLRL}+T^{adbc}_{RRLR}-T^{adbc}_{RRRR})\,,$
$\displaystyle C^{8}_{abcd}=$ $\displaystyle\leavevmode\nobreak\
c^{8}_{LLRR}(T^{abcd}_{LLRR}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RRLL}+T^{adcb}_{LRRL}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RLLR})+$
(A.5) $\displaystyle\leavevmode\nobreak\
c^{8}_{LRLR}(T^{abcd}_{LRLR}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RLRL}+T^{adcb}_{LRLR}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RLRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8}_{LLLR}(T^{abcd}_{LLLR}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RRRL}+T^{adcb}_{LRLL}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{8}_{LRRL}(T^{abcd}_{LRRL}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RLLR}+T^{adcb}_{LLRR}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RRLL})+$
$\displaystyle\leavevmode\nobreak\
c^{8}_{LLRL}(T^{abcd}_{LLRL}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RRLR}+T^{adcb}_{LLRL}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{8}_{LRLL}(T^{abcd}_{LRLL}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RLRR}+T^{adcb}_{LLLR}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RRRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8}_{LLLL}(T^{abcd}_{LLLL}-T^{abcd}_{LRRR}-T^{abcd}_{RLLL}+T^{abcd}_{RRRR}+T^{adcb}_{LLLL}-T^{adcb}_{LRRR}-T^{adcb}_{RLLL}+T^{adcb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LRRL}(T^{abdc}_{LLRR}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RRLL}+T^{acdb}_{LRRL}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LRLR}(T^{abdc}_{LRLR}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RLRL}+T^{acdb}_{LRLR}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RLRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LRLL}(T^{abdc}_{LLLR}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RRRL}+T^{acdb}_{LRLL}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LLRR}(T^{abdc}_{LRRL}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RLLR}+T^{acdb}_{LLRR}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RRLL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LLRL}(T^{abdc}_{LLRL}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RRLR}+T^{acdb}_{LLRL}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LLLR}(T^{abdc}_{LRLL}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RLRR}+T^{acdb}_{LLLR}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RRRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime}_{LLLL}(T^{abdc}_{LLLL}-T^{abdc}_{LRRR}-T^{abdc}_{RLLL}+T^{abdc}_{RRRR}+T^{acdb}_{LLLL}-T^{acdb}_{LRRR}-T^{acdb}_{RLLL}+T^{acdb}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LRRL}(T^{acbd}_{LLRR}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RRLL}+T^{adbc}_{LRRL}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LRLR}(T^{acbd}_{LRLR}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RLRL}+T^{adbc}_{LRLR}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RLRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LRLL}(T^{acbd}_{LLLR}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RRRL}+T^{adbc}_{LRLL}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LLRR}(T^{acbd}_{LRRL}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RLLR}+T^{adbc}_{LLRR}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RRLL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LLRL}(T^{acbd}_{LLRL}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RRLR}+T^{adbc}_{LLRL}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LLLR}(T^{acbd}_{LRLL}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RLRR}+T^{adbc}_{LLLR}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RRRL})+$
$\displaystyle\leavevmode\nobreak\
c^{8\prime\prime}_{LLLL}(T^{acbd}_{LLLL}-T^{acbd}_{LRRR}-T^{acbd}_{RLLL}+T^{acbd}_{RRRR}+T^{adbc}_{LLLL}-T^{adbc}_{LRRR}-T^{adbc}_{RLLL}+T^{adbc}_{RRRR})\,,$
$\displaystyle C^{9}_{abcd}=$ $\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LRRR}(-T^{acbd}_{LRRR}+T^{acbd}_{RLLL}+T^{adbc}_{LRRR}-T^{adbc}_{RLLL})+$
(A.6) $\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LRLR}(-T^{acbd}_{LRLR}+T^{acbd}_{RLRL}+T^{adbc}_{LRLR}-T^{adbc}_{RLRL})+$
$\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LLRR}(-T^{acbd}_{LLRR}-T^{acbd}_{LRRL}+T^{acbd}_{RLLR}+T^{acbd}_{RRLL}+T^{adbc}_{LLRR}+T^{adbc}_{LRRL}-T^{adbc}_{RLLR}-T^{adbc}_{RRLL})+$
$\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LLRL}(-T^{acbd}_{LLRL}+T^{acbd}_{RRLR}+T^{adbc}_{LLRL}-T^{adbc}_{RRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LLLR}(-T^{acbd}_{LLLR}-T^{acbd}_{LRLL}+T^{acbd}_{RLRR}+T^{acbd}_{RRRL}+T^{adbc}_{LLLR}+T^{adbc}_{LRLL}-T^{adbc}_{RLRR}-T^{adbc}_{RRRL})+$
$\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LLLL}(-T^{acbd}_{LLLL}+T^{acbd}_{RRRR}+T^{adbc}_{LLLL}-T^{adbc}_{RRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{9\prime}_{LRRR}(T^{abcd}_{LRRR}-T^{abcd}_{RLLL}-T^{abdc}_{LRRR}+T^{abdc}_{RLLL}+T^{acdb}_{LRRR}-T^{acdb}_{RLLL}-T^{adcb}_{LRRR}+T^{adcb}_{RLLL})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LLRR}(T^{abcd}_{LLRR}-T^{abcd}_{RRLL}-T^{abdc}_{LLRR}+T^{abdc}_{RRLL}+T^{acdb}_{LRRL}-T^{acdb}_{RLLR}-T^{adcb}_{LRRL}+T^{adcb}_{RLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LRLR}(T^{abcd}_{LRLR}-T^{abcd}_{RLRL}-T^{abdc}_{LRLR}+T^{abdc}_{RLRL}+T^{acdb}_{LRLR}-T^{acdb}_{RLRL}-T^{adcb}_{LRLR}+T^{adcb}_{RLRL})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LLLR}(T^{abcd}_{LLLR}-T^{abcd}_{RRRL}-T^{abdc}_{LLLR}+T^{abdc}_{RRRL}+T^{acdb}_{LRLL}-T^{acdb}_{RLRR}-T^{adcb}_{LRLL}+T^{adcb}_{RLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LRRL}(T^{abcd}_{LRRL}-T^{abcd}_{RLLR}-T^{abdc}_{LRRL}+T^{abdc}_{RLLR}+T^{acdb}_{LLRR}-T^{acdb}_{RRLL}-T^{adcb}_{LLRR}+T^{adcb}_{RRLL})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LLRL}(T^{abcd}_{LLRL}-T^{abcd}_{RRLR}-T^{abdc}_{LLRL}+T^{abdc}_{RRLR}+T^{acdb}_{LLRL}-T^{acdb}_{RRLR}-T^{adcb}_{LLRL}+T^{adcb}_{RRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LRLL}(T^{abcd}_{LRLL}-T^{abcd}_{RLRR}-T^{abdc}_{LRLL}+T^{abdc}_{RLRR}+T^{acdb}_{LLLR}-T^{acdb}_{RRRL}-T^{adcb}_{LLLR}+T^{adcb}_{RRRL})+$
$\displaystyle\leavevmode\nobreak\
c^{9}_{LLLL}(T^{abcd}_{LLLL}-T^{abcd}_{RRRR}-T^{abdc}_{LLLL}+T^{abdc}_{RRRR}+T^{acdb}_{LLLL}-T^{acdb}_{RRRR}-T^{adcb}_{LLLL}+T^{adcb}_{RRRR})\,,$
$\displaystyle C^{10}_{abcde}=$ $\displaystyle\leavevmode\nobreak\
c^{10}_{1}(T^{abdec}_{RRLRR}-T^{abdec}_{RRRLR}+T^{abedc}_{RRLRR}-T^{abedc}_{RRRLR}-T^{adbce}_{LLLRL}+T^{adbce}_{LLRLL}+T^{adbce}_{RRLRR}-T^{adbce}_{RRRLR}-$
(A.7) $\displaystyle\qquad
T^{adcbe}_{LLLRL}+T^{adcbe}_{LLRLL}+T^{adcbe}_{RRLRR}-T^{adcbe}_{RRRLR}+T^{bcdae}_{RLLLL}+T^{cbdae}_{RLLLL}+T^{cdeba}_{RLRRR}+T^{cedba}_{RLRRR}+$
$\displaystyle\qquad
T^{dbace}_{LLLLR}-T^{dbace}_{RLLLL}+T^{dcabe}_{LLLLR}-T^{dcabe}_{RLLLL}-T^{debac}_{RLRRR}+T^{ebacd}_{LLLLR}-T^{ebacd}_{RLLLL}-T^{ebcda}_{LLRLL}+$
$\displaystyle\qquad
T^{ebcda}_{RLRRR}-T^{ebcda}_{RRLRR}+T^{ecabd}_{LLLLR}-T^{ecabd}_{RLLLL}-T^{ecbda}_{LLRLL}+T^{ecbda}_{RLRRR}-T^{ecbda}_{RRLRR}-T^{edbac}_{RLRRR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{2}(T^{abdce}_{LLLRL}-T^{abdce}_{LLRLL}-T^{abdce}_{RRLRR}+T^{abdce}_{RRRLR}-T^{abecd}_{LLRLL}-T^{abecd}_{RRLRR}+T^{abecd}_{RRRLR}+T^{acdbe}_{LLLRL}-$
$\displaystyle\qquad
T^{acdbe}_{LLRLL}-T^{acdbe}_{RRLRR}+T^{acdbe}_{RRRLR}-T^{acebd}_{LLRLL}-T^{acebd}_{RRLRR}+T^{acebd}_{RRRLR}+T^{bdace}_{RLLLL}-T^{bdcae}_{RLLLL}+$
$\displaystyle\qquad
T^{cdabe}_{RLLLL}-T^{cdbae}_{RLLLL}+T^{dbeca}_{LLRLL}-T^{dbeca}_{RLRRR}+T^{dbeca}_{RRLRR}+T^{dceba}_{LLRLL}-T^{dceba}_{RLRRR}+T^{dceba}_{RRLRR}-$
$\displaystyle\qquad
T^{ebadc}_{LLLLR}+T^{ebdca}_{LLRLL}-T^{ebdca}_{RLRRR}+T^{ebdca}_{RRLRR}-T^{ecadb}_{LLLLR}+T^{ecdba}_{LLRLL}-T^{ecdba}_{RLRRR}+T^{ecdba}_{RRLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{3}(T^{abdce}_{RLLRL}-T^{abdce}_{RLRLL}-T^{abecd}_{RLRLL}+T^{acdbe}_{RLLRL}-T^{acdbe}_{RLRLL}-T^{acebd}_{RLRLL}-T^{badce}_{RLRLR}-T^{baecd}_{RLRLR}+$
$\displaystyle\qquad
T^{bdace}_{RLRLL}-T^{bdcae}_{RLLRL}+T^{bdcae}_{RLRLR}+T^{becad}_{RLRLR}-T^{cadbe}_{RLRLR}-T^{caebd}_{RLRLR}+T^{cdabe}_{RLRLL}-T^{cdbae}_{RLLRL}+$
$\displaystyle\qquad
T^{cdbae}_{RLRLR}+T^{cebad}_{RLRLR}-T^{dabec}_{RLRLR}-T^{daceb}_{RLRLR}+T^{dbeac}_{RLRLR}+T^{dbeca}_{LLRLR}+T^{dceab}_{RLRLR}+T^{dceba}_{LLRLR}-$
$\displaystyle\qquad
T^{eabdc}_{RLRLR}-T^{eacdb}_{RLRLR}-T^{ebadc}_{LLRLR}+T^{ebdac}_{RLRLR}+T^{ebdca}_{LLRLR}-T^{ecadb}_{LLRLR}+T^{ecdab}_{RLRLR}+T^{ecdba}_{LLRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{4}(-T^{adbce}_{RLLRL}+T^{adbce}_{RLRLL}-T^{adcbe}_{RLLRL}+T^{adcbe}_{RLRLL}+T^{bacde}_{RLRLR}+T^{baced}_{RLRLR}+T^{bcdae}_{RLLRL}-T^{bcdae}_{RLRLR}-$
$\displaystyle\qquad
T^{bcead}_{RLRLR}+T^{cabde}_{RLRLR}+T^{cabed}_{RLRLR}+T^{cbdae}_{RLLRL}-T^{cbdae}_{RLRLR}-T^{cbead}_{RLRLR}+T^{daebc}_{RLRLR}+T^{daecb}_{RLRLR}+$
$\displaystyle\qquad
T^{dbace}_{LLRLR}-T^{dbace}_{RLRLL}+T^{dcabe}_{LLRLR}-T^{dcabe}_{RLRLL}-T^{debac}_{RLRLR}-T^{decab}_{RLRLR}+T^{eadbc}_{RLRLR}+T^{eadcb}_{RLRLR}+$
$\displaystyle\qquad
T^{ebacd}_{LLRLR}-T^{ebacd}_{RLRLL}-T^{ebcda}_{LLRLR}+T^{ecabd}_{LLRLR}-T^{ecabd}_{RLRLL}-T^{ecbda}_{LLRLR}-T^{edbac}_{RLRLR}-T^{edcab}_{RLRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{5}(T^{abcde}_{RLLRL}-T^{abcde}_{RLRLL}+T^{abced}_{RLLRL}+T^{acbde}_{RLLRL}-T^{acbde}_{RLRLL}+T^{acbed}_{RLLRL}-T^{badec}_{RLRLR}-T^{baedc}_{RLRLR}+$
$\displaystyle\qquad
T^{bcade}_{RLRLL}+T^{bdeac}_{RLRLR}+T^{bedac}_{RLRLR}-T^{cadeb}_{RLRLR}-T^{caedb}_{RLRLR}+T^{cbade}_{RLRLL}+T^{cdeab}_{RLRLR}+T^{cedab}_{RLRLR}-$
$\displaystyle\qquad
T^{dabce}_{RLRLR}-T^{dacbe}_{RLRLR}-T^{dbcae}_{RLLRL}+T^{dbcae}_{RLRLR}-T^{dcbae}_{RLLRL}+T^{dcbae}_{RLRLR}-T^{eabcd}_{RLRLR}-T^{eacbd}_{RLRLR}-$
$\displaystyle\qquad
T^{ebcad}_{RLLRL}+T^{ebcad}_{RLRLR}-T^{ecbad}_{RLLRL}+T^{ecbad}_{RLRLR}-T^{edabc}_{LLRLR}-T^{edacb}_{LLRLR}+T^{edbca}_{LLRLR}+T^{edcba}_{LLRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{6}(-T^{abcde}_{LLLRL}+T^{abcde}_{LLRLL}+T^{abcde}_{RRLRR}-T^{abcde}_{RRRLR}-T^{abced}_{LLLRL}+T^{abced}_{RRLRR}-T^{abced}_{RRRLR}-T^{acbde}_{LLLRL}+$
$\displaystyle\qquad
T^{acbde}_{LLRLL}+T^{acbde}_{RRLRR}-T^{acbde}_{RRRLR}-T^{acbed}_{LLLRL}+T^{acbed}_{RRLRR}-T^{acbed}_{RRRLR}-T^{bcade}_{RLLLL}-T^{cbade}_{RLLLL}+$
$\displaystyle\qquad
T^{dbcae}_{RLLLL}+T^{dcbae}_{RLLLL}+T^{debca}_{RLRRR}+T^{decba}_{RLRRR}+T^{ebcad}_{RLLLL}-T^{ebcad}_{RLRRR}+T^{ecbad}_{RLLLL}-T^{ecbad}_{RLRRR}+$
$\displaystyle\qquad
T^{edabc}_{LLLLR}+T^{edacb}_{LLLLR}-T^{edbca}_{LLRLL}+T^{edbca}_{RLRRR}-T^{edbca}_{RRLRR}-T^{edcba}_{LLRLL}+T^{edcba}_{RLRRR}-T^{edcba}_{RRLRR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{7}(T^{abcde}_{LLLLR}-T^{abcde}_{RLRRR}+T^{abced}_{LLLLR}-T^{abced}_{RLRRR}+T^{acbde}_{LLLLR}-T^{acbde}_{RLRRR}+T^{acbed}_{LLLLR}-T^{acbed}_{RLRRR}+$
$\displaystyle\qquad
T^{adebc}_{LLLLR}-T^{adebc}_{RLRRR}+T^{adecb}_{LLLLR}+T^{aedbc}_{LLLLR}-T^{aedbc}_{RLRRR}+T^{aedcb}_{LLLLR}-T^{bcdea}_{RLLLL}-T^{bceda}_{RLLLL}-$
$\displaystyle\qquad
T^{cbade}_{RRRLR}-T^{cbaed}_{RRRLR}-T^{cbdea}_{RLLLL}+T^{cbdea}_{RRRLR}-T^{cbeda}_{RLLLL}+T^{cbeda}_{RRRLR}+T^{deabc}_{RLRRR}-T^{debca}_{RLLLL}-$
$\displaystyle\qquad
T^{decba}_{RLLLL}+T^{ebcad}_{RRLRR}+T^{ecbad}_{RRLRR}+T^{edabc}_{RLRRR}-T^{edbca}_{RLLLL}+T^{edbca}_{RRRLR}-T^{edcba}_{RLLLL}+T^{edcba}_{RRRLR})$
$\displaystyle\leavevmode\nobreak\
c^{10}_{8}(-T^{abdec}_{LLLLR}-T^{abedc}_{LLLLR}-T^{acdeb}_{LLLLR}-T^{acedb}_{LLLLR}-T^{adbce}_{LLLLR}+T^{adbce}_{RLRRR}-T^{adcbe}_{LLLLR}+T^{adcbe}_{RLRRR}-$
$\displaystyle\qquad
T^{aebcd}_{LLLLR}+T^{aebcd}_{RLRRR}-T^{aecbd}_{LLLLR}+T^{aecbd}_{RLRRR}+T^{bdeca}_{RLLLL}+T^{bedca}_{RLLLL}+T^{cdeba}_{RLLLL}+T^{cedba}_{RLLLL}-$
$\displaystyle\qquad
T^{dbace}_{RLRRR}+T^{dbace}_{RRRLR}+T^{dbcea}_{RLLLL}-T^{dbcea}_{RRRLR}-T^{dcabe}_{RLRRR}+T^{dcabe}_{RRRLR}+T^{dcbea}_{RLLLL}-T^{dcbea}_{RRRLR}-$
$\displaystyle\qquad
T^{ebacd}_{RLRRR}+T^{ebacd}_{RRRLR}+T^{ebcda}_{RLLLL}-T^{ebcda}_{RRRLR}-T^{ecabd}_{RLRRR}+T^{ecabd}_{RRRLR}+T^{ecbda}_{RLLLL}-T^{ecbda}_{RRRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{9}(-T^{abdce}_{LLLLR}+T^{abdce}_{RLRRR}-T^{abecd}_{LLLLR}+T^{abecd}_{RLRRR}-T^{acdbe}_{LLLLR}+T^{acdbe}_{RLRRR}-T^{acebd}_{LLLLR}+T^{acebd}_{RLRRR}-$
$\displaystyle\qquad
T^{adbec}_{LLLLR}+T^{adbec}_{RLRRR}-T^{adceb}_{LLLLR}-T^{aebdc}_{LLLLR}+T^{aebdc}_{RLRRR}-T^{aecdb}_{LLLLR}+T^{bdcea}_{RLLLL}+T^{becda}_{RLLLL}-$
$\displaystyle\qquad
T^{cdabe}_{RLRRR}+T^{cdbea}_{RLLLL}-T^{ceabd}_{RLRRR}+T^{cebda}_{RLLLL}+T^{dbaec}_{RRRLR}-T^{dbeac}_{RRLRR}+T^{dbeca}_{RLLLL}-T^{dbeca}_{RRRLR}+$
$\displaystyle\qquad
T^{dceba}_{RLLLL}-T^{dceba}_{RRRLR}+T^{ebadc}_{RRRLR}-T^{ebdac}_{RRLRR}+T^{ebdca}_{RLLLL}-T^{ebdca}_{RRRLR}+T^{ecdba}_{RLLLL}-T^{ecdba}_{RRRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{10}(T^{abcde}_{LLRLR}-T^{abcde}_{RLRLR}+T^{abced}_{LLRLR}-T^{abced}_{RLRLR}+T^{acbde}_{LLRLR}-T^{acbde}_{RLRLR}+T^{acbed}_{LLRLR}-T^{acbed}_{RLRLR}+$
$\displaystyle\qquad
T^{adebc}_{LLRLR}-T^{adebc}_{RLRLR}+T^{adecb}_{LLRLR}-T^{adecb}_{RLRLR}+T^{aedbc}_{LLRLR}-T^{aedbc}_{RLRLR}+T^{aedcb}_{LLRLR}-T^{aedcb}_{RLRLR}-$
$\displaystyle\qquad
T^{bcdea}_{RLRLL}+T^{bcdea}_{RLRLR}-T^{bceda}_{RLRLL}+T^{bceda}_{RLRLR}-T^{cbdea}_{RLRLL}+T^{cbdea}_{RLRLR}-T^{cbeda}_{RLRLL}+T^{cbeda}_{RLRLR}-$
$\displaystyle\qquad
T^{debca}_{RLRLL}+T^{debca}_{RLRLR}-T^{decba}_{RLRLL}+T^{decba}_{RLRLR}-T^{edbca}_{RLRLL}+T^{edbca}_{RLRLR}-T^{edcba}_{RLRLL}+T^{edcba}_{RLRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{11}(T_{LLRLR}^{abdce}+T_{LLRLR}^{abecd}+T_{LLRLR}^{acdbe}+T_{LLRLR}^{acebd}+T_{LLRLR}^{adbec}+T_{LLRLR}^{adceb}+T_{LLRLR}^{aebdc}+T_{LLRLR}^{aecdb}-$
$\displaystyle\qquad
T_{RLRLL}^{bdcea}-T_{RLRLL}^{becda}-T_{RLRLL}^{cdbea}-T_{RLRLL}^{cebda}-T_{RLRLL}^{dbeca}-T_{RLRLL}^{dceba}-T_{RLRLL}^{ebdca}-T_{RLRLL}^{ecdba}-$
$\displaystyle\qquad
T_{RLRLR}^{abdce}-T_{RLRLR}^{abecd}-T_{RLRLR}^{acdbe}-T_{RLRLR}^{acebd}-T_{RLRLR}^{adbec}-T_{RLRLR}^{adceb}-T_{RLRLR}^{aebdc}-T_{RLRLR}^{aecdb}+$
$\displaystyle\qquad
T_{RLRLR}^{bdcea}+T_{RLRLR}^{becda}+T_{RLRLR}^{cdbea}+T_{RLRLR}^{cebda}+T_{RLRLR}^{dbeca}+T_{RLRLR}^{dceba}+T_{RLRLR}^{ebdca}+T_{RLRLR}^{ecdba})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{12}(T^{abdce}_{RLLLR}+T^{abecd}_{RLLLR}+T^{acdbe}_{RLLLR}+T^{acebd}_{RLLLR}+T^{adbec}_{RLLLR}+T^{adceb}_{RLLLR}+T^{aebdc}_{RLLLR}+T^{aecdb}_{RLLLR}-$
$\displaystyle\qquad
T^{bdcea}_{RLLLR}-T^{becda}_{RLLLR}-T^{cadbe}_{RLLRR}-T^{caebd}_{RLLRR}+T^{cdabe}_{RLLRR}-T^{cdbea}_{RLLLR}+T^{ceabd}_{RLLRR}-T^{cebda}_{RLLLR}-$
$\displaystyle\qquad
T^{dabec}_{RLLRR}-T^{daceb}_{RLLRR}+T^{dbaec}_{RLLRR}-T^{dbaec}_{RRLLR}+T^{dbeac}_{RRLLR}-T^{dbeca}_{RLLLR}+T^{dcaeb}_{RLLRR}-T^{dceba}_{RLLLR}-$
$\displaystyle\qquad
T^{eabdc}_{RLLRR}-T^{eacdb}_{RLLRR}+T^{ebadc}_{RLLRR}-T^{ebadc}_{RRLLR}+T^{ebdac}_{RRLLR}-T^{ebdca}_{RLLLR}+T^{ecadb}_{RLLRR}-T^{ecdba}_{RLLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{13}(-T^{abdec}_{RLLLR}-T^{abedc}_{RLLLR}-T^{acdeb}_{RLLLR}-T^{acedb}_{RLLLR}-T^{adbce}_{RLLLR}-T^{adcbe}_{RLLLR}-T^{aebcd}_{RLLLR}-T^{aecbd}_{RLLLR}+$
$\displaystyle\qquad
T^{bdeca}_{RLLLR}+T^{bedca}_{RLLLR}-T^{cbdae}_{RRLLR}-T^{cbead}_{RRLLR}-T^{cdaeb}_{RLLRR}+T^{cdeba}_{RLLLR}-T^{ceadb}_{RLLRR}+T^{cedba}_{RLLLR}+$
$\displaystyle\qquad
T^{daebc}_{RLLRR}+T^{daecb}_{RLLRR}-T^{dbace}_{RLLRR}+T^{dbace}_{RRLLR}+T^{dbcea}_{RLLLR}-T^{dcabe}_{RLLRR}+T^{dcabe}_{RRLLR}+T^{dcbea}_{RLLLR}+$
$\displaystyle\qquad
T^{eadbc}_{RLLRR}+T^{eadcb}_{RLLRR}-T^{ebacd}_{RLLRR}+T^{ebacd}_{RRLLR}+T^{ebcda}_{RLLLR}-T^{ecabd}_{RLLRR}+T^{ecabd}_{RRLLR}+T^{ecbda}_{RLLLR})$
$\displaystyle\leavevmode\nobreak\
c^{10}_{14}(T^{abdec}_{LLRLR}-T^{abdec}_{RLRLR}+T^{abedc}_{LLRLR}-T^{abedc}_{RLRLR}+T^{acdeb}_{LLRLR}-T^{acdeb}_{RLRLR}+T^{acedb}_{LLRLR}-T^{acedb}_{RLRLR}+$
$\displaystyle\qquad
T^{adbce}_{LLRLR}-T^{adbce}_{RLRLR}+T^{adcbe}_{LLRLR}-T^{adcbe}_{RLRLR}+T^{aebcd}_{LLRLR}-T^{aebcd}_{RLRLR}+T^{aecbd}_{LLRLR}-T^{aecbd}_{RLRLR}-$
$\displaystyle\qquad
T^{bdeca}_{RLRLL}+T^{bdeca}_{RLRLR}-T^{bedca}_{RLRLL}+T^{bedca}_{RLRLR}-T^{cdeba}_{RLRLL}+T^{cdeba}_{RLRLR}-T^{cedba}_{RLRLL}+T^{cedba}_{RLRLR}-$
$\displaystyle\qquad
T^{dbcea}_{RLRLL}+T^{dbcea}_{RLRLR}-T^{dcbea}_{RLRLL}+T^{dcbea}_{RLRLR}-T^{ebcda}_{RLRLL}+T^{ebcda}_{RLRLR}-T^{ecbda}_{RLRLL}+T^{ecbda}_{RLRLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{15}(T^{abdce}_{RLLRR}+T^{abecd}_{RLLRR}+T^{acdbe}_{RLLRR}+T^{acebd}_{RLLRR}+T^{adbec}_{RLLRR}+T^{adceb}_{RLLRR}+T^{aebdc}_{RLLRR}+T^{aecdb}_{RLLRR}-$
$\displaystyle\qquad
T^{badce}_{RLLLR}-T^{baecd}_{RLLLR}+T^{bdcae}_{RLLLR}+T^{becad}_{RLLLR}-T^{cadbe}_{RLLLR}-T^{caebd}_{RLLLR}+T^{cdbae}_{RLLLR}+T^{cebad}_{RLLLR}-$
$\displaystyle\qquad
T^{dabec}_{RLLLR}-T^{daceb}_{RLLLR}+T^{dbeac}_{RLLLR}-T^{dbeac}_{RLLRR}-T^{dbeca}_{RRLLR}+T^{dceab}_{RLLLR}-T^{dceab}_{RLLRR}-T^{dceba}_{RRLLR}-$
$\displaystyle\qquad
T^{eabdc}_{RLLLR}-T^{eacdb}_{RLLLR}+T^{ebdac}_{RLLLR}-T^{ebdac}_{RLLRR}-T^{ebdca}_{RRLLR}+T^{ecdab}_{RLLLR}-T^{ecdab}_{RLLRR}-T^{ecdba}_{RRLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{16}(-T^{abcde}_{RLLLR}-T^{abced}_{RLLLR}-T^{acbde}_{RLLLR}-T^{acbed}_{RLLLR}-T^{adebc}_{RLLLR}-T^{adecb}_{RLLLR}-T^{aedbc}_{RLLLR}-T^{aedcb}_{RLLLR}+$
$\displaystyle\qquad
T^{bcdea}_{RLLLR}+T^{bceda}_{RLLLR}+T^{cadeb}_{RLLRR}+T^{caedb}_{RLLRR}+T^{cbade}_{RRLLR}+T^{cbaed}_{RRLLR}+T^{cbdea}_{RLLLR}+T^{cbeda}_{RLLLR}+$
$\displaystyle\qquad
T^{dabce}_{RLLRR}+T^{dacbe}_{RLLRR}-T^{dbcae}_{RRLLR}-T^{dcbae}_{RRLLR}-T^{deabc}_{RLLRR}-T^{deacb}_{RLLRR}+T^{debca}_{RLLLR}+T^{decba}_{RLLLR}+$
$\displaystyle\qquad
T^{eabcd}_{RLLRR}+T^{eacbd}_{RLLRR}-T^{ebcad}_{RRLLR}-T^{ecbad}_{RRLLR}-T^{edabc}_{RLLRR}-T^{edacb}_{RLLRR}+T^{edbca}_{RLLLR}+T^{edcba}_{RLLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{17}(T^{abcde}_{RLLRR}+T^{abced}_{RLLRR}+T^{acbde}_{RLLRR}+T^{acbed}_{RLLRR}+T^{adebc}_{RLLRR}+T^{adecb}_{RLLRR}+T^{aedbc}_{RLLRR}+T^{aedcb}_{RLLRR}-$
$\displaystyle\qquad
T^{badec}_{RLLLR}-T^{baedc}_{RLLLR}+T^{bdeac}_{RLLLR}+T^{bedac}_{RLLLR}-T^{cadeb}_{RLLLR}-T^{caedb}_{RLLLR}-T^{cbdea}_{RRLLR}-T^{cbeda}_{RRLLR}+$
$\displaystyle\qquad
T^{cdeab}_{RLLLR}-T^{cdeab}_{RLLRR}+T^{cedab}_{RLLLR}-T^{cedab}_{RLLRR}-T^{dabce}_{RLLLR}-T^{dacbe}_{RLLLR}+T^{dbcae}_{RLLLR}+T^{dcbae}_{RLLLR}-$
$\displaystyle\qquad
T^{eabcd}_{RLLLR}-T^{eacbd}_{RLLLR}+T^{ebcad}_{RLLLR}-T^{ebcad}_{RLLRR}+T^{ecbad}_{RLLLR}-T^{ecbad}_{RLLRR}-T^{edbca}_{RRLLR}-T^{edcba}_{RRLLR})+$
$\displaystyle\leavevmode\nobreak\
c^{10}_{18}(-T^{abdec}_{RLLRR}-T^{abedc}_{RLLRR}-T^{acdeb}_{RLLRR}-T^{acedb}_{RLLRR}-T^{adbce}_{RLLRR}-T^{adcbe}_{RLLRR}-T^{aebcd}_{RLLRR}-T^{aecbd}_{RLLRR}+$
$\displaystyle\qquad
T^{bacde}_{RLLLR}+T^{baced}_{RLLLR}-T^{bcdae}_{RLLLR}-T^{bcead}_{RLLLR}+T^{cabde}_{RLLLR}+T^{cabed}_{RLLLR}-T^{cbdae}_{RLLLR}-T^{cbead}_{RLLLR}+$
$\displaystyle\qquad
T^{daebc}_{RLLLR}+T^{daecb}_{RLLLR}+T^{dbcea}_{RRLLR}+T^{dcbea}_{RRLLR}-T^{debac}_{RLLLR}+T^{debac}_{RLLRR}-T^{decab}_{RLLLR}+T^{decab}_{RLLRR}+$
$\displaystyle\qquad
T^{eadbc}_{RLLLR}+T^{eadcb}_{RLLLR}+T^{ebcda}_{RRLLR}+T^{ecbda}_{RRLLR}-T^{edbac}_{RLLLR}+T^{edbac}_{RLLRR}-T^{edcab}_{RLLLR}+T^{edcab}_{RLLRR})\,,$
$\displaystyle C^{11}_{abcde}=$ $\displaystyle\leavevmode\nobreak\
c^{11}_{1}(-T^{cadbe}_{LLLLL}+T^{cadbe}_{RRRR}+T^{caebd}_{LLLLL}-T^{caebd}_{RRRR}+T^{cbdae}_{LLLLL}-T^{cbdae}_{RRRR}-T^{cbead}_{LLLLL}+T^{cbead}_{RRRR}-$
(A.8) $\displaystyle\qquad
T^{dabce}_{LLLLL}+T^{dabce}_{RRRR}+T^{dabec}_{LLLLL}-T^{dabec}_{RRRR}+T^{dacbe}_{LLLLL}-T^{dacbe}_{RRRR}-T^{daebc}_{LLLLL}+T^{daebc}_{RRRR}+$
$\displaystyle\qquad
T^{dbace}_{LLLLL}-T^{dbace}_{RRRR}-T^{dbaec}_{LLLLL}+T^{dbaec}_{RRRR}-T^{dbcae}_{LLLLL}+T^{dbcae}_{RRRR}+T^{dbeac}_{LLLLL}-T^{dbeac}_{RRRR}-$
$\displaystyle\qquad
T^{dcabe}_{LLLLL}+T^{dcabe}_{RRRR}+T^{dcbae}_{LLLLL}-T^{dcbae}_{RRRR}+T^{eabcd}_{LLLLL}-T^{eabcd}_{RRRR}-T^{eabdc}_{LLLLL}+T^{eabdc}_{RRRR}-$
$\displaystyle\qquad
T^{eacbd}_{LLLLL}+T^{eacbd}_{RRRR}+T^{eadbc}_{LLLLL}-T^{eadbc}_{RRRR}-T^{ebacd}_{LLLLL}+T^{ebacd}_{RRRR}+T^{ebadc}_{LLLLL}-T^{ebadc}_{RRRR}+$
$\displaystyle\qquad
T^{ebcad}_{LLLLL}-T^{ebcad}_{RRRR}-T^{ebdac}_{LLLLL}+T^{ebdac}_{RRRR}+T^{ecabd}_{LLLLL}-T^{ecabd}_{RRRR}-T^{ecbad}_{LLLLL}+T^{ecbad}_{RRRR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{2}(T^{bacde}_{RLLLL}-T^{baced}_{RLLLL}-T^{badce}_{RLLLL}+T^{badec}_{RLLLL}+T^{baecd}_{RLLLL}-T^{baedc}_{RLLLL}+T^{bcade}_{RLLLL}-T^{bcaed}_{RLLLL}-$
$\displaystyle\qquad
T^{bcdae}_{RLLLL}+T^{bcdae}_{RLRRR}+T^{bcead}_{RLLLL}-T^{bcead}_{RLRRR}-T^{bdace}_{RLLLL}+T^{bdcae}_{RLLLL}-T^{bdcae}_{RLRRR}+T^{bdeac}_{RLRRR}+$
$\displaystyle\qquad
T^{becad}_{RLRRR}-T^{bedac}_{RLRRR}-T^{cabde}_{RLLLL}+T^{cabed}_{RLLLL}-T^{cadbe}_{LLLLR}+T^{cadbe}_{RLLLL}-T^{cadeb}_{RLLLL}+T^{caebd}_{LLLLR}-$
$\displaystyle\qquad
T^{caebd}_{RLLLL}+T^{caedb}_{RLLLL}-T^{cbade}_{RLLLL}+T^{cbaed}_{RLLLL}+T^{cbdae}_{RLLLL}-T^{cbdae}_{RLRRR}-T^{cbead}_{RLLLL}+T^{cbead}_{RLRRR}+$
$\displaystyle\qquad
T^{cdabe}_{RLLLL}-T^{cdbae}_{RLLLL}+T^{cdbae}_{RLRRR}-T^{cdeab}_{RLRRR}-T^{cebad}_{RLRRR}+T^{cedab}_{RLRRR}-T^{dabce}_{LLLLR}+T^{dabce}_{RLLLL}+$
$\displaystyle\qquad
T^{dabec}_{LLLRL}-T^{dabec}_{RLLLL}+T^{dacbe}_{LLLLR}-T^{dacbe}_{RLLLL}+T^{daceb}_{RLLLL}-T^{daebc}_{LLLLR}+T^{daebc}_{RLLLL}+T^{daecb}_{LLLLR}-$
$\displaystyle\qquad
T^{daecb}_{RLLLL}+T^{dbace}_{LLLLR}+T^{dbace}_{RLLLL}-T^{dbaec}_{RLLLL}-T^{dbcae}_{RLLLL}+T^{dbcae}_{RLRRR}+T^{dbeac}_{RLLLL}-T^{dbeac}_{RLRRR}-$
$\displaystyle\qquad
T^{dcabe}_{LLLLR}-T^{dcabe}_{RLLLL}+T^{dcbae}_{RLLLL}-T^{dcbae}_{RLRRR}+T^{dceab}_{RLRRR}+T^{debac}_{RLRRR}-T^{decab}_{RLRRR}+T^{eabcd}_{LLLLR}+$
$\displaystyle\qquad
T^{eabcd}_{LLLRL}-T^{eabcd}_{RLLLL}-T^{eabdc}_{LLLLR}-T^{eabdc}_{LLLRL}+T^{eabdc}_{RLLLL}-T^{eacbd}_{LLLLR}-T^{eacbd}_{LLLRL}+T^{eacbd}_{RLLLL}+$
$\displaystyle\qquad
T^{eacdb}_{LLLLR}-T^{eacdb}_{RLLLL}+T^{eadbc}_{LLLLR}-T^{eadbc}_{RLLLL}-T^{eadcb}_{LLLLR}+T^{eadcb}_{RLLLL}-T^{ebacd}_{LLLLR}-T^{ebacd}_{RLLLL}+$
$\displaystyle\qquad
T^{ebadc}_{LLLLR}+T^{ebadc}_{RLLLL}+T^{ebcad}_{RLLLL}-T^{ebcad}_{RLRRR}-T^{ebdac}_{RLLLL}+T^{ebdac}_{RLRRR}+T^{ecabd}_{LLLLR}+T^{ecabd}_{RLLLL}-$
$\displaystyle\qquad
T^{ecadb}_{LLLLR}-T^{ecbad}_{RLLLL}+T^{ecbad}_{RLRRR}-T^{ecdab}_{RLRRR}-T^{edabc}_{LLLLR}+T^{edacb}_{LLLLR}-T^{edbac}_{RLRRR}+T^{edcab}_{RLRRR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{3}(-T^{abcde}_{RLLLL}+T^{abced}_{RLLLL}+T^{abdce}_{RLLLL}-T^{abecd}_{RLLLL}+T^{acbde}_{RLLLL}-T^{acbed}_{RLLLL}-T^{acdbe}_{RLLLL}+T^{acebd}_{RLLLL}-$
$\displaystyle\qquad
T^{adbce}_{RLLLL}+T^{adcbe}_{RLLLL}-T^{cadbe}_{RLRRR}+T^{caebd}_{RLRRR}+T^{cbdae}_{RRRLR}-T^{cbead}_{RRRLR}-T^{dabce}_{RLRRR}+T^{dabec}_{RLRRR}+$
$\displaystyle\qquad
T^{dacbe}_{RLRRR}-T^{daebc}_{RLRRR}-T^{dbace}_{LLRLL}+T^{dbace}_{RRLRR}-T^{dbaec}_{RRLRR}-T^{dbcae}_{RRRLR}+T^{dbeac}_{RRRLR}+T^{dbeca}_{LLLLR}+$
$\displaystyle\qquad
T^{dcabe}_{LLRLL}-T^{dcabe}_{RRLRR}+T^{dcbae}_{RRRLR}-T^{dceba}_{LLLLR}+T^{eabcd}_{RLRRR}-T^{eabdc}_{RLRRR}-T^{eacbd}_{RLRRR}+T^{eadbc}_{RLRRR}+$
$\displaystyle\qquad
T^{ebacd}_{LLRLL}-T^{ebacd}_{RRLRR}+T^{ebadc}_{RRLRR}-T^{ebcad}_{LLLRL}+T^{ebcad}_{RRRLR}+T^{ebcda}_{LLLLR}-T^{ebdac}_{RRRLR}-T^{ebdca}_{LLLLR}-$
$\displaystyle\qquad
T^{ecabd}_{LLRLL}+T^{ecabd}_{RRLRR}+T^{ecbad}_{LLLRL}-T^{ecbad}_{RRRLR}-T^{ecbda}_{LLLLR}+T^{ecdba}_{LLLLR}+T^{edbca}_{LLLLR}-T^{edcba}_{LLLLR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{4}(-T^{bacde}_{RLLLL}+T^{baced}_{RLLLL}+T^{badce}_{RLLLL}-T^{baecd}_{RLLLL}+T^{cabde}_{RLLLL}-T^{cabed}_{RLLLL}-T^{cadbe}_{LLRLL}-T^{cadbe}_{RLLLL}+$
$\displaystyle\qquad
T^{cadbe}_{RRLRR}-T^{cadeb}_{RRLRR}+T^{caebd}_{LLRLL}+T^{caebd}_{RLLLL}-T^{caebd}_{RRLRR}+T^{caedb}_{RRLRR}+T^{cbade}_{RRRLR}-T^{cbaed}_{RRRLR}+$
$\displaystyle\qquad
T^{cbdae}_{LLLLR}+T^{cbdae}_{LLRLL}-T^{cbdae}_{RRLRR}-T^{cbead}_{LLLLR}-T^{cbead}_{LLRLL}+T^{cbead}_{RRLRR}-T^{cdabe}_{RLRRR}+T^{cdaeb}_{RLRRR}+$
$\displaystyle\qquad
T^{ceabd}_{RLRRR}-T^{ceadb}_{RLRRR}-T^{dabce}_{LLRLL}-T^{dabce}_{RLLLL}+T^{dabce}_{RRLRR}+T^{dabec}_{RLLLL}-T^{dabec}_{RRLRR}+T^{dacbe}_{LLRLL}+$
$\displaystyle\qquad
T^{dacbe}_{RLLLL}-T^{dacbe}_{RRLRR}+T^{daceb}_{RRLRR}-T^{daebc}_{LLRLL}-T^{daebc}_{RLLLL}+T^{daebc}_{RRLRR}-T^{daecb}_{RRLRR}-T^{dbace}_{RLRRR}-$
$\displaystyle\qquad
T^{dbace}_{RRRLR}-T^{dbaec}_{LLLRL}+T^{dbaec}_{RLRRR}+T^{dbaec}_{RRRLR}-T^{dbcae}_{LLLLR}-T^{dbcae}_{LLRLL}+T^{dbcae}_{RRLRR}+T^{dbeac}_{LLLLR}+$
$\displaystyle\qquad
T^{dbeac}_{LLRLL}-T^{dbeac}_{RRLRR}+T^{dcabe}_{RLRRR}+T^{dcabe}_{RRRLR}-T^{dcaeb}_{RLRRR}+T^{dcbae}_{LLLLR}+T^{dcbae}_{LLRLL}-T^{dcbae}_{RRLRR}-$
$\displaystyle\qquad
T^{dceab}_{LLLLR}-T^{deabc}_{RLRRR}+T^{deacb}_{RLRRR}+T^{eabcd}_{LLRLL}+T^{eabcd}_{RLLLL}-T^{eabcd}_{RRLRR}-T^{eabdc}_{RLLLL}+T^{eabdc}_{RRLRR}-$
$\displaystyle\qquad
T^{eacbd}_{LLRLL}-T^{eacbd}_{RLLLL}+T^{eacbd}_{RRLRR}-T^{eacdb}_{RRLRR}+T^{eadbc}_{LLRLL}+T^{eadbc}_{RLLLL}-T^{eadbc}_{RRLRR}+T^{eadcb}_{RRLRR}-$
$\displaystyle\qquad
T^{ebacd}_{LLLRL}+T^{ebacd}_{RLRRR}+T^{ebacd}_{RRRLR}+T^{ebadc}_{LLLRL}-T^{ebadc}_{RLRRR}-T^{ebadc}_{RRRLR}+T^{ebcad}_{LLLLR}+T^{ebcad}_{LLRLL}-$
$\displaystyle\qquad
T^{ebcad}_{RRLRR}-T^{ebdac}_{LLLLR}-T^{ebdac}_{LLRLL}+T^{ebdac}_{RRLRR}+T^{ecabd}_{LLLRL}-T^{ecabd}_{RLRRR}-T^{ecabd}_{RRRLR}+T^{ecadb}_{RLRRR}-$
$\displaystyle\qquad
T^{ecbad}_{LLLLR}-T^{ecbad}_{LLRLL}+T^{ecbad}_{RRLRR}+T^{ecdab}_{LLLLR}+T^{edabc}_{RLRRR}-T^{edacb}_{RLRRR}+T^{edbac}_{LLLLR}-T^{edcab}_{LLLLR})+$
$\displaystyle
c^{11}_{5}(T^{abcde}_{RLLLR}-T^{abced}_{RLLLR}-T^{abdce}_{RLLLR}+T^{abdec}_{RLLLR}+T^{abecd}_{RLLLR}-T^{abedc}_{RLLLR}-T^{acbde}_{RLLLR}+T^{acbed}_{RLLLR}+$
$\displaystyle\qquad
T^{acdbe}_{RLLLR}-T^{acdeb}_{RLLLR}-T^{acebd}_{RLLLR}+T^{acedb}_{RLLLR}+T^{adbce}_{RLLLR}-T^{adbec}_{RLLLR}-T^{adcbe}_{RLLLR}+T^{adceb}_{RLLLR}+$
$\displaystyle\qquad
T^{adebc}_{RLLLR}-T^{adecb}_{RLLLR}-T^{aebcd}_{RLLLR}+T^{aebdc}_{RLLLR}+T^{aecbd}_{RLLLR}-T^{aecdb}_{RLLLR}-T^{aedbc}_{RLLLR}+T^{aedcb}_{RLLLR}+$
$\displaystyle\qquad
T^{bcdea}_{RLLLR}-T^{bceda}_{RLLLR}-T^{bdcea}_{RLLLR}+T^{bdeca}_{RLLLR}+T^{becda}_{RLLLR}-T^{bedca}_{RLLLR}+T^{cadbe}_{RLLRR}-T^{cadeb}_{RLLRR}-$
$\displaystyle\qquad
T^{caebd}_{RLLRR}+T^{caedb}_{RLLRR}+T^{cbade}_{RRLLR}-T^{cbaed}_{RRLLR}-T^{cbdae}_{RRLLR}-T^{cbdea}_{RLLLR}+T^{cbead}_{RRLLR}+T^{cbeda}_{RLLLR}-$
$\displaystyle\qquad
T^{cdabe}_{RLLRR}+T^{cdaeb}_{RLLRR}+T^{cdbea}_{RLLLR}-T^{cdeba}_{RLLLR}+T^{ceabd}_{RLLRR}-T^{ceadb}_{RLLRR}-T^{cebda}_{RLLLR}+T^{cedba}_{RLLLR}+$
$\displaystyle\qquad
T^{dabce}_{RLLRR}-T^{dabec}_{RLLRR}-T^{dacbe}_{RLLRR}+T^{daceb}_{RLLRR}+T^{daebc}_{RLLRR}-T^{daecb}_{RLLRR}-T^{dbace}_{RLLRR}-T^{dbace}_{RRLLR}+$
$\displaystyle\qquad
T^{dbaec}_{RLLRR}+T^{dbaec}_{RRLLR}+T^{dbcae}_{RRLLR}+T^{dbcea}_{RLLLR}-T^{dbeac}_{RRLLR}-T^{dbeca}_{RLLLR}+T^{dcabe}_{RLLRR}+T^{dcabe}_{RRLLR}-$
$\displaystyle\qquad
T^{dcaeb}_{RLLRR}-T^{dcbae}_{RRLLR}-T^{dcbea}_{RLLLR}+T^{dceba}_{RLLLR}-T^{deabc}_{RLLRR}+T^{deacb}_{RLLRR}+T^{debca}_{RLLLR}-T^{decba}_{RLLLR}-$
$\displaystyle\qquad
T^{eabcd}_{RLLRR}+T^{eabdc}_{RLLRR}+T^{eacbd}_{RLLRR}-T^{eacdb}_{RLLRR}-T^{eadbc}_{RLLRR}+T^{eadcb}_{RLLRR}+T^{ebacd}_{RLLRR}+T^{ebacd}_{RRLLR}-$
$\displaystyle\qquad
T^{ebadc}_{RLLRR}-T^{ebadc}_{RRLLR}-T^{ebcad}_{RRLLR}-T^{ebcda}_{RLLLR}+T^{ebdac}_{RRLLR}+T^{ebdca}_{RLLLR}-T^{ecabd}_{RLLRR}-T^{ecabd}_{RRLLR}+$
$\displaystyle\qquad
T^{ecadb}_{RLLRR}+T^{ecbad}_{RRLLR}+T^{ecbda}_{RLLLR}-T^{ecdba}_{RLLLR}+T^{edabc}_{RLLRR}-T^{edacb}_{RLLRR}-T^{edbca}_{RLLLR}+T^{edcba}_{RLLLR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{6}(-T^{abcde}_{RLRLR}+T^{abced}_{RLRLR}+T^{abdce}_{RLRLR}-T^{abdec}_{RLRLR}-T^{abecd}_{RLRLR}+T^{abedc}_{RLRLR}+T^{acbde}_{RLRLR}-T^{acbed}_{RLRLR}-$
$\displaystyle\qquad
T^{acdbe}_{RLRLR}+T^{acdeb}_{RLRLR}+T^{acebd}_{RLRLR}-T^{acedb}_{RLRLR}-T^{adbce}_{RLRLR}+T^{adbec}_{RLRLR}+T^{adcbe}_{RLRLR}-T^{adceb}_{RLRLR}-$
$\displaystyle\qquad
T^{adebc}_{RLRLR}+T^{adecb}_{RLRLR}+T^{aebcd}_{RLRLR}-T^{aebdc}_{RLRLR}-T^{aecbd}_{RLRLR}+T^{aecdb}_{RLRLR}+T^{aedbc}_{RLRLR}-T^{aedcb}_{RLRLR}-$
$\displaystyle\qquad
T^{bacde}_{RLLRL}+T^{baced}_{RLLRL}+T^{badce}_{RLLRL}+T^{bcade}_{RLLRL}-T^{bcaed}_{RLLRL}-T^{bcdae}_{RLRLL}-T^{bcdea}_{RLRLR}+T^{bcead}_{RLRLL}+$
$\displaystyle\qquad
T^{bceda}_{RLRLR}-T^{bdace}_{RLLRL}+T^{bdcae}_{RLRLL}+T^{bdcea}_{RLRLR}-T^{bdeca}_{RLRLR}-T^{becda}_{RLRLR}+T^{bedca}_{RLRLR}+T^{cabde}_{RLLRL}-$
$\displaystyle\qquad
T^{cabed}_{RLLRL}-T^{cadbe}_{LLRLR}-T^{cadbe}_{RLLRL}+T^{caebd}_{LLRLR}-T^{cbade}_{RLLRL}+T^{cbaed}_{RLLRL}+T^{cbdae}_{RLRLL}+T^{cbdea}_{RLRLR}-$
$\displaystyle\qquad
T^{cbead}_{RLRLL}-T^{cbeda}_{RLRLR}+T^{cdabe}_{RLLRL}-T^{cdbae}_{RLRLL}-T^{cdbea}_{RLRLR}+T^{cdeba}_{RLRLR}+T^{cebda}_{RLRLR}-T^{cedba}_{RLRLR}-$
$\displaystyle\qquad
T^{dabce}_{LLRLR}-T^{dabce}_{RLLRL}+T^{dabec}_{RLLRL}+T^{dacbe}_{LLRLR}+T^{dacbe}_{RLLRL}-T^{daebc}_{LLRLR}+T^{daecb}_{LLRLR}+T^{dbace}_{RLLRL}-$
$\displaystyle\qquad
T^{dbaec}_{RLLRL}-T^{dbcae}_{RLRLL}-T^{dbcea}_{RLRLR}+T^{dbeac}_{RLRLL}+T^{dbeca}_{RLRLR}-T^{dcabe}_{RLLRL}+T^{dcbae}_{RLRLL}+T^{dcbea}_{RLRLR}-$
$\displaystyle\qquad
T^{dceba}_{RLRLR}-T^{debca}_{RLRLR}+T^{decba}_{RLRLR}+T^{eabcd}_{LLRLR}+T^{eabcd}_{RLLRL}-T^{eabdc}_{LLRLR}-T^{eabdc}_{RLLRL}-T^{eacbd}_{LLRLR}-$
$\displaystyle\qquad
T^{eacbd}_{RLLRL}+T^{eacdb}_{LLRLR}+T^{eadbc}_{LLRLR}-T^{eadcb}_{LLRLR}-T^{ebacd}_{RLLRL}+T^{ebadc}_{RLLRL}+T^{ebcad}_{RLRLL}+T^{ebcda}_{RLRLR}-$
$\displaystyle\qquad
T^{ebdac}_{RLRLL}-T^{ebdca}_{RLRLR}+T^{ecabd}_{RLLRL}-T^{ecbad}_{RLRLL}-T^{ecbda}_{RLRLR}+T^{ecdba}_{RLRLR}+T^{edbca}_{RLRLR}-T^{edcba}_{RLRLR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{7}(T^{bacde}_{RLRLL}-T^{badce}_{RLRLL}+T^{baecd}_{RLRLL}+T^{bcade}_{RLRLR}-T^{bcaed}_{RLRLR}-T^{bdace}_{RLRLR}+T^{bdaec}_{RLRLR}+T^{beacd}_{RLRLR}-$
$\displaystyle\qquad
T^{beadc}_{RLRLR}-T^{cabde}_{RLRLL}+T^{cadbe}_{RLRLL}-T^{caebd}_{RLRLL}-T^{cbade}_{RLRLR}+T^{cbaed}_{RLRLR}-T^{cbdae}_{LLRLR}+T^{cbead}_{LLRLR}+$
$\displaystyle\qquad
T^{cdabe}_{RLRLR}-T^{cdaeb}_{RLRLR}-T^{ceabd}_{RLRLR}+T^{ceadb}_{RLRLR}+T^{dabce}_{RLRLL}-T^{dacbe}_{RLRLL}+T^{daebc}_{RLRLL}+T^{dbace}_{RLRLR}-$
$\displaystyle\qquad
T^{dbaec}_{RLRLR}+T^{dbcae}_{LLRLR}-T^{dbeac}_{LLRLR}-T^{dcabe}_{RLRLR}+T^{dcaeb}_{RLRLR}-T^{dcbae}_{LLRLR}+T^{dceab}_{LLRLR}+T^{deabc}_{RLRLR}-$
$\displaystyle\qquad
T^{deacb}_{RLRLR}-T^{eabcd}_{RLRLL}+T^{eacbd}_{RLRLL}-T^{eadbc}_{RLRLL}-T^{ebacd}_{RLRLR}+T^{ebadc}_{RLRLR}-T^{ebcad}_{LLRLR}+T^{ebdac}_{LLRLR}+$
$\displaystyle\qquad
T^{ecabd}_{RLRLR}-T^{ecadb}_{RLRLR}+T^{ecbad}_{LLRLR}-T^{ecdab}_{LLRLR}-T^{edabc}_{RLRLR}+T^{edacb}_{RLRLR}-T^{edbac}_{LLRLR}+T^{edcab}_{LLRLR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{8}(-T^{abcde}_{RLLRL}-T^{abcde}_{RLRLL}+T^{abced}_{RLLRL}+T^{abdce}_{RLLRL}+T^{abdce}_{RLRLL}-T^{abecd}_{RLRLL}+T^{acbde}_{RLLRL}+T^{acbde}_{RLRLL}-$
$\displaystyle\qquad
T^{acbed}_{RLLRL}-T^{acdbe}_{RLLRL}-T^{acdbe}_{RLRLL}+T^{acebd}_{RLRLL}-T^{adbce}_{RLLRL}-T^{adbce}_{RLRLL}+T^{adcbe}_{RLLRL}+T^{adcbe}_{RLRLL}-$
$\displaystyle\qquad
T^{bacde}_{RLRLR}+T^{baced}_{RLRLR}+T^{badce}_{RLRLR}-T^{badec}_{RLRLR}-T^{baecd}_{RLRLR}+T^{baedc}_{RLRLR}-T^{bcade}_{RLRLL}+T^{bcdae}_{RLLRL}-$
$\displaystyle\qquad
T^{bcdae}_{RLRLR}+T^{bcead}_{RLRLR}+T^{bdace}_{RLRLL}-T^{bdcae}_{RLLRL}+T^{bdcae}_{RLRLR}-T^{bdeac}_{RLRLR}-T^{becad}_{RLRLR}+T^{bedac}_{RLRLR}+$
$\displaystyle\qquad
T^{cabde}_{RLRLR}-T^{cabed}_{RLRLR}-T^{cadbe}_{RLRLR}+T^{cadeb}_{RLRLR}+T^{caebd}_{RLRLR}-T^{caedb}_{RLRLR}+T^{cbade}_{RLRLL}-T^{cbdae}_{RLLRL}+$
$\displaystyle\qquad
T^{cbdae}_{RLRLR}-T^{cbead}_{RLRLR}-T^{cdabe}_{RLRLL}+T^{cdbae}_{RLLRL}-T^{cdbae}_{RLRLR}+T^{cdeab}_{RLRLR}+T^{cebad}_{RLRLR}-T^{cedab}_{RLRLR}-$
$\displaystyle\qquad
T^{dabce}_{RLRLR}+T^{dabec}_{RLRLR}+T^{dacbe}_{RLRLR}-T^{daceb}_{RLRLR}-T^{daebc}_{RLRLR}+T^{daecb}_{RLRLR}-T^{dbace}_{LLRLR}-T^{dbace}_{RLRLL}+$
$\displaystyle\qquad
T^{dbcae}_{RLLRL}-T^{dbcae}_{RLRLR}+T^{dbeac}_{RLRLR}+T^{dbeca}_{LLRLR}+T^{dcabe}_{LLRLR}+T^{dcabe}_{RLRLL}-T^{dcbae}_{RLLRL}+T^{dcbae}_{RLRLR}-$
$\displaystyle\qquad
T^{dceab}_{RLRLR}-T^{dceba}_{LLRLR}-T^{debac}_{RLRLR}+T^{decab}_{RLRLR}+T^{eabcd}_{RLRLR}-T^{eabdc}_{RLRLR}-T^{eacbd}_{RLRLR}+T^{eacdb}_{RLRLR}+$
$\displaystyle\qquad
T^{eadbc}_{RLRLR}-T^{eadcb}_{RLRLR}+T^{ebacd}_{LLRLR}+T^{ebacd}_{RLRLL}-T^{ebadc}_{LLRLR}-T^{ebcad}_{RLLRL}+T^{ebcad}_{RLRLR}+T^{ebcda}_{LLRLR}-$
$\displaystyle\qquad
T^{ebdac}_{RLRLR}-T^{ebdca}_{LLRLR}-T^{ecabd}_{LLRLR}-T^{ecabd}_{RLRLL}+T^{ecadb}_{LLRLR}+T^{ecbad}_{RLLRL}-T^{ecbad}_{RLRLR}-T^{ecbda}_{LLRLR}+$
$\displaystyle\qquad
T^{ecdab}_{RLRLR}+T^{ecdba}_{LLRLR}+T^{edabc}_{LLRLR}-T^{edacb}_{LLRLR}+T^{edbac}_{RLRLR}+T^{edbca}_{LLRLR}-T^{edcab}_{RLRLR}-T^{edcba}_{LLRLR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{9}(-T^{bacde}_{RLLLR}+T^{bacde}_{RRLLR}+T^{baced}_{RLLLR}-T^{baced}_{RRLLR}+T^{badce}_{RLLLR}-T^{badce}_{RRLLR}-T^{badec}_{RLLLR}+T^{badec}_{RRLLR}-$
$\displaystyle\qquad
T^{baecd}_{RLLLR}+T^{baecd}_{RRLLR}+T^{baedc}_{RLLLR}-T^{baedc}_{RRLLR}-T^{bcdae}_{RLLLR}+T^{bcdae}_{RLLRR}+T^{bcead}_{RLLLR}-T^{bcead}_{RLLRR}+$
$\displaystyle\qquad
T^{bdcae}_{RLLLR}-T^{bdcae}_{RLLRR}-T^{bdeac}_{RLLLR}+T^{bdeac}_{RLLRR}-T^{becad}_{RLLLR}+T^{becad}_{RLLRR}+T^{bedac}_{RLLLR}-T^{bedac}_{RLLRR}+$
$\displaystyle\qquad
T^{cabde}_{RLLLR}-T^{cabde}_{RRLLR}-T^{cabed}_{RLLLR}+T^{cabed}_{RRLLR}-T^{cadbe}_{RLLLR}+T^{cadbe}_{RRLLR}+T^{cadeb}_{RLLLR}-T^{cadeb}_{RRLLR}+$
$\displaystyle\qquad
T^{caebd}_{RLLLR}-T^{caebd}_{RRLLR}-T^{caedb}_{RLLLR}+T^{caedb}_{RRLLR}+T^{cbdae}_{RLLLR}-T^{cbdae}_{RLLRR}-T^{cbead}_{RLLLR}+T^{cbead}_{RLLRR}-$
$\displaystyle\qquad
T^{cdbae}_{RLLLR}+T^{cdbae}_{RLLRR}+T^{cdeab}_{RLLLR}-T^{cdeab}_{RLLRR}+T^{cebad}_{RLLLR}-T^{cebad}_{RLLRR}-T^{cedab}_{RLLLR}+T^{cedab}_{RLLRR}-$
$\displaystyle\qquad
T^{dabce}_{RLLLR}+T^{dabce}_{RRLLR}+T^{dabec}_{RLLLR}-T^{dabec}_{RRLLR}+T^{dacbe}_{RLLLR}-T^{dacbe}_{RRLLR}-T^{daceb}_{RLLLR}+T^{daceb}_{RRLLR}-$
$\displaystyle\qquad
T^{daebc}_{RLLLR}+T^{daebc}_{RRLLR}+T^{daecb}_{RLLLR}-T^{daecb}_{RRLLR}-T^{dbcae}_{RLLLR}+T^{dbcae}_{RLLRR}+T^{dbeac}_{RLLLR}-T^{dbeac}_{RLLRR}+$
$\displaystyle\qquad
T^{dcbae}_{RLLLR}-T^{dcbae}_{RLLRR}-T^{dceab}_{RLLLR}+T^{dceab}_{RLLRR}-T^{debac}_{RLLLR}+T^{debac}_{RLLRR}+T^{decab}_{RLLLR}-T^{decab}_{RLLRR}+$
$\displaystyle\qquad
T^{eabcd}_{RLLLR}-T^{eabcd}_{RRLLR}-T^{eabdc}_{RLLLR}+T^{eabdc}_{RRLLR}-T^{eacbd}_{RLLLR}+T^{eacbd}_{RRLLR}+T^{eacdb}_{RLLLR}-T^{eacdb}_{RRLLR}+$
$\displaystyle\qquad
T^{eadbc}_{RLLLR}-T^{eadbc}_{RRLLR}-T^{eadcb}_{RLLLR}+T^{eadcb}_{RRLLR}+T^{ebcad}_{RLLLR}-T^{ebcad}_{RLLRR}-T^{ebdac}_{RLLLR}+T^{ebdac}_{RLLRR}-$
$\displaystyle\qquad
T^{ecbad}_{RLLLR}+T^{ecbad}_{RLLRR}+T^{ecdab}_{RLLLR}-T^{ecdab}_{RLLRR}+T^{edbac}_{RLLLR}-T^{edbac}_{RLLRR}-T^{edcab}_{RLLLR}+T^{edcab}_{RLLRR})$
$\displaystyle\leavevmode\nobreak\
c^{11}_{10}(T^{bcade}_{RLLLR}-T^{bcaed}_{RLLLR}-T^{bcdea}_{RLLRR}+T^{bceda}_{RLLRR}-T^{bdace}_{RLLLR}+T^{bdaec}_{RLLLR}+T^{bdcea}_{RLLRR}-T^{bdeca}_{RLLRR}+$
$\displaystyle\qquad
T^{beacd}_{RLLLR}-T^{beadc}_{RLLLR}-T^{becda}_{RLLRR}+T^{bedca}_{RLLRR}-T^{cbade}_{RLLLR}+T^{cbaed}_{RLLLR}+T^{cbdea}_{RLLRR}-T^{cbeda}_{RLLRR}+$
$\displaystyle\qquad
T^{cdabe}_{RLLLR}-T^{cdaeb}_{RLLLR}-T^{cdbea}_{RLLRR}+T^{cdeba}_{RLLRR}-T^{ceabd}_{RLLLR}+T^{ceadb}_{RLLLR}+T^{cebda}_{RLLRR}-T^{cedba}_{RLLRR}+$
$\displaystyle\qquad
T^{dbace}_{RLLLR}-T^{dbaec}_{RLLLR}-T^{dbcea}_{RLLRR}+T^{dbeca}_{RLLRR}-T^{dcabe}_{RLLLR}+T^{dcaeb}_{RLLLR}+T^{dcbea}_{RLLRR}-T^{dceba}_{RLLRR}+$
$\displaystyle\qquad
T^{deabc}_{RLLLR}-T^{deacb}_{RLLLR}-T^{debca}_{RLLRR}+T^{decba}_{RLLRR}-T^{ebacd}_{RLLLR}+T^{ebadc}_{RLLLR}+T^{ebcda}_{RLLRR}-T^{ebdca}_{RLLRR}+$
$\displaystyle\qquad
T^{ecabd}_{RLLLR}-T^{ecadb}_{RLLLR}-T^{ecbda}_{RLLRR}+T^{ecdba}_{RLLRR}-T^{edabc}_{RLLLR}+T^{edacb}_{RLLLR}+T^{edbca}_{RLLRR}-T^{edcba}_{RLLRR})\,.$

### A.3 Solution of Wess-Zumino conditions in the bosonic sector

#### A.3.1 P-even

$\displaystyle c^{3}_{RLL}-ic^{0}-c^{3}_{LLR}=0$ (A.9) $\displaystyle
c^{3}_{LLL}=ic^{0}$ $\displaystyle
c^{2}_{LLR}-4c^{6}+c^{2}_{LLL}+2c^{2}_{RLL}=0$ $\displaystyle
c^{3}_{LLR}=-ic^{0}+2c^{6}$ $\displaystyle c^{5}+c^{4}-2c^{6}=0$
$\displaystyle c^{7}_{LRRR}=-ic^{6}+\frac{i}{2}c^{2}_{RLL}+c^{7}_{LRLL}$
$\displaystyle c^{7}_{LLRL}=-\frac{i}{2}c^{2}_{RLL}+c^{7}_{LLLR}$
$\displaystyle c^{7}_{LRRR}=2ic^{6}-ic^{2}_{RLL}+c^{7\prime}_{LLRL}$
$\displaystyle
c^{8\prime}_{LRLL}=4ic^{6}-ic^{2}_{RLL}-c^{8}_{LLLR}+c^{8}_{LLRL}+c^{8\prime}_{LLRL},$
$\displaystyle c^{8\prime\prime}_{LLLL}=c^{8\prime}_{LLLL}$ $\displaystyle
c^{8\prime\prime}_{LLLR}-2ic^{4}+2ic^{6}+ic^{2}_{RLL}-2c^{7}_{LRLL}+c^{8}_{LLRL}-c^{8}_{LRLL}$
$\displaystyle
c^{8\prime}_{LLLL}-2ic^{6}+4c^{7}_{LLLR}+2c^{7}_{LLRR}+4c^{7}_{LRLL}+2c^{7}_{LRLR}+2c^{7}_{LRRL}$
$\displaystyle
c^{8}_{LRRL}=-4ic^{6}+2ic^{2}_{RLL}-4c^{7\prime}_{LLRL}-c^{8}_{LLLL}-c^{8}_{LLLR}-c^{8}_{LLRR}-c^{8}_{LRLL}-c^{8}_{LRLR}$
$\displaystyle
c^{8}_{LLLL}+4ic^{6}-2ic^{2}_{RLL}+4c^{7\prime}_{LLLR}+4c^{7\prime}_{LLRL}+4c^{7\prime}_{LLRR}+2c^{7\prime}_{LRLR}=0$
$\displaystyle
c^{8\prime\prime}_{LLRL}-2ic^{4}+2Ic^{2}_{RLL}-4c^{7}_{LLLR}+c^{8\prime}_{LLLR}$
$\displaystyle
c^{8\prime\prime}_{LLRR}+2ic^{4}-4c^{7}_{LRRL}+c^{8\prime}_{LRRL}=0$
$\displaystyle c^{8\prime\prime}_{LRLL}-4c^{7}_{LLLR}+c^{8\prime}_{LLRL}=0$
$\displaystyle c^{8}_{LRLR}=2c^{7\prime}_{LRLR}$ $\displaystyle
c^{8\prime\prime}_{LRRL}-4c^{7}_{LLRR}+c^{8\prime}_{LLRR}=0$ $\displaystyle
c^{8}_{LRLL}=-2ic^{4}+4c^{7\prime}_{LLLR}-c^{8}_{LLLR}$ $\displaystyle
c^{8\prime\prime}_{LRLR}-4c^{7}_{LRLR}+c^{8\prime}_{LRLR}=0$ $\displaystyle
2c^{8}_{LLLR}=-2ic^{2}_{RLL}+4c^{7\prime}_{LLLR}$ $\displaystyle
c^{8\prime}_{LRRL}=-8ic^{6}+4c^{7}_{LLLR}+2c^{7}_{LLRR}+2c^{7}_{LRLL}+2c^{7}_{LRLR}+2c^{7}_{LRRL}+4c^{7\prime}_{LLLR}$
$\displaystyle\qquad\qquad-2c^{8}_{LLRL}-c^{8\prime}_{LLLR}-2c^{8\prime}_{LLRL}-c^{8\prime}_{LLRR}-c^{8\prime}_{LRLR}$
$\displaystyle 4c^{7\prime}_{LLRL}=-8ic^{6}+2ic^{2}_{RLL}+4c^{7\prime}_{LLLR}$
$\displaystyle 2c^{7}_{LRLL}=2ic^{6}-ic^{2}_{RLL}+2c^{7}_{LLLR}$
$\displaystyle c^{8\prime}_{LLLR}=2ic^{4}-ic^{2}_{RLL}+2c^{7}_{LLLR}$
$\displaystyle
c^{8\prime}_{LLRL}+4ic^{6}-2c^{7}_{LLLR}-2c^{7\prime}_{LLLR}+c^{8}_{LLRL}$
$\displaystyle c^{8\prime}_{LRLR}=2c^{7}_{LRLR}$ $\displaystyle
c^{8\prime}_{LLRR}=-2ic^{4}+2Ic^{6}+2c^{7}_{LLRR}-2c^{7\prime}_{LLRR}+c^{8}_{LLRR}$
$\displaystyle c^{8}_{LLRL}=-4ic^{6}+ic^{2}_{RLL}+2c^{7\prime}_{LLLR}$
$\displaystyle c^{8}_{LLRR}=2ic^{4}-ic^{2}_{RLL}+2c^{7\prime}_{LLRR}$
$\displaystyle 4c^{10}_{11}-ic^{7\prime}_{LRLR}=0$ $\displaystyle
4c^{10}_{8}+ic^{7}_{LLLR}=0$ $\displaystyle
c^{10}_{7}=c^{10}_{13}=-c^{10}_{8}$ $\displaystyle
c^{10}_{1}=c^{10}_{2}=c^{10}_{3}=c^{10}_{4}=c^{10}_{5}=c^{10}_{6}=0$
$\displaystyle 8c^{10}_{9}=c^{2}_{RLL}+2ic^{7\prime}_{LLLR}=0$ $\displaystyle
4c^{10}_{10}=4c^{10}_{14}=-ic^{7}_{LRLR}=0$ $\displaystyle
4c^{10}_{16}-ic^{7}_{LLLR}=0$ $\displaystyle 4c^{10}_{17}+ic^{7}_{LLRR}=0$
$\displaystyle 4c^{10}_{12}+\frac{1}{2}c^{2}_{RLL}+ic^{7\prime}_{LLLR}=0$
$\displaystyle 4c^{10}_{18}=c^{4}-\frac{1}{2}c^{2}_{RLL}+ic^{7}_{LRRL}$
$\displaystyle 4c^{10}_{15}=c^{4}-\frac{1}{2}c^{2}_{RLL}-ic^{7\prime}_{LLRR}$

#### A.3.2 P-odd

$\displaystyle c^{9\prime}_{LLLL}=ic^{1}_{LLL}-c^{9}_{LLLL}$ (A.10)
$\displaystyle c^{9\prime}_{LRRR}=c^{9}_{LLLR}$ $\displaystyle
c^{9\prime}_{LLLR}=ic^{1}_{RLL}+c^{9}_{LRRR}$ $\displaystyle
c^{9}_{LRRR}=-ic^{1}_{RLL}-c^{9}_{LLRL}$ $\displaystyle
c^{9\prime}_{LLRL}=ic^{1}_{LLR}-c^{9}_{LRLL}$ $\displaystyle
c^{9}_{LRRL}=ic^{1}_{LLR}$ $\displaystyle c^{9\prime}_{LLRR}=-ic^{1}_{LLR}$
$\displaystyle c^{9\prime}_{LRLR}=c^{9}_{LRLR}$ $\displaystyle
2c^{9}_{LLLL}-ic^{1}_{LLL}=0$ $\displaystyle
c^{9}_{LLRL}+ic^{1}_{RLL}+c^{9}_{LLLR}=0$ $\displaystyle
c^{9}_{LRLL}+ic^{1}_{LLR}-c^{9}_{LLLR}=0$ $\displaystyle
c^{1}_{RLL}+2c^{1}_{LLR}=0$ $\displaystyle c^{9}_{LLRR}=0$ $\displaystyle
c^{11}_{1}=c^{11}_{3}=c^{11}_{7}=c^{11}_{10}=c^{11}_{8}=c^{11}_{2}=c^{11}_{9}=0$
$\displaystyle 12c^{1}_{4}+ic^{9}_{LLLR}=0$ $\displaystyle
12c^{1}_{6}+ic^{9}_{LRLR}=0$ $\displaystyle 12c^{1}_{5}-ic^{9}_{LLLR}=0$

##  Appendix B Heat kernel

The heat kernel method was pioneered by Schwinger and then developed by De
Witt and Seeley. For a lucid review, we refer the reader to Ref. [51].

This method allows us to write the matrix element of ${\not{D}}^{-2}$ in
position space as

$\displaystyle\langle x|\frac{1}{{\not{D}}^{2}}|y\rangle=i\int
dt\leavevmode\nobreak\ \langle
x|e^{-i{\not{D}}^{2}t}|y\rangle=i\int_{0}^{\infty}dt\leavevmode\nobreak\
H(x,y;t).$ (B.1)

The solution $H(x,y;t)$, which is referred to as the heat kernel, can be
calculated perturbatively in the limit $t\to 0$. We write the ansatz
$H(x,y;t)=H_{0}(x,y;t)U(x,y;t)$, with the “free” solution $H_{0}$ being the
solution of (B.1) with ${\not{D}}^{2}$ replaced by $\partial^{2}$, namely

$\displaystyle H_{0}(x,y;t)=\frac{i}{(4\pi
it)^{d/2}}e^{i\frac{(x-y)^{2}}{4t}-\varepsilon t},$ (B.2)

and

$\displaystyle U(x,y;t)=\sum_{n=0,1,2,\cdots}a_{n}(x,y)(it)^{n}.$ (B.3)

The heat kernel coefficients $a_{n}(x,y)$ are smooth in the limit $y\to x$,
and satisfy the boundary condition $a_{0}(x,y)=1$, i.e.
$H(x,y;0)=\delta^{(d)}(x-y)$. The parameter $\varepsilon>0$ follows from the
$i\varepsilon$ prescription in the Feynman propagator and should not be
confused with $\epsilon=(4-d)/2$.

Employing the above expansion we get

$\displaystyle{\rm
Tr}\left[\alpha_{a}{T}^{a}_{A}\left\\{{\not{D}},\gamma_{5}\right\\}\frac{1}{{\not{D}}}\right]$
$\displaystyle=$ $\displaystyle{\rm
Tr}\left[\alpha_{a}{T}^{a}_{A}\left\\{{\not{D}},\gamma_{5}\right\\}{\not{D}}\frac{1}{{\not{D}}^{2}}\right]$
$\displaystyle=$ $\displaystyle i\int d^{d}x\leavevmode\nobreak\
\alpha_{a}(x)\lim_{y\to x}\int_{0}^{\infty}dt\leavevmode\nobreak\ {\rm
tr}\left[T_{A}^{a}\left(\left\\{{\not{D}},\gamma_{5}\right\\}{\not{D}}\right)_{y}H_{0}(y,x)U(x,y)\right]$
$\displaystyle=$ $\displaystyle i\int d^{d}x\leavevmode\nobreak\
\alpha_{a}(x)\lim_{y\to x}\int_{0}^{\infty}dt\leavevmode\nobreak\ {\rm
tr}\left[T_{A}^{a}\left(2\gamma^{\hat{\mu}}\gamma_{5}\gamma^{\nu}\right)\partial_{\hat{\mu},y}[(\partial_{\nu,y}H_{0})U+H_{0}D_{\nu,y}U]\right]$
$\displaystyle=$ $\displaystyle i\int d^{d}x\leavevmode\nobreak\
\alpha_{a}(x)\lim_{y\to x}\int_{0}^{\infty}dt\leavevmode\nobreak\ {\rm
tr}\left[T_{A}^{a}\left(2\gamma^{\hat{\mu}}\gamma_{5}\gamma^{\nu}\right)(\partial_{\hat{\mu},y}\partial_{\nu,y}H_{0})U\right]+{\rm
Eva}$ $\displaystyle=$ $\displaystyle-(d-4)\int d^{d}x\leavevmode\nobreak\
\alpha_{a}(x)\lim_{y\to
x}\int_{0}^{\infty}\frac{dt}{t}H_{0}(x,y)\leavevmode\nobreak\ {\rm
tr}\left[T_{A}^{a}\gamma_{5}U(x,y)\right]+{\rm Eva}.$

In the second equality of (B) we merely used the definition of heat kernel,
and in the third applied the derivatives. In the fourth step we took advantage
of the fact that all terms with a single derivative of $H_{0}$ vanish in the
limit ${y\to x}$. The non-vanishing contributions come from the second
derivative $\lim_{y\to
x}\partial_{\mu}\partial_{\nu}H_{0}=+ig_{\mu\nu}H_{0}/(2t)$ as well as
evanescent terms proportional to $D_{\hat{\nu}}U$. These latter can be
neglected, as explained around Eq. (4.15). The last equality follows from the
identity $\gamma^{\hat{\mu}}\gamma_{\hat{\mu}}=(d-4)$. Finally, the integral
in $dt$ can be performed explicitly for any order $n$ of the perturbative
expansion of $U$, and is proportional to $\Gamma(d/2-n)$.242424IR-divergences
at large $t$ are cutoff by the factor $e^{-\varepsilon t}$ in $H_{0}(x,y)$.
There is a unique contribution that survives the $d\to 4$ limit. This emerges
from an UV divergence $t\to 0$ that results in a factor $\Gamma(d/2-n)\sim
1/(d/2-n)$. The latter can exactly compensate the $(d-4)$ in front of the
integral, and thus lead to a non-trivial result, only at the $n=2$ order of
the expansion (B.3). The existence of a compensation between the evanescent
$(d-4)$ factor and UV divergent Feynman integrals is typical of quantum
anomalies. The result is

$\displaystyle\lim_{d\to 4}{\rm
Tr}\left[\alpha_{a}{T}^{a}_{A}\left\\{{\not{D}},\gamma_{5}\right\\}\frac{1}{{\not{D}}}\right]$
$\displaystyle=$ $\displaystyle-\frac{i}{8\pi^{2}}\int
d^{d}x\leavevmode\nobreak\ \alpha_{a}(x)\leavevmode\nobreak\ {\rm
tr}\left[T_{A}^{a}\gamma_{5}a_{2}(x,x)\right],$ (B.5)

where the $4$-dimensional limit is formally defined such that $\lim_{d\to
4}{\rm Eva}=0$ for all evanescent operators.

The $4-$dimensional limit of the heat-kernel coefficients $a_{n}(x,x)$ can be
obtained recursively. We first observe that ${\not{D}}^{2}={\cal D}_{\mu}{\cal
D}^{\mu}+X$, with ${\cal D}_{\mu}=\partial_{\mu}+iP_{\mu}$, and where
$P_{\mu},X$ explicitly read

$\displaystyle P^{\mu}$ $\displaystyle=$ $\displaystyle{\cal
V}^{\mu}+\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]\gamma_{5}{\cal A}_{\nu}$ (B.6)
$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]\left({\cal
V}_{\mu\nu}+\gamma_{5}{\cal
A}_{\mu\nu}\right)+i\gamma_{5}\gamma^{\mu}\gamma^{\nu}\left(\partial_{\nu}{\cal
A}_{\mu}+i[{\cal V}_{\nu},{\cal A}_{\mu}]\right)-2{\cal A}_{\alpha}{\cal
A}^{\alpha}.$

The field strengths of the vector and axial components were previously
introduced in (4.31). Now, the heat kernel, defined in (B.1), satisfies
$i\frac{d}{dt}H(x,y;t)={\not{D}}^{2}_{x}H(x,y;t)$. Inserting the ansatz
$H=H_{0}U$ this becomes $idU/dt=-i(x-y)_{\mu}{\cal D}^{\mu}U/t+[{\cal
D}^{2}+X]U$. Equating order by order in $t^{n}$ gives the recursive relations

$\displaystyle(x-y)_{\mu}{\cal{D}}^{\mu}_{x}a_{0}(x,y)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (B.7)
$\displaystyle[n+1+(x-y)_{\mu}{\cal{D}}^{\mu}_{x}]a_{n+1}(x,y)=-[{\cal
D}_{x}^{2}+X]a_{n}(x,y)\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ (n>0).$

The first definition, along with $\lim_{y\to x}a_{0}(x,y)=1$ defines
$a_{0}(x,y)$. We are interested in $a_{2}(x,x)=-\frac{1}{2}\lim_{y\to
x}[{\cal{D}}^{2}_{x}+X]a_{1}(x,y)$, but to find its explicit expression we
need $a_{1}(x,y)$ and its second derivative:

$\displaystyle\lim_{y\to x}a_{1}(x,y)=-\lim_{y\to x}[{\cal
D}^{2}_{x}+X]a_{0}(x,y)$ (B.8) $\displaystyle\lim_{y\to x}{\cal
D}^{2}_{x}a_{1}(x,y)=-\frac{1}{3}\lim_{y\to x}{\cal D}^{2}_{x}[{\cal
D}^{2}_{x}+X]a_{0}(x,y)$

The first relation follows directly from the second equation in (B.7).
Differentiating twice the same relation with $n=0$ with respect to ${\cal D}$
we obtain the other one. Similarly, differentiating the first relation in
(B.7) we derive $\lim_{y\to x}{\cal D}_{\alpha}a_{0}(x,y)=\lim_{y\to x}{\cal
D}^{2}a_{0}(x,y)=0$. This leads us to

$\displaystyle a_{2}(x,x)$ $\displaystyle=$ $\displaystyle\lim_{y\to
x}\frac{1}{6}{\cal D}_{x}^{2}{\cal D}_{x}^{2}a_{0}(x,y)+\frac{1}{6}{\cal
D}^{2}X+\frac{1}{2}X^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{12}[{\cal
D}_{\mu},{\cal D}_{\nu}][{\cal D}^{\mu},{\cal D}^{\nu}]+\frac{1}{6}{\cal
D}^{2}X+\frac{1}{2}X^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{12}P_{\mu\nu}P^{\mu\nu}+\frac{1}{6}{\cal
D}^{2}X+\frac{1}{2}X^{2},$

where
$P_{\mu\nu}=\partial_{\mu}P_{\nu}-\partial_{\nu}P_{\mu}+i[P_{\mu},P_{\nu}]$.
In evaluating $a_{2}$ we used the linearity of the derivative, namely ${\cal
D}_{\mu}[Xa_{0}]=[{\cal D}_{\mu}X]a_{0}+X[{\cal D}_{\mu}a_{0}]$. The relation
${\cal D}_{x}^{2}{\cal D}_{x}^{2}a_{0}(x,y)=\frac{1}{2}[{\cal D}_{\mu},{\cal
D}_{\nu}][{\cal D}^{\mu},{\cal D}^{\nu}]a_{0}(x,y)$ is proven differentiating
four times the first equation in (B.7), and contracting with the metric
tensor. Because $[{\cal D}_{\mu},{\cal D}_{\nu}]$ is not a differential
operator, the limit ${y\to x}$ can be performed trivially and (B) follows.

## References

  * [1] R. Stora, Lagrangian field theory, contribution to Les Houches Summer School on Theoretical Physics, pp. 1-80, $21^{st}$ conference in the Les Houches Summer School series, 29 July-1 September 1971, Les Houches, France.
  * [2] C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42 (1975), 127-162 doi:10.1007/BF01614158
  * [3] C. Becchi, A. Rouet and R. Stora, Annals Phys. 98 (1976), 287-321 doi:10.1016/0003-4916(76)90156-1
  * [4] C. Becchi, A. Rouet and R. Stora, Renormalizable Theories with Symmetry Breaking, in Field Quantization and Statistical Physics, In Memory of Bernard Jouvet, p. 3, E. Tirapegui (Ed.), D. Reidel Publishing Company, Dordrecht, Holland (1981), ISBN 90-277-1128-3
  * [5] O. Piguet and S. P. Sorella, Lect. Notes Phys. Monogr. 28 (1995), 1-134 doi:10.1007/978-3-540-49192-7
  * [6] E. Kraus, Annals Phys. 262 (1998), 155-259 doi:10.1006/aphy.1997.5746 [arXiv:hep-th/9709154 [hep-th]].
  * [7] R. Ferrari and P. A. Grassi, Phys. Rev. D 60 (1999), 065010 doi:10.1103/PhysRevD.60.065010 [arXiv:hep-th/9807191 [hep-th]].
  * [8] P. A. Grassi, T. Hurth and M. Steinhauser, Annals Phys. 288 (2001), 197-248 doi:10.1006/aphy.2001.6117 [arXiv:hep-ph/9907426 [hep-ph]].
  * [9] P. A. Grassi, T. Hurth and M. Steinhauser, Nucl. Phys. B 610 (2001), 215-250 doi:10.1016/S0550-3213(01)00303-0 [arXiv:hep-ph/0102005 [hep-ph]].
  * [10] S. Weinberg, Phys. Rev. 118 (1960), 838-849 doi:10.1103/PhysRev.118.838
  * [11] W. Zimmermann, Commun. Math. Phys. 11 (1968), 1-8 doi:10.1007/BF01654298
  * [12] J. H. Lowenstein, Commun. Math. Phys. 24 (1971), 1-21 doi:10.1007/BF01907030
  * [13] Y. M. P. Lam, Phys. Rev. D 6 (1972), 2145-2161 doi:10.1103/PhysRevD.6.2145
  * [14] T. E. Clark and J. H. Lowenstein, Nucl. Phys. B 113 (1976), 109-134 doi:10.1016/0550-3213(76)90457-0
  * [15] F. Brennecke and M. Duetsch, doi:10.1007/978-3-7643-8736-5_11 [arXiv:0801.1408 [hep-th]].
  * [16] O. Piguet and A. Rouet, Phys. Rept. 76 (1981), 1 doi:10.1016/0370-1573(81)90066-1
  * [17] S. L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177 (1969), 2426-2438.
  * [18] J. S. Bell and R. Jackiw, “A PCAC puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$ model,” Nuovo Cim. A 60 (1969), 47-61.
  * [19] G. Bonneau, Int. J. Mod. Phys. A 5 (1990), 3831-3860 doi:10.1142/S0217751X90001641
  * [20] C. P. Martin and D. Sanchez-Ruiz, Nucl. Phys. B 572 (2000), 387-477 doi:10.1016/S0550-3213(99)00453-8 [arXiv:hep-th/9905076 [hep-th]].
  * [21] D. Sanchez-Ruiz, Phys. Rev. D 68 (2003), 025009 doi:10.1103/PhysRevD.68.025009 [arXiv:hep-th/0209023 [hep-th]].
  * [22] H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, JHEP 08 (2020) no.08, 024 doi:10.1007/JHEP08(2020)024 [arXiv:2004.14398 [hep-ph]].
  * [23] H. Bélusca-Maïto, A. Ilakovac, P. Kühler, M. Mađor-Božinović and D. Stöckinger, JHEP 11 (2021), 159 doi:10.1007/JHEP11(2021)159 [arXiv:2109.11042 [hep-ph]].
  * [24] J. Wess and B. Zumino, “Consequences of anomalous Ward identities,” Phys. Lett. B 37 (1971), 95-97.
  * [25] C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B 12 (1972), 20-26 doi:10.1007/BF02895558
  * [26] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44 (1972), 189-213 doi:10.1016/0550-3213(72)90279-9
  * [27] Q. Bonnefoy, L. Di Luzio, C. Grojean, A. Paul and A. N. Rossia, JHEP 05 (2021), 153 doi:10.1007/JHEP05(2021)153 [arXiv:2012.07740 [hep-ph]].
  * [28] F. Feruglio, JHEP 03 (2021), 128 doi:10.1007/JHEP03(2021)128 [arXiv:2012.13989 [hep-ph]].
  * [29] G. Passarino, Acta Phys. Polon. B 52 (2021) no.6-7, 533 doi:10.5506/APhysPolB.52.533 [arXiv:2104.13569 [hep-ph]].
  * [30] H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12 (1975), 467-481 doi:10.1103/PhysRevD.12.467
  * [31] H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12 (1975), 482-488 doi:10.1103/PhysRevD.12.482
  * [32] L. F. Abbott, Acta Phys. Polon. B 13 (1982), 33 CERN-TH-3113.
  * [33] S. Ichinose and M. Omote, Nucl. Phys. B 203 (1982), 221-267 doi:10.1016/0550-3213(82)90029-3
  * [34] D. M. Capper and A. MacLean, Nucl. Phys. B 203 (1982), 413-422 doi:10.1016/0550-3213(82)90321-2
  * [35] P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977), 11-38 doi:10.1007/BF01609069
  * [36] P. Breitenlohner and D. Maison, MPI-PAE-PTH-26-75.
  * [37] P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977), 39 doi:10.1007/BF01609070
  * [38] P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977), 55 doi:10.1007/BF01609071
  * [39] W. Grimus and M. N. Rebelo, Phys. Rept. 281 (1997), 239-308 doi:10.1016/S0370-1573(96)00030-0 [arXiv:hep-ph/9506272 [hep-ph]].
  * [40] H. Georgi and S. L. Glashow, Phys. Rev. D 6 (1972), 429 doi:10.1103/PhysRevD.6.429
  * [41] G. Durieux, J. Gu, E. Vryonidou and C. Zhang, “Probing top-quark couplings indirectly at Higgs factories,” Chin. Phys. C 42 (2018) no.12, 123107 [arXiv:1809.03520 [hep-ph]].
  * [42] C. Degrande, G. Durieux, F. Maltoni, K. Mimasu, E. Vryonidou and C. Zhang, “Automated one-loop computations in the SMEFT,” [arXiv:2008.11743 [hep-ph]].
  * [43] J. G. Korner, D. Kreimer and K. Schilcher, Z. Phys. C 54, 503-512 (1992) doi:10.1007/BF01559471
  * [44] H. Nicolai and P. K. Townsend, Phys. Lett. B 93, 111-115 (1980) doi:10.1016/0370-2693(80)90106-9
  * [45] A. P. Balachandran, G. Marmo, V. P. Nair and C. G. Trahern, Phys. Rev. D 25, 2713 (1982) doi:10.1103/PhysRevD.25.2713
  * [46] M. S. Chanowitz, M. Furman and I. Hinchliffe, Nucl. Phys. B 159 (1979), 225-243 doi:10.1016/0550-3213(79)90333-X
  * [47] M. S. Chanowitz, M. Furman and I. Hinchliffe, Nucl. Phys. B 159 (1979), 225-243 doi:10.1016/0550-3213(79)90333-X
  * [48] D. Kreimer, Phys. Lett. B 237 (1990), 59-62 doi:10.1016/0370-2693(90)90461-E
  * [49] J. G. Korner, D. Kreimer and K. Schilcher, Z. Phys. C 54 (1992), 503-512 doi:10.1007/BF01559471
  * [50] F. Jegerlehner, Eur. Phys. J. C 18 (2001), 673-679 doi:10.1007/s100520100573 [arXiv:hep-th/0005255 [hep-th]].
  * [51] B. S. DeWitt, “Dynamical Theory of Groups and Fields”, Gordon and Breach.