& -\frac{9}{8}g_f^2  \left(\bar{l}i\slashed{D}l+\bar{q}i\slashed{D}q\right),
\end{align}

where we defined $L\equiv \log \left(m_V^2/Q^2\right)$ with $Q$ being the matching scale and $\kappa = 1/(16\pi^2)$. Introducing the canonically normalised fields in the EFT as

\begin{align}
\hat{\varphi} &= \left(1+\frac{1}{2}\delta Z_\varphi\right) \varphi,\, \varphi \in \{H,l,q,W^A_\mu\}
\end{align}

we find

\begin{align}
\delta Z_H &= \kappa\frac{g_{H}^2(-60 m_V^2 - \mu_h^2 + 6(12 m_V^2 + \mu_h^2) \reslog)}{32 m_V^2},\\
\delta Z_l &= \delta Z_q = -\frac{9}{8}\kappa g_{f}^2, \\
\delta Z_{W^A_\mu} &= \kappa\left(-4(g_2 - 2g_{2VW}) g_{2VW} + 3 g_2 (g_2 + 4 g_{2VW})
\reslog\right),
\end{align}

which gives rise to the following contributions to one-loop Wilson coefficients

\begin{align}
\label{eq: Wilson_from_field_redef_1}
C^{1\Loop,\delta Z}_{H} &=\lambda\frac{g_{H}^4(-45  + 54 \reslog)}{16 m_V^2}, \\
C^{1\Loop,\delta Z}_{H\Box} &= \frac{g_{H}^4(-45   + 54   \reslog)}{32m_V^2},\\
C^{1 \Loop ,\delta Z ij}_{\psi H}& = \left(\frac{3g_{H}^2(-15  + 18
\reslog)}{16}-\frac{9}{16}g_f^2\right)\frac{g_{H}^2 }{4 m_V^2}y_{\psi ij}, \\
C^{1\Loop, \delta Z ,ij}_{H\Psi^{(3)}} &= \frac{g_{f}g_{H} }{4 m_V^2}\left(-\frac{g_{H}^2(-15  + 18\reslog)}{8}+\frac{9}{8}g_{f}^2\right)\delta^{ij},\\
C^{1\Loop, \delta Z ijkl}_{ll} &= \frac{9}{32 m_V^2}g_{f}^4 (\delta^{ij} \delta^{kl} -
2 \delta^{il} \delta^{jk}),\\
\label{eq: Wilson_from_field_redef_last}
C^{1\Loop,\delta Z ijkl}_{qq^{(3)}}&= -\frac{9}{32 m_V^2}g_{f}^4 \delta^{ij}\delta^{kl},\\
C^{1\Loop,\delta Z ijkl}_{lq^{(3)}}&= -\frac{9}{16 m_V^2}g_{f}^4 \delta^{ij}\delta^{kl},
\end{align}

where $\psi$ runs over all $SU(2)_L$ singlet fermions and $\Psi$ over all $SU(2)_L$ doublet fermions and we define

\begin{align}
C_Q = C^{\text{tree}}_Q +  \kappa C^{1\Loop}_Q,
\end{align}

that is we extract a factor of $1/(16\pi^2)$ from the one-loop part of each Wilson coefficient. The remainder of the renormalisable part of the Lagrangian reads

\begin{align}
\frac{1}{\kappa} \LagNoT_{\text{EFT},d \leq 4}^{1\Loop}  ={}& 3g_{VH} m_V^2 |H|^2 (1 - 3\reslog)
\nonumber \\
&-\big[\frac{-216 g_2^2 g_{H}^2 m_V^2 + 3456 g_{VH}^2 m_{V}^2 + 144 g_2^4 \mu_h^2 +
229 g_2^2 g_{H}^2 \mu_h^2 - 90 g_{H}^4 \mu_h^2}{576 m_V^2} \nonumber \\
&+ \frac{2088 g_{H}^2 g_{VH} \mu_h^2 -
192 g_2^2 g_{H} g_{3V} \mu_h^2 - 720 g_{H}^3 g_{3V} \mu_h^2 + 216 g_{H}^2 g_{3V}^2 \mu_h^2}{576 m_V^2}
\nonumber \\
\frac{384 g_2^3 g_{2VW} \mu_h^2 - 1152 g_2 g_{H} g_{3V} g_{2VW} \mu_h^2 + 960 g_2^2 g_{2VW}^2 \mu_h^2 +
18 g_{H}^2 \lambda \mu_h^2}{576 m_V^2} \nonumber \\
& +\frac{- 1350 g_{H}^2 \lambda \mu_h^2 -
9 g_1^2 g_{H}^2 (8 m_V^2 - 15\mu_h^2) + 6(24 g_2^4 \mu_h^2 + 96g_2^3 g_{2VW} \mu_h^2}{576 m_V^2} \nonumber \\
& + \frac{-96 g_2 g_{H} (g_{H} + 6 g_{3V})g_{2VW} \mu_h^2 + 9 g_1^2 g_{H}^2 (4 m_V^2 - 3 \mu_h^2)}{576 m_V^2}\nonumber \\
\frac{18 (48 g_{VH}^2 m_V^2 - 16 g_{H}^2 g_{VH} \mu_h^2 +
g_{H}^2 (g_{H}^2 + 8 g_{H} g_{3V} - 12 g_{3V}^2 +14 \lambda)\mu_h^2)}{576 m_V^2} \nonumber \\
\frac{g_2^2(192 g_{H} g_{3V} \mu_h^2 - 32 g_{2VW}^2 \mu_h^2 +
g_{H}^2 (108 m_V^2 - 17 \mu_h^2)))\reslog}{576 m_V^2}\big] |H|^4 \nonumber
\\ &+ g_{H} \mu_h^2 \frac{12g_{f} - g_{H} + 6(-4g_{f} + g_{H})\reslog}
{64 m_V^2} \left(\bar{q}y_d d H + \bar{l}y_e e H + \bar{q}y_u u \tilde{H} + \text{h.c.} \right).
\end{align}

In addition to the ten operators present at tree level, the one-loop matching induces 33 further effective operators. Thus, at the one-loop level there are 43 dimension six operators induced in total. The Wilson coefficients, without the inclusion of Eqs. (<ref>)–(<ref>), are given below. They are organised according to the classes defined in Ref. <cit.>, again replacing $\varphi$ by $H$.

\begin{align}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{X^3} \nonumber \\
C_W^{1\Loop} ={}& \frac{(g_2^2 + 4 g_{2VW}^2)(g_{2VW}(4 - 6\reslog) + g_2(2 + 3\reslog))}{24 m_V^2},\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{H^6}\nonumber \\
C^{1\Loop}_{H} ={}& \frac{(-3456 g_{VH}^3 - 72 g_{H}^2 g_{VH} \lambda (-29 + 24\reslog) +
\lambda (144 g_2^4 (1 + \reslog)}{576 m_V^2} \nonumber \\
& + \frac{192 g_2^3 g_{2VW} (2 + 3\reslog) +
g_2^2 (g_{H}^2 (229 - 102 \reslog) - 192 g_{2VW}^2 (-5 + \reslog)}{576 m_V^2}
\nonumber \\ &+
\frac{192 g_{H} g_{3V} (-1 + 6\reslog)) - 576 g_2 g_{H} g_{2VW}
(g_{H} \reslog + g_{3V} (2 + 6\reslog))}{576 m_V^2}\nonumber  \\
\frac{9 g_{H}^2 (-3 g_1^2(-5 + 6 \reslog) + 2 (g_{3V}^2 (12 - 72\reslog) +
g_{H}^2 (-5 + 6\reslog)}{576 m_V^2}\nonumber  \\
& + \frac{8 g_{H} g_{3V} (-5 + 6 \reslog) +
2 \lambda (-37 + 42 \reslog)))))}{576 m_V^2},\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{H^4D^2}\nonumber \\
C^{1\Loop}_{H \Box} ={}& \frac{(2 g_1^2 g_{H}^2 (47 - 42\reslog) + 144 g_2^4 (1 + \reslog) +
192 g_2^3 g_{2VW} (2 + 3 \reslog)}{768 m_V^2} \nonumber \\
&+ \frac{-576 g_2 g_{H} g_{2VW}
(g_{H} \reslog + g_{3V} (2 + 6 \reslog)) -
2 g_2^2 (96 g_{2VW}^2 (-5 + \reslog)}{768 m_V^2} \nonumber \\
& +\frac{- 96 g_{H} g_{3V} (-1 + 6 \reslog) +
g_{H}^2 (-137 + 78 \reslog))}{768 m_V^2}
\nonumber  \\ &+ \frac{9 (-384 g_{VH}^2 \reslog +
g_{H}^4 (-5 + 6 \reslog) + 16 g_{H}^3 g_{3V} (-5 + 6 \reslog)}{768 m_V^2}\nonumber  \\
8 g_{H}^2 (3 \lambda (5 - 6 \reslog) + 4 g_{VH} (-4 + 3 \reslog) +
3 g_{3V}^2 (-1 + 6 \reslog))))}{768 m_V^2},\\
C^{1\Loop}_{HD} ={}& \frac{g_1^2 g_{H}^2 (-7 + 12\reslog)}{24 m_V^2},\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{X^2H^2}\nonumber \\
C^{1\Loop}_{HW} ={}&-\frac{(24 g_{H}^2 g_{2VW}^2 (3 - 2\reslog) +
6 g_2 g_{2VW}(96 g_{VH} + g_{H}^2 (-11 + 6 \reslog))}{96 m_V^2}\nonumber  \\
&- \frac{g_2^2 (72 g_{VH} + g_{H}^2(5 + 9 \reslog)))}{96 m_V^2},\\
C^{1\Loop}_{HB} ={}& \frac{-5 g_1^2 g_{H}^2}{64 m_V^2},\\
C^{1\Loop}_{HWB} ={}& g_1 \Big(\frac{(9 g_2^3 (1 + \reslog) + 12 g_2^2 g_{2VW}(2 + 3 \reslog)}{288 m_V^2}\nonumber  \\
\frac{18 g_{H} g_{2VW}(g_{3V} - 3 g_{3V} \reslog + g_{H}(-1 + 2\reslog))}{288 m_V^2}\nonumber  \\
&+ \frac{2 g_2 (3 g_{H} g_{3V} (3 + \reslog) + 3 g_{2VW}^2 (3 + \reslog) +
g_{H}^2 (-8 + 6 \reslog))}{288 m_V^2} \Big), \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{\psi^2XH}\nonumber \\
C^{1\Loop,ij}_{e/d W} ={}&   \big(\frac{g_{f}(-72(g_{f} - g_{H})g_{2VW}(-3 + 2 \reslog) + 18 g_2 g_{H} (-9 + 4 \reslog)}{576m_V^2}\nonumber \\ &+ \frac{g_2 g_{f}(-85 + 12 \reslog))}{576m_V^2}\big)y_{e/d}^{ij}, \\
C^{1\Loop,ij}_{u W} ={}&   \big(\frac{g_{f}(-72(g_{f} + g_{H})g_{2VW}(-3 + 2 \reslog) + 18 g_2 g_{H} (-9 + 4 \reslog)}{576m_V^2}\nonumber \\
&+ \frac{g_2 g_{f}(-85 + 12 \reslog))}{576m_V^2}\big)y_u^{ij}, \\
C^{1\Loop,ij}_{eB} ={}& \frac{g_1 g_{f} (4 g_{f} - 3 g_{H})}{32 m_V^2} y_e^{ij}, C^{1\Loop,ij}_{dB} = -\frac{g_1 g_{f} (4 g_{f} + 9 g_{H})}{96 m_V^2} y_d^{ij}, \\
C^{1\Loop,ij}_{uB} ={}& -\frac{g_1 g_{f} (4 g_{f} - 9 g_{H})}{96 m_V^2} y_u^{ij}, \\
C_{dG}^{1\Loop,ij} ={}& -\frac{g_3 g_{f}^2}{4m_V^2}y_d^{ij},\hspace{1mm} C_{uG}^{1\Loop,ij} = -\frac{g_3 g_{f}^2}{4m_V^2}y_u^{ij}, \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{\psi^2 H^3}\nonumber \\
C^{1\Loop,ij}_{eH} ={}&  \frac{(-144 g_{f}^2 (y_e y_e^\dagger y_e)_{ij} + y_{e}^{ij}
(144 g_2^4 (1 + \reslog) + 192 g_2^3 g_{2VW} (2 + 3 \reslog)} {1152 m_V^2}\nonumber  \\
\frac{g_2^2 (g_{H}^2 (229 - 102 \reslog) - 192 g_{2VW}^2 (-5 + \reslog) +
192 g_{H} g_{3V}(-1 + 6 \reslog))}{1152 m_V^2}\nonumber  \\
&+\frac{- 576g_2 g_{H} g_{2VW}
(g_{H} \reslog + g_{3V} (2 + 6 \reslog))}{1152 m_V^2}\nonumber  \\
\frac{9 g_{H} (-3 g_1^2 g_{H} (-5 + 6 \reslog) + 2 (12 g_{f} \lambda (-1 + 2 \reslog) +
g_{H}^3(-5 + 6 \reslog)}{1152 m_V^2}\nonumber  \\
& +\frac{8 g_{H}^2 g_{3V} (-5 + 6 \reslog) +
g_{H} (g_{VH} (116 - 96 \reslog)}{1152 m_V^2}\nonumber  \\
&+ \frac{g_{3V}^2 (12 - 72 \reslog) +
\lambda (-73 + 78 \reslog)))))}{1152 m_V^2}, \\
C_{dH}^{1\Loop,ij} ={}& \frac{-144 g_{f}^2 (y_d y_d^\dagger y_d)_{ij} + 72 g_{f}^2 (-11 + 6 \reslog)
(y_u y_u^\dagger y_d)_{ij}}{1152 m_V^2}\nonumber  \\
&+ \frac{(144 g_2^4(1 + \reslog) +
192 g_2^3 g_{2VW} (2 + 3 \reslog) + g_2^2 (g_{H}^2 (229 - 102 \reslog)}{1152 m_V^2}\nonumber \\
192 g_{2VW}^2 (-5 + \reslog) + 192 g_{H} g_{3V} (-1 + 6 \reslog))}{1152 m_V^2}\nonumber \\
&+ \frac{-
576 g_2 g_{H} g_{2VW} (g_{H} \reslog + g_{3V} (2 + 6 \reslog))}{1152 m_V^2}\nonumber  \\
9 g_{H} (-3 g_1^2 g_{H} (-5 + 6 \reslog) + 2 (12 g_{f} \lambda (-1 + 2 \reslog) +
g_{H}^3(-5 + 6 \reslog)}{1152 m_V^2}\nonumber  \\
&+ \frac{8 g_{H}^2 g_{3V} (-5 + 6 \reslog) +
g_{H} (g_{VH} (116 - 96 \reslog) + g_{3V}^2 (12 - 72 \reslog)}{1152 m_V^2}\nonumber  \\
\lambda (-73 + 78 \reslog)))))y^{ij}_d)}{1152 m_V^2},\\
C_{uH}^{1\Loop,ij} =&{} \frac{-144 g_{f}^2 (y_u y_u^\dagger y_u)_{ij} + 72 g_{f}^2 (-11 + 6 \reslog)
(y_d y_d^\dagger y_u)_{ij}}{1152 m_V^2} \nonumber \\
&+ \frac{(144 g_2^4(1 + \reslog) +
192 g_2^3 g_{2VW} (2 + 3 \reslog) + g_2^2 (g_{H}^2 (229 - 102 \reslog)}{1152 m_V^2}\nonumber \\
192 g_{2VW}^2 (-5 + \reslog) + 192 g_{H} g_{3V} (-1 + 6 \reslog))}{1152 m_V^2}\nonumber \\
&+ \frac{-
576 g_2 g_{H} g_{2VW} (g_{H} \reslog + g_{3V} (2 + 6 \reslog))}{1152 m_V^2}\nonumber  \\
9 g_{H} (-3 g_1^2 g_{H} (-5 + 6 \reslog) + 2 (12 g_{f} \lambda (-1 + 2 \reslog) +
g_{H}^3(-5 + 6 \reslog)}{1152 m_V^2}\nonumber  \\
&+ \frac{8 g_{H}^2 g_{3V} (-5 + 6 \reslog) +
g_{H} (g_{VH} (116 - 96 \reslog) + g_{3V}^2 (12 - 72 \reslog)}{1152 m_V^2}\nonumber  \\
\lambda (-73 + 78 \reslog)))))y^{ij}_u)}{1152 m_V^2},\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{\psi^2 H^2 D}\nonumber \\
C_{He}^{1\Loop,ji} ={}& -\frac{g_1^2 g_H^2 (19 + 6 \reslog)}{384 m_V^2} \delta^{ij} +\frac{-
12 g_f(9 g_H - 6 g_H \reslog + g_f (-7 + 6\reslog))
(y_e^\dagger y_e)_{ji}}{384 m_V^2},\\
C_{Hu}^{1\Loop,ji} ={}&\frac{g_1^2 g_H^2(19 + 6 \reslog)}{576 m_V^2} \delta^{ij} +\frac{ 18 g_f (9 g_H - 6 g_H \reslog + g_f(-7 + 6\reslog))}{576 m_V^2}(y_u^\dagger y_u)_{ji}, \\
C_{Hd}^{1\Loop,ji} ={}&  \frac{-g_1^2 g_H^2(19 + 6 \reslog)}{1152 m_V^2}\delta^{ij} -\frac{36 g_f(9 g_H - 6 g_H \reslog + g_f (-7 + 6 \reslog))}{ 1152 m_V^2}(y_d^\dagger y_d)_{ji}, \\
C_{Hud}^{1\Loop,ji} =&{} -\frac{g_f(g_H(9 - 6\reslog) + g_f(-7 + 6 \reslog))}
{16 m_V^2}(y_u^\dagger y_d)_{ji},\\
C_{Hl^{(1)}}^{1\Loop,ij} ={}& -\frac{g_1^2(g_{H}^2(19 + 6\reslog) + g_{f}^2(58 + 60 \reslog))}{768 m_V^2} \delta^{ij} \nonumber \\
\frac{36 g_{f}(g_{f} + g_{H}(7 - 10 \reslog) + 6 g_{f} \reslog)
(y_e y_e^\dagger)_{ij}}{768 m_V^2},\\
C_{Hl^{(3)}}^{1\Loop,ij}  ={}&
\Big(\frac{((288 g_2^4 (1 + \reslog) + 384 g_2^3 g_{2VW} (2 + 3 \reslog)
g_2^2(-384 g_{2VW}^2 (-5 + \reslog)}{2304 m_V^2} \nonumber \\
& + \frac{192 g_{H}g_{3V}(-1 + 6\reslog) +
6 g_{f}^2 (-17 + 54 \reslog) + g_{H}^2 (49 + 114\reslog)}{2304 m_V^2}\nonumber \\
48 g_{f} (g_{H} (45 - 54 \reslog) + 4 g_{3V} (-1 + 6 \reslog)))}{2304 m_V^2}\nonumber \\
&+ \frac{-
288 g_2 g_{2VW} (g_{f}^2 (1 + 4 \reslog) +
2 g_{H} (g_{H} \reslog + g_{3V} (2 + 6 \reslog))}{2304 m_V^2}\nonumber \\
g_{f} (g_{H} (5 - 6 \reslog) + 4 (g_{3V} + 3 g_{3V} \reslog)))}{2304 m_V^2}\nonumber  \\
&+\frac{-72 g_{f} g_{H} (3 g_{f}^2 + 12 g_f g_{3V}+ g_f g_H (5 - 6 \reslog) +
4 g_{VH} (-5 + 6 \reslog)}{2304 m_V^2}\nonumber  \\
&+ \frac{2 g_{3V}(g_H (5 - 6\reslog) +
3 g_{3V}(-1 + 6 \reslog)))}{2304 m_V^2}\Big) \delta^{ij}
\nonumber \\&+
\frac{36 g_{f} (g_{f} + 6 g_{f} \reslog + g_{H} (-1 + 6 \reslog))}{2304 m_V^2}(y_e y_e^\dagger)_{ij}, \\
C_{Hq^{(1)}}^{1\Loop,ij} ={}& \frac{g_1^2 (g_H^2 (19 + 6 \reslog) + g_f^2(58 + 60\reslog))}{2304 m_V^2}\delta^{ij}  \nonumber \\
&+ \frac{108 g_f (g_f + g_H (7 - 10 \reslog) + 6 g_f \reslog)}{2304 m_V^2} (y_u y_u^\dagger - y_d y_d^\dagger)_{ij}, \\
C_{Hq^{(3)}}^{1\Loop,ij} ={}& \Big(\frac{288 g_2^4 - 102 g_2^2 g_f^2 + 2160 g_2^2 g_f g_H - 216 g_f^3 g_H + 49 g_2^2 g_H^2 - 360 g_f^2 g_H^2}{2304 m_V^2}\nonumber  \\
&+ \frac{1440 g_f g_H g_{VH}-
192 g_2^2 g_f g_{3V} - 192 g_2^2 g_H g_{3V} - 864 g_{f}^2 g_H g_{3V}}{2304 m_V^2}\nonumber  \\
720 g_f g_H^2 g_{3V} + 432 g_f g_H g_{3V}^2 + 768 g_2^3 g_{2VW}-
288 g_2 g_f^2 g_{2VW}}{2304 m_V^2} \nonumber \\
&+\frac{- 1440 g_2 g_f g_H g_{2VW} - 1152 g_2 g_f g_{3V}g_{2VW} - 1152 g_2 g_H g_{3V} g_{2VW}}{2304 m_V^2} \nonumber \\
&+ \frac{1920 g_2^2 g_{2VW}^2 + (288 g_2^4 +
324 g_2^2 g_f^2 - 2592 g_2^2 g_f g_H  +
114 g_2^2 g_H^2 + 432 g_f^2 g_H^2}{2304 m_V^2}\nonumber \\
1728 g_f g_H g_{VH} + 1152 g_2^2 g_f g_{3V} +
1152 g_2^2 g_H g_{3V} +
864 g_f g_H^2 g_{3V}}{2304 m_V^2} \nonumber \\
&+\frac{- 2592 g_f g_H g_{3V}^2 +
1152 g_2^3 g_{2VW} - 1152 g_2 g_f^2 g_{2VW} +
1728 g_2 g_f g_H g_{2VW}}{2304 m_V^2} \nonumber \\
&+\frac{- 576 g_2 g_H^2 g_{2VW}-
3456 g_2 g_f g_{3V} g_{2VW}}{2304 m_V^2} \nonumber \\
&+\frac{- 3456 g_2 g_H g_{3V} g_{2VW} -
384 g_2^2 g_{2VW}^2)\reslog}{2304 m_V^2}\Big) \delta^{ij}\nonumber  \\
&+\frac{36 g_f (g_f (1 + 6\reslog) +
g_H (-1 + 6\reslog))}{2304 m_V^2}((y_d y_d^\dagger)_{ij}+ (y_u y_u^\dagger)_{ij}),\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{(\bar{L}L)(\bar{L}L)}\nonumber \\
C^{1\Loop,ijkl}_{ll} ={}& \Big(\frac{-24 g_{2}^4 (1 + \reslog) - 32 g_{2}^3 g_{2VW} (2 + 3\reslog)}{384m_{V}^2} \nonumber \\ & +
\frac{g_{f}^2(144g_{3V}g_{f} + g_{1}^2(137 - 42L) + 2g_{f}^2(127 - 36\reslog) +
36g_{3V}^2(-1 + 6\reslog))}{384m_{V}^2} \nonumber \\ & +\frac{96g_{2} g_{2VW} g_{f} (-(g_{f} (-3 + \reslog)) + g_{3V} (2 + 6\reslog))}{384m_{V}^2} \nonumber \\ & + \frac{
g_{2}^2 (32 g_{2VW}^2 (-5 + \reslog) + g_{f} (32 g_{3V} - 795 g_{f} - 192 g_{3V} \reslog + 594 g_{f} \reslog))}{384m_{V}^2}\Big) \delta^{ij} \delta^{kl} \nonumber \\
2 24 g_{2}^4 (1 + \reslog) + 32 g_{2}^3 g_{2VW} (2 + 3\reslog)}{384m_{V}^2} \nonumber \\ & +
\frac{96 g_{2} g_{2VW} g_{f} (g_{f} (-3 + \reslog) - 2 g_{3V} (1 + 3 \reslog))}{384m_{V}^2} \nonumber \\ & +
\frac{2 g_{f}^2 (-72 g_{3V} g_{f} + 18 g_{1}^2 (-3 + 2\reslog) - 18 g_{3V}^2 (-1 + 6 \reslog) +
g_{f}^2 (-73 + 36 \reslog))}{384m_{V}^2} \nonumber \\ & + \frac{g_{2}^2 (-32 g_{2VW}^2 (-5 + \reslog) +
g_{f} (471 g_{f} - 378 g_{f} \reslog + 32 g_{3V} (-1 + 6 \reslog)))}{384m_{V}^2}\Big)
\delta^{il} \delta^{jk},\\
C^{1\Loop,ijkl}_{qq^{(1)}}={}& \frac{g_f^2(474 g_3^2 + 972 g_f^2 - 612 g_3^2 \reslog + 972 g_2^2 (-3 + 2 \reslog) +
g_1^2 (29 + 30 \reslog))}{3456 m_V^2}\delta^{ij} \delta^{kl} \nonumber \\
& +\frac{9 g_f^2 g_3^2 (-79 + 102 \reslog)}{3456 m_V^2}\delta^{il} \delta^{jk}, \\
C^{1\Loop,ijkl}_{qq^{(3)}}={}&  \Big(\frac{24 g_{2}^4 (1 + \reslog) + 32 g_{2}^3 g_{2VW} (2 + 3 \reslog)}{384m_{V}^2} \nonumber \\ & +
\frac{96 g_{2} g_{2VW} g_{f} (g_{f} (-3 + \reslog) - 2 g_{3V} (1 + 3 \reslog))}{384m_{V}^2}\nonumber\\  & +
\frac{2 g_{f}^2 (-72 g_{3V} g_{f} + g_{1}^2 (-6 + 4 \reslog) - 18 g_{3V}^2 (-1 + 6\reslog) +
g_{f}^2 (-73 + 36 \reslog))}{384m_{V}^2} \nonumber \\ & + \frac{g_{2}^2 (-32 g_{2VW}^2 (-5 +\reslog) +
g_{f} (471 g_{f} - 378 g_{f} \reslog + 32 g_{3V}(-1 + 6 \reslog)))}{384m_{V}^2}\Big)
\delta^{ij}\delta^{kl} \nonumber \\ &+ \frac{g_{3}^2 g_{f}^2 (-79 + 102 \reslog)}{384m_{V}^2}
\delta^{il} \delta^{jk} \\
C^{1\Loop,ijkl}_{lq^{(1)}} ={}& \frac{g_f^2(-(g_1^2(29 + 30 \reslog)) + 324(g_f^2 + g_2^2(-3 + 2 \reslog)))}{576m_V^2} \delta^{ij} \delta^{kl}, \\
C^{1\Loop,ijkl}_{lq^{(3)}}  ={}& \Big(\frac{48 g_{2}^4 (1 + \reslog) + 64 g_{2}^3 g_{2VW} (2 + 3 \reslog) -
48 g_{2} g_{2VW} (8 g_{3V} g_{f} (1 + 3 \reslog)}{384 m_{V}^2} \nonumber \\ &
+\frac{2 g_{f}^2 (1 + 4 \reslog) -
2 g_{f}^2 (-5 + 6 \reslog)) - 4 g_{f}^2 (72 g_{3V} g_{f} + 73 g_{f}^2 - 36 g_{f}^2 \reslog}{384 m_{V}^2} \nonumber \\& +
\frac{6 g_{1}^2 (-3 + 2 \reslog) + 18 g_{3V}^2 (-1 + 6 \reslog)) +
g_{2}^2 (942 g_{f}^2 - 64 g_{2VW}^2 (-5 + \reslog)}\nonumber \\ &+ \frac{- 756 g_{f}^2 \reslog +
64 g_{3V} g_{f} (-1 + 6 \reslog))}{384 m_{V}^2}\Big)\delta^{ij} \delta^{kl}, \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{(\bar{L}L)(\bar{R}R)}\nonumber \\
C_{le}^{1\Loop,ijkl} ={}& \frac{g_1^2 g_f^2 (29 + 30 \reslog)\delta^{ij} \delta^{kl}}{192 m_V^2}\nonumber  \\ & + \frac{3 (12 g_f g_H (1 - 2\reslog) +
4 g_f^2(-7 + 6\reslog) + g_H^2(-1 + 6\reslog))
y_e^{il} y_e^{\dagger kj}}{192 m_V^2},\\
C^{1\Loop,ijkl}_{lu} ={}& -\frac{g_1^2 g_f^2 (29 + 30 \reslog)}{288 m_V^2} \delta^{ij} \delta^{kl},  \\
C^{1\Loop,ijkl}_{ld} ={}& \frac{g_1^2 g_f^2 (29 + 30 \reslog)}{576 m_V^2} \delta^{ij} \delta^{kl}, \\
C^{1\Loop,ijkl}_{qe} ={}& -\frac{g_1^2 g_f^2 (29 + 30 \reslog)}{576 m_V^2} \delta^{ij} \delta^{kl}, \\
C_{qd^{(1)}}^{1\Loop,ijkl} ={}& -\frac{g_1^2 g_f^2 (29 + 30\reslog)  \delta^{ij} \delta^{lk}}{1728 m_V^2} \nonumber \\ &
+ \frac{9(12 g_f g_H(1 - 2\reslog) +
4 g_f^2 (-7 + 6 \reslog) + g_H^2 (-1 + 6 \reslog))
y_d^{il} y_d^{\dagger kj}}{1728 m_V^2},\\
C^{1\Loop,ijkl}_{qd^{(8)}} ={}& \frac{g_3^2 g_f^2(29 + 30 \reslog)}{96 m_V^2} \delta^{ij} \delta^{kl} \nonumber \\
&+ \frac{3 (12 g_f g_H (1 - 2 \reslog) +
4 g_f^2 (-7 + 6\reslog) + g_H^2 (-1 + 6\reslog))
y_d^{il} y_d^{\dagger jk}
}{96 m_V^2},\\
C_{qu^{(1)}}^{1\Loop,ijkl} ={}& \frac{2 g_1^2 g_f^2 (29 + 30\reslog)  \delta^{ij} \delta^{lk}}{1728 m_V^2} \nonumber \\ &
+ \frac{9(12 g_f g_H(1 - 2\reslog) +
4 g_f^2 (-7 + 6 \reslog) + g_H^2 (-1 + 6 \reslog))
y_u^{il} y_u^{\dagger kj}}{1728 m_V^2}, \\
C^{1\Loop,ijkl}_{qu^{(8)}}={}& \frac{g_3^2 g_f^2(29 + 30 \reslog) \delta^{ij} \delta^{lk}}{96 m_V^2}\nonumber  \\
&+\frac{3(12g_f g_H (1 - 2 \reslog) + 4 g_f^2 (-7 + 6 \reslog) +
g_H^2 (-1 + 6 \reslog)) y_u^{il} y_u^{\dagger kj}}{96 m_V^2 },\\
&~~~~~~~~~~~~~~~~~~~~~~\boxed{(\bar{L}R)(\bar{R}L)\text{ and }(\bar{L}R)(\bar{L}R)}\nonumber \\
C_{ledq}^{1\Loop,ijlk} ={}& \frac{((g_H^2(1 - 6\reslog) + 12g_f g_H (-1 + 2 \reslog) -
4 g_f^2 (-7 + 6 \reslog))}{32 m_V^2} y_e^{ij} y_d^{\dagger lk},\\
C_{quqd^{(1)}}^{1\Loop,ijkl} ={}& \frac{-8 g_f^2 (-3 + 2 \reslog) y_d^{il} y_u^{kj}}{32 m_V^2} \nonumber \\
&+ \frac{(g_H^2 (1 - 6 \reslog) + 12 g_f g_H (-1 + 2 \reslog) +
4 g_f^2 (-11 + 6 \reslog))y_d^{kl} y_u^{ij}}{32 m_V^2},\\
C_{lequ^{(1)}}^{1\Loop,ijkl} ={}& \frac{8 g_f^2 + 12 g_f g_H(1 - 2 \reslog) + g_H^2(-1 + 6 \reslog)}{32 M_V^2}y_e^{ij} y_u^{kl},\\
C^{1\Loop,ijkl}_{lequ^{(3)}} ={}& \frac{3 g_f^2 (-3 + 2 \reslog)}{32 m_V^2} y_e^{ij} y_u^{kl},\\
C^{1\Loop,ijkl}_{quqd^{(8)}} ={}& \frac{-3g_f^2 (-3 + 2 \reslog)}{2 m_V^2} y_d^{il} y_u^{kj}.
\end{align}

As is clear from the results presented here, the matching of the model defined by the Lagrangian of Eq. (<ref>) is a non-trivial task. Yet, the model is still very simple as it only extends the SM by one new field. Comparing this to SM extensions such as supersymmetry it becomes clear that for the study of such models in the context of the SMEFT or other low-energy EFTs a tool such as is unavoidable.

In the near future the Wilson coefficients computed here will be used in a phenomenological study aimed at investigating SMEFT fits in the context of explicit extensions of the SM. Such a study, however, is beyond the scope of this thesis.

§ A REMARK ON RENORMALISATION

As discussed in Chapter <ref>, the matching condition used to derive all of the results presented in this thesis is the equality of renormalised correlation functions. However, in UOLEA applications the renormalisation is never performed explicitly. Indeed, the final result contains divergences represented by $1/\epsilon$-poles, which in principle can be subtracted systematically by adding counterterms to the tree-level Lagrangians and carrying them over to the final result. In practice, however, the $1/\epsilon$-poles are simply dropped corresponding to $\overline{\text{MS}}$-renormalisation. In cases where one starts from a renormalisable theory and matches it to an EFT this is clearly justified. The model considered in this chapter, however, does not belong to this class of theories as it has to be interpreted as an EFT itself. Indeed, in similar models restrictions on the couplings arise from the condition of renormalisability in the sense of EFTs, as pointed out for the case of the $\rho$-meson in Ref. <cit.>. In order to assure that no further constraints on the couplings arise from this condition one has to show that all terms appearing in the renormalisation of the model can be absorbed in terms corresponding to higher dimensional operators constructed from the fields, and constrained by the symmetries, of the model. Such an analysis is beyond the scope of this thesis.

The non-renormalisability of the UV model also has practical implications for the matching, as it affects the structure of logarithmic contributions to one-loop Wilson coefficients and in principle requires the addition of higher dimensional operators in the UV theory for a complete matching. This can be seen as follows. For a renormalisable UV theory, one would expect logarithmic one-loop matching contributions only in Wilson coefficients which have tree-level contributions or mix with tree-level induced Wilson coefficients under the renormalisation group, for the following reason. Assume that a certain correlation function in the UV theory, which yields a contribution to a given Wilson coefficient in the matching, is divergent at one loop. Then, this correlation function must have a non-zero tree-level contribution as this is needed to provide the counterterms that cancel the divergence. Therefore, there is a tree-level contribution to the Wilson coefficient. A divergence in the corresponding correlation function of the EFT can only arise due to a tree-level induced operator that mixes with the operator whose Wilson coefficient is fixed by the divergent correlation function. Since divergences and logarithms are in one-to-one correspondence this justifies the statement. For the Wilson coefficients presented here, the statement is not true. For example, the Wilson coefficient $C_W$ contains logarithmic terms although $Q_W$ does not mix with any of the tree-level induced operators. These logarithmic contributions can be traced to the non-renormalisabiltiy of the UV theory. For example, the contribution to the Wilson coefficient $C_W$ proportional to $g_{2VW}^3$ arises from the vertex of the form

\begin{align}
-\frac{g_{2VW}}{2}f^{ABC}V^{\mu B} V^{\nu C} W_{\mu \nu}^A,
\label{eq: relevant_coupling}
\end{align}

which yields a triangle diagram contributing to $Q_W$, schematically depicted in Fig. <ref> below.

[scale = 1.5]
[thick, double, snake it] (3.5,2.7) – (4.3,2.7);
[thick, double, snake it] (5.3,2.7) – (6.2,2.7);
[thick, double, snake it] (4.8,3.5) – (4.8,4.3);
[thick, snake it] (4.3,2.7) – (5.3,2.7);
[thick, snake it] (4.3,2.7) – (4.8,3.5);
[thick, snake it] (4.8,3.5) – (5.3,2.7);

Schematic depiction of the $g_{2VW}^3$ contribution to $C^{1\Loop}_W$. The double line corresponds to the $SU(2)_L$ field strength tensor and the single line to the vector triplet.

The momentum integral appearing in this diagram, at zero external momentum, is given by

\begin{align}
\int \frac{\rd^d q}{(2 \pi)^d} \frac{1}{(q^2-m_V^2)^3}\left(g_{\mu \nu}-\frac{q_\mu q_\nu}{m_V^2}\right)\left(g_{\rho \sigma}-\frac{q_\rho q_\sigma}{m_V^2}\right)\left(g_{\kappa \lambda}-\frac{q_\kappa q_\lambda}{m_V^2}\right),
\end{align}

and yields divergent contributions due to the parts of the propagator, which are not momentum-suppressed. If these parts were absent, the integral would be finite. Thus, the divergences come exactly from the part of the vector field propagator that renders the theory non-renormalisable in the strict sense. It was checked that in the UOLEA approach, the logarithmic contributions to $C_W$ proportional to $g_{2VW}^3$ arise from operators of the form $X_{VV}^3$, where the relevant part of $X_{VV}$ is precisely due to the coupling of Eq. (<ref>). These contributions can be interpreted as the diagram in Fig. <ref>, with the field strength tensors constructed from background fields. Clearly, divergent contributions to several correlation functions arise in this way. In order to absorb these divergences one has to add to the Lagrangian of Eq. (<ref>) a Lagrangian containing all necessary higher dimensional operators. This means that these higher dimensional operators are present in the UV theory at tree level and in principle contribute to the matching, both at tree level and at the one-loop level. This introduces a plethora of input parameters in the UV theory, which have been set to zero in the matching presented in this thesis. In this sense, the matching presented here is incomplete. However, one has to also consider the physical origin of these higher dimensional operators. They arise from the remaining NP sector when it is integrated out. Therefore, if the mass hierarchy between the lightest state in this NP sector, represented by the resonance $V^{\mu A}$, and the next-to-lightest state of the NP sector is sufficiently large the results presented here can be used. Sufficiently large in this case means that the suppression of higher dimensional operators by the mass of the next-to-lightest state is larger than the suppression by the loop factor, since we are including one-loop effects in the matching. Note that if the full one-loop renormalisation up to mass dimension six of the model is performed, one can use that result to check the logarithmic terms appearing in the one-loop Wilson coefficients.

The various shortcomings of the SM combined with exclusion limits on the masses of new particles which often lie in the TeV range suggest the presence of a mass hierarchy between the SM and NP. EFTs are an excellent tool for the study of such models, as they allow for the resummation of large logarithmic corrections as well as the systematic inclusion of NP effects in terms of a perturbative expansion in inverse masses of the new particles. In order to construct a suitable EFT from a given NP model, a matching computation has to be performed. To capture most of the relevant NP effects it is necessary to perform this matching computation at the one-loop level including operators up to mass dimension six in the EFT. To avoid redundant steps associated with such a matching computation the UOLEA was introduced in Refs. <cit.>. The expressions presented there represent a master formula for the effective action including operators up to mass dimension six and can be applied to scalar theories.

In this thesis we introduced the BSUOLEA, which is an extension of the results of Refs. <cit.>, that allows for the matching of any renormalisable, Lorentz invariant UV theory, containing scalar fields, vector fields and spin-$1/2$ fermions, to an EFT including operators up to mass dimension six. The specific contributions that were calculated for that purpose are summarised in Table <ref>, where they are marked by BS.

Heavy-only     heavy-light     derivative couplings
Bosonic     <cit.>     <cit.>     BS

Fermionic (pure)     BS , <cit.>, <cit.>     BS     NR

Fermionic (mixed)     BS     BS     NR

Mixed statistics     BS     BS     NR

Overview of the different contributions to the UOLEA divided into contributions arising from heavy-only and heavy-light loops as well as loops depending on derivative couplings. Fermionic (pure) denotes loops with only one kind of fermion, i.e. either Dirac or Majorana. Fermionic (mixed) denotes loops with at least one Dirac and one Majorana fermion. Mixed statistics denotes loops containing at least one boson and one fermion. Contributions that were computed as a part of this thesis are marked by BS. Contributions marked by NR only arise when the UV theory is non-renormalisable.

In order to facilitate the use of the BSUOLEA, it was implemented into the currently private Mathematica package . Supplemented with the code , which was also developed for the purpose of this thesis and allows for the translation of a set of redundant SMEFT operators into the Warsaw basis, this solves the problem of matching a generic, renormalisable, Lorentz invariant UV theory to the dimension six SMEFT at the one-loop level. In addition, allows for the matching of UV theories to more general low-energy EFTs. It is our ambition to make these matching tools publicly available in the near future.

Currently it is not possible to apply the BSUOLEA to the matching of a non-renormalisable UV theory to a low-energy EFT. Such scenarios can be of interest when the NP model itself has a limited range of validity as is the case for the model that was considered in Chapter <ref>. In fact, Section <ref> can be regarded as a first step towards the inclusion of the terms necessary to treat such models using the BSUOLEA. It is desirable to extend the BSUOLEA to fully cover such models as this would also allow for the computation of anomalous dimensions in generic EFTs as discussed in Refs. <cit.>. Furthermore, the functional methods employed throughout this work can be used to compute the divergent parts of 1PI correlation functions of a given model at the one-loop level. They could therefore be used to study which higher dimensional operators are needed for the renormalisation of the model considered in Chapter <ref> at mass dimension six. Combined with the possibility of computing the anomalous dimensions for this model, this would allow for the confirmation of the logarithmic terms of the one-loop Wilson coefficients presented in Section <ref>.

Appendix A

CHAPTER: GENERATING FUNCTIONAL OF 1LPI CORRELATION FUNCTIONS

We here briefly discuss the generating functional of 1LPI correlation functions for the case of a theory containing heavy fields $\Phi$ and light fields $\phi$. The results presented here are well-known and can be found in several textbooks such as in Chapters 9 and 11 of Ref. <cit.>. In Chapter <ref> the generating functional of 1LPI correlation functions was given as

\begin{align}
\Gamma_\text{L}[\bg{\phi}]=-i \log Z[J_\Phi=0,J_\phi]-\int \rd^d x \, J_\phi(x) \bg{\phi}(x),
\label{eq: 1LPI_functional_def}
\end{align}

with $Z$ being the generating functional of all correlation functions given by

\begin{align}
Z[J_\Phi,J_\phi]=\int \measure \Phi \measure \phi \, \exp\left \{i \int \rd^d x \, \big[\LagNoT[\Phi,\phi]+J_{\Phi}(x) \Phi(x)+J_{\phi}(x) \phi(x) \big]\right\},
\end{align}

where $J_\Phi$ and $J_\phi$ are the sources for the heavy and light fields, respectively. The generating functional of 1PI correlation functions, $\Gamma[\bg{\Phi},\bg{\phi}]$, is obtained from $Z[J_\Phi,J_\phi]$ as

\begin{align}
\Gamma[\bg{\Phi},\bg{\phi}] &= -i \log Z[J_\Phi,J_\phi]- \left(J_\phi\right)_x \left(\bg{\phi}\right)_x- \left(J_\Phi\right)_x \left(\bg{\Phi}\right)_x,
\end{align}

where we remind the reader that $W[J_\Phi,J_\phi]\equiv -i \log Z[J_\Phi,J_\phi]$ is the generating functional of connected correlation functions. Generally, $\bg{\phi}$ and $\bg{\Phi}$ are defined as

\begin{align}
\bg{\Phi} &= \frac{\delta W}{\delta J_\Phi}, \\
\bg{\phi} &= \frac{\delta W}{\delta J_\phi},
\end{align}

and depend on both $J_\Phi$ and $J_\phi$. However, setting $J_\Phi = 0$ the light background field appearing in Eq. (<ref>) satisfies

\begin{align}
\bg{\phi} = \bg{\phi}[J_\Phi = 0, J_\phi] = \left . \frac{\delta W}{\delta J_\phi}\right \vert_{J_\Phi=0}.
\label{eq: light_bg_field}
\end{align}

Denoting $W[J_\Phi=0,J_\phi]$ by $\tilde{W}[J_\phi]$ we find

\begin{align}
\frac{\delta \Gamma_L[\bg{\phi}]}{\delta \left(\bg{\phi}\right)_y} &= \frac{\delta \tilde{W}[J_\phi]}{\delta \left(\bg{\phi}\right)_y}-\frac{\delta \left(J_\phi\right)_x}{\delta \left(\bg{\phi}\right)_y}\left(\bg{\phi}\right)_x-\left(J_\phi\right)_y \nonumber \\
&= \frac{\delta \tilde{W}[J_\phi]}{\delta \left(J_\phi\right)_x} \frac{\delta \left(J_\phi\right)_x}{\delta \left(\bg{\phi}\right)_y}-\frac{\delta \tilde{W}[J_\phi]}{\delta \left(J_\phi\right)_x} \frac{\delta \left(J_\phi\right)_x}{\delta \left(\bg{\phi}\right)_y}-\left(J_\phi\right)_y \nonumber \\
&= -\left(J_\phi\right)_y,
\end{align}

where in the second line we used the chain rule and Eq. (<ref>). It follows that

\begin{align}
\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \left(J_\phi\right)_y \delta \left(\bg{\phi}\right)_x} = -\delta(x-y).
\label{eq: delta_eq}
\end{align}

On the other hand, using the chain rule and Eq. (<ref>) again we find

\begin{align}
\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \left(J_\phi\right)_y \delta \left(\bg{\phi}\right)_x} = \frac{\delta^2 \tilde{W}[J_\phi]}{\delta \left(J_\phi\right)_y \delta \left(J_\phi\right)_z} \frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \left(\bg{\phi}\right)_z \delta \left(\bg{\phi}\right)_x},
\end{align}

which together with Eq. (<ref>) implies that

\begin{align}
\left(\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi}}\right)_{xy} = -\left(\frac{\delta^2 \tilde{W}[J_\phi]}{\delta J_\phi \delta J_\phi}\right)_{xy}^{-1}.
\end{align}

In our conventions

\begin{align}
\left(\frac{\delta^2 \tilde{W}[J_\phi]}{\delta J_\phi \delta J_\phi}\right)_{xy} = i D_{xy},
\end{align}

is the connected two-point function of the light field and hence $D_{xy}$ is the full propagator of the light field. It follows that

\begin{align}
\left(\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi}}\right)_{xy} =  i D^{-1}_{xy}.
\end{align}

Now consider the connected three-point function with light external fields given by

\begin{align}
-\frac{\delta^3 \tilde{W}[J_\phi]}{\delta \left(J_\phi\right)_x \delta \left(J_\phi\right)_y \delta \left(J_\phi\right)_z} &= \frac{\delta}{\delta \left(J_\phi\right)_x} \left(\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi}}\right)_{yz}^{-1} \nonumber \\
&= i D_{xw}\frac{\delta}{\delta \left(\bg{\phi}\right)_w} \left(\frac{\delta^2 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi}}\right)_{yz}^{-1} \nonumber \\
&= iD_{xw} (-1) iD_{yu} \left(\frac{\delta^3 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi} \delta \bg{\phi}}\right)_{wuv} iD_{vz} \nonumber \\
&= i D_{xw} D_{yu} D_{vz} \left(\frac{\delta^3 \Gamma_L[\bg{\phi}]}{\delta \bg{\phi} \delta \bg{\phi} \delta \bg{\phi}}\right)_{wuv}.
\label{eq: three_point_func}
\end{align}

This is most easily interpreted diagrammatically.

[scale = 1.5]
[thick] (4,1.8) – (4.4,2.5);
[thick] (5.2,2.5) – (5.6,1.8);
[thick] (4.8,3.2) – (4.8,4);
[thick, fill=lightgray] (4.8,2.7) circle [radius=0.5];;
[circle, fill, inner sep=1pt,label=below:x] at (4,1.8)
[circle, fill, inner sep=1pt,label=below:y] at (5.6,1.8)
[circle, fill, inner sep=1pt,label=above:z] at (4.8,4)
[align=left, above] at (6,2.7)
[thick] (6.8,1.8) – (7.4,2.4);
[thick] (8.2,2.4) – (8.8,1.8);
[thick] (7.8,3.2) – (7.8,4);
[thick] (7.8,2.7) circle [radius=0.5];;
[thick, fill=lightgray] (7.8,3.6) circle [radius=0.17];;
[thick,fill=lightgray] (8.5,2.1) circle [radius=0.17];;
[thick,fill=lightgray] (7.1,2.1) circle [radius=0.17];;
[circle, fill, inner sep=1pt,label=below:x] at (6.8,1.8)
[circle, fill, inner sep=1pt,label=below:y] at (8.8,1.8)
[circle, fill, inner sep=1pt,label=above:z] at (7.8,4)

Diagrammatic representation of Eq. (<ref>).

In Figure <ref>, the gray circles represent the sum of connected correlation functions, whereas the white circle depicts the third derivative of $i\Gamma_\text{L}[\bg{\phi}]$. We see that this is the connected correlation function of three external light fields, with all three full propagators removed. That means that it is 1PI with respect to light fields. That it is not 1PI with respect to heavy fields can be seen by the fact that the diagram in Figure <ref>, where the dashed line represents the heavy field, is part of the third derivative of $i\Gamma_\text{L}[\bg{\phi}]$. Hence, the white circle is the 1LPI three-point function. This generalises to higher derivatives of $\Gamma_\text{L}[\bg{\phi}]$, which therefore, is the generating functional of 1LPI correlation functions.

[scale = 1.5]
[thick] (4.8,1.8) – (4.4,1.4);
[thick] (4.8,1.8) – (5.2,1.4);
[thick] (4.8,3) – (4.8,3.6);
[thick,dashed] (4.8,2.4) – (4.8,1.8);
[thick, dashed] (4.8,2.7) circle [radius=0.3];;
[circle, fill, inner sep=1pt,label=below:x] at (4.4,1.4)
[circle, fill, inner sep=1pt,label=below:y] at (5.2,1.4)
[circle, fill, inner sep=1pt,label=above:z] at (4.8,3.6)
[align=left, above] at (6,2.7)

Example of a diagram that is 1LPI, but not 1PI. The dashed line represents the heavy field and the solid line represents the light field.

Appendix B

CHAPTER: BSUOLEA INGREDIENTS

We here collect the different matrices defined in Section <ref>, which appear in the BSUOLEA after the factors of $\tilde{\mathds{1}}_B$, $\tilde{\mathds{1}}_F$ and $\cc$ have been absorbed and relate them to the symbols appearing in the Mathematica file found at <cit.>. We define

\begin{align*}
X[\text{S},\text{F}] &\equiv \mathbf{X}_{\Phi \Xi}= \begin{pmatrix}
X_{\Sigma ^* \Omega} & X_{\Sigma ^* \bar{\Omega}} \cc ^{-1} & X_{\Sigma ^* \Lambda} \\
-X^{\mu}_{W ^* \Omega} & -X^{\mu}_{W ^* \bar{\Omega}} \cc ^{-1} & -X_{W ^* \Lambda} \\
X_{\Sigma \Omega} & X_{\Sigma \bar{\Omega}} \cc ^{-1} & X_{\Sigma \Lambda} \\
-X^{\mu}_{W \Omega} & -X^{\mu}_{W \bar{\Omega}} \cc ^{-1} & -X_{W \Lambda} \\
X_{\Theta \Omega} & X_{\Theta \bar{\Omega}} \cc ^{-1} & X_{\Theta \Lambda} \\
-X^{\mu}_{V \Omega} & -X^{\mu}_{V \bar{\Omega}} \cc ^{-1} & -X_{V \Lambda}
\end{pmatrix}, \nonumber \\
X[\text{s},\text{F}] &\equiv \mathbf{X}_{\phi \Xi}= \begin{pmatrix}
X_{\sigma ^* \Omega} & X_{\sigma ^* \bar{\Omega}} \cc ^{-1} & X_{\sigma ^* \Lambda} \\
-X^\mu_{w ^* \Omega} & -X^\mu_{w ^* \bar{\Omega}} \cc ^{-1} & -X^\mu_{w ^* \Lambda} \\
X_{\sigma \Omega} & X_{\sigma \bar{\Omega}} \cc ^{-1} & X_{\sigma \Lambda} \\
-X^\mu_{w \Omega} & -X^\mu_{w \bar{\Omega}} \cc ^{-1} & -X^\mu_{w \Lambda} \\
X_{\theta \Omega} & X_{\theta \bar{\Omega}} \cc ^{-1} & X_{\theta \Lambda} \\
-X^\mu_{v \Omega} & -X^\mu_{v \bar{\Omega}} \cc ^{-1} & -X^\mu_{v \Lambda} \\
\end{pmatrix}, \nonumber \\
X[\text{S},\text{f}] &\equiv \mathbf{X}_{\Phi \xi}=\begin{pmatrix}
X_{\Sigma ^* \omega} & X_{\Sigma ^* \bar{\omega}} \cc ^{-1} & X_{\Sigma ^* \lambda} \\
-X^\mu_{W ^* \omega} & -X^\mu_{W ^* \bar{\omega}} \cc ^{-1} & -X^\mu_{W ^* \lambda} \\
X_{\Sigma \omega} & X_{\Sigma \bar{\omega}} \cc ^{-1} & X_{\Sigma \lambda} \\
-X^\mu_{W \omega} & -X^\mu_{W \bar{\omega}} \cc ^{-1} & -X^\mu_{W \lambda} \\
X_{\Theta \omega} & X_{\Theta \bar{\omega}} \cc ^{-1} & X_{\Theta \lambda}\\
-X^\mu_{V \omega} & -X^\mu_{V \bar{\omega}} \cc ^{-1} & -X^\mu_{V \lambda}
\end{pmatrix}, \nonumber \\
X[\text{s},\text{f}] &\equiv \mathbf{X}_{\phi \xi}= \begin{pmatrix}
X_{\sigma ^* \omega} & X_{\sigma ^* \bar{\omega}} \cc ^{-1} & X_{\sigma ^* \lambda} \\
-X^\mu_{w ^* \omega} & -X^\mu_{w ^* \bar{\omega}} \cc ^{-1} & -X^\mu_{w ^* \lambda} \\
X_{\sigma \omega} & X_{\sigma \bar{\omega}} \cc ^{-1} & X_{\sigma \lambda} \\
-X^\mu_{w \omega} & -X^\mu_{w \bar{\omega}} \cc ^{-1} & -X^\mu_{w \lambda} \\
X_{\theta \omega} & X_{\theta \bar{\omega}} \cc ^{-1} & X_{\theta \lambda}\\
-X^\mu_{v \omega} & -X^\mu_{v \bar{\omega}} \cc ^{-1} & -X^\mu_{v \lambda}
\end{pmatrix}, \nonumber \\
X[\text{F},\text{S}] &\equiv \mathbf{X}_{\Xi \Phi} = \begin{pmatrix}
X_{\bar{\Omega} \Sigma} & X^\mu_{\bar{\Omega} W} &  X_{\bar{\Omega} \Sigma ^*} & X^\mu_{\bar{\Omega} W^*} &  X_{\bar{\Omega} \Theta} & X^\mu_{\bar{\Omega} V} \\
\cc^{-1} X_{\Omega \Sigma} & \cc^{-1} X^\mu_{\Omega W} & \cc^{-1} X_{\Omega \Sigma^{*}} & \cc^{-1} X^\mu_{\Omega W^*} & \cc^{-1} X_{\Omega \Theta} & \cc^{-1} X^\mu_{\Omega V} \\
\cc^{-1} X_{\Lambda \Sigma} & \cc^{-1} X^\mu_{\Lambda W} & \cc^{-1} X_{\Lambda \Sigma ^*} & \cc^{-1} X^\mu_{\Lambda W^*} & \cc^{-1} X_{\Lambda \Theta} & \cc^{-1} X^\mu_{\Lambda V}
\end{pmatrix}, \nonumber \\
X[\text{f},\text{S}] &\equiv \mathbf{X}_{\xi \Phi} = \begin{pmatrix}
X_{\bar{\omega} \Sigma} & X^\mu_{\bar{\omega} W} &  X_{\bar{\omega} \Sigma ^*} & X_{\bar{\omega} W^*} &  X_{\bar{\omega} \Theta} & X_{\bar{\omega} V} \\
\cc^{-1} X_{\omega \Sigma} & \cc^{-1} X^\mu_{\omega W} & \cc^{-1} X_{\omega \Sigma^{*}} & \cc^{-1} X^\mu_{\omega W ^{*}} &  \cc^{-1} X_{\omega \Theta} & \cc^{-1} X^\mu_{\omega V}\\
\cc^{-1} X_{\lambda \Sigma} & \cc^{-1} X^\mu_{\lambda W} & \cc^{-1} X_{\lambda \Sigma ^*} & \cc^{-1} X^\mu_{\lambda W ^*} & \cc^{-1} X_{\lambda \Theta} & \cc^{-1} X^\mu_{\lambda V}
\end{pmatrix}, \nonumber \\
X[\text{F},\text{s}] &\equiv \mathbf{X}_{\Xi \phi} = \begin{pmatrix}
X_{\bar{\Omega} \sigma} & X^\mu_{\bar{\Omega} w} &  X_{\bar{\Omega} \sigma ^*} & X^\mu_{\bar{\Omega} w^*} &  X_{\bar{\Omega} \theta} & X^\mu_{\bar{\Omega} v} \\
\cc^{-1} X_{\Omega \sigma} & \cc^{-1} X^\mu_{\Omega w} & \cc^{-1} X_{\Omega \sigma^{*}} & \cc^{-1} X^\mu_{\Omega w^*} &  \cc^{-1} X_{\Omega \theta} & \cc^{-1} X^\mu_{\Omega v} \\
\cc^{-1} X_{\Lambda \sigma} & \cc^{-1} X^\mu_{\Lambda w} & \cc^{-1} X_{\Lambda \sigma ^*} & \cc^{-1} X^\mu_{\Lambda w^*} & \cc^{-1} X_{\Lambda \theta} & \cc^{-1} X^\mu_{\Lambda v}
\end{pmatrix}, \nonumber \\
X[\text{f},\text{s}] &\equiv \mathbf{X}_{\xi \phi} = \begin{pmatrix}
X_{\bar{\omega} \sigma} & X^\mu_{\bar{\omega} w} &  X_{\bar{\omega} \sigma ^*} & X^\mu_{\bar{\omega} w^*} &  X_{\bar{\omega} \theta} & X^\mu_{\bar{\omega} v} \\
\cc^{-1} X_{\omega \sigma} & \cc^{-1} X^\mu_{\omega w} & \cc^{-1} X_{\omega \sigma^{*}} & \cc^{-1} X^\mu_{\omega w^*} & \cc^{-1} X_{\omega \theta} & \cc^{-1} X^\mu_{\omega v} \\
\cc^{-1} X_{\lambda \sigma} & \cc^{-1} X^\mu_{\lambda w} & \cc^{-1} X_{\lambda \sigma ^*} & \cc^{-1} X^\mu_{\lambda w^*} & \cc^{-1} X_{\lambda \theta} & \cc^{-1} X^\mu_{\lambda v}
\end{pmatrix}, \nonumber \\
X[\text{F},\text{F}] &\equiv \mathbf{X}_{\Xi \Xi}=\begin{pmatrix}
X_{\bar{\Omega} \Omega} & X_{\bar{\Omega} \bar{\Omega}} \cc^{-1} &  X_{\bar{\Omega} \Lambda} \\
\cc^{-1} X_{\Omega \Omega} &  \cc ^{-1} X_{\Omega \bar{\Omega}}\cc^{-1} & \cc ^{-1} X_{\Omega \Lambda} \\
\cc ^{-1} X_{\Lambda \Omega} & \cc ^{-1} X_{\Lambda \bar{\Omega}}\cc ^{-1} &  \cc ^{-1} X_{\Lambda \Lambda}
\end{pmatrix}, \nonumber \\
X[\text{f},\text{f}] &\equiv \mathbf{X}_{\xi \xi}=\begin{pmatrix}
X_{\bar{\omega} \omega} & X_{\bar{\omega} \bar{\omega}} \cc ^{-1} &  X_{\bar{\omega} \lambda} \\
\cc ^{-1} X_{\omega \omega} &  \cc ^{-1} X_{\omega \bar{\omega}}\cc^{-1} & \cc ^{-1} X_{\omega \lambda} \\
\cc ^{-1} X_{\lambda \omega} & \cc ^{-1} X_{\lambda \bar{\omega}}\cc ^{-1} &  \cc ^{-1} X_{\lambda \lambda}
\end{pmatrix}, \nonumber \\
X[\text{F},\text{f}] &\equiv \mathbf{X}_{\Xi \xi}=\begin{pmatrix}
X_{\bar{\Omega} \omega} & X_{\bar{\Omega} \bar{\omega}} \cc^{-1} &  X_{\bar{\Omega} \lambda} \\
\cc ^{-1} X_{\Omega \omega} & \cc ^{-1} X_{\Omega \bar{\omega}}\cc ^{-1} & \cc^{-1} X_{\Omega \lambda} \\
\cc ^{-1} X_{\Lambda \omega} & \cc ^{-1} X_{\Lambda \bar{\omega}} \cc ^{-1} &  \cc^{-1} X_{\Lambda \lambda}
\end{pmatrix}, \nonumber \\
X[\text{f},\text{F}] &\equiv \mathbf{X}_{\xi \Xi}=\begin{pmatrix}
X_{\bar{\omega} \Omega} & X_{\bar{\omega} \bar{\Omega}} \cc ^{-1} &  X_{\bar{\omega} \Lambda} \\
\cc ^{-1} X_{\omega \Omega} & \cc ^{-1} X_{\omega \bar{\Omega}}\cc ^{-1} & \cc^{-1} X_{\omega \Lambda} \\
\cc ^{-1} X_{\lambda \Omega} & \cc ^{-1} X_{\lambda \bar{\Omega}} \cc ^{-1} &  \cc^{-1} X_{\lambda \Lambda}
\end{pmatrix}, \nonumber \\
X[\text{S},\text{S}] &\equiv \mathbf{X}_{\Phi \Phi}=\begin{pmatrix}
X_{\Sigma ^* \Sigma} & X_{\Sigma ^* W}^\nu & X_{\Sigma ^* \Sigma ^*} & X_{\Sigma ^* W^*}^\nu & X_{\Sigma^* \Theta} & X_{\Sigma ^* V}^\nu \\
-X^\mu_{W ^* \Sigma} & -X^{\mu\nu}_{W ^* W} & -X^\mu_{W ^* \Sigma ^*} & -X^{\mu \nu}_{W ^* W^*} & -X^\mu_{W^* \Theta} & -X^{\mu \nu}_{W ^* V} \\
X_{\Sigma \Sigma} & X^\nu_{\Sigma W} & X_{\Sigma \Sigma ^{*}} & X^\nu_{\Sigma W^*} &  X_{\Sigma \Theta} & X^\nu_{\Sigma V} \\
-X^\mu_{W \Sigma} & -X^{\mu\nu}_{W W} & -X^\mu_{W \Sigma ^*} & -X^{\mu \nu}_{W W^*} & -X^\mu_{W \Theta} & -X^{\mu \nu}_{W V} \\
X_{\Theta \Sigma} & X^\nu_{\Theta W} &  X_{\Theta \Sigma ^*} & X^\nu_{\Theta W^*} & X_{\Theta \Theta} & X^\nu_{\Theta V}\\
-X^\mu_{V \Sigma} & -X^{\mu\nu}_{V W} & -X^\mu_{V \Sigma ^*} & -X^{\mu \nu}_{V W^*} & -X^\mu_{V \Theta} & -X^{\mu \nu}_{V V}
\end{pmatrix}, \nonumber \\
X[\text{S},\text{s}] &\equiv \mathbf{X}_{\Phi \phi}=\begin{pmatrix}
X_{\Sigma ^* \sigma} & X_{\Sigma ^* w}^\nu & X_{\Sigma ^* \sigma ^*} & X_{\Sigma ^* w^*}^\nu & X_{\Sigma^* \theta} & X_{\Sigma ^* v}^\nu \\
-X^\mu_{W ^* \sigma} & -X^{\mu\nu}_{W ^* w} & -X^\mu_{W ^* \sigma ^*} & -X^{\mu \nu}_{W ^* w^*} & -X^\mu_{W^* \theta} & -X^{\mu \nu}_{W ^* v} \\
X_{\Sigma \sigma} & X^\nu_{\Sigma w} & X_{\Sigma \sigma ^{*}} & X^\nu_{\Sigma w^*} &  X_{\Sigma \theta} & X^\nu_{\Sigma v} \\
-X^\mu_{W \sigma} & -X^{\mu\nu}_{W w} & -X^\mu_{W \sigma ^*} & -X^{\mu \nu}_{W w^*} & -X^\mu_{W \theta} & -X^{\mu \nu}_{W v} \\
X_{\Theta \sigma} & X^\nu_{\Theta w} &  X_{\Theta \sigma ^*} & X^\nu_{\Theta w^*} & X_{\Theta \theta} & X^\nu_{\Theta v}\\
-X^\mu_{V \sigma} & -X^{\mu\nu}_{V w} & -X^\mu_{V \sigma ^*} & -X^{\mu \nu}_{V w^*} & -X^\mu_{V \theta} & -X^{\mu \nu}_{V v}
\end{pmatrix}, \nonumber \\
X[\text{s},\text{S}] &\equiv \mathbf{X}_{\phi \Phi}=\begin{pmatrix}
X_{\sigma ^* \Sigma} & X_{\sigma ^* W}^\nu & X_{\sigma ^* \Sigma ^*} & X_{\sigma ^* W^*}^\nu & X_{\sigma^* \Theta} & X_{\sigma ^* V}^\nu \\
-X^\mu_{w ^* \Sigma} & -X^{\mu\nu}_{w ^* W} & -X^\mu_{w ^* \Sigma ^*} & -X^{\mu \nu}_{w ^* W^*} & -X^\mu_{w^* \Theta} & -X^{\mu \nu}_{w ^* V} \\
X_{\sigma \Sigma} & X^\nu_{\sigma W} & X_{\sigma \Sigma ^{*}} & X^\nu_{\sigma W^*} &  X_{\sigma \Theta} & X^\nu_{\sigma V} \\
-X^\mu_{w \Sigma} & -X^{\mu\nu}_{w W} & -X^\mu_{w \Sigma ^*} & -X^{\mu \nu}_{w W^*} & -X^\mu_{w \Theta} & -X^{\mu \nu}_{w V} \\
X_{\theta \Sigma} & X^\nu_{\theta W} &  X_{\theta \Sigma ^*} & X^\nu_{\theta W^*} & X_{\theta \Theta} & X^\nu_{\theta V}\\
-X^\mu_{v \Sigma} & -X^{\mu\nu}_{v W} & -X^\mu_{v \Sigma ^*} & -X^{\mu \nu}_{v W^*} & -X^\mu_{v \Theta} & -X^{\mu \nu}_{v V}
\end{pmatrix}, \nonumber \\
X[\text{s},\text{s}] &\equiv \mathbf{X}_{\phi \phi}=\begin{pmatrix}
X_{\sigma ^* \sigma} & X_{\sigma ^* w}^\nu & X_{\sigma ^* \sigma ^*} & X_{\sigma ^* w^*}^\nu & X_{\sigma^* \theta} & X_{\sigma ^* v}^\nu \\
-X^\mu_{w ^* \sigma} & -X^{\mu\nu}_{w ^* w} & -X^\mu_{w ^* \sigma ^*} & -X^{\mu \nu}_{w ^* w^*} & -X^\mu_{w^* \theta} & -X^{\mu \nu}_{w ^* v} \\
X_{\sigma \sigma} & X^\nu_{\sigma w} & X_{\sigma \sigma ^{*}} & X^\nu_{\sigma w^*} &  X_{\sigma \theta} & X^\nu_{\sigma v} \\
-X^\mu_{w \sigma} & -X^{\mu\nu}_{w w} & -X^\mu_{w \sigma ^*} & -X^{\mu \nu}_{w w^*} & -X^\mu_{w \theta} & -X^{\mu \nu}_{w v} \\
X_{\theta \sigma} & X^\nu_{\theta w} &  X_{\theta \sigma ^*} & X^\nu_{\theta w^*} & X_{\theta \theta} & X^\nu_{\theta v}\\
-X^\mu_{v \sigma} & -X^{\mu\nu}_{v w} & -X^\mu_{v \sigma ^*} & -X^{\mu \nu}_{v w^*} & -X^\mu_{v \theta} & -X^{\mu \nu}_{v v}
\end{pmatrix}, \nonumber \\
\end{align*}

\begin{align}
\begin{cases}
\mathbf{X}_{A B} = \mathbf{U}_{A B} + P_\mu \mathbf{Z}_{AB}^\mu+\mathbf{Z}^{\dagger\mu}_{AB} P_\mu, \text{ if } A \wedge B \in \{\phi, \Phi\} \\
\mathbf{X}_{A B} = \mathbf{U}_{A B} \text{ otherwise}
\end{cases}.
\end{align}

In the operators published at <cit.> we further defined

\begin{align}
\mathbf{U}_{A B} &= U[\text{A},\text{B}], \\
\mathbf{Z}_{AB}^\mu &= Z[\text{A},\text{B}][\text{lor}], \\
\mathbf{Z}_{AB}^{\dagger \mu} &= ZT[\text{A},\text{B}][\text{lor}].
\end{align}

Finally, covariant derivatives and field strength tensors are denoted by  $P[\text{lor}]$ and  $G[\text{A},\text{A}][\text{lor1},\text{lor2}]$, respectively. The field type A in the field strength tensor defines which covariant derivative is to be used in its construction. For example, $G[\text{F},\text{F}]$ is obtained from the commutator of two covariant derivatives acting on the heavy fermion multiplet $\Xi$. A commutator is denoted by $c$, for instance $c[P[\text{lor}],U[\text{A},\text{B}]]=[P^\mu,\mathbf{U}_{A B}]$, where the apropriate covariant derivative is the one acting on multiplet $A$.

Appendix C

CHAPTER: PROOF OF CONSISTENCY OF SHIFTS

In this appendix we discuss the consistency of the shifts given in Eqs.(<ref>) and (<ref>). The treatment of the shifts given in
Eqs. (<ref>) and (<ref>) is analogous. Since
$\xi$ is a multiplet of Majorana-like component spinors,
for the shifts

\begin{align}
\label{eq:xishift2}
\delta \xi' &= \delta \xi+\mathbf{\Delta}_\xi^{-1}\left[\tilde{\mathbf{X}}_{\xi \Xi} \delta \Xi-\tilde{\mathbf{X}}_{\xi \Phi} \delta \Phi-\tilde{\mathbf{X}}_{\xi \phi} \delta \phi\right], \\
\delta \xi'^T &= \delta \xi^T+\left[\delta \Xi^T \tilde{\mathbf{X}}_{\Xi \xi}+\delta \Phi^T \tilde{\mathbf{X}}_{\Phi \xi}+\delta \phi^T \tilde{\mathbf{X}}_{\phi \xi}\right]\overleftarrow{\mathbf{\Delta}}_\xi^{-1},
\label{eq:xishift2_T}
\end{align}

to be consistent it is necessary and sufficient that

\begin{align}
\left(\mathbf{\Delta}_\xi^{-1}\left[\tilde{\mathbf{X}}_{\xi \Xi} \delta \Xi-\tilde{\mathbf{X}}_{\xi \Phi} \delta \Phi-\tilde{\mathbf{X}}_{\xi \phi} \delta \phi\right]\right)^{T}=\left[\delta \Xi^T \tilde{\mathbf{X}}_{\Xi \xi}+\delta \Phi^T \tilde{\mathbf{X}}_{\Phi \xi}+\delta \phi^T \tilde{\mathbf{X}}_{\phi \xi}\right]\overleftarrow{\mathbf{\Delta}}_\xi^{-1}.
\label{eq:shiftCondition}
\end{align}

In the following we show that (<ref>) holds.

We first construct $\mathbf{\Delta}_\xi ^{-1}$ in position space through
its Neumann series

\begin{align}
\mathbf{\Delta}_\xi ^{-1}(x,y) &= \sum _{n=0} ^{\infty} \left(\prod _{\substack{i=1 \\ n>0}} ^{n} \int \rd^d x_i \; \mathbf{S}(x_{i-1},x_i) \left(-\mathbf{X}_{\xi \xi}(x_i)\right)\right) \mathbf{S}(x_n,y) \tilde{\mathds{1}}_F \cc ^{-1} \nonumber \\
& \equiv \sum _{n=0} ^{\infty} \left(\prod _{\substack{i=1 \\ n>0}} ^{n}  \mathbf{S}_{x_{i-1} x_i} \left(-\mathbf{X}_{\xi \xi x_i}\right)\right) \mathbf{S}_{x_n y} \tilde{\mathds{1}}_F \cc ^{-1},
\end{align}

where $x_0\equiv x$ and $\mathbf{S}(x,y)$ is the matrix-valued Green's
function for $(\slashed{P}-M_\xi)$, which itself can be expressed
through a Neumann series. We may write
$(\slashed{P}-M_\xi) = (i\slashed{\partial}-M_\xi-\mathbf{A})$ with

\begin{align}
\mathbf{A}=i\sum_j g_j\slashed{A}_j^a T_j^a,
\end{align}

where we sum over all factors of the gauge group for a direct product
group and $T_j^a$ is a block-diagonal matrix which generates the
reducible representation of $\xi$. Due to the fact that $\xi$ contains
$\omega$, $\ccfield{\omega}$ and $\lambda$ (see Table <ref>),
the generator is of the form

\begin{align}
T^a _{R(\omega)} && 0 && 0 \\
0 && T^a _{\bar{R}(\omega)} && 0 \\
0 && 0 && T^a_{R( \lambda )}
\end{pmatrix},
\end{align}

where $R(\omega)$ is the representation under which $\omega$
transforms, $\bar{R}(\omega)$ its conjugate representation and
$R(\lambda)$ is the representation of $\lambda$, which is necessarily
real. We then have

\begin{align}
\mathbf{S}_{x y}=\sum _{k=0} ^{\infty} \left(\prod _{\substack{i=1 \\ k>0}} ^{k}  \mathbf{S}_{f,x_{i-1} x_i} \mathbf{A}_{x_i}\right) \mathbf{S}_{f, x_k y},
\end{align}

where again $x_0 \equiv x$ and $\mathbf{S}_{f,x y}$ is the matrix
containing the Green's function of the free Dirac equation on its
diagonal. It can be verified by explicit calculation that

\begin{align}
\mathbf{S}_{x y} \left(-i\overleftarrow{\slashed{\partial}_y}-M_\xi-\mathbf{A}_y \right) &= \delta_{x y},
\end{align}

which means that

\begin{align}
\mathbf{\Delta}_{\xi,x y} ^{-1}\overleftarrow{\mathbf{\Delta}}_{\xi, y}=\delta_{x y}
\end{align}

and therefore
^{-1}=\mathbf{\Delta}_{\xi,yx}^{-1}$. Hence (<ref>)

\begin{align}
\left(\mathbf{\Delta}_{\xi, x y}^{-1}  \left[\tilde{\mathbf{X}}_{\xi \Xi} \delta \Xi-\tilde{\mathbf{X}}_{\xi \Phi} \delta \Phi-\tilde{\mathbf{X}}_{\xi \phi} \delta \phi\right]_y\right)^{T}=\left[\delta \Xi^T \tilde{\mathbf{X}}_{\Xi \xi}+\delta \Phi^T \tilde{\mathbf{X}}_{\Phi \xi}+\delta \phi^T \tilde{\mathbf{X}}_{\phi \xi}\right]_y\mathbf{\Delta}_{\xi, yx}^{-1}.
\end{align}

It is then useful to calculate

\begin{align}
\label{eq: id_needed}
\cc \tilde{\mathds{1}}_F \mathbf{S}^T_{xy}&=\cc \tilde{\mathds{1}}_F\sum _{k=0} ^{\infty} \mathbf{S}^T_{f,x_k y} \left(\prod _{\substack{i=k \\ k>0}} ^{1}  \mathbf{A}^T_{x_i}\mathbf{S}^T_{f,x_{i-1}x_i} \right) \nonumber  \\
&=  \cc \tilde{\mathds{1}}_F\sum _{k=0} ^{\infty} \cc \mathbf{S}_{f,y x_k} \cc^{-1} \left(\prod _{\substack{i=k \\ k>0}} ^{1}  \mathbf{A}^T _{x_i}\cc \mathbf{S}_{f,x_i x_{i-1}}\cc^{-1} \right) \nonumber \\
&=  -\sum _{k=0} ^{\infty}  \tilde{\mathds{1}}_F \mathbf{S}_{f,y x_k} \tilde{\mathds{1}}_F \tilde{\mathds{1}}_F \left(\prod _{\substack{i=k \\ k>0}} ^{1}   (-\mathbf{A}^t _{x_i})\tilde{\mathds{1}}_F \tilde{\mathds{1}}_F \mathbf{S}_{f,x_i x_{i-1}} \tilde{\mathds{1}}_F \tilde{\mathds{1}}_F \right)\cc^{-1} \nonumber
\\
&=  -\sum _{k=0} ^{\infty}  \mathbf{S}_{f, y x_k} \left(\prod _{\substack{i=k \\ k>0}} ^{1} (-\tilde{\mathds{1}}_F\mathbf{A}^t_{x_i}\tilde{\mathds{1}}_F)\mathbf{S}_{f,x_i x_{i-1}} \right)\tilde{\mathds{1}}_F\cc^{-1} \nonumber \\
&=  -\sum _{k=0} ^{\infty}  \mathbf{S}_{f,y x_k} \left(\prod _{\substack{i=k \\ k>0}} ^{1}  \mathbf{A}_{x_i} \mathbf{S}_{f,x_i x_{i-1}} \right)\tilde{\mathds{1}}_F\cc^{-1}  \nonumber \\
&= -\mathbf{S}_{yx}\tilde{\mathds{1}}_F\cc ^{-1},
\end{align}

where $\mathbf{A}^t$ means taking the transpose of the gauge group
generators only and we used that

\begin{align}
\tilde{\mathds{1}}_F\begin{pmatrix}
A && 0 && 0 \\
0 && B && 0 \\
0 && 0 && C
\end{pmatrix}
\tilde{\mathds{1}}_F=\begin{pmatrix}
B && 0 && 0 \\
0 && A && 0 \\
0 && 0 && C
\end{pmatrix}.
\end{align}

We note in passing that Eq. (<ref>) also holds if the generators contain projectors in spinor space, since the charge conjugation operator commutes with these projectors in the chiral representation, which we have adopted. Thus, this also covers the construction of Section <ref>. We now find

\begin{align}
\left(\mathbf{\Delta}_{\xi, xy} ^{-1}\right)^T &= \cc \tilde{\mathds{1}}_F\sum _{n=0} ^{\infty} \mathbf{S}_{x_n,y}^T \left(\prod _{\substack{i=1 \\ n>0}} ^{n}   \left(-\mathbf{X}_{\xi \xi, x_i}\right)^T \mathbf{S}_{x_{i-1}x_i}^T \right) \nonumber  \\
&= \sum _{n=0} ^{\infty} \mathbf{S}_{y x_n} \tilde{\mathds{1}}_F\cc \left(\prod _{\substack{i=1 \\ n>0}} ^{n}  \left(-\mathbf{X}_{\xi \xi, x_i}\right)^T \mathbf{S}_{x_{i-1} x_i}^T \right) \nonumber  \\
&= \sum _{n=0} ^{\infty} \mathbf{S}_{y x_n} \tilde{\mathds{1}}_F\cc \left(\prod _{\substack{i=1 \\ n>0}} ^{n} \left(-\mathbf{X}_{\xi \xi, x_i}\right)^T \tilde{\mathds{1}}_F\cc ^{-1} \tilde{\mathds{1}}_F\cc  \mathbf{S}_{x_{i-1} x_i}^T \tilde{\mathds{1}}_F\cc ^{-1} \tilde{\mathds{1}}_F\cc \right) \nonumber  \\
&= \sum _{n=0} ^{\infty} \mathbf{S}_{y x_n} \tilde{\mathds{1}}_F\cc \left(\prod _{\substack{i=1 \\ n>0}} ^{n}    \left(-\mathbf{X}_{\xi \xi, x_i}\right)^T \tilde{\mathds{1}}_F\cc ^{-1}   \mathbf{S}_{x_i x_{i-1}}  \tilde{\mathds{1}}_F\cc \right) \nonumber   \\
&= -\sum _{n=0} ^{\infty} \mathbf{S}_{y x_n}  \left(\prod _{\substack{i=1 \\ n>0}} ^{n} \left(-\mathbf{X}_{\xi \xi, x_i}\right)    \mathbf{S}_{x_i x_{i-1}} \right)  \tilde{\mathds{1}}_F\cc ^{-1} \nonumber  \\
&= -\mathbf{\Delta}_{\xi yx} ^{-1},
\end{align}

where we used that

\begin{align}
\cc \tilde{\mathds{1}}_F \mathbf{X}^T_{\xi \xi} \tilde{\mathds{1}} _F\cc^{-1}=\mathbf{X}_{\xi \xi}.
\end{align}

Noting that

\begin{align}
\tilde{\mathbf{X}}^T_{\xi \Xi}&=-\tilde{\mathbf{X}}_{\Xi \xi}, \\
\tilde{\mathbf{X}}^T_{\xi \Phi}&=\tilde{\mathbf{X}}_{\Phi \xi}, \\
\tilde{\mathbf{X}}^T_{\xi \phi}&=\tilde{\mathbf{X}}_{\phi \xi},
\end{align}

the validity of (<ref>) follows immediately.

Appendix D

CHAPTER: LOOP FUNCTIONS

The integrals $\ZI [q^{2n_c}]^{n_i n_j \dots n_L} _{i j \dots 0}$ are defined as in Ref. <cit.>, that is

\begin{align}
\int \frac{\rd^d q}{(2\pi)^d} \frac{q^{\mu_1} q^{\mu_2} \dots q^{\mu_{2n_c}}}{(q^2-M_i^2)^{n_i}(q^2-M_j^2)^{n_j}\dots (q^2)^{n_L}}\equiv \frac{i}{16 \pi ^2} g^{\mu_1 \mu_2 \dots \mu_{2n_c}} \ZI [q^{2n_c}]^{n_i n_j \dots n_L} _{i j \dots 0},
\end{align}

where $g^{\mu_1 \mu_2 \dots \mu_{2n_c}}$ is the completely symmetric
combination of metric tensors with $2n_c$ indices, for instance
$g^{\mu \nu \rho \sigma}=g^{\mu \nu}g^{\rho \sigma}+g^{\mu \rho}g^{\nu
\sigma}+g^{\mu \sigma}g^{\nu \rho}$.  For $n_c=0$ we define the
shorthand notation
$\ZI[q^0]^{n_i n_j \dots n_L} _{i j \dots 0} \equiv \ZI^{n_i n_j \dots
n_L} _{i j \dots 0}$.  The integrals can be reduced to basis
integrals using the following reduction relations, which were already stated in Ref. <cit.>

\begin{align}
\label{eq: reduction_formula_1}
\ZI[q^{2n_c}] ^{n_i n_j \dots n_L} _{i j \dots 0} &=
\frac{1}{\Delta^2 _{ij}}\left(\ZI[q^{2n_c}]^{n_i n_j-1 \dots n_L}-\ZI[q^{2n_c}]^{n_i-1 \, n_j \dots n_L}\right), \\
\label{eq: reduction_formula_2}
\ZI[q^{2n_c}] ^{n_i n_j \dots n_L} _{i j \dots 0} &=
\frac{1}{M_i^2}\left(\ZI[q^{2n_c}]^{n_i n_j \dots n_L-1}-\ZI[q^{2n_c}]^{n_i-1 \, n_j \dots n_L}\right),
\end{align}

where $\Delta^2 _{ij}=M_i^2-M_j^2$. The basis integrals can be computed using the formula

\begin{align}
\ZI[q^{2n_c}]^{n_i n_L}_{i0} ={}& \left(-M_i^2\right)^{2+n_c-n_i-n_L} \frac{1}{2^{n_c}(n_i-1)!}\frac{\Gamma(\frac{\epsilon}{2}-2-n_c+n_i+n_L)}{\Gamma(\frac{\epsilon}{2})}\nonumber \\ &\times \frac{\Gamma(-\frac{\epsilon}{2}+2+n_c-n_L)}{\Gamma(-\frac{\epsilon}{2}+2+n_c)}\left(\frac{2}{\bar{\epsilon}}-\log \frac{M_i^2}{\mu^2}\right),
\label{eq: basis_integrals}
\end{align}

where $\epsilon = 4-d$ and $\frac{2}{\bar{\epsilon}}=\frac{2}{\epsilon}-\gamma+\log 4\pi$.

In the lists of BSUOLEA operators found at <cit.> the integrals are denoted by

\begin{align}
J[n_c, \{\{M_i,n_i\}, \{M_j,n_j\}, \ldots\}, n_L] \equiv \ZI [q^{2n_c}]^{n_i n_j \dots n_L} _{i j \dots 0},
\end{align}

and in the code the insertion of the correct integrals is performed automatically.

Appendix E

CHAPTER: THE BEREZIN ALGEBRA

§ GENERALITIES

In this appendix we introduce some properties of the Berezin algebra which are used throughout this thesis. For more details we refer the reader to Appendix D of Ref. <cit.> and to Ref. <cit.>. We will define the Berezin algebra $B$ as the algebra over the field $\mathds{C}$ generated by the elements $z^a = (\theta^i,\xi^\alpha)$, where the generating elements $\xi^\alpha$ satisfy

\begin{align*}
\xi^\alpha \xi^\beta + \xi^\beta \xi^\alpha = 0,
\end{align*}

and any $b \in B$ may be written as

\begin{align}
b = f_0(\theta)+f_\alpha(\theta)\xi^\alpha+\ldots+f_{\alpha_1 \ldots \alpha_n}(\theta)\xi^{\alpha_1} \cdots \xi^{\alpha_n},
\label{eq: general_b_elem}
\end{align}

where the functions $f_{{\alpha_1 \ldots \alpha_n}}$ are anti-symmetric in their indices. Any $b \in B$ may be written as an expansion in $z^a$ by Taylor expanding these functions. We say that an element of $B$ is Grassmann odd if its representation of the form given in Eq. (<ref>) only contains odd powers of $\xi$. Similarly, any element of $B$ whose representation only contains even powers of $\xi$ will be referred to as Grassmann even. For an element $b \in B$ we introduce the parity $P_b$ as

\begin{align}
\begin{cases}
P_b = 1, \text{ if $b$ is Grassmann odd}\\
P_b = 0, \text{ if $b$ is Grassmann even}
\end{cases},
\end{align}

whenever this is well-defined. The only function of the parity $P$ relevant for this thesis is $(-1)^P$ and hence the parity of $b_3=b_1b_2$ can be taken to be $P_{b_3}=P_{b_1}+P_{b_2}$, whenever $b_1$ and $b_2$ have definite parities. We also introduce the parity of the index $a$, $P_{(a)}$, corresponding to the element $z^a$, as

\begin{align}
\end{align}

This definition allows for the introduction of supermatrices as follows. A supermatrix $M$ is a set of matrix elements $M_{ab}\in B$, with definite parity, together with a set of parities of its indices $\left(P_{(a)},P_{(b)}\right)$ such that

\begin{align}
P_{M_{ab}} = P_{(a)}+P_{(b)}.
\end{align}

For any supermatrix $M$ we introduce the normal form $M^{(N)}$, which is obtained from $M$ by simultaneous permutation of rows and columns so that $M^{(N)}$ has a definite order of indices: first come all Grassmann even and then all Grassmann odd indices. This matrix can be represented as

\begin{align}
M^{(N)} = \begin{pmatrix}
\left(M_1\right)_{ij} & \left(M_2\right)_{i \beta} \\
\left(M_3\right)_{\alpha j} & \left(M_4\right)_{\alpha \beta}
\end{pmatrix},
\end{align}

with $P_{(i)}=P_{(j)}=0$ and $P_{(\alpha)}=P_{(\beta)}=1$. Given any supermatrix $M$ we may define its supertrace $\Str \, M$ and its superdeterminant $\Sdet \, M$ as

\begin{align}
\Str \, M &= \sum_a (-1)^{P_{(a)}} M_{aa},\\
\Sdet \, M &= \exp \Str\log \, M.
\end{align}

Several properties of these quantities can be found in Appendix D of Ref. <cit.>. The relevant ones for this thesis are

\begin{align}
\Sdet \, M = \Sdet \, M^{(N)} &= \det M_1 \det(M_4-M_3 M_1^{-1}M_2)^{-1} \\
&= \det(M_1-M_2 M_4^{-1}M_3) \det M_4^{-1}.
\end{align}

We finally define the left and right derivatives w.r.t. generating elements. It is sufficient to define these on a product of generating elements. The left derivative is thus defined through

\begin{align}
\frac{\partial _l}{\partial z^a}z^{a_1}z^{a_2}\ldots z^{a_n}=\sum_{i=1}^n (-1)^{\sum_{j=1}^{i-1}P_{(a)}P_{(a_j)}}\delta^{a_i}_a z^{a_1}\ldots z^{a_{i-1}}z^{a_{i+1}}\ldots z^{a_n},
\end{align}

whereas the right derivative is defined through

\begin{align}
\frac{\partial _r}{\partial z^a}z^{a_1}z^{a_2}\ldots z^{a_n}=\sum_{i=1}^n (-1)^{\sum_{j=n}^{i+1}P_{(a)}P_{(a_j)}}\delta^{a_i}_a z^{a_1}\ldots z^{a_{i-1}}z^{a_{i+1}}\ldots z^{a_n}.
\end{align}

§ THE POISSON BRACKET IN THE BEREZIN ALGEBRA

In order to formulate a classical theory in which some of the degrees of freedom, collectively denoted by $q^i$, are Grassmann even and some Grassmann odd one has to generalise the Poisson bracket. Taking $q^i$ and the associated conjugate momenta $p_i$ as the generators of the Berezin algebra, any function $\mathcal{F}$ of these variables is itself an element of the generated Berezin algebra. Then, considering the functions $\mathcal{F}$ and $\mathcal{G}$ of definite parity, we define their Poisson bracket to be

\begin{align}
\{\mathcal{F},\mathcal{G}\}= \frac{\partial _r \mathcal{F}}{\partial q^i} \frac{\partial _l \mathcal{G}}{\partial p_i}-(-1)^{P_\mathcal{F} P_\mathcal{G}}\frac{\partial _r \mathcal{G}}{\partial q^i} \frac{\partial _l \mathcal{F}}{\partial p_i}.
\label{eq: PB_def}
\end{align}

In Eq. (<ref>) a summation over the index $i$ is implied. In the case of a field theory, the partial derivatives are to be interpreted as functional derivatives taken at a fixed time and the summation over $i$ includes an integration over space. Two useful properties of this generalised Poisson bracket are

\begin{align}
\label{eq: PB_prop1}
\{\mathcal{F},\mathcal{G}\} &= -(-1)^{P_\mathcal{F} P_\mathcal{G}} \{\mathcal{G},\mathcal{F}\}, \\
\label{eq: PB_prop2}
\{\mathcal{F},\mathcal{G}\mathcal{K}\} &= \{\mathcal{F},\mathcal{G}\}\mathcal{K}+(-1)^{P_\mathcal{F} P_\mathcal{G}} \mathcal{G} \{\mathcal{F},\mathcal{K}\}.
\end{align}

Appendix F

CHAPTER: SOME DETAILS OF THE CONSTRAINT ANALYSIS

In this section we show the validity of Eq. (<ref>), that is, we prove that

\begin{align}
\{\Phi_{1(2)}^A,H_T\} \approx -g f^{ABC} \left(W^{0B} \Phi^C_{1(2)}+V^{0B} \Phi^C_{2(2)}\right).
\end{align}

For convenience we repeat the two relevant constraints,

\begin{align}
\label{eq: secondary_gauge_constraint_convenience}
\Phi_{1(2)}^A ={}&  D^i \pi(W)_i^A + g f^{DAC} \pi(V)^D_\mu V^{\mu C} +\frac{i}{2}g\Hdagg \tau^A \pi(\Hdagg) -\frac{i}{2}g\pi(H) \tau^A H \nonumber \\ & -\frac{i}{2} g \bar{\pi}(f) \tau^A f - \frac{i}{2}g \bar{f} \tau^A \pi(\bar{f}), \\
\label{eq: secondary_resonance_constraint_convenience}
\Phi_{2(2)}^A  ={}& D^i \pi(V)_i^A+\frac{i}{2}g_H \Hdagg \tau^A \pi(\Hdagg)-\frac{i}{2}g_H \pi(H) \tau^A H \nonumber \\
&+ V^{0A} |H|^2 \left(g_{VH}-\frac{g_H^2}{2}\right)+ g_f J_0^{fA} + m_V^2 V^{0A}.
\end{align}

Using the linearity of the PB we may write

\begin{align}
\{\Phi_{1(2)}^A,H_T\} ={}& \{D^i \pi(W)_i^A,H_T\}+gf^{DAC} \{\pi(V)^D_\mu V^{\mu C},H_T\} \nonumber \\
&+\frac{i}{2}g \{\Hdagg \tau^{A} \pi(\Hdagg),H_T\}-\frac{i}{2}g \{\pi(H) \tau^{A} H,H_T\} \nonumber \\
&-\frac{i}{2}g\{\bar{\pi}(f) \tau^{A} f,H_T\}-\frac{i}{2}g\{\bar{f} \tau^{A} \pi(\bar{f}),H_T\}.
\label{eq: full_PB}
\end{align}

One may then compute the different contributions independently using Eq. (<ref>) and the definition as given in Eq. (<ref>). It should be noted that the PBs have to be interpreted as functionals and the equalities below hold when acting upon appropriate test functions. Consider the first PB of Eq. (<ref>), which can be written as

\begin{align}
\{D^i \pi(W)^A_i(x),H_T\} ={}& \{\partial^i \pi(W)^A_i(x),H_T\} + g f^{ABC} W^{iB}(x)\{\pi(W)_i^C(x),H_T\} \nonumber \\
&+ g f^{ABC} \pi(W)_i^C(x) \{W^{iB}(x),H_T\},
\label{eq: full_PB_CD_part}
\end{align}

where we have written out the dependence of the field on the spacetime point explicitly. The first term in the first line is given by

\begin{align}
\{\partial^i \pi(W)^A_i(x),H_T\} = - \int \rd^3 y \int \rd^3 z \left(\partial^i_x \delta_{xy}\right) \frac{\delta_t\Ham (z)}{\delta W^{iA}(y)},
\end{align}

which acting on a test-function $\varphi^A(x)$ yields

\begin{align}
-\int \rd^3 x \int \rd^3 y \int \rd^3 z \, \varphi^A(x) \left(\partial^i_x \delta_{xy}\right) \frac{\delta_t\Ham (z)}{\delta W^{iA}(y)} = -\int \rd^3 y \int \rd^3 z \, \varphi^A(y) \partial^i_y \frac{\delta_t\Ham (z)}{\delta W^{iA}(y)}.
\label{eq: full_PB_partial_part}
\end{align}

To arrive at Eq. (<ref>) we first used the delta function, $\delta_{xy}\equiv \delta(\mathbf{x}-\mathbf{y})$, to perform the $x$-integral and then integrated by parts in the $y$-integral. It follows from this that

\begin{align}
\{\partial^i \pi(W)^A_i(x),H_T\} = - \int \rd^3 z \, \partial^i_x \frac{\delta_t\Ham (z)}{\delta W^{iA}(x)},
\end{align}

and in a similar fashion one can show that

\begin{align}
g f^{ABC} W^{iB}(x) \{\pi(W)_i^C(x),H_T\} = - g f^{ABC} \int \rd^3 z \, W^{iB}(x) \frac{\delta_t\Ham (z)}{\delta W^{iA}(x)},
\end{align}

so that the first line of Eq. (<ref>) is given by

\begin{align}
\{\partial^i \pi(W)^A_i(x),H_T\} + g f^{ABC} W^{iB}(x)\{\pi(W)_i^C(x),H_T\} = - \int \rd^3 z \, D^i_x \frac{\delta_t\Ham (z)}{\delta W^{iA}(x)}.
\label{eq: first_line}
\end{align}

The relevant functional derivative of the Hamiltonian density can be computed from Eq. (<ref>) and reads

\begin{align}
-\frac{\delta_t\Ham (z)}{\delta W^{iA}(x)} ={}& \Big(-g f^{DAF} \pi(W)_i^D(z) W^{0F}(z)-g f^{DAF} \pi(V)_i^D(z) V^{0F}(z)\nonumber \\
&+\frac{g_H g}{4} V_i^B(z) \Hdagg (z) [\tau^B,\tau^A]_{+} H(z) + \frac{g}{2}\bar{f}(z)\tau^A \gamma_i f(z) \nonumber \\
&+ \frac{ig}{2}\Hdagg(z) \tau^A \overleftrightarrow{D}_i H(z) + g f^{ABC} V^B_{ij}(z)V^{jC}(z) + g g_M f^{ABC} W_{ij}^{B}(z) V^{jC}(z) \nonumber \\
& +W^B_{ik}(z)D^{kBA}_z+g_M V^B_{ik}(z)D^{kBA}_z\Big) \delta_{zx},
\label{eq: full_PB_CD_kernel}
\end{align}

\begin{align}
D^{kBA}_z = \partial_z^k \delta^{AB} + g f^{BCA} W^{kC}(z).
\end{align}

For the first three lines of Eq. (<ref>) the delta function ensures that we obtain a total covariant derivative of each individual term upon insertion into Eq. (<ref>). For the last line of Eq. (<ref>) the delta function allows us to move the covariant derivative to the field strength tensors at the cost of a minus sign. We may therefore write Eq. (<ref>) as

\begin{align}
\{\partial^i \pi(W)^A_i(x),H_T\} + g f^{ABC} W^{iB}(x)\{\pi(W)_i^C(x),H_T\} = D^i \mathcal{R}_i^A -\frac{g g_M}{2}f^{ABC} W^{ikB}V_{ik}^C  ,
\label{eq: full_PB_CD_kernel_final}
\end{align}

where the r.h.s. is evaluated at the spacetime point $x$. We here defined

\begin{align}
\mathcal{R}_i^A ={}& -g f^{DAF} \pi(W)_i^D W^{0F}-g f^{DAF} \pi(V)_i^D V^{0F}\nonumber \\
&+\frac{g_H g}{4} V_i^B \Hdagg  [\tau^B,\tau^A]_{+} H + \frac{g}{2}\bar{f}\tau^A \gamma_i f \nonumber \\
&+ \frac{ig}{2}\Hdagg \tau^A \overleftrightarrow{D}_i H + g g_M f^{ABC} W_{ij}^B V^{jC} +  g f^{ABC} V^B_{ij}V^{jC},
\end{align}

and used that

\begin{align}
-D^i D^k V_{ik}^B = -\frac{g}{2} f^{BCD}W^{ikC} V_{ikB}.
\end{align}

A similar term arises from $-D^i D^k W_{ik}^B$ and vanishes due the antisymmetry of the structure constants. To Eq. (<ref>) we have to add

\begin{align}
g f^{ABC} \pi(W)_i^C \{W^{iB},H_T\} ={}& -\frac{g_M}{g_M^2-1}g f^{ABC} \pi(W)_i^C \pi(V)^{iB} \nonumber \\
&+ g f^{ABC} \pi(W)_i^C(x) D^i W^{0B},
\label{eq: full_PB_CD_part_gauge_cont}
\end{align}

to obtain a full expression for the first term contributing to Eq. (<ref>). It should be noted that combining the first term of $D^i \mathcal{R}^A_i$ with the last line of Eq. (<ref>) one obtains

\begin{align}
-g f^{ABC} W^{0B}D^i \pi(W)^C_i \subset -g f^{ABC} W^{0B} \Phi_{1(2)}^C.
\label{eq: Phi_12_1}
\end{align}

Having treated the first term contributing to Eq. (<ref>) in detail we simply state the results for the remaining ones. We find

\begin{align}
gf^{DAC} \{\pi(V)^D_\mu V^{\mu C},H_T\} \approx {}& gf^{DAC}\Big(\frac{-g_M}{g_M^2-1}\pi(V)_i^D \pi(W)^{iC}+\pi(V)_i^D D^i V^{0C} \nonumber \\
&~~~~~~~~~~- g f^{CFE} \pi(V)_i^D W^{0F}V^{iE}+g f^{EBD} \pi(V)_i^E V^{iC} W^{0B}\nonumber \\
&~~~~~~~~~~+g_H V^{iC}J^{HD}_i+g_f V^{iC}J^{fD}_i \nonumber \\
\label{eq: full_PB_resonances_contr}
\end{align}

where the first term in Eq. (<ref>) cancels the first term in Eq. (<ref>) and the second term combines with the second term of $D^i \mathcal{R}^A_i$ to yield

\begin{align}
-g f^{ABC} V^{0B}D^i \pi(V)^C_i \subset -g f^{ABC} V^{0B} \Phi_{2(2)}^C.
\label{eq: Phi_2_2_1}
\end{align}

Furthermore, the last term in the last line in the definition of $\mathcal{R}^A_i$, when acted on with a covariant derivative, combines with the first term in the last line of Eq. (<ref>) to yield a contribution that is proportional to $f^{ABC}V_{ij}^{B}V^{ijC}=0$. Similarly, the second term in the last line of Eq. (<ref>) combines with the second to last term of $D^i \mathcal{R}_i^A$ to yield $\frac{gg_M}{2}f^{ABC}V^{jiC}W_{ji}^B$, which then cancels the second term on the r.h.s. of Eq. (<ref>). Using the Jacobi identity, the second line of Eq. (<ref>) can be brought into the form

\begin{align}
-g f^{AFC}W^{0F} \left(gf^{DCE} \pi(V)^D_iV^{iE}\right) \subset -g f^{AFC} W^{0F} \Phi^C_{1(2)}.
\label{eq: Phi_12_2}
\end{align}

This eliminates all contributions to Eqs. (<ref>) and (<ref>) that do not depend on at least one fermionic or one scalar field. The second line of Eq. (<ref>) is given by

\begin{align}
\frac{i}{2}g \{\Hdagg \tau^{A} \pi(\Hdagg),H_T\}-\frac{i}{2}g \{\pi(H) \tau^{A} H,H_T\} \approx{}& -\frac{ig}{2}f^{ABC} \Hdagg \tau^C \pi(\Hdagg) (gW_0^B+g_H V_0^B) \nonumber \\ &+\frac{ig}{2}f^{ABC} \pi(H) \tau^C H (gW_0^B+g_H V_0^B) \nonumber \\
&-\frac{g g_H}{4} \Hdagg \tau^A \tau^B \left(2V^{iB}D_iH+HD_iV^{iB}\right) \nonumber \\
&-\frac{g g_H}{4} \left(2V^{iB}D_i\Hdagg+\Hdagg D_iV^{iB}\right) \tau^B \tau^A H \nonumber \\
& -\frac{ig}{2} \Hdagg \tau^A D_iD^i H +\frac{ig}{2} \left(D_iD^i\Hdagg\right) \tau^A H \nonumber \\
& +\frac{ig}{2} \Hdagg \tau^A \frac{\partial \Lag{rest}}{\partial \Hdagg}-\frac{ig}{2} \frac{\partial \Lag{rest}}{\partial H} \tau^A H,
\label{eq: full_PB_scalar_contr}
\end{align}

the first two lines of which are recognised as

\begin{align}
-gf^{ABC}W_0^B \left(\frac{ig}{2} \Hdagg \tau^C \pi(\Hdagg) - \frac{ig}{2} \pi(H) \tau^C H \right) \subset  -gf^{ABC}W_0^B \Phi^C_{1(2)},
\label{eq: Phi_12_3}
\end{align}

\begin{align}
-gf^{ABC}V_0^B \left(\frac{ig_H}{2} \Hdagg \tau^C \pi(\Hdagg) - \frac{ig_H}{2} \pi(H) \tau^C H \right) \subset  -gf^{ABC}V_0^B \Phi^C_{2(2)},
\label{eq: Phi_22_2}
\end{align}

completing the first line of the constraints of Eqs. (<ref>) and (<ref>). Noting that

\begin{align}
D^i\left(\frac{ig}{2}\Hdagg \tau^A \overleftrightarrow{D}_i H\right) = \frac{ig}{2}\left(-\left(D_iD^i \Hdagg\right)\tau^A H + \Hdagg \tau^A D^iD_i H \right),
\end{align}

we find that this contribution to $D^i \mathcal{R}_i^A$ cancels the second to last line of Eq. (<ref>). In order to treat the third line of Eq. (<ref>) we act on a test function $\varphi^A$ and integrate one of the two terms where the covariant derivative acts on the scalar doublet by parts

\begin{align}
&-\frac{g g_H}{4} \int \rd^3 x\, \Hdagg \tau^A \tau^B \left(2V^{iB}D_iH+HD_iV^{iB}\right) \varphi^A \nonumber \\
&= \frac{g g_H}{4} \int \rd^3 x \, V^{iB} \left[\left(D_i\Hdagg\right) \tau^A \tau^BH\varphi^A+\Hdagg \tau^A \tau^BHD_i\varphi^A-\Hdagg \tau^A \tau^B\left(D_iH\right)\varphi^A\right].
\end{align}

Performing the corresponding operation on the fourth line of Eq. (<ref>) and combining the two results we find

\begin{align}
&-\frac{g g_H}{4} \! \int \rd^3x \! \left[\Hdagg \tau^A \tau^B \left(2V^{iB}D_iH+HD_iV^{iB}\right)+\left(2V^{iB}D_i\Hdagg+\Hdagg D_iV^{iB}\right) \tau^B \tau^A H\right] \varphi^A \nonumber \\
&= -\frac{g g_H}{4} \int \rd^3 x \left[V^{iB}\Hdagg [\tau^A,\tau^B]\left(D_iH\right)-V^{iB}\left(D_i\Hdagg\right) [\tau^A,\tau^B]H\right]\varphi^A \nonumber \\
&~~~~ -\frac{g g_H}{4} \int \rd^3 x\, D_i \left(V^{iB} \Hdagg [\tau^A,\tau^B]_{+}H\right)\varphi^A,
\label{eq: full_PB_scalar_contr_intermediate_step}
\end{align}

where $[\tau^A,\tau^B]$ is the commutator of $\tau^A$ and $\tau^B$. Removing the test function we note that the last line of Eq. (<ref>) cancels with the first term of the second line of $D^i \mathcal{R}_i^A$. The first line of Eq. (<ref>) yields

\begin{align}
-\frac{g g_H}{4} \left[V^{iB}\Hdagg [\tau^A,\tau^B]\left(D_iH\right)-V^{iB}\left(D_i\Hdagg\right) [\tau^A,\tau^B]H\right] = -g g_H f^{CAB}V^{iB}J^{HC}_i,
\end{align}

which cancels the first term of the third line of Eq. (<ref>). This only leaves the last line of Eq. (<ref>), which we will comment on shortly. First, we address the last line of Eq. (<ref>), which is given by

\begin{align}
-\frac{i}{2}g\{\bar{\pi}(f) \tau^{A} f,H_T\}-\frac{i}{2}g\{\bar{f} \tau^{A} \pi(\bar{f}),H_T\} \approx{}& -g g_F f^{ABC} V^{\mu B} J_\mu^{fC}-g^2 f^{ABC} W^{\mu B} J_\mu^{fC} \nonumber \\
&-ig \frac{\partial_r \Lag{rest}}{\partial f}\frac{\tau^A}{2}f-ig \frac{\partial _r \Lag{rest}}{\partial \bar{f}} \frac{\tau^A}{2} \bar{f} \nonumber \\
&-\frac{g}{2}\partial_i\left(\bar{f}\tau^A\gamma^i f\right).
\label{eq: full_PB_fermionic_contr}
\end{align}

The spatial part of the first term of Eq. (<ref>) cancels the last remaining term of Eq. (<ref>) and the time component yields

\begin{align}
-g g_F f^{ABC} V^{0 B} J_0^{fC} \subset -g f^{ABC} V^{0B} \Phi^C_{2(2)}.
\label{eq: Phi_22_3}
\end{align}

The time component of the second term of Eq. (<ref>) is given by

\begin{align}
-g^2 f^{ABC}W^{0B} \bar{f}\gamma^0 \frac{\tau^A}{2}f \approx \frac{i}{2} g^2 f^{ABC} W^{0B} \left( \bar{\pi}(f) \tau^A f + \bar{f} \tau^A \pi(\bar{f})\right) \subset -g f^{ABC}W^{0B}\Phi^C_{1(2)},
\label{eq: Phi_12_4}
\end{align}

where we used the fermionic constraints. The spatial part of this term combines with the last line of Eq. (<ref>) to yield

\begin{align}
\end{align}

which cancels the last remaining term of $D^i \mathcal{R}_i^A$. This only leaves the terms

\begin{align}
\frac{ig}{2} \Hdagg \tau^A \frac{\partial \Lag{rest}}{\partial \Hdagg}-\frac{ig}{2} \frac{\partial \Lag{rest}}{\partial H} \tau^A H-ig \frac{\partial_r \Lag{rest}}{\partial f}\frac{\tau^A}{2}f-ig \frac{\partial _r \Lag{rest}}{\partial \bar{f}} \frac{\tau^A}{2} \bar{f},
\end{align}

which vanish as long as $\Lag{rest}$ is gauge invariant. For a proof see Ref. <cit.>. Collecting Eqs. (<ref>), (<ref>), (<ref>) and (<ref>) we find precisely $-gf^{ABC}W^{0B}\Phi_{1(2)}^C$. Similarly, collecting Eqs. (<ref>), (<ref>) and (<ref>) one finds $-gf^{ABC}V^{0B}\Phi_{2(2)}^C$ since $f^{ABC}V^{0B}V^{0C}=0$.

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volumeB 709

note[Erratum: JHEP 07, 066 (2020)]
titleSinglet night in Feynman-ville: one-loop matching of a real scalar

titleComplete One-Loop Matching for a Singlet Scalar in the Standard Model EFT

journaltitleNucl. Phys.
titleThe Effective One Loop Lagrangian With Derivative Couplings
volumeB 268

journaltitleNucl. Phys.
titleEffective Action for the Standard Model With Large Higgs Mass
volumeB 297

titleHow to use the Standard Model effective field theory

titleThe Universal One-Loop Effective Action

titleIntegrating out heavy particles with functional methods: a simplified framework

titleOne-loop Matching and Running with Covariant Derivative Expansion

titleExtending the Universal One-Loop Effective Action: Heavy-Light Coefficients

titleCompleting the scalar and fermionic Universal One-Loop Effective Action

journaltitlePhys. Lett.
titleSupersymmetric Dimensional Regularization via Dimensional Reduction
volumeB 84

titleRegularization by dimensional reduction: consistency, quantum action principle, and supersymmetry

journaltitleNucl. Phys.
titleRegularization and Renormalization of Gauge Fields
volumeB 44

titleCovariant diagrams for one-loop matching

journaltitleEur. Phys. J.
titleOne-loop effective lagrangians after matching
volumeC 76

Reading, USA

titleAn Introduction to quantum field theory

journaltitlePhys. Rept.
titleChiral Gauge Theory

Berlin, Boston

De Gruyter

journaltitleJ. Reine Angew. Math
titleÜber Potenzreihen, die im Innern des Einheitskreises beschränkt sind.

journaltitleNucl. Phys.
titleAsymptotic expansion of Feynman integrals near threshold
volumeB 522

journaltitleSpringer Tracts Mod. Phys.
titleApplied asymptotic expansions in momenta and masses

titleFoundation and generalization of the expansion by regions

journaltitlearXiv e-prints
titleIntegrating Out New Fermions at One Loop
High Energy Physics - Phenomenology
family=Hoa Vuong,

journaltitlearXiv e-prints
titleThe Fermionic Universal One-Loop Effective Action
High Energy Physics - Phenomenology,High Energy Physics - Theory

titleExtending the Universal One-Loop Effective Action by Regularization Scheme Translating Operators

journaltitleActa Phys. Polon.
titleIntroduction to the Background Field Method
volumeB 13

American Physical Society

journaltitlePhys. Rev.
titleQuantum Theory of Gravity. II. The Manifestly Covariant Theory

booktitle12th Annual Winter School of Theoretical Physics
titleThe Background Field Method in Gauge Field Theories

booktitleOxford Conference on Quantum Gravity
titleA gauge invariant effective action

journaltitlePhys. Rev.
titleGauge Dependence of the Effective Action
volumeD 23

journaltitleNucl. Phys.
titleThe Background Field Method Beyond One Loop
volumeB 185

journaltitlePhys. Lett.
titleFeynman Diagrams for the Yang-Mills Field
volumeB 25

journaltitleNucl. Phys.
titleApplication of the background field method to the electroweak standard model
volumeB 440

titleOn one-loop corrections in the standard model effective field theory; the $\Gamma(h \rightarrow \gamma \, \gamma)$ case

titleLectures on elementary particles and quantum field theory. VOLUME 1. 1970 Brandeis University Summer Institute in Theoretical Physics.

American Physical Society

journaltitlePhys. Rev.
title$g=2$ as the natural value of the tree-level gyromagnetic ratio of elementary particles
volumeD 46

journaltitleActa Phys. Polon.
titleA Simple Method of Calculating Effective Operators
volumeB 43

seriesTexts and monographs in physics
titlePath Integral Approach to Quantum Physics: An Introduction

journaltitlePhys. Rept.
titleTwo-component spinor techniques and Feynman rules for quantum field theory and supersymmetry

journaltitlePhys. Lett.
titleThe one loop functional as a Berezinian
volumeB 438

journaltitleGitHub repository

journaltitleJ. Phys.
titleSupersymmetry in the Four-Dimensional Limit
volumeG 1

journaltitlePhys. Lett.
titleRegularization dependence of running couplings in softly broken supersymmetry
volumeB 318

journaltitlePhys. Lett.
titleTwo-loop parameter relations between dimensional regularization and dimensional reduction applied to SUSY-QCD

journaltitleComput. Phys. Commun.
titleFeynArts model file for MSSM transition counterterms from DREG to DRED

journaltitleNucl. Phys.
titleRegularization by Dimensional Reduction of Supersymmetric and Nonsupersymmetric Gauge Theories
volumeB 167

journaltitleNucl. Phys.
titleThree-loop MSSM Higgs-boson mass predictions and regularization by dimensional reduction
volumeB 935

journaltitlePhys. Rev.
titleDecoupling of the epsilon scalar mass in softly broken supersymmetry
volumeD 50

note[Erratum: JHEP 09, 019 (2019)]
title2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT

titleOperator bases, $S$-matrices, and their partition functions

journaltitlePhys. Rev.
titleConstructing effective field theories via their harmonics
volumeD 100

journaltitlearXiv e-prints
titleOn the systematic construction of basis invariants
High Energy Physics - Phenomenology

journaltitleCommun. Math. Phys.
titleHilbert series and operator bases with derivatives in effective field theories

journaltitleEur. Phys. J.
titleBasisGen: automatic generation of operator bases
volumeC 79

titleDEFT: A program for operators in EFT

journaltitleNucl. Phys.
titleEffective Lagrangian Analysis of New Interactions and Flavor Conservation
volumeB 268

titleDimension-Six Terms in the Standard Model Lagrangian

titleTheory and phenomenology of sparticles: An account of four-dimensional N=1 supersymmetry in high energy physics

titleHiggs Mass and Unnatural Supersymmetry

journaltitleEur. Phys. J.
titleImproved determination of the Higgs mass in the MSSM with heavy superpartners
volumeC 77

journaltitlePhys. Rev.
titleEffective Field Theory of Integrating out Sfermions in the MSSM: Complete One-Loop Analysis
volumeD 97

journaltitleEur. Phys. J.
titleThe MSSM without Gluinos; an Effective Field Theory for the Stop Sector
volumeC 77

titleDirac, Majorana, Weyl in 4d

titleEffective theories of universal theories

journaltitlePhys. Rev.
titleVices and virtues of Higgs effective field theories at large energy
volumeD 91

titleHeavy Vector Triplets: Bridging Theory and Data

journaltitlePhys. Rev.
titleA Gauge Model With Light $W$ and $Z$ Bosons
volumeD 22

titleOn the effect of resonances in composite Higgs phenomenology

titleStrong Higgs Interactions at a Linear Collider

Springer Berlin Heidelberg

seriesSpringer Series in Nuclear and Particle Physics
titleQuantization of Fields with Constraints

titleQuantization of gauge systems

journaltitleAnn. Phys.
titleInconsistency of the local field theory of charged spin 3/2 particles

journaltitlePhys. Rev.
titlePropagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential

journaltitleInt. J. Mod. Phys.
titlePath integral quantization for massive vector bosons
volumeA 25

journaltitleYad. Fiz.
titleThe Equivalence theorem and gauge invariance in renormalizable theories

titleElectroweak Limits on General New Vector Bosons

journaltitlePhys. Rev. Lett.
titleUniversality of the $\rho$ Meson Coupling in Effective Field Theory

seriesMathematical Physics and Applied Mathematics
titleIntroduction to Superanalysis

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleExtending the Universal One-Loop Effective Action by Regularization Scheme Translating Operators

titleCompleting the scalar and fermionic Universal One-Loop Effective Action

journaltitlePhys. Lett.
titleObservation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
volumeB 716

journaltitlePhys. Lett.
titleObservation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC
volumeB 716

American Physical Society

journaltitlePhys. Rev.
titleInfrared singularities and massive fields
volumeD 11

journaltitlePhys. Rev.
titleA new approach to effective field theories
volumeD 21

journaltitlePhys. Lett.
titleThe Decoupling Theorem and Minimal Subtraction
volumeB 100

journaltitleNucl. Phys.
titleGauge Theories With Minimal Subtraction and the Decoupling Theorem
volumeB 179

journaltitlePhys. Rev.
titleLow-Energy Manifestations of Heavy Particles: Application to the Neutral Current
volumeD 18

titlePhenomenological Lagrangians
volumeA 96

journaltitlePhys. Rev.
titleStructure of phenomenological Lagrangians. 1.

journaltitlePhys. Rev.
titleStructure of phenomenological Lagrangians. 2.

journaltitleNucl. Phys.
titleShort Distance Analysis of Weak Interactions
volumeB 122

abstractStrong interaction corrections to K0K̄0 mixing in the six quark model are computed in quantum chromodynamics using the leading logarithm approximation. The corrections to the real and imaginary parts of the off-diagonal K0-K0 mass matrix elements can be large.
journaltitlePhys. Lett.
titleStrong interaction corrections to $K^0-K^0$ mixing in the six quark model
volumeB 93

abstractA method is described for dealing with gauge field theories that contain very heavy particles, by constructing a gauge-invariant effective field theory, in which the heavy particles do not appear. For theories in which a simple group is spontaneously broken to the strong and electroweak gauge groups, the running strong and electroweak effective couplings become equal to the original gauge coupling at different renormalization scales, given as weighted geometric averages of the super-heavy particle masses. The method is also applied to quantum chromodynamics with heavy quarks.
journaltitlePhys. Lett.
titleEffective gauge theories
volumeB 91

journaltitleAnn. Rev. Nucl. Part. Sci.
titleEffective field theory

journaltitleLect. Notes Phys.
titleEffective field theories

booktitleTheoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles
titleEffective field theory and the Fermi surface

booktitleLes Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions
titleEffective field theory: Course

journaltitlearXiv e-prints
titleTASI Lectures on Effective Field Theories
High Energy Physics - Phenomenology

booktitleTheoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small
titleEffective Field Theory and Precision Electroweak Measurements

journaltitleAnn. Rev. Nucl. Part. Sci.
titleIntroduction to Effective Field Theory

booktitle7th Summer School in Nuclear Physics Symmetries
titleEffective field theories

journaltitlePhys. Rept.
titleThe Standard Model as an Effective Field Theory

journaltitleInt. J. Mod. Phys.
titleThe Chiral approach to the electroweak interactions
volumeA 8

journaltitleEur. Phys. J.
titleNonstandard Higgs couplings from angular distributions in $h\to Z \ell^+\ell^-$
volumeC 74

titleDisentangling a dynamical Higgs

titleHiggs ultraviolet softening

titleOn the renormalization of the electroweak chiral Lagrangian with a Higgs

titleCP violation with a dynamical Higgs

titleSigma Decomposition

journaltitlePhys. Lett.
titleNote on Anomalous Higgs-Boson Couplings in Effective Field Theory
volumeB 750

titleSigma Decomposition: The CP-Odd Lagrangian

journaltitleEur. Phys. J.
titleFitting Higgs Data with Nonlinear Effective Theory
volumeC 76

journaltitleEur. Phys. J.
titleThe complete HEFT Lagrangian after the LHC Run I
volumeC 76

journaltitleEur. Phys. J.
titleThe linear–non-linear frontier for the Goldstone Higgs
volumeC 76

journaltitleEur. Phys. J.
titleBaryon Non-Invariant Couplings in Higgs Effective Field Theory
volumeC 77

journaltitlePhys. Rev.
titleDistinguishing A Higgs-Like Dilaton Scenario With A Complete Bosonic Effective Field Theory Basis
volumeD 96

titleEffective Theory of a Dynamically Broken Electroweak Standard Model at NLO

titleThe Gauge-Higgs Legacy of the LHC Run I

journaltitleSciPost Phys.
titleThe Gauge-Higgs Legacy of the LHC Run II

titleO new physics, where art thou? A global search in the top sector

journaltitlearXiv e-prints
titlePutting SMEFT Fits to Work
High Energy Physics - Phenomenology

titleUpdated Global SMEFT Fit to Higgs, Diboson and Electroweak Data

booktitle11th International Conference on Supersymmetry and the Unification of Fundamental Interactions
titleImplications of supersymmetry breaking with a little hierarchy between gauginos and scalars

journaltitleNucl. Phys.
note[Erratum: Nucl.Phys.B 706, 487–487 (2005)]
titleSplit supersymmetry
volumeB 699

journaltitleNucl. Phys.
titleAspects of split supersymmetry
volumeB 709

note[Erratum: JHEP 07, 066 (2020)]
titleSinglet night in Feynman-ville: one-loop matching of a real scalar

titleComplete One-Loop Matching for a Singlet Scalar in the Standard Model EFT

journaltitleNucl. Phys.
titleThe Effective One Loop Lagrangian With Derivative Couplings
volumeB 268

journaltitleNucl. Phys.
titleEffective Action for the Standard Model With Large Higgs Mass
volumeB 297

titleHow to use the Standard Model effective field theory

titleThe Universal One-Loop Effective Action

titleIntegrating out heavy particles with functional methods: a simplified framework

titleOne-loop Matching and Running with Covariant Derivative Expansion

titleExtending the Universal One-Loop Effective Action: Heavy-Light Coefficients

titleCompleting the scalar and fermionic Universal One-Loop Effective Action

journaltitlePhys. Lett.
titleSupersymmetric Dimensional Regularization via Dimensional Reduction
volumeB 84

titleRegularization by dimensional reduction: consistency, quantum action principle, and supersymmetry

journaltitleNucl. Phys.
titleRegularization and Renormalization of Gauge Fields
volumeB 44

titleCovariant diagrams for one-loop matching

journaltitleEur. Phys. J.
titleOne-loop effective lagrangians after matching
volumeC 76

Reading, USA

titleAn Introduction to quantum field theory

journaltitlePhys. Rept.
titleChiral Gauge Theory

Berlin, Boston

De Gruyter

journaltitleJ. Reine Angew. Math
titleÜber Potenzreihen, die im Innern des Einheitskreises beschränkt sind.

journaltitleNucl. Phys.
titleAsymptotic expansion of Feynman integrals near threshold
volumeB 522

journaltitleSpringer Tracts Mod. Phys.
titleApplied asymptotic expansions in momenta and masses

titleFoundation and generalization of the expansion by regions

journaltitlearXiv e-prints
titleIntegrating Out New Fermions at One Loop
High Energy Physics - Phenomenology
family=Hoa Vuong,

journaltitlearXiv e-prints
titleThe Fermionic Universal One-Loop Effective Action
High Energy Physics - Phenomenology,High Energy Physics - Theory

titleExtending the Universal One-Loop Effective Action by Regularization Scheme Translating Operators

journaltitleActa Phys. Polon.
titleIntroduction to the Background Field Method
volumeB 13

American Physical Society

journaltitlePhys. Rev.
titleQuantum Theory of Gravity. II. The Manifestly Covariant Theory

booktitle12th Annual Winter School of Theoretical Physics
titleThe Background Field Method in Gauge Field Theories

booktitleOxford Conference on Quantum Gravity
titleA gauge invariant effective action

journaltitlePhys. Rev.
titleGauge Dependence of the Effective Action
volumeD 23

journaltitleNucl. Phys.
titleThe Background Field Method Beyond One Loop
volumeB 185

journaltitlePhys. Lett.
titleFeynman Diagrams for the Yang-Mills Field
volumeB 25

journaltitleNucl. Phys.
titleApplication of the background field method to the electroweak standard model
volumeB 440

titleOn one-loop corrections in the standard model effective field theory; the $\Gamma(h \rightarrow \gamma \, \gamma)$ case

titleLectures on elementary particles and quantum field theory. VOLUME 1. 1970 Brandeis University Summer Institute in Theoretical Physics.

American Physical Society

journaltitlePhys. Rev.
title$g=2$ as the natural value of the tree-level gyromagnetic ratio of elementary particles
volumeD 46

journaltitleActa Phys. Polon.
titleA Simple Method of Calculating Effective Operators
volumeB 43

seriesTexts and monographs in physics
titlePath Integral Approach to Quantum Physics: An Introduction

journaltitlePhys. Rept.
titleTwo-component spinor techniques and Feynman rules for quantum field theory and supersymmetry

journaltitlePhys. Lett.
titleThe one loop functional as a Berezinian
volumeB 438

journaltitleGitHub repository

journaltitleJ. Phys.
titleSupersymmetry in the Four-Dimensional Limit
volumeG 1

journaltitlePhys. Lett.
titleRegularization dependence of running couplings in softly broken supersymmetry
volumeB 318

journaltitlePhys. Lett.
titleTwo-loop parameter relations between dimensional regularization and dimensional reduction applied to SUSY-QCD

journaltitleComput. Phys. Commun.
titleFeynArts model file for MSSM transition counterterms from DREG to DRED

journaltitleNucl. Phys.
titleRegularization by Dimensional Reduction of Supersymmetric and Nonsupersymmetric Gauge Theories
volumeB 167

journaltitleNucl. Phys.
titleThree-loop MSSM Higgs-boson mass predictions and regularization by dimensional reduction
volumeB 935

journaltitlePhys. Rev.
titleDecoupling of the epsilon scalar mass in softly broken supersymmetry
volumeD 50

note[Erratum: JHEP 09, 019 (2019)]
title2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT

titleOperator bases, $S$-matrices, and their partition functions

journaltitlePhys. Rev.
titleConstructing effective field theories via their harmonics
volumeD 100

journaltitlearXiv e-prints
titleOn the systematic construction of basis invariants
High Energy Physics - Phenomenology

journaltitleCommun. Math. Phys.
titleHilbert series and operator bases with derivatives in effective field theories

journaltitleEur. Phys. J.
titleBasisGen: automatic generation of operator bases
volumeC 79

titleDEFT: A program for operators in EFT

journaltitleNucl. Phys.
titleEffective Lagrangian Analysis of New Interactions and Flavor Conservation
volumeB 268

titleDimension-Six Terms in the Standard Model Lagrangian

titleTheory and phenomenology of sparticles: An account of four-dimensional N=1 supersymmetry in high energy physics

titleHiggs Mass and Unnatural Supersymmetry

journaltitleEur. Phys. J.
titleImproved determination of the Higgs mass in the MSSM with heavy superpartners
volumeC 77

journaltitlePhys. Rev.
titleEffective Field Theory of Integrating out Sfermions in the MSSM: Complete One-Loop Analysis
volumeD 97

journaltitleEur. Phys. J.
titleThe MSSM without Gluinos; an Effective Field Theory for the Stop Sector
volumeC 77

titleDirac, Majorana, Weyl in 4d

titleEffective theories of universal theories

journaltitlePhys. Rev.
titleVices and virtues of Higgs effective field theories at large energy
volumeD 91

titleHeavy Vector Triplets: Bridging Theory and Data

journaltitlePhys. Rev.
titleA Gauge Model With Light $W$ and $Z$ Bosons
volumeD 22

titleOn the effect of resonances in composite Higgs phenomenology

titleStrong Higgs Interactions at a Linear Collider

Springer Berlin Heidelberg

seriesSpringer Series in Nuclear and Particle Physics
titleQuantization of Fields with Constraints

titleQuantization of gauge systems

journaltitleAnn. Phys.
titleInconsistency of the local field theory of charged spin 3/2 particles

journaltitlePhys. Rev.
titlePropagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential

journaltitleInt. J. Mod. Phys.
titlePath integral quantization for massive vector bosons
volumeA 25

journaltitleYad. Fiz.
titleThe Equivalence theorem and gauge invariance in renormalizable theories

titleElectroweak Limits on General New Vector Bosons

journaltitlePhys. Rev. Lett.
titleUniversality of the $\rho$ Meson Coupling in Effective Field Theory

seriesMathematical Physics and Applied Mathematics
titleIntroduction to Superanalysis
e references (bibliography) information are stored in the file named "Bibliography.bib"

First of all I would like to thank Prof. Michael Krämer for making this thesis possible by accepting me as a PhD student in his group. I appreciate that I was given the opportunity to follow my interests while also being provided with guidance when needed.

I also thank Prof. Robert Harlander for agreeing to be my second supervisor.

For encouraging me to work on this project and for countless hours of useful discussions, and useful comments on earlier versions of this thesis, I would like to thank Dr. Alexander Voigt. I also thank Jonas Klappert and Fabian Lange for useful discussions, and Jonas Klappert in particular for comments on earlier versions of this thesis. Furthermore, I thank Jonas Klappert and Fabian Lange for making my time in the office a very pleasant experience.

A special thank you goes to everybody who supported me with encouragement throughout the completion of this work. This in particular includes my parents Roland and Sabine, without whom I would not be the person I am today, as well as my sister Alice. Furthermore I thank Hesam and Moncef, not only for encouragement, support and friendship throughout the completion of this thesis, but also the five years of studies that led up to it. I thank Jonas for his friendship and support and for the many rounds of Swedish basketball that we played together.

Last but not least I thank Physics itself for being such a rich subject that never seizes to surprise me. Thank you for the smile created by the feeling that a new layer of reality has revealed itself.

Ich, Benjamin Summ erkläre hiermit, dass diese Dissertation und die darin dargelegten Inhalte die eigenen sind und selbstständig, als Ergebnis der eigenen originären Forschung, generiert wurden.
Hiermit erkläre ich an Eides statt

* Diese Arbeit wurde vollständig oder größtenteils in der Phase als Doktorand dieser Fakultät und Universität angefertigt;

* Sofern irgendein Bestandteil dieser Dissertation zuvor für einen akademischen Abschluss oder eine andere Qualifikation an dieser oder einer anderen Institution verwendet wurde, wurde dies klar angezeigt;

* Wenn immer andere eigene- oder Veröffentlichungen Dritter herangezogen wurden, wurden diese klar benannt;

* Wenn aus anderen eigenen- oder Veröffentlichungen Dritter zitiert wurde, wurde stets die Quelle hierfür angegeben. Diese Dissertation ist vollständig meine eigene Arbeit, mit der Ausnahme solcher Zitate;

* Alle wesentlichen Quellen von Unterstützung wurden benannt;

* Wenn immer ein Teil dieser Dissertation auf der Zusammenarbeit mit anderen basiert, wurde von mir klar gekennzeichnet, was von anderen und was von mir selbst erarbeitet wurde;

* Ein Teil oder Teile dieser Arbeit wurden zuvor veröffentlicht und zwar in:

[BS1] Summ:2018oko.

[BS2] Kramer:2019fwz.