\frac{1}{\kappa}c^{LR} _{62} &=- \frac{12}{9}d(d+2) g_3^6 m^2_{\gluino{}} \ZI[q^4] ^{33} _{\gluino{}0}-\frac{10}{9}d(d^2+6d+8)g_3^6  \ZI[q^6] ^{33} _{\gluino{}0}, \\
\frac{1}{\kappa}c^{LR} _{63} &= -\frac{1}{6}d(d+2) g_3^6 m^2_{\gluino{}} \ZI[q^4] ^{33} _{\gluino{}0}-\frac{14}{9}d(d^2+6d+8)g_3^6  \ZI[q^6] ^{33} _{\gluino{}0}, \\
\frac{1}{\kappa}c^{LR} _{64} &= \frac{2}{9}d(d^2+6d+8) g_3^6  \ZI[q^6] ^{33} _{\gluino{}0}, \\
\frac{1}{\kappa}c^{RL} _{61} &= -\frac{1}{9}d(d+2) g_3^6 m^2_{\gluino{}} \ZI[q^4] ^{33} _{\gluino{}0}-\frac{40}{27}d(d^2+6d+8)g_3^6  \ZI[q^6] ^{33} _{\gluino{}0},\\
\frac{1}{\kappa}c^{RL} _{62} &=- \frac{12}{9}d(d+2) g_3^6 m^2_{\gluino{}} \ZI[q^4] ^{33} _{\gluino{}0}+\frac{8}{9}d(d^2+6d+8)g_3^6  \ZI[q^6] ^{33} _{\gluino{}0}, \\
\frac{1}{\kappa}c_{51} ^{LR\text{,1\Loop}}&=\frac{1}{\kappa}c_{51} ^{RL\text{,1\Loop}}=-\frac{g_3^4}{3} m_{\gluino{}} \ZI ^{12}_{\gluino{}0}, \\
\frac{1}{\kappa}c_{52} ^{LR\text{,1\Loop}}&=\frac{1}{\kappa}c_{52} ^{RL\text{,1\Loop}}=-\frac{8}{3} g_3^4 x_t m_{\gluino{}} \ZI ^{12}_{\gluino{}0}, \\
\frac{1}{\kappa}c_G&=-\frac{g_3^2}{2} \ZI ^{2} _{\gluino{}}.
\end{align}

The threshold corrections for the two stop mass parameters were previously computed in Ref. <cit.> and the results agree when the effect of the sbottom quarks is neglected.

As in Section <ref> it is again convenient
to use DRED as a regulator in the MSSM. Since supersymmetry is explicitly broken once the gluino is
integrated out from the theory, it is natural to regularise the EFT in
DREG. This switch in regularisation again introduces further
contributions to the couplings of the EFT coming from the epsilon
scalars, which can be computed using Eq. (<ref>).

We obtain the following
additional contributions

\begin{align}
\frac{1}{\kappa}(\delta m^2 _{\sq})_\eps &=\frac{1}{\kappa}(\delta m ^2 _{\su})_\eps=-\frac{4}{3} g_3^2 m_\eps ^2, \label{eq:delta_m2_eps} \\
\frac{1}{\kappa}(c_{t_L})_\eps &= \frac{1}{\kappa}(c_{t_R})_\eps=\frac{4}{3}g_3^2, \\
\frac{1}{\kappa}(c^L_{41})_\eps &= \frac{1}{72}g_3^4, \\
\frac{1}{\kappa}(c^L_{42})_\eps&=\frac{7}{24}g_3^4, \\
\frac{1}{\kappa}(c^R_{4})_\eps &= \frac{11}{36}g_3^4, \\
\frac{1}{\kappa}(c^{LR}_{41})_\eps &= \frac{1}{36}g_3^4, \\
\frac{1}{\kappa}(c^{LR}_{42})_\eps &= \frac{7}{12}g_3^4, \\
\frac{1}{\kappa}(c^{LL}_{51})_\eps &= \frac{1}{\kappa}(c^{LL}_{52})_\eps = \frac{1}{\kappa}(c^{RR}_{51})_\eps=\frac{1}{\kappa}(c^{RR}_{52})_\eps = \frac{3g_3^4 }{2 m_{\gluino{}}}d, \\
\frac{1}{\kappa}(c^{LR}_{51})_\eps &= \frac{1}{\kappa}(c^{RL}_{52})_\eps = -\frac{3g_3^4 }{m_{\gluino{}}}d, \\
\frac{1}{\kappa}(c_G)_\eps &= -\frac{g_3^2}{4}.
\end{align}

Notice that the one-loop DRED–DREG conversion corrections to the
coefficients of the dimension five operators arise from the term

\begin{align}
\left(\epsdim{U}_{\eps t}\right)^\mu \gamma ^\nu U_{\bar{t} \gluino{\dot{A}}} \gamma_{\nu} \left(\epsdim{U}_{\gluino{\dot{A}} \eps}\right)_\mu.
\end{align}

Here $\left(\epsdim{U}_{\gluino{\dot{A}} \eps}\right)^\mu$ has an explicit
dependence on the gluino spinor $\gluino{\dot{A}}$,

\begin{align}
\left(\epsdim{U}_{\gluino{\dot{A}} \eps}\right)^\mu=ig_3\epsdim{\gamma}^\mu f^{\dot{A}\dot{B}\dot{C}}\gluino{\dot{C}},
\end{align}

which must be eliminated by inserting the classical field from Eq.(<ref>).

§ MATCHING OF THE SSM TO THE SMEFT

In this section we consider the SSM, which is the SM with an additional real scalar field $S$, transforming as a singlet under the SM gauge group. We assume the singlet to be heavy and integrate it out of the theory using the BSUOLEA to obtain SMEFT operators up to mass dimension six. The final result is then expressed in the Warsaw basis. This particular matching was first computed in Ref. <cit.> using the UOLEA presented in Ref. <cit.> supplemented by Feynman diagrammatic techniques for loops of mixed statistics and loops including gauge bosons. The matching was also re-computed using Feynman diagrammatic techniques in Ref. <cit.> for the purpose of studying the effect of SMEFT operators on one-loop probes of pseudo Nambu-Goldstone boson dark matter. We here, for the first time, perform the full matching using the UOLEA approach, illustrating that the extensions that were presented in this thesis are needed even for one of the simplest SM extensions of interest to particle phenomenology. Furthermore, this section can be viewed as both a check of the BSUOLEA and its implementation into as well as an illustration of how to consistently treat a UV theory that contains the SM field content and gauge group in the BSUOLEA framework.

As the SSM is a chiral gauge theory it is appropriately described using Weyl fermions when working in the unbroken phase. Strictly speaking, we only considered Dirac and Majorana fermions in this thesis. However, for the purpose of a perturbative calculation Weyl fermions may be implemented as either Dirac fermions or Majorana fermions, as the chiral nature of the interactions will always project out the physically relevant components of these four-component spinors.[A discussion of this and related issues can be found in Ref. <cit.>. Interestingly, in non-perturbative considerations Weyl fermions cannot be treated as either Dirac fermions or Majorana fermions.] We here choose to treat them as Majorana fermions. To this end we introduce the left-handed two-component spinors $q,\bar{u},\bar{d},l$ and $\bar{e}$, where the bar on $u,d,e$ is part of the name and does not signify any kind of conjugation. These spinors all carry flavour indices. The charges under the SM gauge group $SU(3)_c \times SU(2)_L \times U(1)_Y$ carried by these fields are summarised in Table <ref> below, where we also include the Higgs doublet $H$ and the singlet $S$.

$SU(2)_L$     $U(1)_Y$     $SU(3)_c$
1|c|$q$      $2$     $1/6$     $3$
1|c|$\bar{u}$     $1$     $-2/3$     $\bar{3}$
1|c|$\bar{d}$     $1$     $1/3$     $\bar{3}$
1|c|$l$      $2$     $-1/2$     $1$
1|c|$\bar{e}$     $1$     $1$     $1$
1|c|$H$     $2$     $1/2$     $1$
1|c|$S$     $1$     $0$     $1$

Gauge charges of the fields in the SSM.

Note that the $U(1)_Y$ charges of the barred fields are opposite to those of the corresponding SM fields since the barred fields are left-handed and the corresponding SM fields are related to these by Hermitian conjugation. We then introduce the Majorana spinors

\begin{align}
\label{eq: singletExtension: Majorana_spinor_definitions}
\Psi_q = \begin{pmatrix}
q \, \\ q^\dagger
\end{pmatrix}, \Psi_u = \begin{pmatrix}
\bar{u} \; \\ \bar{u}^\dagger \end{pmatrix},
\Psi_d = \begin{pmatrix}
\bar{d} \; \\ \bar{d}^\dagger
\end{pmatrix},
\Psi_l = \begin{pmatrix}
l \, \\ l^\dagger
\end{pmatrix},
\Psi_e = \begin{pmatrix}
\bar{e} \; \\ \bar{e}^\dagger \end{pmatrix},
\end{align}

which transform in chiral representations of the SM gauge group. Observe that the fields $e = \bar{e}^\dagger$, $d = \bar{d}^\dagger$ and $u = \bar{u}^\dagger$ are the usual right-handed spinors introduced in the SM. The generators for the $U(1)_Y$- and $SU(2)_L$-transformation of $\Psi_f$ can be written as

\begin{align}
Y_{\Psi_f} = Y_f P_L-Y_f P_R \text{\; and \;} & T^A_{\Psi_f} = T^A_f P_L + \bar{T}^A_f P_R,
\end{align}

respectively, where $T^A_f$ is the usual generator of $SU(2)_L$ in the representation under which the fermion $f\in \{q,d,u,l,e\}$ transforms and $\bar{T}^A_f$ is the generator of the corresponding conjugate representation. Furthermore, $Y_f$ is the hypercharge of the left-handed component of $\Psi_f$. As already mentioned, the details of this construction are inconsequential to the computation since the SSM interactions are chirality preserving and therefore always project out the physical components of these spinors.

With these preparations we may write the Lagrangian of the SSM as

\begin{align}
\label{eq: SSM_Lag}
\LagNoT &= \LagNoT_S + (D_\mu H)^\dagger (D^\mu H) + \mu_h^2 |H|^2 - \frac{1}{2} \lambda_h |H|^4 - \frac{1}{4} B^{\mu \nu} B_{\mu \nu} - \frac{1}{4} W^{A \mu \nu} W^A_{\mu \nu} \nonumber \\ & \quad  +\frac{i}{2} \sum \limits_{i \in \{q,u,d,l,e\}}\bar{\Psi}_i \slashed{D} \Psi_i - \left(\bar{\Psi}_q y_u \tilde{H} P_R \Psi_u + \bar{\Psi}_q y_d H P_R \Psi_d + \bar{\Psi}_l y_e H P_R \Psi_e + \text{h.c.} \right),
\end{align}

\begin{align}
\LagNoT_S = \frac{1}{2}\left(\partial_\mu S\right)\left(\partial^\mu S\right)-\frac{1}{2} M^2 S^2-A |H|^2 S-\frac{\kappa}{2}|H|^2 S^2 -\frac{\mu}{3!}S^3-\frac{\lambda_S}{4!}S^4.
\end{align}

In Eq. (<ref>) we introduced $\tilde{H}_a = \epstensor_{ab} H^*_b$ with $\epstensor_{ab}$ the totally antisymmetric symbol with two indices and $\epstensor_{12}=1$. The field strength tensors $W^A_{\mu \nu}$ and $B_{\mu \nu}$ correspond to the gauge-kinetic terms of the $SU(2)_L$ and $U(1)_Y$ gauge bosons, respectively. The corresponding gauge couplings will be denoted by $g_1$ and $g_2$. We ignore QCD as it is not relevant for the matching presented here.

We start by briefly describing the tree-level matching. To perform this matching we need to compute the classical field $\classical{S}$, which is determined from the equation of motion of the singlet

\begin{align}
\label{eq: singlet_eom}
\left(-\Box - M^2\right) S = A |H|^2 + \kappa |H|^2 S + \frac{\mu}{2} S^2 + \frac{\lambda_S}{3!} S^3.
\end{align}

This can be solved iteratively in an expansion in $1/M$. Care has to be taken as the parameters $A$ and $\mu$ are dimensionful and are taken to be of $\mathcal{O}(M)$. The lowest order solution is

\begin{align}
\classical{S}^{\left(1\right)} = -\frac{A}{M^2} |H|^2,
\end{align}

which inserted into the r.h.s. of Eq. (<ref>) allows for the determination of $\classical{S}^{\left(2\right)}$

\begin{align}
\left(-\Box-M^2\right) \classical{S}^{\left(2\right)} &= A |H^2| -\frac{A \kappa}{M^2} |H|^4 + \mathcal{O}\left(M^{-3}\right) \nonumber \\
\Rightarrow \classical{S}^{\left(2\right)} &= -\frac{A}{M^2} |H|^2 + \frac{A}{M^4} \Box |H|^2+ \frac{A \kappa}{M^4} |H|^4.
\end{align}

By repeatedly inserting the obtained solution into the r.h.s. of Eq. (<ref>) one can determine the classical solution to the desired order in $M^{-1}$. The final result is given by

\begin{align}
\label{eq: full_classical_singlet}
\classical{\hat{S}} &= -\frac{A}{M^2}|H|^2+\left(\frac{A\kappa}{M^4}-\frac{A^2\mu}{2M^6}\right)|H|^4 + \frac{A}{M^4} \Box |H|^2 - \left(\frac{A\kappa}{M^6}-\frac{A^2\mu}{M^8} \right)|H|^2\Box |H|^2 \nonumber \\
& \quad - \left(\frac{A\kappa}{M^6}-\frac{A^3\lambda_\phi+9A^2\kappa \mu}{6M^8}+\frac{A^3 \mu^2}{2M^10}\right)|H|^6,
\end{align}

which was already reported in Ref. <cit.>. Inserting this into the UV Lagrangian and truncating at mass dimension six one finds the tree-level results reported in that reference. The computation of the classical heavy fields and the tree-level matching is also fully automated in .

We now move to the one-loop matching. The multiplets defined in Table <ref> containing the different kinds of fields are in the case at hand

\begin{align}
\label{eq: singletApp_Phi_def}
\Phi &= \begin{pmatrix}
\end{pmatrix}, \\
\label{eq: singletApp_xi_def}
\xi &= \begin{pmatrix}
\Psi_{qi}^{\dot{\alpha} a} & \Psi_{ui}^{\dot{\alpha}} & \Psi_{di}^{\dot{\alpha}} & \Psi_{li}^{\dot{\alpha} a} & \Psi_{ei}^{\dot{\alpha}}
\end{pmatrix}^T,\\
\label{eq: singletApp_phi_def}
\phi &= \begin{pmatrix}
H_a & H^*_a & B_\mu & W^A_\mu
\end{pmatrix}^T,
\end{align}

where there is no multiplet $\Xi$ since there are no heavy fermions present. Also, since we are treating all of the fermions as Majorana fermions we have $\xi=\lambda$. Next, the relevant derivative matrices have to be computed. The matrices associated with at least one derivative w.r.t. a heavy field originate from $\LagNoT_S$ and are given by

\begin{align}
\label{eq: singletExtension_X_Phi_Phi}
\mathbf{U}_{\Phi \Phi} &= \begin{pmatrix}
\kappa |H|^2 + \mu \classical{\hat{S}} + \frac{\lambda_S}{2} \classical{\hat{S}}
\end{pmatrix}, \\
\label{eq: singletExtension_X_phi_Phi}
\mathbf{U}_{\phi \Phi} &= \begin{pmatrix}
A H_a + \kappa H_a \classical{\hat{S}} \\
A H^*_a + \kappa H^*_a \classical{\hat{S}} \\
0 \\
\end{pmatrix}, \\
\label{eq: singletExtension_X_Phi_phi}
\mathbf{U}_{\Phi \phi} &= \begin{pmatrix}
A H^*_a + \kappa H^*_a \classical{\hat{S}} &
A H_a + \kappa H_a \classical{\hat{S}} &
0 &
\end{pmatrix}.
\end{align}

In Eqs. (<ref>)–(<ref>) the expression of Eq. (<ref>) has to be inserted. Many of the contributions to the remaining matrices originate entirely from the SM part of the Lagrangian and will be present in any application in which the UV theory contains the SM. We therefore present these matrices in their entirety starting with $\mathbf{U}_{\phi \phi}$, which is given by

\begin{align}
\mathbf{U}_{\phi \phi} &= \begin{pmatrix}
\left(U_{H^* H}\right)_{ab} & \left(U_{H^* H^*}\right)_{ab} & \left(U_{H^* B}\right)_{a \nu} & \left(U_{H^* W}\right)_{a \nu B} \\
\left(U_{H H}\right)_{ab} & \left(U_{H H^*}\right)_{ab} & \left(U_{H B}\right)_{a \nu} & \left(U_{H W}\right)_{a \nu B} \\
-\left(U_{B H}\right)_{\mu b} & -\left(U_{B H^*}\right)_{\mu b} & -\left(U_{B B}\right)_{\mu \nu} & -\left(U_{B W}\right)_{\mu \nu B} \\
-\left(U_{W H}\right)_{\mu A b} & -\left(U_{W H^*}\right)_{\mu A b} & -\left(U_{W B}\right)_{\mu A \nu} & -\left(U_{W W}\right)_{\mu A \nu B}
\end{pmatrix},
\end{align}

\begin{align}
\left(U_{H^* H}\right)_{ab} &= \lambda_h |H|^2 \delta_{ab}+\lambda_h H_a H^*_b+A \delta_{ab} \classical{\hat{S}} +\frac{\kappa}{2}\delta_{ab} \classical{\hat{S}}^2,\\
\left(U_{H^* H^*}\right)_{ab} &=  \lambda_h H_a H_b,\\
\left(U_{H^* B}\right)_{a\mu} &= \left(U_{B H^*}\right)_{\mu a} = \frac{g_1}{2} [P_\mu,H_a], \\
\left(U_{H^* W}\right)_{a\mu A} &= \left(U_{W H^*}\right)_{\mu A a} = g_2 T^A_{ab} [P_\mu, H_b], \\
U_{H H} &= \lambda_h H^*_a H^*_b, \\
\left(U_{H H^*}\right)_{a b} &= \lambda_h |H|^2 \delta_{ab}+\lambda_h H_b H^*_a+A \delta_{ab} \classical{\hat{S}} +\frac{\kappa}{2}\delta_{ab} \classical{\hat{S}}^2,\\
\left(U_{H B}\right)_{a \mu} &= \left(U_{B H}\right)_{\mu a} = -\frac{g_1}{2} [P_\mu,H^*_a], \\
\left(U_{H W}\right)_{a \mu A} &= \left(U_{W H}\right)_{\mu A a} =  -g_2 T^A_{ca} [P_\mu,H^*_c], \\
\left(U_{B B}\right)_{\mu \nu} &= -\frac{g_1^2}{2} |H|^2 g_{\mu \nu}, \\
\left(U_{B W}\right)_{\mu \nu A} & = \left(U_{W B}\right)_{\mu A \nu} = -g_1 g_2 H^*_a T^A_{ab} H_b g_{\mu \nu}, \\
\left(U_{W W}\right)_{\mu A \nu B} & = -g_2^2 H^*_a\{T^A,T^B\}_{ab} H_b g_{\mu \nu}.
\end{align}

The matrices with derivatives of mixed statistics are

\begin{align}
\mathbf{U}_{\xi \phi} &= \cc^{-1}_{\dot{\alpha}\dot{\beta}}
\begin{pmatrix}
\left(U_{\Psi_q H}\right)_{\dot{\beta}aib} & \left(U_{\Psi_q H^*}\right)_{\dot{\beta}aib} & \left(U_{\Psi_q B}\right)_{\dot{\beta}ai\mu} & \left(U_{\Psi_q W}\right)_{\dot{\beta}ai\mu A} \\
\left(U_{\Psi_u H}\right)_{\dot{\beta}ib} & \left(U_{\Psi_u H^*}\right)_{\dot{\beta}ib} & \left(U_{\Psi_u B}\right)_{\dot{\beta}i\mu} & \left(U_{\Psi_u W}\right)_{\dot{\beta}ai\mu A} \\
\left(U_{\Psi_d H}\right)_{\dot{\beta}ib} & \left(U_{\Psi_d H^*}\right)_{\dot{\beta}ib} & \left(U_{\Psi_d B}\right)_{\dot{\beta}i\mu} & \left(U_{\Psi_d W}\right)_{\dot{\beta}ai\mu A} \\
\left(U_{\Psi_l H}\right)_{\dot{\beta}aib} & \left(U_{\Psi_l H^*}\right)_{\dot{\beta}aib} & \left(U_{\Psi_l B}\right)_{\dot{\beta}ai\mu} & \left(U_{\Psi_l W}\right)_{\dot{\beta}ai\mu A} \\
\left(U_{\Psi_e H}\right)_{\dot{\beta}ib} & \left(U_{\Psi_e H^*}\right)_{\dot{\beta}ib} & \left(U_{\Psi_e B}\right)_{\dot{\beta}i\mu} & \left(U_{\Psi_e W}\right)_{\dot{\beta}ai\mu A}
\end{pmatrix},
\end{align}

\begin{align}
\mathbf{U}_{\phi \xi} &=
\begin{pmatrix}
\left(U_{H^* \Psi_q}\right)_{a\dot{\beta}bi} & \left(U_{H^* \Psi_u}\right)_{a\dot{\beta}i} & \left(U_{H^* \Psi_d}\right)_{a\dot{\beta}i} & \left(U_{H^* \Psi_l}\right)_{a\dot{\beta}bi} & \left(U_{H^* \Psi_e}\right)_{a\dot{\beta}i} \\
\left(U_{H \Psi_q}\right)_{a\dot{\beta}bi} & \left(U_{H \Psi_u}\right)_{a\dot{\beta}i} & \left(U_{H \Psi_d}\right)_{a\dot{\beta}i} & \left(U_{H \Psi_l}\right)_{a\dot{\beta}bi} & \left(U_{H \Psi_e}\right)_{a\dot{\beta}i} \\
-\left(U_{B \Psi_q}\right)^{\mu}_{ \dot{\beta}bi} & -\left(U_{B \Psi_u}\right)^{\mu}_{\dot{\beta}i} & -\left(U_{B \Psi_d}\right)^\mu_{\dot{\beta}i} & -\left(U_{B \Psi_l}\right)^\mu_{\dot{\beta}bi} & -\left(U_{B \Psi_e}\right)^\mu_{\dot{\beta}i} \\
-\left(U_{W \Psi_q}\right)^{\mu A}_{\dot{\beta}bi} & -\left(U_{W \Psi_u}\right)^{\mu A}_{\dot{\beta}i} & -\left(U_{W \Psi_d}\right)^{\mu A}_{\dot{\beta}i} & -\left(U_{W \Psi_l}\right)^{\mu A}_{\dot{\beta}bi} & -\left(U_{W \Psi_e}\right)^{\mu A}_{\dot{\beta}i}
\end{pmatrix},
\end{align}

\begin{align}
\left(U_{\Psi_q H}\right)_{\dot{\beta}akb} &= \left(U_{H \Psi_q }\right)_{b\dot{\beta}ak} = -\left(y^*_u\right)^{ki} \Psi_{ui}^{\dot{\alpha}} \cc_{\dot{\alpha}\dot{\rho}} \epsilon_{ab}P_L^{\dot{\rho}\dot{\beta}}+y_d^{ki} \cc^{\dot{\beta}\dot{\rho}}P_R^{\dot{\rho}\dot{\sigma}} \Psi_{di}^{\dot{\sigma}}\delta_{ab}, \\
\left(U_{\Psi_q H^*}\right)_{\dot{\beta}akb} &= \left(U_{H^* \Psi_q}\right)_{b\dot{\beta}ak} = y_u^{kj} \cc_{\dot{\beta}\dot{\rho}} \epsilon_{ab}P_R^{\dot{\rho}\dot{\sigma}}\Psi_{uj}^{\dot{\sigma}}-\left(y_d^*\right)^{ki} \Psi_{di}^{\dot{\alpha}} \cc^{\dot{\alpha}\dot{\rho}}P_L^{\dot{\rho}\dot{\beta}} \delta_{ab}, \\
\left(U_{\Psi_u H}\right)_{\dot{\beta}kb} &= \left(U_{H \Psi_u}\right)_{b\dot{\beta}k} =  \left(y^*_u\right)^{ik}  \cc_{\dot{\beta}\dot{\rho}}\epsilon_{ab}P_L^{\dot{\rho}\dot{\sigma}}\Psi_{qi}^{\dot{\sigma}a},\\
\left(U_{\Psi_u H^*}\right)_{\dot{\beta}kb} &= \left(U_{H^* \Psi_u}\right)_{b\dot{\beta}k} = -y_u^{jk} \Psi_{qj}^{\dot{\alpha}a}\cc_{\dot{\alpha}\dot{\rho}} \epsilon_{ab}P_R^{\dot{\rho}\dot{\beta}},\\
\left(U_{\Psi_d H}\right)_{\dot{\beta}kb} &= \left(U_{H \Psi_d}\right)_{b\dot{\beta}k} = -y_d^{jk} \Psi_{qj}^{\dot{\alpha}b}\cc_{\dot{\alpha}\dot{\rho}}P_R^{\dot{\rho}\dot{\beta}},\\
\left(U_{\Psi_d H^*}\right)_{\dot{\beta}kb} &= \left(U_{H^*\Psi_d}\right)_{b\dot{\beta}k} = \left(y^*_d\right)^{ik}  \cc_{\dot{\beta}\dot{\rho}}P_L^{\dot{\rho}\dot{\sigma}}\Psi_{qi}^{\dot{\sigma}b},\\
\left(U_{\Psi_l H}\right)_{\dot{\beta}akb} &= \left(U_{H \Psi_l}\right)_{b\dot{\beta}ak} = y_e^{ki} \cc^{\dot{\beta}\dot{\rho}}P_R^{\dot{\rho}\dot{\sigma}} \Psi_{ei}^{\dot{\sigma}}\delta_{ab}, \\
\left(U_{\Psi_l H^*}\right)_{\dot{\beta}akb} &= \left(U_{H^* \Psi_l }\right)_{b \dot{\beta}ak} = -\left(y_e^*\right)^{ki} \Psi_{ei}^{\dot{\alpha}} \cc^{\dot{\alpha}\dot{\rho}}P_L^{\dot{\rho}\dot{\beta}} \delta_{ab}, \\
\left(U_{\Psi_f B}\right)_{\dot{\beta}ai\mu} &= \left(U_{B \Psi_f}\right)_{\mu \dot{\beta}ai} = -\frac{g_1}{2}\cc_{\dot{\beta}\dot{\rho}}\gamma^\mu_{\dot{\rho}\dot{\sigma}}Y_{\Psi_f}^{\dot{\sigma}\dot{\lambda}}\Psi_{fi}^{\dot{\lambda}a}+\frac{g_1}{2}\Psi_{fi}^{\dot{\lambda}a}\cc_{\dot{\lambda}\dot{\rho}}\gamma^\mu_{\dot{\rho}\dot{\sigma}}Y_{\Psi_f}^{\dot{\sigma}\dot{\beta}}, \\
\left(U_{\Psi_f W}\right)_{\dot{\beta}ai\mu A} &= \left(U_{W \Psi_f}\right)_{\mu A\dot{\beta}ai} =  -\frac{g_2}{2}\cc_{\dot{\beta}\dot{\rho}}\gamma^\mu_{\dot{\rho}\dot{\sigma}}\left(T^A_{\Psi_f}\right)^{ab}_{\dot{\sigma}\dot{\lambda}}\Psi_{fi}^{\dot{\lambda}b}+\frac{g_2}{2}\Psi_{fi}^{\dot{\lambda}b}\cc_{\dot{\lambda}\dot{\rho}}\gamma^\mu_{\dot{\rho}\dot{\sigma}}\left(T^A_{\Psi_f}\right)_{\dot{\sigma}\dot{\beta}}^{ba}.
\end{align}

Finally we have the matrix of purely fermionic derivatives

\begin{align}
\mathbf{U}_{\xi \xi} & =
\cc^{-1}_{\dot{\alpha}\dot{\beta}}
\begin{pmatrix}
0 & \left(U_{\Psi_q \Psi_u}\right)_{\dot{\beta}ai \dot{\rho}j} &
\left(U_{\Psi_q \Psi_d}\right)_{\dot{\beta}ai \dot{\rho}j} &
0 &
0 \\
\left(U_{\Psi_u \Psi_q}\right)_{\dot{\beta}i \dot{\rho}bj} & 0 &
0 &
0 &
0 \\
\left(U_{\Psi_d \Psi_q}\right)_{\dot{\beta}i \dot{\rho}bj} & 0 &
0 &
0 &
0 \\
0 & 0 &
0 &
0 &
\left(U_{\Psi_l \Psi_e}\right)_{\dot{\beta}ai \dot{\rho}j} \\
0 & 0 & 0 &
\left(U_{\Psi_e \Psi_l}\right)_{\dot{\beta}i \dot{\rho}bj} &
\end{pmatrix},
\end{align}

where the nonzero entries are given by

\begin{align}
% \left(X_{\Psi_q \Psi_q}\right)_{\dot{\beta}ai \dot{\rho}bj} &= 0, \\
\left(U_{\Psi_q \Psi_u}\right)_{\dot{\beta}ai \dot{\rho}j} &= -\left(U_{\Psi_u \Psi_q}\right)_{\dot{\rho}j \dot{\beta}ai} = -y_u^{ij} \cc_{\dot{\beta}\dot{\sigma}} \epsilon_{ab} H^*_b P_R^{\dot{\sigma}\dot{\rho}}+(y^*_u)^{ij} \cc_{\dot{\rho}\dot{\sigma}}\epsilon_{ab}H_b P_L^{\dot{\sigma}\dot{\beta}},\\
\left(U_{\Psi_q \Psi_d}\right)_{\dot{\beta}ai \dot{\rho}j} &= -\left(U_{\Psi_d \Psi_q}\right)_{\dot{\rho}j \dot{\beta}ai} = -y_d^{ij} \cc_{\dot{\beta}\dot{\sigma}} H_a P_R^{\dot{\sigma}\dot{\rho}}+(y^*_d)^{ij} \cc_{\dot{\rho}\dot{\sigma}}H^*_a P_L^{\dot{\sigma}\dot{\beta}},\\
%  \left(X_{\Psi_q \Psi_l}\right)_{\dot{\beta}ai \dot{\rho}bj} &= -\left(X_{\Psi_l \Psi_q}\right)_{\dot{\rho}bj \dot{\beta}ai} = 0,\\
% \left(X_{\Psi_q \Psi_e}\right)_{\dot{\beta}ai \dot{\rho}j} &= -\left(X_{\Psi_e \Psi_q}\right)_{\dot{\rho}j \dot{\beta}ai} = 0,\\
%  \left(X_{\Psi_u \Psi_u}\right)_{\dot{\beta}ai \dot{\rho}bj} &= 0\\
%  \left(X_{\Psi_u \Psi_d}\right)_{\dot{\beta}i \dot{\rho}j} &= -\left(X_{\Psi_d \Psi_u}\right)_{\dot{\rho}j \dot{\beta}i} = 0\\
%  \left(X_{\Psi_u \Psi_l}\right)_{\dot{\beta}i \dot{\rho}bj} &= -\left(X_{\Psi_l \Psi_u}\right)_{\dot{\rho}bj \dot{\beta}i} = 0\\
% \left(X_{\Psi_u \Psi_e}\right)_{\dot{\beta}i \dot{\rho}j} &= -\left(X_{\Psi_e \Psi_u}\right)_{\dot{\rho}j \dot{\beta}i} = 0\\
%  \left(X_{\Psi_d \Psi_d}\right)_{\dot{\beta}i \dot{\rho}bj} &= 0\\
% \left(X_{\Psi_d \Psi_l}\right)_{\dot{\beta}i \dot{\rho}bj} &= -\left(X_{\Psi_l \Psi_d}\right)_{\dot{\rho}bj \dot{\beta}i} = 0\\
% \left(X_{\Psi_d \Psi_e}\right)_{\dot{\beta}i \dot{\rho}j} &= -\left(X_{\Psi_e \Psi_d}\right)_{\dot{\rho}j \dot{\beta}i} = 0\\
\left(U_{\Psi_l \Psi_e}\right)_{\dot{\beta}ai \dot{\rho}j} &= -\left(U_{\Psi_e \Psi_l}\right)_{\dot{\rho}j \dot{\beta}ai} = -y_e^{ij} \cc_{\dot{\beta}\dot{\sigma}} H_a P_R^{\dot{\sigma}\dot{\rho}}+(y^*_e)^{ij} \cc_{\dot{\rho}\dot{\sigma}}H^*_a P_L^{\dot{\sigma}\dot{\beta}}.
\end{align}

In the bosonic sector there are also contributions from open covariant derivatives. The nonzero matrix entries of $\mathbf{Z}_{\phi \phi}^\mu$ are

\begin{align}
\left(Z^\mu_{H^* B}\right)_{a \nu} &= -\frac{g_1}{2} H_a g^\mu _\nu, \\
\left(Z^\mu_{H^* W}\right)_{a \nu A} &= -g_2 T^A_{ab} H_b g^\mu _\nu, \\
\left(Z^\mu_{H B}\right)_{a \nu} &= \frac{g_1}{2} H^*_a g^\mu _\nu, \\
\left(Z^\mu_{H W}\right)_{a \nu A} &= g_2 T^A_{ba} H^*_b g^\mu _\nu.
\end{align}

As discussed in Section <ref> computes these matrices automatically and in the matching step inserts the classical fields where appropriate. For this particular model the matrices were also computed by hand as a check. There are 500 BSUOLEA operators contributing to the matching, highlighting that hand calculation is not feasible. The reduction of the redundant effective Lagrangian to a set of non-redundant dimension six operators was performed in two different ways.

In the first approach the feature of was used which allows for the identification of custom operators and the implementation of replacement rules among these. This is most easily realised by writing the SM Lagrangian in terms of the Majorana spinors $\Psi_f$ as in the SM part on the r.h.s. of Eq. (<ref>) and then applying the equations of motion compatible with this Lagrangian to the matching result. These equations of motion read

\begin{align}
&-D^2 H + \mu_h^2 H - \lambda_h |H|^2 H - \Psi_q y_u \epsilon P_R \Psi_u - \bar{\Psi}_d y_d^\dagger P_L \Psi_q - \bar{\Psi}_e y_e^\dagger P_L \Psi_l = 0, \\
& \left(D^\mu W_{\mu \nu}\right)^A + g_2 \left(\bar{\Psi}_q T^A_{\Psi_q} \Psi_q + \bar{\Psi}_l T^A_{\Psi_l} \Psi_l +\frac{i}{2}H^\dagger \overleftrightarrow{D}^AH \right) = 0, \\
& D^\mu B_{\mu \nu} + g_1 \sum \limits_{f \in \{u,d,q,e,l\}} \bar{\Psi}_f Y_{\Psi_f} \gamma_\nu \Psi_f + \frac{i}{2}H^\dagger \overleftrightarrow{D}_\nu   H = 0,\\
& i \cc \slashed{D}\Psi_{q} - H y_d \cc P_R \Psi_d + \bar{\Psi}_d  y_d^\dagger  P_L H^\dagger - \tilde{H} y_u \cc P_R \Psi_u + \bar{\Psi}_u  y_u^\dagger  P_L \tilde{H}^\dagger = 0, \\
& i \cc \slashed{D}\Psi_{u} + \bar{\Psi}_q y_u \tilde{H} P_R - \tilde{H}^\dagger y_u^\dagger \cc P_L \Psi_q = 0, \\
& i \cc \slashed{D}\Psi_{d} + \bar{\Psi}_qy_d H P_R - H^\dagger y_d^\dagger \cc P_L \Psi_q = 0,\\
& i \cc \slashed{D}\Psi_{l} - H y_e \cc P_R \Psi_e + \bar{\Psi}_e  y_e^\dagger  P_L H^\dagger  = 0, \\
& i \cc \slashed{D}\Psi_{e} + \bar{\Psi}_ly_e H P_R - H^\dagger y_e^\dagger \cc P_L \Psi_l = 0,
\end{align}

and the dimension six operators that are generated in the matching are

\begin{align}
Q_{H\Box} &= |H|^2\Box |H|^2, \\
Q_H &= |H|^6, \\
Q_{HD} &= \left(H^\dagger D_\mu H\right)^\dagger \left(H^\dagger D^\mu H\right), \\
Q_{HB} &= |H|^2 B_{\mu \nu}B^{\mu \nu}, \\
Q_{HW} &= |H|^2 W^A_{\mu \nu}W^{\mu \nu A}, \\
Q_{HWB} &= \left(H^\dagger \tau^A H \right) W^A_{\mu \nu}B^{\mu \nu}, \\
Q_{uH} &= |H|^2 \left(\bar{q}\tilde{H}u\right) = |H|^2 \left(\bar{\Psi}_q \tilde{H} P_R \Psi_u\right),\\
Q_{dH} &= |H|^2 \left(\bar{q} H d\right) = |H|^2 \left(\bar{\Psi}_q \tilde{H} P_R \Psi_d\right),\\
Q_{eH} &= |H|^2 \left(\bar{l} H e\right) = |H|^2 \left(\bar{\Psi}_l \tilde{H} P_R \Psi_e\right),\\
Q_{Hu} & = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{u}\gamma^\mu u\right) = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{\Psi}_u\gamma^\mu P_R \Psi_u\right), \\
Q_{Hd} & = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{d}\gamma^\mu d\right) = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{\Psi}_d\gamma^\mu P_R \Psi_d\right), \\
Q_{Hud} &= \left(\tilde{H}^\dagger i D_\mu H\right)\left(\bar{u}\gamma^\mu d\right) = \left(\tilde{H}^\dagger i D_\mu H \right)\left(\bar{\Psi}_u\gamma^\mu P_R \Psi_d\right),\\
Q_{He} & = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{e}\gamma^\mu e\right) = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{\Psi}_e\gamma^\mu P_R \Psi_e\right), \\
Q_{Hq}^{\left(1\right)} & = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{q}\gamma^\mu q\right) = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{\Psi}_q\gamma^\mu P_L \Psi_q
\right), \\
Q_{Hq}^{\left(3\right)} & = \left(H^\dagger i \overleftrightarrow{D}^A_\mu H\right)\left(\bar{q} \sigma^A \gamma^\mu q\right) = \left(H^\dagger i \overleftrightarrow{D}^A_\mu H\right)\left(\bar{\Psi}_q \sigma^A \gamma^\mu \Psi_q\right),\\
Q_{Hl}^{\left(1\right)} & = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{l}\gamma^\mu l\right) = \left(H^\dagger i \overleftrightarrow{D}_\mu H\right)\left(\bar{\Psi}_l\gamma^\mu P_L \Psi_l\right), \\
Q_{Hl}^{\left(3\right)} & = \left(H^\dagger i \overleftrightarrow{D}^A_\mu H\right)\left(\bar{l} \sigma^A \gamma^\mu l\right) = \left(H^\dagger i \overleftrightarrow{D}^A_\mu H\right)\left(\bar{\Psi}_l \sigma^A \gamma^\mu \Psi_l\right), \\
Q_{2y} &= |\bar{q}^b y_u u \epsilon_{ba} + \bar{d}y_d^\dagger q^a + \bar{e} y_e^\dagger l^a|^2 = |\bar{\Psi}^b_q P_R y_u \Psi_u \epsilon_{ba} + \bar{\Psi}_d y_d^\dagger P_L \Psi_q^a + \bar{\Psi}_e y_e^\dagger \Psi_l^a|^2,
\end{align}

where the last operator $Q_{2y}$ is a combination of several four-fermion operators in the Warsaw basis, see Ref. <cit.>. We here introduced the Pauli matrices $\tau^A$ and the Hermitian derivatives

\begin{align}
H^\dagger i\overleftrightarrow{D}^\mu H &=  H^\dagger iD^\mu H - \left(D^\mu H\right)^\dagger iH,\\
H^\dagger i\overleftrightarrow{D}^A_\mu H &=  H^\dagger i\tau^A D_\mu H - \left(D_\mu H\right)^\dagger i\tau^A H.
\end{align}

The Wilson coefficients corresponding to these dimension six operators are found to be in agreement with the ones cited in Ref. <cit.> and are therefore not repeated here, with one exception, namely the presence of the additional operator $Q_{Hud}$ and its Hermitian conjugate. This operator was originally missed both in Ref. <cit.> and in Ref. <cit.> since it is obtained from correlation functions contributing to $HH\rightarrow\bar{f}^{'}f$, which were not calculated in the matching of those references. The Wilson coefficient at $Q_\text{match} = M$ is given by

\begin{align}
C_{Hud} =  -\frac{5A^2}{4M^4} y_u^\dagger y_d,
\end{align}

which has been confirmed by the authors of Ref. <cit.>. The fact that this operator was missed twice using diagrammatic techniques highlights a strength of the UOLEA approach: It is impossible to miss a contribution to the matching once a sufficiently general UOLEA has been computed. This is due to the fact that the effective Lagrangian is fully generated from the UV Lagrangian through a matching condition which, by construction, includes all correlation functions relevant to the matching. It is not necessary to know the effective Lagrangian prior to the matching and to construct a matching condition that captures all possible contributions to the different operators contained in that effective Lagrangian.

In the second approach was again used to perform the matching and the raw, redundant result was passed to , which first translates operators containing the Majorana fermions $\Psi_f$ into the usual SM notation and then reduces the set of structures to the Warsaw basis. The result of this computation matches the result of the previous calculation once $Q_{2y}$ is translated into the Warsaw basis. Note that this illustrates the main difference between the two approaches. The first approach is flexible in the sense that one can use custom operators or operator bases. This also implies more work since an algorithm has to be devised which translates redundant structures into these bases without being circular. The benefit of using is that it is guaranteed to work and terminate. The cost is that one always ends up in the Warsaw basis and that this is limited to the SMEFT.

Matching of a triplet benchmark model

CHAPTER: MATCHING OF A TRIPLET BENCHMARK MODEL

In this chapter we apply the functional matching formalism to a specific simplified model, which is not covered by the BSUOLEA, described by the Lagrangian

\begin{align}
\LagNoT ={}& -\frac{1}{4}V^{\mu \nu A} V_{\mu \nu}^A + \frac{m_V^2}{2}V^{\mu A}V_{\mu}^A+V^{\mu A}\left(\gamma_H g_{*}J_\mu^{HA}+\gamma_V\frac{g_2}{g_{*}}J_\mu^{A}+\sum_{f \in \{l,q\}}\gamma_f g_{*}J_{\mu}^{fA}\right) \nonumber
\\&+ \Lag{SM},
\label{eq: model_lag}
\end{align}

where $\Lag{SM}$ is the SM Lagrangian, $V^{\mu A}$ is a $SU(2)_L$ triplet vector field and the currents are given by

\begin{align}
J_\mu^{HA} &= \frac{i}{2} H^\dagger \overleftrightarrow{D}^A_\mu H,\\
J_\mu^{A} &= \left(D^\nu W_{\nu \mu}\right)^{A},\\
J_\mu^{fA} &=\frac{1}{2} \bar{f} \gamma_\mu \tau^A f,
\end{align}

with $\tau^A$ $A=1,2,3$ being the Pauli matrices. The field strength tensor $V^{\mu\nu A}$ is defined as

\begin{align}
V^{\mu\nu A} = D^{[\mu}V^{\nu]A} = D^\mu V^{\nu A} - D^\nu V^{\mu A},
\end{align}

and the SM conventions are the same as in Section <ref>. This model was introduced in Ref. <cit.> and is a special case of the model discussed in Ref. <cit.>. The explicit mapping is given in Section <ref> and the matching presented in that section also applies to the more general model. Phenomenologically, these models are interesting for two reasons. Firstly, the model of Ref. <cit.> acts as a simplified model that accurately describes several UV completions of the SM. In Ref. <cit.> it was shown explicitly that both the weakly coupled UV completion of the SM discussed in Ref. <cit.> and the Minimal Composite Higgs model discussed in Refs. <cit.> can be mapped onto this simplified model under the assumption that the remainder of the NP sector is heavy compared to the vector field $V^{\mu A}$. In Ref. <cit.> the model defined by Eq. (<ref>) was studied in the context of SMEFT fits. It was argued that the extended energy reach of the LHC can lead to an increased sensitivity to some higher dimensional operators, the effects of which are enhanced by powers of $E/\Lambda$, where $E$ is some energy scale in the process under consideration and $\Lambda$ is the scale of NP. This enhancement is most pronounced in kinematic regions where one might suspect the EFT expansion to have broken down, rendering any derived constraints inconsistent. However, whether or not this is the case cannot be inferred by only considering the general SMEFT Lagrangian since the Wilson coefficients cannot be disentangled from the unknown NP scale $\Lambda$. Nevertheless, in specific UV completions the scale $\Lambda$ is known and has a clear physical meaning. It is the pole mass of the lightest particle in the NP sector. The question then becomes whether there are specific UV completions of the SM for which the apparently inconsistent constraints are meaningful. One way of realising an energy enhancement without breaking the EFT assumptions is to have large Wilson coefficients, which can be seen from the following argument presented in Ref. <cit.>. Assume that the dimension six operator $\op$ with Wilson coefficient $C_\op$ is enhanced by $E^2/\Lambda^2$ and let $\Lambda$ be the pole mass of the lightest particle in the NP sector. With the naive expectation of $C_\op \sim 1$ the EFT expansion indeed breaks down as $E/\Lambda \to 1$ and no enhancement arises. However, if $C_\op = g_e^2 \hat{C}_\op$, where $\hat{C}_\op \sim 1$ and the enhancement factor $g_e^2 > 1$, we get an effective suppression scale $f = \Lambda/g_e$ such that $C_\op \op/\Lambda^2 = \hat{C}_\op \op / f^2$ and the energy enhancement becomes $E^2/f^2$ rather than $E^2/\Lambda^2$. Nothing special happens as $E/f \to 1$ and the EFT expansion is perfectly valid in the region where the operator is enhanced (as long as $E < \Lambda$ is satisfied). Clearly, this argument crucially depends on the fact that the scale $\Lambda$ has a well defined physical meaning, which it only carries once a certain UV completion has been specified. It was precisely this effect that was studied at tree level in Ref. <cit.> with $g_{*}$ playing the role of the enhancement. Since it is necessary that $g_{*}>1$ Eq. (<ref>) was interpreted as the effective description of a composite sector and it was shown that in this case an enhancement does occur for certain operator combinations that contribute to Higgs physics.

Given the fact that the model defined by Eq. (<ref>) (or rather its generalisation as given in Ref. <cit.>) can be interpreted as the effective description of several interesting UV completions of the SM at an intermediate scale it is an interesting benchmark model to study. Taking into account that parametrically large Wilson coefficients can occur when the model is interpreted as the effective description of a composite sector, this latter scenario becomes particularly interesting. Adding that the one-loop matching is expected to induce operators that are not present at tree level, which might have a dramatic impact on the SMEFT fit similar to the RGE induced operators in Ref. <cit.>, we here focus on the one-loop matching of this model using the methods described in Chapters <ref> and <ref>. There are two complications that arise when trying to directly apply the results of Chapter <ref> to the model studied in this section. Both of these issues are connected to the interpretation of the vector field as a resonance arising from a strongly coupled UV completion. Firstly, an explicit vector resonance comprises a system with second-class constraints. It is well-known that the generating functional of correlation functions in such theories might have to be modified as compared to the naive expectation to take these constraints into account.[See for example Eq. (3.4.26) of Ref. <cit.>.] Since the starting point of the UOLEA derivation is the generating functional, it is imperative to check whether any relevant modifications arise in this particular instance. As will be shown in Section <ref> this is not the case and one can proceed with the naive expectation for the generating functional. However, it turns out that for a self-consistent field theory the Lagrangian of Eq. (<ref>) is insufficient and a coupling of the form $V^{\mu A}V_{\mu}^A |H|^2$ has to be added. Secondly, as the resonance $V^{\mu A}$ is not a gauge boson, it does not have a gauge-fixing term and the gauge choice $\xi = 1$ is not possible. This leads to a different propagator as compared to the one used in Chapter <ref> and to extra terms with two open covariant derivatives. This means that a dedicated computation of the relevant UOLEA operators has to be performed. We will briefly discuss this computation in Section <ref>.

§ CONSTRAINT ANALYSIS AND DERIVATION OF THE GENERATING FUNCTIONAL

It is well-known that a massless vector field has two physical degrees of freedom and a massive vector field has three, yet in four dimensions each vector field $V^{\mu A}$ is described by four components, which are needed to make Lorentz invariance of the theory manifest. This mismatch between the number of physical degrees of freedom and the total number of degrees of freedom means that constraints are present in the theory. In the case of gauge theories[For a thorough discussion of the quantisation of gauge theories see Ref. <cit.> and for a general discussion of quantisation of constrained systems see Ref. <cit.>.], one way of dealing with these constraints is the Fadeev-Popov method, which introduces ghosts into the generating functional of the theory. More generally it is expected that the presence of constraints in a theory introduces unphysical degrees of freedom into the generating functional, which have to be taken into account when computing correlation functions. The systematic construction of the generating functional in these theories is well understood and reviewed in Ref. <cit.>, which we follow closely here. Moreover, it is known that the quantisation of an interacting field theory is not necessarily equivalent to the free field quantisation of the individual fields comprising the theory if constraints are present, as pointed out in Refs. <cit.> in the case of spin-$3/2$ fields. It may very well be the case that a consistent formulation of the interacting field theory, even at the classical level, restricts the couplings of the theory. This fact, together with the requirement of renormalisability in the sense of EFTs, was used in Ref. <cit.> to show that the self-couplings of the $\rho$-meson are restricted to the Yang-Mills form. Hence, performing a constraint analysis of the model under consideration here is imperative to check its self-consistency.

We next introduce the main algorithm and concepts needed to derive the generating functional and refer the interested reader to Ref. <cit.> for more details. A collection of useful definitions and relations can also be found in Appendix <ref>. In what follows we will make a clear distinction between the Lagrangian function $L$ and the Lagrangian density $\LagNoT$ as well as the Hamiltonian function $H$ and the Hamiltonian density $\mathcal{H}$. Consider a theory whose Lagrangian function depends on the fields $q^i(t,\mathbf{x})$ with $i=1,\dots,n$ and the corresponding time derivatives[In a Lorentz covariant theory we expect only derivatives of the form $\partial_\mu q^i$ to appear. However, in the Hamiltonian formalism the time derivative plays a special role and we separate it from the spatial derivatives.], which we will refer to as velocities, $\dot{q}^i(t,\mathbf{x})$. In going from the Lagrangian formulation in terms of $L[q^i,\dot{q}^i]$ to the Hamiltonian formulation one introduces the canonical momenta $\pi_i(t,\mathbf{x})$ defined by

\begin{align}
\pi_i(t,\mathbf{x}) \equiv \frac{\delta_t L}{\delta \dot{q}^i},
\label{eq: canonical_momentum_def}
\end{align}

where the functional derivative $\frac{\delta_t}{\delta \dot{q}^i}$ is taken at a fixed time. One then uses Eq. (<ref>) to express the velocities in terms of the canonical fields and their corresponding momenta. As shown in Ref. <cit.>, in general, one might only be able to solve for $N \leq n$ velocities, $\dot{q}^I(t,\mathbf{x})$, expressed through corresponding fields $q^I(t,\mathbf{x})$ and their conjugate momenta $\pi^I(t,\mathbf{x})$, where $I=1,\dots,N$. The remaining $n-N$ equations then yield constraints on the possible values of the fields and conjugate momenta. These constraints can be brought into the form

\begin{align}
\Phi^a(q^i,\pi_i) = 0, \,a = 1,\dots,n-N,
\end{align}

where $\Phi^a$ is some function. Such constraints, arising from the inability to determine certain velocities, will be referred to as primary constraints in the following. For a system with such constraints the classical equations of motion for all degrees of freedom $\eta = (q^i,\pi_i)$ can be expressed in terms of a modified Hamiltonian

\begin{align}
H_T = H(\eta)+\lambda^a \Phi^a(\eta),
\end{align}

where $H(\eta)$ is given by

\begin{align}
H(\eta) = \left(\frac{\delta_t L}{\delta \dot{q}^i_\mathbf{x}}\dot{q}_\mathbf{x}^i-L\right)_{\dot{q}^I=\dot{q}^I(q^i,\Pi_I,\dot{q}^a)},
\label{eq: Hamiltonian_from_Lagrangian}
\end{align}

and the fields $\lambda^a = \lambda^a(t,\mathbf{x})$ can be thought of as Lagrange multiplier fields. In Eq. (<ref>) the repeated boldface indices imply an integration over space. The usual equations of motion have to be supplemented by the constraint equations, i.e. by the requirement that the constraints are satisfied on the phase-space trajectory. For a consistent solution, this property has to hold at all times and hence

\begin{align}
0 &\overset{!}{\approx} \{\Phi^a,H_T\} = \{\Phi^a,H\}+ \lambda^b \{\Phi^a,\Phi^b\},
\label{eq: conservation_condition}
\end{align}

where we use $\approx$ to mean weak equality, that is equality holds after the constraints have been imposed.[However, Poisson brackets have to be computed before imposing the constraints.] The curly brackets denote the Poisson bracket (PB). For any given constraint Eq. (<ref>) may lead to one of three outcomes. The first possibility is that the equation is identically satisfied and no further constraints arise due to the conservation of the constraint under consideration. The second possibility is that one of the Lagrange multipliers can be solved for, after which the equation becomes an identity and no further constraints arise. The final possibility is that the equation is neither identically satisfied, nor does it allow for the determination of a Lagrange multiplier. In this case, the conservation of the constraint on the trajectory has to be imposed as a further constraint. We will refer to constraints arising from the requirement that other constraints be conserved as secondary constraints. If secondary constraints are present, these again have to be conserved on the trajectory. One thus proceeds in this manner, imposing the conservation of the new constraints, thereby determining a subset of the Lagrange multipliers and obtaining new constraints, until no new constraints appear. At this point some of the Lagrange multipliers may still be undetermined. Indeed, for a primary constraint $\Phi$ whose PB with any other constraint vanishes weakly, the corresponding Lagrange multiplier remains undetermined and introduces an arbitrariness into the solutions of the equations of motion. In general, a constraint whose PB with any other constraint vanishes weakly is referred to as a first-class constraint and primary first-class constraints introduce a gauge redundancy into the theory, see Refs. <cit.> for thorough discussions. A constraint that is not first-class is referred to as second-class. In the model under consideration in this chapter first-class constraints arise due to the presence of the gauge fields and second-class constraints due to the presence of the vector resonances.[There are also second-class constraints associated with the fermionic degrees of freedom.]

In order to systematically construct the generating functional of the corresponding quantum theory one first canonically quantises the theory and then passes from the Hamiltonian path integral to the Lagrangian path integral. This is desirable in the case under consideration as the inclusion of constraints into the canonical formalism and the Hamiltonian path integral is well understood. As is well-known, for a gauge theory one has to break the gauge redundancy of the theory in order to obtain meaningful results from the generating functional. One way to accomplish this in the canonical formalism is to introduce supplementary constraints $\Phi_{G}$, which turn the first-class constraints associated with the gauge freedom into second-class ones, thereby fixing the gauge. Indeed the Hamiltonian generating functional in the presence of constraints $\mathbf{\Phi}$ can be generally written as (c.f. Eq. (3.4.17) of Ref. <cit.>)

\begin{align}
Z[J] = \int \measure \eta \; \Sdet^{1/2} \{\mathbf{\Phi}, \mathbf{\Phi}\} \delta(\mathbf{\Phi}) \exp \left[i\left(S+J_x \eta_x\right)\right],
\label{eq: constrained_Hamiltonian_generating_functional}
\end{align}

where $\mathbf{\Phi}$ is the vector of all constraints, including the supplementary constraints $\Phi_{G}$ and any secondary constraints that may arise from their conservation, $\{\mathbf{\Phi},\mathbf{\Phi}\}$ is the matrix that results from taking the PBs of the different constraints and $\delta(\mathbf{\Phi})$ is a delta functional enforcing the constraints. As before, the symbol $\eta$ in Eq. (<ref>) runs over all of the canonical variables and their conjugate momenta, including the unphysical ones, and the action $S$ is given by

\begin{align}
S = \int dt \; \left[\pi_{i}^{\mathbf{x}} \dot{q}^i_\mathbf{x} - H\right].
\end{align}

Both the superdeterminant and the delta functional in Eq. (<ref>) can be brought into the exponential by introducing new, unphysical degrees of freedom such as ghosts, which then have to be included in the computation of correlation functions.

We now show that for the model defined by Eq. (<ref>) these new degrees of freedom are irrelevant, except for the usual Fadeev-Popov ghosts associated with the gauge fixing, by explicitly deriving the generating functional of the model in the Hamiltonian formalism. As it will turn out in the analysis, the Lagrangian of Eq. (<ref>) does not define a self-consistent theory. We therefore add an extra coupling term, which is allowed by the symmetries of the theory. Furthermore, it is more convenient to convert the $V^{\mu A}J_\mu^A$ coupling into a kinetic mixing term through integration by parts. It should be mentioned that at this point the kinetic mixing term cannot be removed by a field redefinition since we are working with a classical theory that is yet to be quantised. The field redefinition is not a canonical transformation and therefore not allowed classically. It is only at the level of the path integral that such field redefinitions are valid. We thus consider the Lagrangian

\begin{align}
\LagNoT ={}& -\frac{1}{4}V^{\mu \nu A} V_{\mu \nu}^A - \frac{g_M}{2}V^{\mu \nu A} W_{\mu \nu}^A -\frac{1}{4}W^{\mu \nu A} W_{\mu \nu}^A +  \frac{m_V^2}{2}V^{\mu A}V_{\mu}^A+g_{H} V^{\mu A} J_\mu^{HA} \nonumber \\ &+g_{f} V^{\mu A} J_{\mu}^{fA}+ \bar{f}i\slashed{D}f+ \left(D_\mu H\right)^\dagger \left(D^\mu H\right)+ \frac{g_{VH}}{2}|H|^2V^{\mu A}V_{\mu}^A + \Lag{rest},
\end{align}

where we focus on the $SU(2)_L$ part of the SM since the $SU(3)_c$ and $U(1)_Y$ contributions are inconsequential. We also only consider one kind of fermion as the introduction of several fermions does not alter the treatment significantly and we simplified the notation with the identifications

\begin{align}
g_H = \gamma_H g_{*}, \; g_M=\gamma \frac{g}{g_{*}},\; g_f=\gamma_f g_{*},
\end{align}

where in this section we denote the $SU(2)_L$ gauge coupling by $g$. We further introduced the term

\begin{align}
\frac{g_{VH}}{2}|H|^2V^{\mu A}V_{\mu}^A,
\end{align}

which, as will be shown, is needed for the self-consistency of the theory. The Lagrangian density $\Lag{rest}$ contains an $SU(2)_L$ invariant polynomial in $H$, $H^{\dagger}$ and the fermionic fields. We find the following expressions for the conjugate momenta

\begin{align}
\label{eq: conj_mom_1}
\pi(W)^{\mu A} &= -W^{0 \mu A}-g_M V^{0 \mu A}, \\
\label{eq: conj_mom_2}
\pi(V)^{\mu A} &= -V^{0 \mu A}-g_M W^{0 \mu A}, \\
\pi(\Hdagg) &= D^0H-\frac{i}{2}g_H V^{0A}\tau^A H,\\
\pi(H) &= (D^0H)^\dagger+\frac{i}{2}g_H V^{0A}\Hdagg \tau^A, \\
\bar{\pi}(f) &= i\bar{f}\gamma^0, \\
\label{eq: conj_mom_last}
\pi(\bar{f}) &= 0,
\end{align}

where we denote the conjugate momentum corresponding to the field $q$ by $\pi(q)$. As there are four conjugate momenta that are entirely independent of velocities, there are four primary constraints in the system, which read

\begin{align}
\label{eq: constr_1}
\Phi_1^A &= \pi(W)^{0 A}, \\
\Phi_2^A &= \pi(V)^{0 A}, \\
\Phi_3 &= \bar{\pi}(f) - i\bar{f}\gamma^0, \\
\label{eq: constr_last}
\Phi_4 &= \pi(\bar{f}).
\end{align}

For the remaining fields the corresponding velocities can be expressed through fields and conjugate momenta using Eqs. (<ref>)–(<ref>), provided that $g_M \neq \pm 1$ in which case Eqs. (<ref>) and (<ref>) become linearly dependent and lead to three further constraints. This renders the theory inconsistent as the number of independent degrees of freedom will not match the number of physical degrees of freedom. Therefore, the exclusion of these values of $g_M$ is a consistency condition. For the determinable velocities one finds

\begin{align}
\partial_0 W^{i A} &= \frac{1}{g_M^2-1}\pi(W)^{iA}-\frac{g_M}{g_M^2-1}\pi(V)^{iA}-g f^{ABC}W_{0}^B W^{iC}+\partial ^i W^{0A},\\
\partial_0 V^{i A} &= \frac{1}{g_M^2-1}\pi(V)^{iA}-\frac{g_M}{g_M^2-1}\pi(W)^{iA}-g f^{ABC} W^{[0|B|}V^{i]C}+\partial ^i V^{0 A},\\
\partial_0 H &= \pi(\Hdagg)+\frac{i}{2}g W_0^A \tau^A H +\frac{i}{2} g_H V_0^A \tau^A H, \\
\partial_0 \Hdagg &= \pi(H)-\frac{i}{2}g W_0^A \Hdagg \tau^A -\frac{i}{2} g_H V_0^A \Hdagg \tau^A,
\end{align}

where lower case Roman indices $i$, $j$, $k$ etc. take the values $1,2,3$ and denote spatial components of a four-vector.[We use a metric with the mostly minus signature so one has to be careful when raising and lowering spatial indices.]
Using these expressions as well as the constraints in Eqs. (<ref>)–(<ref>) we can construct the modified Hamiltonian density $\mathcal{H}_T$ as

\begin{align}
\mathcal{H}_T ={}& \frac{1}{2(g_M^2-1)}\pi(W)^A_i \pi(W)^{iA}+\frac{1}{2(g_M^2-1)}\pi(V)^A_i \pi(V)^{iA}-\frac{g_M}{g_M^2-1}\pi(W)^A_i \pi(V)^{iA}\nonumber \\ &+\pi(W)^A_i D^i W^{0A} + \pi(V)^A_i \left(D^i V^{0A}-g f^{ABC}W^{0B}V^{iC}\right)\nonumber \\
&-\frac{i}{2}\Hdagg \tau^A \pi(\Hdagg) \left(g W^{0A}+g_H V^{0A}\right)+\frac{i}{2}\pi(H) \tau^A H \left(g W^{0A}+g_H V^{0A}\right)\nonumber \\
&+\frac{1}{4}g_H^2V_0^{A}V_0^{B} \Hdagg \tau^A \tau^B H - \frac{g_{VH}}{2}|H|^2 V^{\mu A} V^{A}_{\mu} -g_H V^{iA}J^{HA}_i-g_f V^{\mu A}J_\mu^{f A} \nonumber \\ &+\frac{1}{4}W^{ijA} W_{ij}^A +\frac{1}{4}V^{ijA} V_{ij}^A +\frac{g_M}{2}W^{ijA} V_{ij}^A -\frac{m_V^2}{2}V_{\mu}^A V^{\mu A} - \left(D_i H\right)^\dagger \left(D^i H\right) \nonumber \\
&-\bar{f}i \gamma^i \partial_i f -g \bar{f}\slashed{W}f - \Lag{rest} \nonumber \\ &+\lambda_1^A \pi(W)^{0A} +\lambda_2^A \pi(V)^{0A} +\lambda_3 \left(\bar{\pi}(f)-i\bar{f}\gamma^0\right) +\lambda_4 \pi(\bar{f}),
\label{eq: modified_Hamiltonian_density}
\end{align}

from which the modified Hamiltonian is obtained as

\begin{align}
H_T = \int d^3x \, \mathcal{H}_T(x).
\end{align}

In Eq. (<ref>) we defined $\slashed{W}=W^{A\mu}\frac{\tau^A}{2}\gamma_\mu$. Given the Hamiltonian we may now impose the conservation of constraints according to Eq. (<ref>) for each constraint. Starting with $\Phi_4$ we find

\begin{align}
\{\Phi_4,H_T\} &= -\frac{g_f}{2} V^{\mu A} \gamma_\mu \tau^{A} f- i\gamma^i \partial_i f - g \slashed{W} f + \frac{\partial_r \Lag{rest}}{\partial \bar{f}} + i \gamma^0 \lambda_3 \overset{!}{\approx} 0,
\end{align}

which determines the Lagrange multiplier $\lambda_3$ for all field values.[Here the subscript $r$ on the partial derivative indicates that we are taking the right derivative. For a definition see Appendix <ref>.] Hence, no secondary constraint arises in this case. Similarly

\begin{align}
\{\Phi_3,H_T\} &= \frac{g_f}{2} V^{\mu A} \bar{f} \gamma_\mu \tau^{A} + \bar{f} i\gamma^i \overleftarrow{\partial}_i + g \bar{f} \slashed{W} + \frac{\partial_r \Lag{rest}}{\partial f} - i  \lambda_4 \gamma^0 \overset{!}{\approx} 0
\end{align}

determines $\lambda_4$ without the appearance of secondary constraints. This concludes the discussion of the constraints arising in the fermionic sector. For the constraints associated with the gauge field we find

\begin{align}
\label{eq: secondary_gauge_constraint_old}
\{\Phi_1^A,H_T\} ={}& D^i \pi(W)_i^A + g f^{DAC} \pi(V)^D_i V^{iC}+\frac{i}{2}g\Hdagg \tau^A \pi(\Hdagg) -\frac{i}{2}g\pi(H) \tau^A H \nonumber \\ &+ \frac{g}{2} \bar{f}\gamma^0 \tau^A f,
\end{align}

which does not depend on any Lagrange multiplier and hence requiring it to vanish weakly constitutes a secondary constraint. We may add to Eq. (<ref>) any function that vanishes weakly since the resulting constraint is equivalent to the original one. It is customary to write the secondary constraint as

\begin{align}
\Phi_{1(2)}^A &= \{\Phi_1^A,H_T\} - \frac{ig}{2} \Phi_3 \tau^A f - \frac{ig}{2} \bar{f} \tau^A \Phi_4 + g f^{DAC} \Phi_2^{D} V^{0C} \nonumber \\
&=  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 \\ &\quad - \frac{ig}{2} \bar{\pi}(f) \tau^A f -\frac{ig}{2} \bar{f} \tau^A \pi(\bar{f}).
\label{eq: secondary_gauge_constraint}
\end{align}

Similarly, the conservation of $\Phi_{2}^A$ gives rise to a secondary constraint $\Phi_{2(2)}^A$ with

\begin{align}
\Phi_{2(2)}^A =\{\Phi^A_2,H_T\} ={}&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 \\ &-\frac{1}{4}g_H^2 V^{0B} \Hdagg [\tau^A,\tau^B]_{+} H+g_{VH}|H|^2 V^{0A} + g_f J_0^{fA} + m_V^2 V^{0A}\nonumber \\
={}& 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}

where $[A,B]_{+}=AB+BA$ is the anticommutator and we used that $[\tau^A,\tau^B]_{+}=2I\delta^{AB}$, where $I$ is the $2\times 2$ identity matrix. Since non-trivial secondary constraints exist we have to consider their conservation in time. The computation of $\{\Phi_{1(2)}^A,H_T\}$ is rather lengthy and we simply quote the result. As we were unable to find the computation in the literature, even for the case of a gauge theory without vector resonances, we present some of the details of the computation in Appendix <ref>. The result of the calculation is

\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) \approx 0,
\label{eq: conservation_secondary_gauge_constraint}
\end{align}

which means that no further constraints arise from the gauge bosons. This is expected from the known analysis of Yang-Mills theory with scalar and spin-$1/2$ matter fields, see Ref. <cit.>. However, as is clear from Eq. (<ref>) the presence of the resonances leads to modifications of the computation. We finally turn to the conservation of $\Phi^A_{2(2)}$, which is expected to determine $\lambda_2^A$ in order to yield the correct number of degrees of freedom for a massive vector field. To see that this is indeed the case we note that $\Phi^A_{2(2)}$ contains terms involving $V^{0A}$ which in the PB with $H_T$ will provide terms proportional to $\lambda_2^A$. These are also the only terms that give rise to such contributions. Furthermore, $\Phi^A_{2(2)}$ is independent of $W^{0A}$ and hence $\lambda_1^A$ will indeed remain undetermined. Noting that

\begin{align}
\{\Phi^A_{2(2)},H_T\} ={}&\{D^i \pi(V)_i^A,H_T\}+\frac{i}{2}g_H\{\Hdagg \tau^A \pi(\Hdagg),H_T\}-\frac{i}{2}g_H\{\pi(H) \tau^A H,H_T\} \nonumber \\
&+ g_f\{J_0^{fA},H_T\}+V^{0A}\left(g_{VH}-\frac{g_H^2}{2}\right) \{|H|^2,H_T\} \nonumber \\
&+ \left(m_v^2 +g_{VH} |H|^2 -\frac{g_H^2}{2}|H|^2\right) \{V^{0A},H_T\},
\end{align}

we define

\begin{align}
\mathcal{F}^A \equiv{}& \{D^i \pi(V)_i^A,H_T\}+\frac{i}{2}g_H\{\Hdagg \tau^A \pi(\Hdagg),H_T\}-\frac{i}{2}g_H\{\pi(H) \tau^A H,H_T\} \nonumber \\
&+ g_f\{J_0^{fA},H_T\}+V^{0A}\left(g_{VH}-\frac{g_H^2}{2}\right) \{|H|^2,H_T\},
\end{align}

which is independent of $\lambda_2^A$ and write

\begin{align}
\{\Phi^A_{2(2)},H_T\}\overset{!}{\approx}0
\end{align}

\begin{align}
\left(m_v^2 +g_{VH} |H|^2 -\frac{g_H^2}{2}|H|^2\right) \lambda_2^A = - \mathcal{F}^A.
\end{align}

As pointed out before, self-consistency requires this equation to determine $\lambda_2^A$ for all field values, which in turn requires that

\begin{align}
m_V^2+g_{VH}|H|^2-\frac{g_H^2}{2}|H|^2 \neq 0
\end{align}

for all values of $|H|^2$. For this condition to be satisfied it is necessary and sufficient that $g_{VH} \geq \frac{g_H^2}{2}$. We thus see that it is pivotal to introduce the $V^{\mu A}V_{\mu}^A |H|^2$-coupling into the Lagrangian from the point of view of consistency.

At this point of the analysis no further constraints arise and all Lagrange multipliers except $\lambda_1^A$, which is associated with the gauge freedom, are determined. In order to fix the gauge we impose the constraint

\begin{align}
\Phi_G^A = \partial_i W^{iA},
\end{align}

which may be recognised as the Coulomb gauge condition. Like the previous constraints, this constraint has to be conserved in time. Its PB with the modified Hamiltonian is given by

\begin{align}
\{\Phi^A_{G},H_T\} = \frac{1}{g_M^2-1}\partial_i \pi(W)^{iA} -\frac{g_M}{g_M^2-1}\partial_i \pi(V)^{iA}+D^i\partial_i W^{0A} \equiv \Phi_{G(2)}^A,
\end{align}

which yields a secondary constraint. Imposing the conservation of this secondary constraint we obtain

\begin{align}
\{\Phi^A_{G(2)},H_T\} = -\Delta \lambda_1^A + g f^{ABC} W^{iB}\partial_i \lambda_1^C + \mathcal{G}^A \overset{!}{\approx} 0,
\label{eq: lambda_1_cond}
\end{align}

where $\mathcal{G}^A$ is independent of $\lambda_1^A$ and $\Delta$ is the Laplace operator. Eq. (<ref>) determines the final Lagrange multiplier through a differential equation.[One might be concerned that the differential operator depends on the spatial gauge field components, which means that there might be field values for which the differential operator is not invertible. However, the particular differential operator arising here is elliptic independent of the values of the field $W^{iA}$ and hence always invertible.] We are now in a position to formulate the generating functional in the Hamiltonian formalism using Eq. (<ref>). To this end we need to compute the matrix of PBs of all constraints. It is sufficient to evaluate the result on the constraint surface since the constraints are enforced by the delta functional. Defining

\begin{align*}
\mathbf{\Phi} = \begin{pmatrix}
\Phi^A_{G(2)} &
\Phi_1^A &
\Phi_2^A &
\Phi_{G}^A &
\Phi_{1(2)}^A &
\Phi_{2(2)}^A &
\Phi_3 &
\Phi_4
\end{pmatrix}^T
\end{align*}

we find

\begin{align*}
\{\mathbf{\Phi},\mathbf{\Phi}\} &= \begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\end{align*}

\begin{align}
A &= \begin{pmatrix}
-\frac{gf^{ABD}}{g_M^2-1} \Delta W^{0D} & \left(D_i\partial^i\right)^{AB}&  0& \frac{\delta^{AB}}{g_M^2-1}\Delta & \{\Phi_{G(2)}^A,\Phi_{1(2)}^B\} & \{\Phi_{G(2)}^A,\Phi_{1(2)}^B\}  \\
-\left(D_i\partial^i\right)^{AB}& 0 & 0 & 0 & 0 & 0  \\
0 & 0 & 0 & 0 & 0 & -M_V^2 \delta^{AB}  \\
-\frac{\delta^{AB}}{g_M^2-1}\Delta & 0 & 0 & 0 & \left(D_i\partial^i\right)^{AB}&0  \\
\{\Phi_{1(2)}^A,\Phi_{G(2)}^B\} & 0 & 0& -\left(D_i\partial^i\right)^{AB}& 0 & 0 \\
\{\Phi_{2(2)}^A,\Phi_{G(2)}^B\} & 0 & M_V^2\delta^{AB} & 0 & 0 & \{\Phi_{2(2)}^A,\Phi_{2(2)}^B\}
\end{pmatrix},\\
B&= \begin{pmatrix}
0 & 0\\
0 & 0\\
0 & 0\\
0 & 0\\
0 & 0\\
\frac{g_f}{2} \bar{f} \gamma^{0} \tau^{A} & -\frac{g_f}{2} \gamma^{0} \tau^{A} f
\end{pmatrix},\text{ }
0 & 0 & 0 & 0 & 0 & -\frac{g_f}{2} \bar{f} \gamma^{0} \tau^{A} \\
0 & 0 & 0 & 0 & 0 & \frac{g_f}{2} \gamma^{0} \tau^{A} f
\end{pmatrix},\text{ }
D = \begin{pmatrix}
0 & -i \gamma^0 \\
i\gamma^0 & 0
\end{pmatrix},
\end{align}

\begin{align}
M_V^2 = m_V^2+g_{VH}|H|^2-\frac{g_H^2}{2}|H|^2.
\end{align}

We only care about field dependent contributions to the superdeterminant since any constant factors can be absorbed in the normalisation of the generating functional. Using the definition of the superdeterminant, see Appendix <ref>, we obtain

\begin{align}
\Sdet \{\mathbf{\Phi},\mathbf{\Phi}\} &= \det\left(A-BD^{-1}C\right) \det(D)^{-1} \propto \det(-M_V^2)^2 \det(D_i \partial^i)^4.
\end{align}

The second factor is precisely what is expected in a gauge theory without resonances, whereas the first factor is entirely due to the presence of the resonances. We next bring the generating functional into the Lagrangian form following Refs. <cit.>. In this discussion we omit the source terms since these can always be adjusted by the equivalence theorem of Ref. <cit.>. We thus write

\begin{align}
Z \equiv Z[0] = \int \measure \mu \exp{\left\{i\int d^4x \left[\mathcal{P}_I \partial^0 \mathcal{Q}^I - \mathcal{H} \right]\right\}},
\end{align}

\begin{align*}
\mathcal{P} = \begin{pmatrix}
\pi(W)^A_\mu & \pi(V)^A_\mu & \bar{\pi}(f) & \pi(\bar{f}) & \pi(H) & \pi(\Hdagg)
\end{pmatrix}^T, \text{ }
\mathcal{Q} = \begin{pmatrix} W^{\mu A} & V^{\mu A} & f & \bar{f} & H & \Hdagg
\end{pmatrix}^T,
\end{align*}

and the integration measure is given by

\begin{align*}
\measure \mu ={}& \measure \mathcal{Q} \, \measure \mathcal{P} \, \delta\left(\pi(W)^{0A}\right) \delta\left(\pi(V)^{0A}\right) \delta\left(\bar{\pi}(f)-i\bar{f}\gamma^0\right)\delta\left(\pi(\bar{f})\right)\delta\left(\partial_i W^{iA}\right)\nonumber \\
&\times \delta\left(\frac{1}{g_M^2-1}\partial_i \pi(W)^{iA}-\frac{g_M}{g_M^2-1}\partial_i \pi(V)^{iA}+D^i\partial_i W^{0A}\right) \nonumber \\
& \times \delta\left(\Phi^A_{1(2)}\right) \delta\left(\Phi^A_{2(2)}\right) \det\left(-M_V^2\right) \det\left(D_i \partial^i\right)^2.
\end{align*}

Due to the first four delta functionals in the measure the integrals over $\bar{\pi}(f)$, $\pi(\bar{f})$, $\pi(W)^{0A}$ and $\pi(V)^{0A}$ can be trivially performed. For the $W^{0A}$ integration we define $\tilde{W}^{0A}$ through

\begin{align*}
W^{0A}   = (\tilde{D}_i \partial^i)^{-1}\left[\tilde{W}^{0A}-\frac{1}{g_M^2-1}\partial_j \tilde{\pi}(W)^{jA}+\frac{g_M}{g_M^2-1}\partial_j \tilde{\pi}(V)^{jA}\right],
\end{align*}

where we also define $\tilde{\pi}(W)^{jA} =\pi(W)^{jA}$, $\tilde{\pi}(V)^{jA} =\pi(V)^{jA}$ and $\tilde{W}^{jA} =W^{jA}$ where the latter appears in $\tilde{D}^i$. The Jacobian of the change of variables from the fields without tilde to the fields with tilde is $\det \left(D_i\partial^i\right)^{-1}$, which removes one of the determinants in the measure. After performing the integration over $\tilde{W}^{0A}$ using the delta functional and renaming the integration variables to ones without tilde, the argument of the exponential contains

\begin{align}
&{}\left(D_{i} \partial^{i}\right)^{-1} \left[-\frac{1}{g_M^2-1}\partial_{j} \tilde{\pi}(W)^{jA}+\frac{g_M}{g_M^2-1}\partial_{j} \tilde{\pi}(V)^{jA}\right] \nonumber \\  &\times \left[D^{k}\pi(W)^{A}_{k}+g f^{DAC}\pi(V)^D_{k} V^{kC}+\frac{i}{2}g\Hdagg\tau^{A}\pi(\Hdagg)-\frac{i}{2}g\pi(H)\tau^{A} H+\frac{g}{2}\bar{f}\gamma^{0} W^{A}_{0} \tau^{A} f\right].
\end{align}

Comparing this to

\begin{multline}
\delta\left(\Phi^{A}_{1(2)}\right) = \int \measure W_{0}^{A} \exp \Big\{i\int \rd^{d} x\, W_{0}^{A} \Big[D^{i} \pi(W)_{i}^{A}+g f^{DAC}\pi(V)^{D}_{k} V^{kC} \\ +\frac{i}{2}g\Hdagg\tau^{A}\pi(\Hdagg)-\frac{i}{2}g\pi(H)\tau^{A} H+\frac{g}{2}\bar{f}\gamma^{0} W^{A}_{0} \tau^{A} f\Big]\Big\},
\end{multline}

where we have used that some of the integrations over delta functionals have already been performed, we see that introducing this constraint into the exponential we may shift the field $W_{0}^{A}$ to absorb the previously generated term. Up to this point the treatment presented here is textbook procedure as outlined in Ref. <cit.> and produces the generating functional

\begin{align}
Z = \int \measure \mu' \exp{\left\{i\int \rd^d x\, \mathcal{B} \right\}},
\end{align}

\begin{align}
\label{eq: B}
\mathcal{B} ={}& \pi(W)_i^A \partial_0 W^{iA} + \pi(V)_i^A \partial_0 V^{iA} + \pi(H) \partial_0 H  + \pi(\Hdagg) \partial_0 \Hdagg + \Lag{rest} \nonumber \\
&-\frac{1}{2(g_M^2-1)}\pi(W)^{iA}\pi(W)_i^{A}-\frac{1}{2(g_M^2-1)}\pi(V)^{iA}\pi(V)_i^{A}+\frac{g_M}{g_M^2-1}\pi(W)^{iA}\pi(V)_i^{A} \nonumber \\
& - \pi(H)\pi(\Hdagg) + \frac{g_{VH}}{2}V_i^A V^{iA}|H|^2 -\frac{1}{4}W^A_{ij} W^{ijA} -\frac{1}{4}V^A_{ij} V^{ijA} -\frac{g_M}{2}V^A_{ij} W^{ijA} \nonumber \\
&+ \frac{m_V^2}{2} V_i^A V^{iA} + g_H V^{iA} J_i^{HA} + g_f V^{iA} J_i^{fA} + \bar{f} i \slashed{D} f + \left(D_i H\right)^\dagger \left(D^i H\right) \nonumber \\
&+ W^{0A} \left[D^i \pi(W)^A_i + \frac{i}{2}g\Hdagg \tau^A \pi(\Hdagg)-\frac{i}{2}g \pi(H) \tau^A H+gf^{BAC}\pi(V)^B_i V^{iC}\right] \nonumber \\
&+ V^{0A} \left[D^i \pi(V)^A_i + \frac{i}{2}g_H\Hdagg \tau^A \pi(\Hdagg)-\frac{i}{2}g_H \pi(H) \tau^A H+g_f J_0^{fA}+\frac{M_V^2}{2}V^{0A}\right],
\end{align}

\begin{align}
\measure \mu' ={}& \measure \mathcal{Q} \, \measure \mathcal{P'} \delta\left(\partial_i W^{iA}\right) \delta\left(\Phi^A_{2(2)}\right) \det\left(-M_V^2\right) \det\left(D_i \partial^i\right).
\end{align}

In the measure $\mathcal{Q}$ is unchanged and

\begin{align*}
\mathcal{P'} &= \begin{pmatrix}
\pi(W)^A_i & \pi(V)^A_i& \pi(H) & \pi(\Hdagg)
\end{pmatrix}^T.
\end{align*}

We now bring $\Phi^A_{2(2)}$ into the exponential by writing

\begin{align}
\delta\left(\Phi^A_{2(2)}\right) = \int \measure \lambda^A \exp\Big\{i\int \rd^dx \, \lambda^A \Big[&D^i \pi(V)_i^A+\frac{i}{2}g_H\Hdagg\tau^A\pi(\Hdagg)\nonumber \\ &-\frac{i}{2}g_H\pi(H)\tau^A H+g_f J_0^{fA}+M_V^2V_0^A\Big]\Big\},
\end{align}

which we may combine with the last line of Eq. (<ref>) to yield the following contribution to the argument of the exponential

\begin{align}
&\left(\lambda^A+V^{0A}\right) \Big[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+g_f J_0^{fA}\Big] \nonumber \\
&+ M_V^2 \lambda^A V_0^A + \frac{M_V^2}{2} V^{0A}V^{0A}.
\end{align}

Shifting $V^{0A}\rightarrow V^{0A}+\lambda^A$ we can arrange for only this field to couple to the terms in the brackets giving rise to the remaining terms

\begin{align}
-\frac{M_V^2}{2}\lambda^A \lambda^A + \frac{M_V^2}{2} V^{0A} V^{0A},
\end{align}

which leaves us with a term quadratic in the field $\lambda^A$. The fact that this is the only $\lambda^A$-dependent term will become important later. We now integrate over the remaining canonical momenta, starting with the scalar ones. The terms involving the scalar momenta can be arranged as

\begin{align}
&-\left[\pi(H)+\partial_0 \Hdagg+\frac{i}{2}g_H V^{0B}\Hdagg \tau^B +\frac{i}{2}g W^{0B}\Hdagg \tau^B \right] \nonumber \\
&\times\left[\pi(\Hdagg)+\partial_0 H-\frac{i}{2}g_H V^{0A}\tau^A H-\frac{i}{2}g W^{0A}\tau^A H\right] \nonumber \\
&+ \left[\left(D_0 H\right)^\dagger + \frac{i}{2}g_H \Hdagg \tau^A V^{0A}\right]\left[\left(D_0 H\right) - \frac{i}{2}g_HV^{0B} \tau^B H\right],
\end{align}

where we may perform a constant shift of the conjugate momenta to completely decouple them. The Gaussian path integral over these variables then yields an irrelevant constant. This only leaves the second term which, using that $V^{0A}V^{0B} [\tau^A,\tau^B]_{+} = 2 V^{0A} V^{0A} I$, yields

\begin{align}
\left(D_0 H\right)^\dagger \left(D_0 H\right) + g_H V^{0A} J_0^{HA}+\frac{g_H^2}{4}V^{0A}V^{0A}|H|^2.
\end{align}

The first and second term combine with $\left(D_i H\right)^\dagger \left(D^i H\right)$ and $g_H V^{iA} J_i^{HA}$, respectively, to yield $\left(D_\mu H\right)^\dagger \left(D^\mu H\right)$ and $g_H V^{\mu A} J_\mu^{HA}$. The last term cancels the corresponding contribution to $\frac{M_V^2}{2}V^{0A}V^{0A}$ leaving only  $\frac{m_V^2}{2}V^{0A}V^{0A}$ and $\frac{g_{VH}}{2}|H|^2 V^{0A}V^{0A}$, which combine with $\frac{m_V^2}{2}V^{iA}V_i^{A}$ and $\frac{g_{VH}}{2}|H|^2 V^{iA}V_i^{A}$, respectively, to yield the corresponding Lorentz invariant expressions. At this point the scalar and fermionic contributions to the argument of the exponential are of the same form as in the original Lagrangian and there remains an integration over $\pi(W)^{iA}$ and $\pi(V)^{iA}$. To perform this final integration we decouple  $\pi(W)^{iA}$ and $\pi(V)^{iA}$ by changing variables to $F^{iA}$ and $G^{iA}$ defined by

\begin{align}
\begin{pmatrix}
F^{iA} \\
\end{pmatrix}
\frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 1 \\
-1 & 1
\end{pmatrix}
\begin{pmatrix}
\pi(W)^{iA} \\
\pi(V)^{iA}
\end{pmatrix}.
\end{align}

The Jacobian of the transformation is unity and the terms involving $\pi(W)^{iA}$ and $\pi(V)^{iA}$ can be written as

\begin{align}
&\frac{1}{2(g_M+1)}\left[F_i^A+\frac{g_M+1}{\sqrt{2}}\left(W_{0i}^A+V_{0i}^A\right)\right]^2   -\frac{1}{2(g_M-1)}\left[G_i^A-\frac{g_M-1}{\sqrt{2}}\left(V_{0i}^A-W_{0i}^A\right)\right]^2 \nonumber \\
&-\frac{g_M+1}{4} \left(W^{0iA}+V^{0iA}\right)^2+ \frac{g_M-1}{4} \left(V^{0iA}-W^{0iA}\right)^2,
\end{align}

where for the spatial components of any Lorentz vector $v^i$ we write $(v^i)^2 \equiv v_i v^i$. After shifting $F^{iA}$ and $G^{iA}$ the integrals over these variables yield irrelevant constants and the remaining terms add

\begin{align}
\end{align}

to the argument of the exponential. These terms combine with

\begin{align}
\end{align}

to yield the corresponding fully Lorentz invariant terms. The generating functional at this point is of the form

\begin{align}
Z = \int \measure \mathcal{Q}\, \measure \lambda^A \, \delta\left(\partial_i W^{iA}\right) \det\left(-M_V^2\right) \det\left(D_i \partial^i\right) \exp{\left\{i\int \rd^d x \, \left[\LagNoT - \frac{M_V^2}{2} \lambda^A \lambda^A \right]\right\}},
\end{align}

and introducing ghost fields $\bar{v}^A$ and $v^A$ we may bring $M_V^2$ up into the exponential and write

\begin{align}
Z = \int \measure \eta\, \delta\left(\partial_i W^{iA}\right) \det\left(D_i \partial^i\right) \exp{\left\{i\int \rd^dx \, \left[\LagNoT - \frac{M_V^2}{2} \left(\lambda^A \lambda^A + \bar{v}^A v^A\right) \right]\right\}},
\end{align}

where $\eta = \begin{pmatrix} \mathcal{Q} & \lambda^A & \bar{v}^A & v^A \end{pmatrix}^T$. Comparing this functional to Eq. (4.3.30) of Ref. <cit.> we note that it matches the naive expectation for a theory quantised in Coulomb gauge up to terms involving the unphysical degrees of freedom $\lambda^A$, $\bar{v}^A$ and $v^A$. It should be noted that these degrees of freedom can never be produced and hence only appear as internal lines in any Feynman diagram. Furthermore, they only appear quadratically meaning that whenever any of these fields is present as an internal line it has to be part of a closed loop. That is, whenever there is a $\lambda^A$ present internally it has to be part of a closed $\lambda^A$-loop and similarly for $v^A$. Furthermore, none of these fields has a kinetic term, which implies that any closed loop of such fields yields a scaleless integral. Hence, working in , these fields never contribute to any correlation function, and therefore the theory described by the generating functional omitting these fields is equivalent to the one described by the generating functional containing the fields. We may therefore drop these extra degrees of freedom altogether. The generating functional is then of the form that is naively expected for the quantisation of the theory in Coulomb gauge, which in Ref. <cit.> is shown to be equivalent to the generating functional obtained through the Fadeev-Popov procedure quantising the theory in a manifestly Lorentz invariant way. We have thus shown that we may proceed with a naively constructed generating functional, as long as we work in and obey the consistency conditions

\begin{align}
\label{eq: coupling_constr_1}
g_M&=\gamma \frac{g}{g_{*}} \neq \pm 1, \\
\label{eq: coupling_constr_2}
g_{VH}&\geq \frac{g_H^2}{2} = \frac{\gamma_H^2 g_{*}^2}{2}.
\end{align}

§ COMPUTATION OF UOLEA OPERATORS

In this section we briefly discuss the computation of the necessary UOLEA operators. The situation is very similar to the one encountered in Section <ref>, but instead of $\epsilon$-scalars we want to integrate out the heavy resonances. It is clear that the appropriate second variation for this model can be directly obtained from Eq. (<ref>) by simply replacing $\epsilon_\mu$ by $V_\mu$ and forgetting about the fact that certain matrices were projected onto $Q\epsilon S$. It should be kept in mind that the term quadratic in fluctuations of the resonance fields is not of the form given in Eq. (<ref>), but rather takes the form

\begin{align}
\Delta_V^{\mu \nu} =  P^2 g^{\mu \nu}-P^\nu P^\mu  - m^2_V g^{\mu \nu} + X_{VV} ^ {\mu \nu},
\label{eq:resonance_prop}
\end{align}

where, as has been mentioned before, there is no $R_\xi$ gauge-fixing contribution. Due to this fact, the quadratic term of the resonances cannot be brought into the scalar form by a choice of gauge and it is sensible to not treat the Lorentz indices of resonances as internal indices as was done in Section <ref> for gauge bosons.[A possible way of still treating these indices as internal indices is to introduce a projector onto the subspace of resonances for the terms that deviate from the scalar form. This, however, turns out to be very tedious and the treatment presented here is more straightforward.] Furthermore, there are two slightly different ways of including light scalars and gauge bosons into the computation. The choice concerns the contributions with open covariant derivatives. First consider a scalar coupling to resonances. The only way to obtain an open covariant derivative is through a coupling of the resonance to the scalar current as

\begin{align*}
i V^\mu \phi D_\mu \phi,
\end{align*}

where we omitted gauge indices as they do not affect the argument. This gives rise to the open covariant derivative term

\begin{align*}
P_\mu Z_{\phi V}^{\mu \rho},
\end{align*}

\begin{align*}
Z_{\phi V}^{\mu \rho} = -\phi g^{\mu \rho},
\end{align*}

so that effectively we may write

\begin{align*}
P_\mu Z_{\phi V}^{\mu \rho} = -P^{\rho} \phi = P^{\rho} \tilde{Z}_{\phi V}.
\end{align*}

In the case of the gauge bosons the corresponding couplings come from the gauge-kinetic term of the resonances and one finds contributions to $\frac{\delta^2 \mathcal{L}}{\delta W_{\rho} \delta V_{\sigma}}$ which mix Lorentz indices of gauge bosons and resonances such as

\begin{align*}
-D_\mu g^{\rho \sigma} V^\mu = iP_\mu g^{\rho \sigma} V^\mu.
\end{align*}

It is clear that, in order to capture such contributions in $Z_{\phi V}$, in general one has to attribute three Lorentz indices to the object. Alternatively, one can separate $Z_{W V}$ from $Z_{\phi V}$. In the computation performed here we choose the second path for the following reasons. First of all, the mass dimension of $Z_{\phi V}$ is one and hence this object can appear up to six times in UOLEA operators. If each copy carries three Lorentz indices a vast number of possible contractions is created, which in the end turn out to be equivalent as they are simply contractions of metric tensors. If instead $Z_{\phi V}$ does not carry any Lorentz indices, this issue is avoided. Secondly, the mass dimension of $Z_{W V}$ is three. This is due to the fact that $Z^{\mu \rho \sigma}_{W V} \propto V^{\mu A}$ and, since $V^{\mu A}$ is a heavy field it will be replaced by the corresponding classical field. As discussed in Chapter <ref> this is the inverse mass expansion of the solution of the classical equations of motion, whose leading contribution for the model under consideration comes from the fermionic and scalar currents, which both are of mass dimension three. Hence, separating $Z_{\phi V}$ from $Z^{\mu \rho \sigma}_{W V}$ has the benefit of avoiding a large number of Lorentz contractions, which turns out to simplify the computation significantly. In practice one may perform the derivation of the functional trace keeping the Lorentz indices of the gauge bosons as internal indices and then perform the split of $Z_{\phi V}$ into a scalar-resonance part and a gauge-boson-resonance part at the level of the expansion of the functional trace. This is how the computation has been performed. Whereas this will affect the explicit computation of the functional trace, it is entirely inconsequential for the derivation of $\Gamma^{1 \Loop}_\text{L,UV}$, which means that all of the steps leading to Eq. (<ref>) go through without modification. Also, the soft-region cancellation is clearly covered by the proof of Section <ref>. It then follows that the one-loop contribution to the effective Lagrangian is given by

\begin{align}
\Lag{EFT}^{1\Loop} = \frac{i}{2} \left. \Tr \log \left[\left(M_{V V}\right)^{\mu \nu}_\xi - \left(M_{V \phi}\right)^\mu_\xi \left(M_{\phi \phi}\right)_\xi^{-1}  \left(M_{\phi V}\right)^\nu_\xi \right]\right|^{P_\mu \to P_\mu-q_\mu}_\text{hard,trunc},
\label{eq: resonance_EFT_lag}
\end{align}

with the usual notation, see in particular Eq. (<ref>) . Here we are of course interested in all contributions up to and including mass dimension six and we may not set $X_{\phi V}^\mu$ and $X_{V \phi}^\mu$ to zero as was possible for the corresponding matrices in Section <ref>. Writing the argument of the logarithm as

\begin{align}
(P^2 +q^2 -2q\cdot P - m^2_V) g^{\mu \nu} - P^\nu P^\mu - q^\nu P^\mu - P^\nu q^\mu - q^\mu q^\nu   + F_{VV} ^ {\mu \nu},
\end{align}

\begin{align}
F_{VV} ^ {\mu \nu} = \left. \left( X_{VV}^{\mu \nu}-\mathbf{U}_{V \xi}^\mu \mathbf{\Delta}_\xi^{-1} \mathbf{U}_{\xi V}^\nu - \left(M_{V \phi}\right)^\mu_\xi \left(M_{\phi \phi}\right)_\xi^{-1}  \left(M_{\phi V}\right)^\nu_\xi \right) \right|^{P_\mu \to P_\mu - q_\mu},
\end{align}

one can expand the logarithm as

\begin{align}
&\log \left[(P^2 +q^2 -2q\cdot P - m^2_V) g^{\mu \nu} - P^\nu P^\mu - q^\nu P^\mu - P^\nu q^\mu - q^\mu q^\nu   + F_{VV} ^ {\mu \nu}\right] \nonumber \\
&= \log \left[-g^{\mu \sigma} \left(-q^2+m_V^2\right)-q^\mu q^\sigma\right] \nonumber \\
&~~~+ \log \left[\delta^\nu_\sigma - (q^2-m_V^2)^{-1}T_{\lambda \sigma}\left(g^{\nu \lambda} \left(2q \cdot P-P^2\right)+P^\nu P^\lambda - q^{(\nu} P^{\lambda)} - F^{\lambda \nu}\right)\right] \nonumber \\
&= \log \left[-g^{\mu \sigma} \left(-q^2+m_V^2\right)-q^\mu q^\sigma\right] \nonumber \\
&~~~-\sum_{n=1}^{\infty}\frac{1}{n} \left[(q^2-m_V^2)^{-1}T_{\lambda \sigma}\left(g^{\nu \lambda} \left(2q \cdot P-P^2\right)+P^\nu P^\lambda - q^{(\nu} P^{\lambda)} - F^{\lambda \nu}\right)\right]^n,
\end{align}

\begin{align}
T_{\lambda \sigma} = g_{\lambda \sigma} - q_\sigma q_\lambda m_V^{-2}, \\
\intertext{and} q^{(\nu}P^{\lambda)} = q^\nu P^\lambda + q^\lambda P^\nu.
\end{align}

Inserting this into Eq. (<ref>) and ignoring the infinite constant as usual, we find the following expression for the effective Lagrangian at one loop

\begin{align}
\Lag{EFT}^{1\Loop} = -\frac{i}{2} \sum_{n=1}^{\infty} \frac{1}{n} \Tr \left[(q^2-m_V^2)^{-1}T_{\lambda \sigma}\left(g^{\nu \lambda} \left(2q \cdot P-P^2\right)+P^\nu P^\lambda - q^{(\nu} P^{\lambda)} - F^{\lambda \nu}\right)\right]^n,
\end{align}

which has to be evaluated in the hard region and truncated at the desired mass dimension. The differences between this trace and the one computed in Section <ref> are the extra $P^\mu$-dependent terms and the modified propagator. Both modifications introduce extra dependencies on the loop momentum, thus modifying the universal coefficients as compared to the previous computation. Furthermore, the presence of the extra terms leads to further UOLEA operators which were not present before. Whereas the evaluation of this trace is computationally more intensive as there are more possible Lorentz structures in the final result, the computation can be performed using the same Mathematica code that was used to perform the calculation of Section <ref>. The evaluation was executed as a part of this thesis and the resulting operators were implemented into . The result of the matching based on this implementation is presented in the next section.

§ MATCHING RESULTS

It is most convenient to remove the kinetic mixing term in order to perform the matching utilising the pre-computed UOLEA operators. This can be achieved by starting from the Lagrangian

\begin{align}
\label{eq: og_lag}
\LagNoT ={}& -\frac{1}{4}\tilde{V}^{\mu \nu A} \tilde{V}_{\mu \nu}^A - \frac{\tilde{g}_M}{2}\tilde{V}^{\mu \nu A} \tilde{W}_{\mu \nu}^A  +  \frac{\tilde{m}_V^2}{2}\tilde{V}^{\mu A}\tilde{V}_{\mu}^A+\tilde{g}_{H} \tilde{V}^{\mu A} J_\mu^{HA}\nonumber \\ & +\tilde{V}^{\mu A} \sum_{f} \tilde{g}_{f}  J_{\mu}^{fA} + \frac{\tilde{g}_{VH}}{2}|H|^2\tilde{V}^{\mu A}\tilde{V}_{\mu}^A + \Lag{SM},
\end{align}

and performing the following field redefinitions, see Refs. <cit.>

\begin{align}
\begin{cases}
\tilde{W}^{\mu A} =  W^{\mu A} -\frac{\tilde{g}_M}{\sqrt{1-\tilde{g}_M^2}} V^{\mu A} \\
\tilde{V}^{\mu A} = \frac{1}{\sqrt{1-\tilde{g}_M^2}} V^{\mu A}
\end{cases},
\label{eq: field_redefs}
\end{align}

which removes the kinetic mixing and brings the Lagrangian into the form

\begin{align}
\label{eq: redef_lag}
\LagNoT ={}& -\frac{1}{4}V^{\mu \nu A} V_{\mu \nu}^A  +  \frac{m_V^2}{2}V^{\mu A}V_{\mu}^A+g_{H} V^{\mu A} J_\mu^{HA}\nonumber \\ & + V^{\mu A} \sum_{f} g_{f} J_{\mu}^{fA}+ \frac{g_{VH}}{2}|H|^2V^{\mu A}V_{\mu}^A + \frac{g_{3V}}{2} f^{ABC} V^{\mu A}  V^{\nu B}  V_{\mu \nu}^C \nonumber \\
&- \frac{g_{2VW}}{2} f^{ABC} V^{\mu B}  V^{\nu C}  W_{\mu \nu}^A+ \Lag{SM}.
\end{align}

The new couplings expressed through the old ones are then given by[Note that the gauge coupling appears in these relations since the redefinition of the gauge field affects the gauge-kinetic terms.]

\begin{align}
\label{eq: coupling_rel_1}
m_V^2 &= \frac{1}{1-\tilde{g}_M^2}\tilde{m}_V^2, \\
g_H  &=\frac{1}{\sqrt{1-\tilde{g}_M^2}}(\tilde{g}_H-g_2\tilde{g}_M), \\
g_f  &=\frac{1}{\sqrt{1-\tilde{g}_M^2}}(\tilde{g}_f-g_2 \tilde{g}_M), \\
g_{VH}  &=\frac{1}{1-\tilde{g}_M^2}\left(\tilde{g}_{VH}+g_2^2 \tilde{g}_M^2-g_2 \tilde{g}_H \tilde{g}_M\right), \\
g_{2VW} &= -\frac{1}{1-\tilde{g}_M^2}\tilde{g}_M^2g_2, \\
g_{3V}  &=\frac{\tilde{g}_M g_2}{(1-\tilde{g}_M^2)^{3/2}}\left(2+\tilde{g}_M^2-3\tilde{g}_M\right)
\label{eq: coupling_rel_-1}
\end{align}

and we express the result of the matching through these couplings. Notice that we have assumed a universal coupling to fermions $g_f = g_q = g_l$ for the matching below. If the coupling relations Eqs. (<ref>)-(<ref>) are not enforced, the result presented below amounts to a general matching of the simplified model proposed in Ref. <cit.>. However, one has to be careful when interpreting this result since there is no guarantee that the model is self-consistent. There might very well be constraints on certain couplings as derived in Section <ref> for the more restricted case. We leave the constraint analysis of the more general model for future work. The interpretation in terms of the Lagrangian in Eq. (<ref>) is unproblematic as long as the conditions represented by Eqs. (<ref>) and (<ref>) are taken into account. In what follows the SM conventions are the same as in Section <ref> with the exception that we drop the subscript $h$ on the quartic Higgs coupling.
The tree-level matching gives rise to the following effective Lagrangian

\begin{align}
\Lag{EFT}^{\text{tree}} ={}& \Lag{SM} + \frac{g_{H}^2 \mu_h^2 |H|^4}{2 m_V^2}
-\frac{g_{H}^2 \lambda Q_H}{2 m_V^2}
-\frac{3g_{H}^2Q_{H\Box}}{8 m_V^2} \nonumber \\
&-\frac{g_{H}^2 Q^{ij}_{eH} y_{eij}}{4 m_V^2}
-\frac{g_{H}^2 Q^{ij}_{dH} y_{dij}}{4 m_V^2}
-\frac{g_{H}^2 Q^{ij}_{uH} y_{uij}}{4 m_V^2} \nonumber \\
&-\frac{g_{H}^2Q^{\dagger ij}_{eH} y^\dagger_{eij}}{4 m_V^2}
-\frac{g_{H}^2 Q^{\dagger ij}_{dH} y^\dagger_{dij}}{4 m_V^2}
-\frac{g_{H}^2 Q^{\dagger ij}_{uH} y^\dagger_{uij}}{4 m_V^2} \nonumber \\
&-\frac{g_{f}g_{H} Q^{(3)ii}_{Hl}}{4 m_V^2}
-\frac{g_{f} g_{H} Q^{(3) ii}_{Hq}}{4 m_V^2}
+\frac{g_{f}^2 (Q^{{iijj}}_{ll} -
2 Q^{{ij}ji}_{ll})}{8 m_V^2}
-\frac{g_{f}^2 Q^{(3) iijj}_{qq}}{8 m_V^2}-\frac{g_f^2 Q_{lq}^{(3) iijj}}{4 m_V^2},
\end{align}

where repeated flavour indices are summed over and we use the naming conventions of Ref. <cit.> with the modification that the Higgs doublet is denoted by $H$ rather than $\varphi$. We see that, at tree level, ten independent operators are induced and there is a shift in $\lambda$ which is suppressed by $\mu_h^2/m_V^2$. Due to this suppression this shift is inconsequential for the application of equations of motion in the translation to Warsaw basis. This tree-level matching was already performed in Ref. <cit.>, where the four fermion operators were omitted. The result presented here agrees with that of Ref. <cit.> for the operators present in both results.

Next we consider the renormalisable part of the effective action induced by the one-loop matching. Since corrections to the kinetic terms give rise to one-loop contributions to Wilson coefficients of effective operators induced at tree level once the fields of the EFT are canonically normalised, the gauge-kinetic terms are particularly interesting. These read

\begin{align}
\frac{1}{\kappa}\LagNoT_{\text{EFT,kin.}}^{1\Loop}  ={}& \frac{g_{H}^2(-60 m_V^2 - \mu_h^2 + 6(12 m_V^2 + \mu_h^2) \reslog)}
{32 m_V^2} \left(D_\mu H\right)^\dagger \left(D^\mu H\right) \nonumber \\
&-\frac{1}{4} \left(-4 (g_2 - 2g_{2VW}) g_{2VW} + 3 g_2 (g_2 + 4 g_{2VW})
\reslog\right) W^{A\mu \nu} W_{\mu \nu} ^A \nonumber \\
& -\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.

Conclusions and outlook

CHAPTER: CONCLUSIONS AND OUTLOOK

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>.

In this chapter we present several applications of the BSUOLEA. In Section <ref>, we start with a very simple example in which the top quark is integrated out from the SM in order to illustrate how the BSUOLEA can be applied. This application also serves as a check for the treatment of Dirac fermions in the BSUOLEA. Next, in Section <ref>, we reproduce the threshold correction to the quartic Higgs coupling when matching the Minimal Supersymmetric Standard Model[Familiarity with the MSSM is not required to follow the applications presented in this chapter. For the reader interested in learning about supersymmetry we recommend Ref. <cit.>.] (MSSM) to the SM, which serves as a check for purely fermionic operators, including those with Majorana fermions, as well as operators that translate between DRED and DREG. Afterwards, in Section <ref>, we consider an application in which the gluino and the stops are integrated out from the MSSM. In this matching, contributions from operators with mixed statistics arise. Since we again reproduce known results this serves as a check of these operators. We then modify the scenario slightly, in Section <ref>, keeping the stops and integrating out only the gluino. In this application operators with one Majorana fermion, namely the gluino, and one Dirac fermion, the top quark, contribute to the known mass corrections of the stops. As these corrections are reproduced correctly this indicates that the treatment of such operators is also correct. In the process previously unknown Wilson coefficients are computed. These four applications were already presented in Ref. <cit.> as a part of this thesis and we closely follow that reference here. The scalar contributions to the threshold correction of the quartic Higgs coupling were computed by Dr. Alexander Voigt. For all of these applications the effective Lagrangian was computed by hand. For our final application, in Section <ref>, the matching of the SSM to the SMEFT, this is not possible anymore since 500 BSUOLEA operators contribute. Therefore the code was used to perform this matching. Given that the result is also known from previous computations, this serves as a check of the code itself.

As it is unavoidable to handle many different kinds of indices when applying the BSUOLEA we introduce the index conventions used throughout this chapter in Table <ref>. Note that all spinor indices carry dots to distinguish these from Lorentz indices. We do not use dotted and undotted spinor indices to distinguish left- and right-handed spinors. We also suppress indices whenever they do not add to the clarity of presentation. Finally, lower case letters from the middle of the alphabet, $i,\, j, \ldots$ are used as both flavour indices and multi-indices. The latter only appear in expressions that show the BSUOLEA operators which contribute to a certain computation. Therefore, this should not cause any confusion. Repeated indices are always summed over.

Index type     Example indices
Flavour     $i$, $j$, $k$, $l$
$SU(2)_L$ fundamental     $a$, $b$, $c$, $d$
$SU(2)_L$ adjoint     $A$, $B$, $C$, $D$
$SU(3)_c$ fundamental     $I$, $J$, $K$, $L$
$SU(3)_c$ adjoint     $\dot{A}$, $\dot{B}$, $\dot{C}$, $\dot{D}$
Lorentz     $\mu$, $\nu$, $\rho$, $\sigma$
Spinor     $\dot{\alpha}$, $\dot{\beta}$, $\dot{\rho}$, $\dot{\sigma}$

Index naming conventions used throughout this chapter.

§ INTEGRATING OUT THE TOP QUARK FROM THE SM

We consider the corrections to the Higgs
tadpole and mass parameter that arise when integrating out the top
quark from the SM. This example is purposely chosen to be very simple in order to illustrate the use of the BSUOLEA. We restrict the interaction Lagrangian
to only one coupling

\begin{align}
\Lag{SM} \supset -\frac{g_t}{\sqrt{2}}h \bar{t}t,
\end{align}

where $h$ denotes the physical Higgs field, $t$ is the top quark and
$g_t$ is the top Yukawa coupling.  The relevant operators of the BSUOLEA of Eq.(<ref>) are given by

\begin{align}
\frac{1}{\kappa} \Lag{EFT}^{1\Loop} = \tr \Bigg\lbrace & \frac{1}{4} M_{\Xi i} M_{\Xi j}^3 \ZI ^{13} _{ij} [P_\mu,(\mathbf{U}_{\Xi \Xi})_{ij}][P^\mu,(\mathbf{U}_{\Xi \Xi})_{ji}]
\nonumber \\ &  -\frac{1}{2} \ZI[q^4] ^{22} _{ij} \gamma^\nu [P_\mu,(\mathbf{U}_{\Xi \Xi})_{ij}]\gamma_\nu[P^\mu,(\mathbf{U}_{\Xi \Xi})_{ji}]
\nonumber \\ &  - \ZI[q^4] ^{22} _{ij} \gamma^\nu [P_\nu,(\mathbf{U}_{\Xi \Xi})_{ij}]\gamma_\mu[P^\mu,(\mathbf{U}_{\Xi \Xi})_{ji}]
\nonumber \\ & +\frac{1}{2} M_{\Xi i} \ZI ^1 _i (\mathbf{U}_{\Xi \Xi})_{ii}
\nonumber \\ & -\frac{1}{4} M_{\Xi i} M_{\Xi j} \ZI ^{11} _{ij} (\mathbf{U}_{\Xi \Xi})_{ij} (\mathbf{U}_{\Xi \Xi})_{ji}
\nonumber \\ & -\frac{1}{4} \ZI[q^2] ^{11} _{ij} \gamma ^\mu (\mathbf{U}_{\Xi \Xi})_{ij} \gamma_\mu (\mathbf{U}_{\Xi \Xi})_{ji}
\Bigg\rbrace
\label{eq:UOLEALAG-topout},
\end{align}

where $M_{\Xi i}$ denotes the $i$th component of $M_\Xi = \begin{pmatrix}
m_t & m_t
\end{pmatrix}^T$ with $m_t$ being the mass of the top quark and $\Xi = \begin{pmatrix}
t_{\dot{\alpha} I} & \ccfield{t}_{\dot{\alpha} I}
\end{pmatrix}^T$. Note that $M_{\Xi1} = M_{\Xi2}$ and thus we are in the case of a degenerate mass as mentioned in Section <ref>. The matrix $\mathbf{U}_{\Xi \Xi}$ is given by

\begin{align}
(\mathbf{U}_{\Xi \Xi})_{\dot{\alpha} \dot{\beta} IJ} = \begin{pmatrix}
(U_{\bar{t}t})_{\dot{\alpha} \dot{\beta} IJ} & 0 \\
0 & \cc^{-1} _{\dot{\alpha} \dot{\rho}} (U_{t\bar{t}})_{\dot{\rho} \dot{\sigma} IJ} \cc^{-1} _{\dot{\sigma} \dot{\beta}}
\end{pmatrix}
-\frac{g_t}{\sqrt{2}}h \delta_{\dot{\alpha} \dot{\beta}} \delta_{IJ} \mathbf{1}_{2},
\label{eq:top-derivative}
\end{align}

where $\mathbf{1}_{2}$ is the $2\times 2$ identity matrix. In Eq. (<ref>) we
included terms with two covariant derivatives in order to obtain the
field-redefinition of the Higgs field that is necessary to canonically
normalise the corresponding Higgs field, $\hat{h}$, in the effective
theory.  Since this redefinition arises from the correction to the
kinetic term only, we can set $P^\mu = i\partial ^\mu$. In order to illustrate the handling of spinor indices, the appearance of derivatives and mass-degenerate coefficients we compute the contribution of the operator in the third line of Eq. (<ref>) in detail. Since the masses are degenerate we have that

\begin{align}
\ZI[q^4] ^{22} _{11} =  \ZI[q^4] ^{22} _{12} =  \ZI[q^4] ^{22} _{21} = \ZI[q^4] ^{22} _{22} =  \ZI[q^4] ^{4} _{t},
\label{eq: baby_example_coeff}
\end{align}

meaning that the integral coefficient is independent of the indices. For the operator structure we find

\begin{align}
-\tr \lbrace \gamma^\nu [P_\nu,(\mathbf{U}_{\Xi \Xi})_{ij}]\gamma_\mu[P^\mu,(\mathbf{U}_{\Xi \Xi})_{ji}]\rbrace &= \frac{g_t^2}{2}\tr \lbrace \gamma^\nu_{\dot{\alpha} \dot{\beta}} \partial_\nu \left(h \delta_{\dot{\beta} \dot{\rho}} \delta_{IJ} \mathbf{1}_{2} \right) \gamma^\mu_{\dot{\rho} \dot{\sigma}} \partial_\mu \left(h \delta_{\dot{\sigma} \dot{\alpha}} \delta_{JI} \mathbf{1}_{2} \right) \rbrace \nonumber \\
&= 12g_t^2 \partial_\mu h \partial^\mu h,
\end{align}

which together with Eq. (<ref>) yields the contribution

\begin{align}
12 g_t^2 \ZI[q^4] ^{4} _{t} (\partial_\mu h) (\partial^\mu h) = -\frac{1}{2}g_t^2 \log\frac{m_t^2}{\mu^2} \partial_\mu h \partial^\mu h,
\end{align}

where $\mu$ is the matching scale and we used

\begin{align}
\ZI[q^4]^4_t = \frac{1}{12\eps}-\frac{1}{24}\log\frac{m_t^2}{\mu^2}.
\end{align}

We are here allowed to drop the $\eps$-dependence of the integral since no $\eps$-dependence arises from the operator structure. This is not the case for the second line of Eq. (<ref>), where the contraction of the gamma matrices produces a $d=4-\eps$. Proceeding in a similar fashion for the remaining contributions to Eq. (<ref>) we find

\begin{align}
\frac{1}{\kappa}\Lag{EFT} ^{1\Loop}  ={}& -3g_t^2 \left(m_t^4 \ZI ^4 _t-2d\ZI [q^4]^4 _{t}-4 \ZI [q^4] ^4 _{t} \right) (\partial_\mu h) (\partial^\mu h) \nonumber \\
& -3g_t^2 \left(\ZI ^2_t m_t^2+d \ZI [q^2]^2_t \right)h^2-\frac{12}{\sqrt{2}} g_t m_t \ZI^1_t h.
\label{eq: HiggsEFT}
\end{align}

Introducing the canonically normalised field $\hat{h}$ which is related to $h$

\begin{align}
\hat{h}=\left(1+\frac{1}{2}\delta Z_h \right) h,
\end{align}

one can read off $\delta Z_h$ to be

\begin{align}
\delta Z_h = -6g_t^2\left(m_t^4 \ZI ^4 _t-2d\ZI [q^4]^4 _{t}-4 \ZI [q^4] ^4 _{t}\right)
=  -6g_t^2\left(m_t^4 \ZI ^4_t - 12 \ZI [q^4] ^4_{t} + \frac{1}{6}\right).
\label{eq:delta_Zh_top}
\end{align}

The loop functions that appear in Eqs. (<ref>) and
(<ref>) can be calculated using Eqs. (<ref>)-(<ref>) and are given by

\begin{align}
\ZI^1_t &= 2 \ZI[q^2]^2_t = m_t^2 \left(\frac{2}{\eps} + 1 - \log\frac{m_t^2}{\mu^2}\right), \\
\ZI^2_t &= 24 \ZI[q^4]^4_t = \frac{2}{\eps}-\log\frac{m_t^2}{\mu^2}, \\
\ZI^4_t &= \frac{1}{6 m_t^4}.
\end{align}

§ MSSM THRESHOLD CORRECTION TO THE QUARTIC HIGGS COUPLING

In this section we reproduce the one-loop threshold
correction of the quartic Higgs coupling $\lambda$ when matching the MSSM to the
SM at the one-loop level in the unbroken phase. This result was first reported in Ref. <cit.> and as
discussed in that reference there are several contributions
of distinct origins. The scalar contribution
$\Delta \lambda^{1\Loop,\phi}$ arises from interactions of the SM-like
Higgs with heavy Higgs bosons, squarks and sleptons,  and the relevant
interaction Lagrangian is given by

\begin{align}
\LagNoT_{\phi} ={}&  - \frac{g_t^2}{2} h^2 (\st{L}^* \st{L} + \st{R}^*\st{R})-\frac{1}{\sqrt{2}} g_t X_t h (\st{L}^* \st{R} + \st{L}\st{R}^*) \nonumber \\ & -\frac{1}{8} c_{2\beta} h^2 \left[\left(g_2^2 - \frac{g_1^2}{5}\right)  \su^* _{Li} \su _{Li}+ \frac{4}{5} g_1^2 \su_{Ri}^* \su_{Ri}- \left(g_2^2 + \frac{g_1^2}{5}\right) \sd_{Li}^* \sd_{Li}- \frac{2}{5} g_1^2 \sd_{Ri}^* \sd_{Ri}\right]
\nonumber \\ &-\frac{1}{8} c_{2\beta} h^2 \left[\left(g_2^2 + \frac{3}{5}g_1^2\right)  \sneu^* _{Li} \sneu _{Li}- \left(g_2^2 - \frac{3}{5}g_1^2\right) \sel_{Li}^* \sel_{Li}- \frac{6}{5} g_1^2 \sel_{Ri}^* \sel_{Ri}\right]
\nonumber \\ & +\frac{1}{16} c_{2\beta}^2 \left(\frac{3}{5} g_1^2 + g_2^2\right) h^2 A^2- \frac{1}{8} \left((1 + s_{2\beta}^2) g_2^2 - \frac{3}{5} g_1^2 c_{2\beta}^2\right) h^2 H^{-} H^{+} \nonumber \\ & - \frac{1}{16} \left(\frac{3}{5} g_1^2 + g_2^2\right) (3 s_{2\beta}^2 - 1) h^2 H^2- \frac{1}{8} \left(\frac{3}{5} g_1^2 + g_2^2\right) s_{2\beta} c_{2\beta} h^3 H \nonumber \\ & + \frac{1}{8} \left(\frac{3}{5} g_1^2 + g_2^2\right) s_{2\beta} c_{2\beta} h^2 (G^{-} H^{+} + H^{-} G^{+})+ \frac{1}{8} \left(\frac{3}{5} g_1^2 + g_2^2\right) s_{2\beta} c_{2\beta} h^2 G^0 A.
\end{align}

Here the three generations
of left- and right-handed squarks are denoted as
$\su_{Li}$, $\su_{Ri}$, $\sd_{Li}$, $\sd_{Ri}$, where
$\st{L} \equiv \su_{L3}$ and $\st{R} \equiv \su_{R3}$ are the left-
and right-handed stops. Correspondingly, the three generations of left- and right-handed sleptons are denoted as $\sel_{Li}$,
$\sel_{Ri}$, $\sneu_{Li}$. The couplings $g_1$ and $g_2$ are the GUT-normalised electroweak gauge
couplings, $X_t$ is the stop mixing parameter, and $g_t = y_t s_\beta$
with $y_t$ being the MSSM top Yukawa coupling and $s_\beta=\sin (\beta)$. The angle $\beta$ should not be
regarded as a ratio of vacuum expectation values, but as the
fine-tuned mixing angle which rotates the two MSSM Higgs doublets,
$H_u$ and $H_d$, into two new doublets, $\mathcal{H}$ and $\mathcal{A}$, with

\begin{align}
\label{eq:rot_H}
\mathcal{H} &= - c_\beta \epstensor H^{*}_d + s_\beta H_u,\\
\label{eq:rot_A}
\mathcal{A} &= s_\beta \epstensor H^{*}_d + c_\beta H_u,
\end{align}

since we are working in the unbroken phase. See Ref. <cit.> for details. In Eqs. (<ref>) and (<ref>) $\epstensor$ is the antisymmetric tensor with $\epstensor_{12}=1$  and $c_\beta = \cos(\beta)$, $s_{2\beta} = \sin(2\beta)$ and $c_{2\beta} = \cos(2\beta)$. The SM-like Higgs doublet $\mathcal{H}$ gives rise to the SM-like physical Higgs field $h=\sqrt{2}\, \Re (\mathcal{H}^0)$, where $\mathcal{H}^0$ is the neutral component of $\mathcal{H}$ and $\Re (\mathcal{H}^0)$ its real part. The fields $G^0$ and $G^{\pm}$ are Goldstone bosons arising from the same Higgs doublet. The heavy Higgs
bosons of the MSSM, $H$, $A$ and $H^\pm$, originate from the heavy doublet

The fermionic contribution $\Delta \lambda ^{1\Loop,\chi}$ to the
threshold correction of $\lambda$ originates from interactions of the
Higgs boson with charginos $\tilde{\chi}^{+}_i$ ($i=1,2$) and
neutralinos $\tilde{\chi}^0_i$ ($i=1,\ldots,4$) described by the
interaction Lagrangian

\begin{align}
\LagNoT_\chi ={}& - \frac{g_2}{\sqrt{2}} h c_\beta (\overline{\tilde{\chi}^{+}_1} P_R \tilde{\chi}^{+} _2  + \overline{\tilde{\chi}^{+}_2}  P_L \tilde{\chi}^{+} _1)- \frac{g_2}{\sqrt{2}} h s_\beta (\overline{\tilde{\chi}^{+}_2} P_R \tilde{\chi}^{+} _1 + \overline{\tilde{\chi}^{+}_1} P_L \tilde{\chi}^{+}_2   )\nonumber
\\ &  +i  \frac{g_Y}{2\sqrt{2}} (c_\beta - s_\beta) h \overline{\tilde{\chi}^0_1} \gamma^5  \tilde{\chi}^0_3-\frac{g_Y}{2\sqrt{2}} (c_\beta + s_\beta)h \overline{\tilde{\chi}^0_1} \tilde{\chi}^0_4 \nonumber
\\ & -i \frac{g_2}{2\sqrt{2}} (c_\beta - s_\beta) h \overline{\tilde{\chi}^0_2} \gamma^5  \tilde{\chi}^0_3+  \frac{g_2}{2\sqrt{2}}  (c_\beta + s_\beta) h \overline{\tilde{\chi}^0_2} \tilde{\chi}^0_4 ,
\end{align}

where $\overline{\tilde{\chi}^0_i} = (\tilde{\chi}^0_i)^T \cc$ and
$g_Y = \sqrt{3/5}\, g_1$.

To calculate the one-loop threshold correction for $\lambda$, the
following contributions with purely scalar and purely fermionic
operators of the BSUOLEA are relevant,