\section{Introduction}

Despite its enormous success in accounting for almost all experimental
data to date, the Standard Model (SM) of particle physics still has no
explanation for a number of other key observations, such as neutrino
masses, the baryon asymmetry of the universe, dark matter, etc. For
this reason, it is widely believed that there must exist physics
beyond the SM. And since the LHC has not discovered any new particles
up to a scale of $O({\rm TeV})$, this new physics (NP) is likely to be
very massive.

When the NP is integrated out, one obtains an effective field theory
(EFT), of which it is now generally believed that the SM is simply the
leading part. This EFT must obey the SM gauge symmetry $SU(3)_C \times
SU(2)_L \times U(1)_Y$. Since the discovery of the Higgs boson, the
default assumption is that this symmetry is realized linearly, i.e.,
the symmetry is broken via the Higgs mechanism, resulting in the
Standard Model EFT, or SMEFT (see, {\it e.g.},
Refs.~\cite{Buchmuller:1985jz, Brivio:2017vri}). The SMEFT has been
studied extensively: a complete and non-redundant list of dimension-6
operators is given in Ref.~\cite{Grzadkowski:2010es}, the dimension-7 operators can be found in Ref.~\cite{Lehman}, and the dimension-8 operators are tabulated in Refs.~\cite{Li:2020gnx,
  Murphy:2020rsh}.

The LEFT (low-energy effective field theory) describes the physics
below the $W$ mass, and is produced when the heavy SM particles ($W$,
$Z$, $t$, $H$) are also integrated out. (This is also called the WET
(weak effective field theory).) In Ref.~\cite{Jenkins:2017jig},
Jenkins, Manohar and Stoffer (JMS) present a complete and
non-redundant basis of LEFT operators up to dimension 6, including
those that violate $B$ and $L$. The matching to dimension-6 SMEFT
operators at tree level is also given. The one-loop contributions of
dimension-6 SMEFT operators can be taken into account through the
renormalization-group running of the coefficients of the LEFT. This is
computed in Refs.~\cite{Alonso:2013hga, Aebischer:2017gaw, Jenkins:2017dyc,
  Dekens:2019ept}. With this information, if a discrepancy with the SM
is observed in a process that uses a particular LEFT operator, we will
know which dimension-6 SMEFT operators are involved.

However, this is not always sufficient. Information about the
contributions from higher-dimension operators may be important if the
process in question is suppressed in the SM and/or is very precisely
measured. Examples of observables for which dimension-8 SMEFT effects must be taken into account include electroweak precision data from
LEP \cite{Corbett:2021eux}, lepton-flavour-violating processes
\cite{Ardu:2021koz, Ardu:2022pzk}, meson-antimeson mixing ($\Delta F = 2$)
\cite{Bobeth:2017xry, Silva:2022tln}, and electric dipole moments
\cite{Panico:2018hal}. (Dimension-8 SMEFT operators have also been
discussed in the context of high-energy processes, see
Refs.~\cite{Chala:2018ari, Hays:2018zze, Ellis:2019zex,
  Alioli:2020kez, Hays:2020scx, Ellis:2020ljj, Boughezal:2021tih,
  Gu:2020ldn}.)

The analysis of Refs.~\cite{Ardu:2021koz, Ardu:2022pzk} involves (i) the
identification of all LEFT operators relevant for lepton flavour
violation, and (ii) the computation of the tree-level matching of
these operators to SMEFT operators up to dimension 8. In the present
paper, we apply step (ii) to all LEFT operators. The idea is simply
that, when it is necessary to consider the contributions of
dimension-8 SMEFT operators in the analysis of an experimental result,
this information can be found here.

In our analysis, we follow closely the approach of
Ref.~\cite{Jenkins:2017jig}, and extend it to include dimension-8
SMEFT operators.  Below, we often refer to this paper simply by the
initials of its authors, as JMS.

We begin in Sec.~2 with some preliminary remarks comparing our
analysis with that of JMS, and discuss in general terms how matching
conditions are computed. In Sec.~3, we present the setup, showing how
the presence of dimension-8 SMEFT operators affects the symmetry
breaking, the generation of masses, and the couplings of the gauge and
Higgs bosons to fermions. The computations required to derive the complete matching conditions are described in Sec.~4; the results are presented in Appendix D. 
We conclude in Sec.~5. Appendices A, B, C 
give a variety of information relevant to the details of the analysis.

\section{Preliminaries}
\label{Preliminaries}

In Ref.~\cite{Jenkins:2017jig}, JMS compute the tree-level SMEFT matching conditions for the LEFT operators. The matching conditions for operators that conserve both $B$ and $L$ involve only even-dimension SMEFT operators, and are given up to dimension 6. For operators that violate $B$ and/or $L$, the matching conditions can involve even- or odd-dimension SMEFT operators (but not both), depending on the operator, and are computed to dimension 6 or dimension 5. In the present paper, we extend this analysis: we compute these matching conditions up to dimension 8 (dimension 7) if even-dimension (odd-dimension) SMEFT operators are involved. (In this paper, when we refer to ``computing the matching conditions up to dimension 8,'' both of these possibilities are understood.) If one eliminates the dimension-8 or dimension-7 contributions, the results of JMS are reproduced. This makes it easy to compare the results. Also, we present the elements of our analysis in much the same order as JMS.

In the LEFT Lagrangian, we consider only operators up to dimension 6 (like JMS):
\begin{equation}
\mathcal L_{\text{LEFT}}=\mathcal L_{\text{LEFT}}^{\text{Neutrino mass}}+\mathcal L_{\text{QCD}+\text{QED}}+\sum_{n=5}^6 \, \sum_{\mathcal O\in\dim n}\dfrac{C_{\mathcal O}} {\Lambda^{n-4}}\,\mathcal O ~.
\end{equation}
For the SMEFT, all operators up to dimension 8 are included:
\begin{equation}
\mathcal L_{\text{SMEFT}}=\mathcal L_{\text{SM}}+ \sum_{n=5}^8 \, \sum_{Q\in\dim n}\dfrac{C_Q}{\Lambda^{n-4}}\,Q ~.
\end{equation}

Still, there are two differences in our notation:
\begin{itemize}

\item Our convention is to have dimensionless Wilson coefficients
  (WCs). For instance, for the dimension-6 SMEFT lagrangian, we write
\begin{equation}
\mathcal L^{(6)}_{\text{SMEFT}}=\sum_{\mathcal O\in\dim6}\dfrac{C_{\mathcal O}}
{\Lambda^2}\,\mathcal O ~.
\end{equation}
This convention is different from that of JMS, which uses dimensionful
WCs.

\item In the unbroken phase, the SM lagrangian is
\begin{eqnarray}
\mathcal L_{\text{SM}} &=& \, -\dfrac14 G^A_{\mu\nu} G^{A\mu\nu} - \dfrac14 W^I_{\mu\nu} W^{I\mu\nu}
- \dfrac14 B_{\mu\nu} B^{\mu\nu} \nonumber\\
&& +~\sum_{\psi=q,u,d,l,e} \overline\psi i\slashed D\psi + (D_\mu H)^\dagger (D^\mu H)
- \lambda \left(H^\dagger H-\dfrac12v^2\right)^2 \nonumber\\
&& -~\left[\overline l_pe_r(Y_e)_{pr}H+\overline q_pu_r(Y_u)_{pr}\tilde H+\overline q_pd_r(Y_d)_{pr}H+\text{h.c.}\right]
\nonumber\\
&& +~\dfrac{\theta_3{g_s^2}}{32\pi^2}G^A_{\mu\nu}\tilde G^{A\mu\nu}
+\dfrac{\theta_2g^2}{32\pi^2}W^I_{\mu\nu}\tilde W^{I\mu\nu}+\dfrac{\theta_1{g'}^2}{32\pi^2}B_{\mu\nu}\tilde B^{\mu\nu} ~.
\label{SMLagrangian}
\end{eqnarray}
This uses the same notation as JMS, with one exception: our Yukawa
matrices (the $Y$s) are the hermitian conjugates of those of JMS.

\end{itemize}

In Eq.~(\ref{SMLagrangian}), the fields $q_r$ and $l_r$ are
(left-handed) $SU(2)_L$ doublets, while $u_r$, $d_r$ and $e_r$ are
(right-handed) $SU(2)_L$ singlets, where $r = 1,~2,~3$ is a generation
(weak-eigenstate) index. The physical (mass-eigenstate) states are the
same for the charged leptons, the left- and right-handed $u$-type
quarks, and the right-handed $d$-type quarks. For the left-handed
$d$-type quarks, the relation between the weak and mass eigenstates is
\begin{equation}
d_{Lr} = V_{rd} d_L + V_{rs} s_L + V_{rb} b_L \equiv V_{rx} d_{Lx} ~,
\end{equation}
where the left-hand side is a weak eigenstate, and the right-hand side
is a linear combination of mass eigenstates. The $V_{rx}$ are elements
of the unitary mixing matrix, which is the Cabibbo-Kobayashi-Maskawa
(CKM) matrix in the SM. Note: as in JMS, our LEFT matching conditions are given in the weak eigenstate basis. They can be written in terms of the physical states by using the above relation.

In our analysis, we make reference to several different sets of
operators. The LEFT operators are taken from JMS
\cite{Jenkins:2017jig}, the dimension-6 SMEFT operators are found in
Ref.~\cite{Grzadkowski:2010es}, and we use Ref.~\cite{Murphy:2020rsh}
for the dimension-8 SMEFT operators. In all cases, we use the same
notation for the operators and their WCs as is used in the references.
For the dimension-7 SMEFT operators, we use a basis that is equivalent that of Ref.~\cite{Lehman}, but with a different notation. For convenience, in the Appendices, we present tables of all the operators
used in this paper. These include LEFT operators (Appendix A), along with
dimension-5 to 8 SMEFT operators (Appendix B).

It is useful to give an example that illustrates the various issues
involved in deriving matching conditions. Consider the charged-current
four-fermion operator
\begin{equation}
{\mathcal O}_{\nu edu}^{V,LL} = ({\bar\nu}_{Lp}\gamma^\mu e_{Lr})({\bar d}_{Ls}\gamma_\mu u_{Lt}) + {\rm h.c.},
~~~~~ {\rm coefficient:} \dfrac1{\Lambda^2}C_{\substack{\nu edu\\prst}}^{V,LL} ~.
\end{equation}
We begin by examining the matching to the SM. That is, ${\mathcal
  O}_{\nu edu}^{V,LL}$ is taken to be an operator of the Fermi theory,
whose coefficient has magnitude $4 G_F / \sqrt{2}$. The SM Lagrangian
consists only of operators of at most dimension 4. This four-fermion
operator can be generated in the SM when a $W$ is exchanged between
the two fermion currents, and the $W$ is integrated out. The SM
matching condition is then
\begin{equation}
\dfrac1{\Lambda^2}C_{\substack{\nu edu\\prst}}^{V,LL} = -\dfrac{g^2}{2M_W^2}\,[W_l]_{pr}[W_q]_{ts}^* ~.
\label{SMmatch}
\end{equation}
Here, $W_l$ and $W_q$ are the couplings of the $W$ to the lepton and
quark pair, respectively. In the SM, $[W_l]_{pr} = \delta_{pr}$ and
$[W_q]_{ts} = \delta_{ts}$. Knowing that the coefficient has magnitude
$4 G_F / \sqrt{2}$, this leads to the well-known relation
\begin{equation}
\frac{G_F}{\sqrt{2}} = \dfrac{g^2}{8M_W^2} ~.
\end{equation}

The matching to SMEFT at dimension 6 was computed by JMS. It is
\begin{equation}
\dfrac1{\Lambda^2} C_{\substack{\nu edu\\prst}}^{V,LL} \, + {\rm h.c.} = \dfrac2{\Lambda^2} C^{(3)}_{\substack{lq\\prst}}
- \dfrac{\overline g^2}{2M_W^2} \, [W_l]^{\text{eff}}_{pr}{[W_q]^{\text{eff}}_{ts}}^* + \, {\rm c.c.}
\label{SMEFTdim6match}
\end{equation}
Since the SMEFT includes dimension-6 terms, it contains the
four-fermion operator. That is, there is a {\it direct} contribution
to the matching conditions, $C^{(3)}_{\substack{lq\\prst}}$. As was
the case in the SM, $C_{\substack{\nu edu\\prst}}^{V,LL}$ can also be
generated by the exchange of a $W$ between the two fermion
currents. This is represented by the second term above. Although this
resembles the term in Eq.~(\ref{SMmatch}), there are several
differences:
\begin{enumerate}

\item In the presence of dimension-6 SMEFT operators, the coupling
  constant is modified: $g \to {\bar g}$. This is due to the fact that, when one adds dimension-6 corrections to the
  kinetic terms of the gauge bosons, these fields and the coupling constants must be redefined in
  order to ensure that the kinetic term is properly normalized.

\item In the SM, the $W$ coupling to fermions is fixed by the fermion
  kinetic term, ${\bar\psi} {\slashed D} \psi$. In SMEFT, there are
  dimension-6 corrections, such as $H^\dagger H {\bar\psi} {\slashed
    D} \psi$. These will change the magnitudes of the couplings, and
  permit inter-generational couplings, hence the `eff' superscript on
  $W_l$ and $W_q$.

\end{enumerate}
The bottom line is that many dimension-6 SMEFT operators are
implicitly present in the second term of Eq.~(\ref{SMEFTdim6match})
above. Collectively, these operators form the {\it indirect}
contributions. They must be carefully taken into account in the
matching conditions. (Note that, if one expands the effective parameters appearing in the matching conditions, many terms will appear; those that are of higher order than dimension 8 are to be ignored.)

\section{Setup}
\label{Setup}

The Lagrangian for the SM in the unbroken phase is given in
Eq.~(\ref{SMLagrangian}). When the Higgs field acquires a vev, given
by the minimum of the Higgs potential, the symmetry is broken, and
masses are generated for the $W^\pm$, the $Z^0$ and the fermions. One
can easily compute the masses of the physical gauge bosons, as well as
their couplings to the physical fermions, in terms of the parameters
of $\mathcal L_{\text{SM}}$, in particular $g$, $g'$ and $v$.

When one includes higher-order SMEFT operators of dimension 6, 8,
etc., this whole process must be recalculated in order to take into
account these new operators. One must make field redefinitions so that
the kinetic terms are properly normalized, the minimum of the Higgs
potential (i.e., the Higgs vev) must be recomputed, corrections to
$\sin\theta_W$ must be taken into account, etc. One sees the effects
of these additional operators in the redefinitions of the coupling
constants, the couplings of gauge bosons to fermions, and other
quantities that appear in both the direct and indirect contributions
to the matching conditions.

In this section, we present the main effects of including SMEFT
operators up to dimension 8. We emphasize those results that are
important for the matching conditions.

\subsection{Higgs sector}

After the Higgs acquires a vev, we redefine the Higgs field as
follows:
\begin{equation}
H\to\dfrac1{\sqrt2}\begin{pmatrix} 0 \\ [1+c_{H,\text{kin}}]h+v_T \end{pmatrix} ~.
\end{equation}
Here, $v_T$ and $c_{H,\text{kin}}$ are respectively determined by
minimizing the Higgs potential and by normalizing the Higgs kinetic
term.

\subsubsection{ Higgs vev}

In the presence of SMEFT operators up to dimension 8, the Higgs
potential is
\begin{equation}
V(H) = \lambda\left(H^2-\dfrac12v^2\right)^2-\dfrac1{\Lambda^2}C_{H}H^6-\dfrac1{\Lambda^4}C_{H^8}H^8 ~,
\end{equation}
where only the real part of the second component of the Higgs doublet,
$H$, is taken to be nonzero. We define the physical Higgs vev, $v_T$,
to be $v_T \equiv \sqrt2\,H^{\text{min}}$, where $H^{\text{min}}$
minimizes the Higgs potential.  This implies that
\begin{equation}
\label{Higgs' vev dimension 8}
v_T = v \left(1+\dfrac{3v^2}{8\lambda\Lambda^2}\,C_{H}
+\dfrac{v^4}{4\lambda\Lambda^4}\left[\dfrac{63}{32\lambda}\,[C_H]^2+C_{H^8}\right]\right) ~.
\end{equation}
$v_T$ is the physical parameter that appears in the matching
relations, and whose value can in principle be determined by a fit to
the data.

\subsubsection{ Higgs kinetic term}

Including SMEFT contributions up to dimension 8, the Higgs kinetic
term is
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Higgs kinetic}} = \,\,\dfrac12\left[1+\dfrac{2 v_T^2}{\Lambda^2}
  \left(\dfrac14C_{HD}-C_{H\Box}\right)+\dfrac{v_T^4}{4\Lambda^4}
  \left(C_{H^6}^{(1)}+C_{H^6}^{(2)}\right)\right](1+c_{H,\text{kin}})^2(\partial_\mu h)(\partial^\mu h) ~.
\end{equation}
In order for this term to be properly normalized, one must have
\begin{equation}
\label{Higgs scalar field term normalization}
c_{H,\text{kin}}=\dfrac{v_T^2}{\Lambda^2}\left(C_{H\Box}-\dfrac14C_{HD}\right)-\dfrac{v_T^4}{8\Lambda^4}\left(C_{H^6}^{(1)}+C_{H^6}^{(2)}\right)+\dfrac{3 v_T^4}{2\Lambda^4}\left(C_{H\Box}-\dfrac14C_{HD}\right)^2 ~.
\end{equation}
This is essentially a redefiniton of the normalization of the Higgs
field.

\subsubsection{Higgs mass}

Taking into account the SMEFT contributions up to dimension 8, the Higgs boson mass term is
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Higgs mass}}=\dfrac12\left[\lambda v^2-3\lambda{v_T^2}+\dfrac{15{v_T^4}}{4\Lambda^2}\,C_{H}+\dfrac{7{v_T^6}}{2\Lambda^4}\,C_{H^8}\right](1+c_{H,\text{kin}})^2h^2 ~.
\end{equation}
This gives the following expression for the Higgs boson mass:
\begin{equation}
\label{Higgs boson masses dimension 8}
{m_h}^2=(1+c_{H,\text{kin}})^2{v_T^2}\left[2\lambda-\dfrac{3{v_T^2}}{\Lambda^2}\,C_{H}-\dfrac{3{v_T^4}}{\Lambda^4}\,C_{H^8}\right] ~.
\end{equation}

\subsection{Fermion mass matrices \& Yukawa couplings}

Before symmetry breaking, the SMEFT Lagrangian up to dimension 8
contains the following terms for charged leptons and quarks:
\begin{equation}
-(Y_\psi)_{pr} \, {\bar\chi}_{p} \psi_{r} \, \overline H
+ \frac{1}{\Lambda^2} C_{\substack{\psi H\\pr}} \, {\bar\chi}_{p} \psi_{r} \, \overline H (H^\dagger H)
+ \frac{1}{\Lambda^4} C_{\substack{\chi\psi H^5\\pr}} \, {\bar\chi}_{p} \psi_{r} \, \overline H (H^\dagger H)^2 + {\rm h.c.} ~,
\end{equation}
where $\psi\in\{e,u,d\}$ (right-handed $SU(2)_L$ singlets),
$\chi\in\{l,q\}$ (left-handed $SU(2)_L$ doublets), $\overline H=H$ if $\psi\in\{e,d\}$ and $\overline H=\tilde H$ if $\psi\in\{u\}$. Here, the first
term (dimension 4) belongs to the SM and the last two terms are SMEFT
operators (respectively dimension 6 and 8). Lepton-number-violating terms are also present:
\begin{equation}
\frac{1}{\Lambda} C_{\substack{5\\pr}} \, \epsilon^{ij}\epsilon^{kl}(l^T_{ip}Cl_{kr}) \, H_jH_l
+ \frac{1}{\Lambda^3} C_{\substack{l^2H^4\\pr}} \, \epsilon^{ij}\epsilon^{kl}(l^T_{ip}Cl_{kr}) \, H_jH_l (H^\dagger H) + {\rm h.c.} ~.
\end{equation}
When the Higgs gets a vev,
both mass matrices and Yukawa coupling terms are generated.

\subsubsection{Fermion mass matrices}

The SMEFT mass terms for charged leptons and quarks up to dimension 8 are
\begin{eqnarray}
\mathcal L_{\text{SM}}^{\text{Fermion mass}} &=& -\dfrac{v_T}{\sqrt2} \left[
(Y_e)_{pr}\overline e_{Lp}e_{Rr} + (Y_u)_{pr}\overline u_{Lp}u_{Rr} + (Y_d)_{pr}\overline d_{Lp}d_{Rr} \right] + \mathrm{h.c.}, \nonumber\\
\mathcal L_{\text{SMEFT},6}^{\text{Fermion mass}} &=& \dfrac{{v_T^3}}{2\sqrt2\,\Lambda^2} \left[
  C_{\substack{eH\\pr}}\overline e_{Lp}e_{Rr} + C_{\substack{uH\\pr}}\overline u_{Lp}u_{Rr}
  + C_{\substack{dH\\pr}}\overline d_{Lp}d_{Rr} \right] + \mathrm{h.c.}, \\
\mathcal L_{\text{SMEFT},8}^{\text{Fermion mass}} &=& \dfrac{{v_T^5}}{4\sqrt2\,\Lambda^4} \left[
C_{\substack{leH^5\\pr}}\overline e_{Lp}e_{Rr} + C_{\substack{quH^5\\pr}}\overline u_{Lp}u_{Rr}
+ C_{\substack{qdH^5\\pr}}\overline d_{Lp}d_{Rr} \right] + \mathrm{h.c.} \nonumber
\end{eqnarray}
This gives the following mass matrices:
\begin{equation}
\label{mass matrices dimension 8}
[M_\psi]_{pr}=\dfrac{v_T}{\sqrt2}\left[(Y_\psi)_{pr}
- \dfrac{{v_T^2}}{2\Lambda^2}\,C_{\substack{\psi H\\pr}}-\dfrac{{v_T^4}}{4\Lambda^4}\,C_{\substack{\chi\psi H^5\\pr}}\right] ~.
\end{equation}
The SMEFT neutrino mass terms are
\begin{eqnarray}
\mathcal L_{\text{SMEFT}}^{\text{Neutrino mass}} &=& \dfrac{{v_T}^2}{2\Lambda} \left[C_{\substack{5\\pr}} + \dfrac{{v_T}^2}{2\Lambda^2}C_{\substack{l^2H^4\\pr}}\right]\overline\nu^T_{Lp}C\nu_{Lr} + \mathrm{h.c.} ~.
\end{eqnarray}
This gives the following mass matrices:
\begin{equation}
\label{neutrino mass matrices dimension 8}
[M_\nu]_{pr}=-\dfrac{{v_T}^2}{\Lambda}\left[C_{\substack{5\\pr}}+\dfrac{{v_T}^2}{2\Lambda^2}C_{\substack{l^2H^4\\pr}}\right] ~.
\end{equation}

\subsubsection{Yukawa couplings}

The SM, dimension-6 and dimension-8 SMEFT Yukawa coupling terms for charged leptons and quarks are
\begin{eqnarray}
\mathcal L_{\text{SM}}^{\text{Yukawa}} &=& -\dfrac{(1+c_{H,\text{kin}})}{\sqrt2} \left[
  (Y_e)_{pr}\overline e_{Lp}e_{Rr}h + (Y_u)_{pr}\overline u_{Lp}u_{Rr}h + (Y_d)_{pr}\overline d_{Lp}d_{Rr}h \right]
+ \mathrm{h.c.}, \nonumber\\
\mathcal L_{\text{SMEFT},6}^{\text{Yukawa}} &=& \dfrac{3(1+c_{H,\text{kin}}) v_T^2}{2\sqrt2\,\Lambda^2} \left[
  C_{\substack{eH\\pr}}\overline e_{Lp}e_{Rr}h + C_{\substack{uH\\pr}}\overline u_{Lp}u_{Rr}h
+ C_{\substack{dH\\pr}}\overline d_{Lp}d_{Rr}h \right] + \mathrm{h.c.}, \\
\mathcal L_{\text{SMEFT},8}^{\text{Yukawa}} &=& \dfrac{5(1+c_{H,\text{kin}}) v_T^4}{4\sqrt2\,\Lambda^4} \left[
  C_{\substack{leH^5\\pr}}\overline e_{Lp}e_{Rr}h + C_{\substack{quH^5\\pr}}\overline u_{Lp}u_{Rr}h
  + C_{\substack{qdH^5\\pr}}\overline d_{Lp}d_{Rr}h \right] + \mathrm{h.c.} \nonumber
\end{eqnarray}
This gives the following Yukawa couplings (up to dimension 8):
\begin{eqnarray}
\label{Yukawa couplings dimension 8}
(Y_\psi)^{\text{eff}}_{pr} = \dfrac{1+c_{H,\text{kin}}}{\sqrt2}\left[\dfrac{\sqrt2}{v_T}\,[M_\psi]_{pr}-\dfrac{{v_T^2}}{\Lambda^2}\,C_{\substack{\psi H\\pr}}-\dfrac{{v_T^4}}{\Lambda^4}\,C_{\substack{\chi\psi H^5\\pr}}\right] ~.
\end{eqnarray}
There are also momentum-dependent Yukawa couplings (i.e., ordinary Yukawa couplings with additional derivatives) occurring at dimension 6 in SMEFT. However, as we will see in Sec. 4.1.3, these can be neglected in our analysis.

The SMEFT Yukawa coupling terms for neutrinos are
\begin{eqnarray}
\mathcal L_{\text{SMEFT}}^{\text{Neutrino Yukawa}} &=& \dfrac{v_T}{\Lambda}(1+c_{H,\text{kin}})\left[C_{\substack{5\\pr}} + \dfrac{{v_T}^2}{\Lambda^2}C_{\substack{l^2H^4\\pr}}\right]h\,\overline\nu^T_{Lp}C\nu_{Lr} + \mathrm{h.c.} ~.
\end{eqnarray}
This gives the following Yukawa couplings:
\begin{eqnarray}
\label{neutrino Yukawa couplings dimension 8}
(Y_\nu)^{\text{eff}}_{pr} = (1+c_{H,\text{kin}})\left[\dfrac1{v_T}\,[M_\nu]_{pr}-\dfrac{{v_T}^3}{2\Lambda^3}C_{\substack{l^2H^4\\pr}}\right] ~.
\end{eqnarray}
These Yukawa couplings enter the matching conditions of certain
four-fermion operators in LEFT. 

\subsection{Electroweak gauge boson masses \& mixing and coupling constants}

\subsubsection{Kinetic terms}
\label{GBkinetic}

Including the SMEFT contributions up to dimension 8, the kinetic terms
of the electroweak gauge bosons after symmetry breaking are
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Electroweak kin}}=-\dfrac14\left\{\begin{aligned}
& \left[1-\dfrac{2 v_T^2}{\Lambda^2}\,C_{HW}-\dfrac{ v_T^4}{\Lambda^4}\,C_{W^2H^4}^{(1)}\right]W^I_{\mu\nu}W^{I\mu\nu} \\
& -\dfrac{ v_T^4}{\Lambda^4}\,C_{W^2H^4}^{(3)}\,W^3_{\mu\nu}W^{3\mu\nu} \\
& +\left[1-\dfrac{2 v_T^2}{\Lambda^2}\,C_{HB}-\dfrac{ v_T^4}{\Lambda^4}\,C_{B^2H^4}^{(1)}\right]B_{\mu\nu}B^{\mu\nu} \\
& +\left[\dfrac{2 v_T^2}{\Lambda^2}\,C_{HWB}+\dfrac{ v_T^4}{\Lambda^4}\,C_{WBH^4}^{(1)}\right]W^3_{\mu\nu}B^{\mu\nu}
\end{aligned}\right\} ~.
\label{gaugemassterms}
\end{equation}
Here there are two issues that must be resolved. First, the kinetic
terms must be properly normalized. Second, the
$W^3_{\mu\nu}B^{\mu\nu}$ mixing term must be removed.

Proper normalization of the kinetic terms can be achieved by
redefining the coupling constants and the normalization of the gauge
fields:
\begin{eqnarray}
\label{ccredefs}
\overline g &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HW}
  +\dfrac{{v_T^4}}{2\Lambda^4}\,C_{W^2H^4}^{(1)}+\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HW}]^2\right]g ~,  \nonumber\\
\overline g' &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HB}
  +\dfrac{{v_T^4}}{2\Lambda^4}\,C_{B^2H^4}^{(1)}+\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HB}]^2\right]g' ~, \\
\label{gbredefs}
W^I_\mu &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HW}+\dfrac{{v_T^4}}{2\Lambda^4}\,C_{W^2H^4}^{(1)}
  +\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HW}]^2\right]\mathcal W^I_\mu ~, \nonumber\\
B_\mu &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HB}+\dfrac{{v_T^4}}{2\Lambda^4}\,C_{B^2H^4}^{(1)}
  +\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HB}]^2\right]\mathcal B_\mu ~.
\end{eqnarray}

At this stage, there is still a $\mathcal W^3_{\mu\nu}\mathcal
B^{\mu\nu}$ mixing term, as well as a separate
$W^3_{\mu\nu}W^{3\mu\nu}$ term. These can both be removed by defining
\begin{eqnarray}
\begin{bmatrix}
\mathcal W^3_\mu \\ \mathcal B_\mu
\end{bmatrix}=\begin{bmatrix}
1+\dfrac{{v_T}^4}{2\Lambda^4}\,C_{W^2H^4}^{(3)}+\dfrac{3{v_T}^4}{8\Lambda^4}\,[C_{HWB}]^2 & -\dfrac{{v_T}^2}{2\Lambda^2}\left(\begin{aligned}
& C_{HWB}+\dfrac{{v_T}^2}{2\Lambda^2}\,C_{WBH^4}^{(1)} \\
& +\dfrac{{v_T}^2}{\Lambda^2}\,C_{HWB}C_{HW} \\
& +\dfrac{{v_T}^2}{\Lambda^2}\,C_{HWB}C_{HB}
\end{aligned}\right) \\
-\dfrac{{v_T}^2}{2\Lambda^2}\left(\begin{aligned}
& C_{HWB}+\dfrac{{v_T}^2}{2\Lambda^2}\,C_{WBH^4}^{(1)} \\
& +\dfrac{{v_T}^2}{\Lambda^2}\,C_{HWB}C_{HW} \\
& +\dfrac{{v_T}^2}{\Lambda^2}\,C_{HWB}C_{HB}
\end{aligned}\right) & 1+\dfrac{3{v_T}^4}{8\Lambda^4}\,[C_{HWB}]^2
\end{bmatrix}\begin{bmatrix}
\overline{\mathcal W}^3_\mu \\ \overline{\mathcal B}_\mu
\end{bmatrix}.
\end{eqnarray}
With this, we have
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Electroweak kin}} = -\dfrac12\mathcal W^+_{\mu\nu}\mathcal W^{-\mu\nu}
-\dfrac14\overline{\mathcal W}^3_{\mu\nu}\overline{\mathcal W}^{3\mu\nu}
-\dfrac14\overline{\mathcal B}_{\mu\nu}\overline{\mathcal B}^{\mu\nu} ~,
\end{equation}
where $\overline{\mathcal W}^\pm_{\mu\nu} \equiv \partial_\mu\overline{\mathcal W}^\pm_\nu-\partial_\nu\overline{\mathcal W}^\pm_\mu$, $\overline{\mathcal W}^\pm_\mu \equiv \dfrac1{\sqrt2}(\overline{\mathcal W}^1_\mu\mp i\overline{\mathcal W}^2_\mu)$, $\overline{\mathcal W}^3_{\mu\nu} \equiv
\partial_\mu\overline{\mathcal W}^3_\nu-\partial_\nu\overline{\mathcal
  W}^3_\mu$, $\overline{\mathcal B}_{\mu\nu} \equiv
\partial_\mu\overline{\mathcal B}_\nu-\partial_\nu\overline{\mathcal
  B}_\mu$, and we have dropped the cubic and quartic self-coupling
terms of the gauge bosons.

Note that we still have the freedom to perform the following rotation:
\begin{equation}
\label{thetaWdef}
\begin{bmatrix}
\overline{\mathcal W}^3_\mu \\ \overline{\mathcal B}_\mu
\end{bmatrix}=\begin{bmatrix}
\cos\overline\theta_W & \sin\overline\theta_W \\
-\sin\overline\theta_W & \cos\overline\theta_W
\end{bmatrix}\begin{bmatrix}
\mathcal Z_\mu \\ \mathcal A_\mu
\end{bmatrix} ~.
\end{equation}
In terms of the new fields, we have
\begin{equation}
\label{physicalGBs}
\mathcal L_{\text{SMEFT}}^{\text{Electroweak kin}}=-\dfrac12\mathcal W^+_{\mu\nu}\mathcal W^{-\mu\nu}
-\dfrac14\mathcal Z_{\mu\nu}\mathcal Z^{\mu\nu}-\dfrac14\mathcal F_{\mu\nu}\mathcal F^{\mu\nu} ~.
\end{equation}

For completeness, we also present the results for gluons. Including the
SMEFT contributions up to dimension 8, the gluon kinetic term is
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Gluons kin}}=-\dfrac14\left[1-\dfrac{2{v_T^2}}{\Lambda^2}\,C_{HG}-\dfrac{{v_T^4}}{\Lambda^4}\,C_{G^2H^4}^{(1)}\right]G^A_{\mu\nu}G^{A\mu\nu} ~.
\end{equation}
In order to properly normalize this kinetic term, we make
redefinitions similar to those in Eqs.~(\ref{ccredefs}) and
(\ref{gbredefs}):
\begin{eqnarray}
\label{strong coupling constant dim 8}
\overline g_s &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HG}+\dfrac{{v_T^4}}{2\Lambda^4}\,C_{G^2H^4}^{(1)}
  +\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HG}]^2\right]g_s ~, \\
\label{gluon field normalization dim 8}
G^A_\mu &=& \left[1+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HG}+\dfrac{{v_T^4}}{2\Lambda^4}\,C_{G^2H^4}^{(1)}
  +\dfrac{3{v_T^4}}{2\Lambda^4}\,[C_{HG}]^2\right]\mathcal G^A_\mu ~.
\end{eqnarray}

\subsubsection{Mass terms}

The SMEFT contributions up to dimension 8 to the mass terms of the
electroweak gauge bosons after symmetry breaking are
\begin{equation}
\mathcal L_{\text{SMEFT}}^{\text{Electroweak mass}}=\dfrac{ v_T^2}8\left\{\begin{aligned}
& \left[1+\dfrac{ v_T^4}{4\Lambda^4}\left(C_{H^6}^{(1)}-C_{H^6}^{(2)}\right)\right]g^2(W_\mu^1W^{1\mu}+W_\mu^2W^{2\mu}) \\
& \left[\begin{aligned}
& 1+\dfrac{ v_T^2}{2\Lambda^2}\,C_{HD} \\
& +\dfrac{ v_T^4}{4\Lambda^4}\left(C_{H^6}^{(1)}+C_{H^6}^{(2)}\right)
\end{aligned}\right](gW_\mu^3-g'B_\mu)(gW^{3\mu}-g'B^\mu)
\end{aligned}\right\} ~.
\end{equation}
We can write $W_\mu^1$ and $B_\mu$ in terms of $\mathcal W^\pm_\mu$,
$\mathcal Z_\mu$ and $\mathcal A_\mu$ using the transformations
described in Sec.~\ref{GBkinetic}. In order to ensure a massless
photon, we require that the mixing angle ${\overline\theta}_W$ of
Eq.~(\ref{thetaWdef}) satisfy 
\begin{equation}
\label{mixing angle}
\begin{split}
\cos\overline\theta_W & =\dfrac1{\sqrt{\overline g^2+\overline g'^2}}\left[\begin{aligned}
& \overline g+\dfrac{\overline g{v_T^4}}{8\Lambda^4}\dfrac{(6\overline g^2\overline g'^2-\overline g^4-5\overline g'^4)}{(\overline g^2+\overline g'^2)^2}\,[C_{HWB}]^2+\dfrac{{v_T^4}}{2\Lambda^4}\dfrac{\overline g\overline g'^2}{\overline g^2+\overline g'^2}\,C_{W^2H^4}^{(3)} \\
& -\dfrac{\overline g'{v_T^2}}{2\Lambda^2}\dfrac{\overline g^2-\overline g'^2}{\overline g^2+\overline g'^2}\left(C_{HWB}+\dfrac{{v_T^2}}{2\Lambda^2}\,C_{WBH^4}^{(1)}+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HWB}[C_{HW}+C_{HB}]\right)
\end{aligned}\right] \\
\sin\overline\theta_W & =\dfrac1{\sqrt{\overline g^2+\overline g'^2}}\left[\begin{aligned}
& \overline g'+\dfrac{\overline g'{v_T^4}}{8\Lambda^4}\dfrac{(6\overline g^2\overline g'^2-\overline g'^4-5\overline g^4)}{(\overline g^2+\overline g'^2)^2}\,[C_{HWB}]^2-\dfrac{{v_T^4}}{2\Lambda^4}\dfrac{\overline g^2\overline g'}{\overline g^2+\overline g'^2}\,C_{W^2H^4}^{(3)} \\
& +\dfrac{\overline g{v_T^2}}{2\Lambda^2}\dfrac{\overline g^2-\overline g'^2}{\overline g^2+\overline g'^2}\left(C_{HWB}+\dfrac{{v_T^2}}{2\Lambda^2}\,C_{WBH^4}^{(1)}+\dfrac{{v_T^2}}{\Lambda^2}\,C_{HWB}[C_{HW}+C_{HB}]\right)
\end{aligned}\right]
\end{split}
\end{equation}
up to dimension 8.

Note that, while in the SM we have $\sin\theta_W = g'/\sqrt{g^2 +
  {g'}^2}$ and $\cos\theta_W = g/\sqrt{g^2 + {g'}^2}$, these relations
no longer hold in the presence of SMEFT operators. Similalry, in the
SM, $e = gg'/\sqrt{g^2 + {g'}^2}$. Including SMEFT operators, this
becomes
\begin{equation}\label{electric charge}
\overline e=\dfrac{\overline g\overline g'}{\sqrt{\overline g^2+\overline g'^2}}\left[\begin{aligned}
& 1-\dfrac{\overline g\overline g' v_T^2C_{HWB}}{(\overline g^2+\overline g'^2)\Lambda^2}-\dfrac{\overline g\overline g' v_T^4C^{(1)}_{WBH^4}}{2(\overline g^2+\overline g'^2)\Lambda^4}+\dfrac{\overline g'^2 v_T^4C^{(3)}_{W^2H^4}}{2(\overline g^2+\overline g'^2)\Lambda^4} \\
& -\dfrac{\overline g\overline g' v_T^4C_{HWB}(C_{HW}+C_{HB})}{(\overline g^2+\overline g'^2)\Lambda^4}+\dfrac{3\overline g^2\overline g'^2 v_T^4[C_{HWB}]^2}{2(\overline g^2+\overline g'^2)^2\Lambda^4}
\end{aligned}\right] ~.
\end{equation}

The masses of the $W$ and $Z$ are given by
\begin{eqnarray}
\label{W boson mass}
      M_W^2 &=& \dfrac{\overline g^2 v_T^2}4\left[1+\dfrac{ v_T^4}{4\Lambda^4}
        \left(C_{H^6}^{(1)}-C_{H^6}^{(2)}\right)\right] ~, \\
\label{Z boson mass}
M_Z^2 &=& \dfrac{{\overline g_Z}^2 v_T^2}4\left[1+\dfrac{ v_T^2}{2\Lambda^2}\,C_{HD}+\dfrac{ v_T^4}{4\Lambda^4}\left(C_{H^6}^{(1)}+C_{H^6}^{(2)}\right)\right] ~,
\end{eqnarray}
where
\begin{eqnarray}
\label{Z coupling constant}
\overline g_Z &=& \sqrt{\overline g^2+\overline g'^2} \left[
  1 + \dfrac{\overline g\overline g' v_T^2C_{HWB}}{(\overline g^2+\overline g'^2)\Lambda^2}
  + \dfrac{\overline g\overline g' v_T^4C^{(1)}_{WBH^4}}{2(\overline g^2+\overline g'^2)\Lambda^4}
  + \dfrac{\overline g^2 v_T^4C^{(3)}_{W^2H^4}}{2(\overline g^2+\overline g'^2)\Lambda^4} \right. \nonumber\\
  && \left. + ~\dfrac{\overline g\overline g' v_T^4C_{HWB}(C_{HW}+C_{HB})}{(\overline g^2+\overline g'^2)\Lambda^4}
  + \left(1-\dfrac{\overline g^2\overline g'^2}{(\overline g^2+\overline g'^2)^2}\right)\dfrac{[C_{HWB}]^2}{2\Lambda^4}
\right] ~.
\end{eqnarray}

In the SM, the ``charge'' to which the $Z^0$ couples is $I_{3L} -
Q_{em} \sin^2 \theta_W$. When one adds SMEFT operators up to dimension
6, the mixing angle is changed, $\theta_W \to {\bar\theta_W}$, but the
$Z^0$ coupling still has the same form: it couples to $I_{3L} - Q_{em}
\sin^2 {\bar\theta_W}$ \cite{Jenkins:2017jig}. However, when SMEFT
operators up to dimension 8 are included, this no longer
holds. Instead, the $Z^0$ couples to $I_{3L} - Q_{em} \sin^2
{\bar\theta}_Z$, where
\begin{equation}
\label{new angle}
  \sin^2\overline\theta_Z=\sin^2\overline\theta_W
  +\dfrac{ v_T^4}{4\Lambda^4}[C_{HWB}]^2(\sin^2\overline\theta_W-\cos^2\overline\theta_W) ~.
\end{equation}
(This was also noted in Ref.~\cite{Hays:2018zze}.)

\subsection{Couplings of electroweak gauge bosons to fermions}

As shown in Eq.~(\ref{physicalGBs}), the physical electroeak gauge
bosons are $\mathcal A_\mu$, $\mathcal W_\mu^\pm$ and $\mathcal
Z_\mu$. Their effective couplings to fermions, as well as those of
the gluon $\mathcal G^A_\mu$, take the following form:
\begin{equation}
\label{fermion - gauge bosons interactions terms}
\mathcal L = -\overline g_s\mathcal G^A_\mu j_{\mathcal G}^{A\mu} -\overline e\mathcal A_\mu j_{\mathcal A}^\mu
-\dfrac{\overline g}{\sqrt2}\{\mathcal W^+_\mu j_{\mathcal W}^{+\mu}+\mathcal W^-_\mu j_{\mathcal W}^{-\mu}\}
-\overline g_Z\mathcal Z_\mu j_{\mathcal Z}^\mu ~,
\end{equation}
in which the corresponding currents are 
\begin{eqnarray}
\label{fermion - gauge bosons currents}
j_{\mathcal G}^{A\mu} &=& \overline u_{Lp}\gamma^\mu T^Au_{Lr}+\overline d_{Lp}\gamma^\mu T^Ad_{Lr}
+\overline u_{Rp}\gamma^\mu T^Au_{Rr}+\overline d_{Rp}\gamma^\mu T^Ad_{Rr} ~, \nonumber\\
j_{\mathcal A}^\mu &=& -\overline e_{Lp}\gamma^\mu e_{Lr}+\dfrac23\overline u_{Lp}\gamma^\mu u_{Lr}
-\dfrac13\overline d_{Lp}\gamma^\mu d_{Lr}-\overline e_{Rp}\gamma^\mu e_{Rr}
+\dfrac23\overline u_{Rp}\gamma^\mu u_{Rr}-\dfrac13\overline d_{Rp}\gamma^\mu d_{Rr} ~, \nonumber\\
j_{\mathcal W}^{+\mu} &=& [W_l]^{\text{eff}}_{pr}\overline\nu_{Lp}\gamma^\mu e_{Lr}
+[W_q]^{\text{eff}}_{pr}\overline u_{Lp}\gamma^\mu d_{Lr}+[W_R]^{\text{eff}}_{pr}\overline u_{Rp}\gamma^\mu d_{Rr} ~, \\
j_{\mathcal W}^{-\mu} &=& {[W_l]^{\text{eff}}_{rp}}^*\overline e_{Lp}\gamma^\mu\nu_{Lr}
+{[W_q]^{\text{eff}}_{rp}}^*\overline d_{Lp}\gamma^\mu u_{Lr}
+{[W_R]^{\text{eff}}_{rp}}^*\overline d_{Rp}\gamma^\mu u_{Rr} ~, \nonumber\\
j_{\mathcal Z}^\mu &=& \left[\begin{aligned}
& [Z_{\nu_L}]^{\text{eff}}_{pr}\overline\nu_{Lp}\gamma^\mu\nu_{Lr}+[Z_{e_L}]^{\text{eff}}_{pr}\overline e_{Lp}\gamma^\mu e_{Lr}
+[Z_{u_L}]^{\text{eff}}_{pr}\overline u_{Lp}\gamma^\mu u_{Lr}+[Z_{d_L}]^{\text{eff}}_{pr}\overline d_{Lp}\gamma^\mu d_{Lr} \\
& +[Z_{e_R}]^{\text{eff}}_{pr}\overline e_{Rp}\gamma^\mu e_{Rr}+[Z_{u_R}]^{\text{eff}}_{pr}\overline u_{Rp}\gamma^\mu u_{Rr}+[Z_{d_R}]^{\text{eff}}_{pr}\overline d_{Rp}\gamma^\mu d_{Rr}
\end{aligned}\right] ~. \nonumber
\end{eqnarray}

Since $SU(3)_C \times U(1)_{em}$ remains unbroken, the currents involving gluons and photons are fully
determined by QCD and QED. This is not the case for the $\mathcal
W_\mu^\pm$ and $\mathcal Z_\mu$ gauge bosons: the fermion currents to which the $\mathcal
W_\mu^\pm$ and $\mathcal Z_\mu$ couple are given by the following (up
to dimension 8):
\begin{equation}
\label{gauge boson - fermions couplings}
\begin{split}
[W_l]^{\text{eff}}_{pr} & =\delta_{pr}+\dfrac{ v_T^2}{\Lambda^2}\,C_{\substack{Hl\\pr}}^{(3)}+\dfrac{ v_T^4}{2\Lambda^4}\left(C_{\substack{l^2H^4D\\pr}}^{(2)}-iC_{\substack{l^2H^4D\\pr}}^{(3)}\right) ~, \\
[W_q]^{\text{eff}}_{pr} & =\delta_{pr}+\dfrac{ v_T^2}{\Lambda^2}\,C_{\substack{Hq\\pr}}^{(3)}+\dfrac{ v_T^4}{2\Lambda^4}\left(C_{\substack{q^2H^4D\\pr}}^{(2)}-iC_{\substack{q^2H^4D\\pr}}^{(3)}\right) ~, \\
[W_R]^{\text{eff}}_{pr} & =\dfrac{ v_T^2}{2\Lambda^2}\,C_{\substack{Hud\\pr}}^{(3)}+\dfrac{ v_T^4}{4\Lambda^4}\,C_{\substack{udH^4D\\pr}} ~, \\
[Z_{\nu_L}]^{\text{eff}}_{pr} & =\dfrac12\delta_{pr}-\dfrac{ v_T^2}{2\Lambda^2}\left(C_{\substack{Hl\\pr}}^{(1)}-C_{\substack{Hl\\pr}}^{(3)}\right)-\dfrac{ v_T^4}{4\Lambda^4}\left(C_{\substack{l^2H^4D\\pr}}^{(1)}-2C_{\substack{l^2H^4D\\pr}}^{(2)}\right) ~, \\
[Z_{e_L}]^{\text{eff}}_{pr} & =\dfrac12g^e_L\delta_{pr}-\dfrac{ v_T^2}{2\Lambda^2}\left(C_{\substack{Hl\\pr}}^{(1)}+C_{\substack{Hl\\pr}}^{(3)}\right)-\dfrac{ v_T^4}{4\Lambda^4}\left(C_{\substack{l^2H^4D\\pr}}^{(1)}+2C_{\substack{l^2H^4D\\pr}}^{(2)}\right) ~, \\
[Z_{u_L}]^{\text{eff}}_{pr} & =\dfrac12g^u_L\delta_{pr}-\dfrac{ v_T^2}{2\Lambda^2}\left(C_{\substack{Hq\\pr}}^{(1)}-C_{\substack{Hq\\pr}}^{(3)}\right)-\dfrac{ v_T^4}{4\Lambda^4}\left(C_{\substack{q^2H^4D\\pr}}^{(1)}-2C_{\substack{q^2H^4D\\pr}}^{(2)}\right) ~, \\
[Z_{d_L}]^{\text{eff}}_{pr} & =\dfrac12g^d_L\delta_{pr}-\dfrac{ v_T^2}{2\Lambda^2}\left(C_{\substack{Hq\\pr}}^{(1)}+C_{\substack{Hq\\pr}}^{(3)}\right)-\dfrac{ v_T^4}{4\Lambda^4}\left(C_{\substack{q^2H^4D\\pr}}^{(1)}+2C_{\substack{q^2H^4D\\pr}}^{(2)}\right) ~, \\
[Z_{e_R}]^{\text{eff}}_{pr} & =\dfrac12g^e_R\delta_{pr}-\dfrac{ v_T^2}{2\Lambda^2}\,C_{\substack{He\\pr}}-\dfrac{ v_T^4}{4\Lambda^4}\,C_{\substack{e^2H^4D\\pr}}^{(1)} ~, \\
[Z_{u_R}]^{\text{eff}}_{pr} & =\dfrac12g^u_R\delta_{pr}-\dfrac{v^2}{2\Lambda^2}\,C_{\substack{Hu\\pr}}-\dfrac{ v_T^4}{4\Lambda^4}\,C_{\substack{u^2H^4D\\pr}}^{(1)} ~, \\
[Z_{d_R}]^{\text{eff}}_{pr} & =\dfrac12g^d_R\delta_{pr}-\dfrac{v^2}{2\Lambda^2}\,C_{\substack{Hd\\pr}}-\dfrac{ v_T^4}{4\Lambda^4}\,C_{\substack{d^2H^4D\\pr}}^{(1)} ~.
\end{split}
\end{equation}
Here, we have defined $g^e_L\equiv-1+2\sin^2\overline\theta_Z$,
$g^u_L\equiv1-\dfrac43\sin^2\overline\theta_Z$,
$g^d_L\equiv-1+\dfrac23\sin^2\overline\theta_Z$,
$g^e_R\equiv2\sin^2\overline\theta_Z$,
$g^u_R\equiv-\dfrac43\sin^2\overline\theta_Z$, and
$g^d_R\equiv\dfrac23\sin^2\overline\theta_Z$, where
$\sin^2\overline\theta_Z$ is defined in Eq.~(\ref{new angle}).

\section{Matching conditions}

There are four categories of LEFT operators up to dimension 6: (i) four-fermion operators, (ii) magnetic dipole moment operators, (iii) three-gluon operators, and (iv) neutrino mass terms. The matching conditions for operators that conserve both $B$ and $L$ involve only even-dimension SMEFT operators, and are given up to dimension 6 in JMS. For operators that violate $B$ and/or $L$, the matching conditions involve either even- or odd-dimension SMEFT operators, depending on the operator; these are given to dimension 6 or dimension 5 in JMS. In this section, we present the tree-level matching conditions for all of these operators up to dimension 8 in SMEFT.

\subsection{Four-fermion operators}

As was described in Sec.~\ref{Preliminaries}, there are generally two
types of contributions to the SMEFT matching conditions of
four-fermion LEFT operators: direct and indirect contributions.

\subsubsection{Direct contributions}

The dimension-6 SMEFT direct contribution is the LEFT operator itself, in
which all left- and right-handed particles are replaced by the left-handed
$SU(2)_L$ doublets and right-handed $SU(2)_L$ singlets to which they respectively belong.  The dimension-8
contributions involve the dimension-6 SMEFT operator multiplied by a
pair of Higgs fields. When the Higgs gets a vev, this generates the
four-fermion LEFT operator.

The details of the computation are best illustrated with an example.
Consider the LEFT operator
\begin{equation}
\mathcal O^{V,LL}_{\substack{\nu\nu\\prst}} \equiv (\overline\nu_{Lp}
\gamma_\mu \nu_{Lr}) (\overline\nu_{Ls} \gamma^\mu \nu_{Lt}) ~.
\label{LEFT4nu}
\end{equation}
It is generated by
the dimension-6 SMEFT operator $Q_{\substack{ll\\prst}} \equiv (\overline
l_p\gamma_\mu l_r)(\overline l_s\gamma^\mu l_t)$. This can be seen by
separating the SMEFT operator into components:
\begin{equation}
\dfrac1{\Lambda^2} C_{\substack{ll\\prst}} Q_{\substack{ll\\prst}} \to\dfrac1{\Lambda^2}
C_{\substack{ll\\prst}}\left[\begin{aligned}
& (\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt}) \\
& +(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt})
\end{aligned}\right] ~.
\end{equation}
The first term is $\mathcal O^{V,LL}_{\substack{\nu\nu\\prst}}$.

One dimension-8 SMEFT operator that is among the matching conditions
is $Q^{(1)}_{\substack{l^4H^2\\prst}} \equiv (\overline l_p\gamma_\mu l_r)
(\overline l_s\gamma^\mu l_t)(H^\dagger H)$. Because the $SU(2)_L$
doublets $l$ and $H$ are involved, there are two additional
dimension-8 SMEFT operators that must be included:
$Q^{(2)}_{\substack{l^4H^2\\prst}}\equiv(\overline l_p\gamma_\mu
l_r)(\overline l_s\gamma^\mu\tau^Il_t)(H^\dagger\tau^IH)$ and
$Q^{(2)}_{\substack{l^4H^2\\stpr}}\equiv(\overline
l_p\gamma_\mu\tau^Il_r)(\overline l_s\gamma^\mu
l_t)(H^\dagger\tau^IH)$. When the Higgs gets a vev, these three
operators can also generate $\mathcal
O^{V,LL}_{\substack{\nu\nu\\prst}}$:
\begin{eqnarray}
\dfrac1{\Lambda^4}C^{(1)}_{\substack{l^4H^2\\prst}}Q^{(1)}_{\substack{l^4H^2\\prst}} &\to&
\dfrac{{v_T^2}}{2\Lambda^4}C^{(1)}_{\substack{l^4H^2\\prst}}\left[\begin{aligned}
& (\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt}) \\
& +(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt})
\end{aligned}\right] ~, \nonumber\\
\dfrac1{\Lambda^4}C^{(2)}_{\substack{l^4H^2\\prst}}Q^{(2)}_{\substack{l^4H^2\\prst}} &\to&
\dfrac{{v_T^2}}{2\Lambda^4}C^{(2)}_{\substack{l^4H^2\\prst}}\left[\begin{aligned}
& -(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt}) \\
& -(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt})
\end{aligned}\right] ~, \nonumber\\
\dfrac1{\Lambda^4}C^{(2)}_{\substack{l^4H^2\\stpr}}Q^{(2)}_{\substack{l^4H^2\\stpr}} &\to&
\dfrac{{v_T^2}}{2\Lambda^4}C^{(2)}_{\substack{l^4H^2\\stpr}}\left[\begin{aligned}
& -(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})-(\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt}) \\
& +(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline\nu_{Ls}\gamma^\mu\nu_{Lt})+(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline e_{Ls}\gamma^\mu e_{Lt})
  \end{aligned}\right] ~.
\end{eqnarray}

We therefore see that the direct contribution to the matching
condition of the LEFT operator $\dfrac1{\Lambda^2}\mathcal
O^{V,LL}_{\substack{\nu\nu\\prst}}$, up to dimension 8, is
\begin{equation}
\dfrac1{\Lambda^2}\left[C_{\substack{ll\\prst}}+\dfrac{{v_T^2}}{2\Lambda^2}
  \left(C^{(1)}_{\substack{l^4H^2\\prst}}-C^{(2)}_{\substack{l^4H^2\\prst}}-C^{(2)}_{\substack{l^4H^2\\stpr}}\right)\right] ~.
\end{equation}

The direct contributions to the matching conditions of the other LEFT
four-fermion operators are calculated similarly.

\subsubsection{Indirect contributions}

A four-fermion operator can also be generated when a boson is
exchanged between two fermion currents and this boson is integrated
out. This produces an indirect contribution [e.g., see
  Eq.~(\ref{SMmatch})].

Consider once again the LEFT operator $\mathcal
O^{V,LL}_{\substack{\nu\nu\\prst}}$ of Eq.~(\ref{LEFT4nu}). The
indirect contributions arise from the $Z$-exchange diagrams of
Fig.~\ref{4nuZexchange}, when the $Z^0$ is integrated out. We note
that (i) there is a relative minus sign between the two diagrams, and
(ii) when one Fierz transforms (see Appendix \ref{Fierz}) the amplitude of the second diagram,
one obtains the amplitude of the first diagram, but with an exchange
of generation indices $r \leftrightarrow t$. The indirect contribution
to the matching condition of this operator, up to dimension 8, is
\begin{equation}
-\dfrac{\overline{g}_Z^2}{4 M_Z^2} \left([Z_{\nu_L}]^{\text{eff}}_{pr}[Z_{\nu_L}]^{\text{eff}}_{st}
+[Z_{\nu_L}]^{\text{eff}}_{pt}[Z_{\nu_L}]^{\text{eff}}_{sr}\right) ~,
\end{equation}
where $\overline{g}_Z$ and $[Z_{\nu_L}]^{\text{eff}}_{pr}$ are defined
in Eqs.~(\ref{Z coupling constant}) and (\ref{gauge boson - fermions
  couplings}), respectively.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.3\textwidth]{4nuZ1.png}
~~~~~~~
\includegraphics[width=0.3\textwidth]{4nuZ2.png}
\end{center}
\vskip-5truemm
\caption{\small $Z$-exchange contributions to $\mathcal O^{V,LL}_{\nu\nu}$ with flavour indices $prst$.}
\label{4nuZexchange}
\end{figure}

Another example is the LEFT operator $\mathcal O^{V,LL}_{\substack{\nu
    e\\prst}} \equiv (\overline\nu_{Lp}\gamma_\mu\nu_{Lr})(\overline
e_{Ls}\gamma^\mu e_{Lt})$. Here the indirect contributions arise from
the $Z$- and $W$-exchange diagrams of Fig,~\ref{2nu2eZWexchange}, when
the heavy gauge bosons are integrated out. The indirect contribution
to the matching condition of this operator, up to dimension 8, is
\begin{equation}
-\dfrac{\overline{g}_Z^2}{M_Z^2}[Z_{\nu_L}]^{\text{eff}}_{pr}[Z_{e_L}]^{\text{eff}}_{st}
-\dfrac{\overline g^2}{2 M_W^2}[W_l]^{\text{eff}}_{pr}{[W_l]^{\text{eff}}_{st}}^* ~,
\end{equation}
where $\overline{g}$ and $[W_l]^{\text{eff}}_{pr}$ are defined in
Eqs.~(\ref{ccredefs}) and (\ref{gauge boson - fermions couplings}),
respectively.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.3\textwidth]{2nu2eZ.png}
~~~~~~~
\includegraphics[width=0.3\textwidth]{2nu2eW.png}
\end{center}
\vskip-5truemm
\caption{\small $Z$- and $W$-exchange contributions to $\mathcal O^{V,LL}_{\nu e}$ with flavour indices $prst$.}
\label{2nu2eZWexchange}
\end{figure}

The indirect contributions from gauge-boson exchange to the matching
conditions of the other LEFT four-fermion operators are calculated
similarly. Most such operators can be generated via diagrams with the
exchange of a $Z^0$. A small subset of these also involve $W$-exchange
diagrams. And a few LEFT operators can be generated only via the
exchange of a $W$.

Finally, the matching conditions of certain LEFT operators receive
indirect contributions from Higgs exchange. As an example, consider
the operator $\mathcal O^{V,LR}_{\substack{ee\\prst}} \equiv
(\overline e_{Lp}\gamma_\mu e_{Lr})(\overline e_{Rs}\gamma^\mu
e_{Rt})$. 

The indirect contributions come from the diagrams of
Fig.~\ref{eLeRZhexchange}. The $Z$- and $h$-exchange contributions are
computed similarly to the previous examples. The indirect contribution
to the matching condition is
\begin{equation}
-\dfrac{\overline{g_Z}^2}{{M_Z}^2}[Z_{e_L}]^{\text{eff}}_{pr}[Z_{e_R}]^{\text{eff}}_{st}
-\dfrac1{2{m_h}^2}\,(Y_e)^{\text{eff}}_{pt}{(Y_e)^{\text{eff}}_{rs}}^* ~.
\end{equation}
The Yukawa coupling is [Eq.~(\ref{Yukawa couplings dimension 8}),
  repeated for convenience]
\[
(Y_e)^{\text{eff}}_{pr} = \dfrac{1+c_{H,\text{kin}}}{\sqrt2}\left[\dfrac{\sqrt2}{v_T}\,[M_e]_{pr}-\dfrac{{v_T^2}}{\Lambda^2}\,C_{\substack{e H\\pr}}-\dfrac{{v_T^4}}{\Lambda^4}\,C_{\substack{\chi e H^5\\pr}}\right] ~.
\]
The first term is $\sim m_e/v_T$ and is negligible. For this reason,
JMS, which works only to dimension 6, argues that the $h$-exchange
indirect contributions to the matching conditions are unimportant.
However, when one works to dimension 8, there is a non-negligible
contribution resulting from the square of the second term.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.3\textwidth]{eLeRZ.png}
~~~~~~~
\includegraphics[width=0.3\textwidth]{eLeRh.png}
\end{center}
\vskip-5truemm
\caption{\small $Z$- and $h$-exchange contributions to $\mathcal O^{V,LL}_{\nu e}$ with flavour indices $prst$.}
\label{eLeRZhexchange}
\end{figure}

\subsubsection{Subdominant contributions}

When one computes the matching conditions of LEFT operators to higher dimensions in SMEFT, one necessarily must take into account smaller contributions. For example, up to dimension-4 SMEFT operators, the smallest coefficient of a four-fermion LEFT operator is $O(1/v^2)$. At dimension 6, it is $O(1/\Lambda^2)$, and it falls to $O(v^2/\Lambda^4)$ at dimension 8 (technically, $O(v_T^2/\Lambda^4)$). 

In the above indirect contributions, we have approximated the gauge-boson (or Higgs) propagators as $1/v^2$. However, there are corrections proportional to $q^2 / v^2$, giving a contribution to the coefficient of $O(q^2/v^4)$. For $q \sim m_b$ (the largest low-energy scale), this is still $\ll 1/\Lambda^2$, so that it is unimportant for matching conditions up to dimension 6 (which is why JMS do not mention it). However, it {\it can be} of the same order as $O(v^2/\Lambda^4)$, the typical size of dimension-8 SMEFT contributions.

Now, this contribution is momentum-dependent, which means that it is process-dependent. For example, it may be important for $q \sim m_b$, but is not for $q \sim m_\mu$. For this reason, it is not included in our matching conditions, which are process-independent. Still, it should be included in any analysis of particular observables that considers dimension-8 SMEFT contributions. (Similarly, loop-level dimension-6 SMEFT contributions must also be taken into account \cite{Alonso:2013hga, Aebischer:2017gaw, Jenkins:2017dyc,
  Dekens:2019ept}.)

But this raises the question: what about momentum-dependent SMEFT contributions? Indirect contributions involve a propagator, approximated as $1/v^2$, and two vertices. If both vertices in a given diagram contribute the factor $v^2/\Lambda^2$, this saturates the dimension-8 contribution, so any momentum-dependent correction will be smaller. But if one vertex is $\sim 1$ (the SM) and the other vertex is $\sim v^2/\Lambda^2$, this yields a net contribution of $O(1/\Lambda^2)$, which contributes to the matching conditions at dimension 6. In this case, a subdominant contribution to the second vertex of $\sim q v/\Lambda^2$ would lead to a momentum-dependent SMEFT contribution, and could potentially be important. 

It turns out that such a subdominant contribution does not arise in diagrams with the exchange of a gauge boson. It is only diagrams with Higgs exchange that contain such a vertex term. (For example, these can be generated from operators in the dimension-6 SMEFT class $\psi^2H^2D$ (see Appendix \ref{evenSMEFTops}).) The subdominant SMEFT vertex is $\sim m v/\Lambda^2$ (the $m$ appears due to the Dirac equation), while the other vertex is the SM Higgs contribution, $\sim m/v$. The net contribution is
\begin{equation}
\dfrac1{v^2} \, \dfrac{v m}{\Lambda^2} \, \dfrac{m}{v} \sim \dfrac{m^2}{v^2\Lambda^2}
\ll\dfrac{v^2}{\Lambda^4} ~.
\end{equation}
The bottom line is that there are no sizeable momentum-dependent SMEFT contributions to the matching conditions. 

Finally, another possibility is in the Higgs-exchange contribution to the LEFT operator $\mathcal O^{S,LL}_{\substack{\nu\nu\\prst}}$ through lepton-number-violating Yukawa couplings. This corresponds to the processes shown in Fig.~\ref{4nuhexchange}, which have the total amplitude
\begin{equation}
\dfrac{i}{{m_h}^2} \, [(Y_\nu)^{\text{eff}}_{pr} \, (Y_\nu)^{\text{eff}}_{st} + (Y_\nu)^{\text{eff}}_{pt} \, (Y_\nu)^{\text{eff}}_{sr}] \, (\nu^T_{Lp}C\nu_{Lr})(\nu^T_{Ls}C\nu_{Lt}) ~.
\end{equation}
The key point here is that the Higgs coupling $(Y_\nu)^{\text{eff}}_{pr}$ is not proportional to the neutrino mass (which is tiny). There is also a contribution from the dimension-7 term, whose coefficient $\sim {v}^3/\Lambda^3$ [see Eq.~(\ref{neutrino Yukawa couplings dimension 8})]. This then leads to a coefficient 
\begin{equation}
\dfrac1{{v}^2} \, \dfrac{{v}^3}{\Lambda^3} \, \dfrac{{v}^3}{\Lambda^3} = \dfrac{{v}^4}{\Lambda^6} \ll \dfrac{{v}^2}{\Lambda^4} ~,
\end{equation}
i.e., it is still negligible.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.3\textwidth]{4nuh1.png}
~~~~~~~
\includegraphics[width=0.3\textwidth]{4nuh2.png}
\end{center}
\vskip-5truemm
\caption{\small $h$-exchange contributions to $\mathcal O^{S,LL}_{\nu\nu}$ with flavour indices $prst$.}
\label{4nuhexchange}
\end{figure}

\subsection{Results}

The matching conditions for all four-fermion LEFT operators up to dimension 8 in SMEFT are determined using the techniques described above for computing the direct and indirect contributions. For the LEFT magnetic dipole moment operators, three-gluon operators and neutrino mass terms, the calculations are straightforward, as there are no indirect contributions. The SMEFT matching conditions for all LEFT operators up to dimension 8 are given in the tables in Appendix \ref{Matchings}.

In the literature, the matching conditions of LEFT operators to dimension-8 SMEFT operators has only been performed in Refs.~\cite{Ardu:2021koz, Ardu:2022pzk}, where the focus was on LEFT operators that lead to lepton flavour violation. Our results agree with this analysis. Matching conditions to dimension-8 SMEFT operators have also been computed in Ref.~\cite{Boughezal:2021tih}, but in the context of high-energy processes. Although LEFT operators were not involved, there is still some overlap, and we agree here as well. Finally, the contributions of dimension-8 SMEFT operators to the SM parameters, as described in Sec.~\ref{Setup}, was also examined in Ref.~\cite{Ardu:2021koz}, and we are in agreement.


\section{Conclusions}

The modern thinking is that the Standard Model is the leading part of an effective field theory, produced when the heavy new physics is integrated out. This EFT is usually assumed to be the SMEFT, which includes the Higgs boson. The SMEFT has been well-studied – all operators up to dimension 8
have been worked out. 

When the heavy particles of the SM ($W^\pm$, $Z^0$, $H$, $t$) are also integrated out, one obtains the LEFT (low-energy EFT), applicable at scales $\le m_b$. In order to establish how low-energy measurements are affected by the underlying NP, it is necessary to determine how the LEFT operators depend on the SMEFT operators (the matching conditions). 

In Ref.~\cite{Jenkins:2017jig}, Jenkins, Manohar and Stoffer (JMS) present a complete and non-redundant basis of LEFT operators up to dimension 6, and compute the matching to SMEFT operators up to dimension 6. However, if the low-energy observable in question is suppressed in the SM and/or is very precisely measured, this may not be sufficient. Indeed, it has been pointed out that dimension-8 SMEFT contributions may be important for electroweak precision data from LEP, lepton-flavour-violating processes, meson-antimeson mixing, and electric dipole moments. 

In this paper, we extend the analysis of JMS: for all LEFT operators, we work out the complete matching conditions to SMEFT operators up to dimension 8. There are direct contributions to these matching conditions for all LEFT operators, and four-fermion operators also receive indirect contributions due to the exchange of a $W^\pm$, $Z^0$ and/or $H$. 

Should the analysis of a LEFT observable require information about dimension-8 SMEFT contributions, that information can be found here.

\section*{Acknowledgements}
We thank many people for helpful discussions about their work: M. Ardu
and S. Davidson (2103.07212), C.W. Murphy (2005.00059), E.E. Jenkins,
A.V. Manohar and P. Stoffer (1709.04486), C. Hays, A. Martin, V. Sanz
and J. Setford (1808.00442), R. Boughezal, E. Mereghetti and
F. Petriello (2106.05337).  This work was partially financially
supported by NSERC of Canada (SH, DL). JK is financially supported by
a postdoctoral research fellowship of the Alexander von Humboldt
Foundation.

