# Impact of dimension-eight SMEFT operators in the EWPO and Triple Gauge
Couplings analysis in Universal SMEFT

Tyler Corbett<EMAIL_ADDRESS>Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Wien, Austria  Jay Desai<EMAIL_ADDRESS>C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony
Brook New York 11794-3849, USA  O. J. P. Éboli<EMAIL_ADDRESS>Instituto de
Física, Universidade de São Paulo, São Paulo – SP, Brazil.  Departament de
Fisíca Quàntica i Astrofísica and Institut de Ciencies del Cosmos, Universitat
de Barcelona, Diagonal 647, E-08028 Barcelona, Spain  M. C. Gonzalez-Garcia
<EMAIL_ADDRESS>C.N. Yang Institute for Theoretical
Physics, Stony Brook University, Stony Brook New York 11794-3849, USA
Departament de Fisíca Quàntica i Astrofísica and Institut de Ciencies del
Cosmos, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys
23, 08010 Barcelona, Spain.  Matheus Martines<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, São Paulo – SP, Brazil.  Peter
Reimitz<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, São
Paulo – SP, Brazil.  Particle Theory Department, Fermilab, P.O. Box 500,
Batavia, IL 60510, USA

###### Abstract

We perform a complete study of the electroweak precision observables and
electroweak gauge boson pair production in terms of the SMEFT up to ${\cal
O}(1/\Lambda^{4})$ under the assumption of universal, C and P conserving new
physics. We show that the analysis of data from those two sectors allows us to
obtain closed constraints in the relevant parameter space in this scenario. In
particular we find that the Large Hadron Collider data can independently
constrain the Wilson coefficients of the dimension-six and -eight operators
directly contributing to the triple gauge boson vertices. Our results show
that the impact of dimension-eight operators in the study of triple gauge
couplings is small.

††preprint: YITP-SB-2023-04, FERMILAB-PUB-23-134-V,UWThPh 2023-13

##  I Introduction

During the last decade the Large Hadron Collider (LHC) has accumulated a large
amount of data that lead to further tests of the Standard Model (SM) and the
search for Physics beyond the Standard Model (BSM). Presently there is no
smoking gun indication of any extension of the SM. Therefore, one can assume
that there is a mass gap between the electroweak scale and the BSM scale. In
this scenario, the use of Effective Field Theory (EFT) Weinberg (1979); Georgi
(1984); Donoghue et al. (2014) as the tool to search for hints of new Physics
has become customary.

The EFT approach is suited for model–independent analyses since it is based
exclusively on the low-energy accessible states and symmetries. Assuming that
the scalar particle observed in 2012 Aad et al. (2012); Chatrchyan et al.
(2012) belongs to an electroweak doublet, we can realize the $SU(2)_{L}\otimes
U(1)_{Y}$ symmetry linearly. The resulting model is the so-called Standard
Model EFT (SMEFT). There have been many analyses of the LHC data using
dimension-six SMEFT; see for instance Corbett et al. (2012, 2013); Ellis et
al. (2015); Corbett et al. (2015); Butter et al. (2016); Baglio et al. (2017);
Barducci et al. (2018); Ellis et al. (2018); da Silva Almeida et al. (2019);
Brivio et al. (2020); Ellis et al. (2021); Dawson et al. (2020); Ethier et al.
(2021); Almeida et al. (2021) and references therein. In order to access the
convergence of the $1/\Lambda$ expansion, as well as avoid the appearance of
phase space regions where the cross section is negative Baglio et al. (2017),
it is important to perform the full calculation at order $1/\Lambda^{4}$. The
consistent calculation at order $1/\Lambda^{4}$ requires the introduction of
the contributions stemming from dimension-eight operators. In the most general
scenario, the number of dimension-eight operators contributing to the present
observables is extremely large Henning et al. (2017) and that precludes a
complete general analysis including all effects up to order $1/\Lambda^{4}$.
Due to its complexity, the systematic study of the ${\cal O}(1/\Lambda^{4})$
effects is still in its early stages. To date there have been a few case
studies for Drell-Yan Alioli et al. (2020); Boughezal et al. (2021, 2022); Kim
and Martin (2022); Allwicher et al. (2023), $\bar{t}tH$ production Dawson et
al. (2021), the production of electroweak gauge boson pairs Degrande and Li
(2023), Higgs boson processes Hays et al. (2019); Corbett et al. (2021a);
Martin and Trott (2022); Corbett (2021); Hays et al. (2020), and the
electroweak precision data Hays et al. (2020); Corbett et al. (2021b).

With this motivation, we perform a complete study of the electroweak precision
observables (EWPO) and electroweak diboson (EWDB) production at order
$1/\Lambda^{4}$ including all relevant dimension-six and dimension-eight
operators under the assumption of universal New Physics with conservation of
$C$ and $P$ Wells and Zhang (2016) so that the EFT contains only bosonic
operators after field redefinitions. In this case, we show that the analysis
of existing data from those two sectors allows one to obtain closed
constraints on the the full relevant parameter space. Furthermore, we argue
that it is still possible to perform the analysis sequentially, obtaining
first the constraints on four effective combinations of Wilson coefficients
using the EWPO, and then apply those bounds to reduce the number of Wilson
coefficients which are relevant for the the diboson analysis. Besides
demonstrating the feasibility of the analysis and deriving the corresponding
bounds, our main result is to show that in this scenario the impact of
dimension-eight operators in our present determination of the triple gauge
couplings (TGC) is small.

This work is organized as follows . The analysis framework employed is
presented in Sec. II. Sections III and IV contain the results of the analysis
of EWPO and of the EWDB data respectively. In Sec. V we summarize our
conclusions. We present in the appendices the full expressions of the
couplings of the electroweak gauge bosons to fermions and TGC to order ${\cal
O}(1/\Lambda^{4})$ in this scenario.

##  II Analysis framework

Following Wells and Zhang (2016), we consider a theory as universal if its EFT
can be expressed exclusively in terms of bosonic operators via field
redefinitions. We will also assume conservation of $C$ and $P$. The
requirement of the EFT to be universal limits the number of operators that
have to be considered and in Ref. Wells and Zhang (2016) the independent set
of dimension-six operators for universal theories is explicitly worked out in
several bases. In this work we use the Hagiwara, Ishihara, Szalapski, and
Zeppenfeld (HISZ) dimension-six basis Hagiwara et al. (1993, 1997). The
relevant set of operators left in HISZ basis for universal theories can be
straightforwardly adapted from the results in Ref. Wells and Zhang (2016) for
the SILH basis Giudice et al. (2007) taking into account the different choice
of two of the bosonic operators left in the basis. With this, one finds that
in the HISZ basis universal theories are described by 11 bosonic operators and
5 fermionic operators. The 11 bosonic operators are:

$\displaystyle{\cal
O}_{\Phi,1}=(D_{\mu}\Phi)^{\dagger}\Phi\Phi^{\dagger}(D^{\mu}\Phi)\;,$
$\displaystyle{\cal
O}_{\Phi,2}=\frac{1}{2}\partial^{\mu}\left(\Phi^{\dagger}\Phi\right)\partial_{\mu}\left(\Phi^{\dagger}\Phi\right)\;,$
$\displaystyle\hskip 5.69046pt{\cal
O}_{\Phi^{6}}=(\Phi^{\dagger}\Phi)^{3}\;,\;$ $\displaystyle{\cal
O}_{WW}=\Phi^{\dagger}\widehat{W}_{\mu\nu}\widehat{W}^{\mu\nu}\Phi\;,$
$\displaystyle{\cal
O}_{BB}=\Phi^{\dagger}\widehat{B}_{\mu\nu}\widehat{B}^{\mu\nu}\Phi\;,$
$\displaystyle{\cal
O}_{BW}=\Phi^{\dagger}\widehat{B}_{\mu\nu}\widehat{W}^{\mu\nu}\Phi\;,\;$
$\displaystyle{\cal
O}_{W}=(D_{\mu}\Phi)^{\dagger}\widehat{W}^{\mu\nu}(D_{\nu}\Phi)\;,$
$\displaystyle{\cal
O}_{B}=(D_{\mu}\Phi)^{\dagger}\widehat{B}^{\mu\nu}(D_{\nu}\Phi)\;,$
$\displaystyle{\cal O}_{WWW}={\rm
Tr}[\widehat{W}_{\mu}^{\nu}\widehat{W}_{\nu}^{\rho}\widehat{W}_{\rho}^{\mu}]\;,\;$
$\displaystyle{\cal O}_{GG}=\Phi^{\dagger}\Phi\;G_{\mu\nu}^{a}G^{a\mu\nu}\;,$
$\displaystyle{\cal
O}_{GGG}=g_{s}^{3}f^{abc}{G}_{\mu}^{a\,\nu}{G}_{\nu}^{b\,\rho}{G}_{\rho}^{c\,\mu}\;,$
(1)

where $\Phi$ stands for the SM Higgs doublet and we have defined
$\widehat{B}_{\mu\nu}\equiv i(g^{\prime}/2)B_{\mu\nu}$ and
$\widehat{W}_{\mu\nu}\equiv i(g/2)\sigma^{a}W^{a}_{\mu\nu}$, with $g_{s}$, $g$
and $g^{\prime}$ being the $SU(3)_{C}$, $SU(2)_{L}$ and $U(1)_{Y}$ gauge
couplings, respectively. $\sigma^{a}$ stands for the Pauli matrices while
$f^{abc}$ are the $SU(3)_{C}$ structure constants.

Five four-fermion operators are generated when applying the equations of
motion (EOM) to eliminate bosonic operators involving the square of
derivatives of the gauge strength tensors and four Higgs fields in total
analogy with the SILH basis in Ref. Wells and Zhang (2016):

$\displaystyle{\cal O}_{y}=|\Phi|^{2}(\Phi_{\alpha}J_{y}^{\alpha}+{\rm
h.c.})\;,$ $\displaystyle{\cal O}_{2y}=J^{\dagger}_{y\alpha}J^{\alpha}_{y}\;,$
$\displaystyle{\cal
O}_{2JW}=\sum_{f,f^{\prime}\in\\{Q,L\\}}\left(\bar{f}\gamma_{\mu}\frac{\sigma^{a}}{2}f\right)\left(\bar{f^{\prime}}\gamma^{\mu}\frac{\sigma^{a}}{2}f^{\prime}\right)\;,$
$\displaystyle{\cal
O}_{2JB}=\sum_{f,f^{\prime}\in\\{Q,L,u,d,e\\}}\left(Y_{f}\bar{f}\gamma_{\mu}f\right)\left(Y_{f^{\prime}}\bar{f^{\prime}}\gamma^{\mu}f^{\prime}\right)\;,$
$\displaystyle{\cal
O}_{2JG}=\sum_{f,f^{\prime}\in\\{Q,u,d\\}}\left(\bar{f}\gamma_{\mu}T^{a}f\right)\left(\bar{f^{\prime}}\gamma^{\mu}T^{a}f^{\prime}\right)\;,$
(2)

where $Y_{f}$ are the hypercharges, $Q$ and $L$ are the quark and lepton
doublets and $u$, $d$ and $e$ represent the fermion singlets. In addition,
$T^{a}$ are the Gell-Mann matrices, $y_{f}$ are the Yukawa matrices, and

$J_{y}^{\alpha}=\bar{u}y_{u}^{\dagger}Q_{\beta}\epsilon^{\alpha\beta}+\bar{Q}^{\alpha}y_{d}d+\bar{L}^{\alpha}y_{e}e\;.$

For the dimension-eight operators, we will work in the basis defined in Ref.
Murphy (2020). For universal theories the potentially relevant bosonic
operators for our analyses belong to the classes $\Phi^{6}D^{2}$,
$X^{3}\Phi^{2}$, $X^{2}\Phi^{4}$, and $X\Phi^{4}D^{2}$ with $X$ standing for a
field strength tensor. This includes:
$\bullet$ two operators in the class $\Phi^{6}D^{2}$ related to the dimension-
six ${\cal O}_{\Phi,1}$ and ${\cal O}_{\Phi,2}$:

${\cal
O}^{(1)}_{D^{2}\Phi^{6}}=(\Phi^{\dagger}\Phi)^{2}(D_{\mu}\Phi)^{\dagger}D^{\mu}\Phi\qquad\hbox{and}\qquad{\cal
O}^{(2)}_{D^{2}\Phi^{6}}=(\Phi^{\dagger}\Phi)(\Phi^{\dagger}\sigma^{I}\Phi)(D_{\mu}\Phi)^{\dagger}\sigma^{I}D^{\mu}\Phi\;,$
(3)

$\bullet$ two $CP$ conserving operators in class $X^{3}\Phi^{2}$ that
contribute to the EWDB analysis are

${\cal O}^{(1)}_{W^{3}\Phi^{2}}=(\Phi^{\dagger}\Phi){\rm
Tr}[\widehat{W}^{\mu}_{\nu}\widehat{W}^{\nu}_{\rho}\widehat{W}^{\rho}_{\mu}]\qquad\hbox{and}\qquad{\cal
O}^{(1)}_{W^{2}B\Phi^{2}}=\frac{g^{3}s_{W}}{8c_{W}}\epsilon^{IJK}\Phi^{\dagger}\sigma^{I}\Phi
B^{\mu}_{\nu}W^{J\,\nu}_{\rho}W^{K\,\rho}_{\mu}\;,$ (4)

$\bullet$ two operators in class $X\Phi^{4}D^{2}$ contributing to anomalous
TGC are siblings of dimension-six operators ${\cal O}_{B}$ and ${\cal O}_{W}$:

${\cal
O}^{(1)}_{B\Phi^{4}D^{2}}=(\Phi^{\dagger}\Phi)(D_{\mu}\Phi)^{\dagger}\widehat{B}^{\mu\nu}D_{\nu}\Phi\qquad\hbox{and}\qquad{\cal
O}^{(1)}_{W\Phi^{4}D^{2}}=(\Phi^{\dagger}\Phi)(D_{\mu}\Phi)^{\dagger}\hat{W}^{\mu\nu}D_{\nu}\Phi\;,$
(5)

$\bullet$ four operators in the $X^{2}\Phi^{4}$ class

$\displaystyle{\cal
O}^{(1)}_{W^{2}\Phi^{4}}=(\Phi^{\dagger}\Phi)\Phi^{\dagger}\widehat{W}_{\mu\nu}\widehat{W}^{\mu\nu}\Phi\;,$
$\displaystyle{\cal
O}^{(1)}_{B^{2}\Phi^{4}}=(\Phi^{\dagger}\Phi)^{2}\widehat{B}_{\mu\nu}\widehat{B}^{\mu\nu}\;,$
(6) $\displaystyle{\cal
O}^{(1)}_{BW\Phi^{4}}=(\Phi^{\dagger}\Phi)\Phi^{\dagger}\widehat{W}_{\mu\nu}\Phi\widehat{B}^{\mu\nu}\;,$
$\displaystyle{\cal
O}^{(3)}_{W^{2}\Phi^{4}}=\Phi^{\dagger}\widehat{W}_{\mu\nu}\Phi\Phi^{\dagger}\widehat{W}^{\mu\nu}\Phi\;.$
(7)

In addition some dimension-eight fermionic operators will be generated by the
EOM in analogy to the dimension-six case. Presently there there is no study of
the fermionic operators compatible with universal theories for the dimension-
eight basis. So in what follows, we assume that only four-fermion operators
are generated in exchanging a subset of the purely bosonic operators defining
the universal basis for fermionic operators.

It is important to notice that not all operators listed above appear in the
analysis of EWPO and EWDB data even after accounting for their finite
renormalization contribution to the SM parameters. In this work, we adopt as
input parameters $\\{\widehat{\alpha}_{\rm
em}\;,\;\widehat{G}_{F}\;,\;\widehat{M}_{Z}\\}$ and consider the following
three relations to define the renormalized parameters

$\displaystyle\hat{e}=\sqrt{4\pi\hat{\alpha}_{\rm em}}\;,$
$\displaystyle\hat{v}^{2}=\frac{1}{\sqrt{2}\,\hat{G}_{F}}\;,$ (8)
$\displaystyle\hat{c}^{2}\hat{s}^{2}=\frac{\pi\widehat{\alpha}_{\rm
em}}{\sqrt{2}\,\widehat{G}_{F}\widehat{M}^{2}_{Z}}\;,$

where $\hat{s}$ ($\hat{c}$) is the sine (cosine) of the weak mixing angle
$\hat{\theta}$.

The predictions of SMEFT at order $1/\Lambda^{4}$ and the input parameters in
Eq. (8) allow us to obtain the SM mixing angle, electric charge, and the Higgs
vev as a function of the input parameters and some of dimension-six and -eight
Wilson coefficients. In this process the operators ${\cal O}_{WW}$, ${\cal
O}_{BB}$, ${\cal O}^{(1)}_{W^{2}\Phi^{4}}$, and ${\cal
O}^{(1)}_{B^{2}\Phi^{4}}$ induce an overall renormalization of the $W^{a}$ and
$B$ field wave functions that can be absorbed by a redefinition of the
coupling constants. Furthermore, the contribution of ${\cal
O}^{(1)}_{D^{2}\Phi^{6}}$ to the Higgs vev cancels against its contribution to
the renormalization of the $W^{a}$ and $B$ field wave functions. Consequently
their coefficients drop out of any of the predictions in the EWPO and EWDB
data (see the appendix for the explicit expressions).

Altogether the effective Lagrangian considered in this work reads:

$\displaystyle{\cal L}_{\rm eff}=$ $\displaystyle{\cal
L}_{SM}+\frac{f_{WWW}}{\Lambda^{2}}{\cal
O}_{WWW}+\frac{f_{W}}{\Lambda^{2}}{\cal O}_{W}+\frac{f_{B}}{\Lambda^{2}}{\cal
O}_{B}+\frac{f_{BW}}{\Lambda^{2}}{\cal
O}_{BW}+\frac{f_{\Phi,1}}{\Lambda^{2}}{\cal
O}_{\Phi,1}+\frac{f_{4F}}{\Lambda^{2}}{\cal O}_{4F}$ (9)
$\displaystyle+\frac{f^{(2)}_{D^{2}\Phi^{6}}}{\Lambda^{4}}{\cal
O}^{(2)}_{D^{2}\Phi^{6}}+\frac{f^{(1)}_{W^{3}\Phi^{2}}}{\Lambda^{4}}{\cal
O}^{(1)}_{W^{3}\Phi^{2}}+\frac{f^{(1)}_{W^{2}B\Phi^{2}}}{\Lambda^{4}}{\cal
O}^{(1)}_{W^{2}B\Phi^{2}}+\frac{f^{(1)}_{B\Phi^{4}D^{2}}}{\Lambda^{4}}{\cal
O}^{(1)}_{B\Phi^{4}D^{2}}$
$\displaystyle+\frac{f^{(1)}_{W\Phi^{4}D^{2}}}{\Lambda^{4}}{\cal
O}^{(1)}_{W\Phi^{4}D^{2}}+\frac{f^{(3)}_{W^{2}\Phi^{4}}}{\Lambda^{4}}{\cal
O}^{(3)}_{W^{2}\Phi^{4}}+\frac{f^{(1)}_{BW\Phi^{4}}}{\Lambda^{4}}{\cal
O}^{(1)}_{BW\Phi^{4}}+\frac{\Delta^{(8)}_{4F}}{\Lambda^{4}}{\cal
O}_{4F}^{(8)}\;,$

where ${\cal O}_{4F}$ stands for the part of ${\cal O}_{2JW}$ that contributes
to the muon decay while ${\cal O}_{4F}^{(8)}$ is the corresponding dimension-
eight operator. They have been defined so that their contribution to the Higgs
field vacuum expectation value in the SM Lagrangian reads

$\left[2\langle\Phi^{\dagger}\Phi\rangle-\frac{1}{\sqrt{2}\hat{G}_{F}}\right]_{\rm
fermionic}\equiv\frac{\hat{v}^{4}}{\Lambda^{2}}\Delta_{4F}+\frac{\hat{v}^{6}}{\Lambda^{4}}\Delta^{(8)}_{4F}\;.$
(10)

The predictions for observables at order $1/\Lambda^{4}$ require evaluating
the SM contributions, the interference between the $1/\Lambda^{2}$ amplitude
(${\cal M}^{(6)}$) with the SM amplitude, the square of the dimension-six
amplitude, as well as the interference of the dimension-eight amplitude ${\cal
M}^{(8)}$ with the SM one, that we represent as:

$|M_{\rm SM}|^{2}+{\cal M}_{\rm SM}^{\star}{\cal M}^{(6)}+|{\cal
M}^{(6)}|^{2}+{\cal M}_{\rm SM}^{\star}{\cal M}^{(8)}\;.$ (11)

Notice that ${\cal M}^{(8)}$ includes dimension-eight vertices as well as the
contribution of the insertion of two dimension-six couplings in the amplitude.

##  III EWPO analysis

Our EWPO analysis includes 14 observables of which 12 are $Z$ observables
Schael et al. (2006):

$\displaystyle\Gamma_{Z}\;\;,\;\;\sigma_{h}^{0}\;\;,\;\;{\cal
A}_{\ell}(\tau^{\rm pol})\;\;,\;\;R^{0}_{\ell}\;\;,\;\;{\cal A}_{\ell}({\rm
SLD})\;\;,\;\;A_{\rm FB}^{0,l}\;\;,\;\;$ $\displaystyle
R^{0}_{c}\;\;,\;\;R^{0}_{b}\;\;,\;\;{\cal A}_{c}\;\;,\;\;{\cal
A}_{b}\;\;,\;\;A_{\rm FB}^{0,c}\;\;,\;\;\hbox{ and}\;\;A_{\rm FB}^{0,b}\hbox{
(SLD/LEP-I)}\;\;\;,$

supplemented by two $W$ observables

$M_{W}\;\;,\;\;\Gamma_{W}\;\;$

that are, respectively, its average $W$-boson mass taken from Patrignani et
al. (2016)111In order to be conservative we did not take into account the
recent CDF measurement of the $W$ mass Aaltonen et al. (2022)., its width from
LEP2/Tevatron Group (2010)222We do not include the average leptonic $W$
branching ratio because it does not include any additional constraint for
universal EFT.. The correlations among these inputs Schael et al. (2006) are
taken into consideration in the analyses. The SM predictions and their
uncertainties due to variations of the SM parameters were extracted from de
Blas et al. (2022).

The statistical analysis of the EWPO data is made by means of a binned log-
likelihood function defining a $\chi^{2}$ function which depends on seven
Wilson coefficients,

$\displaystyle\chi^{2}_{\rm EWPO}$ $\displaystyle\equiv$
$\displaystyle\chi^{2}_{\rm
EWPO}\left(f_{BW},f_{\Phi,1},\Delta_{4F},f^{(1)}_{BW\Phi^{4}},f^{(2)}_{D^{2}\Phi^{6}},\Delta_{4F}^{(8)},f^{(3)}_{W^{2}\Phi^{4}}\right)\;.$
(12)

In fact, EWPO cannot constrain the seven Wilson coefficients independently.
This is so because, as described in the Appendix A, the corrections to the $Z$
interaction to fermions to order $\Lambda^{-4}$ can be expressed in terms of
the following three combinations of Wilson coefficients:

$\displaystyle\widetilde{\Delta}_{4F}=\Delta_{4F}+\frac{\hat{v}^{2}}{\Lambda^{2}}\Delta_{4F}^{(8)}\;,$
$\displaystyle\widetilde{f}_{BW}=f_{BW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{BW\Phi^{4}}\;,$
(13)
$\displaystyle\widetilde{f}_{\Phi,1}=f_{\Phi,1}+\frac{\hat{v}^{2}}{\Lambda^{2}}f^{(2)}_{D^{2}\Phi^{6}}\;.$

The corrections to the $W$ mass and coupling to fermions further involve the
addition of only one operator ${\cal O}^{(3)}_{W^{2}\Phi^{4}}$. Using these
variables we incorporate in our calculation some higher order terms in the
$1/\Lambda$ expansion in the spirit of geometric SMEFT Helset et al. (2020);
Corbett et al. (2021b).

These three coefficient combinations and $f^{(3)}_{W^{2}\phi^{4}}$ are
directly related to the contributions to the oblique $S$, $T$, $U$ parameters
Peskin and Takeuchi (1990), and $\delta G_{F}$ at linear order in Wilson
coefficients of operators up to dimension-eight:

$\alpha
S=-\hat{e}^{2}\frac{\hat{v}^{2}}{\Lambda^{2}}\widetilde{f}_{BW}\;,\;\;\;\;\alpha
T=-\frac{\hat{v}^{2}}{2\Lambda^{2}}\widetilde{f}_{\Phi,1}\;,\;\;\;\;\alpha
U=\hat{e}^{2}\frac{\hat{v}^{4}}{\Lambda^{4}}f^{(3)}_{W^{2}\Phi^{4}}\;\;\;,\;\;\frac{\delta
G_{F}}{\hat{G}_{F}}=\frac{\hat{v}^{2}}{\Lambda^{2}}\widetilde{\Delta}_{4F}\;.$
(14)

It is interesting to notice that there is a contribution to the oblique
parameter $U$ at dimension-eight. Thus, effectively the EWPO chi-squared
function is:

$\displaystyle\tilde{\chi}^{2}_{\rm EWPO}$ $\displaystyle\equiv$
$\displaystyle\tilde{\chi}^{2}_{\rm
EWPO}(\tilde{f}_{BW},\tilde{f}_{\Phi,1},f^{(3)}_{W^{2}\Phi^{4}},\widetilde{\Delta}_{4F})\;.$
(15)

Figure 1: One- and two-dimensional projections of $\Delta\tilde{\chi}^{2}_{\rm
EWPO}$ for the coefficients $\tilde{f}_{BW}\hat{v}^{2}/\Lambda^{2}$,
$\tilde{f}_{\Phi,1}\hat{v}^{2}/\Lambda^{2}$, $\delta G_{F}/\hat{G}_{F}$, and
$f^{(3)}_{W^{2}\Phi^{4}}\hat{v}^{4}/\Lambda^{4}$, as indicated in each panel
after marginalizing over the undisplayed parameters.

Figure 1 shows the one- and two-dimensional projections of
$\Delta\tilde{\chi}^{2}_{\rm EWPO}$ as a function of the coefficients
$\widetilde{f}_{BW}\hat{v}^{2}/\Lambda^{2}$,
$\widetilde{f}_{\Phi,1}\hat{v}^{2}/\Lambda^{2}$, $\delta G_{F}/\hat{G}_{F}$,
and $f^{(3)}_{W^{2}\Phi^{4}}\hat{v}^{4}/\Lambda^{4}$. The panels in the top
row contain the one-dimension marginalized projection of
$\Delta\tilde{\chi}^{2}_{\rm EWPO}$, where the dashed line stands for the
${\cal O}(1/\Lambda^{2})$ analysis while the green solid one also contains the
dimension-six squared contribution. The full analysis that includes the
dimension-eight contribution is represented by the blue line. As seen in the
figure, the results at linear dimension-six and dimension-six squared are
identical, which is expected given the precision of the data.

From Fig. 1 we also see that once the dimension-eight coefficient
$f^{(3)}_{W^{2}\Phi^{4}}$ is included the bounds on
$\widetilde{f}_{\Phi,1}\hat{v}^{2}/\Lambda^{2}$ and $\delta G_{F}/\hat{G}_{F}$
weaken by about a factor 3–4. The main reason is that when
$f^{(3)}_{W^{2}\Phi^{4}}$ is also included in the analysis cancellations can
occur. In particular as can be seen in Eqs. (25)–(27) for

$\widetilde{f}_{\Phi,1}=-2\widetilde{\Delta}_{4F}=\frac{\hat{e}^{2}}{2\hat{s}^{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}f^{(3)}_{W^{2}\Phi^{4}}\;$
(16)

the linear contributions from $\widetilde{f}_{\Phi,1}$ (i.e. $T$),
$\widetilde{\Delta}_{4F}$ ($\delta G_{F}/\hat{G}_{F}$) and
$f^{(3)}_{W^{2}\Phi^{4}}$ ($U$), cancel both in the Z observables and in
$M_{W}$. Therefore, along this direction in the parameter space, the bounds on
these three quantities dominantly come from the contribution of $\Gamma_{W}$
in Eq. (29), but this observable is less precisely determined. Hence the
strong correlations we observe in the corresponding two-dimensional allowed
regions in Fig. 1. Nevertheless, the limits are still quite stringent; see
Table 1.

| EWPO 95% CL allowed range
---|---
Coupling | dimension 6 | dimension 8
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,\tilde{f}_{BW}$ | $[-0.018,0.044]$ | $[-0.018,0.044]$
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,\tilde{f}_{\Phi,1}$ | $[-0.0028,0.0018]$ | $[-0.080,0.081]$
$\frac{\delta G_{F}}{\hat{G}_{F}}$ | $[-0.0016,0.0017]$ | $[-0.038,0.044]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f^{(3)}_{W^{2}\Phi^{4}}$ |  — | $[-0.40,0.36]$
Table 1: 95% CL allowed ranges for the effective couplings entering in the
EWPO with the analysis done including only the dimension-six contributions
(left column) and also the dimension-eight contributions (right column).

##  IV Diboson analysis

The electroweak production of $WZ$, $WW$ and $W\gamma$ pairs, as well as the
vector boson fusion production of $Z$’s ($Zjj$), collectively denoted by EWDB,
allow us to study the triple couplings of electroweak gauge bosons. In this
work we consider the EWDB data shown in Table 2 which comprise a total of 73
data points.

| Channel ($a$) | Distribution | # bins |  Data set | Int Lum
---|---|---|---|---|---
| $WZ\to\ell^{+}\ell^{-}\ell^{\prime\pm}$ | $M(WZ)$ | 7 | CMS 13 TeV, | 137.2 fb-1 CMS Collaboration (2021a)
EWDB data | $WW\to\ell^{+}\ell^{(\prime)-}+0/1j$ | $M(\ell^{+}\ell^{(\prime)-})$ | 11 | CMS 13 TeV, | 35.9 fb-1 Sirunyan et al. (2020)
$W\gamma\to\ell\nu\gamma$ | $\frac{d^{2}\sigma}{dp_{T}d\phi}$ | 12 | CMS 13 TeV, | 137.1 fb-1 CMS Collaboration (2021b)
$WW\rightarrow e^{\pm}\mu^{\mp}+/\\!\\!\\!E_{T}\;(0j)$ | $m_{T}$ | 17 (15) | ATLAS 13 TeV, | 36.1 fb-1 Aaboud et al. (2018)
$WZ\rightarrow\ell^{+}\ell^{-}\ell^{(\prime)\pm}$ | $m_{T}^{WZ}$ | 6 | ATLAS 13 TeV, | 36.1 fb-1 ATLAS Collaboration (2018)
$Zjj\to\ell^{+}\ell^{-}jj$ | $\frac{d\sigma}{d\phi}$ | 12 | ATLAS 13 TeV, | 139 fb-1 Aad et al. (2021a)
$WW\rightarrow\ell^{+}\ell^{(\prime)-}+/\\!\\!\\!E_{T}\;(1j)$ | $\frac{d\sigma}{dm_{\ell^{+}\ell^{-}}}$ | 10 | ATLAS 13 TeV, | 139 fb-1 Aad et al. (2021b)
Table 2: EWDB data from LHC used in the analyses. For the $W^{+}W^{-}$ results
from ATLAS run 2 Aaboud et al. (2018) we combine the data from the last three
bins into one to ensure gaussianity.

The theoretical predictions needed for the EWDB data are obtained by
simulating at leading order the $W^{+}W^{-}$, $W^{\pm}Z$, $W^{\pm}\gamma$, and
$Zjj$ channels that receive contributions from TGC. To this end, we use
MadGraph5_aMC@NLO Frederix et al. (2018) with the UFO files for our effective
Lagrangian generated with FeynRules Christensen and Duhr (2009); Alloul et al.
(2014). We employ PYTHIA8 Sjostrand et al. (2008) to perform the parton shower
and hadronization, while the fast detector simulation is carried out with
Delphes de Favereau et al. (2014). Jet analyses are performed using FASTJET
Cacciari et al. (2012).

The results of the analysis can be qualitatively understood in terms of the
effective $\gamma W^{+}W^{-}$ and $ZW^{+}W^{-}$ TGC introduced in Ref.
Hagiwara et al. (1987)

$\displaystyle{\cal L}_{WWV}=$ $\displaystyle-
ig_{WWV}\Big{\\{}g_{1}^{V}\Big{(}W^{+}_{\mu\nu}W^{-\,\mu}V^{\nu}-W^{+}_{\mu}V_{\nu}W^{-\,\mu\nu}\Big{)}$
(17)
$\displaystyle+\kappa_{V}W_{\mu}^{+}W_{\nu}^{-}V^{\mu\nu}+\frac{\lambda_{V}}{\hat{M}_{W}^{2}}W^{+}_{\mu\nu}W^{-\,\nu\rho}V_{\rho}^{\;\mu}\Big{\\}}\;\;,$

where $g_{WW\gamma}=\hat{e}$, $g_{WWZ}=\hat{e}\hat{c}/\hat{s}$, and
$\hat{M}_{W}=\hat{e}\hat{v}/2\hat{s}$. In the SM
$g_{1}^{\gamma}=g_{1}^{Z}=\kappa_{\gamma}=\kappa_{Z}=1$ and
$\lambda_{Z}=\lambda_{\gamma}=0$. After including the direct contribution from
the dimension-six and dimension-eight operators, electromagnetic gauge
invariance still enforces $g_{1}^{\gamma}=1$, while the other effective TGC
couplings read:

$\displaystyle\Delta
g_{1}^{Z}=\frac{\hat{e}^{2}}{\hat{s}^{2}\hat{c}^{2}}\left[\frac{1}{8}\frac{\hat{v}^{2}}{\Lambda^{2}}\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)\right]\;,$
$\displaystyle\Delta\kappa_{\gamma}=\frac{\hat{e}^{2}}{\hat{s}^{2}}\left[\frac{1}{8}\frac{\hat{v}^{2}}{\Lambda^{2}}\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}+f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)\right]\;,$
$\displaystyle\Delta\kappa_{Z}=\frac{\hat{e}^{2}}{\hat{s}^{2}}\left[\frac{1}{8}\frac{\hat{v}^{2}}{\Lambda^{2}}\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)-\frac{\hat{s}^{2}}{8\hat{c}^{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}\left(f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)\right]\;,$
(18)
$\displaystyle\lambda_{\gamma}=\frac{3\hat{e}^{2}}{2\hat{s}^{2}}\frac{\hat{M}_{W}^{2}}{\Lambda^{2}}\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W^{3}\Phi^{2}}\right]-\frac{\hat{M}_{W}^{4}}{2\Lambda^{4}}f^{(1)}_{W^{2}B\Phi^{2}}\;,$
$\displaystyle\lambda_{Z}=\frac{3\hat{e}^{2}}{2\hat{s}^{2}}\frac{\hat{M}_{W}^{2}}{\Lambda^{2}}\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W^{3}\Phi^{2}}\right]+\frac{\hat{M}_{W}^{4}}{2\Lambda^{4}}\frac{\hat{s}^{2}}{\hat{c}^{2}}f^{(1)}_{W^{2}B\Phi^{2}}\;.$

The complete analysis of diboson production at fixed order $1/\Lambda^{4}$
depends on not only the direct SMEFT contributions to TGC in (18), but also on
the indirect contributions from ${\cal O}_{BW}$, ${\cal O}_{\Phi,1}$, ${\cal
O}_{4F}$, ${\cal O}^{(1)}_{BW\Phi^{4}}$, ${\cal O}^{(2)}_{D^{2}\Phi^{6}}$, and
${\cal O}_{4F}^{(8)}$, through renormalization of the SM gauge couplings to
fermions and TGC. In appendix B we list the complete expressions and show
that, in fact, the indirect effects involve the same there combinations (13)
and are therefore bounded by the EWPO. In light of the constraints derived in
the previous section, in what follows we will neglect the effect of those
operators in the EWDB data analysis.

For the direct effects, Eq. (18) explicitly shows that the contributions of
the dimension-eight operators ${\cal O}^{(1)}_{W\Phi^{4}D^{2}}$, ${\cal
O}^{(1)}_{B\Phi^{4}D^{2}}$ and ${\cal O}^{(1)}_{W^{3}\Phi^{2}}$ to the TGC
couplings have the same structure of the contributions from the dimension-six
operators ${\cal O}_{W}$, ${\cal O}_{B}$ and ${\cal O}_{WWW}$, respectively.
Conversely, ${\cal O}^{(1)}_{W^{2}B\Phi^{2}}$ contributes a purely ${\cal
O}(1/\Lambda^{4})$ to $\lambda_{\gamma}\neq\lambda_{Z}$.

Following an approach equivalent to that employed for the analysis of EWPO we
can define three effective coefficients,

$\displaystyle\tilde{f}_{W}=f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\;,$
$\displaystyle\tilde{f}_{B}=f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\;,$
(19)
$\displaystyle\tilde{f}_{WWW}=f_{WWW}+\frac{\hat{v}^{2}}{2}f^{(1)}_{W^{3}\Phi^{2}}\;.$

which, together with $f^{(1)}_{W^{2}B\Phi^{2}}$, effectively parametrize the
relevant contributions to the EWDB analysis.

Following this approach we perform the statistical analysis of the EWDB data
using a binned chi-squared function defined in terms of these effective
coefficients

$\tilde{\chi}^{2}_{\rm
EWDB}\left(\tilde{f}_{W},\tilde{f}_{B},\tilde{f}_{WWW},f^{(1)}_{W^{2}B\Phi^{2}}\right)\;.$
(20)

Fig. 2 depicts the one- and two-dimensional marginalized 68% and 95% CL
allowed regions for $\tilde{f}_{W}\hat{v}^{2}/\Lambda^{2}$,
$\tilde{f}_{B}\hat{v}^{2}/\Lambda^{2}$,
$\tilde{f}_{WWW}\hat{v}^{2}/\Lambda^{2}$, and
$f^{(1)}_{W^{2}B\Phi^{2}}\hat{v}^{4}/\Lambda^{4}$ after marginalizing over the
remaining fit parameters. The light pink (blue) regions in these panels
correspond to the 68% (95%) CL allowed regions of the ${\cal
O}(1/\Lambda^{2})$ analysis; see the three lower panels. This analysis yields
the marginalized 95% CL, allowed intervals for the Wilson coefficients of the
three relevant dimension-six operators displayed in the left column of Table
3.

Figure 2:  One- and two-dimensional projections of $\Delta\tilde{\chi}^{2}_{\rm EWDB}$ for the effective coefficients $\tilde{f}_{W}\hat{v}^{2}/\Lambda^{2}$, $\tilde{f}_{B}\hat{v}^{2}/\Lambda^{2}$, $\tilde{f}_{WWW}\hat{v}^{2}/\Lambda^{2}$, and $f^{(1)}_{W^{2}B\Phi^{2}}\hat{v}^{4}/\Lambda^{4}$ as indicated in each panel after marginalizing over the undisplayed parameters.  | EWDB 95% CL allowed range
---|---
Coupling | dimension 6 | (dimension 6)2 | dimension 8
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,\tilde{f}_{B}$ | $[-3.3,1.8]$ | $[-0.75,0.83]$ | $[-0.73,0.86]$
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,\tilde{f}_{W}$ | $[-0.11,0.085]$ | $[-0.079,0.16]$ | $[-0.080,0.16]$
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,\tilde{f}_{WWW}$ | $[-0.22,0.16]$ | $[-0.049,0.045]$ | $[-0.048,0.049]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f^{(1)}_{W^{2}B\Phi^{2}}$ |  — |  — | $[-1.9,4.2]$
Table 3: 95% CL allowed ranges for the effective couplings entering in the
EWDB analysis including only up to the dimension-six contributions (left
column), up to the dimension-six squared contributions (central column) and
including also the dimension-eight contributions (right column).

The dark red (blue) shaded regions in Fig. 2 represent the two-dimensional
allowed regions at 68% (95%) C.L. including also the dimension-six squared and
the dimension-eight contributions. The corresponding one dimensional
projections are given in the blue lines in the upper panels. For the sake of
comparison we also show the corresponding results including only the
dimension-six squared contributions. These are the black dashed lines in the
one-dimensional projections in the upper panels and the dotted line contours
in the three lower panels. From the figure we see that including the
$1/\Lambda^{4}$ effects lead to stronger bounds on the effective coefficients
$\tilde{f}_{B}$ and $\tilde{f}_{WWW}$ while the bound for $\tilde{f}_{W}$ is
slightly looser and shifted; see also the central and right columns of Table
3. We traced the counter-intuitive behaviour of the bounds on $\tilde{f}_{W}$
to the $WZ$ datasets. Removing $WZ$ production from the fit leads to stronger
limits at the ${\cal O}(1/\Lambda^{4})$ also for $\tilde{f}_{W}$.

The results in Fig. 2 also show that the dimension-six squared terms are
dominant over the dimension-eight one. Or in other words, the inclusion of the
relevant dimension-eight operator in this analysis, ${\cal
O}^{(1)}_{W^{2}B\Phi^{2}}$, has very little impact on the results. The
physical reason for this can be traced to the different dependence on the
partonic center-of-mass energy ($\hat{S}$) of the contribution to the relevant
squared amplitudes from dimension-six squared and dimension-eight terms. As it
is well-known, the anomalous TGCs spoil the cancellations that take place in
the SM allowing the scattering amplitudes to grow with the partonic center-of-
mass energy. The fastest growing amplitudes are (for $\hat{S}\gg m_{W,Z}$):

$\displaystyle{\cal M}\left(d_{-}\bar{d}_{+}\to
W^{+}_{0}W^{-}_{0}\right)=-i\frac{\hat{e}^{2}}{24\hat{s}^{2}\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[3\hat{c}^{2}\,f_{W}-\hat{s}^{2}\,f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(3\hat{c}^{2}\,f^{(1)}_{W\Phi^{4}D^{2}}-\hat{s}^{2}\,f^{(1)}_{B\Phi^{4}D^{2}}\right)\right]\;,$
$\displaystyle{\cal M}\left(d_{+}\bar{d}_{-}\to
W^{+}_{\pm}W^{-}_{\pm}\right)={i\frac{\hat{e}^{2}}{48\hat{s}^{2}\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\,\frac{\hat{v}^{2}}{\Lambda^{2}}f^{(1)}_{W^{2}B\Phi^{2}}}\;,$
$\displaystyle{\cal M}\left(d_{-}\bar{d}_{+}\to
W^{+}_{\pm}W^{-}_{\pm}\right)=-i\frac{3\hat{e}^{4}}{8\hat{s}^{4}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(f^{(1)}_{W^{3}\Phi^{2}}+{\frac{\hat{s}^{2}}{18\hat{c}^{2}}}f^{(1)}_{W^{2}B\Phi^{2}}\right)\right]\;,$
$\displaystyle{\cal M}\left(d_{+}\bar{d}_{-}\to
W^{+}_{0}W^{-}_{0}\right)=-i\frac{\hat{e}^{2}}{12\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left(f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)\;,$
$\displaystyle{\cal M}\left(u_{-}\bar{u}_{+}\to
W^{+}_{0}W^{-}_{0}\right)=i\frac{\hat{e}^{2}}{24\hat{s}^{2}\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[3\hat{c}^{2}\,f_{W}+\hat{s}^{2}\,f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(3\hat{c}^{2}\,f^{(1)}_{W\Phi^{4}D^{2}}+\hat{s}^{2}\,f^{(1)}_{B\Phi^{4}D^{2}}\right)\right]\;,$
$\displaystyle{\cal M}\left(u_{+}\bar{u}_{-}\to
W^{+}W^{-}\right)=i\frac{\hat{e}^{2}}{6\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left(f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)\;,$
(21) $\displaystyle{\cal M}\left(u_{+}\bar{u}_{-}\to
W^{+}_{\pm}W^{-}_{\pm}\right)={-i\frac{\hat{e}^{4}}{24\hat{s}^{2}\hat{c}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\,\frac{\hat{v}^{2}}{\Lambda^{2}}f^{(1)}_{W^{2}B\Phi^{2}}}\;,$
$\displaystyle{\cal M}\left(u_{-}\bar{u}_{+}\to
W^{+}_{\pm}W^{-}_{\pm}\right)={i\frac{3\hat{e}^{4}}{8\hat{s}^{4}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(\,f^{(1)}_{W^{3}\Phi^{2}}-\frac{\hat{s}^{2}}{18\hat{c}^{2}}f^{(1)}_{W^{2}B\Phi^{2}}\right)\right]}\;,$

as well as

$\displaystyle{\cal M}\left(d_{-}\bar{u}_{+}\to
Z_{\pm}W^{-}_{\pm}\right)=i\frac{3\hat{c}\hat{e}^{4}}{4\sqrt{2}\hat{s}^{4}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(f^{(1)}_{W^{3}\Phi^{2}}+{\frac{\hat{s}^{2}}{6\hat{c}^{2}}f^{(1)}_{W^{2}B\Phi^{2}}}\right)\right]\;,$
$\displaystyle{\cal M}\left(d_{-}\bar{u}_{+}\to
Z_{0}W^{-}_{0}\right)=i\frac{\hat{e}^{2}}{4\sqrt{2}\hat{s}^{2}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)\;,$
(22) $\displaystyle{\cal
M}\left(d_{-}\bar{u}_{+}\to\gamma_{\pm}W^{-}_{\pm}\right)=i\frac{3\hat{e}^{4}}{4\sqrt{2}\hat{s}^{3}}\frac{\hat{S}}{\Lambda^{2}}\sin\theta\left[f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}\left(f^{(1)}_{W^{3}\Phi^{2}}-{\frac{1}{6}f^{(1)}_{W^{2}B\Phi^{2}}}\right)\right]\;,$

where we indicated the particle polarization as a subscript and we denoted by
$\theta$ the polar scattering angle in the center of mass system. Therefore,
dimension-six squared contribution to the amplitude squared grows as
$\hat{S}^{2}$, while the dimension-eight contribution – which enters in the
interference with the SM amplitude – grows as $\hat{S}$.

Notice that, unlike EWPO, which correspond to squared amplitudes for fixed
center-of-mass energy (either $M_{Z}$ or $M_{W}$), EWDB data correspond to
squared amplitudes at different center-of-mass energies. Thus, since at order
$1/\Lambda^{4}$, the dimension-six squared and the dimension-eight
contributions exhibit different energy dependence, the approximate analysis
performed in terms of the effective couplings (19), does not exhaust the
potential of the data to constrain the Wilson coefficients of all the
operators involved. It is then possible to perform an analysis in terms of the
seven Wilson coefficients contributing to the amplitudes of the EWDB data
because in fact to order $1/\Lambda^{4}$ the $\chi^{2}$ function depends
independently on them:

$\chi^{2}_{\rm
EWDB}\left(f_{W},f_{B},f_{WWW},f^{(1)}_{W^{2}B\Phi^{2}},f^{(1)}_{W\Phi^{4}D^{2}},f^{(1)}_{B\Phi^{4}D^{2}},f^{(1)}_{W^{3}\Phi^{2}}\right)\;.$
(23)

We present in Fig. 3 the one- and two-dimensional marginalized 68% and 95%
C.L. allowed regions for the seven Wilson coefficients in Eq. (23) for several
analyses differing by the order in $1/\Lambda$ used in the calculations. We
list the corresponding 95% CL allowed ranges in Table 4. Notice that the
${\cal O}(1/\Lambda^{2})$, [${\cal O}(1/\Lambda^{4})$ (dim-6)2] analysis is
identical to the one described above as dimension-6 [(dimension-6)2] and leads
to the limits on the Wilson coefficients $f_{W}$, $f_{B}$ and $f_{WWW}$ given
on left (central) column in Table 3 and the light shaded regions in Fig. 2
(dotted contours) in the three lowest panels. We reproduce these regions and
ranges in Fig. 3 and Table 4 for clarity and completeness.

From the figure we see that including the ${\cal O}(1/\Lambda^{4})$ terms
strengthens the constraints obtained at order $1/\Lambda^{2}$ for the
operators ${\cal O}_{B}$ and ${\cal O}_{WWW}$ while it weakens the bounds on
${\cal O}_{W}$, as expected from the results obtained in the approximate
analysis in Fig. 2.

The comparison with the analysis performed including only dimension-six
squared terms in the evaluation of the $1/\Lambda^{4}$ contribution (see
dashed lines) shows that the dimension-six Wilson coefficient whose
determination is most quantitatively affected by the inclusion of the
independent effects of the four dimension-eight Wilson coefficients is
$f_{W}$. The reason for this is the anti-correlation between $f_{W}$ and
$f^{(1)}_{W\Phi^{4}D^{2}}$ that is apparent in the second panel of the fourth
row; see Eq. (18). In other words, the EWDB data analyzed provides a weaker
discrimination between the dimension-six and the dimension-eight contribution
to $\tilde{f}_{W}$. On the contrary the corresponding two-dimensional plots in
Fig. 3 show that no large correlations are present between $f_{B}$ and
$f^{(1)}_{B\Phi^{4}D^{2}}$, nor between $f_{WWW}$ and $f_{W^{3}B\Phi^{2}}$.
The only other large correlation is observed between the dimension-eight
coefficients $f^{(1)}_{W^{2}B\Phi^{2}}$ and $f_{W^{3}\Phi^{2}}$ both
contributing at the same order to $\lambda_{\gamma}$ and $\lambda_{Z}$. At the
linear order on these coefficients the stronger sensitivity comes from the
$W\gamma$ channel which bounds the combination
$6f_{W^{3}\Phi^{2}}-f^{(1)}_{W^{2}B\Phi^{2}}$ (see Eq. (18)) leading to the
positive correlation observed.

Figure 3: One- and two-dimensional 68% and 95% CL projections of $\Delta\chi^{2}_{\rm EWDB}$ for $f_{B}\hat{v}^{2}/\Lambda^{2}$, $f_{W}\hat{v}^{2}/\Lambda^{2}$, $f_{WWW}\hat{v}^{2}/\Lambda^{2}$, $f^{(1)}_{B\Phi^{4}D^{2}}\hat{v}^{2}/\Lambda^{4}$, $f^{(1)}_{W\Phi^{4}D^{2}}\hat{v}^{2}/\Lambda^{4}$, $f^{(1)}_{W^{3}\Phi^{2}}\hat{v}^{2}/\Lambda^{4}$, and $f^{(1)}_{W^{2}B\Phi^{2}}\hat{v}^{2}/\Lambda^{4}$ as indicated in the panels after marginalizing over the remaining fit parameters.  Coefficient | EWPB 95% CL allowed range
---|---
| ${\cal O}(\Lambda^{-2})$ |  ${\cal O}(\Lambda^{-4})$ (dim-6)2 |  ${\cal O}(\Lambda^{-4})$ (dim-6)2 \+ dim-8
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,f_{B}$ | $[-3.3,1.8]$ | $[-0.75,0.83]$ |  $[-0.89,0.89]$
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,f_{W}$ | $[-0.11,0.085]$ | $[-0.079,0.16]$ |  $[-0.18,0.18]$
$\frac{\hat{v}^{2}}{\Lambda^{2}}\,{f}_{WWW}$ | $[-0.22,0.16]$ | $[-0.049,0.045]$ |  $[-0.05,0.05]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f^{(1)}_{W^{2}B\Phi^{2}}$ |  — |  — |  $[-2.6,5.0]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f_{B\Phi^{4}D^{2}}$ |  — |  — |  $[-6.48,7.8]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f_{W\Phi^{4}D^{2}}$ |  — |  — |  $[-0.33,0.66]$
$\frac{\hat{v}^{4}}{\Lambda^{4}}\,f_{W^{3}\Phi^{2}}$ |  — |  — |  $[-0.47,0.51]$
Table 4: 95% C.L. allowed range for the Wilson coefficients present in the
EWDB data analysis performed with predictions obtained at different orders in
the $1/\Lambda^{2}$ expansion.

##  V Summary and Conclusions

We have studied the impact of ${\cal O}(1/\Lambda^{4})$ corrections in the
EWPO and EWDB data analyses assuming a universal, C and P conserving new
physics scenario. The universality assumption reduces the number of dimension-
eight operators contributing to the processes making a complete analysis
possible. As described in Sec. II, in the HISZ basis for the dimension-six
SMEFT the universal theories are described by 11 bosonic operators and five
fermionic operators, the latter being generated by the application of the EOM
in the reduction of the basis. At dimension eight there are ten potentially
relevant bosonic operators and one expects a fermionic operator generated by
the EOM. Of those, we find that there are six (nine) dimension-six (-eight)
operators contributing to the EWPO and EWDB observables (see Eq. (9)).

The analysis of EWPO involves three dimension-six and four dimension-eight
operators whose Wilson coefficients cannot be independently bound. However, we
find that it is possible to eliminate the blind directions by redefining three
effective coefficients which are just a shift of the three Wilson coefficients
of the dimension-six operators corrected by their corresponding dimension-
eight siblings – see Eq. (13)– and which contain the sibling dimension-eight
contribution to the universal parameters S, T, and $\Delta G_{F}$. In addition
the analysis contains a purely dimension-eight contribution to the universal
parameter $U$. The fit to EWPO performed in terms of these four parameters
results in strong constraints on $\tilde{f}_{BW}$, $\tilde{f}_{\Phi,1}$,
$\tilde{\Delta}_{4F}$, and $f^{(3)}_{W^{2}\Phi^{4}}$; see Table 1.

At ${\cal O}(1/\Lambda^{4})$ EWDB analysis involves six (seven) dimension-six
(-eight) operators of which three (four) contribute directly to the TGC while
three (three) enter indirectly via the finite renormalization of the SM
parameters. The indirect contributions can be cast in terms of three effective
couplings bounded by the EWPO (see Appendix B) allowing us to neglect them in
the EWDB analysis.

The direct contributions to the TGC can be expressed in terms of three
effective coefficients which are just a shift of the three Wilson coefficients
of the corresponding three dimension-six operators corrected by their
corresponding dimension-eight siblings (i.e. rescaled by
$\Phi^{\dagger}\Phi$); see Eq. (19). In addition there is a genuine dimension-
eight contribution to the difference between the $\lambda_{\gamma}$ and
$\lambda_{Z}$ couplings. We performed an effective analysis of the EWDB data
in terms of these four coefficients and showed that the bulk of ${\cal
O}(1/\Lambda^{4})$ impact on the analysis is due to the dimension-six squared
contribution $|{\cal M}^{(6)}|^{2}$; see Fig. 2 and Table 3. This is so
because of the different dependence on the partonic center-of-mass energy of
dimension-six squared terms which give a pure quadratic TGC contribution to
the amplitude squared, and the dimension-eight contribution which enters in
the interference with the SM amplitude.

Profiting from the different energy dependence of the dimension-six squared
and the dimension-eight contributions it is possible to perform an analysis of
the EWDB data which allows us to constrain the seven Wilson coefficients
independently. The result of this analysis is presented in Fig. 3 and Table 4.
The results show that the bounds on the three Wilson coefficients of the
dimension-six operators are only slightly looser than in the effective four-
parameter analysis, while the bounds on the four Wilson coefficients of the
dimension-eight operators are all of similar order and all substantially
weaker than those on their dimension-six siblings.

In summary, we have shown that, for the universal scenario, the analysis of
the EWPO and the EWDB data allows us to constrain the full parameter space of
operators up to ${\cal O}(\Lambda^{-4})$. Within the present precision of EWDB
data and with our choice of basis, it is still consistent to perform the
analysis sequentially: first obtain the constraints on the relevant Wilson
coefficients using the EWPO and then apply those bounds to reduce the number
of Wilson coefficients which are relevant for the the diboson analysis. That
said, the LHC continues to accumulate data on EWDB production, and
consequently, we anticipate stronger bounds on the TGC couplings in the
future. At some point, it will be necessary to perform a combined analysis of
EWPO+EWDB data taking into account the indirect contributions due to the
finite renormalization to the TGC in analogy with study of TGC and possible
anomalous fermionic couplings Alves et al. (2018).

###### Acknowledgements.

OJPE thanks the hospitality of the Departament de Fisica Quantica i
Astrofisica, Universitat de Barcelona, where part of this work was carried
out. OJPE is partially supported by CNPq grant number 305762/2019-2 and FAPESP
grants 2019/04837-9 and 2022/05332-0. M.M. is supported by FAPESP grant
2021/08669-3 while P.R. acknowledges support by FAPESP grants 2020/10004-7 and
2021/12305-7. This project is funded by USA-NSF grant PHY-1915093. It has also
received support from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No
860881-HIDDeN, and Horizon Europe research and innovation programme under the
Marie Skłodowska-Curie Staff Exchange grant agreement No 101086085 –
ASYMMETRY”. It also receives support from grants PID2019-105614GB-C21, and
“Unit of Excellence Maria de Maeztu 2020-2023” award to the ICC-UB
CEX2019-000918-M, funded by MCIN/AEI/10.13039/501100011033, and from grant
2021-SGR-249 (Generalitat de Catalunya). This manuscript has been authored by
Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the
U.S. Department of Energy, Office of Science, Office of High Energy Physics.

##  Appendix A Corrections to the $Z$ and $W$ couplings

We parametrize the $Z$ coupling to fermion ($f$) pairs as

$\frac{\hat{e}}{\hat{s}\hat{c}}~{}\left(\hat{g}^{f}\,(1+\Delta
g_{1})+Q^{f}\,\Delta g_{2}\right)$ (24)

where $\hat{g}^{f}=T_{3}^{f}-\hat{s}^{2}Q^{f}$, $T_{3}^{f}$ is the fermion
third component of isospin and $Q^{f}$ is its charge. After the
renormalization of the SM parameters, we obtain at order $1/\Lambda^{4}$

$\displaystyle\Delta g_{1}=$
$\displaystyle-\frac{1}{4}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[2\left(\Delta_{4F}+\frac{\hat{v}^{2}}{\Lambda^{2}}\Delta^{(8)}_{4F}\right)+f_{\Phi,1}+\frac{\hat{v}^{2}}{\Lambda^{2}}f^{(2)}_{D^{2}\Phi^{6}}\right]$
$\displaystyle-\frac{1}{32}\frac{\hat{v}^{4}}{\Lambda^{4}}\left[-12(\Delta_{4F})^{2}+4\Delta_{4F}f_{\Phi,1}-3(f_{\Phi,1})^{2}\right]$
$\displaystyle\simeq$
$\displaystyle-\frac{1}{4}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[2\tilde{\Delta}_{4F}+\tilde{f}_{\Phi,1}\right]-\frac{1}{32}\frac{\hat{v}^{4}}{\Lambda^{4}}\left[-12(\tilde{\Delta}_{4F})^{2}+4\tilde{\Delta}_{4F}\tilde{f}_{\Phi,1}-3(\tilde{f}_{\Phi,1})^{2}\right]\;.$
(25)

In the last line we have used that to order $1/\Lambda^{4}$ the corrections
linear in the Wilson coefficients depend on the four combinations in Eq. (13)
and therefore neglecting terms of ${\cal O}(1/\Lambda^{6})$ we can rewrite
$\Delta g_{1}$ in terms of those combinations. In the same way we find:

$\displaystyle\Delta g_{2}=$
$\displaystyle\frac{\hat{v}^{2}}{\Lambda^{2}}\frac{1}{2\hat{c}_{2}}\Big{[}-\hat{s}^{2}\hat{c}^{2}\left(2\tilde{\Delta}_{4F}+\tilde{f}_{\Phi,1}\right)+\frac{\hat{e}^{2}}{2}\tilde{f}_{BW}\Big{]}$
(26)
$\displaystyle+\frac{\hat{v}^{4}}{\Lambda^{4}}\frac{1}{8\hat{c}_{2}^{3}}\Bigg{\\{}\frac{\hat{s}_{2}^{2}}{4}\Big{[}(1+3\hat{c}_{4})\left((\tilde{\Delta}_{4F})^{2}+\frac{1}{4}(\tilde{f}_{\phi,1})^{2}\right)-(3+\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{\phi,1}\Big{]}$
$\displaystyle-\frac{\hat{e}^{2}}{2}\left(\hat{c}_{4}\tilde{f}_{BW}\tilde{f}_{\phi,1}-2\tilde{\Delta}_{4F}\tilde{f}_{BW}+\hat{e}^{2}(\tilde{f}_{BW})^{2}\right)\Bigg{\\}}$

with $\hat{c}_{n}=\cos(n\hat{\theta})$ and $\hat{s}_{n}=\sin(n\hat{\theta})$.

As for the $W$ observables

$\displaystyle\frac{\Delta M_{W}}{\hat{M}_{W}}=$
$\displaystyle\frac{1}{4\hat{c}_{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[\hat{e}^{2}\tilde{f}_{BW}-2\hat{s}^{2}\tilde{\Delta}_{4F}-\hat{c}^{2}\tilde{f}_{\Phi,1}\right]+\frac{\hat{e}^{2}}{8\hat{s}^{2}}\frac{\hat{v}^{4}}{\Lambda^{4}}f^{(3)}_{W^{2}\Phi^{4}}\;$
(27)
$\displaystyle+\frac{1}{8\hat{c}^{3}_{2}}\frac{\hat{v}^{4}}{\Lambda^{4}}\Big{[}-\hat{s}^{4}(2+3\hat{c}_{2})(\tilde{\Delta}_{4F})^{2}+\frac{1}{4}\hat{c}^{4}(-2+5\hat{c}_{2})(\tilde{f}_{\Phi,1})^{2}-\frac{1}{16}\hat{e}^{4}\frac{(7-6\hat{c}_{2}+3\hat{c}_{4})}{\hat{s}^{2}}(\tilde{f}_{BW})^{2}$
$\displaystyle-\frac{\hat{c}^{2}}{4}(9-6\hat{c}_{2}+5\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{\Phi,1}+\frac{1}{4}\hat{e}^{2}(7-2\hat{c}_{2}+3\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{BW}-\frac{1}{2}\hat{e}^{2}\hat{c}^{2}(-2+3\hat{c}_{2})\tilde{f}_{\Phi,1}\tilde{f}_{BW}\Big{]}$

where $\hat{M}_{W}=\frac{\displaystyle\hat{e}\hat{v}}{\displaystyle
2\hat{s}}$. And we parametrize the $W$ coupling to left-handed fermions as

$\frac{\hat{e}}{\hat{s}}~{}(1+\Delta g_{W})$ (28)

where

$\displaystyle\Delta g_{W}=$
$\displaystyle\frac{1}{4\hat{c}_{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[\hat{e}^{2}\tilde{f}_{BW}-2\hat{c}^{2}\tilde{\Delta}_{4F}-\hat{c}^{2}\tilde{f}_{\Phi,1}\right]$
(29)
$\displaystyle+\frac{1}{8\hat{c}_{2}^{3}}\frac{\hat{v}^{4}}{\Lambda^{4}}\Big{[}\hat{e}^{2}\frac{\hat{c}_{2}^{3}}{\hat{s}^{2}}f^{(3)}_{W^{2}\Phi^{4}}+\hat{c}^{4}(-2+5\hat{c}_{2})(\tilde{\Delta}_{4F})^{2}-\frac{1}{16}\frac{(7-6\hat{c}_{2}+3\hat{c}_{4})}{\hat{s}^{2}}\hat{e}^{4}(\tilde{f}_{BW})^{2}$
$\displaystyle+\frac{1}{4}\hat{c}^{4}(-2+5\hat{c}_{2})(\tilde{f}_{\Phi,1})^{2}-\frac{1}{4}\hat{c}^{2}(7-6\hat{c}_{2}+3\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{\Phi,1}+\frac{1}{4}\hat{e}^{2}(5-2\hat{c}_{2}+\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{BW}$
$\displaystyle-\frac{1}{2}\hat{e}^{2}\hat{c}^{2}(-2+3\hat{c}_{2})\tilde{f}_{\Phi,1}\tilde{f}_{BW}\Big{]}$

##  Appendix B Corrections to TGC

The renormalization of the SM parameters give rise to indirect contributions
to TGC in addition to the direct contributions from the dimension-six and
-eight operators to the TGC. Using the parametrization for the $\gamma
W^{+}W^{-}$ and $ZW^{+}W^{-}$ TGC given in Eq. (17), we find that up to order
$1/\Lambda^{4}$ (and neglecting terms of ${\cal O}(1/\Lambda^{6})$) the
coupling to $W^{+}_{\mu\nu}W^{-\,\mu}Z^{\nu}$ reads

$\displaystyle g_{1}^{Z}=$ $\displaystyle
1+\frac{1}{2}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[\frac{\hat{e}^{2}}{4\hat{s}^{2}\hat{c}^{2}}\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)-\frac{1}{\hat{c}_{2}}\tilde{\Delta}_{4F}+\frac{1}{2}\frac{\hat{e}^{2}}{\hat{c}^{2}\hat{c}_{2}}\tilde{f}_{BW}-\frac{1}{2\hat{c}_{2}}\tilde{f}_{\Phi,1}\right]$
(30)
$\displaystyle+\frac{1}{16\hat{c}^{3}_{2}}\frac{\hat{v}^{4}}{\Lambda^{4}}\Big{[}(1+2\hat{c}_{2}+3\hat{c}_{4})\left((\tilde{\Delta}_{4F})^{2}+\frac{1}{4}(\tilde{f}_{\Phi,1})^{2}\right)-\frac{\hat{e}^{4}}{\hat{c}^{2}}(\tilde{f}_{BW})^{2}$
$\displaystyle+2\frac{\hat{e}^{2}}{\hat{c}^{2}}\tilde{\Delta}_{4F}\tilde{f}_{BW}-(3-2\hat{c}_{2}+\hat{c}_{4})\tilde{\Delta}_{4F}\tilde{f}_{\Phi,1}-\hat{e}^{2}\frac{\hat{c}_{4}}{\hat{c}^{2}}\tilde{f}_{BW}\tilde{f}_{\Phi,1}\Big{]}$
$\displaystyle-\frac{\hat{e}^{2}}{4\hat{s}\hat{c}\hat{s}_{4}}\frac{\hat{v}^{4}}{\Lambda^{4}}\left(\tilde{\Delta}_{4F}-\hat{e}^{2}\tilde{f}_{BW}+\frac{1}{2}(1+2\hat{c}_{2})\tilde{f}_{\Phi,1}\right)f_{W}\;,$

The couplings to $W_{\mu}^{+}W_{\nu}^{-}V^{\mu\nu}$ are respectively

$\displaystyle\kappa_{\gamma}=$ $\displaystyle
1+\frac{1}{8}\frac{\hat{e}^{2}}{\hat{s}^{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[\left(f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)+\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)-2\tilde{f}_{BW}\right]$
$\displaystyle-\frac{\hat{e}^{2}}{32}\frac{\hat{v}^{4}}{\Lambda^{4}}\frac{1}{\hat{s}^{2}\hat{c}_{2}}\Big{(}2(1-\hat{c}_{2})\tilde{\Delta}_{4F}-2\hat{e}^{2}\tilde{f}_{BW}+(1+\hat{c}_{2})\tilde{f}_{\Phi,1}\Big{)}(f_{B}+f_{W}-2\tilde{f}_{BW})$
$\displaystyle+\frac{\hat{e}^{2}}{4\hat{s}^{2}}\frac{\hat{v}^{4}}{\Lambda^{4}}f^{(3)}_{W^{2}\Phi^{4}}\,$

and

$\displaystyle\kappa_{Z}=$ $\displaystyle
1+\frac{1}{8}\frac{\hat{e}^{2}}{\hat{s}^{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}\left[\left(f_{W}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W\Phi^{4}D^{2}}\right)-\frac{\hat{s}^{2}}{\hat{c}^{2}}\left(f_{B}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{B\Phi^{4}D^{2}}\right)+\frac{4\hat{s}^{2}}{\hat{c}_{2}}\tilde{f}_{BW}-\frac{4\hat{s}^{2}}{\hat{e}^{2}\hat{c}_{2}}\tilde{\Delta}_{4F}-\frac{2\hat{s}^{2}}{\hat{e}^{2}\hat{c}_{2}}\tilde{f}_{\Phi,1}\right]$
$\displaystyle+\frac{1}{16\hat{s}^{2}\hat{c}_{2}^{3}}\frac{\hat{v}^{4}}{\Lambda^{4}}\left[\hat{s}^{2}(1+2\hat{c}_{2}+3\hat{c}_{4})\left((\tilde{\Delta}_{4F})^{2}+\frac{1}{4}(\tilde{f}_{\Phi,1})^{2}\right)-\hat{e}^{4}(2-2\hat{c}_{2}+\hat{c}_{4})(\tilde{f}_{BW})^{2}\right.$
$\displaystyle+\hat{s}^{2}(3-2\hat{c}_{2}+\hat{c}_{4})\left(2\hat{e}^{2}\tilde{\Delta}_{4F}\tilde{f}_{BW}-\tilde{\Delta}_{4F}\tilde{f}_{\phi,1}\right)-\hat{e}^{2}\hat{s}^{2}(-1+2\hat{c}_{2}+\hat{c}_{4})\tilde{f}_{BW}\tilde{f}_{\Phi,1}$
$\displaystyle+f_{W}\left(\hat{e}^{2}\hat{c}_{2}^{2}(-2+\hat{c}_{2})\tilde{\Delta}_{4F}+\hat{e}^{2}\frac{\hat{c}_{2}^{2}}{2}(2+\hat{c}_{2})\left(\frac{\hat{e}^{2}}{\hat{c}^{2}}\tilde{f}_{BW}-\tilde{f}_{\Phi,1}\right)\right)$
$\displaystyle\left.+f_{B}\hat{e}^{2}\frac{\hat{s}^{2}\hat{c}_{2}^{3}}{2\hat{c}^{2}}\left(\tilde{f}_{\Phi,1}-2\tilde{\Delta}_{4F}+\frac{\hat{e}^{2}}{\hat{s}^{2}}\tilde{f}_{BW}\right)+4\hat{e}^{2}\hat{c}_{2}^{3}f^{(3)}_{W^{2}\Phi^{4}}\right]\,$

And the couplings to $W^{+}_{\mu\nu}W^{-\,\nu\rho}V_{\rho}^{\;\mu}$ are:

$\displaystyle\lambda_{\gamma}=$
$\displaystyle\frac{3}{2}\frac{\hat{e}^{2}}{\hat{s}^{2}}\frac{\hat{M}_{W}^{2}}{\Lambda^{2}}\Big{[}{\left(f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W^{3}\Phi^{2}}\right)}$
$\displaystyle+\frac{1}{2\hat{c}_{2}}\frac{\hat{v}^{2}}{\Lambda^{2}}f_{WWW}\left(\hat{e}^{2}\tilde{f}_{BW}-\left(2\tilde{\Delta}_{4F}+\tilde{f}_{\Phi,1}\right)\hat{c}^{2}\right)\Big{]}-\frac{\hat{M}_{W}^{4}}{2\Lambda^{4}}f^{(1)}_{W^{2}B\Phi^{2}}\;,$
$\displaystyle\lambda_{Z}=$
$\displaystyle\frac{3}{2}\frac{\hat{e}^{2}}{\hat{s}^{2}}\frac{\hat{M}_{W}^{2}}{\Lambda^{2}}\Big{[}{\left(f_{WWW}+\frac{\hat{v}^{2}}{2\Lambda^{2}}f^{(1)}_{W^{3}\Phi^{2}}\right)}$
$\displaystyle-\frac{\hat{v}^{2}}{\Lambda^{2}}\frac{\hat{s}^{2}(2+\hat{c}_{2})}{\hat{s}_{2}\hat{s}_{4}}f_{WWW}\Big{(}4\tilde{\Delta}_{4F}\hat{c}^{2}+2\tilde{f}_{\Phi,1}\hat{c}^{2}-2\hat{e}^{2}\tilde{f}_{BW}\Big{)}\Big{]}+\frac{\hat{M}_{W}^{4}}{2\Lambda^{4}}\frac{\hat{s}^{2}}{\hat{c}^{2}}f^{(1)}_{W^{2}B\Phi^{2}}\;,$

where $\hat{M}_{W}=\hat{e}\hat{v}/2\hat{s}$.

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Zizheng Zhang1, Chen Chen2, Xiang Liu1, Yuchen Hu2, Eng Siong Chng2

# Noise-aware Speech Separation with Contrastive Learning

###### Abstract

Recently, speech separation (SS) task has achieved remarkable progress driven
by deep learning technique. However, it is still challenging to separate
target signals from noisy mixture, as neural model is vulnerable to assign
background noise to each speaker. In this paper, we propose a noise-aware SS
method called NASS, which aims to improve the speech quality of separated
signals in noisy conditions. Specifically, NASS views background noise as an
independent speaker and predicts it with other speakers in a mask-based
manner. Then we conduct patch-wise contrastive learning on feature level to
minimize the mutual information between the predicted noise-speaker and other
speakers, which suppresses the noise information in separated signals. The
experimental results show that NASS effectively improves the noise-robustness
for different mask-based separation backbones with less than 0.1M parameter
increase. Furthermore, SI-SNRi results demonstrate that NASS achieves state-
of-the-art performance on WHAM! dataset.

Index Terms: noisy speech separation, contrastive learning

##  1 Introduction

Speech separation (SS) aims to separate speech signals from the overlapping
speech mixture[1], which can serve as a pre-processor for downstream speech
applications[2, 3, 4, 5]. Recently, deep learning-based methods have developed
various neural networks for SS[6, 7, 8, 9], and achieved remarkable
performances on public datasets[10, 11] mixed by clean speech. However, it is
still challenging to separate target speech from noisy mixture, e.g., _noisy
speech separation_ , since noise signal usually has a wide distribution on
frequency domain to interfere with the human voice.

For noisy SS, the mainstream mask-based method[6, 7, 8, 9] is vulnerable to
assign background noise to the target speaker[12]. We also validate this
phenomenon with a related experiment. One intuitive solution is utilizing the
speech enhancement (SE)[13, 14, 15] technique as a pre-processor to remove
noise information from the mixture in a multi-task learning manner[16, 17].
Despite the slight improvement, this method may lead to an over-suppression
problem[18] — SE module would inevitably remove some helpful information when
it attempts to denoise, thus resulting in a sub-optimal performance for SS.

To alleviate the influence of noise, our basic idea is to view background
noise as an independent speaker, which can be simultaneously predicted along
with other speakers. In addition to avoiding the over-suppression problem, the
estimated noise signal can benefit the separated speech from a mutual
information[19] perspective: we aim to minimize the mutual information between
predicted noise and separated speech, which can prevent the noise from
existing in the separated signal.

In this paper, we propose a noise-aware speech separation method called NASS,
which follows a typical encoder-separator-decoder pipeline[6]. Unlike previous
works, NASS learns to predict the noise signal and leverage it to improve the
speech quality of each speaker. Specifically, we conduct patch-wise
contrastive learning[20, 21, 22] on different representations: 1) We first
sample hundreds of patches from the speaker's representations. 2) The
corresponding patches from ground-truth representations are viewed as positive
examples, while 3) other patches from noise representation are all viewed as
negative examples. Reshaped by a two-layer MLP, the positive and negative
training examples are calculated with cosine similarities. By optimizing the
cross-entropy loss, we minimize the mutual information between the
representation of each speaker and noise representation, which significantly
suppresses the noise information from the separated speech signals.

To evaluate the effectiveness of NASS, we conduct intensive experiments on
noisy WHAM![23] and LibriMix[11] datasets, and select three milestone works
Conv-TasNet[6], DPRNN[7], and Sepformer[8] as the separator's backbone. The
experimental results show that NASS can effectively improve the noise-
robustness for all these separation models with less than 0.1M parameter
increase. Furthermore, NASS achieves the state-of-the-art (SOTA) performance
on WHAM! in terms of SI-SNRi[24] and SDRi[25].

Figure 1: The overall pipeline (2-speaker version) of NASS. $x_{n}$ and
$\hat{n}$ denotes the input noisy mixture and predicted noise. $\hat{s}_{1}$
and $\hat{s}_{2}$ are the separated speech signals while $s_{1}$ and $s_{2}$
are the ground-truth. $h_{\hat{s}_{1}}$, $h_{\hat{s}_{2}}$ and $h_{\hat{n}}$
in dashed boxes are trainable predicted representations, while
$h_{\hat{s}_{1}}$ and $h_{\hat{s}_{2}}$ in solid boxes are the ground-truth.
``+" denotes the mutual information between separated and ground-truth speech
is maximized while ``-" denotes the mutual information between separated
speech and noise is minimized. Utterance-level permutation-invariant training
(uPIT)[26] is employed to determine the order of speakers for corresponding
comparisons.

##  2 NASS Method

We now introduce our proposed NASS method, as shown in Figure 1, which
consists of mask-based architecture and the patch-wise contrastive learning
strategy.

###  2.1 Mask-based Architecture

In this work, we follow the encoder-separator-decoder pipeline, where the mask
of each speaker is predicted as shown in Figure 1. Since NASS is theoretically
applicable to any mask-based method, we select Sepformer as an example in this
part due to its remarkable performance in SS.

####  2.1.1 Encoder and Decoder

The encoder takes the input noisy mixture ${x_{n}}\in{\mathbb{R}^{1\times T}}$
in time domain and then generates a STFT-like[27] representation
${h_{x_{n}}}\in{\mathbb{R}^{N\times L}}$, where $T$ is the signal length, $N$
is the number of filters and $L$ is the number of vectors:

${h_{x_{n}}}{\rm{=}}{\mathop{\rm ReLU}\nolimits}({\mathop{\rm
Conv1d}\nolimits}({x_{n}}))$ (1)

The decoder acts as an inverse operation of the encoder, which takes all
predicted representations ${h_{k}}\in{\mathbb{R}^{N\times L}}$ and
reconstructs the separated signals ${\hat{y}_{k}}\in{\mathbb{R}^{1\times T}}$
in time domain:

${\hat{y}_{k}}={\mathop{\rm Conv1dTranspose}\nolimits}({h_{k}})$ (2)

In our work, additionally, the ground-truth speech signal
${s_{t}}\in{\mathbb{R}^{1\times T}}$ is encoded as
${h_{s_{t}}}\in{\mathbb{R}^{N\times L}}$, which is used together with
${h_{k}}$ in the subsequent contrastive learning, where
${t}\in\left\\{{{1},{2},\ldots,{C}}\right\\}$ and $C$ is the number of human
speakers.

Figure 2: The structure of masking network in Sepformer. $h_{x_{n}}$ is the
encoded representation of noisy input and $m_{k}$ is the generated trainable
mask. ``Processing Block" is the main component, where Sepformer uses the dual
path (intra-Transformer and inter-Transformer)[8] for long sequence modeling.

####  2.1.2 Masking Network and Noise-speaker

The masking network takes $h_{x_{n}}$ and learns a mask
${m_{k}}\in{\mathbb{R}^{N\times L}}$ for each of $G$ sources, then yielding
$h_{k}$:

${m_{k}}={\mathop{\rm MaskingNet}\nolimits}({h_{{x_{n}}}})$ (3)
${h_{k}}={m_{k}}\cdot{h_{{x_{n}}}}$ (4)

As shown in Figure 2. First $h_{x_{n}}$ is processed by layer
normalization[28] and a linear layer, then chunked on the time axis with 50%
overlap, resulting in an output $h^{\prime}\in{\mathbb{R}^{N\times K\times
S}}$, where $K$ is the chunk size and $S$ is the number of chunks. Next
$h^{\prime}$ is sent into the processing block, which learns the local and
global information by permuting. Then the output of processing block
$h^{\prime\prime}\in{\mathbb{R}^{N\times K\times S}}$ is processed by a
PReLU[29] activation and a linear layer, denoted as
$h^{\prime\prime\prime}\in{\mathbb{R}^{(N\times G)\times K\times S}}$. The
overlap-add operation, acting as the inverse of chunking, is employed to
obtain $h^{\prime\prime\prime\prime}\in{\mathbb{R}^{(N\times G)\times L}}$
from $h^{\prime\prime\prime}$. Finally $h^{\prime\prime\prime\prime}$ goes
through a two-layer MLP and a ReLU[30] activation, then generating all $m_{k}$
at once.

As mentioned before, we count noise as an additional speaker, which can make
use of the noise information within the existing framework. The noise-speaker
has its own supervision and prediction like human speaker. From Equation 2, 3
and 4, we have the predicted noise signal ${\hat{n}}\in{\mathbb{R}^{1\times
T}}$, the noise mask ${m_{n}}\in{\mathbb{R}^{N\times L}}$ and the predicted
noise representation $h_{\hat{n}}\in{\mathbb{R}^{N\times L}}$, respectively.
Thus far we have
${m_{k}}\in\left\\{{{m_{1}},{m_{2}},\ldots,{m_{C}},{m_{n}}}\right\\}$,
${h_{k}}\in\left\\{{{h_{\hat{s}_{1}}},{h_{\hat{s}_{2}}},\ldots,{h_{\hat{s}_{C}}},h_{\hat{n}}}\right\\}$,
etc., for a total of $C+1$ sources.

Figure 3: The diagram of patch-wise contrastive learning. The query example
$r^{i}_{q}$, positive example $r^{i}_{p}$ and negative examples $r^{i}_{n}$
are sampled from the predicted speech representation $h_{\hat{s}_{t}}$,
ground-truth speech representation $h_{s_{t}}$ and predicted noise
representation $h_{\hat{n}}$, respectively. The classification is based on the
results of their cosine similarity.

###  2.2 Patch-wise Contrastive Learning

Based on the noise-speaker, we can further denoise the separated speech
utilizing predicted noise. As shown in Figure 1, patch-wise contrastive
learning is conducted on $h_{s_{t}}$ and $h_{k}$, where details can be found
in Figure 3.

####  2.2.1 Patch Sampler

We performed 256 comparisons in the contrastive learning. In each comparison,
we first randomly sample a small patch on the predicted speech representation
$h_{\hat{s}_{t}}$, which serves as a query example $r_{q}^{i}$. Then the patch
of corresponding position on its ground-truth representation $h_{s_{t}}$ is
sampled, which serves as a positive example $r_{p}^{i}$. Finally other $M$
patches randomly sampled on predicted noise representation $h_{\hat{n}}$ serve
as negative examples $r_{n}^{j}$. The whole process is implemented by patch
sampler, which learns to project query, positive and negative patches into
examples in a shared 3-D embedding unit space, which prevents the space from
collapsing or expanding:

$r_{p}^{i},r_{q}^{i},r_{n}^{j}={\mathop{\rm
PatchSampler}\nolimits}({h_{{s_{t}}}},{h_{{{\hat{s}}_{t}}}},{h_{\hat{n}}})$
(5)

where ${r_{q}^{i},r_{p}^{i},r_{n}^{j}}\in{\mathbb{R}^{P\times P\times Q}}$ and
${i,j}\in\left\\{{{1},{2},\ldots,{M}}\right\\}$. $P$ denotes the patch size
and $Q$ denotes the features of MLP in patch sampler. In this work, $M$, $P$
and $Q$ is set to 256, 1 and 256, respectively.

####  2.2.2 Contrastive Loss

From the perspective of mutual information, query examples should be similar
to corresponding positive examples but dissimilar to all negative examples,
which provides an $M+1$ classification problem. By calculating their cosine
similarities and optimizing the cross-entropy loss, the mutual information
between predicted speech and noise would be minimized, thus suppressing the
noise from separated speech. The contrastive loss is conducted from each of
$C$ human speakers, which can be formulated as:

$\mathcal{L}_{PCL}=\frac{{\rm{1}}}{C}\sum\limits_{t=1}^{C}{\sum\limits_{i=1}^{M}{-\ln}}\left[{\frac{{{e^{r_{q}^{i}\cdot
r_{p}^{i}/\tau}}}}{{{e^{r_{q}^{i}\cdot
r_{p}^{i}/\tau}}+\sum\nolimits_{j=1}^{M}{{e^{r_{q}^{i}\cdot
r_{n}^{j}/\tau}}}}}}\right]$ (6)

where $\tau$ denotes the temperature parameter[31], and is set to 0.07 in this
work.

###  2.3 Training Objective

The main separation loss $\mathcal{L}_{si-snr}$ is to maximize SI-SNR between
separated signals ${\hat{y}_{k}}$ and ground-truth signals ${y_{k}}$ for $G$
sources:

$\mathcal{L}_{si-
snr}=\frac{1}{G}\sum\limits_{k=1}^{G}{-10{{\log}_{10}}\left({\frac{{{{\|{\frac{{{{\hat{y}}^{T}}_{k}{y_{k}}}}{{{{\|{{y_{k}}}\|}^{2}}}}{y_{k}}}\|}^{2}}}}{{{{\|{\frac{{{{\hat{y}}^{T}}_{k}{y_{k}}}}{{{{\|{{y_{k}}}\|}^{2}}}}{y_{k}}-{{\hat{y}}_{k}}}\|}^{2}}}}}\right)}$
(7)

Thus far, the total loss of proposed NASS method is formulated as:

$\mathcal{L}_{Total}=\mathcal{L}_{si-snr}+\lambda\mathcal{L}_{PCL}$ (8)

where $\lambda$ is the parameter to balance SS loss $\mathcal{L}_{si-snr}$ and
contrastive loss $\mathcal{L}_{PCL}$, which is set to 2 or 3 in this work.

The model is trained with uPIT. It is worthy noting that $\mathcal{L}_{PCL}$
utilizes the permutation result of $\mathcal{L}_{si-snr}$, which reduces
double-counting and ensures the correct comparisons.

##  3 Experiments

###  3.1 Datasets

We evaluate NASS and existing methods on two common noisy datasets: WHAM! and
LibriMix. WHAM! is a noisy version of WSJ0-2Mix[10], which is added noise
samples recorded in coffee shops, restaurants and bars. SNR between the
loudest speaker and noise varies from -6 to +3 dB. WHAM! follows the same
structure as WSJ0-2Mix, which has 119 speakers and 43 hours of speech (30h,
8h, 5h for training, validation, test, respectively). LibriMix contains
Libri2Mix and Libri3Mix for noisy multi-speaker SS tasks. In our chosen
version of LibriMix, the clean mixture is selected from LibriSpeech
train-100[32] and mixed between -25 and -33 dB of LUFS[33]. Noise samples from
WHAM! are added to the mixture between -38 and -30 dB of LUFS. With 331
speakers, Libri2Mix has 80 hours of speech (58h, 11h, 11h for training,
validation, test, respectively) and Libri3Mix has 62 hours (40h, 11h, 11h for
training, validation, test, respectively).

###  3.2 Experimental Setup

To ensure reproducibility, we conduct the experiments on SpeechBrain[34], an
open-source AI toolkit for SS tasks. The network configurations of existing
methods are the same as WSJ0-Mix recipes on SpeechBrain, which follows the
prior works where more details can be found.

We train 200 epochs for all methods on NVIDIA V100 GPUs, using Adam
optimizer[35] with initial learning rate of $1.5\times 10^{-4}$ and automatic
mixed precision[36]. After training 85 epochs (5 epochs for Conv-TasNet), the
learning rate is halved if with no improvement of validation for 3 epochs.
Speed perturbation[37] is applied for data augmentation. There's no dynamic
mixing[38] in our experiments. The batch size and number of workers are set to
1 and 4. The training signal length is 4 seconds long and loss threshold is
set to -30. Gradient clipping is applied to limit the $L_{2}$ norm of
gradients to 5. All the hyperparameters are adjusted on the validation set.

Table 1: The ablation study of noise-speaker on LibriMix. ``NS" denotes the
setting option of noise-speaker. ``∗" denotes the baseline results is self-
reproduced.

Method | Spk | NS | SI-SNRi (dB) | SDRi (dB) | Params (M)
---|---|---|---|---|---
ConvTasNet∗ | 2 | $\times$ | 8.8 | 9.4 | 6.8
✓ | 9.2 | 9.8 | 6.8
3 | $\times$ | 6.2 | 6.7 | 6.9
✓ | 7.3 | 7.9 | 7.0
Sepformer∗ | 2 | $\times$ | 12.9 | 13.5 | 25.9
✓ | 13.3 | 13.8 | 25.9
3 | $\times$ | 10.0 | 10.5 | 26.0
✓ | 10.4 | 11.0 | 26.1

Table 2: The ablation study of patch-wise contrastive learning on Libri2Mix
with Sepformer. ``S$\xrightarrow{}$N" denotes
$\mathcal{L}^{S\xrightarrow{}N}_{PCL}$ while ``N$\xrightarrow{}$S" denotes
$\mathcal{L}^{N\xrightarrow{}S}_{PCL}$. ``$\lambda$" indicates the balancing
parameter. ``$M$" indicates the number of negative patches in each comparison.

S$\xrightarrow{}$N | N$\xrightarrow{}$S | $\lambda$ | $M$ | SI-SNRi (dB) | SDRi (dB) | Params (M)
---|---|---|---|---|---|---
$\times$ | $\times$ | - | - | 13.3 | 13.8 | 25.9
✓ | ✓ | 1 | 256 | 13.4 | 14.0 | 25.9
$\times$ | ✓ | 1 | 256 | 13.5 | 14.0 | 25.9
✓ | $\times$ | 1 | 256 | 13.5 | 14.1 | 25.9
✓ | $\times$ | 1 | 1 | 13.4 | 14.0 | 25.9
✓ | $\times$ | 2 | 256 | 13.6 | 14.1 | 25.9
✓ | $\times$ | 2 | 512 | 13.5 | 14.1 | 25.9
✓ | $\times$ | 3 | 256 | 13.4 | 14.0 | 25.9

###  3.3 Metrics and Baselines

We use the results of SI-SNRi and SDRi in the test set to evaluate all
experiments, which measure the degree of improvement in clarity and fidelity
of these separated audios. For all metrics, higher score indicates better
performance.

To assess the effectiveness of proposed NASS method, we set three baselines
from mask-based methods for comparisons:

  * •

ConvTasNet[6]: A fully-convolutional network without dual path but good
performance in clean SS tasks.

  * •

DPRNN[7]: An RNN-based dual-path model with relatively large receptive field
for long sequence modeling.

  * •

Sepformer[8]: A Transformer-based dual-path network with relatively large size
but excellent performance, allowing parallel computations.

##  4 Results

###  4.1 Effect of Noise-speaker

The effect of noise-speaker is shown in Table 1. Note that all the following
test results of SI-SNR include human speakers only. Compared to the baselines,
benefit from noise supervision, the two models with noise-speaker perform
better in the separation of either 2 or 3 speakers, especially in the case of
3 speakers, and within 0.1M additional parameters.

###  4.2 Effect of Patch-wise Contrastive Learning

Based on noise-speaker, we study the effect of patch-wise contrastive learning
in Table 2. As mentioned in Equation 6, the query example $r_{q}^{i}$ and
negative examples $r_{n}^{j}$ come from speech and noise, respectively,
denoting the loss as $\mathcal{L}^{S\xrightarrow{}N}_{PCL}$. However, the
query example can be sampled from noise representation and compared to the
negative examples on speech representations, denoting the loss as
$\mathcal{L}^{N\xrightarrow{}S}_{PCL}$. The results show that both the choices
work but conflict to each other in a degree, and the best is with
$\mathcal{L}^{S\xrightarrow{}N}_{PCL}$ only. We also conduct the ablation
study from the balancing parameter $\lambda$ and the number of negative
patches $M$ in each comparison. The results show that the best settings are as
described previously.

Figure 4: The comparison results of spectrum on Libri2mix with Sepformer.
Subplot (a) denotes the mixture; (b), (d), (f) denotes the separated result of
speaker 1 in Sepformer baseline, noise-speaker, patch-wise contrastive
learning, respectively; (c), (e), (g) denotes the same as (b), (d), (f) in
speaker 2, respectively; (h), (i) denotes the result of predicted noise in
noise-speaker, patch-wise contrastive learning, respectively.

###  4.3 Benchmark against Competitive Methods

Table 3 and Table 4 show the comparison results on noisy LibriMix and WHAM!,
respectively. Not only self-reproduced but other-reported results indicate
NASS can effectively improve the noise-robustness of prior works in LibriMix
and WHAM! datasets. It's worth pointing out that NASS can make the previous
DPRNN and Sepformer surpass the current SOTA model TDANet in WHAM!, within
0.1M additional parameters.

In addition, to further illustrate the effectiveness of NASS, Figure 4
visualizes the comparison results of spectrum for 2-speaker SS. We can see
some spectrum of noise in (b), (c) from the Sepformer baseline. While from
(d), (e) with noise-speaker, and (f), (g) with additional patch-wise
contrastive learning, we can see the spectrum of noise gradually fades away in
a degree, thus separating the two sources better. Besides, NASS yields the
prediction of noise, which can be used in other speech tasks. In (h), (i), we
can also see an improvement from the patch-wise contrastive learning to noise-
speaker.

Table 3: The competitive results of proposed NASS method on LibriMix. ``∗"
denotes the baseline results is self-reproduced.

Method | Spk | SI-SNRi (dB) | SDRi (dB) | Params (M)
---|---|---|---|---
ConvTasNet∗ | 2 | 8.8 | 9.4 | 6.8
3 | 6.2 | 6.7 | 6.9
DPRNN∗ | 2 | 12.8 | 13.4 | 14.8
3 | 11.1 | 11.6 | 14.9
Sepformer∗ | 2 | 12.9 | 13.5 | 25.9
3 | 10.0 | 10.5 | 26.0
ConvTasNet (NASS) | 2 | 9.6 | 10.2 | 6.8
3 | 7.9 | 8.5 | 7.0
DPRNN (NASS) | 2 | 13.1 | 13.6 | 14.9
3 | 11.5 | 12.0 | 15.0
Sepformer (NASS) | 2 | 13.6 | 14.1 | 25.9
3 | 10.9 | 11.4 | 26.1

Table 4: The competitive results of proposed NASS method on WHAM!. The
baseline results are reported in the original paper[9], where all can be
found.

Method | SI-SNRi (dB) | SDRi (dB) | Params (M)
---|---|---|---
DPRNN | 13.7 | 14.1 | 2.7
Sepformer | 14.4 | 15.0 | 26.0
TDANet | 15.2 | 15.4 | 2.3
DPRNN (NASS) | 15.6 | 15.9 | 14.9
Sepformer (NASS) | 15.9 | 16.2 | 25.9

##  5 Conclusions

In this paper, we propose NASS method to improve the noise-robustness of
previous SS works. Specifically, noise-speaker is set up to make use of the
noise supervision, and the prediction of noise can be used by other speech
tasks. Patch-wise contrastive learning is conducted on predicted speech and
noise representations, which minimize the mutual information beetween them,
thus separating to each other in detail. Experimental results show that NASS
can significantly suppress the noise from separated speech, with less than
0.1M additional parameters and achieves the current SOTA on WHAM! dataset.

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