aainstitutetext: Tsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, China

#  $N_{\rm eff}$ as a new physics probe in the precision era of cosmology

Yong Du<EMAIL_ADDRESS>
###### Abstract

We perform a global fit to the electroweak vertices and 4-fermion operators of
the standard model effective field theory in this work using $N_{\rm eff}$
from cosmological probes, as well as data sets from colliders and low-energy
experiments. Assuming flavor universality for both leptons and quarks, we find
$N_{\rm eff}$ from CMB-S4/HD can improve the fit for several operators by a
factor of a few. Without making any flavor assumption instead, $N_{\rm eff}$
can probe a different direction compared with the well-known flat one
presented in the neutrino trident process in the $4\mu$ sector. Fresh
sensitivity to two extra operators in the $\tau$ sector is also identified at
the 10% level. In addition, we find reducing the uncertainty in primordial
helium abundance from metal-poor galaxies down below 0.2% could play an
important role in deepening our understanding on the free neutron lifetime
anomaly.

##  1 Introduction

The standard model (SM) of particle physics has been very successful in
explaining almost all experimental data up to now, yet it is also broadly
accepted that the SM cannot be the complete theory as it does not provide any
explanation for several puzzles: the nature of dark matter, the masses of
neutrinos, the origin of nucleon mass and spin, the quantum nature of gravity
etc.

While many well-motivated new models can be constructed to explain the
unresolved aforementioned puzzles, it is unclear which one would be realized
in nature due to the null observation of any new particles at the LHC since
the discovery of the Higgs boson in 2012 ATLAS:2012yve ; CMS:2012qbp . With
its increasing luminosity and center-of-mass energy in particular, the LHC is
gradually pushing the new physics scale above around 1 TeV, a scale much
larger than the weak one. This, in turn, suggests that at the weak scale, we
can integrate the underlying new heavy physics out and work in its low-energy
effective theory that contains the standard matter contents of the SM only.
The apparent difference with the SM is that we now have contact interactions
with mass dimension larger than 4. Generically, we can parameterize the
corresponding Lagrangian of this kind of effective field theories as

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm
SM}+\sum_{n=5}C_{4-n}\mathcal{O}_{n},$ (1)

with $n$ the mass dimension of the operator $\mathcal{O}_{n}$, and $(4-n)$
that of the Wilson coefficient $C_{4-n}$. In this work, we will work in the SM
Effective Field Theory (SMEFT) that respects the same local gauge group of the
SM with the same field components, and only consider the leading order
corrections from these operators. Recall that the dimension-5 operators only
contribute to neutrino masses Weinberg:1979sa , this approximation thus means
that we will only consider dimension-6 SMEFT operators Buchmuller:1985jz ;
Grzadkowski:2010es  in this work.

These dimension-6 operators will, on the one hand, change the SM interaction
strength, and on the other hand, introduce contact interactions that are
absent in the SM Lagrangian. One well-known example is the semi-leptonic
4-fermion operator $\bar{u}de\bar{\nu}$ responsible for beta decay as proposed
by Fermi in 1933 Fermi:1934sk . In a similar pattern and in the SMEFT
framework, pure leptonic 4-fermion operators like $\bar{l}l\bar{l}l$ will be
present. These 4-fermion operators, together with those modifying the SM
vertices, could affect lepton pair production measured at LEP ALEPH:2013dgf
for instance. Provided these corrections from the SMEFT operators are large,
these deviations from the SM prediction will have the chance to be observed
experimentally. Alternatively, if no significant deviation is captured by
detectors, constraints can be put on the corresponding Wilson coefficients.

Due to the high energy and/or the high luminosity, tremendous studies
utilizing the SMEFT framework have been carried out given the very rich data
from LEP and/or the LHC, see for example Efrati:2015eaa ; Falkowski:2015krw ;
Berthier:2016tkq ; Falkowski:2017pss ; Barklow:2017suo ; Barklow:2017awn ;
Cirigliano:2018dyk ; DeBlas:2019qco ; Ellis:2020unq ; Gu:2020ldn ;
Ethier:2021bye ; Cirigliano:2021yto ; Durieux:2022cvf ; Ellis:2022zdw ;
Aoude:2022deh ; Schwienhorst:2022yqu ; Breso-Pla:2023tnz  and references
therein. As mentioned earlier, since the SM has been working so well to fit
these data, results are generically presented as constraints on the Wilson
coefficients. While it is a common practice in literature to consider one
operator at a time in this procedure, a model-independent study shall include
all contributing operators since in the underlying model(s), different
operators are likely interdependent. This in turn motivates us to perform a
global fit to the SMEFT operators when doing such an exercise.

In this work, we will combine colliders with cosmological probes and consider
SMEFT global fit during neutrino decoupling in the early Universe. At the
beginning, neutrinos were in thermal equilibrium with the other components in
the thermal plasma through weak interactions. However, as the Universe
expands, the temperature drops such that when the temperature is below about 2
MeV Dolgov:2002wy , this equilibrium could not be maintained any longer and
neutrinos then decoupled from the rest of the plasma and started free
streaming. In the SMEFT, several types of new interactions between neutrinos,
electrons, as well as neutrinos and electrons, are present. As a result, the
evolution history of neutrinos in the early Universe will be modified and
neutrinos could decouple from the plasma at a different epoch than that in the
standard scenario. Consequently, $N_{\rm eff}$, the effective number of
relativistic species in the early Universe, will be predicted differently, and
this difference may then be detected from precision cosmology. Theoretically,
$N_{\rm eff}$ has recently been calculated with a precision of
$\mathcal{O}(10^{-4})$ Akita:2020szl ; Froustey:2020mcq . Experimentally,
though the current Planck uncertainty is still at the 10% level, the planned
next generation experiments such as SPT-3G Benson:2014qhw , CMB-S4
Abazajian:2016yjj , CORE DiValentino:2016foa , the Simon Observatory
Ade:2018sbj , PICO Hanany:2019lle  and CMB-HD Sehgal:2019ewc ; CMB-HD:2022bsz
would be able to eventually reduce the uncertainty down below the percent
level. This therefore motivates us to perform the global fit of the SMEFT
mentioned at the beginning of this paragraph.

By including $N_{\rm eff}$ in the global fit, we then find that

  * •

ignoring the two operators in the next bullet, the well-known flat direction
from neutrino trident production in the $4\mu$ sector can be lifted due to
muon depletion during the decoupling as shown in figure 1;

  * •

fresh 10% level sensitivity to two extra operators, i.e.,
$[\mathcal{O}_{ll}]_{2233,3333}$ as seen in figure 2, can be obtained as we do
not make any flavor assumption during the fit;

  * •

the global fit results of five operators — two electroweak vertices plus three
4-lepton operators — can be improved significantly in the flavor universal
scenario as shown in eq. (38). While in the flavor general case, the new
direction probed by $N_{\rm eff}$ will be constrained at the 10% level as
indicated by the last line of eq. (47);

  * •

improving the extraction precision of the primordial abundance of helium down
below 0.2% experimentally could help answer if new ingredients beyond the
standard big-bang nucleosynthesis are needed in understanding the free neutron
lifetime anomaly, as illustrated in figure 3.

In obtaining these results, we firstly discuss $N_{\rm eff}$ calculation
within the SM and in the presence of SMEFT operators in section 2, we discuss
the input parameters used in this study in the end, and then show our results
in section 3. In presenting our results, we show the numbers in both the
flavor universal scenario and the flavor general scenario. Then in section 4,
we discuss how the global fit could affect the prediction of the primordial
abundance of helium, as well as its connection to the free neutron lifetime
anomaly. We then conclude in section 5.

##  2 $N_{\rm eff}$ in the SMEFT

In this section, we will firstly briefly review the precision calculation of
$N_{\rm eff}$ in the SM, and then introduce corrections from the dimension-6
operators in the SMEFT. Given our treatment on lepton flavors, a brief comment
on $\nu_{\mu,\tau}$ will also be discussed.

###  2.1 Brief review of $N_{\rm eff}$ in the SM

Neutrinos, electrons and positrons, as well as photons are in thermal
equilibrium in the very early Universe with an equal temperature, i.e.,
$T_{\gamma}=T_{e}=T_{\nu}$. However, as the Universe expands, weak
interactions start to decouple from the thermal plasma at around $T_{\rm
dec}=2$ MeV Dolgov:2002wy  when the weak interaction strength falls below the
Hubble parameter. At the same time, photons and electrons are still tightly
coupled with each other with $T_{\gamma}=T_{e}$ thanks to quantum
electrodynamics (QED). The neutrinos will simply undergo dilution after the
decoupling, while the photons can still receive energy injection from the
plasma even when the temperature of the Universe drops below $m_{e}$ from
$e^{+}e^{-}\to\gamma\gamma$.111Note that when the temperature of the Universe
is below $m_{e}$, the inverse process $\gamma\gamma\to e^{+}e^{-}$ will be
suppressed by $e^{-m_{e}/T}$. As a result, the photon temperature will be
slightly higher than that of the neutrino, which can be parameterized as
Shvartsman:1969mm ; Steigman:1977kc ; Mangano:2001iu :

$\displaystyle\rho_{R}=\left[1+\frac{7}{8}\left(\frac{4}{11}\right)^{\frac{4}{3}}N_{\rm
eff}\right]\rho_{\gamma}.$ (2)

Here, $\rho_{\gamma}=(\pi^{2}/30)g_{\gamma}T_{\gamma}^{4}$ is the photon
energy density, $\rho_{R}=\rho_{\gamma}+3\rho_{\nu}+3\rho_{\bar{\nu}}$ with
$\rho_{\nu}=\rho_{\bar{\nu}}=(7\pi^{2}/240)g_{\nu}T_{\nu}^{4}$ is the total
energy density of all relativistic species at the considered time,222We ignore
neutrino masses due to their tininess. whose effective number is denoted as
$N_{\rm eff}$, and $g_{i}$ is the intrinsic degrees of freedom of particle
$i$. In the instantaneous neutrino decoupling limit, it is well-known that
$N_{\rm eff}=3$, while in general, it is straightforward to solve from the
above equations that

$\displaystyle N_{\rm
eff}=3\times\left(\frac{11}{4}\right)^{\frac{4}{3}}\times\left(\frac{T_{\nu}}{T_{\gamma}}\right)^{4}.$
(3)

Taking the neutrino non-instantaneous decoupling effects, neutrino
oscillations during neutrino decoupling Hannestad:2001iy ; Dolgov:2002ab ;
Mangano:2005cc ; deSalas:2016ztq ; Gariazzo:2019gyi , as well as finite
temperature QED corrections Heckler:1994tv ; Fornengo:1997wa ; Bennett:2019ewm
into account, it has been calculated that $N_{\rm eff}=3.044$ Akita:2020szl ;
Froustey:2020mcq , with higher-order theoretical corrections at
$\mathcal{O}(10^{-4})$ and can be safely neglected for this study.

Therefore, at this stage, it is clear that to precisely predict $N_{\rm eff}$
at any time during the evolution of the Universe, one shall track the
evolution of $T_{\gamma}$ and $T_{\nu}$ and then make use of eq. (3). On the
other hand, the evolution of $T_{\gamma}$ and $T_{\nu}$ can be obtained by
solving the Boltzmann equations, which read

$\displaystyle\frac{dn}{dt}+3Hn$ $\displaystyle=\frac{\delta n}{\delta
t}\equiv\int g\frac{d^{3}p}{(2\pi)^{3}}\mathcal{C}[f],$ (4)
$\displaystyle\frac{d\rho}{dt}+3H(\rho+p)$
$\displaystyle=\frac{\delta\rho}{\delta t}\equiv\int
gE\frac{d^{3}p}{(2\pi)^{3}}\mathcal{C}[f],$ (5)

where $g$ is the intrinsic degrees of freedom of the particle under
consideration, $E$ is its energy, and $n$ and $\rho$ are the number and the
energy densities, respectively. The collision term $\mathcal{C}$ is given by

$\displaystyle\mathcal{C}\left[f_{i}\right]\equiv$
$\displaystyle\frac{1}{2E_{i}}\sum_{X,Y}\int\prod_{i,j}d\Pi_{X_{i}}d\Pi_{Y_{j}}(2\pi)^{4}\delta^{4}\left(p_{i}+p_{X}-p_{Y}\right)$
$\displaystyle\times\left(\langle\mathcal{M}^{2}\rangle_{Y\rightarrow
i+X}\prod_{i,j}f_{Y_{j}}\left[1\pm f_{i}\right]\left[1\pm
f_{X_{i}}\right]-\langle\mathcal{M}^{2}\rangle_{i+X\rightarrow
Y}\prod_{i,j}f_{i}f_{X_{i}}\left[1\pm f_{Y_{j}}\right]\right),$ (6)

with $d\Pi_{{i}}\equiv d^{3}p_{i}/[(2\pi)^{3}2E_{i}]$, and “+ ($-$)” for
bosonic (fermionic) particles. The evolution of $T_{\gamma}$ and $T_{\nu}$ can
then be obtained through applying chain rules to eq. (4-5) to give
Escudero:2020dfa

$\displaystyle\frac{dT_{\gamma}}{dt}=$
$\displaystyle\,-\frac{4H\rho_{\gamma}+3H\left(\rho_{e}+p_{e}\right)+3HT_{\gamma}\frac{dP_{\text{int
}}}{dT_{\gamma}}+\frac{\delta\rho_{\nu e}}{\delta
t}+\frac{\delta\rho_{\nu\mu}}{\delta t}+\frac{\delta\rho_{\nu\tau}}{\delta
t}}{\frac{\partial\rho_{\gamma}}{\partial
T_{\gamma}}+\frac{\partial\rho_{e}}{\partial
T_{\gamma}}+T_{\gamma}\frac{d^{2}P_{\text{int }}}{dT_{\gamma}^{2}}},$ (7)
$\displaystyle\frac{dT_{\nu_{\alpha}}}{dt}=$
$\displaystyle\,-HT_{\nu_{\alpha}}+\frac{\delta\rho_{\nu_{\alpha}}}{\delta
t}/\frac{\partial\rho_{\nu_{\alpha}}}{\partial
T_{\nu_{\alpha}}},\quad\quad\alpha=e,\mu,\tau.$ (8)

Here, $P_{\rm int}$ and $\rho_{\rm int}\equiv-P_{\rm int}+dP_{\rm int}/d\ln
T_{\gamma}$, calculated in Bennett:2019ewm , are finite temperature
corrections to the electromagnetic pressure and energy density, respectively.
Solving eqs. (7-8) is in general quite challenging and time-consuming since,
for any $2\to 2$ processes, the collision term integrals are 12-fold ones
whose analytical expressions are not known. In the EFT language as we adopt in
this work, a complete, analytical, and generic dictionary for the collision
term integrals have been built in Du:2021idh  up to dimension 7, rendering
numerically solving eqs. (7 -8) or $N_{\rm eff}$ efficient when including
dimension-6 operators in the SMEFT.

Scattering/Annihilation Processes
---
Label |  $\nu_{e}$ case ($j={\mu,\tau}$) |  $\nu_{i}$ case ($i={\mu,\tau}$; $j=i\cup{e}$, $i\neq j$)
1 | $\nu_{e}+\bar{\nu}_{e}\to\nu_{e}+\bar{\nu}_{e}$ | $\nu_{i}+\bar{\nu}_{i}\to\nu_{i}+\bar{\nu}_{i}$
2 | $\nu_{e}+\nu_{e}\to\nu_{e}+\nu_{e}$ | $\nu_{i}+\nu_{i}\to\nu_{i}+\nu_{i}$
3 | $\nu_{e}+\bar{\nu}_{e}\to\nu_{j}+\bar{\nu}_{j}$ | $\nu_{i}+\bar{\nu}_{i}\to\nu_{j}+\bar{\nu}_{j}$
4 | $\nu_{e}+\bar{\nu}_{j}\to\nu_{e}+\bar{\nu}_{j}$ | $\nu_{i}+\bar{\nu}_{j}\to\nu_{i}+\bar{\nu}_{j}$
5 | $\nu_{e}+\nu_{j}\to\nu_{e}+\nu_{j}$ | $\nu_{i}+\nu_{j}\to\nu_{i}+\nu_{j}$
6 | $\nu_{e}+\bar{\nu}_{e}\to e^{+}+e^{-}$ | $\nu_{i}+\bar{\nu}_{i}\to e^{+}+e^{-}$
7 | $\nu_{e}+e^{-}\to\nu_{e}+e^{-}$ | $\nu_{i}+e^{-}\to\nu_{i}+e^{-}$
8 | $\nu_{e}+e^{+}\to\nu_{e}+e^{+}$ | $\nu_{i}+e^{+}\to\nu_{i}+e^{+}$
Table 1: $\nu_{e}$ and $\nu_{\mu,\tau}$ scattering/annihilating processes
contributing to non-instantaneous neutrino decoupling in the early Universe.
For each process above, the incoming neutrino can be replaced by its anti-
partner, which also contributes to $N_{\rm eff}$ and is implicitly implied in
the table above.

In the SM, the collision term integrals come from the scattering/annihilation
processes summarized in Table 1. All these processes are mediated by an off-
shell $Z/W^{\pm}$, which will be modified in the presence of new physics that
modifies $Z\bar{f}f/Wff^{\prime}$ couplings and/or introduces contact
4-fermion interactions. Specific ultraviolet examples include the $Z^{\prime}$
model or a neutral scalar $\phi$ in the neutral current case, and an
intermediate $W_{R}$ exchange from the left-right symmetric model for example
or a charged scalar $\phi^{\pm}$ in the charged current case. Here in this
work, we are aiming at a model independent analysis of $N_{\rm eff}$ in the
SMEFT, thus we will avoid any specific model discussion in the following even
though this extension is straightforward.

###  2.2 From the SMEFT to $N_{\rm eff}$

As mentioned in last subsection, if new physics modifies
$Z\bar{f}f/Wff^{\prime}$ couplings and/or introduces effective 4-fermion
interactions, the prediction of $N_{\rm eff}$ will be modified in general.
Examples include the $Z^{\prime}$ model and the $W_{R}$ exchange from the
left-right symmetric model discussed earlier. While these models may have very
rich physics at colliders or low-energy experiments like neutrino
oscillations, we will adopt the SMEFT framework in this work to avoid any bias
in model selection. We shall point out that the results presented in this work
can be directly translated onto specific models, provided the new physics is
heavy to ensure the validity of the SMEFT, see deBlas:2022ofj  for example in
this regard. In what follows, we will first review the SMEFT operators
relevant for $N_{\rm eff}$, and then discuss their impact on the latter.

####  2.2.1 The SMEFT framework

In the SMEFT, we parameterize the electroweak vertices and the contact
4-fermion interactions in the Higgs basis
LHCHiggsCrossSectionWorkingGroup:2016ypw , and summarize our notations in eq.
(9) and Table 2, respectively. Generically speaking, $Zqq$ and $Wqq^{\prime}$
vertices are also modified by SMEFT operators and can be parameterized in a
similar way as those in eq. (9). These quark vertices are however irrelevant
for the calculation of $N_{\rm eff}$ due to the approximation we adopt here
and thus not included explicitly. Similar arguments apply to semi-leptonic
4-fermion operators, and we therefore only show the pure leptonic operators in
Table 2.

$\displaystyle{\cal L}_{Vff}=\,$
$\displaystyle{g_{L}\over\sqrt{2}}\left[W^{\mu+}\bar{\nu}_{I}\bar{\sigma}_{\mu}(\delta_{IJ}+[\delta
g^{W\ell}_{L}]_{IJ})e_{J}+h.c.\right]$
$\displaystyle+\sqrt{g_{L}^{2}+g_{Y}^{2}}Z^{\mu}\sum_{f=e,\nu}\bar{f}_{I}\bar{\sigma}_{\mu}\left((T_{3}^{f}-s_{w}^{2}Q_{f})\delta_{IJ}+\left[\delta
g^{Zf}_{L}\right]_{IJ}\right)f_{J}$
$\displaystyle+\sqrt{g_{L}^{2}+g_{Y}^{2}}Z^{\mu}e_{I}^{c}\sigma_{\mu}\left(-s^{2}_{w}Q_{e}\delta_{IJ}+\left[\delta
g^{Ze}_{R}\right]_{IJ}\right)\bar{e}_{J}^{c},$ (9)

4-fermion operators
---
One flavor ($p,r=1,2,3$) |  Two flavors ($p<r=1,2,3$)
$[\mathcal{O}_{ll}]_{pppp}={1\over 2}(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{p})(\bar{\ell}_{p}\bar{\sigma}^{\mu}\ell_{p})$ | $[\mathcal{O}_{ll}]_{pprr}=(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{p})(\bar{\ell}_{r}\bar{\sigma}^{\mu}\ell_{r})$
$[\mathcal{O}_{le}]_{pppp}=(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{p})(e_{p}^{c}\sigma^{\mu}\bar{e}_{p}^{c})$ | $[\mathcal{O}_{ll}]_{prrp}=(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{r})(\bar{\ell}_{r}\bar{\sigma}^{\mu}\ell_{p})$
|
$[\mathcal{O}_{le}]_{pprr}=(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{p})(e_{r}^{c}\sigma^{\mu}\bar{e}_{r}^{c})$
|
$[\mathcal{O}_{le}]_{rrpp}=(\bar{\ell}_{r}\bar{\sigma}_{\mu}\ell_{r})(e_{p}^{c}\sigma^{\mu}\bar{e}_{p}^{c})$
|
$[\mathcal{O}_{le}]_{prrp}=(\bar{\ell}_{p}\bar{\sigma}_{\mu}\ell_{r})(e_{r}^{c}\sigma^{\mu}\bar{e}_{p}^{c})$
Table 2: 4-fermion operators contributing to $N_{\rm eff}$ .

Several comments are in order:

  * •

One extra 4-lepton operator $[\mathcal{O}_{ee}]_{1111}$ is present in the
SMEFT. This operator will contribute to electron scattering/annihilation
during neutrino decoupling, which in general also depends on
$[\mathcal{O}_{ll,le}]_{1111}$. The Bhabha process at LEP ALEPH:2013dgf  with
different energy runs and the Møller process at SLAC-E158 SLACE158:2005uay ,
as well as the next generation parity-violating electron scattering experiment
MOLLER at the Jefferson Laboratory MOLLER:2014iki  with polarized electron
beams, can be used to stringently constrained them. For this study, since
$[\mathcal{O}_{ee}]_{1111}$ neither changes the number nor the energy density
of electrons, it will not modify $N_{\rm eff}$ and can therefore be neglected.

  * •

The two 4-fermion operators in the left column contribute to $\nu_{e}$-$e$ and
$\nu_{e}$-$\nu_{e}$ scattering/annihilation.333Scattering/Annihilation
involving anti-particles is to be understood with our notations here and in
the following. The first process had been measured in the past by several
experiments including the Savannah River Plant Reines:1976pv , LAMPF
Allen:1992qe , CHARM CHARM-II:1994aeb ; CHARM-II:1995xfh , LSND LSND:2001akn ,
and TEXONO Deniz:2010mp , as nicely reviewed in Formaggio:2012cpf . The latter
process is more challenging from homemade targets, but the cosmological probe
$N_{\rm eff}$ can be used to provide this missing information.

  * •

The remaining 4-fermion operators in the right column of Table 2 would
contribute to $\nu_{\mu,\tau}$ scattering/annihilation. Depending on the
strength of the corresponding Wilson coefficients, they can either delay or
advance the decoupling of neutrinos from the plasma, thus modifying the
prediction of $N_{\rm eff}$ at late time. In turn, a precision measurement of
$N_{\rm eff}$ from Planck Aghanim:2018eyx  can be used to constrain these
operators, which will be further improved by next-generation experiments
introduced in the previous section, and will be the main focus of this work.

  * •

Similar to the 4-fermion operators, electroweak vertices in eq. (9) can also
change the prediction of $N_{\rm eff}$ through modifying the interacting
strengths between the SU(2) gauge bosons and the SM fermions. These vertices
are stringently constrained at the $10^{-3}\sim 10^{-4}$ level currently
deBlas:2022ofj , such that one can safely ignore their effects on $N_{\rm
eff}$ during the global fit. For this study, we nevertheless include them in
our analysis, and confirm that their impact on $N_{\rm eff}$ is marginal, as
will become clear in section 3.

  * •

Specific flavor assumption can be made to reduce the number of parameters and
thus simplify the analysis. We will not make any flavor assumption in this
work in order to be generic. Generalization to specific preferred flavor
scenarios will be straightforward at the end of the analysis, and global fit
results in the flavor universal case will be presented for illustration.

  * •

It is common in literature to assume the indistinguishability between
$\nu_{\mu}$ and $\nu_{\tau}$ for the calculation of $N_{\rm eff}$ during the
evolution of the Universe, i.e., completely ignoring the flavor difference
between $\nu_{\mu}$ and $\nu_{\tau}$. This is, however, in contradiction with
the spirit of this work, viz, a flavor general global fit of electroweak
vertices and 4-fermion operators. For this reason, we will track independently
the evolution of $T_{\gamma,\nu_{e},\nu_{\mu},\nu_{\tau}}$ for a precision
calculation of $N_{\rm eff}$ in the SMEFT, which is one of the novel aspects
of this work. As a result, as we shall see shortly, we are able to probe more
SMEFT operators in the $\tau$ sector utilizing $N_{\rm eff}$.

With this, we will then calculate SMEFT corrections to the collision term
integrals in the next subsection.

####  2.2.2 Corrections to the collision terms from the SMEFT

The invariant amplitude for each process in Table 1 receives SMEFT corrections
from eq. (9) and Table 2, which can be generically written as

$\displaystyle\langle\mathcal{M}^{2}\rangle$
$\displaystyle\,=\langle\mathcal{M}_{\rm SM}^{2}\rangle+2\,{\rm
Re}\langle\mathcal{M}_{\rm SM}\cdot\mathcal{M}_{\rm
SMEFT}^{\dagger}\rangle+\langle\mathcal{M}_{\rm SMEFT}^{2}\rangle$
$\displaystyle\,\approx\langle\mathcal{M}_{\rm SM}^{2}\rangle+2\,{\rm
Re}\langle\mathcal{M}_{\rm SM}\cdot\mathcal{M}_{\rm SMEFT}^{\dagger}\rangle,$
(10)

where $\mathcal{M}_{\rm SM}$ and $\mathcal{M}_{\rm SMEFT}$ are the SM and the
SMEFT amplitudes, respectively. As mentioned earlier, in this work, we will
only keep the leading order corrections from the SMEFT operators, thus the
last term in the first line of eq. (10) will be neglected in the following.

For reference, we summarize the leading order SMEFT corrections to the
invariant amplitudes of each process in Table 1 in this subsection. In each
case, the amplitudes are given in the same order as that in each column of
Table 1, with the symmetry factor implicitly included.

  * •

$\nu_{e}$ case ($j=\mu,\tau$, or equivalently, $j=2,3$):

$\displaystyle\langle\mathcal{M}^{2}\rangle_{1}$
$\displaystyle\,=128G_{F}^{2}\cdot
p_{14}^{2}\cdot\left(1-2[c_{ll}]_{1111}+8\delta g_{W}^{e\nu}+8\delta
g_{Z,L}^{ee}\right),$ (11) $\displaystyle\langle\mathcal{M}^{2}\rangle_{2}$
$\displaystyle\,=64G_{F}^{2}\cdot
p_{12}^{2}\cdot\left(1-2[c_{ll}]_{1111}+8\delta g_{W}^{e\nu}+8\delta
g_{Z,L}^{ee}\right),$ (12)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{3}^{j}$
$\displaystyle\,=32G_{F}^{2}\cdot
p_{14}^{2}\cdot\left(1-2[c_{ll}]_{11jj}-2[c_{ll}]_{1jj1}+4\delta
g_{W}^{e\nu}\right.$ $\displaystyle\left.\qquad\qquad\qquad\quad+4\delta
g_{W}^{j\nu_{j}}+4\delta g_{Z,L}^{ee}+4\delta g_{Z,L}^{jj}\right),$ (13)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{4}^{j}$
$\displaystyle\,=\langle\mathcal{M}^{2}\rangle_{3}^{j},$ (14)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{5}^{j}$
$\displaystyle\,=\frac{p_{12}^{2}}{p_{14}^{2}}\cdot\langle\mathcal{M}^{2}\rangle_{3}^{j},$
(15) $\displaystyle\langle\mathcal{M}^{2}\rangle_{6}$
$\displaystyle\,=32G_{F}^{2}\left[2m_{e}^{2}s_{W}^{2}(1+2s_{W}^{2})p_{12}+(1+2s_{W}^{2})^{2}p_{13}^{2}+4s_{W}^{4}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad-2[c_{ll}]_{1111}(p_{13}^{2}+(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{1111}(4p_{14}^{2}s_{W}^{2}+m_{e}^{2}p_{12}(1+2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{13}^{2}+2(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2}+4(m_{e}^{2}p_{12}+p_{13}^{2}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{e\nu}\right.$
$\displaystyle\left.\qquad\qquad\quad+4s_{W}^{2}(m_{e}^{2}p_{12}+2p_{13}^{2}+4(m_{e}^{2}p_{12}+p_{13}^{2}+p_{14}^{2})s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{14}^{2}+2m_{e}^{2}(1+2s_{W}^{2})p_{12})\delta
g_{Z,R}^{ee}\right],$ (16) $\displaystyle\langle\mathcal{M}^{2}\rangle_{7}$
$\displaystyle\,=32G_{F}^{2}\left[(1+2s_{W}^{2})^{2}p_{12}^{2}-2m_{e}^{2}s_{W}^{2}(1+2s_{W}^{2})p_{13}+4s_{W}^{4}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad-2[c_{ll}]_{1111}(p_{12}^{2}+(2p_{12}^{2}-m_{e}^{2}p_{13})s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{1111}(4p_{14}^{2}s_{W}^{2}-m_{e}^{2}p_{13}(1+2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{12}^{2}+2(2p_{12}^{2}-m_{e}^{2}p_{13})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{e\nu}\right.$
$\displaystyle\left.\qquad\qquad\quad+4s_{W}^{2}(2p_{12}^{2}-m_{e}^{2}p_{13}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{14}^{2}-2m_{e}^{2}(1+2s_{W}^{2})p_{13})\delta
g_{Z,R}^{ee}\right],$ (17) $\displaystyle\langle\mathcal{M}^{2}\rangle_{8}$
$\displaystyle\,=32G_{F}^{2}\left[4s_{W}^{4}p_{12}^{2}-2m_{e}^{2}s_{W}^{2}(1+2s_{W}^{2})p_{13}+(1+2s_{W}^{2})^{2}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad-2[c_{ll}]_{1111}(p_{14}^{2}+(2p_{14}^{2}-m_{e}^{2}p_{13})s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{1111}(4p_{12}^{2}s_{W}^{2}-m_{e}^{2}p_{13}(1+2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{14}^{2}+2(2p_{14}^{2}-m_{e}^{2}p_{13})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{e\nu}\right.$
$\displaystyle\left.\qquad\qquad\quad+4s_{W}^{2}(2p_{14}^{2}-m_{e}^{2}p_{13}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{12}^{2}-2m_{e}^{2}(1+2s_{W}^{2})p_{13})\delta
g_{Z,R}^{ee}\right],$ (18)

with

$\displaystyle p_{ij}\equiv p_{i}\cdot p_{j}.$ (19)

  * •

$\nu_{j}$ ($j=\mu,\tau$, or equivalently, $j=2,3$, respectively) case:

$\displaystyle\langle\mathcal{M}^{2}\rangle_{1}^{j}$
$\displaystyle\,=128G_{F}^{2}\cdot
p_{14}^{2}\cdot\left(1-2[c_{ll}]_{jjjj}+8\delta g_{W}^{j\nu_{j}}+8\delta
g_{Z,L}^{jj}\right),$ (20) $\displaystyle\langle\mathcal{M}^{2}\rangle_{2}$
$\displaystyle\,=64G_{F}^{2}\cdot
p_{12}^{2}\cdot\left(1-2[c_{ll}]_{jjjj}+8\delta g_{W}^{j\nu_{j}}+8\delta
g_{Z,L}^{jj}\right),$ (21) $\displaystyle\langle\mathcal{M}^{2}\rangle_{3}$
$\displaystyle\,=32G_{F}^{2}\cdot
p_{14}^{2}\cdot\left(1-2[c_{ll}]_{11jj}-2[c_{ll}]_{1jj1}+4\delta
g_{W}^{e\nu}\right.$ $\displaystyle\left.\qquad\qquad\qquad\quad+4\delta
g_{W}^{j\nu_{j}}+4\delta g_{Z,L}^{ee}+4\delta
g_{Z,L}^{jj}\right),\quad\text{for
}(\nu_{j}\bar{\nu}_{j}\to\nu_{e}\bar{\nu}_{e})$ (22)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{3}$
$\displaystyle\,=32G_{F}^{2}\cdot
p_{14}^{2}\cdot\left(1-2[c_{ll}]_{2233}-2[c_{ll}]_{2332}+4\delta
g_{W}^{\mu\nu}\right.$ $\displaystyle\left.\qquad\qquad\qquad\quad+4\delta
g_{W}^{\tau\nu}+4\delta g_{Z,L}^{\mu\mu}+4\delta
g_{Z,L}^{\tau\tau}\right),\quad\text{for
}(\nu_{j}\bar{\nu}_{j}\to\nu^{\prime}_{j}\bar{\nu}^{\prime}_{j})$ (23)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{4}^{j}$
$\displaystyle\,=\langle\mathcal{M}^{2}\rangle_{3}^{j},$ (24)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{5}^{j}$
$\displaystyle\,=\frac{p_{12}^{2}}{p_{14}^{2}}\cdot\langle\mathcal{M}^{2}\rangle_{3}^{j},$
(25) $\displaystyle\langle\mathcal{M}^{2}\rangle_{6}^{j}$
$\displaystyle\,=32G_{F}^{2}\left[2m_{e}^{2}s_{W}^{2}(2s_{W}^{2}-1)p_{12}+(1-2s_{W}^{2})^{2}p_{13}^{2}+4s_{W}^{4}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad+[c_{ll}]_{11jj}(2p_{13}^{2}-2(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{jj11}(4p_{14}^{2}s_{W}^{2}-m_{e}^{2}p_{12}(1-2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{13}^{2}-2(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2}+4(m_{e}^{2}p_{12}+p_{13}^{2}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{j\nu_{j}}\right.$
$\displaystyle\left.\qquad\qquad\quad-4(p_{13}^{2}-(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{13}^{2}-2(m_{e}^{2}p_{12}+2p_{13}^{2})s_{W}^{2}+4(m_{e}^{2}p_{12}+p_{13}^{2}+p_{14}^{2})s_{W}^{4})\delta
g_{Z,L}^{jj}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{14}^{2}-2m_{e}^{2}(1-2s_{W}^{2})p_{12})\delta
g_{Z,R}^{ee}\right],$ (26)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{7}^{j}$
$\displaystyle\,=32G_{F}^{2}\left[(1-2s_{W}^{2})^{2}p_{12}^{2}+2m_{e}^{2}s_{W}^{2}(1-2s_{W}^{2})p_{13}+4s_{W}^{4}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad+2[c_{ll}]_{11jj}(p_{12}^{2}+m_{e}^{2}s_{W}^{2}p_{13}-2p_{12}^{2}s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{jj11}(4p_{14}^{2}s_{W}^{2}+m_{e}^{2}p_{13}(1-2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{12}^{2}+2(m_{e}^{2}p_{13}-2p_{12}^{2})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{j\nu_{j}}\right.$
$\displaystyle\left.\qquad\qquad\quad-4(p_{12}^{2}+m_{e}^{2}p_{13}s_{W}^{2}-2p_{12}^{2}s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{12}^{2}+2(m_{e}^{2}p_{13}-2p_{12}^{2})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{Z,L}^{jj}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{14}^{2}+2m_{e}^{2}(1-2s_{W}^{2})p_{13})\delta
g_{Z,R}^{ee}\right],$ (27)
$\displaystyle\langle\mathcal{M}^{2}\rangle_{8}^{j}$
$\displaystyle\,=32G_{F}^{2}\left[4s_{W}^{4}p_{12}^{2}+2m_{e}^{2}s_{W}^{2}(1-2s_{W}^{2})p_{13}+(1-2s_{W}^{2})^{2}p_{14}^{2}\right.$
$\displaystyle\left.\qquad\qquad\quad+2[c_{ll}]_{11jj}(p_{14}^{2}+m_{e}^{2}s_{W}^{2}p_{13}-2p_{14}^{2}s_{W}^{2})\right.$
$\displaystyle\left.\qquad\qquad\quad-[c_{le}]_{jj11}(4p_{12}^{2}s_{W}^{2}+m_{e}^{2}p_{13}(1-2s_{W}^{2}))\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{14}^{2}+2(m_{e}^{2}p_{13}-2p_{14}^{2})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{W}^{j\nu_{j}}\right.$
$\displaystyle\left.\qquad\qquad\quad-4(p_{14}^{2}+m_{e}^{2}p_{13}s_{W}^{2}-2p_{14}^{2}s_{W}^{2})\delta
g_{Z,L}^{ee}\right.$
$\displaystyle\left.\qquad\qquad\quad+4(p_{14}^{2}+2(m_{e}^{2}p_{13}-2p_{14}^{2})s_{W}^{2}+4(p_{12}^{2}-m_{e}^{2}p_{13}+p_{14}^{2})s_{W}^{4})\delta
g_{Z,L}^{jj}\right.$
$\displaystyle\left.\qquad\qquad\quad+(8s_{W}^{2}p_{12}^{2}+2m_{e}^{2}(1-2s_{W}^{2})p_{13})\delta
g_{Z,R}^{ee}\right],$ (28)

In each case, we reproduce the corresponding SM result when setting all the
Wilson coefficients to zero. Then, with the results above, the collision term
integrals can be directly obtained from the dictionary derived in Du:2021idh ,
which will serve as the input for us to numerically solve $T_{\gamma}$ and
$T_{\nu_{\alpha}}$ before obtaining $N_{\rm eff}$ through eq. (3). As an
example, we consider SMEFT corrections to
$\nu_{e}+\bar{\nu}_{e}\to\nu_{j}+\bar{\nu}_{j}\,(j=\mu,\tau)$. The invariant
amplitudes are given in eq. (13). Then, from the dictionary in Du:2021idh ,
their eq.(4.19) to be more specific, one immediately finds, for the number and
the energy density respectively,

$\displaystyle\frac{\delta n_{\nu_{e}}}{\delta t}=$
$\displaystyle\,\frac{1}{8\pi^{5}}\left[-{e}^{\frac{2\mu_{\nu_{e}}}{T_{\nu_{e}}}}T_{\nu_{e}}^{8}+{e}^{\frac{2\mu_{\nu_{j}}}{T_{\nu_{j}}}}T_{\nu_{j}}^{8}\right]$
$\displaystyle\,\times
32G_{F}^{2}\cdot\left(1-2[c_{ll}]_{11jj}-2[c_{ll}]_{1jj1}+4\delta
g_{W}^{e\nu}+4\delta g_{W}^{j\nu_{j}}+4\delta g_{Z,L}^{ee}+4\delta
g_{Z,L}^{jj}\right),$ (29) $\displaystyle\frac{\delta\rho_{\nu_{e}}}{\delta
t}=$
$\displaystyle\,\frac{1}{8\pi^{5}}\left[-4{e}^{\frac{2\mu_{\nu_{e}}}{T_{\nu_{e}}}}T_{\nu_{e}}^{9}+4{e}^{\frac{2\mu_{\nu_{j}}}{T_{\nu_{j}}}}T_{\nu_{j}}^{9}\right]$
$\displaystyle\,\times
32G_{F}^{2}\cdot\left(1-2[c_{ll}]_{11jj}-2[c_{ll}]_{1jj1}+4\delta
g_{W}^{e\nu}+4\delta g_{W}^{j\nu_{j}}+4\delta g_{Z,L}^{ee}+4\delta
g_{Z,L}^{jj}\right).$ (30)

The same procedure is applied to all the processes in Table 1 using the
invariant amplitudes in eqs. (11-28), and keeping the leading order SMEFT
corrections to $T_{\gamma}$ and $T_{\nu_{\alpha}}$ only. Then we numerically
solve eqs. (7-8) to get $N_{\rm eff}$ from eq. (3), which we discuss in the
next subsection.

####  2.2.3 Corrections to $N_{\rm eff}$ from the SMEFT

Including the electroweak vertex and the 4-fermion operator corrections
discussed above, we use the Mathematica code EFT2Neff implemented in-house and
based on nudec_BSM of Escudero:2020dfa , to numerically solve the Boltzmann
equations, i.e., eqs. (7-8). Then from eq. (3), we find that in the SMEFT,
$N_{\rm eff}$ is modified to be

$\displaystyle N_{\rm eff}=$
$\displaystyle\,3.04385+3.03784[c_{le}]_{1111}+3.01675[c_{ll}]_{1111}+3.03606[c_{le}]_{2211}+3.03539[c_{le}]_{3311}$
$\displaystyle\,+3.07371\delta g_{W}^{e\nu}+3.03917\delta
g_{Z,L}^{ee}+3.02654\delta g_{Z,R}^{ee}+3.01006\,\hat{c}_{{\rm cosmo}},$ (31)

with

$\displaystyle\hat{c}_{{\rm cosmo}}\equiv$
$\displaystyle\,[c_{ll}]_{1122}+[c_{ll}]_{1221}+[c_{ll}]_{1133}+[c_{ll}]_{1331}+[c_{ll}]_{2332}+[c_{ll}]_{2222}+[c_{ll}]_{2233}+[c_{ll}]_{3333}$
$\displaystyle\,+\delta g_{W}^{\mu\nu}+\delta g_{W}^{\tau\nu}+\delta
g_{Z,L}^{\mu\mu}+\delta g_{Z,L}^{\tau\tau}.$ (32)

Several points merit addressing:

  * •

From eq. (31), it is clear that the SM prediction $N_{\rm eff}=3.044$
Akita:2020szl ; Froustey:2020mcq  is reproduced when all SMEFT corrections are
absent.

  * •

As commented above, we do not make any flavor assumption when calculating
$N_{\rm eff}$ within the SMEFT. To solve the Boltzmann equations for
$T_{\gamma}$ and $T_{\nu_{\alpha}}$, we assume that all three flavor neutrinos
have the same initial conditions with
$T_{\gamma}^{0}=T_{\nu_{\alpha}}^{0}=10\rm\,MeV$, $\mu_{\gamma}=0=\mu_{e}$
since photons and electrons/positions are still tightly coupled, and
$\mu_{\nu_{\alpha}}=-10^{-3}\rm\,MeV$ whose impact on $N_{\rm eff}$ turns out
to be negligible.

  * •

With the input from cosmological observables, we find $N_{\rm eff}$ is able to
probe more operators in the tau sector, the $[c_{ll}]_{2233}$ and the
$[c_{ll}]_{3333}$ operators to be specific. In principle, $[c_{ll}]_{2233}$
can be measured at a future muon collider Ankenbrandt:1999cta ;
Forslund:2022xjq ; deBlas:2022aow , and $[c_{ll}]_{3333}$ can be measured
through $Z\to 4\tau$ or $h\to 4\tau$ CMS:2022kdx  even though the current
statistics is extremely limited.

  * •

We find $N_{\rm eff}$ can only probe the linear combination as shown in eq.
(31) and identify a large degeneracy in $N_{\rm eff}$ as defined in eq. (32).
This degeneracy is due to our assumption that muon neutrinos and tau neutrinos
have identical thermodynamics after muon depletion.444Assuming a slightly
different initial condition will only change the corresponding coefficients in
eq. (32) or the direction $N_{\rm eff}$ is exploring in the full parameter
space. On the other hand, one shall note that the electroweak vertices can be
independently constrained by electroweak precision observables very
stringently, and the 4-fermion operators only involving the first generation
leptons, as well as $[c_{ll}]_{2332}$ in eq. (31), can all be well constrained
thanks to Bhabha scattering and $\mu^{+}\mu^{-}/\tau^{+}\tau^{-}$ pair
production at LEP ALEPH:2013dgf , polarized Møller scattering at SLAC-E158
SLACE158:2005uay , and $G_{F}$ measurement from muon and tau decay
ParticleDataGroup:2014cgo . As a result, the flat direction in eq. (32) is
effectively reduced as

$\displaystyle\hat{c}_{{\rm cosmo,eff}}=$
$\displaystyle\,[c_{ll}]_{2222}+[c_{ll}]_{2233}+[c_{ll}]_{3333}.$ (33)

Figure 1: $N_{\rm eff}$ in eliminating the flat direction in the pure muonic
4-fermion sector with $\delta\chi^{2}\equiv\chi^{2}_{\rm SMEFT}-\chi^{2}_{\rm
SM}\leq 5.99$. The region between the blue dashed lines represents the
constraint from current trident data, that between the purple lines is that
from $N_{\rm eff}$ alone, and the magenta ellipse corresponds to the
constraint from their combination. The red dot indicates the SM prediction,
and the purple star represents the best-fit point.

  * •

The remaining 4-fermion operators in eq. (33) cannot be simultaneously
constrained currently. We recall that the neutrino trident process measured at
CHARM-II CHARM-II:1990dvf , CCFR CCFR:1991lpl , and NuTeV NuTeV:1998khj  can
only constrain the following well-known direction defined as Falkowski:2017pss

$\displaystyle{\left[\hat{c}_{ll}\right]_{2222}}$
$\displaystyle=\left[c_{ll}\right]_{2222}+\frac{2g_{Y}^{2}}{g_{L}^{2}+3g_{Y}^{2}}\left[c_{le}\right]_{2222}.$
(34)

Clearly, the degeneracy on the
$\left[c_{ll}\right]_{2222}$-$\left[c_{le}\right]_{2222}$ plane can be lifted
through a combined analysis of $N_{\rm eff}$ and the trident process since
$N_{\rm eff}$ only depends on $\left[c_{ll}\right]_{2222}$.555$N_{\rm eff}$ is
free of $\left[c_{le}\right]_{2222}$ for the reason that muons do not survive
during neutrino decoupling. Therefore, with $N_{\rm eff}$, we would be able to
provide a simultaneous fit to $\left[c_{ll}\right]_{2222}$ and
$\left[c_{le}\right]_{2222}$ as seen in figure 1.666There is a third 4-muon
operator $[\mathcal{O}_{ee}]_{2222}$, which cannot be probed by $N_{\rm eff}$
but can be constrained by $Z\to 4\mu$ decay, as well as a future muon collider
deBlas:2022ofj . This is currently under investigation, but beyond the scope
of this work. When obtaining the curves in figure 1, we set the other
parameters to zero. Marginalizing over the other parameters instead, we get
similar results as those in figure 1. As indicated by the blue lines, the
trident process can only probe the combination in eq. (34), and $N_{\rm eff}$
is only sensitive to the horizontal direction indicated by purple. Their
combination, represented by the magenta ellipse, can fit these two operators
simultaneously. The best fit point, indicated by the purple star, is
consistent with SM expectation at (0,0), as depicted by the red dot.

  * •

There exist very strong correlations among $[c_{ll}]_{2222}$,
$[c_{ll}]_{2233}$, and $[c_{ll}]_{3333}$, which can be clearly seen in eq.
(32). For this reason, we identify two linear combinations as follows to close
the fit and choose to report the global fit results in terms of these
combinations in the following:

$\displaystyle{\left[\hat{c}_{ll}\right]_{2222}}$
$\displaystyle=\left[c_{ll}\right]_{2222}+\frac{2g_{Y}^{2}}{g_{L}^{2}+3g_{Y}^{2}}\left[c_{le}\right]_{2222},$
(35) $\displaystyle{\left[\hat{c}_{ll}\right]_{2233}}$
$\displaystyle=\left[c_{ll}\right]_{2222}+\left[c_{ll}\right]_{2233}+\left[c_{ll}\right]_{3333}.$
(36)

###  2.3 Experimental input parameters

For the fit without $N_{\rm eff}$, dubbed “current fit” in the following, we
use the same input as summarized in deBlas:2022ofj  except that we do not
include any data from future experiments. This includes fermion-pair
production except the top quark, electroweak precision observables, and low-
energy pure leptonic and semi-leptonic processes. When including $N_{\rm
eff}$, dubbed “+CMB-S4” for example when including the projection of $N_{\rm
eff}$ from CMB-S4, we use the following data/projections from Planck
Aghanim:2018eyx , CMB-S4 Abazajian:2016yjj , and CMB-HD Sehgal:2019ewc ; CMB-
HD:2022bsz , respectively,

$\displaystyle N_{\rm eff}^{\rm Planck}=2.99\pm 0.34,\quad N_{\rm eff}^{\rm
CMB-S4}=2.99\pm 0.027,\quad N_{\rm eff}^{\rm CMB-HD}=2.99\pm 0.014,$ (37)

assuming the central value of $N_{\rm eff}$ stays the same as the current one
from Planck.777The central value of $N_{\rm eff}$ from CMB-S4/HD is unknown at
this stage as the corresponding experiment is not performed yet. A different
central value will only affect the best-fit points, which will not affect our
conclusion in this work.

##  3 Results

A global fit is then performed for the 62 parameters simultaneously,88861 of
them are the same as those in deBlas:2022ofj , plus one extra direction
defined in eq. (33) due to the sensitivity of $N_{\rm eff}$ to operators in
the $\tau$ sector. and the results are presented below in the flavor universal
case in section 3.1 and the flavor general case in section 3.2.

###  3.1 Flavor universal case

In this section, we present the global fit results with the inclusion of
$N_{\rm eff}$ from precision cosmology in the flavor universal case, and the
results are shown in eq. (38). The left column corresponds to the current fit,
and the right column corresponds to the global fit with $N_{\rm eff}$ from
CMB-S4. Impact on the global fit from the current value of $N_{\rm eff}$ from
Planck is marginal compared with the current fit, we thus do not include them
here. Similarly, the global fit results using $N_{\rm eff}$ from CMB-HD are
very close to those in the CMB-S4 case and are therefore not shown explicitly.

$\displaystyle\begin{pmatrix}\\\ \,\delta g_{Wl}\\\ \,\delta g_{Z,L}^{l}\\\
\,\delta g_{Z,R}^{l}\\\ \,\delta g_{Z,L}^{u}\\\ \,\delta g_{Z,R}^{u}\\\
\,\delta g_{Z,L}^{d}\\\ \,\delta g_{Z,R}^{d}\\\ \,c_{ll}\\\ \,c_{le}\\\
\,c_{ee}\\\ \,c_{lq}\\\ \,c_{lq}^{(3)}\\\ \,c_{eu}\\\ \,c_{ed}\\\ \,c_{eq}\\\
\,c_{lu}\\\ \,c_{ld}\\\ \,c_{lequ}\\\ \,c_{lequ}^{(3)}\\\ \,c_{ledq}\\\
\,c_{ledq}^{(3)}\\\ \end{pmatrix}=\begin{pmatrix}{\rm current\,fit}\\\
-0.024(0.036)\\\ -0.023(0.019)\\\ -0.0091(0.021)\\\ -0.14(0.17)\\\
-0.16(0.34)\\\ 0.19(0.14)\\\ 1.5(0.73)\\\ -0.29(0.43)\\\ -0.17(0.13)\\\
0.071(0.13)\\\ 0.15(0.31)\\\ -1.9(0.89)\\\ -0.67(0.52)\\\ 0.26(1.1)\\\
3.8(1.9)\\\ -0.57(2.2)\\\ 1.4(2.5)\\\ 2.7(4.0)\\\ -0.078(0.075)\\\
-0.024(0.19)\\\ -0.078(0.075)\\\ -0.25(7.1)\\\ \end{pmatrix}\times
10^{-2}\to\begin{pmatrix}\text{+CMB-S4}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-0.0032(0.026)}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-0.026(0.018)}\\\
-0.0093(0.021)\\\ -0.14(0.17)\\\ -0.16(0.34)\\\ 0.20(0.14)\\\ 1.5(0.73)\\\
-0.31(0.43)\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-0.079(0.083)}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0.11(0.12)}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-0.0077(0.25)}\\\
-1.9(0.89)\\\ -0.68(0.52)\\\ 0.26(1.1)\\\ 3.9(1.9)\\\ -0.59(2.2)\\\
1.38(2.5)\\\ 2.7(4.0)\\\ -0.078(0.075)\\\ -0.024(0.19)\\\ -0.078(0.075)\\\
-0.081(7.1)\\\ \end{pmatrix}\times 10^{-2},$ (38)

As commented in section 2.2.3, with $N_{\rm eff}$, the Planck data can be used
to constrain ${\left[\hat{c}_{ll}\right]_{2233}}$. However, in the flavor
universe case, the dominant constraints on $c_{ll,le,ee}$ come from the
differential cross section of Bhabha scattering measured at LEP ALEPH:2005ab ,
which will be surpassed by the MOLLER experiment MOLLER:2014iki  that is
planned to start data collection in 2026 mollerexp1 ; mollerexp2 , or future
lepton colliders such as CLIC CLICPhysicsWorkingGroup:2004qvu ; Robson:2018zje
, ILC ILC:2013jhg ; ILCInternationalDevelopmentTeam:2022izu , CEPC
CEPCStudyGroup:2018ghi ; CEPCPhysicsStudyGroup:2022uwl , FCC-ee FCC:2018evy ;
Bernardi:2022hny , or a muon collider Ankenbrandt:1999cta ; Forslund:2022xjq ;
deBlas:2022aow .

From eq. (38), it is clear that inclusion of the CMB-S4/HD data on $N_{\rm
eff}$ can improve the fit for $\delta g_{Wl}$, $c_{le}$, and $c_{lq}$ as shown
in red, whose correlation matrix is accompanied by eq. (45). Note the
reduction in this correlation matrix when including $N_{\rm eff}$ from CMB-S4.
The improvement on $\delta g_{Z,L}^{l}$ and $c_{ee}$, as indicated in blue, is
relatively less significant, and in general, we find the data consistent with
the SM within $\sim 2\sigma$.

$\displaystyle\rho_{\rm current}=\left(\begin{array}[]{ccc}1&0.18&-0.023\\\
&1&-0.003\\\ &0&1\\\ \end{array}\right),\quad\rho_{\rm
CMB-S4}=\left(\begin{array}[]{ccc}1&-0.11&-0.023\\\ &1&0.\\\ &0&1\\\
\end{array}\right).$ (45)

In the meantime, we find the electroweak vertices are constrained at or even
below the per-mille level, thanks to the $Z/W$ pole runs of LEP. The helicity-
conserving operators, i.e., $c_{ll}\to c_{ld}$, are generically constrained at
the percent level, limited by the energy and luminosity of LEP. In contrast,
the helicity-violating ones, i.e., $c_{lequ}\to c_{ledq}^{(3)}$, are also
constrained at or below the per-mille level, due to (semi-)leptonic kaon decay
from flavor studies Gonzalez-Alonso:2016etj .

###  3.2 Flavor general case

Global fit results in the most general flavor scenario are summarized in this
section in eqs. (46-48). Again, the left column summarizes the results from
the current fit, and the right column those by including $N_{\rm eff}$ from
CMB-S4. Results from the inclusion of $N_{\rm eff}$ from the Planck are very
close to the current fit results, we thus do not include them here. The only
exception is ${\left[\hat{c}_{\ell\ell}\right]_{2233}}$, which is
unconstrained from the current fit, and is constrained to be
$[\hat{c}_{ll}]_{2233}=({-2.5}\pm{17})\times 10^{-2}$ using $N_{\rm eff}$ from
Planck. In the near future, with a more precise determination of $N_{\rm eff}$
from CMB-S4, we can further improve the fit for
$[\hat{c}_{ll}]_{2233}\to({-0.69}\pm{13})\times 10^{-2}$, as indicated by the
last line of eq. (47). Finally, as in the flavor universe case, we find the
impact of the projected $N_{\rm eff}$ from CMB-HD on the CMB-S4 fit is
marginal, we thus do not include the numbers in the following, which can also
be clearly seen from figure 2.

In figure 2, we show the 95% CL region by marginalizing over the other 59
parameters in eqs. (46-48). The vertical blue band corresponds to the
constraint using the current trident data from CCFR CCFR:1991lpl , and the
horizontal purple band that from $N_{\rm eff}^{\rm Planck}$ alone.
Individually, each experiment cannot constrain these two directions
simultaneously, while their combination, as shown in orange, can
simultaneously constrain these two direction for the first time. These
constraints can be further improved with the inclusion of the CMB-S4/HD data,
as depicted in green.

$\displaystyle\begin{pmatrix}\\\ \,\delta g_{W}^{e\nu}\\\ \,\delta
g_{W}^{\mu\nu}\\\ \,\delta g_{W}^{\tau\nu}\\\ \,\delta g_{Z,L}^{ee}\\\
\,\delta g_{Z,L}^{\mu\mu}\\\ \,\delta g_{Z,L}^{\tau\tau}\\\ \,\delta
g_{Z,R}^{ee}\\\ \,\delta g_{Z,R}^{\mu\mu}\\\ \,\delta g_{Z,R}^{\tau\tau}\\\
\,\delta g_{Z,L}^{uu}\\\ \,\delta g_{Z,L}^{cc}\\\ \,\delta g_{Z,L}^{tt}\\\
\,\delta g_{Z,R}^{uu}\\\ \,\delta g_{Z,R}^{cc}\\\ \,\delta g_{Z,L}^{dd}\\\
\,\delta g_{Z,L}^{ss}\\\ \,\delta g_{Z,L}^{bb}\\\ \,\delta g_{Z,R}^{dd}\\\
\,\delta g_{Z,R}^{ss}\\\ \,\delta g_{Z,R}^{bb}\\\ \,\delta g^{Wq_{1}}_{R}\\\
\end{pmatrix}=\begin{pmatrix}{\rm current\,fit}\\\ -1.0(0.64)\\\ -1.4(0.59)\\\
2.0(0.79)\\\ -0.023(0.027)\\\ 0.013(0.12)\\\ 0.019(0.059)\\\ -0.031(0.027)\\\
0.0054(0.14)\\\ 0.042(0.062)\\\ -2.1(2.3)\\\ -0.15(0.36)\\\ 0.13(4.1)\\\
-1.1(2.8)\\\ -0.35(0.53)\\\ -2.9(3.4)\\\ 1.6(2.5)\\\ 0.33(0.17)\\\ -3.0(12)\\\
3.3(4.9)\\\ 2.3(0.87)\\\ -1.3(1.7)\\\ \end{pmatrix}\times
10^{-2}\to\begin{pmatrix}\text{+CMB-S4}\\\ -1.0(0.64)\\\ -1.4(0.59)\\\
2.0(0.79)\\\ -0.023(0.027)\\\ 0.013(0.12)\\\ 0.019(0.059)\\\ -0.031(0.027)\\\
0.0054(0.14)\\\ 0.042(0.062)\\\ -2.1(2.3)\\\ -0.15(0.36)\\\ 0.13(4.1)\\\
-1.1(2.8)\\\ -0.35(0.53)\\\ -2.9(3.4)\\\ 1.6(2.5)\\\ 0.33(0.17)\\\ -3.0(12)\\\
3.3(4.9)\\\ 2.3(0.87)\\\ -1.3(1.7)\\\ \end{pmatrix}\times 10^{-2},$ (46)

$\displaystyle\begin{pmatrix}\\\ \,[c_{ll}]_{1111}\\\ \,[c_{le}]_{1111}\\\
\,[c_{ee}]_{1111}\\\ \,[c_{ll}]_{1221}\\\ \,[c_{ll}]_{1122}\\\
\,[c_{le}]_{1122}\\\ \,[c_{le}]_{2211}\\\ \,[c_{ee}]_{1122}\\\
\,[c_{ll}]_{1331}\\\ \,[c_{ll}]_{1133}\\\ \,[c_{le}]_{1133}\\\
\,[c_{le}]_{3311}\\\ \,[c_{ee}]_{1133}\\\ \,[\hat{c}_{ll}]_{2222}\\\
\,[c_{ll}]_{2332}\\\ \,[\hat{c}_{ll}]_{2233}\\\
\end{pmatrix}=\begin{pmatrix}{\rm current\,fit}\\\ 0.91(0.40)\\\
-0.22(0.22)\\\ 0.17(0.40)\\\ -4.8(1.6)\\\ 0.59(1.8)\\\ 1.6(1.9)\\\
-1.4(1.8)\\\ 4.3(2.4)\\\ 1.5(1.3)\\\ -0.097(16)\\\ -2.3(10)\\\ 1.7(10)\\\
-1.3(17)\\\ 10(22)\\\ 3.0(2.3)\\\ \text{---}(\text{---})\\\
\end{pmatrix}\times 10^{-2}\to\begin{pmatrix}\text{+CMB-S4}\\\ 0.91(0.40)\\\
-0.22(0.22)\\\ 0.17(0.40)\\\ -4.8(1.6)\\\ 0.59(1.8)\\\ 1.59(1.9)\\\
-1.5(1.8)\\\ 4.3(2.4)\\\ 1.5(1.3)\\\ -0.097(16)\\\ -2.3(10)\\\ 1.7(10)\\\
-1.3(17)\\\ 10(23)\\\ 3.0(2.3)\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-0.69(13)}\\\
\end{pmatrix}\times 10^{-2},$ (47)

$\displaystyle\begin{pmatrix}\\\ \,[c_{lq}^{(3)}]_{1111}\\\
\,[\hat{c}_{eq}]_{1111}\\\ \,[\hat{c}_{lu}]_{1111}\\\
\,[\hat{c}_{ld}]_{1111}\\\ \,[\hat{c}_{eu}]_{1111}\\\
\,[\hat{c}_{ed}]_{1111}\\\ \,[\hat{c}_{lq}^{(3)}]_{1122}\\\
\,[c_{lu}]_{1122}\\\ \,[\hat{c}_{ld}]_{1122}\\\ \,[c_{eq}]_{1122}\\\
\,[c_{eu}]_{1122}\\\ \,[\hat{c}_{ed}]_{1122}\\\
\,[\hat{c}_{lq}^{(3)}]_{1133}\\\ \,[c_{ld}]_{1133}\\\ \,[c_{eq}]_{1133}\\\
\,[c_{ed}]_{1133}\\\ \,[c_{lq}^{(3)}]_{2211}\\\ \,[c_{lq}]_{2211}\\\
\,[c_{lu}]_{2211}\\\ \,[c_{ld}]_{2211}\\\ \,[\hat{c}_{eq}]_{2211}\\\
\,[c_{lequ}]_{1111}\\\ \,[c_{ledq}]_{1111}\\\ \,[c_{lequ}^{(3)}]_{1111}\\\
\,\epsilon_{P}^{d\mu}[{\rm 2\,GeV}]\\\
\end{pmatrix}=\begin{pmatrix}\text{current}\\\ -1.5(2.9)\\\ 79(180)\\\
0.51(8.8)\\\ 4.3(18)\\\ -4.2(9.4)\\\ -9.4(17)\\\ -50(31)\\\ 1.9(7.6)\\\
-260(128)\\\ -18(25)\\\ -71(45)\\\ 221(134)\\\ -3.7(7.2)\\\ 1.3(10)\\\
-1.9(5.0)\\\ 6.2(18)\\\ -0.45(3.6)\\\ -2.0(5.9)\\\ 9.8(8.1)\\\ 12(26)\\\
0.76(41)\\\ -0.082(0.076)\\\ -0.081(0.076)\\\ -0.024(0.19)\\\ 0.021(0.14)\\\
\end{pmatrix}\times 10^{-2}\to\begin{pmatrix}\text{+CMB-S4}\\\ -1.5(2.9)\\\
79(180)\\\ 0.51(8.8)\\\ 4.3(18)\\\ -4.2(9.4)\\\ -9.4(17)\\\ -50(31)\\\
1.9(7.6)\\\ -260(128)\\\ -18(25)\\\ -71(45)\\\ 221(134)\\\ -3.7(7.2)\\\
1.3(10)\\\ -1.9(5.0)\\\ 6.2(18)\\\ -0.45(3.6)\\\ -2.0(5.9)\\\ 9.8(8.1)\\\
12(26)\\\ 0.76(40)\\\ -0.082(0.076)\\\ -0.081(0.076)\\\ -0.024(0.19)\\\
0.021(0.14)\\\ \end{pmatrix}\times 10^{-2},$ (48)

Figure 2: The plot for the extra direction that can be probed by $N_{\rm eff}$
found in this work. Blue: Direction probed by the trident data from CCFR
CCFR:1991lpl ; Purple: Direction probed by $N_{\rm eff}^{\rm Planck}$
Aghanim:2018eyx ; Orange: Combined fit using trident + $N_{\rm eff}^{\rm
Planck}$ data; Dark green: Combined fit using trident + $N_{\rm eff}^{\rm
Planck}$ \+ $N_{\rm eff}^{\rm CMB-S4/HD}$ data Abazajian:2016yjj ;
Sehgal:2019ewc ; CMB-HD:2022bsz . All shaded regions are at 95% CL,
corresponding to $\delta\chi^{2}\equiv\chi^{2}_{\rm SMEFT}-\chi^{2}_{\rm
SM}\leq 5.99$. The cyan star represents the SM prediction at (0,0), the white
cross corresponds to the best-fit point in the orange case with the 68% CL
error bars shown explicitly, and the magenta cross that in the dark green case
(see text for details).

##  4 $Y_{P}$ and the neutron lifetime anomaly

With the results presented in last section, one can then compute $N_{\rm eff}$
and its uncertainty from the global fit. The uncertainty of $N_{\rm eff}$ also
affects the precision of $Y_{P}$, the primordial abundance of helium, through
error propagation. In addition, the uncertainty of $Y_{P}$ also depends on
that in the free neutron lifetime. Therefore, $N_{\rm eff}$ from the global
fit may provide, from a different angle, information on the free neutron
lifetime anomaly. This section is devoted to this point from the global fit.

###  4.1 Brief overview of the free neutron lifetime anomaly

Free neutrons have a finite lifetime, they can decay weakly through $n\to
p+e^{-}+\bar{\nu}_{e}$, with a lifetime $\tau_{n}$ of about 15 minutes
ParticleDataGroup:2022pth . Due to the flavor- and mass-eigenstate
misalignment, the decay amplitude depends on the $V_{ud}$ element of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv .
Therefore, a precision measurement of $\tau_{n}$ can play a precision role in
terms of SM test through checking the unitarity of the CKM matrix. We refer
the readers to the seminal paper by Sirlin Sirlin:1977sv  in radiative
corrections to $\tau_{n}$ and Marciano:2005ec ; Seng:2018yzq  for the recent
progress.

The leading $\tau_{n}$ measurements come from two different and independent
groups using the beam and the bottle methods, respectively. In the former
case, it is the decay product $p$ of free neutrons that is measured, and in
the latter case, it is the number of surviving free neutrons that is counted.
Currently, the most precise result in the beam case is obtained by Sussex-ILL-
NIST, with

$\displaystyle\tau_{n}=(887.7\pm 1.2\,{\rm[stat.]}\pm
1.9{\rm[syst.]})\,s\quad\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Yue:2013qrc}{\@@citephrase{(}}{\@@citephrase{)}}}}.$ (49)

Note that the current uncertainty in the beam case is still above 1 second. In
the bottle case, this uncertainty has recently been reduced below 1 second in
Pattie:2017vsj ; UCNt:2021pcg  using ultra cold neutrons to reduce wall loss,
and the latest result from UCN$\tau$ is

$\displaystyle\tau_{n}=(877.75\pm
0.28\,{\rm[stat.]}^{+0.22}_{-0.16}\,{\rm[syst.]})\,s\quad\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{UCNt:2021pcg}{\@@citephrase{(}}{\@@citephrase{)}}}}.$ (50)

These two most precise results differ by about 10 seconds in their central
values, which is beyond $3\sigma$ and known as the neutron lifetime anomaly.
For a recent review on this topic, see Wietfeldt:2011suo . This anomaly could
possibly be accounted for by exotic neutron decay(s), though the corresponding
decay branching ratio is highly constrained Fornal:2018eol ; Czarnecki:2018okw
; Dubbers:2018kgh .

As mentioned at the beginning of this section, the uncertainty in $\tau_{n}$
enters that in $Y_{P}$ in the Standard Big-Bang Nucleosynthesis (SBBN)
calculation. The same is true in $N_{\rm eff}$, whose absolute theoretical
uncertainty is currently at the $10^{-4}$ level as discussed earlier. In this
work, we ask ourselves the following question: Given the global fit results
presented in last section, what is their impact on $Y_{P}$ and what light it
can shed on the neutron lifetime anomaly? We will try to answer this question
in the next subsection.

###  4.2 From SMEFT global fit to the primordial helium abundance

Figure 3: The plot for the predicted primordial Helium abundance $Y_{P}$ as a
function of $N_{\rm eff}$. The standard model primordial value
$Y_{P}=0.24709\pm 0.00017$ Pitrou:2018cgg  at 95% CL is indicated by the green
cross. The predicted 95% CL value of $Y_{P}$ from the globally fitted $N_{\rm
eff}$ is given by the diagonal band, with the dashed line representing its
central value. The gray band in the lower left corner is disfavored by BBN.
The other crosses are experimental measurements of $Y_{P}$ from different
groups as indicated by the legends (see the text for more details).

With the $1\sigma$ bounds for the electroweak vertices and the 4-fermion
operators in eqs. (46-48), we randomly varying the dependent Wilson
coefficients in eq. (31) inside the 17-dimensional $1\sigma$ sphere with
$10^{9}$ points to fit $N_{\rm eff}$ and obtain

$\displaystyle N_{\rm eff}=2.990\pm 0.452,$ (51)

which is consistent with the current measurement from Planck Aghanim:2018eyx
and the SM prediction Akita:2020szl ; Froustey:2020mcq  within $1\sigma$. The
fitted $N_{\rm eff}$ in eq. (51) is then used to predict the primordial
abundance of helium $Y_{P}$ using AlterBBN Arbey:2018zfh  with $N_{\rm eff}$
being the only free parameter and fixing the neutron lifetime $\tau_{n}$ and
its uncertainty $\Delta\tau_{n}$ from UCNt:2021pcg . The result is shown in
figure 3 by the diagonal band,999If instead the beam values of $\tau_{n}$ and
$\Delta\tau_{n}$ are used Yue:2013qrc , the width of this band will increase
and the other parts of this plot will remain untouched. whose half-width
corresponds to the uncertainty in $Y_{P}$. The gray part in the lower-left
corner is disfavored by BBN, ignoring the correlation of ${}^{4}{\rm He}$ with
the other light elements. The predicted $Y_{P}=0.24709\pm 0.00017$
Pitrou:2018cgg  from the SBBN is indicated by the green cross, whose absolute
error is given by the vertical bar and is invisible in the plot.
Experimentally, $Y_{P}$ can be measured from metal-poor galaxies by observing
the ${}^{4}{\rm He}$ and hydrogen emission lines as a function of the galaxy
metallicity and extrapolating back to the zero metallicity point for the
estimation. In Izotov:2014fga ; Aver:2015iza ; Peimbert:2016bdg ;
Fernandez:2018xyz ; Hsyu:2020uqb , they obtained, respectively,

$\displaystyle Y_{P}=0.2551\pm 0.0022,$ $\displaystyle\,\quad Y_{P}=0.2449\pm
0.0040,\quad Y_{P}=0.2446\pm 0.0029,$ $\displaystyle\,Y_{P}=0.245\pm
0.007,\quad Y_{P}=0.2436_{-0.0040}^{+0.0039}.$ (52)

In addition, from the intergalactic medium, Cooke:2018qzw  recently found

$\displaystyle Y_{P}=0.250_{-0.025}^{+0.033}.$ (53)

Note that the error in the latter case is much larger than that from metal-
poor galaxies.

All the above results are shown in order in figure 3 as indicated by the
legend. Recall that the uncertainty in $Y_{P}$ is directly related to that in
neutron lifetime Pitrou:2018cgg , we show in figure 3 the uncertainty band of
$Y_{P}$ in hatched orange to show explicitly the region insufficient to
resolve the $\sim 10\,s$ neutron lifetime anomaly. We stress that, to get this
hatched region, we assume the central value of $Y_{P}$ is aligned with the
SBBN prediction in Pitrou:2018cgg , and use the central value of $\tau_{n}$
from Pattie:2017vsj ; UCNt:2021pcg  as its uncertainty is small below 1
s.101010We do not use the world average in ParticleDataGroup:2022pth  here as
we want to account for the difference in $\tau_{n}$ from different
experiments. In addition, Aver:2015iza ; Fernandez:2018xyz ; Cooke:2018qzw
did not report any results on $N_{\rm eff}$, we thus use its SM value when
plotting their data in figure 3. A spurious shift of 0.003 to $N_{\rm eff}$ is
then applied to separate these data, in black, magenta, and cyan respectively,
for a better visibility. Clearly, the SBBN result precludes its possibility in
accounting for the difference between the beam and the bottle results, as is
well-known. On the other hand, an improvement in extracting $Y_{P}$ from
astrophysical observation, reducing the uncertainty in $Y_{P}$ to be within
the orange band to be more quantitative, would deepen our understanding on the
connection between SBBN and the neutron lifetime anomaly.

##  5 Conclusions

In light of the percent level precision target of $N_{\rm eff}$ from future
cosmological probes like CMB-S4, we think it is now proper to combine the
cosmological data together with the collider and the low-energy data sets
available to perform a global fit of the SMEFT in searching for new physics
indirectly. In this work, by utilizing the $N_{\rm eff}$ data from Planck, as
well as its projections for CMB-S4 and CMB-HD, we perform a global fit of the
SMEFT without any flavor assumption in the electroweak vertex and
(semi-)leptonic 4-fermion sectors. From this work, we find that:

  1. 1.

While there are in total three 4$\mu$ operators in the SMEFT, viz,
$[\mathcal{O}_{\ell\ell,\ell e,ee}]_{2222}$, $N_{\rm eff}$ only depends on
$[\mathcal{O}_{\ell\ell}]_{2222}$ due to muon depletion at neutrino
decoupling. Ignoring the $\tau$-related operators in eq. (33), it would allow
us to break the well-known flat direction in the neutrino trident process as
shown in figure 1 when combining these two data sets. This, however, still
leaves $[\mathcal{O}_{ee}]_{2222}$ unconstrained as neither observable has any
sensitivity to it. This is expected to be resolved with the inclusion of $Z\to
4\mu$.

  2. 2.

The inclusion of $N_{\rm eff}$ also enables us to probe two extra operators
$[\mathcal{O}_{\ell\ell}]_{2233,3333}$ in the $\tau$ sector, whose detection
at the LHC and/or the HL-LHC will be rather challenging due to limited
statistics. In this regard, a future muon collider is expected to have a good
precision on $[c_{\ell\ell}]_{2233}$, and a $Z$ factory like CEPC and FCC-ee
could help constrain the $4\tau$ operator $[c_{\ell\ell}]_{3333}$ through
$h/Z\to 4\tau$.

  3. 3.

A global fit using the current data plus $N_{\rm eff}$ from Planck enables us
to constrain the direction defined in eq. (33) at the 10% level as seen in
figure 2 and the last line of eq. (47), which can be further improved in the
future using the input from CMB-S4/HD. The impact from $N_{\rm eff}$ on the
other 61 parameters is found to be marginal, as shown in eqs. (46-48). In
contrast, the fit for $\delta g_{Wl}$, $c_{le}$, and $c_{lq}$, as well as
$\delta g_{Z,L}^{l}$ and $c_{ee}$, can be improved if one ignores the flavor
difference. This is shown in red and blue, respectively, in eq. (38).

  4. 4.

With the global fit results, we find that improving the experimental precision
on $Y_{P}$ from metal-poor galaxies will play an important role in our
understanding on the neutron lifetime anomaly within the SBBN, as shown in
figure 3.

In the flavor universal case, we commented that the precision in
$[\mathcal{O}_{\ell\ell,\ell e,ee}]$ was currently limited by Bhabha
scattering at LEP. Looking ahead into the near future, the MOLLER experiment
at the Jefferson Laboratory, planned to start data collection in early 2026,
will measure the weak mixing angle at a low momentum transfer at the per-mille
level. We expect the global fit in this sector to be improved by at least a
factor of a few in both the flavor universal and the flavor general cases.
Furthermore, in the current work, we only consider $Y_{P}$ after performing
the global fit and leaving out the other light elements from the consideration
of their current uncertainties. It would be desirable to include the full
relevant SMEFT operators during BBN for studying $N_{\rm eff}$, the primordial
abundances of light elements, and its interplay with the neutron lifetime.
This is left for a future work.

###### Acknowledgements.

The author would like to thank Jorge de Blas, Christophe Grojean, Jiayin Gu,
Victor Miralles, Michael E. Peskin, Junping Tian, Marcel Vos and Eleni
Vryonidou for working on SMEFT global fit for Snowmass 2021 deBlas:2022ofj .
This project was funded by China Postdoctoral Science Foundation under grant
number 2023M732255. YD also acknowledges support from the T.D. Lee
Postdoctoral Fellowship at the Tsung-Dao Lee Institute, Shanghai Jiao Tong
University.

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