Preprint of ]Physica C 613, 1354321 (2023), Open Access,
https://doi.org/10.1016/j.physc.2023.1354321

# Can $dd$ excitations mediate pairing ?

Francesco Barantani  Department of Quantum Matter Physics, University of
Geneva, 1211 Geneva, Switzerland  Institute of Physics, École Polytechnique
Fédérale de Lausanne, Lausanne, 1015, Switzerland  Christophe Berthod
Department of Quantum Matter Physics, University of Geneva, 1211 Geneva,
Switzerland  Dirk van der Marel<EMAIL_ADDRESS>Department of
Quantum Matter Physics, University of Geneva, 1211 Geneva, Switzerland

###### Abstract

The Cu-$3d$ states in the high-$T_{c}$ cuprates are often described as a
single band of $3d_{x^{2}-y^{2}}$ states, with the other four $3d$ states
having about 2 to 3 eV higher energy due to the lower-than-octahedral crystal
field at the copper sites. However, excitations to these higher energy states
observed with RIXS show indications of strong coupling to doped holes in the
$3d_{x^{2}-y^{2}}$ band. This relaunches a decades-old question of the
possible role of the orbital degrees of freedom that once motivated Bednorz
and Müller to search for superconductivity in these systems. Here we explore a
direction different from the Jahn-Teller electron-phonon coupling considered
by Bednorz and Müller, namely the interaction between holes mediated by $dd$
excitations.

##  I Introduction

In 1986 it was widely believed that superconductivity above the then-known
limit of 23 Kelvin was not possible. Who could have predicted that, initiated
by Georg Bednorz and Karl Alex Müller’s discovery, superconductivity in the
cuprates would go through the liquid nitrogen ceiling. The consequences for
fundamental science and for applications are numerous and will be detailed
throughout the tributes in this special issue.

In their Nobel lecture [1] Bednorz and Müller provided a captivating account
of the path that led them to their discovery. A remarkable aspect was the
consideration of the 3$d$ orbital degrees of freedom: “For Cu3+ with 3$d^{8}$
configuration, the orbitals transforming as base functions of the cubic
$e_{g}$ group are half-filled, thus a singlet ground state is formed. In the
presence of Cu2+ with 3$d^{9}$ configuration the ground state is degenerate,
and a spontaneous distortion of the octahedron occurs to remove this
degeneracy. This is known as the Jahn-Teller effect.” For a Cu ion surrounded
by O2- ligands with an octahedral crystal field, the 3$d$-levels are grouped
in $\ket{\underline{e}_{g}}$ (i.e. $\ket{\underline{d}_{x^{2}-y^{2}}}$,
$\ket{\underline{d}_{3r^{2}-z^{2}}}$) and $\ket{\underline{t}_{2g}}$ (i.e.
$\ket{\underline{d}_{xy}}$, $\ket{\underline{d}_{yz}}$,
$\ket{\underline{d}_{zx}}$) levels at about 2 eV higher energy. Here the
notation $\ket{\underline{d}_{j}}$ refers to a hole in an otherwise fully
occupied 3$d$-shell. In the cuprate superconductors the crystal field symmetry
at the Cu sites is lower than octahedral. This causes the ground state to
become $\ket{\underline{d}_{x^{2}-y^{2}}}$, with
$\ket{\underline{d}_{3r^{2}-z^{2}}}$ at least 1 eV above the ground state.
This splitting can be directly observed with resonant inelastic x-ray
scattering (RIXS). An example is provided in Fig. 1 [2] for the hole-doped Bi-
based cuprate. Since the Jahn-Teller effect relies on the two states being
degenerate, it is tempting to conclude that this effect does not play a role
here. However, the Jahn-Teller effect is restored — at least in part — if the
$d_{3r^{2}-z^{2}}$ and $d_{x^{2}-y^{2}}$ bands are coupled (citing K. A.
Müller [3]) “via vibronic interactions, i.e. the ions are allowed to move [4].
The latter coupling lowers the upper band to within a few meV of the lower
one.” In the RIXS spectrum this should be revealed by a strong broadening of
the $\ket{\underline{d}_{3r^{2}-z^{2}}}$ state, so that the zero-phonon line
occurs close to zero RIXS excitation energy. This effect was observed by Lee
et al. for the case of Ca2Y2Cu5O10 [5]. However, they located the zero-phonon
line only 0.2 eV below the peak maximum, indicating that the vibrational
coupling is too weak to overcome the splitting between
$\ket{\underline{d}_{3r^{2}-z^{2}}}$ and $\ket{\underline{d}_{x^{2}-y^{2}}}$.

Figure 1: RIXS spectrum of overdoped Bi2Sr2CaCu2O8-x (Bi-2212) ($T_{c}=70$ K).
Labels refer to spin (red), $dd$ (blue) and charge transfer (orange)
excitations. See Ref. [2] for details.

Recently the detailed temperature dependence of the $dd$ excitations in
Bi-2212 has been studied by, among others, the present authors [2] and it was
found that the energies of these modes are influenced by the system becoming
superconducting. Superconductivity-induced shifts of $+30$ meV were observed
on the overdoped side, about $-12$ meV on the underdoped side, and no shift
within the error bar for optimal doping. It was argued that these shifts —
including the sign-change as a function of doping — could be caused by
differences in local spin-correlations of the different phases. These
observations also clearly indicate that there is a non-negligible coupling
between the conducting holes in the cuprates and the $dd$ excitations.

A $dd$ excitation is essentially a local orbital flip from the
$\ket{\underline{d}_{x^{2}-y^{2}}}$ ground state to one of the other
$\ket{\underline{d}_{j}}$ states ($j=xy,yz,zx$ or $3r^{2}-z^{2}$). From this
perspective it is unavoidable that a coupling exists between the holes in the
${d}_{x^{2}-y^{2}}$ band and the $dd$ excitations, and that this coupling is
in fact very strong. We interrupt the flow of logic to point out a commonality
with Bednorz and Müller’s intuition, namely that the 3$d$ orbital degrees of
freedom could be one of the key ingredients. Instead of coupling the $d$
orbital degrees of freedom to vibrations, we will now consider the coupling
between $dd$ excitations and conduction electrons or holes. The question then
is whether such coupling contributes to the pairing interaction, and whether
this contribution is positive or negative.

The possibility that $dd$ excitations in the cuprates could mediate
superconducting pairing has been pioneered by Werner Weber and collaborators
[6, 7] and by William Little and collaborators [8, 9, 10, 11]. Weber
considered a model where conduction is confined to holes in the O-$2p$ band,
where the copper ions are purely $3d^{9}$, and pairing is mediated by $dd$
orbital excitations within the $e_{g}$ manifold of the Cu $3d^{9}$. Little et
al. used the Eliashberg equations to describe the coupling of conduction
electrons or holes to $dd$ excitations. The approach that we explore here
differs from the previous ones. Compared to the approach of Little et al. we
use a more direct description of the coupling to the $3d$ orbital degrees of
freedom; and instead of Weber’s model with separate O-$2p$ and
Cu-$3d_{x^{2}-y^{2}}$ bands, we adopt the model of a single partly filled band
of mixed Cu-$3d_{x^{2}-y^{2}}$–O-$2p$ character.

##  II The model

Figure 2: a Two additional holes present on distant sites (blue spheres) are
renormalized by the interaction with 4 nearest neighbors each (green spheres).
b Two additional holes present on neighboring sites are renormalized by the
interaction with 6 nearest neighbors and by virtual hopping of particles
between the two sites (yellow).

The question that we want to address is whether the coupling to $dd$
excitations could provide an attractive short-range interaction between the
free carriers. We model the problem by considering a 2-dimensional square
lattice of atoms. We then dope two additional holes into this lattice, (i) in
a configuration where the holes are at infinite distance from each other
giving the energy $E_{\infty}$, (ii) where they occupy nearest-neighbor sites
giving the energy $E_{nn}$. The energy difference $E_{nn}-E_{\infty}$ then
constitutes the effective interaction energy of a doped hole pair at a
nearest-neighbor distance.

In the interest of simplicity, we will limit the Hilbert space on each site to
two of the $3d$ orbitals, which is also the minimal model to consider $dd$
excitations. In the hole language, $3d_{x^{2}-y^{2}}$ ($\ket{a}$) is the state
with the lowest energy, and we represent the other four orbitals at 2–3 eV
above the ground state by a single state $\ket{b}$. In this subspace the local
Hamiltonian reads 111A more general expression of this Hamiltonian in the
SU(2) invariant form can be found in Ref. [20].

$H=\sum_{\sigma}\big{(}\varepsilon_{a}a_{\sigma}^{\dagger}a_{\sigma}+\varepsilon_{b}b_{\sigma}^{\dagger}b_{\sigma}\big{)}+U\big{(}a_{\uparrow}^{\dagger}a_{\uparrow}a_{\downarrow}^{\dagger}a_{\downarrow}+b_{\uparrow}^{\dagger}b_{\uparrow}b_{\downarrow}^{\dagger}b_{\downarrow}\big{)}\\\
+(U-2J)\big{(}a_{\uparrow}^{\dagger}a_{\uparrow}b_{\downarrow}^{\dagger}b_{\downarrow}+a_{\downarrow}^{\dagger}a_{\downarrow}b_{\uparrow}^{\dagger}b_{\uparrow}\big{)}\\\
+(U-3J)\big{(}a_{\uparrow}^{\dagger}a_{\uparrow}b_{\uparrow}^{\dagger}b_{\uparrow}+a_{\downarrow}^{\dagger}a_{\downarrow}b_{\downarrow}^{\dagger}b_{\downarrow}\big{)}\\\
-J\big{(}a_{\uparrow}^{\dagger}a_{\downarrow}b_{\downarrow}^{\dagger}b_{\uparrow}+a_{\downarrow}^{\dagger}a_{\uparrow}b_{\uparrow}^{\dagger}b_{\downarrow}+a_{\uparrow}^{\dagger}a_{\downarrow}^{\dagger}b_{\uparrow}b_{\downarrow}+a_{\downarrow}a_{\uparrow}b_{\downarrow}^{\dagger}b_{\uparrow}^{\dagger}\big{)},$
(1)

where $a^{\dagger}$ ($a$) and $b^{\dagger}$ ($b$) are the creation
(annihilation) operators for a hole with energy $\varepsilon_{a}$ and
$\varepsilon_{b}$, respectively, $U=A+4B+3C$, $J=5B/2+2C$ and $A,B,C$ are the
Racah parameters. The $dd$ excitation corresponds to an excitation from
$\ket{a}$ to $\ket{b}$, with excitation energy

$\Omega_{ab}=\varepsilon_{b}-\varepsilon_{a}.$ (2)

The first term of Eq. (1) is the kinetic energy, the second term proportional
to $U$ is the intra-orbital Coulomb interaction, the third and fourth terms
represent inter-orbital Coulomb and Hund’s coupling, and the last term
involves on-site inter-orbital spin-flips and on-site pair-wise orbital flips
[13].

A hole doped into the Mott-insulating state has the possibility of making
virtual excursions to the neighboring sites, which can be either the same
orbital $a$ or the orbital $b$ at a higher energy. When we dope two holes and
keep them at a sufficiently long distance from each other, there is no
interference between their respective virtual excursions, but if they occupy
two neighboring sites this changes the pattern of virtual excitations, as is
schematically illustrated in Fig. 2. In the Appendix, the energies of these
various different configurations are calculated using second-order
perturbation theory and assuming a hopping energy $t_{ab}$ between different
orbitals on neighboring sites. The nearest-neighbor interaction energy
$\Gamma_{nn}$ is defined as the energy gained by two holes when they benefit
from these virtual excitations sitting on two-neighboring sites, rather than
at an infinite distance from each other [see Eq. (28)].

It is convenient to introduce the dimensionless quantities

$u=\frac{U}{\Omega_{ab}};\quad
j=\frac{J}{\Omega_{ab}};\quad\delta_{j}=\frac{\Delta_{J}}{\Omega_{ab}};\quad\gamma_{nn}=\Gamma_{nn}\frac{\Omega_{ab}}{t_{ab}^{2}}.$
(3)

We further note that

$\delta_{j}=\sqrt{1+j^{2}}-1\quad\text{and}\quad\alpha^{2}=\frac{1}{2}+\frac{1}{2\sqrt{1+j^{2}}}.$
(4)

With these definitions, we express the nearest-neighbor interaction, Eq. (28),
as a dimensionless quantity

$\gamma_{nn}=\frac{\alpha^{2}}{1+\delta_{j}-j}+\frac{3\alpha^{2}}{1+\delta_{j}-3j}-\frac{4\alpha^{4}}{1+2\delta_{j}+u-5j}.$
(5)

##  III Discussion

Figure 3: a Exciton mediated hole-hole interaction $\gamma_{nn}$ as function
of the dimensionless $u$ and $j$ using Eq. (5). b Zoom of panel a. The black
dot in the top panel shows a realistic combination of parameters for Bi2212.
The gray and moss green areas indicate where perturbation theory cannot be
applied.

A color map of $\gamma_{nn}$ in the $(u,j)$ plane is shown in Fig. 3. Blue
(red) indicates zones of attractive (repulsive) interaction. The gray color
indicates the region where the ground state of a doped hole becomes unstable
toward a different orbital configuration. This happens for
$j\geqslant(1+\delta_{j})/3$. For the purpose of our discussion we will only
need to consider $j<(1+\delta_{j})/3$. In this parameter range, $\delta_{j}\ll
1$ and $\alpha^{2}\approx 1$.

The experimental values of the $dd$ exciton peak observed by RIXS indicate
that $\Omega_{ab}=2$ eV. Following Ghijsen et al. [14], we find for the
cuprates $U=8.8$ eV and $J=1$ eV, so that $u=4.4$ and $j=0.5$. As pointed out
above, this high value of $j$ is outside the range of validity of Eq. (5).
This should not come as a surprise: it is a manifestation of the second Hund’s
rule that — in the absence of a crystal field — the ground state is a high-
spin state. For $j<1/3$ the crystal field is sufficiently strong to quench the
high-spin state, but for the more realistic estimate $j=0.5$ the ground state
has $S=1$. In the actual cuprate materials this does _not_ happen, because the
energy of introducing a hole in O-$2p$ is lower than that of creating a
$3d^{8}$ state [15]. In other words, if one considers realistic interaction
parameters of the Cu-$3d$ electrons, it is necessary to take into account the
O-$2p$ bands [16, 15].

Meanwhile the question “can $dd$ excitations mediate pairing ?” has a non-
trivial answer: There exists a set of parameters in the $(u,j)$ plane,

$u_{c}=j_{c}\frac{5-9j_{c}}{2-3j_{c}},$ (6)

for which $\gamma_{nn}=0$ 222The expression for $u_{c}(j)$ is obtained by
applying the condition $\gamma_{nn}(u_{c},j)=0$ to Eq. (5). For $u>u_{c}(j)$
the interaction $\gamma_{nn}$ is repulsive; for $5j-1<u<u_{c}(j)$ it is
attractive (always considering $j<1/3$). For $u<5j-1$ (moss green area on the
left of the color map) the state of the two doped holes on nearest-neighbor
sites becomes unstable toward a different orbital configuration. As pointed
out above, this model cannot be directly applied to the cuprates. On the other
hand, as a matter of principle it _is_ of interest that such an attractive
interaction is possible, and it may provide a method of tuning an attractive
interaction in doped multi-band Mott insulators. Let us therefore take a step
back and try to understand what exactly causes the attractive pairing channel.

From Eq. (5) we see that in the parameter range $j<(1+u)/5$ and $u<u_{c}(j)$,
the first and second terms of Eq. (5) are both positive. When $j$ approaches
$(1+u)/5$ from below, the denominator of the third term tends to zero and the
attractive interaction potential diverges. Clearly the attractive interaction
is caused by the third term. In the Appendix, we show that this term
originates in virtual transitions that are unique to having two doped holes on
nearest neighbor sites, i.e.
$\underline{d}(R_{1})^{2}\underline{d}(R_{2})^{2}$. This opens a channel of
virtual transitions to excited states
$\underline{d}(R_{1})^{1}\underline{d}(R_{2})^{3}$. The energy cost of such a
process is $\Omega_{ab}$ plus the effective on-site interaction energy
$U_{\mathrm{eff}}=U-5J$. $U_{\mathrm{eff}}$ is _negative_ in the parameter
range considered, $j<(1+u)/5$. For $j\rightarrow(1+u)/5$ the effective on-site
interaction $U_{\mathrm{eff}}\rightarrow-\Omega_{ab}$. Consequently, the
energies of the two states $\underline{d}(R_{1})^{2}\underline{d}(R_{2})^{2}$
and $\underline{d}(R_{1})^{1}\underline{d}(R_{2})^{3}$ approach the same
value. One can therefore interpret this type of pairing interaction as a
resonant process whereby two doped holes temporarily occupy the same site. It
is not completely obvious if we should regard this as an exciton-mediated
interaction; however, from a broader perspective, this interaction is
effectively mediated by virtual local orbital fluctuations.

We close by making a speculation if this type of behavior could be realized in
a real system, and if so, in which materials. From Fig. 3, we see that in the
region of attractive interaction we have $j\ll 1$, so that $u_{c}(j)\approx
5j/2$. The conditions $u<u_{c}(j)$ and $j<(1+u)/5$ then become

$5J-\Omega_{ab}<U\lesssim 5J/2.$ (7)

It is a well-known fact that the Hund’s coupling $J$ is a relatively robust
atomic parameter that is only weakly screened in a solid-state environment. In
the $3d$ series, $J$ varies from 0.9 eV (Ti) to 1.5 eV (Cu), in the $4d$
series from 0.6 eV (Zr) to 1.0 eV (Ag), and in the $5d$ series from 0.7 eV
(Hf) to 1.0 eV (Au) [18]. The best targets are Nb4+ or W5+, both having a
$d^{1}$ configuration. Taking $J=0.6$ eV and $\Omega_{ab}=2$ eV, $U$ should be
about 1 to 1.5 eV. This is compatible with reported empirical values of the
early $4d$ and $5d$ elements [18]. However, since in this case the role of
electrons and holes is reversed, the electron in a $d^{1}$ configuration
occupies one of the $t_{2g}$ states. For an octahedral coordination the ground
state is then 3-fold degenerate, making it Jahn-Teller active. This degeneracy
can be lifted by the combination of spin-orbit coupling and a tetragonal
crystal field. Electron-doped Sr2NbO4 or K2WO4 and other members of the
Ruddleston-Popper series of these compounds could be candidate materials for
observing the exciton mediated interactions of the type sketched above. To
arrive at a quantitative prediction requires the implementation of spin-orbit
coupling and a tetragonal crystal field.

##  IV Conclusions

We calculated the effective interaction between 2 holes doped in a 2-band
Mott-insulator. We conclude from our study that the coupling to $dd$
excitations (orbital flips) mediates a nearest-neighbor interaction, the sign
of which depends on the relative values of the on-site Coulomb repulsion and
Hund’s-rule exchange. Our analysis demonstrates that exciton-mediated
interactions can in principle contribute positively to the pairing
interaction, but depending on the parameters it may also lead to the opposite
effect. To obtain a precise understanding it is necessary to include in the
model the O-$2p$ states and the full set of Cu-$3d$ states. Better candidate
materials for observing the type of interactions described here are members of
the Ruddleston-Popper series of early transition-metal oxides such as Sr2NbO4
or K2WO4.

###### Acknowledgements.

This work was supported by the Swiss National Science Foundation through
Project No. 179157.

*

##  Appendix A Details of the exciton mediated interaction

###  A.1 The $\underline{d}^{1}$ states

In the limit $U\rightarrow\infty$, the ground state of the undoped Mott
insulator is the direct product of the eigenstates for each site, namely

$\displaystyle\ket{a_{j}}$ $\displaystyle=a^{\dagger}_{j\sigma}\ket{0};$
$\displaystyle E_{a}$ $\displaystyle=\varepsilon_{a}$ (8)
$\displaystyle\ket{b_{j}}$ $\displaystyle=b^{\dagger}_{j\sigma}\ket{0};$
$\displaystyle E_{b}$ $\displaystyle=\varepsilon_{a}+\Omega_{ab}$

where $\Omega_{ab}=\varepsilon_{b}-\varepsilon_{a}$.

###  A.2 The $\underline{d}^{2}$ states

Hole doping causes a finite fraction of the sites to be occupied with 2 holes.
For the case of 2 holes on a single site, the subspace with spin $S=0$ is
spanned by the vectors

$\displaystyle\ket{aa;0}$
$\displaystyle=a^{\dagger}_{\uparrow}a^{\dagger}_{\downarrow}\ket{0}$ (9)
$\displaystyle\ket{ab;0}$
$\displaystyle=\frac{1}{\sqrt{2}}(a^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}-a^{\dagger}_{\downarrow}b^{\dagger}_{\uparrow})\ket{0}$
$\displaystyle\ket{bb;0}$
$\displaystyle=b^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}\ket{0}.$

On this basis, the Hamiltonian for $S=0$ is

$\hat{H}_{S=0}=\begin{pmatrix}2\varepsilon_{a}+U&0&J\\\
0&\varepsilon_{a}+\varepsilon_{b}+U-J&0\\\
J&0&2\varepsilon_{b}+U\end{pmatrix}.$ (10)

Straightforward diagonalization gives the following set of eigenstates

$\displaystyle\ket{\tilde{aa};0}$ $\displaystyle=\big{(}\alpha
a^{\dagger}_{\uparrow}a^{\dagger}_{\downarrow}-\beta
b^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}\big{)}\ket{0}$ (11)
$\displaystyle\ket{\tilde{bb};0}$ $\displaystyle=\big{(}\beta
a^{\dagger}_{\uparrow}a^{\dagger}_{\downarrow}+\alpha
b^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}\big{)}\ket{0}$
$\displaystyle\ket{ab;0}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}a^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}-a^{\dagger}_{\downarrow}b^{\dagger}_{\uparrow}\big{)}\ket{0},$

where

$\alpha^{2}=\frac{1}{2}+\frac{\Omega_{ab}}{2\sqrt{\Omega_{ab}^{2}+J^{2}}},\quad\beta^{2}=\frac{1}{2}-\frac{\Omega_{ab}}{2\sqrt{\Omega_{ab}^{2}+J^{2}}}.$
(12)

The corresponding energies are

$\displaystyle E_{\tilde{aa};0}$ $\displaystyle=2\varepsilon_{a}+U-\Delta_{J}$
(13) $\displaystyle E_{\tilde{bb};0}$
$\displaystyle=2\varepsilon_{a}+2\Omega_{ab}+U+\Delta_{J}$ $\displaystyle
E_{ab;0}$ $\displaystyle=2\varepsilon_{a}+\Omega_{ab}+U-J$

with

$\Delta_{J}=\sqrt{\Omega_{ab}^{2}+J^{2}}-\Omega_{ab}.$ (14)

The basis vectors for the $S=1$ manifold are

$\displaystyle\ket{ab;1,1}$
$\displaystyle=a^{\dagger}_{\uparrow}b^{\dagger}_{\uparrow}\ket{0}$ (15)
$\displaystyle\ket{ab;1,0}$
$\displaystyle=\frac{1}{\sqrt{2}}(a^{\dagger}_{\uparrow}b^{\dagger}_{\downarrow}+a^{\dagger}_{\downarrow}b^{\dagger}_{\uparrow})\ket{0}$
$\displaystyle\ket{ab;1,-1}$
$\displaystyle=a^{\dagger}_{\downarrow}b^{\dagger}_{\downarrow}\ket{0}.$

On this basis, the Hamiltonian for $S=1$ is

$\hat{H}_{S=1}=\begin{pmatrix}E_{ab;1}&0&0\\\ 0&E_{ab;1}&0\\\
0&0&E_{ab;1}\end{pmatrix}$ (16)

with

$E_{ab;1}=2\varepsilon_{a}+\Omega_{ab}+U-3J.$ (17)

###  A.3 The $\underline{d}^{3}$ states

For the evaluation of virtual processes where 2 doped holes occupy nearest-
neighbor sites, we will need the following $\underline{d}^{3}$ states:

$\displaystyle{\ket{aab;1/2,\sigma}}$
$\displaystyle=a^{\dagger}_{\uparrow}a^{\dagger}_{\downarrow}b^{\dagger}_{\sigma}\ket{0}$
(18) $\displaystyle E_{aab}$
$\displaystyle=3\varepsilon_{a}+\Omega_{ab}+3U-5J.$

###  A.4 Dressing of a doped hole by excitons on neighboring sites

When a hole is introduced in the system, it can be “dressed” by creating
virtual excitons on the neighboring sites. To estimate the energy saving by
these processes, we consider a 2-site cluster where one hole is introduced in
addition to the hole already present at each site:

$\displaystyle\ket{\psi_{g}^{1h}}$ $\displaystyle=\big{(}\alpha
a^{\dagger}_{1\uparrow}a^{\dagger}_{1\downarrow}-\beta
b^{\dagger}_{1\uparrow}b^{\dagger}_{1\downarrow}\big{)}a^{\dagger}_{2\uparrow}\ket{0}$
(19) $\displaystyle E_{g}^{1h}$
$\displaystyle=E_{\tilde{aa};0}+E_{a}=3\varepsilon_{a}+U-\Delta_{J},$

where the indices $1,2$ refer to the different sites and the label $1h$ refers
to 1 doped hole. The relevant excited states are

$\displaystyle\ket{\psi_{1}^{1h}}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}a^{\dagger}_{1\uparrow}a^{\dagger}_{2\uparrow}b^{\dagger}_{2\downarrow}-a^{\dagger}_{1\uparrow}a^{\dagger}_{2\downarrow}b^{\dagger}_{2\uparrow}\big{)}\ket{0}$
(20) $\displaystyle E_{1}^{1h}$
$\displaystyle=E_{ab;0}+E_{a}=E_{g}^{1h}+\Delta_{J}+\Omega_{ab}-J$
$\displaystyle\ket{\psi_{2}^{1h}}$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}a^{\dagger}_{1\uparrow}a^{\dagger}_{2\uparrow}b^{\dagger}_{2\downarrow}+a^{\dagger}_{1\uparrow}a^{\dagger}_{2\downarrow}b^{\dagger}_{2\uparrow}\big{)}\ket{0}$
$\displaystyle E_{2}^{1h}$
$\displaystyle=E_{ab;1}+E_{a}=E_{g}^{1h}+\Delta_{J}+\Omega_{ab}-3J$
$\displaystyle\ket{\psi_{3}^{1h}}$
$\displaystyle=a^{\dagger}_{1\downarrow}a^{\dagger}_{2\uparrow}b^{\dagger}_{2\uparrow}\ket{0}$
$\displaystyle E_{3}^{1h}$
$\displaystyle=E_{ab;1}+E_{a}=E_{g}^{1h}+\Delta_{J}+\Omega_{ab}-3J.$

The intersite hopping term $\hat{H}_{t}$ connecting different orbitals $a$ and
$b$ is

$\hat{H}_{t}=t_{ab}\sum_{\sigma}\left(a^{\dagger}_{1\sigma}b_{2\sigma}+b^{\dagger}_{1\sigma}a_{2\sigma}+\mathrm{h.c.}\right).$
(21)

We will need below the matrix elements of the hopping term $\hat{H}_{t}$, for
which we obtain

$\displaystyle|\bra{\psi_{g}^{1h}}\hat{H}_{t}\ket{\psi_{1}^{1h}}|^{2}$
$\displaystyle=|\bra{\psi_{g}^{1h}}\hat{H}_{t}\ket{\psi_{2}^{1h}}|^{2}=\frac{1}{2}\alpha^{2}t_{ab}^{2}$
$\displaystyle|\bra{\psi_{g}^{1h}}\hat{H}_{t}\ket{\psi_{3}^{1h}}|^{2}$
$\displaystyle=\alpha^{2}t_{ab}^{2}.$ (22)

With the help of these expressions, we calculate the energy lowering of a site
with a single doped hole due to virtual hopping to a different 333We do not
consider contributions from virtual hopping to the same orbital on a
neighboring site, since — important as they are — these processes are
accounted for by the one-band Hubbard model. orbital on a neighboring site

$\tilde{E}_{g}^{1h}=E_{g}^{1h}-\frac{\alpha^{2}t_{ab}^{2}/2}{\Omega_{ab}-J+\Delta_{J}}-\frac{3\alpha^{2}t_{ab}^{2}/2}{\Omega_{ab}-3J+\Delta_{J}}.$
(23)

###  A.5 Two holes at nearest-neighbor sites

If we introduce two doped holes on nearest-neighbor sites, as described by the
state vector

$\displaystyle\ket{\psi_{g}^{2h}}$ $\displaystyle=\big{(}\alpha
a^{\dagger}_{1\uparrow}a^{\dagger}_{1\downarrow}-\beta
b^{\dagger}_{1\uparrow}b^{\dagger}_{1\downarrow}\big{)}\big{(}\alpha
a^{\dagger}_{2\uparrow}a^{\dagger}_{2\downarrow}-\beta
b^{\dagger}_{2\uparrow}b^{\dagger}_{2\downarrow}\big{)}\ket{0}$ $\displaystyle
E_{g}^{2h}$ $\displaystyle=2E_{\tilde{aa};0}=4\varepsilon_{a}+2U-2\Delta_{J},$
(24)

the virtual excitations whereby a hole hops from one of these sites to the
other are modified, because the intermediate state on one of the two sites now
has $2$ doped holes in addition to the hole already present in the undoped
state (i.e. $3$ holes in total). Specifically these virtual states are

$\displaystyle\ket{\psi_{1}^{2h}}$
$\displaystyle=a^{\dagger}_{1\uparrow}a^{\dagger}_{2\uparrow}a^{\dagger}_{2\downarrow}b^{\dagger}_{2\downarrow}\ket{0}$
(25) $\displaystyle\ket{\psi_{2}^{2h}}$
$\displaystyle=a^{\dagger}_{1\downarrow}a^{\dagger}_{2\uparrow}a^{\dagger}_{2\downarrow}b^{\dagger}_{2\uparrow}\ket{0}$
$\displaystyle\ket{\psi_{3}^{2h}}$
$\displaystyle=a^{\dagger}_{1\uparrow}a^{\dagger}_{1\downarrow}a^{\dagger}_{2\uparrow}b^{\dagger}_{1\downarrow}\ket{0}$
$\displaystyle\ket{\psi_{4}^{2h}}$
$\displaystyle=a^{\dagger}_{1\uparrow}a^{\dagger}_{1\downarrow}a^{\dagger}_{2\downarrow}b^{\dagger}_{1\uparrow}\ket{0}$

with energy

$\displaystyle E_{j}^{2h}$ $\displaystyle=E_{aab}+E_{a}$ (26)
$\displaystyle=E_{g}^{2h}+2\Delta_{J}+\Omega_{ab}+U-5J\quad(1\leqslant
j\leqslant 4).$

As before, we have to compute the matrix elements of the hopping term
$\hat{H}_{t}$, with the result

$|\bra{\psi_{g}^{2h}}\hat{H}_{t}\ket{\psi_{j}^{2h}}|^{2}=\alpha^{4}t_{ab}^{2}\qquad(1\leqslant
j\leqslant 4).$ (27)

We calculate the correction to the ground-state energy due to virtual hopping
between the two sites sketched in Fig. 2b and subtract the contribution from
the virtual hopping processes that they replace [see Eq. (23)]. This way we
obtain for the energy of bringing two holes from an infinite distance to a
nearest-neighbor distance

$\Gamma_{nn}=\frac{\alpha^{2}t_{ab}^{2}}{\Omega_{ab}-J+\Delta_{J}}+\frac{3\alpha^{2}t_{ab}^{2}}{\Omega_{ab}-3J+\Delta_{J}}\\\
-\frac{4\alpha^{4}t_{ab}^{2}}{\Omega_{ab}+U-5J+2\Delta_{J}}.$ (28)

## References

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  * Note [1] A more general expression of this Hamiltonian in the SU(2) invariant form can be found in Ref. [20].
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  * Note [2] The expression for $u_{c}(j)$ is obtained by applying the condition $\gamma_{nn}(u_{c},j)=0$ to Eq. (5).
  * van der Marel and Sawatzky [1988] D. van der Marel and G. A. Sawatzky, Phys. Rev. B 37, 10674 (1988).
  * Note [3] We do not consider contributions from virtual hopping to the same orbital on a neighboring site, since — important as they are — these processes are accounted for by the one-band Hubbard model.
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# HIGGS TURNS 10: THE CHILDHOOD STORY 111This talk is dedicated to the great
visionaries who made a theoretical breakthrough more than half-a-century ago,
which took so long to appreciate and even longer to confirm experimentally:
François Englert, Carl Hagens, and Peter Higgs, and ad memoriam Robert Brout
(1928–2011), Gerald Guralnik (1936–2014), and Thomas Kibble
(1932–2016).${}^{,}\,$222Original artwork © Julia Landsberg, 2022.

GREG LANDSBERG  Brown University, Department of Physics, 182 Hope St.,
Providence, RI 02912, USA

###### Abstract

In this talk, I give a historical and personal overview of the events that
lead to the discovery of the Higgs boson by the ATLAS and CMS experiments in
2012, and highlight the childhood years of the Higgs boson studies at the
Large Hadron Collider. These recollections are based on my time as the CMS
Physics Coordinator (2012–2013), and because of this they are somewhat biased
toward the CMS side of the story, which I know first-hand.

##  1 Higgs Turns 10!

July 4, 2022 will mark the tenth anniversary of the birth of the Higgs boson
to the happy parents: the ATLAS and CMS Collaborations at the CERN Large
Hadron Collider (LHC). Here are some highlights of the first ten years of the
Higgs boson childhood, which I’ll expand on in the subsequent sections:
• 2012: I am born!
• 2013: First steps
• 2014: Who am I?
• 2015: Why am I alone?
• 2016: Not afraid of dark
! • 2017: Trip to the tau-land
• 2018: From top to bottom
• 2019: I am getting precise
• 2020: Meet the 2nd generation
• 2021: How broad am I?
• 2022: I am charming!

##  2 Long Road to the Higgs Boson Discovery

The 1964 was quite a remarkable year! Many memorable events took place then:
Martin Luther King, Jr. has received a peace Nobel Prize; the Beatles first
arrived to America; Berkeley students stormed the university administration
building, marking the Free Speech Movement; the “Boston Strangler”, Albert
DeSalvo, was arrested; and the first Ford Mustang was introduced at the
World’s Fair in the New York City…

In the meantime, in physics, several remarkable discoveries took place: the
quark model was established in seminal papers by Gell-Mann [1] and Zweig [2];
the $CP$ violation was observed by Christenson, Cronin, Fitch, and Turlay in
neutral kaon decays [3], demonstrating that the time has a preferred
direction; cosmic microwave background was discovered by Penzias and Wilson
[4]. And, hardly noticed at the time, four rather obscure papers on certain
features of quantum field theory, which outlined the basics of what now known
as the Brout–Englert–Higgs (BEH) mechanism, have appeared [5, 6, 7, 8] as
well…

In the first five years since their appearance, these four papers received
only a handful of citations—sixty-four in total—while by now they have been
cited altogether more than 23,000 times, solidly putting them among the most
remarkable papers ever written! The real appreciation of these papers came
only about a decade later, with the realization in late 1966 why the photon
remains massless [9], incorporation of the BEH mechanism in the model of
electroweak interactions [10] by Weinberg and Salam [11, 12] in 1967, and the
proof in 1971–1972 that this model, now known as the standard model (SM) of
particle physics, is renormalizable, via the work of ‘t Hooft and Veltman [13,
14].

The realization that the BEH mechanism must have a particle carrier, the Higgs
boson, was established already in the original papers. The attempts to
calculate its various properties, along with the development of the SM,
spawned early phenomenological work on the Higgs boson production and decays
that took place in the late 1970-ies through late 1980-ies. This work has been
summarized by Gunion, Haber, Kane, and Dawson in “The Higgs Hunter’s Guide”,
first printed in 1989 [15], which instantly became a handbook for the
experimentalists, as was reflected in the 2017 J.J. Sakurai prize given to the
authors. Thus, by mid 1980-ies, the experimental quest for the Higgs boson has
finally started.

Due to the lack of time I’ll skip the last century searches for the Higgs
boson, mainly at DORIS (DESY) and LEP (CERN). After seeing tantalizing hints
of a Higgs boson at the mass of about 115 GeV, LEP was finally switched off in
November 2000, to open way to the construction of the LHC. By then we learned
that the Higgs boson must be rather massive: its mass must exceed 114.4 GeV at
95% confidence level (CL) [16]. Thus, in 2001, at the dawn of the new century,
the baton was passed from Europe to the U.S., and the Fermilab Tevatron became
the next place where the Higgs boson was sought. Meanwhile the construction of
the LHC and the two general-purpose detectors, ATLAS and CMS, went into full
swing at CERN.

The Tevatron Run II started rather slowly, and it took several years for an
instantaneous luminosity to reach the design level. By mid-2005, each of the
CDF and D0 experiments has accumulated an integrated luminosity of 1 fb-1, but
this amount of data was not enough to extend the LEP limits. In the meantime,
the LHC construction and commissioning was completed, and the first proton
beams, still at the injection energy, were circulated in the LHC on September
10, 2008. However, just nine days later, when ramping the LHC dipole magnets
to their design current, a major incident happened because of a defective
superconducting junction between two dipoles, which resulted in an explosion
inside the LHC tunnel and a setback of the LHC startup by a whole year.

By early November 2009, while the LHC was still undergoing the repairs, the
Tevatron finally published the first limits on the Higgs boson that went
beyond the LEP exclusion. Based on an integrated luminosity of between 2.1 and
5.4 fb-1 the CDF and D0 Collaborations managed to exclude a small region of
Higgs boson masses around 165 GeV [17]. Just a few days later, on November 23,
2009, first LHC collisions at the injection energy, after the completion of
the repairs, took place. The Tevatron Run II was officially extended until the
end of 2011, and the race of the two machines to the ultimate goal—the
discovery of the Higgs boson—has started! By March 2010, the LHC reached its
initial collision energy of 7 TeV (limited by the fact that a more serious
mitigation of other possibly defective junctions was required to reach the
design energy).

Meanwhile, the 2010 J.J. Sakurai prize in theoretical physics was given to
Brout, Englert, Hagen, Higgs, Guralnik, and Kibble “for elucidation of the
properties of spontaneous symmetry breaking in four-dimensional relativistic
gauge theory and of the mechanism for the consistent generation of vector
boson masses”.

By summer 2011, the LHC experiments accumulated the first fb-1 of 7 TeV data,
which dramatically changed the landscape of the Higgs boson searches. Both the
ATLAS and CMS Collaborations [18, 19] made public their first limits on the
Higgs boson mass and were able to exclude a broad range of masses from about
135 to 450 GeV. The Tevatron, in the meantime, has expanded on their exclusion
to 156–177 GeV [20]. It became clear that the spectacular LHC performance in
2011 put it way ahead of the Tevatron in the race for the Higgs boson. As a
result, the request by the CDF and D0 experiments to extend the Tevatron run
till 2013 was declined, and on September 30, 2011, in an emotional “Tevatr-
off” event, the Tevatron legacy of two decades has finally ended, and the
machine was permanently switched off. The Higgs baton was back in the European
hands.

In December 2011, ATLAS and CMS made public the last limits on the Higgs
boson, based on an integrated luminosity of up to 4.9 fb-1. The excluded range
of masses was between 127 and 600 GeV [22, 21] . More importantly both
collaborations saw some excesses in the mass range of 124–126 GeV, but it was
not clear if they are compatible between different channels analyzed and
between the two experiments. From then on, ATLAS and CMS focused on the
discovery of the Higgs boson in the low-mass range, around 115–130 GeV.

In CMS, we decided to bet on the fact that the machine energy will be
increased to 8 TeV for 2012 running, and started producing large Monte Carlo
event samples for this energy as early as in November 2011, which were
necessary for proper optimization of the Higgs boson search and detailed
understanding of the backgrounds. This bet played out very well and actually
allowed CMS to show all five main decay channels of the Higgs boson: $H\to
ZZ\to 4\ell$, $H\to\gamma\gamma$, $H\to WW$, $H\to\tau\tau$, and $H\to
b\bar{b}$ at the time of the discovery. The decision to increase the machine
energy was not easy. The LHC machine physicists wanted to err on the side of
caution and run at the same 7 TeV energy as before; the ATLAS and CMS
experiments were pushing for a safe increase in energy, as this would minimize
the amount of time needed for the discovery. At the end, the experiments
prevailed, and at the beginning of February 2012, at the annual Chamonix LHC
workshop, it was decided to operate the machine at 8 TeV. The rest is now
history!

The LHC startup in 2012 was very smooth and already early in the run it became
clear that the machine will double the amount of the delivered integrated
luminosity by early summer. Half of this integrated luminosity was to be
delivered at higher energy, with the dominant Higgs boson gluon fusion cross
section 30% higher than at 7 TeV. It was therefore decided to plan for the
next major update on the Higgs boson searches for the ICHEP conference,
scheduled to start on July 4, 2012 in Melbourne. It was anticipated that if
the $\sim$125 GeV excess seen by both ATLAS and CMS experiments is real, the
new data would suffice to establish the existence of the Higgs boson at the
discovery level, i.e., five standard deviations in each experiment.

The analysis of the complexity of the Higgs boson search was performed in a
“blind” way, i.e., the data in the mass region of 110–140 GeV was kept hidden
in order not to subconsciously bias the results. The analysis was fully
optimized, and the backgrounds and all the systematic uncertainties were
estimated with the data in the region of interest still being blinded. By mid-
June 2012, CMS was ready to finally unblind the analyses in all five main
decay channels.

The unblinding of the data was done as a formal event on Friday, June 15,
2012, in the Filtration Plant auditorium at CERN and via video. The entire
collaboration was invited to join the event. About 250 people were sitting and
standing in the room for about 2 hours it took to show the results in all five
channels, on a hot summer day with the temperature in the room reaching 28∘ C,
but nobody minded! More than 500 more people were connected to the event via
video.

The unblinding was done based on about 80% of the full data set. The most
sensitive $\gamma\gamma$ and $4\ell$ channels were unblinded first, and to the
cheers of the audience we have observed very clear peaks at 125 GeV in both of
these channels with the significances around four standard deviations each! We
knew then that we had the Higgs boson; the rest of the world will learn about
it three weeks later…

The next day, on Saturday June 16, I decided to throw an informal discovery
party at our place in Challex, France. Some 30 people attended, but given that
I had several ATLAS colleagues leaving nearby, we “masqueraded” this party as
a birthday party, complete with a cake proudly displaying “125” with candles.
Still, it was probably a big surprise for the trash collectors the following
Tuesday to find a recycling bin filled with about 30 empty champagne bottles
next to my house. The Higgs was born!

The following three weeks were a mad rush to finalize everything for the
discovery announcement. The rest of the data were added, with the analysis
selections completely frozen at the time of the unblinding, and the five
channels were combined together to yield the overall observed (expected)
significance of 5.0 (5.8) standard deviations. The discovery presentation was
scheduled for July 4th, in the CERN Main Auditorium, simultaneously
broadcasted to Melbourne. The ATLAS and CMS spokespeople, Fabiola Gianotti and
Joe Incandela, presented the results to the world. Each experiment achieved 5
standard deviations—the gold standard for the discovery—independently. At the
end of the presentation, Rolf-Dieter Heuer, the CERN Director General, said:
“I think we have it!” The Higgs was now born officially! Given the
Independence Day holiday in the U.S., the fireworks came for free…

The two discovery papers were submitted to the Physics Letters B journal on
July 31, 2012 and appeared online on August 18 [23, 24].

##  3 Higgs at Moriond

More data were delivered by the LHC in 2012, and the discovery sample was more
than doubled. The next big update on the Higgs boson was to be announced at
the Moriond series of conferences, in March 2013. However, in CMS, we saw that
the significance of the Higgs boson signal in the very sensitive diphoton
channel did drop despite significantly more data analyzed. This was very
puzzling, as nothing has changed in the analysis, and the significance in the
most sensitive four-lepton channel kept increasing.

It took us several months to decisively prove that there were no mistakes in
the analysis and that the drop in the significance was a result of a
statistical fluctuation. The calibration of the electromagnetic calorimeter,
which is crucial for the high-resolution diphoton channel, has been improved,
and that was the main change that resulted in the statistical fluctuation.
When dealing with the case of a rather large signal on top of an overwhelming
background, the efficiency of the optimal selection is fairly low. Even small
changes in the kinematic parameters of the events, such as the change in the
photon energy due to the new calibration, could result in completely different
signal candidates to be selected. Indeed, when the new calibration was applied
to the discovery sample, we noticed that the overlap between the signal
candidates in the discovery analysis and in the new analysis was not that
large; in fact we were dealing with largely statistically independent event
candidate samples in the two analyses. In such a situation, large statistical
fluctuation are possible. Using a special statistical technique, known as the
jack-knife resampling [25], we were able to show that the two results were in
fact consistent with one the other within 2 standard deviations and approved
the new diphoton result for presentation at the Moriond QCD conference. We
practiced the talk after the dinner on the night before the presentation, well
beyond midnight, since we have expected a lot of questions about this new
result. The observed (expected) significance of the Higgs boson signal was 3.2
(4.2) standard deviations, a big change from the 4.1 (2.8) standard deviations
at the time of the discovery. Statistics could play funny tricks! As a result
of the late-night practice talk, some of us, including myself, were half-
asleep during the morning session where the new result was shown for the first
time, which was captured by the sharp eye of the Moriond photographer!

That Moriond conference was quite memorable, as the host of new results
released by the ATLAS and CMS experiments demonstrated with an increased
precision that the new particle was indeed consistent with the SM Higgs boson.
The CERN press release stated: “New results indicate that a new particle is a
Higgs boson”! Since then we dropped the term “Higgs-like particle”, which was
originally used to denote the new boson, and started calling it “a Higgs
boson” and, a bit later, “the Higgs boson”. Incidentally, the same week the
new pope—Pope Francis—was confirmed in Vatican. The Simon Fraser University (a
member of the ATLAS Collaboration) newsletter connected these two events on
March 14, 2013 under the title “New boson and new pope confirmed”.

##  4 Dazzling Precision

We have learned a lot about the Higgs boson in the past ten years, thanks to
the spectacular performance of the LHC machine, which delivered 15 times the
discovery sample, most of it at much higher energy of 13 TeV. This rapid
progress would not have been possible without significant theoretical work on
the Higgs physics, captured in four Yellow Reports of the LHC Higgs Cross
Section Working Group, as well as in more recent papers. Many new results
based on this large data set have been shown at this conference.

At the time of the discovery the Higgs boson production cross section in
various decay channels in each experiment was measured with the precision of
about 25%. The precision has now reached 6% [26]. By 2015, just a couple of
years after the discovery, we have firmly established that the Higgs boson has
quantum numbers of the vacuum $J^{PC}=0^{++}$ and ruled out pure pseudoscalar
or tensor hypotheses at $>$99.9% CL [27, 28]. (The fact that Higgs boson can’t
be a spin-1 particle is evident from its decay to a pair of photons, which is
not possible for a spin-1 particle, according to the Landau–Yang theorem.)

After a slight initial tension between the two experiments on the value of the
Higgs boson mass, it is now measured to a per mil precision. The most precise
measurement to date, coming from the CMS experiment, is $125.38\pm 0.14$ GeV
[29]. Very recently, a two-sided 95% CL interval on the Higgs boson width and
the first evidence for the off-shell Higgs boson production were reported,
with the measured width $\Gamma_{H}=3.2^{+2.4}_{-1.7}$ GeV [30] (fully
consistent with the width of 4.1 MeV predicted in the SM).

By 2018, the coupling of the Higgs boson to $\tau$ leptons has been firmly
established by both ATLAS [31] and CMS [32], with the significance around 6
standard deviations in each experiment, thus establishing the coupling to the
third-generation quarks (indirectly, via the gluon fusion production through a
top quark loop) and leptons. Later the same year, the direct coupling to top
quarks via the $t\bar{t}H$ production was established as well [33, 34]. And
finally, also in 2018, the coupling of the Higgs boson to bottom quarks was
finally measured by both collaborations, primarily via associated production
with vector bosons, with an overall significance of about 5.5 standard
deviations by each experiment [35, 36]. Ironically, this decay with the
largest branching fraction among all other Higgs boson decays took six years
and tour-de-force analyses to be finally observed, since it suffers from very
large backgrounds. The 2018 became the year of the third generation!

In 2019, first evidence for Higgs boson decay to a second-generation fermion
(the muon) was seen by CMS at the 3 standard deviations level [37]; ATLAS also
saw an excess of dimuon events at the Higgs boson mass corresponding to 2
standard deviations [38]. Finally, in 2022, both ATLAS and CMS presented
impressive limits on the coupling of Higgs bosons to charm quarks, which,
while still being far from the SM expectations, showed that it may be possible
to establish this decay mode at the High-Luminosity LHC.

In parallel with this quest for various SM decay channels of the Higgs boson,
both ATLAS and CMS looked for other, exotic decay channels, which are either
prohibited or very suppressed in the SM. Among these decays are the lepton
flavor violating decays, such as $H\to\tau\mu$, as well as invisible decays of
the Higgs boson, which allow to probe potential connection between the Higgs
boson and dark matter. The most recent result from ATLAS [39] has established
a fairly stringent upper limit on the branching fraction of invisible Higgs
boson decays at 14.5% at 95% CL (with the 10.3% expected limit).

Four dedicated experimental talks at this conference, along with three
theoretical ones, showcased some of these results, as well as discussed future
measurements, including the pair production of Higgs bosons.

##  5 Conclusions

Higgs boson had a fun childhood and is now finishing the “primary school”. The
ATLAS and CMS experiments have been caring and watchful parents for all these
childhood years, and will follow the “middle school” years with the large Run
3 and beyond data samples. With the Higgs boson couplings to third-generation
fermions well established, our sights are shifting on the second generation
and on exotic decays, as well as on precision differential measurements and on
the Higgs boson pair production. Happy birthday, the Higgs boson!

## Acknowledgments

I’m grateful to the organizers for inviting me to give this talk, which jogged
a lot of fond memories. This work is partially supported by the U.S.
Department of Energy grant no. DE-SC0010010. Special thanks to my daughter
Julia, who prepared a nice set of drawings for this talk with an artist’s
rendition of its main character, some of which made it to these proceedings.

## References

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