††institutetext: Department of Physics, Carnegie Mellon University,
Pittsburgh, PA 15213

# An effective field theory of damped ferromagnetic systems

Jingping Li<EMAIL_ADDRESS>
###### Abstract

Using the in-in formalism, we generalize the recently constructed
magnetoelastic EFT PavaskarPencoRothstein  to describe the damping dynamics of
ferromagnetic systems at long wavelengths. We find that the standard Gilbert
damping term naturally arises as the simplest leading-order symmetry-
consistent non-conservative contribution within the in-in framework. The EFT
is easily generalized to scenarios with anisotropy and inhomogeneity. In
particular, we find the classic Landau-Lifshitz damping term emerges when
isotropy is broken by a constant external background field. This provides a
first principle explanation for distinguishing the two types of damping
dynamics that were originally constructed phenomenologically. Furthermore, the
EFT framework could also incorporate intrinsic anisotropy of the material in a
straightforward way using the spurion method. For systems with inhomogeneity
such as nontrivial spin textures, we find that the leading order derivative
correction yields the generalized Gilbert damping equations that were found in
condensed matter literature. This shows that the EFT approach enables us to
derive the form of higher-derivative-order corrections in a systematic way.
Lastly, using the phonon-magnon coupling deduced in the magnetoelastic EFT, we
are able to make a prediction for the generic form of the phononic
contribution to the damping equation.

##  1 Introduction

It has long been established that the methodology of coset construction serves
as a powerful tool of relativistic effective field theories (EFTs) of
Goldstone bosons (e.g. pions from spontaneously broken approximate chiral
symmetry) ColemanWessZumino ; Callanetal ; Volkov ; Ogievetsky . In recent
years, many works have demonstrated that its versatility is extendable to
condensed matter systems where we are interested in the macroscopic behavior
which are usually the massless low energy excitations EFT ; CMEFT . Using this
approach, a recent paper PavaskarPencoRothstein  constructed an EFT of
magnetoelastic systems where the phonons and magnons are considered Goldstones
associated with translations spontaneously broken by the ground state location
of the material (the lattice) and an $SO(3)$ symmetry of the magnetic moments
by their ground state orientations. The EFT approach provides a systematic way
to understand phonon-magnon interactions from first principles and predict the
forms of higher-order corrections which has not been done previously.

While the paper was focused on conservative dynamics, there has also been
extensive study on the theoretical description of non-conservative dynamics of
damped magnetic systems since the seminal work of Landau, Lifshitz, and later
Gilbert LandauLifshitz ; Gilbert . However, to our knowledge, the prior works
were mostly model-dependent phenomenological descriptions. It is therefore
desirable to have a first principle derivation from a similar many-body EFT
perspective.

On the other hand, the Schwinger-Keldysh formalism Schwinger ; Keldysh  (and
the related in-in formalism Galley ; GalleyTsangStein ) has been known to
describe the quantum field theory of open systems and hence fully capable of
describing dissipative effects. In recent years, its power has been
successfully extended to the EFT framework of dissipative dynamics in
astrophysics and black holes EndlichPenco ; GoldbergerRothstein ;
GoldbergerLiRothstein  that systematically derives dissipative equations of
motion. Therefore, it is natural to consider its utility in describing
deriving damping equations in condensed matter EFTs.

In this paper, combining the power of the two techniques, we apply the
Schwinger-Keldysh formalism to incorporate dissipative effects into the EFT of
magnons to reproduce the known results of magnetic damping. In section 2, we
review the coset construction of the magnon EFT as well as the techniques in
Schwinger-Keldysh formalism to be applied in this paper. Section 3 derives the
original Gilbert damping equation for homogeneous and isotropic materials. In
section 4, we move on to more general materials and recover the Landau-
Lifshitz damping equation for anisotropic systems and generalized Gilbert
damping for spatially inhomogeneous materials. In section 5, we derive the
damping terms originating from the magnon-phonon interaction.

Conventions: we use natural units where $\hbar=1$. Unless specified, the
uppercase Latin indices $A,B,C\dots$ denote the full internal spin symmetry
space which runs over $1,2,3$ while the lower case ones $a,b,c\dots$ in the
begining alphabet index the broken subspace $1,2$. Those in the middle
alphabet $i,j,k\dots$ run over the three spatial dimensions (to generalize to
higher dimensions, the internal spin symmetry will have to be modified
accordingly).

##  2 Review of the magnon EFT and Schwinger-Keldysh formalism

In this section, we first provide a self-contained review on the symmetries
and the corresponding coset construction of magnon-phonon EFT proposed by
PavaskarPencoRothstein . Furthermore, we summarize the Schwinger-Keldysh
formalism which is the central tool for deriving the dissipative equations of
motion.

###  2.1 Symmetry breaking in magnetoelastic systems

We follow the derivations in [1]. The symmetries under consideration are the
spatial Galilean group (generated by translations $P_{i}$, rotations, $L_{i}$,
boost $K_{i}$) and internal symmetries (internal translations $T_{i}$,
internal rotations $Q_{i}$, spin rotations $S_{A}$). The algebra of the
generators is given by

$\displaystyle\left[L_{i},K_{j}\right]=i\epsilon_{ijk}K_{k},$
$\displaystyle\left[L_{i},P_{j}\right]=i\epsilon_{ijk}P_{k},$ (1)
$\displaystyle\left[K_{i},H\right]=-iP_{k},$
$\displaystyle\left[K_{i},P_{j}\right]=-iM\delta_{ij},$ (2)
$\displaystyle\left[Q_{i},T_{j}\right]=i\epsilon_{ijk}T_{k},$
$\displaystyle\left[Q_{i},Q_{j}\right]=i\epsilon_{ijk}Q_{k},$ (3)
$\displaystyle\left[S_{A},S_{B}\right]=i\epsilon_{ijk}K_{k},$
$\displaystyle\left[L_{i},L_{j}\right]=i\epsilon_{ijk}L_{k}.$ (4)

In particular, the $T_{i}$ and $Q_{i}$ generators generate translations and
rotations on the “comoving” coordinates $\phi^{I}(x)$ (or the Lagrangian
coordinates in continuum mechanics which simply gives an initial labeling to
the continuum)

$\phi^{I}(x)\rightarrow\phi^{I}(x)+a^{I},\ \phi^{I}(x)\rightarrow R_{\
J}^{I}\phi^{J}(x),$ (5)

and $S_{A}$ generates the internal rotation on the orientation of the spin

$\mathcal{N}^{A}\rightarrow O_{\ B}^{A}\mathcal{N}^{B},$ (6)

where $O=e^{i\chi_{a}S^{a}}$ and the Néel vector $\vec{\mathcal{N}}$ is the
order parameter for the spin orientation.

In the ground state, the order parameters gain vacuum expectation values
(VEVs) which we choose to be

$\left\langle\vec{\phi}(x)\right\rangle=\vec{x},\
\left\langle\vec{\mathcal{N}}\right\rangle=\hat{x}_{3}.$ (7)

The VEVs are not invariant under the transformations and hence spontaneously
break some of the symmetries

$Unbroken=\begin{cases}H\\\ P_{i}+T_{i}\equiv\bar{P}_{i}\\\ L_{i}+Q_{i}\\\
S_{3}\\\ M\end{cases},\ Broken=\begin{cases}K_{i}\\\ T_{i}\\\ Q_{i}\\\
S_{1},S_{2}\equiv S_{a}\end{cases}.$ (8)

The parametrization of the ground state manifold which is simply the broken
symmetry transformations plus the unbroken translations is given by

$\Omega=e^{-itH}e^{ix^{i}\bar{P}_{i}}e^{i\eta^{i}K_{i}}e^{i\pi^{i}T_{i}}e^{i\theta^{i}Q_{i}}e^{i\chi^{a}S_{a}},$
(9)

where $\eta^{i}$, $\theta^{i}$, $\chi^{a}$, and $\pi^{i}=\phi^{i}-x^{i}$ are
the corresponding Goldstone fields.

For any Goldstone fields $\psi^{i}$ corresponding to the broken generators
$X_{i}$, their covariant derivatives are the basic building blocks of the low
energy EFT. They are systematically computed by the coset construction using
the Mauer-Cartan forms of the broken group

$\Omega^{-1}\partial_{\mu}\Omega\supset\left(\nabla_{\mu}\psi^{i}\right)X_{i},$
(10)

by computing the coefficients of the broken generator $X_{i}$. In addition, in
the case that one broken generator $X^{\prime}$ appears in the commutation
algebra of the other $X$’ with the unbroken translations

$\left[\bar{P},X\right]\supset X^{\prime},$ (11)

it means that the two Goldstones are not independent and one of them can be
eliminated. This is known as the inverse Higgs phenomenon.

The result of this exercise is that $\eta^{i}$, $\theta^{i}$ are eliminated
and the only independent degrees of freedom are the magnons $\chi^{a}$ and
phonons $\pi^{i}$, and at leading order in derivatives, they appear in the
following combinations:

$\displaystyle\nabla_{(i}\pi_{j)}$
$\displaystyle=(D\sqrt{D^{T}D}D^{-1})_{ij}-\delta_{ij},$ (12)
$\displaystyle\nabla_{t}\chi^{a}$
$\displaystyle=\frac{1}{2}\epsilon^{aBC}\left\\{O^{-1}\left[\partial_{t}-\partial_{t}\pi^{k}(D^{-1})_{k}^{\
j}\partial_{j}\right]O\right\\}_{BC},$ (13) $\displaystyle\nabla_{i}\chi^{a}$
$\displaystyle=\frac{1}{2}\epsilon^{aBC}\left(O^{-1}\partial_{i}O\right)_{BC},$
(14)

where $D_{ij}=\delta_{ij}+\partial_{i}\pi_{j}$. PavaskarPencoRothstein  found
the most general action for ferromagnetic material in the form

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{c_{1}}{2}\det\left(D\right)\epsilon^{ab}\left[\left(O^{-1}\partial_{t}O\right)_{ab}-\partial_{t}\pi^{k}\left(D^{-1}\right)_{k}^{\
j}\left(O^{-1}\partial_{j}O\right)_{ab}\right]$ (15)
$\displaystyle-\frac{1}{2}F_{2}^{ij}(\nabla_{(i}\pi_{j)})\nabla_{i}\chi_{a}\nabla_{j}\chi^{a}-\frac{1}{2}F_{3}(\nabla_{(i}\pi_{j)})\nabla_{t}\chi_{a}\nabla_{t}\chi^{a},$
(16)

where the first term is similar to a Wess-Zumino-Witten (WZW) term that
differs by a total derivative under the symmetry transformation.

###  2.2 The magnon EFT and the conservative equation of motion

To derive the equations of motion for magnons, it is more convenient to
express the magnon fields in the nonlinear form

$\hat{n}=\mathcal{R}(\chi)\hat{x}_{3}=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right),$
(17)

where the two magnon fields are related to the angular fields by

$\chi_{1}=\theta\sin\phi,\quad\chi_{2}=\theta\cos\phi.$ (18)

Physically, this unit vector represents the direction of the magnetic moment.
Under this representation, the pure magnon Lagrangian (in the absence of
phonon excitations) becomes

$\mathcal{L}\rightarrow\frac{c_{2}}{2}\epsilon^{ab}\left(O^{-1}\partial_{t}O\right)_{ab}+\frac{c_{6}}{2}\left(\partial_{t}\hat{n}\right)^{2}-\frac{c_{7}}{2}\left(\partial_{i}\hat{n}\right)^{2},$
(19)

where $F_{3}(0)=c_{6}$ and $F_{2}^{ij}(0)=c_{7}\delta^{ij}$.

The dispersion relation for the quadratic Lagrangian has two solutions
PavaskarPencoRothstein

$\omega_{+}^{2}=\bigg{(}\frac{c_{2}}{c_{6}}\bigg{)}^{2}+O(k^{2}),\quad\omega_{-}^{2}=\bigg{(}\frac{c_{7}}{c_{2}}\bigg{)}^{2}k^{4}+O(k^{6}).$
(20)

For ferromagnetic materials, where $c_{2}~{}(c_{6}c_{7})^{3/4}$, the first
mode is gapped around the EFT cutoff scale, while the second has $\omega\sim
k^{2}$ scaling and exits the EFT. In the long wavelength limit, we may assign
the scaling $\partial_{t}^{1/2}\sim\partial_{i}$ to the derivatives for
ferromagnets.

To derive the equation of motion, we notice that an action of this form has a
symmetry under the infinitesimal spin rotation
$\delta\hat{n}=\vec{\omega}\times\hat{n}$, where $\vec{\omega}$ is the
constant infinitesimal parameter. It can be shown that the Wess-Zumino-Witten
term contributes to a total derivative $\partial_{\mu}\vec{F}^{\mu}$ under
this transformation. Using the equation for Noether current in Lagrangian
mechanics

$\vec{J}^{\mu}=\hat{n}\times\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\hat{n}}-\vec{F}^{\mu},$
(21)

we find the conserved current

$\vec{J}^{0}=-c_{2}\hat{n}-c_{6}\partial_{t}\hat{n}\times\hat{n},\
\vec{J}^{i}=c_{7}\partial_{i}n\times\hat{n}.$ (22)

The continuity equation $\partial_{\mu}\vec{J}^{\mu}=0$ is then explicitly

$c_{2}\partial_{t}\hat{n}=-\left(c_{6}\partial_{t}^{2}\hat{n}-c_{7}\nabla^{2}\hat{n}\right)\times\hat{n},$
(23)

which is the equation of motion for Landau-Lifshitz model of magnetism.

###  2.3 The Schwinger-Keldysh formalism

The appropriate formalism for non-conservative system is the so-called in-in
or Schwinger-Keldysh formalism Schwinger ; Keldysh ; Galley ; GalleyTsangStein
. The basic idea is that there is an external sector $X$ that the energy is
dissipated into since the total energy needs to be conserved. The external
sector could evolve into any final state which we do not observe, so all the
dynamics are inclusive of the final states in the Hilbert space of $X$

$\sum_{X_{out}}\langle X_{in}|\dots|X_{out}\rangle\langle
X_{out}|\dots|X_{in}\rangle\equiv\langle\dots\rangle_{in},$ (24)

and depends only on the initial state (hence the name in-in). We can generate
an effective action for the in-in observables via the Schwinger-Keldysh closed
time path integral

$\exp\bigg{[}i\Gamma[q;\tilde{q}]\bigg{]}=\int_{initial}\mathcal{D}X\mathcal{D}\tilde{X}\exp\bigg{[}iS[q,X]-iS[\tilde{q},\tilde{X}]\bigg{]},$
(25)

where we are integrating over an additional copy of variables $\tilde{X}$
which corresponds to evolving back to the boundary conditions fixed at the
initial time.

The equation of motion for the degrees of freedom in the observed sector $q$
can be derived from the action functional $\Gamma[q;\tilde{q}]$ by

$\frac{\delta}{\delta q}\Gamma[q;\tilde{q}]\Biggr{|}_{q=\tilde{q}}=0.$ (26)

Any external sector operator $\mathcal{O}(X)$ coupled to some operator in the
observable sector $F(q)$ by the interaction term $\int
dx\mathcal{O}\left(X(x)\right)F\left(q\left(x^{\prime}\right)\right)$ (where
$x$ is the corresponding spacetime coordinates) would enter the equations of
motion in terms of

$\langle\mathcal{O}\left(X(x)\right)\rangle_{in}=\int_{initial}\mathcal{D}X\mathcal{D}\tilde{X}\exp\bigg{[}iS[q,X]-iS[\tilde{q},\tilde{X}]\bigg{]}\mathcal{O}(X),$
(27)

where we have abbreviated $\langle\mathcal{O}(X)\rangle_{in}\equiv\langle
X_{in}|\mathcal{O}(X)|X_{in}\rangle$. Just as in the perturbative quantum
field theory correlation functions calculated by Feynman propagators, this can
be similarly calculated using the Schwinger-Keldysh propagators

$\langle\mathcal{O}_{a}(x)\mathcal{O}_{b}(x^{\prime})\rangle=\left(\begin{array}[]{cc}\langle
T\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle&\langle\mathcal{O}(x^{\prime})\mathcal{O}(x)\rangle\\\
\langle\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle&\langle\tilde{T}\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle\end{array}\right),$
(28)

where $T$ and $\tilde{T}$ represent time and anti-time orderings. The sub-
indices label the first and the second copy, which determines the relative
time-ordering of the operators.

Explicitly, the linear response gives

$\langle\mathcal{O}(x)\rangle=i\int dx^{\prime}\\{\langle
T\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle-\langle\mathcal{O}(x^{\prime})\mathcal{O}(x)\rangle\\}F\left(q(x^{\prime})\right)+\mathcal{O}(F^{2}).$
(29)

or equivalently

$\langle\mathcal{O}(x)\rangle=\int
dx^{\prime}G_{R}(x,x^{\prime})F\left(x^{\prime}\right),$ (30)

with the retarded Green’s function given by

$\displaystyle G_{R}(x,x^{\prime})$
$\displaystyle=i\theta(t-t^{\prime})\langle[\mathcal{O}(x),\mathcal{O}(x^{\prime})]\rangle$
$\displaystyle=i(\langle
T\mathcal{O}(x)\mathcal{O}(x^{\prime})\rangle-\langle\mathcal{O}(x^{\prime})\mathcal{O}(x)\rangle).$
(31)

Therefore, the exact form of the damping term in the equation of motion would
depend on the detailed structure of these retarded response functions.

##  3 Gilbert damping term from EFT

###  3.1 Coupling at the leading order

The composite operators $\mathcal{O}^{r}(X)$ that encapsulate the external
sector transform under arbitrary representations (labeled by $r$), provided
they form invariants of the unbroken $SO(2)$ with the magnon $\chi_{a}$ and
derivatives. In order to achieve this, the operators have to be dressed with
the broken $SO(3)/SO(2)$ subgroup parametrized by the Goldstones $T^{R}(\chi)$
in the corresponding representation

$\tilde{\mathcal{O}}^{r}(X)\equiv\mathcal{R}^{r}(\chi)\mathcal{O}^{r}(X),$
(32)

such that they transform covariantly under the unbroken subgroup
Delacretazetal .

In the long wavelength regime, the theory is organized by a spatial derivative
expansion. In fact, the simplest invariant operator at zeroth-order in the
derivative expansion is the singlet aligned along the ground state orientation
$\hat{x}_{3}$

$S_{int}=\int d^{4}x\tilde{\mathcal{O}}^{3}(X)\equiv\int
d^{4}x\hat{x}_{3}\cdot\tilde{\vec{\mathcal{O}}}(X),$ (33)

(note that we are adopting manifestly relativistic notations for spacetime and
energy-momentum for convenience, albeit the system may or may not be
relativistic). Equivalently, we may write

$S_{int}=\int d^{4}x\hat{n}\cdot\vec{\mathcal{O}}(X)$ (34)

where $\hat{n}=O(\chi)\hat{x}_{3}$ as defined previously.

At the same order in this expansion, there could be more operators that can be
added, such as when the operator is a two-index tensor operator
$\mathcal{O}^{(2)}$ and we may have combinations of the form
$\hat{n}\cdot\mathcal{O}^{(2)}\cdot\hat{n}$. However, as will be explained in
the next subsection, due to the constraint $\hat{n}^{2}=1$, these will not
lead to any new contributions, and from the perspective of the EFT, they are
redundant operators. Hence, Eq. (34) is the only non-trivial operator at this
order.

Furthermore, we observe that the form of the operator restricts the possible
external sector that can couple in this way. For example, one can form such
operators from fermions, where

$\vec{\mathcal{O}}(\psi)=\psi^{\dagger}\vec{\sigma}\psi+\psi\vec{\sigma}\psi^{\dagger},$
(35)

where $\vec{\sigma}$ are the Pauli matrices acting as the intertwiner between
the spinor and $SO(3)$ spin space. On the other hand, phonons cannot form
operators in this form, and hence will not contribute in this way.

###  3.2 Gilbert damping

The undamped equation of motion is derived from the continuity equation

$\partial_{\mu}\vec{J}^{\mu}=0$ (36)

corresponding to the spin rotation transformation
$\delta\hat{n}=\vec{\omega}\times\hat{n}$. To derive the non-conservative
equation of motion, we need to find out how the additional term Eq. (34)
affects the continuity equation.

The spin rotation transformation on $\hat{n}$ alone is itself a valid symmetry
for the pure magnon action Eq. (19), but the interaction term Eq. (34) is not
invariant if we keep the external sector fixed. A standard trick of Noether
theorem is that for an arbitrary symmetry transformation
$\phi(x)\mapsto\phi(x)+f(x)\epsilon$, if we promote the global symmetry
variation parameter to be an arbitrary local variation $\epsilon(x)$, the
total variation takes the form

$\delta S=-\int d^{4}xJ^{\mu}\partial_{\mu}\epsilon,$ (37)

such that when $\epsilon$ is a constant, invariance under the symmetry
transformation is guaranteed $\delta S=0$ even off-shell (equation of motion
is not satisfied). Integrate by parts, we find that on-shell

$\delta S=\int d^{4}x\left(\partial_{\mu}J^{\mu}\right)\epsilon.$ (38)

However, for an arbitrary $\epsilon(x)$, this is also the equation of motion,
since

$\delta S=\int d^{4}x\frac{\delta S}{\delta\phi}f(x)\epsilon(x).$ (39)

Therefore, the effect of an additional term in the action $\Delta S$ is adding
a term to the current divergence

$\partial_{\mu}J^{\mu}=0\rightarrow\partial_{\mu}J^{\mu}+\frac{\delta\Delta
S}{\delta\phi}f(x)=0.$ (40)

For the spin rotations, the variation of the pure magnon EFT with local
$\omega(x)$ is given by

$\delta S=\int
d^{4}x\left(\partial_{\mu}\vec{J}^{\mu}\right)\cdot\vec{\omega}.$ (41)

Correspondingly, the addition of Eq. (34) leads to the modification

$\delta S=\int
d^{4}x\left(\partial_{\mu}\vec{J}^{\mu}+\hat{n}\times\vec{\mathcal{O}}(X)\right)\cdot\vec{\omega}.$
(42)

Thus, the (non-)continuity equation becomes

$\partial_{\mu}\vec{J}^{\mu}=-\hat{n}\times\vec{\mathcal{O}}(X).$ (43)

When we focus on the measurements of the magnons, the effect of the external
sector enters as an in-in expectation value
$\left\langle\vec{\mathcal{O}}\right\rangle_{in}$. This may be evaluated via
the in-in formalism, and the leading order contribution is given by

$\vec{\omega}\cdot\int
d^{4}x\partial_{\mu}\vec{J}^{\mu}=-\vec{\omega}\cdot\int
d^{4}x\hat{n}\times\left\langle\vec{\mathcal{O}}\right\rangle_{in}=-\vec{\omega}\cdot\int
d^{4}x\hat{n}\times\int
d^{4}x^{\prime}\left(G_{R}\cdot\hat{n}^{\prime}\right),$ (44)

where $G_{R}(t^{\prime},\vec{x}^{\prime};t,\vec{x})$ is the retarded response
function of the operator $\vec{\mathcal{O}}$.

In frequency space, we have
$G_{R}(t,\vec{x})=\int\frac{d^{3}\vec{k}d\omega}{(2\pi)^{4}}e^{-i\omega
t+\vec{k}\cdot\vec{x}}\tilde{G}_{R}(\omega,\vec{k})$ and furthermore, using
the spectral representation (making the spin space indices explicit
temporarily)

$\tilde{G}_{R}^{AB}(\omega,\vec{k})=\int_{-\infty}^{\infty}\frac{d\omega_{0}}{\pi}\frac{i}{\omega-\omega_{0}+i\epsilon}\rho^{AB}(\omega_{0},\vec{k}),$
(45)

we can separate the prefactor using the identity

$\frac{i}{\omega-\omega_{0}+i\epsilon}=\pi\delta(\omega-\omega_{0})+\mathcal{P}\frac{i}{\omega-\omega_{0}},$
(46)

into a $\delta$-function and a principal part. The dissipative part is
captured by the former

$\tilde{G}_{R,diss}^{AB}(\omega,\vec{k})=\int_{-\infty}^{\infty}d\omega_{0}\delta(\omega-\omega_{0})\rho^{AB}(\omega_{0},\vec{k})=\rho^{AB}(\omega,\vec{k}).$
(47)

The indices of this spectral function lives in the spin $SO(3)$ space and, for
isotropic systems, should be built from invariant tensors $\delta^{AB}$,
$\epsilon^{ABC}$. However, the latter could not neither form a two-index
object nor respect parity invariance by itself, so the only symmetry-
consistent possibility is

$\rho^{AB}(\omega,\vec{k})=f(\omega,|\vec{k}|^{2})\delta^{AB},$ (48)

with $f$ assumed to be an analytic function of its arguments such that it has
a smooth limit as $\omega$ goes to zero. Dissipative dynamics is antisymmetric
under time reversal, so it should be odd under the simultaneous transformation
$\omega\leftrightarrow-\omega$ and $(A,B)\leftrightarrow(B,A)$, meaning that
the leading order contribution is given by

$\rho^{AB}(\omega,\vec{k})=-i\mathscr{C}\omega\delta^{AB}$ (49)

or in real spacetime

$G_{R,diss}^{AB}(t,\vec{x})=\mathscr{C}\frac{\partial}{\partial
t}\delta(t)\delta^{3}(\vec{x})\delta^{AB}.$ (50)

$\mathscr{C}$ could be understood as a Wilson coefficient in this non-
conservative sector.

From this, we arrive at the equation

$\partial_{\mu}\vec{J}^{\mu}=-\mathscr{C}\hat{n}\times\frac{\partial}{\partial
t}\hat{n}.$ (51)

Combining with the conservative part of the continuity equation Eq. (23), we
find the Gilbert damping equation

$\frac{\partial}{\partial
t}\vec{m}=-\gamma\vec{m}\times\frac{\partial}{\partial t}\vec{m}+\dots,$ (52)

where $\gamma=\mathscr{C}/(c_{2}m_{s})$, $m_{s}\hat{n}=\vec{m}$ (for uniform
materials), and the higher-order terms on the right-hand side of the original
equation of motion Eq. (23) are contained in$\dots$ which will be omitted in
the following.

When we have another singlet of the form
$\hat{n}\cdot\mathcal{O}^{(2)}\cdot\hat{n}$, the effect of the extra
$\hat{n}$s is a replacement of the spectral density
$\rho^{AB}\rightarrow\rho^{ACBD}\hat{n}_{C}\hat{n}_{D}$. The tensor basis
still consists of Kronecker delta since the only structure that Levi-Civita
tensors could contract to $\hat{n}$ and would vanish automatically. Therefore,
any additional structures will appear in the form of the inner product
$\hat{n}\cdot\hat{n}$ due to contractions with the Kronecker delta.
Consequently, they do not lead to anything new due to the normalization
condition $\hat{n}^{2}=1$.

We see that using the in-in formalism, the form of the damping equation and
the coefficients are completely fixed by the principles of EFT: the
symmetries, power counting, and Wilson coefficients.

##  4 More general materials

By choosing the spectral function to depend only on invariant tensors and the
frequency, we naturally arrive at the Gilbert damping Eq. (52) which applies
to isotropic and homogeneous systems. However, in generic materials, we may be
interested in situations with more general materials which for instance have
non-trivial spin textures or highly anisotropic lattices and thus
inhomogeneous or anisotropic. The advantage of the EFT framework is that these
generalizations can be systematically incorporated by including additional
couplings. In this section, we explore several possibilities along these
lines.

###  4.1 Anisotropic materials

For homogeneous systems, one can still have anisotropy due to a background
field. The retarded response function can then depend on the background. Given
a homogeneous background vector field $\vec{h}_{eff}$, the Levi-Civita tensor
can now be incorporated into the response $\rho^{AB}=\mathscr{D}\epsilon_{\ \
C}^{AB}h_{eff}^{C}$. Since dissipative effects need to be antisymmetric under
the simultaneous transformation $A\leftrightarrow B$,
$\omega\leftrightarrow-\omega$ and $A\leftrightarrow B$ antisymmetry is
already included in the Levi-Civita tensor structure, the response function
has to be symmetric under $\omega\leftrightarrow-\omega$. This means that the
leading order contribution is now independent of $\omega$. 111This is
analogous to the dissipative EFT of a spinning black hole in which case the
role of this background is played by the direction of the spin vector
GoldbergerLiRothstein . The corresponding response function is

$G_{R,diss}^{A,B}(t,\vec{x})=\mathscr{D}\epsilon^{ABC}h_{eff}^{C}\delta(t)\delta^{3}(\vec{x})$
(53)

and gives rise to the conservation equation

$\frac{\partial}{\partial
t}\vec{m}=-\lambda\vec{m}\times(\vec{m}\times\vec{h}_{eff}).$ (54)

For systems with conserved parity, the external field $\vec{h}_{eff}$ is an
effective magnetic field in the sense of being parity odd to ensure an even-
parity response function.

This damping equation involving an effective external background magnetic
field is known as the Landau-Lifshitz damping equation LandauLifshitz . We
observe that from the EFT point of view, the Landau-Lifshitz and Gilbert
damping terms are distinguished by symmetries.

For completeness, we note that in the literature there are generalizations to
Gilbert damping by introducing anisotropic damping tensors. In field theoretic
language, anisotropy corresponds to explicitly breaking of $SO(3)$ and is thus
straightforwardly realized in the EFT by a spurion condensation. Instead of
introducing a symmetry-breaking VEV to an explicit spurion operator in the
action, one may assume the 2-point functions acquire a VEV and the most
general resulting dissipative response function at LO is given by

$G_{R,diss}^{A,B}(\omega)=S^{AB}\omega+A^{AB},$ (55)

where $S^{AB}$ and $A^{AB}$ are general symmetric and antisymmetric tensors
which are not $SO(3)$ invariant. The generalized damping equation for an
anisotropic but homogeneous magnetic system then become

$\frac{\partial}{\partial
t}\vec{m}=-\vec{m}\times\bm{S}\cdot\frac{\partial}{\partial
t}\vec{m}+\vec{m}\times\bm{A}\cdot\vec{m}.$ (56)

The exact form of these anisotropy tensors can then be extracted from the
microscopic details of the given full theory.

###  4.2 Inhomogeneous materials

For inhomogenous materials, e.g. configurations with background spin textures,
we may have contributions from higher orders in (spatial) derivative
expansion. The simplest possible operator is given by the coupling

$S_{int}=\int d^{4}x\partial_{i}\chi_{a}\tilde{\mathcal{O}}^{ai}(X).$ (57)

In terms of the orientation vector, this is equivalent to

$S_{int}\approx\int
d^{4}x\left(\hat{n}\times\partial_{i}\hat{n}\right)\cdot\vec{\mathcal{O}}^{i}(X),$
(58)

at leading order in $\chi$-field.

Again promoting the spin rotation transformation
$\delta\hat{n}=\vec{\omega}\times\hat{n}$ to a local parameter
$\vec{\omega}(x)$, we find an additional contribution to the current
divergence

$\delta S_{int}=\int
d^{4}x\vec{\omega}\cdot\left(2\partial_{i}\hat{n}\hat{n}+\left(\hat{n}\hat{n}-\delta\right)\cdot\partial_{i}\right)\cdot\vec{\mathcal{O}}^{i},$
(59)

where we have used the fact that
$\hat{n}\cdot\partial_{i}\hat{n}=\frac{1}{2}\partial_{i}\hat{n}^{2}=0$ to
simplify the expression. For the spectral function of the form

$\rho^{iA,jB}(\omega)=\mathscr{E}\omega\delta^{ij}\delta^{AB},$ (60)

this leads to the damping term

$\displaystyle c_{2}\partial_{t}\hat{n}=$
$\displaystyle\mathscr{E}\left(2\partial_{i}\hat{n}\hat{n}\cdot\left(\partial_{t}\hat{n}\times\partial_{i}\hat{n}\right)+\left(\hat{n}\hat{n}-\delta\right)\cdot\partial_{t}\left(\hat{n}\times\vec{\nabla}^{2}\hat{n}\right)\right)$
$\displaystyle=$
$\displaystyle\mathscr{E}\left(2\hat{n}\times\left(\hat{n}\times\partial_{i}\hat{n}\right)\left(\hat{n}\times\partial_{i}\hat{n}\right)\cdot\partial_{t}\hat{n}+\left(\hat{n}\hat{n}-\delta\right)\cdot\partial_{t}\left(\hat{n}\times\vec{\nabla}^{2}\hat{n}\right)\right),$
(61)

where we have used that
$\hat{n}\times\left(\hat{n}\times\partial_{i}\hat{n}\right)=-\partial_{i}\hat{n}$
on the first term. More compactly, this is

$\frac{\partial}{\partial
t}\vec{m}=\vec{m}\times\mathbb{A}\cdot\frac{\partial}{\partial
t}\vec{m}+\frac{\mathscr{E}\left(\hat{n}\hat{n}-\delta\right)}{c_{2}m_{s}}\cdot\partial_{t}\left(\vec{m}\times\vec{\nabla}^{2}\vec{m}\right),$
(62)

where the first term contains the generalized damping tensor

$\mathbb{A}=\frac{2\mathscr{E}}{c_{2}m_{s}}\left(\vec{m}\times\partial_{i}\vec{m}\right)\left(\vec{m}\times\partial_{i}\vec{m}\right).$
(63)

This corresponds to the generalized Gilbert damping in the presence of non-
trivial spin textures (i.e. when $\nabla\vec{m}\neq 0$) ZhangZhang .

We note that Eq. (58) is only the leading order derivative correction to the
damping dynamics. The EFT framework is capable of systematically generating
higher derivative corrections. For example, other types of inhomogeneity may
be attributed to interactions in the lattice model BrinkerDiasLounis  by

$\vec{m}_{i}\times\sum_{ij}\mathcal{G}_{ij}\cdot\vec{m}_{j},$ (64)

where the $i,j$ indices label the lattice sites associated with the magnetic
moments. In the continuum field theory, the lattice variables become
$\vec{m}_{i}\mapsto\vec{m}(\vec{x}_{i})$ and the tensorial structure becomes
the response function $\mathcal{G}_{ij}\mapsto
G_{R,diss}(t,\vec{x}_{i}-\vec{x}_{j})$, except that unlike the one in Eq.
(50), it is non-local (no longer proportional to $\delta(\vec{x})$). In the
simpler case that the long-range coupling falls off sufficiently quickly,
these terms are traded for a series expansion

$\vec{m}\times\sum_{n}\mathcal{A}_{i_{1}\dots i_{n}}\partial_{i_{1}\dots
i_{n}}^{n}\partial_{t}\vec{m},$ (65)

for some coefficient tensors $\mathcal{A}_{i_{1}\dots i_{n}}$. In terms of the
action, this means the spectral functions are now dependent on the wave
vectors

$\rho^{AB}(\omega,\vec{k})=\sum_{n}\tilde{\mathcal{A}}_{i_{1}\dots
i_{n}}k_{i_{1}}\dots k_{i_{n}}\omega\delta^{AB}.$ (66)

##  5 Magnon damping term from phonons

From the magnetoelastic EFT, the generic magnon-phonon couplings are given by
PavaskarPencoRothstein

$\mathcal{L}_{ph}=-\frac{1}{2}F_{2}^{ij}(\nabla_{(i}\pi_{j)})\partial_{i}\hat{n}\cdot\partial_{j}\hat{n}+\frac{1}{2}\rho\tilde{F}_{3}(\nabla_{(i}\pi_{j)})D_{t}\hat{n}\cdot
D_{t}\hat{n},$ (67)

where $D_{t}\equiv\partial_{t}+v^{i}\partial_{i}$ with the velocity of the
material given by $v_{i}=-\partial_{t}\phi(D^{-1})^{\ i}_{j}$. The "full
theory" (technically the EFT at the next level of the hierarchy) action
constrains the form of the couplings between magnons the external sector to be

$\mathcal{L}_{ph}=\frac{1}{2}\partial_{i}\hat{n}\cdot\partial_{j}\hat{n}\mathcal{O}_{2}^{ij}(\pi)+\frac{1}{2}\partial_{t}\hat{n}\cdot\partial_{t}\hat{n}\mathcal{O}_{3}(\pi)+\frac{1}{2}\partial_{t}\hat{n}\cdot\partial_{i}\hat{n}\mathcal{O}_{4}^{i}(\pi),$
(68)

where the last term arises from the linear-in-$v$ contribution in the
expansion $D_{t}\hat{n}\cdot D_{t}\hat{n}$.

For ferromagnets, the dispersion relation dictates the first term to be
dominant. After integrating out the external sector using the Schwinger-
Keldysh method $\int\mathcal{D}\pi\mathcal{D}\tilde{\pi}$, we find its
contribution to the in-in equation of motion is given by

$c_{2}\partial_{t}\hat{n}=\hat{n}\times\partial_{i}\left(\partial_{j}\hat{n}\Bigr{\langle}\mathcal{O}_{2}^{ij}\Bigr{\rangle}\right),$
(69)

where the in-in expectation value is given by

$\Bigr{\langle}\mathcal{O}_{2}^{ij}(x)\Bigr{\rangle}=\int
d^{4}x^{\prime}\mathcal{G}_{R,2}^{ij,kl}(x-x^{\prime})\partial_{k}\hat{n}(x^{\prime})\cdot\partial_{l}\hat{n}(x^{\prime}).$
(70)

For 4-index tensor structures under $SO(3)$, there are two invariant tensors
corresponding to the symmetric-traceless and trace irreps. Therefore, one may
write the leading-order dissipative contribution to the retarded response
function as

$\mathcal{G}_{R,2}^{ij,kl}(x)\simeq\left(\frac{1}{2}\delta^{ij}\delta^{kl}\mathscr{C}_{2}+\delta^{i(k}\delta^{l)j}\mathscr{D}_{2}\right)\delta^{3}(\vec{x})\partial_{t}\delta(t),$
(71)

where $\mathscr{C}_{2}$ and $\mathscr{D}_{2}$ are the independent (Wilson)
coefficients.

Substituting the results, we find the damping equation in a similar form

$\displaystyle
c_{2}\partial_{t}\hat{n}=\frac{1}{2}\mathscr{C}_{2}\hat{n}\times\partial_{i}\left(\partial^{i}\hat{n}\partial_{t}\left(\partial_{j}\hat{n}\cdot\partial^{j}\hat{n}\right)\right)+\mathscr{D}_{2}\hat{n}\times\partial_{i}\left(\partial_{j}\hat{n}\partial_{t}\left(\partial^{i}\hat{n}\cdot\partial^{j}\hat{n}\right)\right)$
$\displaystyle=\mathscr{C}_{2}$
$\displaystyle\hat{n}\times\partial_{i}\left(\partial^{i}\hat{n}\partial^{j}\hat{n}\cdot\partial_{j}\partial_{t}\hat{n}\right)+\mathscr{D}_{2}\hat{n}\times\partial_{i}\left(\partial^{j}\hat{n}\partial^{i}\hat{n}\cdot\partial_{j}\partial_{t}\hat{n}\right)+\mathscr{D}_{2}\hat{n}\times\partial^{j}\left(\partial_{i}\hat{n}\partial^{i}\hat{n}\cdot\partial_{j}\partial_{t}\hat{n}\right),$
(72)

or more compactly

$\frac{\partial}{\partial_{t}}\vec{m}=\vec{m}\times\mathbb{D}\cdot\frac{\partial}{\partial_{t}}\vec{m},$
(73)

where the "damping tensor" $\mathbb{D}$ is given by

$\displaystyle\mathbb{D}=$
$\displaystyle\frac{1}{c_{2}m_{s}}\Big{[}\partial_{i}\left(\mathscr{C}_{2}\partial^{i}\hat{n}\partial^{j}\hat{n}+\mathscr{D}_{2}\partial^{j}\hat{n}\partial^{i}\hat{n}\right)+\mathscr{D}_{2}\partial^{j}\left(\partial_{i}\hat{n}\partial^{i}\hat{n}\right)$
$\displaystyle+\left(\mathscr{C}_{2}\partial^{i}\hat{n}\partial^{j}\hat{n}+\mathscr{D}_{2}\partial^{j}\hat{n}\partial^{i}\hat{n}\right)\partial_{i}+\mathscr{D}_{2}\partial_{i}\hat{n}\partial^{i}\hat{n}\partial^{j}\Big{]}\partial_{j}.$
(74)

We notice that the form of the couplings restricts the damping tensor to
appear at higher-orders in derivative expansions and hence they are expected
to be small compared with contributions from fermions (e.g. electrons) in the
long wavelength limit. However, for insulating materials that have electron-
magnon coupling suppressed, we expect their effects to be more significant.

##  6 Conclusion and discussions

In this paper, we used the in-in (Schwinger-Keldysh) formalism to generalize
the recently constructed EFT of magnetoelasticity PavaskarPencoRothstein  to
describe damped magnetic dynamics. We discover that the Gilbert damping term
naturally arises as the simplest symmetry consistent dissipative correction
within the in-in formalism. Systematic generalizations to anisotropic and
inhomogeneous setups also yield desired results such as the Landau-Lifshitz
magnetic damping equation. Moreover, we are able to predict the form of
phononic contribution to the damping dynamics. Thus we have shown that this is
a useful framework to derive dissipative dynamics from first principles and to
predict the forms of higher-order corrections in a systematic way.

It would be interesting to investigate the explicit full theory model of the
“external sector” such as the fermionic fields in Eq. (35) and extract the
relevant Wilson coefficients by matching the response functions. In this way,
we may gain better insights into what controls the damping parameter and give
more predictive power to the EFT approach. It would also be interesting to
match the relevant coefficients in Eq. (66) to obtain an EFT framework for a
generalized class of models.

Furthermore, various applications of the magnetoelastic EFT EspositoPravaskar
; PavaskarRothstein  have appeared more recently. It would be interesting to
investigate the effects of adding dissipative terms into these problem. There
are also further developments in the technical aspects of such EFTs
NicolisRothstein ; AkyuzGoonPenco . It is natural to consider their
implications on the non-conservative sector. We leave these problems for
future works.

## Acknowledgement

The author thanks Ira Rothstein for advising throughout the project and a
careful reading of the manuscript. The author also thanks Riccardo Penco for
important discussions, Shashin Pavaskar for other useful discussions, and
Witold Skiba for comments on the draft. This work is partially supported by
the grants DE- FG02-04ER41338 and FG02- 06ER41449.

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set of cuts as we used in the CMS MS analysis for displaced jets ($D_{jj}$),
along with the same cuts for the prompt associated objects. We change the
pseudo-rapidity coverage of the FCC-hh MS for both the barrel-endcap and
forward regions as described in the beginning of this section.

Figure 33: Left: Efficiency of various cuts with varying decay lengths for the
10 GeV mediator particle in the FCC-hh barrel and endcap MS along with the
corresponding efficiencies obtained in the CMS MS. Right: Comparison of the
normalised distributions of $\Delta\phi_{\rm max}$ for two decay lengths for
the 10 GeV mediator particle in the 14 TeV and 100 TeV colliders.

To understand how the same set of cuts affect the signal at the 100 TeV
experiment, we revisit the efficiency plot for the 10 GeV mediator particle in
the ggF production mode. The left panel of Fig. 33 shows the efficiencies of
the $n_{\rm dSV}^{\rm ch}$, $\sum p_{T,\,dSV}$ and $\Delta\phi_{\rm max}$ cuts
on the signal. We find on comparing these with the corresponding efficiencies
at the 14 TeV with the one we get at the 100 TeV colliders, the latter has
reduced by a factor of about half. The decomposition of the efficiency in
individual components of the cuts show that this reduction in the efficiency
is mostly due to the $\Delta\phi_{\rm max}$ cut. The left panel of the figure
shows the $\Delta\phi_{\rm max}$ distribution for two benchmark decay lengths
of 0.1 m and 10 m for the mediator particle of 10 GeV mass at the 14 TeV and
100 TeV collider experiments. We find that the 100 TeV distributions fall off
faster than the ones at the 14 TeV, which explains the reduced efficiency.

Figure 34: Comparison of the efficiencies as a function of decay lengths, as
obtained from the $D_{jets}^{S}\geq 1$ vtx (top) and $P_{\rm ggF}^{S}\times
D_{jets}^{S}\geq 1$ vtx (bottom) sets of cuts between the CMS MS at 14 TeV and
FCC-hh MS at 100 TeV, when a mediator of mass 10 GeV (left) and 50 GeV (right)
decays to a pair of $b$ quarks. For the FCC-hh detector, we also show the
efficiencies for the barrel and endcap MS, and the forward MS separately.

Fig. 34 shows the effect of including the forward MS in our analysis observing
displaced MS clusters for two sets of cuts $-$ $D_{jets}^{S}\geq 1$ vtx and
$P_{\rm ggF}^{S}\times D_{jets}^{S}\geq 1$ vtx, for Higgs boson produced in
ggF mode. We find that for both the sets of cuts, the forward MS increases
sensitivity to lower decay lengths, which is explained by revisiting Fig. 13,
where we found that the dSV cut selects more forward LLPs with lower decay
lengths, which can now pass the $\eta$ condition with the forward MS. The
combined efficiencies of barrel-endcap and forward MS is almost similar to the
14 TeV efficiencies for the 50 GeV mediator, however, it is reduced for the 10
GeV mediator, predominantly due to the $\Delta\phi_{max}$ cut, as discussed
earlier.

Figure 35: Comparison of the projected upper limits on the branching fraction,
Br$(h\to\phi\phi)$, for 50 observed decays of long-lived mediator particles
into a pair of $b$ quarks within the CMS and the FCC-hh MS for the
$P^{S}\times D^{S}_{jj}\geq$ 1 vtx set of cuts. The shown limits are obtained
by combining the ggF, VBF and Vh modes for the Higgs boson production.

Nevertheless, the 100 TeV collider experiment gains from increased cross
section as well as luminosity. The cross section for the ggF production is
increased from around 50 pb to 740 pb (enhancement of a factor of $\sim 15$),
and it is expected to collect data with an integrated luminosity of 30 ab-1 (a
factor of 10 increase compared to the HL-LHC collider experiment). We,
therefore, expect the projected sensitivity to increase by a factor of around
$\sim 150$, however, the reduction in efficiency costs us a factor of half and
$0.84$ for the $D^{S}\geq 1$ vtx and $P^{S}\times D^{S}\geq 1$ vtx sets of
cuts respectively for the 10 GeV mediator. We are therefore able to improve
the 14 TeV CMS results by a factor of $\sim$ 120-140 with the $P^{S}\times
D^{S}\geq 1$ vtx cuts for mediator mass of 10 GeV. For the 50 GeV mediator,
there is not much reduction in the efficiencies compared to the 14 TeV
results, as can also be seen from Fig. 34, since it has lesser boost and
effect of $\Delta\phi_{max}$ cut is small $-$ for the $D^{S}\geq 1$ vtx cuts,
we find a reduction by a factor of 0.82 and for the $P^{S}\times D^{S}\geq 1$
vtx cuts, we have a slight improvement by a factor of 1.08. Fig. 35 compares
the 14 TeV HL-LHC CMS limits with the 100 TeV FCC-hh limits on the branching
fraction, Br$(h\to\phi\phi)$, for mediator particle masses of 10 GeV and 50
GeV. This shows the results when all the production modes are combined, and in
each of them, the $P^{S}\times D^{S}_{jj}\geq$ 1 vtx set of cuts are used. We
find the results to improve by around a similar factor which we have just
estimated above.

Figure 36: Projected upper limits on the branching fraction,
Br$(h\to\phi\phi)$, for 50 observed decays of long-lived mediator particles
into $b$ quarks within the FCC-hh MS at the 100 TeV collider experiment for
the four sets of cuts explained in the text. The shown limits are obtained by
combining the ggF, VBF and Vh production modes of the Higgs boson.

Figure 37: Projected upper limits on the branching fraction,
Br$(h\to\phi\phi)$, for 50 observed decays of long-lived mediator particles
into $c$ quarks within the FCC-hh MS at the 100 TeV collider experiment for
the four sets of cuts explained in the text. The shown limits are obtained by
combining the ggF, VBF and Vh production modes of the Higgs boson.

Figs. 36 and 37 show the projected upper limits on the branching fraction of
the Higgs boson into a pair of the long-lived mediator particles,
Br$(h\to\phi\phi)$, which further decays to a pair of $b$ and $c$ quarks
respectively, for all the four combinations of cuts as described earlier, for
a range of mass and decay length values, using the combination of barrel-
endcap and forward MS regions. For decay to $b$-quarks ($c$-quarks), we
achieve a limit of Br$(h\rightarrow\phi\phi)<1.8\times 10^{-7}$ ($2.1\times
10^{-1}$) for 10 GeV mediator at $c\tau=1$ m and
Br$(h\rightarrow\phi\phi)<1.8\times 10^{-7}$ ($2.1\times 10^{-1}$) for 50 GeV
mediator at $c\tau=5$ m, when we use the $D^{H}\geq 1$ vtx set of cuts,
closely followed by limits from $P^{S}\times D^{S}\geq 1$ vtx, which increase
by a factor of around $2$.

Figure 38: Efficiencies of different $\sum p_{T}$ cuts, with and without the
$\Delta\phi_{max}>0.2$ cut in the FCC-hh barrel and endcap MS with varying
decay lengths for the 10 GeV mediator particle.

In the 100 TeV collider, a displaced activity in the MS with energy deposition
of 20 GeV might not be enough to select such events maintaining reasonable
background rates, and we might need to increase this energy threshold. In Fig.
38 shows the efficiencies of different $\sum p_{T}$ cuts, with and without the
$\Delta\phi_{max}>0.2$ cut in the FCC-hh barrel and endcap MS with varying
decay lengths for the 10 GeV mediator particle, produced from the decay of
Higgs boson in the ggF production mode. The $n_{dSV}^{ch}\geq 3$ cut is
imposed for each of the curves shown in Fig. 38. For the $\sum p_{T}>20$ GeV
cut used along with the $\Delta\phi_{max}>0.2$ cut, we have already seen from
the left panel of Fig. 33, a reduction by a factor of 0.4 compared to the 14
TeV efficiency at $c\tau=1$ m. Increasing the $\sum p_{T}$ cut to 50 GeV, 75
GeV and 100 GeV reduces the efficiencies by factors of around 0.23, 0.07 and
0.03 respectively, when the $\Delta\phi_{max}>0.2$ cut is also imposed. We
expected an improvement by a factor of 150 in the projected limits in going
from the 14 TeV to 100 TeV collider provided the efficiencies remained the
same. With the computed efficiency of the $\sum p_{T}>100$ GeV (50 GeV) cut,
we can improve the results only by a factor of $\sim$4.5 (34.5) due to the
loss in efficiency. The future detectors can be expected to have a higher
granularity that can help in relaxing the $\Delta\phi_{max}$ cut and regaining
the improvement in sensitivity. Removing the $\Delta\phi_{max}$ cut, we can
get an improvement of $\sim$ 24 with the $\sum p_{T}>100$ GeV cut, and gain
the full improvement by a factor of $\sim$150 with the $\sum p_{T}>50$ GeV
cut. These can be kept in mind while finalising the design as well as the
analysis strategies for the FCC-hh detector.

###  4.2 DELIGHT: proposal of dedicated LLP detectors at the 100 TeV collider

Similar to the multitude of proposals of dedicated LLP detectors discussed in
the previous section for the HL-LHC experiment, we can also place such
dedicated decay volumes near main general-purpose detectors for the 100 TeV
collider experiment to increase the sensitivity to larger decay length of
LLPs. The advantage of designing such a dedicated detector in the periphery of
the future 100 TeV collider is that the collider, as well as the detectors,
are not yet constructed. Therefore, we can optimise the position as well as
the size of the detector to maximise its sensitivity, rather than finding
empty spaces near the various IPs to place and fit the LLP detectors for the
HL-LHC experiment. This gives us the scope to vary over the various parameters
which specify the position and size of the decay volume of the dedicated LLP
detector to find the one which gives the best sensitivity over a large range
of LLP masses and decay lengths, also being the most ergonomic among them, as
we are not limited by already occupied spaces. We here propose few designs of
a dedicated LLP detector, DELIGHT (Detector for long-lived particles at high
energy of 100 TeV), which is a box-type detector in the periphery of the 100
TeV collider and present their sensitivities.

Figure 39: Illustration defining the parameters denoting the position and size
of the decay volumes for DELIGHT detector, a box-type LLP detector near the IP
of the 100 TeV collider.

Before specifying the dimensions and position of proposed LLP detector for the
100 TeV collider experiment, we define the coordinate system that we use as
follows:

IP: Origin x: Towards the centre of the collider ring y: Vertically upwards z:
Along the beam pipe

We can choose from a box-type geometry such as the MATHUSLA and CODEX-b
detectors, or a cylindrical geometry such as the FASER and ANUBIS detectors.
In this paper, we focus on the former. Then, there are six parameters
specifying the box, which we denote as $(x_{1},y_{1},z_{1})$ and $(\Delta
x,\Delta y,\Delta z)$ denoting the coordinates of the starting position of the
box and the sizes in each direction, respectively. Fig. 39 shows the
illustration of such a box-type detector with the parameters denoting the
position and decay volume of the detector. Since the distribution of the LLP
is uniform in the azimuthal ($\phi$) direction, we can fix it without
affecting the sensitivity, which leaves us with five free parameters. We fix
$y_{1}=0$, implying the detector is horizontal to the collider detector with
$\phi=0$. As we have earlier seen from the pseudo-rapidity distributions of
the mediator particle, they are mostly central for lighter masses. Therefore,
in order to increase the sensitivity, we propose to place the detector such
that it is centered at the IP, i.e. $z_{1}=-\Delta z/2$. The remaining free
parameter that defines the distance of the detector from the IP is $x_{1}$.
The closer the detector is placed near the IP, the sensitivity will be, of
course, more. However, we need to keep it at a distance to provide sufficient
shielding. We therefore choose $x_{1}=$ 25 m motivated from the CODEX-b
detector. Revisiting the proposed design of the FCC-hh detector, we find that
the collider detector extends till $R=9$ m and will mostly have concrete and
shafts. A position starting at around 25 m in the $x$-direction around
$\eta=0$ region can, therefore, be kept empty for placing a dedicated LLP
detector. We understand that the 100 TeV collider will have very high
luminosity, and a distance of 25 m might not seem enough to provide a
background free environment. However, since it is still in the designing
phase, we might be able to use materials that can provide significant
shielding even with a smaller distance from the IP. Moreover, we will be able
to use some active veto components in between the IP and the detector, such as
the one proposed for the CODEX-b detector Gligorov:2017nwh ; Aielli:2019ivi
in order to further reduce backgrounds. Nevertheless, it should be better to
treat our limits as optimistic ones.

Fixing the position of the detector as above, we study three benchmark sizes:

DELIGHT (A): The same as the dimensions of the MATHUSLA detector,
$\displaystyle{\rm i.e.}\,\Delta x\times\Delta y\times\Delta z=25\times
100\times 100\,{\rm m^{3}}.$ DELIGHT (B): Four times bigger than the MATHUSLA
detector, $\displaystyle{\rm i.e.}\,\Delta x\times\Delta y\times\Delta
z=100\times 100\times 100\,{\rm m^{3}}.$ DELIGHT (C): The same decay volume as
the MATHUSLA detector with $\displaystyle\text{different dimensions,
i.e.}\,\Delta x\times\Delta y\times\Delta z=200\times 50\times 50\,{\rm
m^{3}}.$

Figure 40: Projected upper limits on the branching fraction,
Br$(h\to\phi\phi)$, for 4 observed decays of long-lived mediator particles
within DELIGHT (A), (B) and (C) at the 100 TeV collider experiment which are
described in the text. The shown limits are obtained by combining the ggF, VBF
and Vh production modes of the Higgs boson.

The detector volume requires elements for detecting charged decay products of
the LLP with good timing and spatial resolutions. The current technology of
RPCs can achieve a timing resolution of 350 ps per single gas-gap of 1 mm,
which can be further improved by combining several such layers Aielli:2019ivi
. We can also achieve a spatial resolution of at least 0.5 cm Bauer:2019vqk .
Since the long-lived particle can decay to a wide variety of final states, the
presence of some calorimeter can aid in the identification of the particles,
which can enhance the sensitivity of these detectors, extending the search to
LLPs decaying to photons. calorimeters will also enable mass reconstruction,
which can help in reducing the SM background coming from long-lived particles
like $K_{L}$. Placing our detector in the central region also has the added
advantage of reduced backgrounds as compared to the forward direction. The
possibility of integrating the detector with the FCC-hh trigger system can
also be explored since we are proposing it to be placed just 25 m away from
the IP.

Fig. 40 shows the projected upper limits on the branching fraction,
Br$(h\to\phi\phi)$, for 4 observed decays of long-lived mediator particles
within the three decay volumes of the DELIGHT detectors as described above.
Comparing the results from DELIGHT (A) with that of MATHUSLA, we observe an
improvement by a factor of $\sim 540$, of which around $\sim 150$ is accounted
by the increased cross-section and integrated luminosity. Another factor of
$\sim$ 3–4 is gained in the efficiency by moving the detector close to the IP.
The best limits come from DELIGHT (B), which has the highest decay volume
among the three (about four times bigger than the decay volume of MATHUSLA),
and the performance is better by a factor of 2 compared to DELIGHT (A).

DELIGHT (A), and (C) have the same decay volumes, however their different
dimensions lead to different geometric acceptances. Here, the
$\Delta\eta\times\Delta\phi$ coverage of the detector is determined by its
$\Delta y\times\Delta z$, which is smaller for DELIGHT (C) compared to DELIGHT
(A). Reducing $\Delta y$ makes sure that it has some shielding from cosmic
rays, and therefore, this background is easier to be handled. Its detector
volume is compensated by an increased $\Delta x$, and it is like a long tunnel
along the line towards the centre of the collider ring. The limits obtained
from DELIGHT (C) are smaller than that from DELIGHT (A) by a factor of around
$0.8-0.9$. Therefore, if the dimensions of DELIGHT (C) is more ergonomic and
feasible than DELIGHT (A), since the former has smaller $\Delta y\times\Delta
z$, we do not lose on the sensitivity much.

##  5 Summary and conclusion

Motivated by the importance of a light scalar mediator particle connecting
between the dark sector and the SM and also driven by the fact that the mixing
of such a particle with the SM Higgs boson has been constrained to be small by
the LHC experiment, we take a closer look into the long-lived parameter space
of such a mediator particle in this paper. The summary of our work with the
key findings are listed below:

  * •

We have focused on the production of the mediator from the on-shell decay of
the SM Higgs boson and have considered the dominant production modes of the
latter in hadron colliders, viz., ggF, VBF and Vh. We also consider a variety
of decay modes of the mediator particle $-$ muons, pions, Kaons, gluons, $s$
quarks, $c$ quarks, $\tau$ leptons and $b$ quarks. We have examined in this
work how the interplay between a wide variety of production and decay modes of
the long-lived scalar mediator affects the search sensitivity of such a
particle.

  * •

As reasoned in the text earlier, we focus on displaced activities in the CMS
MS detector at the HL-LHC experiment, and use a combination of cuts on the
prompt particles coming from production modes of the Higgs boson and the
displaced SM particles coming from the mediator particle decay for the
analyses. We use trigger cuts proposed for the Phase-II HL-LHC runs, and relax
them when an additional displaced or prompt activity is demanded. Prospects of
using such hybrid cuts could prove very beneficial for designing LLP analyses.

  * •

We begin by presenting the analyses for each of the decay modes individually,
assuming 100% branching ratio to that particular mode. Such a model
independent depiction of the result presented in this work can be translated
and used by various models as well. We present our results in the form of
grids in the mass and $c\tau_{\phi}$ plane, with the exact value of the
projected upper limits on the branching fraction, Br$(h\to\phi\phi)$,
available as well, which is easier to recast than reading from the color plot.
Although we do not quantify the background in this paper, we present a
discussion on some of the possible backgrounds that can arise in the CMS MS
search along with ways of reducing them, and elucidate the minimum number of
observed events we should use for setting the limits.

  * •

Next, we combine the decay modes according to the branching ratio predicted in
the minimal model, and present the results in the $(m_{\phi},\,c\tau_{\phi}$)-
and $(m_{\phi},\,\sin\theta)$-planes. The limits obtained from the displaced
di-muons analysis are at least one order of magnitude better than the ones
obtained for displaced activities in the muon spectrometer from mesons like
pions and Kaons, gluons, and various flavors of quarks. We can achieve a
sensitivity of Br$(h\rightarrow\phi\phi)<1.8\times 10^{-5}$ for 60 GeV
mediator at decay length of 5 m with the $P^{S}\times D^{S}\geq 1$ vtx set of
cuts, described in Section 3.1.2.

  * •

In addition, we study the contribution of dedicated LLP detectors such as the
MATHUSLA and CODEX-b detectors in the search for long-lived scalar mediator
particles coming from the decay of the Higgs boson produced in either of ggF,
VBF or Vh production modes, and their complementarity with the CMS MS search.
We observe that CMS and MATHUSLA together can probe decay lengths till 105 m
for a mediator of 60 GeV mass, without any gap if the branching of Higgs boson
to the mediator particles is at least $\sim$0.1%.

  * •

We study the sensitivity of the muon spectrometer of the proposed 100 TeV
hadronic collider experiment, FCC-hh, and investigate the effect of including
the forward MS of the FCC-hh detector to the barrel and endcap MS stations of
the detector. Apart from the increased cross-section and integrated
luminosity, we discuss how the signal efficiency is affected for the FCC-hh
detector design. The forward MS increases the efficiencies by around 20% for
the displaced di-muons analysis, and opens up sensitivity to lower decay
lengths for the analyses with displaced activities in the muon spectrometer.
The factor responsible for this is the more forward distribution of long-lived
mediators with higher mass and lower decay lengths once the dSV is restricted
within the MS, which are not missed by the forward MS. This is discussed in
detail in the text.

  * •

We propose detector designs for dedicated LLP searches in the periphery of the
100 TeV FCC-hh collider, named DELIGHT. Since the designs for the 100 TeV
collider are under development, we have the freedom to survey the nearby
regions for identifying the positions and sizes of detectors that will be more
effective for LLP signals. Motivated from the distributions of direction of
the scalar mediator particle produced from the decay of the Higgs boson in the
colliders, we investigate the prospects of three benchmarks of detector
designs. Our study suggests that these can provide an improvement by a factor
of 500 or more compared to the MATHUSLA detector.

In conclusion, we believe that our detailed analysis and findings will be
useful for the a range of LLP models. Our analyses for the future 100 TeV
collider can be a starting point for further detailed studies exploring their
prospects for long-lived particles. The proposal put forth by us in this work
for the DELIGHT detector near the FCC-hh collider shows promising results,
motivating further studies in this direction for other LLP models.

## Acknowledgement

BB acknowledges the hospitality of Kavli IPMU, Japan, where the project was
initiated. The work of SM is supported by Grant-in-Aid for Scientific Research
from the Ministry of Education, Culture, Sports, Science, and Technology
(MEXT), Japan (17H02878, 20H01895, 19H05810, 20H00153), by World Premier
International Research Center Initiative (WPI), MEXT, Japan (Kavli IPMU), and
also by JSPS Core-to-Core Program (JPJSCCA20200002). BB and RS thank Prabhat
Solanki and Rahool Kumar Barman for useful discussions.

##  Appendix A Dedicated LLP detectors at HL-LHC - validation, efficiency maps

Figure 41: Validation of the CODEX-b (left) and MATHUSLA (right) detectors
Gligorov:2017nwh ; Alpigiani:2020iam .

Figure 42: Efficiency maps for decays of long-lived mediators within CODEX-b
for the ggF (top), VBF (center) and Vh (bottom) production of SM Higgs.

Figure 43: Efficiency maps for decays of long-lived mediators within MATHUSLA
for the ggF (top), VBF (center) and Vh (bottom) production of SM Higgs.

##  Appendix B Efficiency maps of DELIGHT detector

Figure 44: Efficiency maps for decays of long-lived mediators within DELIGHT
(A) near the 100 TeV collider for the ggF (top), VBF (center) and Vh (bottom)
production of SM Higgs boson.

Figure 45: Efficiency maps for decays of long-lived mediators within DELIGHT
(B) near the 100 TeV collider for the ggF (top), VBF (center) and Vh (bottom)
production of SM Higgs boson.

Figure 46: Efficiency maps for decays of long-lived mediators within DELIGHT
(C) near the 100 TeV collider for the ggF (top), VBF (center) and Vh (bottom)
production of SM Higgs boson.

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