title section subsection

#  TTP-2020-016, P3H-20-014
A Study of New Physics Searches with Tritium and Similar Molecules

Wolfgang Gregor Hollika,b Matthias Linsterb Mustafa Tabetb
aInstitute for Nuclear Physics, Karlsruhe Institute of Technology,
D-76021 Karlsruhe, Germany
bInstitute for Theoretical Particle Physics, Karlsruhe Institute of
Technology,
D-76128 Karlsruhe, Germany

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###### Abstract

Searches for New Physics focus either on the direct production of new
particles at colliders or at deviations from known observables at low
energies. In order to discover New Physics in precision measurements, both
experimental and theoretical uncertainties must be under full control. Laser
spectroscopy nowadays offers a tool to measure transition frequencies very
precisely. For certain molecular and atomic transitions the experimental
technique permits a clean study of possible deviations. Theoretical progress
in recent years allows us to compare _ab initio_ calculations with
experimental data. We study the impact of a variety of New Physics scenarios
on these observables and derive novel constraints on many popular generic
Standard Model extensions. As a result, we find that molecular spectroscopy is
not competitive with atomic spectroscopy and neutron scattering to probe new
electron-nucleus and nucleus-nucleus interactions, respectively. Molecular and
atomic spectroscopy give similar bounds on new electron-electron couplings,
for which, however, stronger bounds can be derived from the magnetic moment of
the electron. In most of the parameter space H2 molecules give stronger
constraints than T2 or other isotopologues.

###### Contents

  1. 1 Introduction
  2. 2 Spectroscopy of Molecular Hydrogen and its Isotopologues
    1. 2.1 A Brief Review of the Current Theoretical Status
    2. 2.2 Experimental Status
  3. 3 New Physics Coupled to Electrons and Nuclei
    1. 3.1 Implementation of the New Physics Corrections
    2. 3.2 Scalar and Pseudoscalar Potentials
    3. 3.3 Vector and Axialvector Exchange Potentials
    4. 3.4 Singular Potentials: Effective Contact Interactions
  4. 4 Conclusions
  5. A Experimental data

##  1 Introduction

So far, no heavy new particles beyond those of the Standard Model of
elementary particle physics have been found. There is, however, observational
evidence of physics not covered in the Standard Model. In the context of Dark
Matter, for example, the interest to search for new sub-$\mathrm{GeV}$
particles has recently gained impetus [1, 2, 3], where molecules have been
identified as good study objects [4]. In particular, molecular spectroscopy is
one possibility to look for new dark forces [5, 6]. Although the laws of
Quantum Mechanics as the physical framework at molecular scales are well
established, it is intrinsically difficult to provide reliable precise
predictions. Theoretical calculations are challenging, CPU-intensive, and
potentially lacking important higher-order contributions that might have been
neglected. Nevertheless, there are precise state-of-the art predictions for
hydrogen-like molecules that can be exploited for a dedicated analysis of New
Physics effects.

Following the early groundbreaking works of Kołos and Wolniewicz in the 1960s
[7, 8, 9, 10, 11], a vast progress in the theoretical determination of energy
levels of hydrogen-like molecules has been made during the last decade. The
crucial improvement was a clear conceptional separation of electronic and
nuclear motion developed in form of nonadiabatic perturbation theory by
Pachucki and Komasa [12, 13, 14, 15]. With this method, a full nonadiabatic
treatment of the system is performed. Furthermore, leading [16, 17, 18] and
higher-order [19] QED corrections have been implemented, as well as
relativistic corrections [20, 21]. Precise theoretical predictions of
rovibrational lines for hydrogen isotopologues have been made publicly
available in the computer code H2Spectre [22, 23] and are nicely reviewed in
Reference [24].

On the experimental side, improved techniques allow for more precise
measurements of these spectra, testing the theoretical prediction to
unprecedented accuracy. As a consequence of the agreement between theory and
experiment, severe constraints can be put on any deviation of known physics
[25, 26, 27].

The simplest modification of the well-known Coulomb potential $V_{\mathrm{C}}$
for the interaction between two point charges $q_{1,2}$ which accounts for
light New Physics is given by the addition of a Yukawa-type potential
$V_{\mathrm{Y}}$,

$V_{\mathrm{int}}(r)=V_{\mathrm{C}}(r)+V_{\mathrm{Y}}(r)=\frac{\alpha_{\mathrm{em}}}{r}\left(q_{1}q_{2}+\frac{g_{\text{NP}}}{4\pi\alpha_{\mathrm{em}}}\exp(-mr)\right)\,,$
(1)

where $\alpha_{\mathrm{em}}\simeq 1/137$ is the electromagnetic fine structure
constant. In this way, a new light particle coupling to known physics leads to
a potential which is exponentially suppressed by this particle’s mass $m$ and
is proportional to the coupling strength $g_{\text{NP}}$. Note that the
coupling strength $g_{\mathrm{NP}}$ might have both signs depending on the
interacting particles. A brief comparison of units yields a mass of
$\order{$\mathrm{keV}$}$ for a typical bond length
$\sim$1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ of a light molecule. For
larger masses the exponential term in $V_{\mathrm{Y}}$ drops too fast to have
an effect on molecular distances, while for lower masses the Yukawa term is
too long-ranged to be distinct from a Coulomb potential and, hence, redefines
$\alpha_{\mathrm{em}}$ by a constant shift [28].

In the simplest case, a Yukawa-like potential as shown in Equation (1) might
originate from a light scalar exchange between two bound fermions but may also
appear as the leading contribution from spin-dependent potentials [29, 30].
Many models include such light particles as carriers for weak long-range
forces, for instance an additional light Higgs Boson as a scalar mediator [31,
32], axions [33] and axion-like particles (ALPs) [34] as examples for a
pseudoscalar exchange, or Dark Photons as a vector particle [35, 36, 28].

The low-energy regime has already been explored in other experiments, for a
review see Reference [37]. Precision QED tests can be performed with atomic
spectroscopy, for example in highly excited Rydberg atoms or isotope shifts in
singly ionized divalent elements [38, 39, 40]. While these measurements give
slightly better constraints than molecular spectroscopy, long-distance inter-
nuclear interactions can only be tested in molecules—although neutron
scattering might be more competitive [41]. Additionally, atomic precision
tests for light scalar couplings have been considered [42], where light ALPs
modify the Coulomb potential by a screening effect and may have impact on the
Lamb Shift in atomic hydrogen [43]. However, there are competing laboratory
techniques with higher sensitivity as pointed out in Reference [43] and by
further dedicated studies on atomic spectroscopy [38, 39, 40]. Especially in
the mass regime below several $\mathrm{MeV}$, there are stringent indirect
constraints from astrophysics [44, 45, 46] and cosmology [47]. With atomic and
molecular spectroscopy, new forces at the $\mathrm{keV}$ scale can be probed
directly by the single-particle interaction in contrast to multi-particle
coherent effects in massive objects.

Alternatively, the New Physics contribution might be interpreted as a test of
gravity and/or deviations from known gravitational interactions, often called
“ _fifth forces_ ” in the literature. Such _fifth forces_ may have different
origin like extra dimensions where the mass parameter $m$ corresponds to the
size of the extra dimension. A tight upper bound on the New Physics coupling
of $g_{\text{NP}}\lesssim 10^{-34}$ can be derived from test of the
gravitational inverse-square law [48, 49, 50, 51]. However, these contraints
apply only to mediator masses up to the $\mathrm{meV}$ regime. In that
respect, atomic and molecular spectroscopy are highly relevant as probes of
new forces in the $\mathrm{\SIUnitSymbolAngstrom}$-regime corresponding to
masses of $\order{$\mathrm{keV}$}$. Such constraints on _fifth forces_ from
molecular spectroscopy have been derived for a Yukawa-like interaction between
nuclei in References [25, 26]. Similarly, _fifth force_ experiments lead to
constraints on light scalars coupled to photons [52].

In this work, we study the impact of New Physics potentials on molecular
spectroscopy of the hydrogen isotopologues H2, D2, T2, HD, DT, and HT. First,
we briefly review the current status of molecular spectroscopy from a
theoretical and experimental perspective in Section 2. Next, we show the
constraints resulting from each type of new interaction in Section 3. Finally,
we conclude in Section 4.

##  2 Spectroscopy of Molecular Hydrogen and its Isotopologues

Atomic and molecular spectroscopy have become fields of research at the
precision frontier. Unlike atoms, diatomic molecules contain a second nucleus
leading to vibrational and rotational excitations of the whole molecule.
Therefore, the spectral lines present in molecules are rather bands comprising
many single lines characterized by vibrational and rotational quantum numbers
$v$ and $J$, respectively. The distance of the individual lines within one
band is much smaller than spectral lines of electronic transitions. As a
consequence, molecular spectra have a richer structure and are sensitive to
phenomena at much smaller energies compared to atomic spectra.

In particular, the accurate determination of rovibrational transitions probes
the internuclear potential and new interactions among electrons and nuclei.
These transitions are classified by the change $\Deltaup J$ in the angular
momentum quantum number $J$ for a certain vibrational transition, where $Q$,
$R$, and $S$ branches refer to $\Deltaup J=0,1,2$, respectively. Thus, the
vibrational transition can be determined with increased statistical
significance by probing such a branch for different values of the angular
momentum $J$. This is especially advantageous for New Physics searches where
effects are rotationally invariant so that lines for different angular
momentum values comprise the same New Physics effect.

Molecular hydrogen and its isotopologues have been thoroughly studied
nowadays. For instance, the fundamental vibrational line corresponding to the
$v=1\rightarrow 0$ transition has been observed in the $Q$ branch, i. e. in
$\Deltaup J=0$ transitions, with high accuracy for H2, HD, and D2 [53, 54,
55]. Moreover, the world’s best spectra of molecules containing Tritium have
been recently obtained using Coherent Anti-Stokes Raman Scattering
Spectroscopy (CARS) for T2 and DT [56, 57, 58]. A relative precision of up to
$\order{${10}^{-10}$}$ has been reached in these measurements. Remarkably,
theory predictions are able to match the experimental sensitivity although
becoming more complex.

In the following, we briefly review the current status of theory calculations
in Section 2.1 and of experimental measurements in Section 2.2.

###  2.1 A Brief Review of the Current Theoretical Status

A full theoretical treatment of the Hydrogen molecule H2 as a four-particle
system is intrinsically difficult. First approaches date back to the 1920s and
have been developed independently by Heitler and London [59], and Born and
Oppenheimer [60]. The key part of the Born–Oppenheimer approximation is that
it is a formal expansion in the small ratio of electron over nucleus mass in
powers of the _4th root_ $\sqrt[4]{m_{\mathrm{e}}/m_{\mathrm{N}}}$, while
Heitler and London neglected the motion of the nuclei in the Hamiltonian. This
effect can be included in the adiabatic approximation using perturbation
theory [61]. A consequent _non-adiabatic_ treatment takes the movement of the
nuclei into account in order to calculate the energy levels of the whole
system [8].

Assuming the nuclei to be at fixed positions $\mathbf{R}_{\mathrm{A}}$ and
$\mathbf{R}_{\mathrm{B}}$, the Hamiltonian for this system reads [59]

$H_{\mathrm{el}}=\frac{\mathbf{P}_{1}^{2}}{2m_{\mathrm{e}}}+\frac{\mathbf{P}_{2}^{2}}{2m_{\mathrm{e}}}+\alpha_{\mathrm{em}}\left\\{-\frac{1}{r_{1\mathrm{A}}}-\frac{1}{r_{2\mathrm{B}}}+\left(\frac{1}{r_{12}}+\frac{1}{R_{\mathrm{AB}}}-\frac{1}{r_{1\mathrm{B}}}-\frac{1}{r_{2\mathrm{A}}}\right)\right\\}\,,$
(2)

with the electromagnetic fine structure constant $\alpha_{\mathrm{em}}\simeq
1/137$ and the distances $r_{12}$ and $R_{\mathrm{AB}}$ between the two
electrons $1$ and $2$ and nuclei A and B, respectively. Correspondingly,
$r_{iX}$ denotes the separation of electron $i$ from nucleus $X$, with $i=1,2$
and $X=\mathrm{A},\mathrm{B}$. The Schrödinger equation for this Hamiltonian
(2) is usually solved using a variational method with a trial wave function
$\psi_{\mathrm{el}}(\mathbf{r}_{1},\mathbf{r}_{2};\mathbf{R}_{\mathrm{A}},\mathbf{R}_{\mathrm{B}})$
expanded in a suitable basis. In the case of the hydrogen ground state,
precise results can be obtained using the symmetric James–Coolidge basis [62,
63],

$\psi_{\mathrm{el}}(\mathbf{r}_{1},\mathbf{r}_{2};\mathbf{R}_{\mathrm{A}},\mathbf{R}_{\mathrm{B}})=\hat{S}\sum_{n_{0},n_{1},n_{2},n_{3},n_{4}}C_{n_{0},\dots,n_{4}}R_{\mathrm{AB}}^{-3-n_{0}-n_{1}-n_{2}-n_{3}-n_{4}}\>\mathrm{e}^{-u(r_{1\mathrm{A}}+r_{1\mathrm{B}}+r_{2\mathrm{A}}+r_{2\mathrm{B}})}\\\
\times
r_{12}^{n_{0}}(r_{1\mathrm{A}}-r_{1\mathrm{B}})^{n_{1}}(r_{2\mathrm{A}}-r_{2\mathrm{B}})^{n_{2}}(r_{1\mathrm{A}}+r_{1\mathrm{B}})^{n_{3}}(r_{2\mathrm{A}}+r_{2\mathrm{B}})^{n_{4}}$
(3)

with variational parameter $u$, non-negative integers $n_{i}$,
$i=0,1,\dots,4$, and the symmetrization operator $\hat{S}$ to satisfy the
Pauli principle.

Now, the effects of the nuclear motion and kinetic interaction between
electrons and nuclei are described by the Hamiltonian

$H_{\mathrm{n}}=-\frac{1}{2\mu_{\mathrm{n}}}\left(\laplacian_{R}+\laplacian_{\text{el}}\right)+\left(\frac{1}{M_{\mathrm{A}}}-\frac{1}{M_{\mathrm{B}}}\right)^{2}\gradient_{R}\cdot\gradient_{\text{el}}\,,$
(4)

where the electron positions are taken relative to the geometric center of the
nuclei. Moreover,
$\mu_{\mathrm{n}}=M_{\mathrm{A}}M_{\mathrm{B}}/(M_{\mathrm{A}}+M_{\mathrm{B}})$
is the reduced nuclear mass for the nuclei A and B; the internuclear distance
is given by
$\mathbf{R}=\mathbf{R}_{\mathrm{AB}}=\mathbf{R}_{\mathrm{A}}-\mathbf{R}_{\mathrm{B}}$
and $\gradient_{\text{el}}=(\gradient_{1}+\gradient_{2})/2$ for the electrons
$1$ and $2$.

A consequent non-adiabatic treatment has been developed in the framework of
the non-adiabatic perturbation theory (NAPT) [64]. Here, the total wave
function is decomposed into an electronic and nuclear part,
$\psi_{\mathrm{el}}$ and $\chi$, respectively, while the non-adiabatic mixing
effects are encoded in a small deviation $\delta\Psi_{\text{na}}$,

$\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{R})=\psi_{\mathrm{el}}(\mathbf{r}_{1},\mathbf{r}_{2};\mathbf{R})\chi(\mathbf{R})+\delta\Psi_{\text{na}}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{R})\,,$
(5)

such that
$\left\langle\delta\Psi_{\text{na}}\right|\left.\\!\psi_{\mathrm{el}}\right\rangle=0$
and $\ket{\psi_{\mathrm{el}}}$ solves the electronic Schrödinger equation for
the Hamiltonian (2),

$H_{\mathrm{el}}\ket{\psi_{\mathrm{el}}}=\mathcal{E}^{(2,0)}(R)\ket{\psi_{\mathrm{el}}}\,.$
(6)

This yields the wave function
$\psi_{\mathrm{el}}(\mathbf{r}_{1},\mathbf{r}_{2};\mathbf{R})$ and the
leading-order eigenvalue $\mathcal{E}^{(2,0)}(R)$, closely following the
notation of Reference [64]. The Born–Oppenheimer energy
$\mathcal{E}^{(2,0)}(R)$ serves as an effective potential in the nuclear
Schrödinger equation

$\left[-\frac{\laplacian_{R}}{2\mu_{\mathrm{n}}}+\mathcal{E}^{(2,0)}(R)\right]\chi(\mathbf{R})=E^{(2,0)}\chi(\mathbf{R})\,,$
(7)

which is solved by the wave function $\chi(\mathbf{R})$ and the leading-order
energy $E^{(2,0)}$ of the full problem. Note that $\chi(\mathbf{R})$ can be
factorized as $\chi(\mathbf{R})=\frac{u(R)}{R}Y_{J}^{m}(\hat{\mathbf{R}})$
with a radial wave function $u(R)$ and the spherical harmonics
$Y_{J}^{m}(\hat{\mathbf{R}})$, resulting in the differential equation of an
anharmonic oscillator for the radial part. Hence, the energy levels and
$\chi(\mathbf{R})$ are characterized by the non-negative integers
$J=0,1,\dots$ for the angular momentum, and $v=0,1,\dots$ for the oscillatory
part.

Finally, non-adiabatic, relativistic, and QED corrections as well as finite
nuclear-size effects are added perturbatively. For instance, the first non-
adiabatic correction reads [64]

$E^{(2,1)}=\bra{\chi_{v,J}}\bra{\psi_{\mathrm{el}}}H_{\mathrm{n}}\ket{\psi_{\mathrm{el}}}\ket{\chi_{v,J}}\,.$
(8)

This leads to an expansion in both the fine structure constant $\alpha$ and
the ratio $m_{\mathrm{e}}/\mu_{\mathrm{n}}$,

$E(\alpha)=\alpha^{2}\left(E^{(2,0)}+\frac{m_{\mathrm{e}}}{\mu_{\mathrm{n}}}\>E^{(2,1)}+\left(\frac{m_{\mathrm{e}}}{\mu_{\mathrm{n}}}\right)^{2}\>E^{(2,2)}+\dots\right)+\alpha^{4}\left(E^{(4,0)}+\frac{m_{\mathrm{e}}}{\mu_{\mathrm{n}}}\>E^{(4,1)}+\dots\right)+\dots\,,$
(9)

where all displayed terms and the leading corrections of
$\order{\alpha^{5},\alpha^{6}}$ are fully known, while the contribution of
$\order{\alpha^{7}}$ is partly available [16, 17, 18, 19, 20, 21, 65, 64].

All existing corrections are implemented in the computer code H2spectre [22],
which returns the energy levels and transition energies of all hydrogen
isotopologues. Moreover, the program H2Solv [66] can be used to determine the
numerical solution of the electronic Schrödinger equation (6).

###  2.2 Experimental Status

During the last decade, different precision spectroscopy methods have been
applied to measure fundamental vibrational modes of hydrogen isotopologues
with high accuracy. For example, Doppler-free laser spectroscopy uses the
principle of two counterpropagating waves of the same frequency resulting in a
cancellation of the Doppler shift effects, see Reference [67] for a review.
The application of this method allowed the measurement of several vibrational
energy levels of H2, D2 and HD up to a relative precision of
$\order{10^{-10}}$ [53, 54, 55].

By contrast, stimulated Raman spectroscopy uses two laser beams of different
frequencies with one frequency being scanned over. If the frequency difference
matches the energy of a physical transition, the intensity of the Stokes line
will be enhanced, as described in Reference [68] and references therein. While
several lines have been measured, for instance in D2 with a relative precision
of $\order{10^{-6}}$ [68], an improvement is given by the CARS spectroscopy
technique [56]. Here instead, the anti-Stokes line is coherently induced
which—although suppressed—leads to a cleaner measurement due to less
background lines. Another advantage of the new technique is the use of smaller
probe volumes, which is especially advantageous when dealing with a
radioactive gas like Tritium. This allowed to record the world best spectrum
of molecular Tritium T2 [56, 57] and recently of DT [58], see Table 1 for the
measurement of T2.

Table 1: Example of the measurement of fundamental vibrational splittings in the T2 molecule for the $Q(J)$ band, which is given by transitions from $v=1$ to $v=0$ for a fixed rotational quantum number $J$ [57]. The central theory values have been extracted from H2spectre [22], while the theory uncertainties are calculated according to Equation (19). All numbers are given in $\mathrm{c}\mathrm{m}^{-1}$. Line | experiment | theory | difference
---|---|---|---
$Q(0)$ | $2464.5052(4)$ | $2464.5042(3)$ | $0.0010$
$Q(1)$ | $2463.3494(3)$ | $2463.3484(3)$ | $0.0010$
$Q(2)$ | $2461.0388(4)$ | $2461.0392(3)$ | $-0.0004$
$Q(3)$ | $2457.5803(4)$ | $2457.5814(3)$ | $-0.0011$
$Q(4)$ | $2452.9817(4)$ | $2452.9821(3)$ | $-0.0004$
$Q(5)$ | $2447.2510(4)$ | $2447.2509(3)$ | $0.0001$

Note that there is a discrepancy between theory and experiment in some lines
of the T2 spectrum, which might be explained by New Physics effects. However,
the deviating sign of the difference $E_{\text{exp}}-E_{\text{theo}}$ in Table
1 makes a New Physics interpretation acting on the inter-particle potentials
more challenging. In the context of DT, the authors of Reference [58]
performed a quick analysis of a pure Yukawa potential among the nuclei,

$V(R;\tilde{\alpha},\lambda)\sim\,\frac{\tilde{\alpha}\mathrm{e}^{-R/\lambda}}{R}\,,$
(10)

where $\tilde{\alpha}$ and $\lambda$ are the coupling strength and interaction
length introduced by a new force. Demanding compatibility within one standard
deviation, they derive an upper bound on the interaction strength
$\tilde{\alpha}<2\times 10^{-8}\alpha_{\mathrm{em}}$ for a distance of
$\lambda=$1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. A more detailed analysis
of New Physics effects is done in this work.

##  3 New Physics Coupled to Electrons and Nuclei

The simplest incarnation of light New Physics is a Yukawa-type exchange
potential of a massive particle, see Equation (1). This appears generically in
many models like the traditional scalar exchange, the leading contribution of
a vector mediator, or in the deconstruction of an extra dimension where the
particle “mass” is replaced by the inverse size of the extra dimension, see
Reference [27]. While the latter is supposed to give a universal contribution
to all massive objects rather like a _fifth force_ extending gravity, scalar
and vector mediators may couple with different charges to electrons and nuclei
(or quarks). Another interesting option is given by the exchange of two
fermions between the two nuclei of a molecule, such as the two-neutrino
exchange [69]. In all these cases, one might expect a measurable effect in
molecular spectra if the new particle’s mass is of $\order{$\mathrm{keV}$}$.

There are very strong but indirect constraints on these types of new
interactions. Star cooling gives an efficient constraint from both the sun and
red giants, excluding a large part of the parameter space in the
$\mathrm{keV}$-regime [44, 45, 46]. However, it is not clear up to which
masses these bounds are valid. It is nevertheless interesting to study this
mass range with direct laboratory experiments since they are exclusively
accessible in molecular and atomic spectroscopy. Thus, they directly probe new
nucleus-nucleus, electron-nucleus and electron-electron interactions.

We assume a generic New Physics potential being present among all particles in
the molecule, that is between all combinations of the two nuclei A and B with
masses $m_{\mathrm{A},\mathrm{B}}$ and two electrons $1$ and $2$ with mass
$m_{\mathrm{e}}$. For instance, adding a new Yukawa interaction of a mediator
of mass $m$ to the Coulomb force, the full potential in the notation of
Equation (2) reads

$\displaystyle V_{\text{NP-
full}}(r_{1},r_{2},r_{\mathrm{A}},r_{\mathrm{B}})=\alpha_{\mathrm{em}}$
$\displaystyle\>\bigg{\\{}$
$\displaystyle\left(-1+\frac{g_{\text{eN}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mr_{\mathrm{1A}}}\right)\frac{1}{r_{\mathrm{1A}}}$
$\displaystyle\>{}+{}\>$
$\displaystyle\left(-1+\frac{g_{\text{eN}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mr_{\mathrm{2B}}}\right)\frac{1}{r_{\mathrm{2B}}}$
(11) $\displaystyle\>{}+{}\>$
$\displaystyle\left(-1+\frac{g_{\text{eN}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mr_{\mathrm{1B}}}\right)\frac{1}{r_{\mathrm{1B}}}$
$\displaystyle\>{}+{}\>$
$\displaystyle\left(-1+\frac{g_{\text{eN}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mr_{\mathrm{2A}}}\right)\frac{1}{r_{\mathrm{2A}}}$
$\displaystyle\>{}+{}\>$
$\displaystyle\left(1+\frac{g_{\text{ee}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mr_{12}}\right)\frac{1}{r_{12}}$
$\displaystyle\>{}+{}\>$
$\displaystyle\left(1+\frac{g_{\text{NN}}}{4\pi\alpha_{\mathrm{em}}}\mathrm{e}^{-mR_{\mathrm{AB}}}\right)\frac{1}{R_{\mathrm{AB}}}\>\bigg{\\}}\,$

with a New Physics coupling between electrons and nuclei $g_{\text{eN}}$,
electrons and electrons $g_{\text{ee}}$, and nuclei and nuclei
$g_{\text{NN}}$. In principle, each $g_{ij}$ can have both signs and thus work
either attractive or repulsive irrespective of the electric charge. However,
$g_{\text{NN}}$ and $g_{\text{ee}}$ should be positive bearing in mind that
all coupling strengths $g_{ij}$ are rather multiplicative couplings if
expressed in terms of fundamental interactions $y_{i}$ as $g_{ij}\sim
y_{i}y^{*}_{j}$.

The _fifth force_ analyses of References [25] and [58] only constrain the last
term of Equation (11) with the replacement
$\frac{g_{\text{NN}}}{4\pi}\to\alpha_{5}$. In the following, we will extent
this analysis by discussing also the other terms in Equation (11) and more
types of potentials.

###  3.1 Implementation of the New Physics Corrections

In order to estimate the full impact of New Physics on the spectra, we set all
but one coupling $g_{ij}$ to zero. For a given New Physics potential
$V_{\mathrm{NP}}$, the energy correction $\Deltaup E^{\mathrm{NP}}_{v,J}$ of a
rovibrational level $(v,J)$ is calculated in first-order perturbation theory
by evaluating the matrix element

$\Deltaup
E^{\mathrm{NP}}_{v,J}=\bra{\chi_{v,J}}\bra{\psi_{\mathrm{el}}}V_{\mathrm{NP}}\ket{\psi_{\mathrm{el}}}\ket{\chi_{v,J}}\,,$
(12)

so that the full energy reads

$E^{\mathrm{NP}}_{v,J}=E^{\mathrm{SM}}_{v,J}+\Deltaup
E^{\mathrm{NP}}_{v,J}\,.$ (13)

Here, $E^{\mathrm{SM}}_{v,J}$ describes the Standard Model prediction which,
including its theoretical uncertainty $\delta E^{\mathrm{SM}}_{v,J}$, can be
extracted from the computer code H2spectre [22]. For the evaluation of the New
Physics shift $\Deltaup E^{\mathrm{NP}}_{v,J}$, we use the same unperturbed
states $\ket{\psi_{\mathrm{el}}}\ket{\chi_{v,J}}$ that also enter the
computation of all corrections in the Standard Model calculation, see Equation
(8).

The case of a pure nuclear force is straightforward. Here, the electronic part
$\ket{\psi_{\mathrm{el}}}$ of the wave function evaluates to $1$, leaving

$\Deltaup
E^{\mathrm{NP}}_{v,J}=\bra{\chi_{v,J}}V_{\mathrm{NP}}\ket{\chi_{v,J}}\,.$ (14)

We extract the nuclear wave function $\chi_{v,J}(R)$ from H2spectre in a
discrete value representation (DVR) with grid spacing $\Deltaup R$.
Analogously to the H2spectre computation of the higher-order Standard Model
corrections [22], we use

$\Deltaup E^{\mathrm{NP}}_{v,J}=\Deltaup
R\cdot\sum_{i}V_{\mathrm{NP}}^{i}\cdot\left(\chi_{v,J}^{i}\right)^{2}\,,$ (15)

with $\chi_{v,J}^{i}=\chi_{v,J}(R_{i})$ and $V_{\mathrm{NP}}^{i}=V(R_{i})$
being the nuclear wave function and potential evaluated at the DVR grid points
$R_{i}$, respectively.

In case of a force that also couples to electrons, the electronic matrix
element

$\mathcal{E}^{\mathrm{el}}_{\mathrm{NP}}(R)=\bra{\psi_{\mathrm{el}}}V_{\mathrm{NP}}\ket{\psi_{\mathrm{el}}}$
(16)

needs to be evaluated first since the electronic wave function depends on the
nuclear separation $R$. We extract this wave function in the symmetric
James–Coolidge basis, as specified in Equation (3), from the publicly
available code H2SOLV [66]. In particular, we fix the nuclear distance $R$ and
minimize the energy expectation value of the wave function as computed by
H2SOLV with respect to the variational parameter $u$ defined in Reference
[66]. Using the coefficients of the basis expansion for the minimal energy
expectation value, we compute the electronic matrix element by numerical
integration. To avoid the time-consuming numerical integration at each
parameter point $(R,m,g)$, we evaluate the electronic matrix element on a grid
in $(R,m)$ only, since the coupling $g$ factorizes in each case. The full
dependence of $\mathcal{E}^{\mathrm{el}}_{\mathrm{NP}}(R)$ on $R$ and $m$ is
afterwards reconstructed by interpolation with splines of degree two.
$\mathcal{E}^{\mathrm{el}}_{\mathrm{NP}}(R)$ obtained in this way serves as an
effective potential for the nuclei in the same manner as the relativistic
corrections do in the H2spectre computation. Consequently, the New Physics
contribution $\Deltaup E_{\mathrm{NP}}$ is calculated using Equation (15) with
$V_{\mathrm{NP}}(R)$ replaced by $\mathcal{E}^{\mathrm{el}}_{\mathrm{NP}}(R)$.

There is an additional complication for spin-dependent potentials when coupled
to nuclei. In order to comply with the Pauli principle, the nuclear spin state
$\ket{f_{1},m_{\mathrm{f},1},f_{2},m_{\mathrm{f},2}}$ depends on the angular
momentum quantum number $J$. Since the leading-order energy is independent of
the magic quantum number $m_{\mathrm{f},i}$ of the nucleus $i$, this leads to
a degeneracy and, hence, we need to use degenerate perturbation theory to
calculate the energy correction. In this case, the New Physics energy shift
$\Deltaup E^{\mathrm{NP}}_{v,J}$ for the ground state is determined by the
minimal eigenvalue of the perturbation in the degenerate subspace.

Finally, the energy $\Deltaup
E^{\mathrm{NP}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}$ for the transition
from a level $(v_{1},J_{1})$ to the level $(v_{2},J_{2})$ is given by

$\Deltaup
E^{\mathrm{NP}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}=E^{\mathrm{NP}}_{v_{2},J_{2}}-E^{\mathrm{NP}}_{v_{1},J_{1}}\,.$
(17)

We expect the size of the New Physics contribution to be of order of the
uncertainty $\delta E^{\mathrm{SM}}_{v,J}$ of the Standard Model calculation.
Since the theoretical error in the New Physics energy shift should be much
smaller than the contribution itself, $\delta E^{\mathrm{NP}}_{v,J}\ll\Deltaup
E^{\mathrm{NP}}_{v_{2},J_{2}}\sim\delta E^{\mathrm{SM}}_{v,J}$, we approximate
the overall uncertainty of the level energy to be $\delta
E^{\mathrm{SM}}_{v,J}$, that is

$\delta E^{\mathrm{NP}}_{v,J}=\delta E^{\mathrm{SM}}_{v,J}+\delta
E^{\mathrm{NP}}_{v,J}\approx\delta E^{\mathrm{SM}}_{v,J}\,.$ (18)

In contrast to the error estimate for transitions in H2spectre, we linearly
add the uncertainties of the two corresponding levels to get a more
conservative theoretical uncertainty $\delta\Deltaup
E^{\mathrm{NP}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}$ for the transition
energy,

$\delta\Deltaup E^{\mathrm{NP}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}=\delta
E^{\mathrm{SM}}_{v_{1},J_{1}}+\delta E^{\mathrm{SM}}_{v_{2},J_{2}}\,.$ (19)

Given an experimental measurement $\Deltaup
E^{\mathrm{exp}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}$ for a transition
$(v_{1},J_{1})\rightarrow(v_{2},J_{2})$ with an uncertainty
$\sigma^{\mathrm{exp}}_{(v_{1},J_{1})\rightarrow(v_{2},J_{2})}$, we require
the theoretical prediction including New Physics effects to lie within the
interval

$[\Deltaup E^{\mathrm{exp}}-3\,\sigma^{\mathrm{exp}}-\delta\Deltaup
E^{\mathrm{NP}},\Deltaup
E^{\mathrm{exp}}+3\,\sigma^{\mathrm{exp}}+\delta\Deltaup E^{\mathrm{NP}}]\,,$
(20)

for each transition, suppressing the indices for clarity. For a given molecule
and mass of the new mediator, this criterion allows to derive upper bounds on
the couplings $g_{ij}$ by combining all measurements listed in Appendix A.

###  3.2 Scalar and Pseudoscalar Potentials

Many New Physics scenarios comprise light scalar or pseudoscalar fields, for
instance an additional light Higgs boson [31, 32] or as remnants of (softly or
spontaneously) broken continuous global symmetries and thus
(Pseudo-)Nambu–Goldstone bosons, like the axion [70, 71, 72, 73, 74] or the
Majoron [75].

Complete spin-dependent potentials for various mediator particles have been
summarized in [29]. In particular, potentials for massive scalar (S) or
pseudoscalar (P) mediators $\phi$ with mass $m$ between two fermions a and b
with masses $m_{\mathrm{a,b}}$, are given by the expressions

$\displaystyle V_{\mathrm{S}}(\mathbf{r})=\;$
$\displaystyle-g^{\mathrm{S}}_{\mathrm{ab}}\frac{\mathrm{e}^{-mr}}{4\pi r}\,,$
(21a) $\displaystyle V_{\mathrm{P}}(\mathbf{r})=\;$
$\displaystyle-g^{\mathrm{P}}_{\mathrm{ab}}\frac{m^{2}}{4m_{\mathrm{a}}m_{\mathrm{b}}}\bigg{[}\left(\bm{\sigma}_{\mathrm{a}}\cdot\bm{\sigma}_{\mathrm{b}}\right)\left(\frac{1}{m^{2}r^{2}}+\frac{1}{mr}+\frac{4\pi
r}{3m^{2}}\delta^{(3)}(\mathbf{r})\right)$
$\displaystyle\qquad-\left(\bm{\sigma}_{\mathrm{a}}\cdot\hat{\mathbf{r}}\right)\left(\bm{\sigma}_{\mathrm{b}}\cdot\hat{\mathbf{r}}\right)\left(1+\frac{3}{m^{2}r^{2}}+\frac{3}{mr}\right)\bigg{]}\frac{\mathrm{e}^{-mr}}{4\pi
r}\,,$ (21b)

with the (pseudo)scalar couplings $g_{\mathrm{a}\mathrm{b}}^{\mathrm{S(P)}}$
and the spin Pauli matrices $\bm{\sigma}_{\mathrm{a,b}}$.

Note that the pseudoscalar interaction is suppressed by the masses of the
interacting particles. For this reason, we do not expect strong limits from
molecular systems for pseudoscalar interactions. Moreover, these potentials
have been derived between spin-$\frac{1}{2}$ fermions. Nevertheless, the spin-
independent scalar potential can also be applied to a force between spin-1
bosons like the deuteron, while the pseudoscalar is to be used for
spin-$\frac{1}{2}$ particles only.

Applying the criterion in Equation (20), we derive upper limits on the
couplings $g^{\mathrm{S,P}}_{\mathrm{ab}}$ shown in the mass–coupling plane,
see Figure 1. For a scalar interaction between electrons, cf. Equation (21),
H2 and HD molecules constrain the scalar coupling
$g^{\mathrm{S}}_{\mathrm{ee}}$ up to $\order{10^{-8}}$, see Figure 1(a). By
contrast, the coupling of a pseudoscalar mediator is weakly constrained,
$g^{\mathrm{P}}_{\mathrm{ee}}\sim\order{10^{-3}}$ for
$m\sim$1\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$$, meeting the expectation of
a suppression by $m^{2}/m_{\mathrm{e}}^{2}\sim 10^{-6}$ relative to the scalar
case, as shown in Figure 1(b). It can be seen that the bounds for the
pseudoscalar coupling become ineffective at about
$7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. This happens when the New Physics
contribution approaches zero and eventually changes its sign as a consequence
of an internal cancellation between the terms with different spin structure.
This is an interesting feature which might be resolved using polarized probes.

(a) Scalar electron–electron interaction.

(b) Pseudoscalar electron–electron interaction.

(c) Scalar nucleus–nucleus interaction.

(d) Scalar electron–nucleus interaction with positive coupling
$g^{\mathrm{S}}_{\mathrm{eN}}>0$.

(e) Scalar electron–nucleus interaction with negative coupling
$g^{\mathrm{S}}_{\mathrm{eN}}<0$.

Figure 1: Exclusion limits in mass–coupling plane on forces mediated by scalar
or pseudoscalar particles. For each molecule, the corresponding upper limit
results from the combination of all available measurements. The area above the
curves is excluded.

For a pure nucleus–nucleus interaction, a pseudoscalar contribution is even
more suppressed by $m^{2}/m_{\mathrm{N}}^{2}\sim 10^{-11}$, thus, we only
consider the scalar potential. The corresponding limits in the mass-coupling
plane are shown in Figure 1(c). Kinks in the plot are an artefact of the
combination of several measured lines indicating that another measurement
becomes more efficient in the exclusion of parameter space. While the dominant
limits of $\order{10^{-8}}$ for masses $\lesssim$10\text{\,}\mathrm{keV}$$
arise from H2 and HD transitions, T2 measurements are most constraining for
larger masses. Note that our limits are weaker than the ones presented in
References [25, 58] as a consequence of a more conservative exclusion
criterion by allowing for $3\sigma$ deviations.

In the case of an electron–nucleus interaction the spin matrix elements for
the electronic ground state vanish and thus only the scalar interaction
survives. Since the relative sign of the electron and nucleus coupling
$g^{\mathrm{S}}_{\mathrm{eN}}$ is not fixed, we show the exclusion limits for
both signs in Figures 1(d) and 1(e). As in the electron–electron case, the
strongest constraints are again given by the transitions measured in H2 and HD
molecules with upper limits on the coupling
$\absolutevalue{g^{\mathrm{S}}_{\mathrm{eN}}}$ up to $\order{10^{-8}}$ and
$\order{10^{-9}}$ for a positive and negative coupling, respectively. Compared
to the electron–electron and nucleus–nucleus case, the slightly better
constraints are expected because of the four possible combinations of
electrons and nuclei. Note that there should be another enhancement due to
$g^{\mathrm{S}}_{\mathrm{eN}}$ also implying a $g^{\mathrm{S}}_{\mathrm{ee}}$
and $g^{\mathrm{S}}_{\mathrm{NN}}$ coupling, however, the order of magnitude
will not change.

###  3.3 Vector and Axialvector Exchange Potentials

There are different options of introducing a new (axial)vector coupling. One
possibility is via kinetic mixing with a “dark” photon, where a new “dark”
$U(1)$ gauge field described by the field-strength tensor
$F^{\prime}_{\mu\nu}=\partial_{\mu}A^{\prime}_{\nu}-\partial_{\nu}A^{\prime}_{\mu}$
mixes with the electromagnetic photon through a Lagrangian of the type [35]

$\mathcal{L}_{\text{kin-
mix}}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}F^{\prime\mu\nu}F^{\prime}_{\mu\nu}-\frac{1}{2}F^{\mu\nu}F^{\prime}_{\mu\nu}\,.$
(22)

Another possibility involves the Stueckelberg mechanism where additionally a
light axion-like field is present [76, 77]. In this case, the heavy vector is
usually referred to a $Z^{\prime}$ boson and thus supposed to have a mass in
the $\mathrm{GeV}$ regime rather than $\mathrm{keV}$.

The presence of a light spin-1 mediator with vector and axialvector couplings
$g^{\mathrm{V,A}}_{\mathrm{ab}}=g^{\mathrm{V,A}}_{\mathrm{a}}g^{\mathrm{V,A}}_{\mathrm{b}}$
of the type
$A_{\mu}^{\prime}\;\bar{\psi}\;\gamma^{\mu}\left(g^{\mathrm{V}}_{\psi}+\mathrm{i}\,\gamma_{5}\;g^{\mathrm{A}}_{\psi}\right)\psi$
and a mass $m$ leads to non-relativistic potentials [29]

$\displaystyle V_{\mathrm{V}}(\mathbf{r})=\;$
$\displaystyle\frac{g^{\mathrm{V}}_{\mathrm{ab}}}{4\pi}\;\frac{\mathrm{e}^{-mr}}{r}\bigg{\\{}1+$
$\displaystyle\qquad\frac{m^{2}}{4m_{\mathrm{a}}m_{\mathrm{b}}}\bigg{[}\bm{\sigma}_{\mathrm{a}}\cdot\bm{\sigma}_{\mathrm{b}}\left(\frac{1}{m^{2}r^{2}}+\frac{1}{mr}+1-\frac{8\pi
r}{3m^{2}}\delta(\mathbf{r})\right)$
$\displaystyle\qquad\qquad\qquad\qquad-(\bm{\sigma}_{\mathrm{a}}\cdot\hat{\mathbf{r}})(\bm{\sigma}_{\mathrm{b}}\cdot\hat{\mathbf{r}})\left(\frac{3}{m^{2}r^{2}}+\frac{3}{mr}+1\right)\bigg{]}\bigg{\\}}\,,$
(23a) $\displaystyle V_{\mathrm{A}}(\mathbf{r})=\;$
$\displaystyle-\frac{g^{\mathrm{A}}_{\mathrm{ab}}}{4\pi}\frac{\mathrm{e}^{-mr}}{r}\bigg{\\{}\;\bm{\sigma}_{\mathrm{a}}\cdot\bm{\sigma}_{\mathrm{b}}\bigg{[}1+\frac{1}{m^{2}r^{2}}+\frac{1}{mr}+\frac{4\pi
r}{3m^{2}}\delta^{(3)}(\mathbf{r})\bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad-(\bm{\sigma}_{\mathrm{a}}\cdot\hat{\mathbf{r}})(\bm{\sigma}_{\mathrm{b}}\cdot\hat{\mathbf{r}})\bigg{[}1+\frac{3}{m^{2}r^{2}}+\frac{3}{mr}\bigg{]}\bigg{\\}}\,.$
(23b)

Here, $\bm{\sigma}_{i}$ and $\hat{\mathbf{r}}$ are the Pauli matrices of
particle $i$ and the unit vector pointing in the direction between the two
fermions a and b, respectively.

The dominant spin-independent effect can be found from the $V_{\mathrm{V}}$
potential above, being exactly the Yukawa-type potential mentioned earlier.
Note that the spin-dependent vector interactions are suppressed by the fermion
masses. Thus, the leading contribution for a vector mediator is given by the
Yukawa potential

$V_{\text{V}}(\mathbf{r})=\pm\frac{g^{\text{V}}_{\mathrm{ab}}}{4\pi}\frac{\mathrm{e}^{-mr}}{r}\,.$
(24)

In contrast to the pseudoscalar case, the axial vector interaction is not
suppressed by the inverse fermion masses. The contribution from the
axialvector potential is expected to be of the same size as the leading vector
contribution, where additionally the dependence on the mediator mass plays a
more significant role. As for the pseudoscalar case, the potentials have been
derived for an interaction among fermions and, hence, we only consider the
leading Yukawa-type contribution for bosonic nuclei.

(a) Vector electron–electron interaction.

(b) Axialvector electron–electron interaction.

(c) Axialvector nucleus–nucleus interaction.

Figure 2: Exclusion limits in mass–coupling plane on forces mediated by vector
or axialvector particles. For each molecule, the corresponding upper limit on
the coupling constants results from the combination of all available
measurements.

Like in the scalar interaction, the strongest constraints for a pure
electronic force are given by the measurement of H2 and HD transition, see
Figures 2(a) and 2(b). We find an upper limit on the coupling
$g^{\mathrm{V,A}}_{\mathrm{ee}}$ of $\order{10^{-8}}$ and $\order{10^{-10}}$
for the vector and axialvector potential, respectively, for masses around
$1\text{\,}\mathrm{keV}$.

Regarding the nucleus–nucleus force, the additional terms in the vector
contribution are suppressed by two powers of the nuclear mass and, therefore,
the limits coincide with the ones for the scalar potential shown in Figure
1(c). Bounds from the axialvector exchange are again stronger by two orders of
magnitude yielding an upper bound on the coupling
$g_{\mathrm{NN}}^{\mathrm{A}}$ of $\order{10^{-10}}$, see Figure 2(c). Since
we do not consider the bosonic nuclei D2 and HD, the best limits are now given
by H2 measurements for masses below
$10\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ and by T2 lines for larger masses.

Analogously to the pseudoscalar electron–nucleus interaction, the spin-
dependent terms vanish for the electronic ground state of H2 isotopologues. As
a consequence, the bounds on the vector potential are the same as for the
scalar case, see Figures 1(d) and 1(e), while an axialvector force vanishes
entirely.

###  3.4 Singular Potentials: Effective Contact Interactions

The Standard Model already comprises a suppressed short-range Yukawa-like
potential mediated by the heavy electroweak vector bosons or the scalar Higgs
boson. According to the decoupling theorem, these interactions should not have
any effect on atomic or molecular scales so that they can be safely ignored.
However, there are claims in the literature that an effective coupling
mediated by heavy $W$, $Z$ or the Higgs boson leads to a measurable two-
particle exchange of a very light mediator. This two-fermion exchange may
induce long-range forces as pointed out in the literature [69, 78, 79, 80, 81,
5, 82, 83, 84, 85], see Figure 3. For instance, the case of an effective Fermi
interaction with massless neutrinos has been first discussed by Feinberg and
Sucher in the late 1960s [69] and was completed by Hsu and Sikivie in the
early 1990s [79]. Their work has been extended by Grifols et al. [80] to the
case of massive Dirac and Majorana-type neutrinos of mass $m_{\nu}$, yielding
the long-range potentials

$\displaystyle V_{\mathrm{M}}(r)$
$\displaystyle=\frac{G_{\text{eff}}^{2}m_{\nu}^{2}}{8\pi^{2}r^{3}}\;K_{2}(2m_{\nu}r)\stackrel{{\scriptstyle
m_{\nu}r\gg
1}}{{\approx}}\frac{G_{\text{eff}}}{16\pi^{2}r^{2}}\sqrt{\frac{m_{\nu}^{3}}{\pi
r^{3}}}\operatorname{e}^{-2m_{\nu}r}\,,$ (25a) $\displaystyle
V_{\mathrm{D}}(r)$
$\displaystyle=\frac{G_{\mathrm{eff}}^{2}m_{\nu}^{3}}{16\pi^{3}r^{2}}\;K_{3}(2m_{\nu}r)\stackrel{{\scriptstyle
m_{\nu}r\gg
1}}{{\approx}}\frac{G_{\text{eff}}}{32\pi^{2}}\sqrt{\frac{m_{\nu}^{5}}{\pi
r^{5}}}\operatorname{e}^{-2m_{\nu}r}\,,$ (25b)

with the modified Bessel functions $K_{n}$. In the Standard Model case, the
effective coupling $G_{\text{eff}}$ is given by the Fermi constant
$G_{\mathrm{F}}$. Both potentials scale like $\sim 1/r^{5}$ in the limit of
vanishing neutrino masses or short distances, reproducing the well-known
result by Feinberg and Sucher [69],

$\displaystyle V(r)=\frac{G_{\mathrm{eff}}^{2}}{16\pi^{3}r^{5}}\,.$ (26)

Due to the highly singular behaviour of the two-neutrino exchange potential,
one needs to be careful in the analysis. Naively, we expect a quadratic
divergence in our integrals from power-counting, matching Stadnik’s
observation for hydrogen atoms in Reference [82]. Assuming a cut-off of the
$Z$ boson mass $M_{\mathrm{Z}}$ for the integrals to account for the limited
validity of the effective theory, Stadnik derives tight bounds on the coupling
$G_{\text{eff}}$ close to the Standard Model value for positronium. However,
we doubt the adequateness of such a cut-off, as remarked by other authors [86,
87]. The strong dependence of a result on the arbitrary value of the cut-off
parameter signals an incorrect treatment of the ultraviolet divergences of the
theory, bearing in mind that already at the inverse Compton wavelength of the
electron much below $M_{\text{Z}}$ the physics of the pure two-neutrino
potential changes.

Figure 3: Long-range force mediated by double neutrino exchange (left) and
double (Pseudo-)Goldstone boson exchange. In both cases, the direct coupling
to the fermions is by a heavy particle ($Z$ or Higgs boson) leading to
effective four-particle interactions.

A correct treatment involving a proper matching in the tower of effective
theories at each scale in the problem is subtle and beyond the scope of this
analysis. In order to get an impression of the magnitude of the neutrino
exchange effects, we calculate the short-range effect to the molecular levels
caused by the exchange of a $W$ boson with mass $M_{\mathrm{W}}$ as depicted
in the diagram in Figure 4. Parametrically, the effects of both this $W$ box
diagram and the neutrino exchange are of order
$\order{G_{\mathrm{F}}^{2}M_{\mathrm{W}}^{2}}$. We are rather expecting a
further suppression, for instance due to small mixing factors in the case of
sterile neutrinos. Evaluating the box diagram, we derive the effective low-
energy potential

$V_{W\mathrm{box}}(\mathbf{r})=\frac{4}{3}\pi
G_{\mathrm{F}}^{2}M_{\mathrm{W}}^{2}\delta^{(3)}(\mathbf{r})\,.$ (27)

The size of this effect is of
$\order{${10}^{-11}\text{\,}\mathrm{c}\mathrm{m}^{-1}$}$ and, hence, far below
the current experimental sensitivity. Due to its smallness, the neutrino
exchange is negligible if the Coulomb force is present and one may rather
expect an effect in cases where the electromagnetic force is absent or
screened like between neutral atoms and molecules, as has been noted in
Reference [69].

Figure 4: Additional contribution to the two-neutrino force induced by a
$W$-box diagram.

The two-neutrino exchange has recently been studied in the context of atomic
parity violation [88, 84]. By using higher angular momentum transitions, the
authors are able to derive limits from wave functions dropping rapidly for
small distances, which seems to be a suitable approach to deal with the
divergence. However, their analysis is missing a full systematic treatment of
all matchings at intermediate scales, which might have an influence on the
results. Nevertheless, the effect is far below the reach of current and future
experiments, similar to our estimate. The study of Reference [88] goes beyond
the known physics properties of electroweak physics and constrains new models
extending the Standard Model, especially with an extra Higgs doublet or a
light $Z^{\prime}$. A recent discussion of the full effective theory framework
of the long-range neutrino potential has been given in [85].

There are further modifications of the intramolecular forces possible due to
Higgs and Goldstone boson exchange, where a long-range potential arises in a
similar manner as for neutrinos [81], see Figure 3. In particular, a massive
(pseudo)scalar boson $a$ of mass $m$ interacting with the Standard Model Higgs
boson $H$ via the Lagrangian
$\mathcal{L}_{\mathrm{Haa}}=g_{\mathrm{Haa}}\>aaH$ yields the potential

$V_{\text{PG}}(r)=-\frac{G\,m}{8\pi^{3}r^{2}}K_{1}(2mr)\,.$ (28)

Here, the coupling strength $G$ is related to the Standard Model Higgs-fermion
interaction $g_{\mathrm{Hff}}$ by [81]

$G=\frac{g_{\mathrm{Hff}}g_{\mathrm{Hf^{\prime}f^{\prime}}}g_{\mathrm{Haa}}^{2}}{m_{\mathrm{H}}^{4}}\sim${10}^{-19}\text{\,}\mathrm{G}\mathrm{eV}^{-4}$g_{\mathrm{Haa}}^{2}\,.$
(29)

Although the potential reduces to a well-behaved $1/r^{3}$ functional form for
small masses $m$, the tiny prefactor renders this process impossible to
observe in molecular spectra.

##  4 Conclusions

In the present work, we have performed the first extensive and systematic
study of New Physics effects on molecular spectra. Starting from available
codes which give precise _ab initio_ predictions in the Standard Model for
transitions in hydrogen-like molecules, we treated a variety of New Physics
potentials as perturbations and derived constraints on the new forces from
direct measurements.

Molecular spectroscopy, as well as atomic spectroscopy, essentially probes the
Coulomb potential and Quantum Electrodynamics with high accuracy. From our
analysis we conclude that New Physics effects are unlikely to be seen in
spectroscopy at all. Spectroscopical observations of sufficiently large
deviations would be in conflict with indirect constraints stemming from
astrophysics and cosmology. Nevertheless, spectroscopy is a complementary and
direct test of the Standard Model in the laboratory and is an important tool
in the context of New Physics searches in this only indirectly excluded
parameter region. Despite the expectation that heavier nuclei as in tritiated
molecules give stronger constraints than hydrogen alone, we do not observe
higher sensitivity in the isotopologues compared to hydrogen.

We have found that constraints on new interactions between electrons and
nuclei from molecular spectroscopy are compatible with atomic spectroscopy,
but the latter derives more stringent bounds of up to three orders of
magnitude. The same is true for probes of the nucleus–nucleus interaction from
rovibrational spectroscopy compared to direct neutron scattering. Furthermore,
in the case of a modified electron–electron coupling, molecular spectroscopy
is competitive with Helium spectroscopy, although there are stronger limits of
approximately two orders of magnitude on the coupling $g_{\mathrm{NP}}$
available from measurements of the anomalous magnetic moment of the electron.

As an advantage to other direct techniques molecular spectroscopy allows to
probe a plethora of New Physics interactions between different types of
particles in one single measurement—assuming that only one type of interaction
is present at the time. Further improvements in both theoretical treatment of
hydrogen-like molecules and experimental precision are going to yield stronger
constraints. Moreover, we have found relatively loose constraints for a
certain mass window of the mediator particle for some potentials. In case of
spin-dependent forces, polarized probes may help to improve the exclusion
limits.

Searches for a new long-range force mediated by an exchange of two light
particles like neutrinos are not promising in spectroscopy since the expected
effect is too small due to parametric suppression. Furthermore, a strong cut-
off dependence appears when divergences in the theory are not properly taken
into account so that the sensitivity is misestimated. A full treatment of the
effective theories at all scales down to very short distances is beyond the
scope of this paper. In any case, we do not expect an effect that is going to
be visible in the next generation experiments.

During the finalization of this work, we got aware of a new set of
measurements including more lines for T2, DT, and HT [89]. This study shows
very good agreement with the theoretical prediction.

## Acknowledgements

We thank Magnus Schlösser for initiation of this study, initial discussion and
intermediate feedback, and Ulrich Nierste for many helpful discussions and
advices. Furthermore, we want to acknowledge Sonia Rani and Aman Sardwal from
the IIT Bombay for their participation in the early stages of the project. WGH
wants to thank Andreas Ringwald for helpful conversions in the beginning of
the project. Additionally, we thank Kirill Melnikov for helpful advice on the
proper treatment of the cut-off with singular potentials, and we thank
Alejandro Segarra for a discussion of the two-neutrino exchange potentials. We
thank Ulrich Nierste for a careful reading of the manuscript and comments. ML
acknowledges the support by the Doctoral School “Karlsruhe School of
Elementary and Astroparticle Physics: Science and Technology” and the
collaborative research center TRR 257 “Particle Physics Phenomenology after
the Higgs Discovery”. MT acknowledges the support of the DFG-funded Research
Training Group 1694, “Elementary particle physics at highest energy and
precision”.

##  Appendix A Experimental data

Here, we list all experimental data that were used in our analysis in Section
3.

Table 2: List of all measurements used in our analysis. The transitions $(v_{1},J_{1})\rightarrow(v_{2},J_{2})$ are characterized by the vibrational quantum numbers $v_{i}$ and the angular momentums $J_{i}$ of the involved levels $(v_{i},J_{i})$. All numbers are given in $\mathrm{c}\mathrm{m}^{-1}$. molecule | transition | energy | reference
---|---|---|---
H2 |  $(3,5)$ $\rightarrow$ $(0,3)$ | $12\,559.749\,52(5)$ | [90]
H2 |  $(1,0)$ $\rightarrow$ $(0,0)$ | $4161.166\,36(15)$ | [54]
H2 |  $(1,1)$ $\rightarrow$ $(0,1)$ | $4155.254\,00(21)$ | [54]
H2 |  $(1,2)$ $\rightarrow$ $(0,2)$ | $4143.465\,53(15)$ | [54]
H2 |  $(11,1)$ $\rightarrow$ $(0,0)$ | $32\,937.7554(16)$ | [91]
H2 |  $(11,3)$ $\rightarrow$ $(0,0)$ | $33\,186.4791(16)$ | [91]
H2 |  $(11,4)$ $\rightarrow$ $(0,0)$ | $33\,380.1025(33)$ | [91]
H2 |  $(11,5)$ $\rightarrow$ $(0,0)$ | $33\,615.5371(18)$ | [91]
HD |  $(1,0)$ $\rightarrow$ $(0,0)$ | $3632.160\,52(22)$ | [54]
HD |  $(1,1)$ $\rightarrow$ $(0,1)$ | $3628.304\,50(22)$ | [54]
HD |  $(2,2)$ $\rightarrow$ $(0,1)$ | $7241.849\,350\,87(67)$ | [55]
HD |  $(2,3)$ $\rightarrow$ $(0,2)$ | $7306.483\,222\,50(93)$ | [55]
HD |  $(2,4)$ $\rightarrow$ $(0,3)$ | $7361.903\,173\,35(93)$ | [55]
D2 |  $(1,0)$ $\rightarrow$ $(0,0)$ | $2993.617\,06(15)$ | [54]
D2 |  $(1,1)$ $\rightarrow$ $(0,1)$ | $2991.507\,06(15)$ | [54]
D2 |  $(1,2)$ $\rightarrow$ $(0,2)$ | $2987.293\,52(15)$ | [54]
D2 |  $(0,2)$ $\rightarrow$ $(0,0)$ | $179.068(2)$ | [68]
D2 |  $(0,3)$ $\rightarrow$ $(0,1)$ | $297.533(3)$ | [68]
D2 |  $(0,4)$ $\rightarrow$ $(0,2)$ | $414.648(2)$ | [68]
T2 |  $(1,0)$ $\rightarrow$ $(0,0)$ | $2464.5052(4)$ | [57]
T2 |  $(1,1)$ $\rightarrow$ $(0,1)$ | $2463.3494(3)$ | [57]
T2 |  $(1,2)$ $\rightarrow$ $(0,2)$ | $2461.0388(4)$ | [57]
T2 |  $(1,3)$ $\rightarrow$ $(0,3)$ | $2457.5803(4)$ | [57]
T2 |  $(1,4)$ $\rightarrow$ $(0,4)$ | $2452.9817(4)$ | [57]
T2 |  $(1,5)$ $\rightarrow$ $(0,5)$ | $2447.2510(4)$ | [57]
DT |  $(1,0)$ $\rightarrow$ $(0,0)$ | $2743.341\,71(40)$ | [58]
DT |  $(1,1)$ $\rightarrow$ $(0,1)$ | $2741.732\,04(330)$ | [58]
DT |  $(1,2)$ $\rightarrow$ $(0,2)$ | $2738.516\,59(40)$ | [58]
DT |  $(1,3)$ $\rightarrow$ $(0,3)$ | $2733.704\,70(40)$ | [58]
DT |  $(1,4)$ $\rightarrow$ $(0,4)$ | $2727.307\,34(40)$ | [58]
DT |  $(1,5)$ $\rightarrow$ $(0,5)$ | $2719.341\,93(40)$ | [58]

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