A Study of New Physics Searches with Tritium and Similar Molecules

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 H_2 molecules give stronger constraints than T_2 or other isotopologues.


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-GeV particles has recently gained impetus [1], where molecules have been identified as good study objects [2]. In particular, molecular spectroscopy is one possibility to look for new dark forces [3,4].
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 [5][6][7][8][9], 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 [10][11][12][13]. With this method, a full nonadiabatic treatment of the system is performed.
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 [23][24][25].
The simplest modification of the well-known Coulomb potential V C for the interaction between two point charges q 1,2 which accounts for light New Physics is given by the addition of a Yukawatype potential V Y , where α em 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 NP . Note that the coupling strength g NP might have both signs depending on the interacting particles. A brief comparison of units yields a mass of O(keV) for a typical bond length ∼ 1 Å of a light molecule. For larger masses the exponential term in V 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 α em by a constant shift [26].
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 [27]. Many models include such light particles as carriers for weak long-range forces, for instance an additional light Higgs Boson as a scalar mediator [28,29], axions [30] and axion-like particles (ALPs) [31] as examples for a pseudoscalar exchange, or Dark Photons as a vector particle [26,32,33].
The low-energy regime has already been explored in other experiments, for a review see Reference [34]. Precision QED tests can be performed with atomic spectroscopy, for example in highly excited Rydberg atoms or isotope shifts in singly ionized divalent elements [35][36][37]. While these measurements give slightly better constraints than molecular spectroscopy, long-distance internuclear interactions can only be tested in molecules-although neutron scattering might be more competitive [38]. Additionally, atomic precision tests for light scalar couplings have been considered [39], where light ALPs modify the Coulomb potential by a screening effect and may have impact on the Lamb Shift in atomic hydrogen [40]. However, there are competing laboratory techniques with higher sensitivity as pointed out in Reference [40] and by further dedicated studies on atomic spectroscopy [35][36][37]. Especially in the mass regime below several MeV, there are stringent indirect constraints from astrophysics [41][42][43] and cosmology [44]. With atomic and molecular spectroscopy, new forces at the keV scale can be probed directly by the single-particle interaction in contrast to multi-particle coherent effects in massive objects.  [45][46][47][48]. However, these contraints apply only to mediator masses up to the meV regime. In that respect, atomic and molecular spectroscopy are highly relevant as probes of new forces in the Å-regime corresponding to masses of O(keV). Such constraints on fifth forces from molecular spectroscopy have been derived for a Yukawa-like interaction between nuclei in References [23,24]. Similarly, fifth force experiments lead to constraints on light scalars coupled to photons [49].
In this work, we study the impact of New Physics potentials on molecular spectroscopy of the hydrogen isotopologues H 2 , D 2 , T 2 , 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.

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. Moreover, the world's best spectra of molecules containing Tritium have been recently obtained using Coherent Anti-Stokes Raman Scattering Spectroscopy (CARS) for T 2 and DT [53][54][55]. A relative precision of up to O 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.

A Brief Review of the Current Theoretical Status
A full theoretical treatment of the Hydrogen molecule H 2 as a four-particle system is intrinsically difficult. First approaches date back to the 1920s and have been developed independently by Heitler and London [56], and Born and Oppenheimer [57]. 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 4 m e /m 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 [58]. A consequent non-adiabatic treatment takes the movement of the nuclei into account in order to calculate the energy levels of the whole system [6].
Assuming the nuclei to be at fixed positions R A and R B , the Hamiltonian for this system reads [56] with the electromagnetic fine structure constant α em 1/137 and the distances r 12 and R 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 = A, B. The Schrödinger equation for this Hamiltonian (2) is usually solved using a variational method with a trial wave function ψ el (r 1 , r 2 ; R A , R 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 [59,60], with variational parameter u, non-negative integers n i , i = 0, 1, . . . , 4, and the symmetrization operatorŜ to satisfy the Pauli principle. Now, the effects of the nuclear motion and kinetic interaction between electrons and nuclei are described by the Hamiltonian where the electron positions are taken relative to the geometric center of the nuclei. Moreover, is the reduced nuclear mass for the nuclei A and B; the internuclear distance is given by R = R AB = R A − R B and ∇ el = (∇ 1 + ∇ 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) [61]. Here, the total wave function is decomposed into an electronic and nuclear part, ψ el and χ, respectively, while the non-adiabatic mixing effects are encoded in a small deviation δΨ na , such that 〈δΨ na | ψ el 〉 = 0 and |ψ el 〉 solves the electronic Schrödinger equation for the Hamiltonian (2), This yields the wave function ψ el (r 1 , r 2 ; R) and the leading-order eigenvalue E (2,0) (R), closely following the notation of Reference [61]. The Born-Oppenheimer energy E (2,0) (R) serves as an effective potential in the nuclear Schrödinger equation which is solved by the wave function χ(R) and the leading-order energy E (2,0) of the full problem.
Note that χ(R) can be factorized as with a radial wave function u(R) and the spherical harmonics Y m J (R), resulting in the differential equation of an anharmonic oscillator for the radial part. Hence, the energy levels and χ(R) are characterized by the non-negative integers J = 0, 1, . . . for the angular momentum, and v = 0, 1, . . . 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 [61] This leads to an expansion in both the fine structure constant α and the ratio m e /µ n , where all displayed terms and the leading corrections of O α 5 , α 6 are fully known, while the contribution of O α 7 is partly available [14-19, 61, 62].
All existing corrections are implemented in the computer code H2spectre [20], which returns the energy levels and transition energies of all hydrogen isotopologues. Moreover, the program H2Solv [63] can be used to determine the numerical solution of the electronic Schrödinger equation (6).

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 Table 1: Example of the measurement of fundamental vibrational splittings in the T 2 molecule for the Q(J) band, which is given by transitions from v = 1 to v = 0 for a fixed rotational quantum number J [54]. The central theory values have been extracted from H2spectre [20], while the theory uncertainties are calculated according to Equation (19). All numbers are given in cm −1 .
Line experiment theory difference (4)  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 [65] and references therein. While several lines have been measured, for instance in D 2 with a relative precision of O 10 −6 [65], an improvement is given by the CARS spectroscopy technique [53].
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 T 2 [53,54] and recently of DT [55], see Table 1 for the measurement of T 2 .
Note that there is a discrepancy between theory and experiment in some lines of the T 2 spectrum, which might be explained by New Physics effects. However, the deviating sign of the dif- Table 1 makes a New Physics interpretation acting on the inter-particle potentials more challenging. In the context of DT, the authors of Reference [55] performed a quick analysis of a pure Yukawa potential among the nuclei, whereα and λ 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 strengthα < 2 × 10 −8 α em for a distance of λ = 1 Å. A more detailed analysis of New Physics effects is done in this work.

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 [25]. 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 [66].
In all these cases, one might expect a measurable effect in molecular spectra if the new particle's mass is of O(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 keV-regime [41][42][43]. 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 A,B and two electrons 1 and 2 with mass m 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) with a New Physics coupling between electrons and nuclei g eN , electrons and electrons g ee , and nuclei and nuclei g NN . In principle, each g i j can have both signs and thus work either attractive or repulsive irrespective of the electric charge. However, g NN and g ee should be positive bearing in mind that all coupling strengths g i j are rather multiplicative couplings if expressed in terms of fundamental interactions y i as g i j ∼ y i y * j . The fifth force analyses of References [23] and [55] only constrain the last term of Equation (11) with the replacement g NN 4π → α 5 . In the following, we will extent this analysis by discussing also the other terms in Equation (11) and more types of potentials.

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 i j to zero. For a given New Physics potential V NP , the energy correction ∆E NP v,J of a rovibrational level (v, J) is calculated in first-order perturbation theory by evaluating the matrix element so that the full energy reads Here, E SM v,J describes the Standard Model prediction which, including its theoretical uncertainty δE SM v,J , can be extracted from the computer code H2spectre [20]. For the evaluation of the New Physics shift ∆E NP v,J , we use the same unperturbed states |ψ el 〉 χ 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 |ψ el 〉 of the wave function evaluates to 1, leaving We extract the nuclear wave function χ v,J (R) from H2spectre in a discrete value representation (DVR) with grid spacing ∆R. Analogously to the H2spectre computation of the higher-order Standard Model corrections [20], we use with χ i v,J = χ v,J (R i ) and V i NP = 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 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 [63]. 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 [63]. 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 E el NP (R) on R and m is afterwards re-constructed by interpolation with splines of degree two. E el 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 ∆E NP is calculated using Equation (15) with V NP (R) replaced by E el 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 f 1 , m f,1 , f 2 , m f,2 depends on the angular momentum quantum number J. Since the leading-order energy is independent of the magic quantum number m 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 ∆E NP v,J for the ground state is determined by the minimal eigenvalue of the perturbation in the degenerate subspace.

Finally, the energy ∆E NP
(v 1 ,J 1 )→(v 2 ,J 2 ) for the transition from a level (v 1 , J 1 ) to the level (v 2 , J 2 ) is given by We expect the size of the New Physics contribution to be of order of the uncertainty δE 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, v,J , we approximate the overall uncertainty of the level energy to be δE SM v,J , that is 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 δ∆E NP for the transition energy, Given an experimental measurement ∆E exp (v 1 ,J 1 )→(v 2 ,J 2 ) for a transition (v 1 , J 1 ) → (v 2 , J 2 ) with an uncertainty σ exp (v 1 ,J 1 )→(v 2 ,J 2 ) , we require the theoretical prediction including New Physics effects to lie within the interval 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 i j by combining all measurements listed in Appendix A.
Complete spin-dependent potentials for various mediator particles have been summarized in [27].
In particular, potentials for massive scalar (S) or pseudoscalar (P) mediators φ with mass m between two fermions a and b with masses m a,b , are given by the expressions with the (pseudo)scalar couplings g S(P) ab and the spin Pauli matrices σ 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- 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-1 2 particles only. Applying the criterion in Equation (20), we derive upper limits on the couplings g S,P ab shown in the mass-coupling plane, see Figure 1. For a scalar interaction between electrons, cf. Equation (21), H 2 and HD molecules constrain the scalar coupling g S ee up to O 10 −8 , see Figure 1a. By contrast, the coupling of a pseudoscalar mediator is weakly constrained, g P ee ∼ O 10 −3 for m ∼ 1 keV, meeting the expectation of a suppression by m 2 /m 2 e ∼ 10 −6 relative to the scalar case, as shown in Figure 1b. It can be seen that the bounds for the pseudoscalar coupling become ineffective at about 7 keV. 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.
For a pure nucleus-nucleus interaction, a pseudoscalar contribution is even more suppressed by m 2 /m 2 N ∼ 10 −11 , thus, we only consider the scalar potential. The corresponding limits in the mass-coupling plane are shown in Figure 1c (e) Scalar electron-nucleus interaction with negative coupling g S eN < 0. For each molecule, the corresponding upper limit results from the combination of all available measurements. The area above the curves is excluded. and 1e. As in the electron-electron case, the strongest constraints are again given by the transitions measured in H 2 and HD molecules with upper limits on the coupling g S eN up to O 10 −8 and O 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 S eN also implying a g S ee and g S NN coupling, however, the order of magnitude will not change.

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- Another possibility involves the Stueckelberg mechanism where additionally a light axion-like field is present [73,74]. In this case, the heavy vector is usually referred to a Z boson and thus supposed to have a mass in the GeV regime rather than keV.
The presence of a light spin-1 mediator with vector and axialvector couplings g of the type A µψ γ µ g V ψ + i γ 5 g A ψ ψ and a mass m leads to non-relativistic potentials [27] Here, σ i andr 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 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 In contrast to the pseudoscalar case, the axial vector interaction is not suppressed by the inverse (a) Vector electron-electron interaction.
(c) Axialvector nucleus-nucleus interaction. 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 1c. Bounds from the axialvector exchange are again stronger by two orders of magnitude yielding an upper bound on the coupling g A NN of O 10 −10 , see Figure 2c. Since we do not consider the bosonic nuclei D 2 and HD, the best limits are now given by H 2 measurements for masses below 10 keV and by T 2 lines for larger masses.
Analogously to the pseudoscalar electron-nucleus interaction, the spin-dependent terms vanish for the electronic ground state of H 2 isotopologues. As a consequence, the bounds on the vector potential are the same as for the scalar case, see Figures 1d and 1e, while an axialvector force vanishes entirely.

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 [3,66,[75][76][77][78][79][80][81], 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 [66] and was completed by Hsu and Sikivie in the early 1990s [76]. Their work has been extended by Grifols et al. [77] to the case of massive Dirac and Majorana-type neutrinos of mass m ν , yielding the long-range potentials with the modified Bessel functions K n . In the Standard Model case, the effective coupling G eff is given by the Fermi constant G F . Both potentials scale like ∼ 1/r 5 in the limit of vanishing neutrino masses or short distances, reproducing the well-known result by Feinberg and Sucher [66], 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 powercounting, matching Stadnik's observation for hydrogen atoms in Reference [79].
Assuming a 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 The size of this effect is of O 10 −11 cm −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 [66].
The two-neutrino exchange has recently been studied in the context of atomic parity violation [81,83]. 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 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 [78], see 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.

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 predic- 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 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 measurementassuming 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 T 2 , DT, and HT [84]. This study shows very good agreement with the theoretical prediction.

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