Bounding quantum dark forces Philippe Brax,1,* Sylvain Fichet,2,† and Guillaume Pignol3,‡ 1Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191 Gif/Yvette Cedex, France 2ICTP-SAIFR & IFT-UNESP, R. Dr. Bento Teobaldo Ferraz 271, São Paulo 01140-070, Brazil 3Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, Grenoble F-38026, France (Received 11 October 2017; published 20 June 2018) Dark sectors lying beyond the Standard Model and containing sub-GeV particles which are bilinearly coupled to nucleons would induce quantum forces of the Casimir-Polder type in ordinary matter. Such new forces can be tested by a variety of experiments over many orders of magnitude. We provide a generic interpretation of these experimental searches and apply it to a sample of forces from dark scalars behaving as 1=r3, 1=r5, 1=r7 at short range. The landscape of constraints on such quantum forces differs from the one of modified gravity with Yukawa interactions and features, in particular, strong short-distance bounds from molecular spectroscopy and neutron scattering. DOI: 10.1103/PhysRevD.97.115034 I. INTRODUCTION When going beyond the StandardModel (SM) of particle physics, it is natural to imagine the existence of other light particles, which would have been so far elusive because of their weak or vanishing interactions with the SM particles. Such speculations on dark sectors could be simply driven by theoretical curiosity although there are more concrete motivations coming from two striking observational facts: dark matter and dark energy. In both cases, theoretical constructions elaborated to explain one or both of these fundamental aspects of the Universe tend to assume the existence of dark sectors of various complexity. Among the many possibilities for the content of the dark sector, our interest in this work lies in dark particles with masses below the GeV scale, where quantum chromody- namics (QCD) reduces to an effective theory of nucleons. If a light scalar coupled to nucleons, it would induce a fifth force of the form V ¼ αe−r=λ=r, with λ ¼ ℏ=mc being the Compton wavelength of the scalar and m its mass. The presence of such a Yukawa-like force is sometimes dubbed “modified gravity.” Experimental searches for such fifth forces between nucleons extend from nuclear to astronomic scales and lead to a landscape of exclusion regions, see summary plots in [1–5]. As noted in [6], even in the absence of a light boson linearly coupled to nucleons, other fifth forces can still arise from the dark sector whenever a sub-GeV particle of any spin is bilinearly coupled to nucleons. Such forces would arise from the double exchange of a particle and are, thus, fundamentally quantum. Moreover, in order to take into account retardation effects, such forces have to be com- puted within relativistic quantum field theory. This kind of computation has been first done by Casimir and Polder for polarizable particles [7], and by Feinberg and Sucher for neutrinos [8]. We will refer to such quantum forces as Casimir-Polder forces. There is a variety of motivations for having a particle of the dark sector coupling bilinearly to nucleons. The dark particle can be for instance charged under a symmetry of the dark sector, can be a symmetron from a dark energy model, or simply a dark fermion sharing a contact inter- action with nucleons. Such Z2 symmetry also can be needed in order to explain the stability of dark matter. In the presence of forces which do not have a Yukawa- like behavior, as is the case of the Casimir-Polder forces we focus on, the landscape of fifth force searches is expected to change drastically. A thorough investigation of the exper- imental fifth force searches then becomes mandatory in order to put bounds on such extra forces in a consistent manner, and thus on the underlying dark particles. This requires revisiting each of the experimental results, a task that will be performed in this paper. In Sec. II, we consider Casimir-Polder forces focussing on the case of a scalar with various effective interactions with nucleons. General features of Casimir-Polder forces are then derived *philippe.brax@ipht.fr †sylvain@ift.unesp.br ‡guillaume.pignol@lpsc.in2p3.fr Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 97, 115034 (2018) 2470-0010=2018=97(11)=115034(17) 115034-1 Published by the American Physical Society https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevD.97.115034&domain=pdf&date_stamp=2018-06-20 https://doi.org/10.1103/PhysRevD.97.115034 https://doi.org/10.1103/PhysRevD.97.115034 https://doi.org/10.1103/PhysRevD.97.115034 https://doi.org/10.1103/PhysRevD.97.115034 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ in Sec. III. A generic interpretation of the most recent and stringent fifth force searches, valid for arbitrary potentials, is given in Sec. IV. The exclusion regions will be displayed and discussed in Sec. V. We emphasize that our approach to constrain dark sectors relies only on virtual dark particles, and is thus independent on whether or not the dark particle is stable. The case where the dark particle is stable and identified as dark matter has been treated in a dedicated companion paper, Ref. [6], where a complementarity with low-mass direct detection bounds (like XQC [9–11]) has been found. Searches for dark sectors via loops of virtual dark particles include Refs. [6,12,13], and are yet under-represented in the literature. II. CASIMIR-POLDER FORCES FROM A DARK SCALAR There are many reasons why the dark sector could feature a scalar with a Z2 symmetry with respect to the Standard Model sector. If such a scalar is charged under a new symmetry such as a Uð1ÞX charge while the SM fields are not, the scalar should interact with the SM via bilinear operators. The scalar can also be the pseudo- Nambu-Goldstone boson (pNGB) of an approximate global symmetry, in which case it couples mostly with derivative couplings to the nucleons. Theories of modified gravity can also feature light scalars with a bilinear coupling to the stress-energy tensor [14]. While the properties of these scalars are often considered to be modified by some screening mechanism, it is certainly relevant to consider scenarios where screening is negligible or absent. This is the most minimal possibility and can also serve as a reference for comparison with the screened models. Moreover, for models like the symmetrons, screening does not happen in vacuum. All these possibilities of the dark sector can be UV- completed in a warped five-dimensional framework where the SM lies on the UV brane while the dark sector is on the IR brane and in the bulk. At energies larger than the KK scale μ, interactions between SM and dark sector become exponentially suppressed by ∼e−E=μ and the SM is mostly ignorant of the dark sector—as first noted in [15] in another context. By AdS=CFT this framework is also equivalent to have a dark sector made of bound states from a strongly interacting theory with large number of color and con- formal in the UV. Some details are given in Appendix A. This model will be the focus of a future publication [16]. It is convenient to use an effective field theory (EFT) approach to describe the interactions of the dark particle. All the measurements we consider occur well below the quantum chromodynamics (QCD) confinement scale; hence, we can readily write down effective interactions with nucleons. The operators we consider have the form OnucODS, where Onuc is bilinear in the nucleon fields and ODS is bilinear in the dark sector field.Onuc has in principle a N̄ΓAN structure, where ΓA can have any kind of Lorentz structure. In the limit of unpolarized nonrelativistic nucle- ons, only the interactions involving Onuc ¼ N̄N; N̄γ0N are relevant, the other being either canceled by averaging over nucleon spins or suppressed by powers of m−1 N . In this paper, we focus on the exchange of a dark scalar. The exchange of dark fermions and dark vectors, either self-conjugate or complex, have been treated in [6], and details of the calculations for all these cases are given in Appendix B. Here, we focus on three types of effective interactions, L ¼ LSM þOi, with O0 a ¼ 1 Λ N̄N ϕ2 2 ; O0 b ¼ 1 Λ2 N̄γμNϕ�i∂↔μϕ; O0 c ¼ 1 Λ3 N̄N ð∂μϕÞ2 2 : ð2:1Þ We assume that only one of these operators is turned on at a time. In the O0 a;c cases, we assume a real scalar, while for O0 b we assume a complex scalar. The O0 a interaction corresponds to the case of a symmetron, the O0 b interaction is typically the one generated from a heavy Z0 exchange, and the O0 c would occur if the scalar is the pNGB of a hidden global symmetry. In the last case, as the pNGBmass explicitly breaks the shift symmetry, an interaction of the form m2 Λ2 O0 a could also be present. However, its effect would be negligible at short distance; hence, we do not take it into account. Similar calculations have been performed for disformal couplings in [17,18]. Higher-dimensional operators are in principle present in the effective Lagrangian, and are suppressed by higher powers of either Λ or ΛQCD. The EFT is valid for momenta below min ðΛ;ΛQCDÞ when coupling constants are Oð1Þ in the UV theory. We will assume a universal coupling to protons and neutrons—all our results are easily generalized for nonuniversal couplings. Also, for simplicity, we do not consider the dark particle coupling to electrons. Including the coupling to electrons would lead typically to stronger forces and thus to enhanced limits. As a result of the Oa;b;c interactions, nucleons can exchange two scalars as shown in the Feynman diagram of Fig. 1. This Feynman diagram induces a Casimir-Polder FIG. 1. The exchange of two scalars inducing a force between the nucleons. BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-2 force (i.e., a relativistic van der Waals force) between the nucleons. The forces induced by the Oa;b;c operators have been computed in [6] and are given by the potentials Va ¼ − 1 32π3Λ2 m r2 K1ð2mrÞ; Vb ¼ 1 8π3Λ4 m2 r3 K2ð2mrÞ; Vc ¼ − 1 32π3Λ6r �� 30m2 r4 þ 6m4 r2 � K2ð2mrÞ þ � 15m3 r3 þm5 r � K1ð2mrÞ � ; ð2:2Þ where Ki is the ith modified Bessel function of the second kind. The Va force is consistent with a previous calculation of [19] after matching to our conventions. The main steps of the general calculation are as follows. One first calculates the amplitude corresponding to the diagram in Fig. 1. In order to calculate loop amplitudes in the EFT, dimensional regularization has to be used in order not to spoil the EFT expansion. The one-loop amplitudes can be decomposed over the basis fn ¼ Z 1 0 dxðxð1 − xÞÞn log � Δ Λ2 � ; ð2:3Þ where Δ ¼ m2 − xð1 − xÞq2. Λ is the scale at which the effective theory is matched on to the UV theory and is also the scale at which the EFT breaks down. Then one takes the nonrelativistic limit of the amplitude and identifies the scattering potential Ṽ as iM ¼ −iṼðjqjÞ4m2 Nδ s1s01δs2s 0 2 ; ð2:4Þ where s1;2 ðs01;2Þ corresponds to the spin polarization of each ingoing (outgoing) nucleons. The spatial potential is given by the three-dimensional Fourier transform of ṼðjqjÞ, VðrÞ ¼ Z d3q ð2πÞ3 ṼðjqjÞe iq·r ¼ −i ð2πÞ2r Z ∞ −∞ dρρṼðρÞeiρr; ð2:5Þ where r ¼ jrj and the momentum has been extended to the complex plane in the last equality, ρ≡ jqj. Using standard complex integration, one obtains VðrÞ ¼ −i ð2πÞ2r Z i∞ i2m dρρ½Ṽ�eiρr ¼ i ð2πÞ2r Z ∞ 2m dλλ½Ṽ�e−λr; ð2:6Þ where [V] is the discontinuity from right to left across the positive imaginary axis, ½V� ¼ Vright − V left, and one has defined ρ ¼ iλ. Notice that λ can also be understood as ffiffi t p , the square root of the tMandelstam variable extended to the complex plane. The discontinuities ½fn� needed to compute the Casimir-Polder force via Eq. (2.6) are given in Appendix B. In the case of the scalar dark particle exchanged via the Oa, Ob or Oc operators, the amplitudes are given in Appendix C. The discontinuities needed to calculate the Va;b;c potentials are ½f0� ¼ iπ 2 λ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 − 4m2 p ; ½f1� ¼ iπ 2m2 þ λ2 3λ3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 − 4m2 p ; ½f2� ¼ iπ 6m4 þ 2m2λ2 þ λ4 15λ5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 − 4m2 p : ð2:7Þ The discontinuity of the nonrelativistic scattering potentials for the three diagrams considered above are ½Ṽa� ¼ ½f0� 32π2Λ2 ; ½Ṽb� ¼ m2½f0�− λ2½f1� 8π2Λ4 ; ½Ṽc� ¼ ð6m4 þm2λ2Þ½f0� þ ð24m2λ2 þ λ4Þ½f1� þ 20λ4½f2� 64π2Λ6 : ð2:8Þ At short distance mr ≪ 1 the forces behave as Va ¼ − 1 64π3Λ2r3 ; Vb ¼ 1 16π3Λ4r5 ; Vc ¼ − 15 32π3Λ6r7 ; ð2:9Þ while at long distance mr ≫ 1, the forces go as Va ¼ − ffiffiffiffi m p e−2mr 64π5=2Λ2r5=2 ; Vb ¼ m3=2e−2mr 16π5=2Λ4r7=2 ; Vc ¼ − m9=2e−2mr 64π5=2Λ6r5=2 : ð2:10Þ As sketched in [6], the broad features of these forces can be understood from general principles. The arguments are given in detail in the next section. III. GENERAL CONSIDERATIONS ON CASIMIR-POLDER FORCES A. Structure of the effective theory Let us first comment on the effective theory giving rise to the Casimir-Polder forces. The four-nucleon loop dia- grams we consider come from higher-dimensional oper- ators and are thus more divergent than the four-nucleon diagrams from the UV theory lying above Λ. This implies BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-3 that four-nucleon local operators (i.e.,. counterterms) of the form ðN̄NÞ2, ð∂μðN̄NÞÞ2;… are also present in the effective Lagrangian to cancel the divergences which are not present in the UV theory. The finite contribution from these local operators is fixed by the UV theory at the matching scale, and is expected to be of same order as the coefficient of the logΛ term in the amplitude by naive dimensional analysis (this situation is analog to renormalization of the nonlinear sigma model; see Ref. [20]). The loop amplitudes have the form M ¼ Fðq2Þ þ Gðq2Þ log � m Λ � ; ð3:1Þ where Fðq2Þ is complex, with Fðq2 ¼ 0Þ ¼ 0, andGðq2Þ is a real polynomial in q2 (both depend also on m, Λ). The log term is a consequence of the divergence. The log term is real and contributes to the running of local four-nucleon operators. The Casimir-Polder force arises from the branch cut of Fðq2Þ and is, thus, independent of the log term. An experiment measuring only the Casimir-Polder force will have the advantage of being insensitive to these four- nucleon operators, which are set by the UV completion and, thus, introduce theoretical uncertainty. This happens either when the experiment is nonlocal by design (e.g.,. measuring the force between nucleons at a nonzero dis- tance), or by construction of the observables as we will see in the case of neutron scattering. All the measurements considered in this paper are either fully or approximately insensitive to local four-nucleon interactions. Certain of the experiments we are considering (namely molecular spectroscopy and bouncing neutrons) rely on systems which are small enough to necessitate a quantum description of the nucleus. The wave functions associated to the quantum states in these systems are computed using QED and are, thus, valid down to an internuclear distance of order the radius of the nucleus r ∼ rnuc, with typically r−1nuc ¼ Oð100 MeVÞ. These wave functions will be convoluted with the potential of the dark force, hence the space integral should be cut off at a value rUV given by the maximum of rnuc and the inverse cutoff of the EFT. Depending on the behavior of the wave function and of the dark force at small r, the observables can be sensitive to rUV or not. Having a cutoff-dependent observable is not inconsistent, and often occurs in high-energy physics, for instance when loop contributions to electroweak observables are com- puted. The r < rUV contribution to the space integrals are not computed. When the observable is independent of rUV, one may expect that the r < rUV contribution to the integral is negligible. On the other hand, when the observable is cutoff dependent, it is likely that physics beyond r < rUV contribute as well. In such case, the predictions from the r > rUV piece of the integral should be understood as mere estimates. B. General features of Casimir-Polder forces The main features of Casimir-Polder forces between two nonrelativistic sources can be understood using dimen- sional analysis and the optical theorem. We focus on the double exchange of a particle having local interactions with the sources, the operators used in Sec. II being examples of such a scenario. We further assume that the sources are identical—a similar approach applies similarly to different sources. We denote by X the dark particle exchanged, X̄ its conjugate, m its mass. We use nucleons as source for concreteness. X can take any spin. The generic operator we consider has the form L ⊃ 1 Λn OðXÞN̄ΓAN; ð3:2Þ where ΓA can be any Lorentz structure. When averaging over the nucleon spins, the first nonvanishing Lorentz structures are N̄N (“scalar channel”), N̄γμN (“vector channel”), and we will focus on those ones. Within the above assumptions we obtain the following properties: (i) Sign. Operators of the form OðXÞN̄N give rise to attractive forces. Operators of the form OμðXÞN̄γμN give rise to repulsive forces. (ii) Short distance. An operator of dimension nþ 4 gives rise to a potential behaving at short distance as VðrÞ ∝ 1 r1þ2n : ð3:3Þ (iii) Long distance. When the square amplitude jMðNN̄ ↔ XX̄Þj2 taken at ffiffiffi s p ∼ 2m is suppressed by a power ðs − 4m2Þp (i.e., velocity-suppressed by v2p), the long range behavior of the force is given by VðrÞ ∝ e−2mr r 5 2 þp : ð3:4Þ Let us prove the above properties. Property 2 is simply a consequence of dimensional analysis. When r ≪ 1=m, the potential can be expanded with respect to rm and at first order, VðrÞ ¼ VðrÞjm¼0ð1þOðmrÞÞ. In this limit the potential depends only on r and on the effective coupling 1=Λn squared. The potential having dimension one, it must have a dependence in 1=r2nþ1 so that dimensions match. Notice that this argument applies similarly for the exchange of a single particle (giving then a 1=r potential) or for the exchange of an arbitrary number of particles. For Properties 1 and 3, let us denote the amplitude of interest (Fig. 1) by iMt, and introduce the amplitude iMs ¼ iMðNN̄ → X�X̄� → NN̄Þ, which is the s ↔ t crossing of iMt. In order to get some insight on iMt, we can study iMs use crossing symmetry. The optical theorem applies to iMs, with BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-4 ImðMsÞ ¼ ImðMðNN̄ → X�X̄� → Nðq1ÞN̄ðq2ÞÞÞ ¼ 1 2 X polar Z d4q1 ð2πÞ3 δðq 2 1Þ d4q2 ð2πÞ3 δðq 2 2Þð2πÞ4δ4ðq1 þ q2 − qÞjMðNN̄ → Xðq1ÞX̄ðq2ÞÞj2 ¼ 1 16π ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 4m2 s r X polar jMðNN̄ → Xðq1ÞX̄ðq2ÞÞj2 ð3:5Þ where in the last line we use the fact that the amplitude arising from local interactions [Eq. (2.6)] depend only on the center-of-mass energy ffiffiffi s p . The optical theorem is of interest because ImðMtÞ is directly related to the disconti- nuity of Mt over its branch cut, which is precisely the quantity needed to calculate nonrelativistic potential. In the formalism of Sec. II, we have ImðMtÞ ¼ −2½Ṽ�m2 Nδ s1s2δs 0 1 s0 2 : ð3:6Þ It turns out that ImðMtÞ > 0 (< 0) corresponds to an attractive (repulsive) force. Let us prove Property 1. For the scalar channel, the crossing of ImðMsÞ stays positive, hence ImðMtÞ > 0 and the force is attractive. For the vector channel, we have MðNN̄ → XX̄Þ ∝ Jμ;NJ μ X where the Jμ are vector currents. The square matrix elements takes the form ðJμ;NJν;NÞðJμ;XJν;XÞ. All the Jμ are conserved currents, Jμqμ ¼ 0. The Jμ;N can be pulled outside of the integral in Eq. (3.5). Conservation of the Jμ;N currents implies that they project out the components proportional to qμ of the quantity they are contracted with. It follows that Jμ;NJν;N X polar ðJμ;XJν;XÞ¼Jμ;NJν;NAðsÞðqμqν−sgμνÞ; ð3:7Þ where we have introduced s ¼ ðq1 þ q2Þ2 and AðsÞ is a positive function. In the nonrelativistic limit, one keeps only the μ ¼ ν ¼ 0 components of the nucleon currents, and the projector reduces to qμqν − sgμν ∼ q2—hence AðsÞ has to be positive to ensure ImðMsÞ > 0. The crossing of ImðMsÞ gives ImðMtÞ ¼ ðJ̃μ;NJ̃ν;NÞAðtÞðqμqν − tgμνÞ; ð3:8Þ where J̃μ;N denotes the crossed nucleon currents. In the nonrelativistic limit, we have J̃μ;NJ̃ν;N∼4m2 Nδ μ0δν0δs1s2δs 0 1 s0 2 , q0 ∼ 0, t ∼ −q2. However, when taking the Fourier trans- form of ṼðqÞ [see Eq. (2.6)], jqj is extended to the complex plane. The nonrelativistic potential is then given by an integral of ImðMtÞ over positive values of the real variable λ, which is related to t by λ≡ ffiffi t p . Hence the t variable in Eq. (3.8) is positive when computing the nonrelativistic potential. This implies that ImðMtÞ is always negative, and thus the Casimir-Polder force between nucleons induced by a vector channel is always repulsive. Let us finally prove Property 3. We first remark that the long distance behavior of the VðrÞ potential amounts to having a steep exponential in R ∞ 2m dλλ½Ṽ�e−λr, see Eq. (2.6). When this is true we are allowed to expand ½Ṽ� as a power series at small values of λ, hence at the point λ ¼ 2m. In order to understand what form this power series takes, let us consider the square amplitude jMðNN̄ ↔ XX̄Þj2, which corresponds to pair production or annihilation of X. This amplitude arises from the local operators of Eq. (2.6) hence it depends only on the center- of-mass energy ffiffiffi s p . We extend s to the complex plane. We can always perform a power series expansion near s ¼ 4m2,1 jMðNN̄ ↔ XX̄Þj2 ¼ 4m2 Nðaþ bðs − 4m2Þ þ cðs − 4m2Þ2 þ…Þ ð3:9Þ where the 4m2 N factor is introduced for further conven- ience and the a, b, c are dimensionful constants. Using the optical theorem, we obtain that ImðMsÞ ¼ m2 N 4π2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 4m2 s r ðaþ bðs − 4m2Þ þ cðs − 4m2Þ2 þ…Þ; ð3:10Þ and crossing then gives ImðMtÞ ¼ m2 N 4π2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 4m2 t r ðaþ bðt − 4m2Þ þ cðt − 4m2Þ2 þ…Þ: ð3:11Þ 1Note that the quantity ffiffiffiffiffiffiffiffiffiffi s−4m2 p 4m ¼ q m ≡ v taken in the center-of- mass frame is the usual velocity of the X particle. It is common to say that the squared matrix-element is “velocity-suppressed” when e.g., a ¼ 0. The nucleons being by assumption heavier than X, neither production nor annihilation of X can physically happen at this threshold. However, formally, nothing forbids us to perform the expansion. BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-5 ImðMtÞ is related to ½Ṽ� by Eq. (3.6) and ½Ṽ� is related to VðrÞ by Eq. (2.6). The potential in the long range limit turns out to be2 VðrÞ¼− 1 32π5=2 e−2mr � a m1=2 r5=2 þb 6m3=2 r7=2 þc 60m5=2 r9=2 þ… � : ð3:13Þ We can see that an extra factor of 1=r in VðrÞ is associated to each factor of s − 4m2 in the expansion of jMðNN̄ ↔ XX̄Þj2. IV. FIFTH FORCE SEARCHES This section describes how to interpret the results of a number of experiments as bounds on an arbitrary fifth force. A. Neutron scattering Progress in measuring the scattering of cold neutrons off nuclei have been recently made and have been used to put bounds on short-distance modified gravity, [21–28]. The cold neutron scattering cross section can be measured at zero angle by “optical” methods, at nonzero angles using Bragg diffraction, or over all angles by the “transmission” method giving then the total cross section [29]. In the following, we adapt the analyses of [27] to the Casimir-Polder forces of Eq. (2.2). At low energies the standard neutron-nuclei interaction is a contact interaction in the sense that it can be described by a four-fermion operator O4N ¼ ðN̄NÞ2.3,4 New physics can in general induce both contact and noncontact contributions to the neutron-nuclei interaction. A noncontact contribution van- ishes at zero momentum, while a contact contribution remains non-null and can be described by O4N. It is convenient to introduce the scattering length ffiffiffiffiffiffiffiffiffiffi σðqÞ 4π r ≡ lðqÞ ¼ lCstd þ lCNP þ lNCNP ðqÞ; ð4:1Þ where the lCstd, l C NP local terms are independent of momen- tum transfer q and lNCNP ðqÞ, which satisfies lNCNP ðq ¼ 0Þ ¼ 0, is the noncontact contribution. The lNCNP ðqÞ term contains the Casimir-Polder force (see Sec. III), and log terms of the form jqj2n logðm=ΛÞ. The new physics contribution lNPðqÞ is related to the scattering potential Ṽ by lNPðqÞ ¼ 2mNṼðqÞ, which is just the Born approximation.5 For the forces described in Eq. (2.2), the new physics contributions are given by laðjqj2Þ ¼ mN 16π2Λ2 f0; ð4:2Þ lbðjqj2Þ ¼ mN 4π2Λ4 ðm2f0 þ jqj2f1Þ; ð4:3Þ lcðjqj2Þ ¼ mN 16π2Λ6 �� 3m4 − m2jqj2 2 � f0 þ �jqj4 2 − 12m2jqj2 � f1 þ 10jqj4f2 � ; ð4:4Þ where the fn are the loop functions defined Eq. (2.3). A convenient way to look for an anomalous interaction is to search for lNCNP ðqÞ by comparing the scattering length obtained by different methods, using for instance lBragg − lopt, ltot − lopt. This approach eliminates the contact contributions lCstd and lCNP, and is, therefore, only sensitive to lNCNP ðqÞ. (i) Opticalþ Bragg.—One approach is to compare the forward and backward scattering lengths measured respectively by optical and Bragg methods. Using the analysis from [27], one has a 95% CL bound 1 2mN ðlið0Þ − liðk2BraggÞÞ < ð0.01 fmÞ2; ð4:5Þ with kBragg ¼ 2 keV. (ii) Opticalþ total cross section.—The total cross section measured by the transmission method pro- vides the average scattering length l̄iðkÞ ¼ 1 2 Z π 0 dθ sinðθÞlið4k2sin2ðθ=2ÞÞ: ð4:6Þ Using information from optical method measure- ment, we have the 95% CL bound lið0Þ − l̄iðkexÞ < 6 × 10−4 fm; ð4:7Þ with kex ¼ 40 keV. For both methods, a dependence on the jqj2n logðm=ΛÞ remains, which turns out to be mild in practice. Hence, our results are still approximatively independent of the local 2The general case is obtained similarly using the identity Z ∞ 2m dλλðλ2 − 4m2Þ12þp ¼ � 4m r � pþ1 Γð3=2þ pÞffiffiffi π p Kpþ1ð2mrÞ: ð3:12Þ 3Various contact operators can be written when differentiating between neutrons and protons. However, this is not crucial for the discussion hence we use only “N,” which refers to both protons and neutrons. 4As described in [27], there is also a small electromagnetic dipole interaction, which is taken into account in the analysis and which we do not discuss here. 5We emphasize that no extra theoretical assumptions have been made above, this formalism is a mere rewriting of the scattering amplitude to highlight the important features and to put it in a form more common to neutron experiments. BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-6 four-nucleon operators, which are fixed by the unspecified UV completion (see Sec. III). B. Molecular spectroscopy Impressive progress on both the experimental [30–37] and the theoretical [38–49] sides of precision molecular spectroscopy have been accomplished in the past decade, opening the possibility of searching for extra forces below the Å scale using transition frequencies of well-understood simple molecular systems. Certain of these results have recently been used to bound short distance modifications of gravity, see Refs. [5,50–52]. The most relevant systems for which both precise measurements and predictions are available are the hydro- gen molecule H2, the molecular hydrogen-deuterium ion HDþ and muonic molecular deuterium ion ddμþ, where d is the deuteron. This last system is exotic in the sense that a heavy particle, the muon, has been substituted for an electron. As a result the internuclear distances are reduced, providing a sensitivity to forces of shorter range, and thus to heavier dark particles. The presence of an extra force shifts the energy levels by ΔEi ¼ Z d3rΨ�ðrÞViðrÞΨðrÞ ð4:8Þ at first order in perturbation theory. We have computed these energy shifts for the transitions between the ðν ¼ 1; J ¼ 0Þ − ðν ¼ 0; J ¼ 0Þ states for H2, the ðν ¼ 4; J ¼ 3Þ − ðν ¼ 0; J ¼ 2Þ states of HDþ, and the binding energy of the (ðν ¼ 1; J ¼ 0Þ) state of ddμþ using the wave functions given in [5,51]. ν and J are respectively the rotational and vibrational quantum numbers. For the quantum states considered here, the typical internuclear distances are ∼1 Å for H2 and HDþ, and ∼0.005–0.08Å for ddμþ. The wave functions of these states are shown in Appendix D. The bounds on the extra forces are obtained by asking that ΔE be smaller than the combined uncer- tainties given by δE ¼ δEexp ⊕ δEth. These experimental and theoretical uncertainties are summarized in Table I, and more details can be found in the original references. For the transition energies, the experimental uncertainties are larger than the theoretical ones by a factor of Oð1Þ to Oð10Þ. For the binding energy of the ddμþ ground state, the exper- imental uncertainty dominates by a Oð10Þ factor. At small r, the wave functions of these states are constant except for ddμþ which behaves as ∼r [53]. In all cases but for the ðν ¼ 1; J ¼ 0Þ states of ddμþ, the wave function at small r is so small that it is neglected—and its value is even difficult to obtain numerically [51,53]. On the other hand for ddμþ the tail at small r is not negligible. For a force in 1=r5 (resp. 1=r7) the predicted energy shift depends on logðrUVÞ (resp. 1=r2UV). We have taken rUV to be the radius of the deuteron, which is of order 2 × 10−5 Å. To get a concrete idea of the dependence on the choice of the cut-off scale rUV, one can compare with the result obtained when cutting the wave function at a larger distance, taking as an example rUV ¼ 10−3 Å. In terms of the sensitivity to Λ, we get that the change is Oð10%Þ for the 1=r5 force, and of a factor ∼4 for the 1=r7 force. In the latter case, this follows from the 1=r2UVΛ6 dependence of the short distance contribution to energy levels, which is the dominant one, unlike in the other cases considered, and for which a change of rUV by a factor of 50 leads to a change in Λ of order 501=3 ∼ 4. C. Experiments with effective planar geometry Avariety of experiments searching for new forces at sub- millimeter scales are measuring the attraction between two dense objects with typically planar or spherical geometries. Whenever the distance between the objects is small with respect to their size, these objects can be effectively approximated as infinite plates, and the force becomes proportional to the potential energy between the plates. This is the proximity force (or Derjaguin’s) approximation [54]. An important subtlety is that most of the experiments are using objects coated with various layers of dense materials, that should be taken into account in the compu- tation of the force. Thus, we end up with calculating the potential between two plates with various layers of density for each. The effective plane-on-plane geometries are summarized in Table II. It is convenient to describe all these configurations at once using a piecewise mass density function describing n layers over a bulk with density ρ, γnðzÞ ¼ 8>>>>>< >>>>>: ρn if 0 < z < Δn ρn−1 if 0 < z < Δn þ Δn−1 .. . ρ if z > P n i Δi: : ð4:9Þ In this notation, the layer labelled n is the closest to the other plate. The potential between an infinite plate of density structure γaðzÞ and a plate with area A and density structure γbðzÞ at a distance s is then given by Vplate i ¼ 2πA Z ∞ 0 dρρ Z ∞ 0 dzaγðzaÞ Z ∞ 0 dzbγðzbÞ × Vi � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 þ ðsþ za þ zbÞ2 q � : ð4:10Þ TABLE I. Experimental and theoretical uncertainties for molecular observables (transition or bin-ding energies) consid- ered in this work. δEexp δEth H2 ðν ¼ 1; J ¼ 0Þ − ðν ¼ 0; J ¼ 0Þ [50,51] 3.0 neV 1.5 neV HDþ ðν ¼ 4; J ¼ 3Þ − ðν ¼ 0; J ¼ 2Þ [50] 0.33 neV 0.044 neV ddμþ ðν ¼ 1; J ¼ 0Þ [5] 0.7 meV < 0.1 meV BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-7 In practice, most of these sub-millimeter experiments have released their results as bounds on a Yukawa-like force. In order to obtain consistent bounds on the strength of the Casimir-Polder forces Λ as a function of the scalar mass m, we have to compare the plane-on-plane potentials from the Casimir-Polder forces to the plane-on-plane potential from the Yukawa force. Bounds on the ðα; mÞ parameters of the Yukawa force can be then translated into bounds on the ðΛ; mÞ parameters of the Casimir-Polder forces, using the limit-setting procedure provided by each experiment. The plane-on-plane potential for the Yukawa force is straightforward to compute analytically and reads Vplate Yuk ¼2πA 1 m3 e−msKa nKb n0 ; Kn¼ρnþ Xn l¼1 ðρl−1−ρlÞexp � −m Xl i¼1 Δn−iþ1 � ð4:11Þ with ρ0 ¼ ρ. In the case of the Casimir-Polder forces shown in Eq. (2.2), the triple integral of Eq. (4.10) are much less trivial to carry on analytically. A numerical integration is, however, easily done. It is worth noticing that the z-integrals on the Casimir- Polder potentials can be realized using a different repre- sentation for the potentials, which naturally occurs when calculating the diagram of Fig. 1 in a mixed position- momentum space formalism, which we will extensively use in future work [62]. D. Bouncing neutrons New forces can also be probed using bouncing ultracold neutrons (i.e., neutrons with velocities of a few m/s) [63–67]. The vertical motion of a neutron bouncing above a mirror nicely realizes the situation of a quantum point particle confined in a potential well, the gravitational potential mNgz pulling the neutron down, and the mirror pushing the neutron up. The properties of the discrete stationary quantum states for the bouncing neutron can be calculated exactly. The wave function of the kth state reads ψkðzÞ ¼ CkAiðz=z0 − ϵkÞ; ð4:12Þ where Ai is the Airy function, ϵk is the sequence of the negative zeros of Ai and z0 ¼ ð2m2 Ng=ℏ 2Þ−1=3 ≈ 6 μm. The first wave functions are shown in Appendix D. The theoretical energies of the quantum states are Ek¼mNgz0ϵk¼f1.41;2.46;3.21;4.08;…g peV: ð4:13Þ Recently, a measurement of the energy difference E3 − E1 was performed at the Institut Laue Langevin in Grenoble using a resonance technique [68]. The result is in agreement with the theoretical predictions. From this experiment a bound can be set on any new force which would modify the energy levels, the experimental precision being δðE3 − E1Þ < 10−14 eV: ð4:14Þ Let us calculate the energy shift due to the new Casimir- Polder dark force. The additional potential of a neutron at a height z above a semi-infinite glass mirror is given by Vi;zðzÞ ¼ 2π ρglass mN Z ∞ z dz0 Z ∞ 0 ρdρViðrÞ ð4:15Þ TABLE II. Summary of the fifth forces experiments with effective planar geometry used in this work. The reported densities which differ from the nominal ones given in Table III are indicated in parentheses. Experiment Plane a Separation Plane b Stanford [55] − Au, 30 μm 25 μm Au IUPUI [56] Sapphire Cr, 10 nm Au, 250 nm ½30–8000� nm Au, 250 nm Cr, 10 nm Si, 2.1 μm SiO2Au, 2.1 μm Lamoreaux [54,57] SiOð2.23Þ 2 Cu, 0.5 μm Au, 0.5 μm ½0.6; 6� μm Au, 0.5 μm Cu, 0.5 μm SiOð2.40Þ 2 AFM [54,58] Polystyrene Auð18.88Þ, 86.6 nm [62, 350] nm Auð18.88Þ, 86.6 nm Sapphire μ-oscillator [54,59,60] Sapphireð4.1Þ Cr, 10 nm Au, 180 nm [180, 450] nm Au, 210 nm Cr, 10 nm Si Casimirless [54,60,61] Sapphire Cr, 1 nm Au, 200 nm [150, 500] nm Au, 150 nm Pt, 1 nm Ge, 200 nm Ti, 1 nm SiAu, 200 nm TABLE III. Densities of the materials used in the fifth force experiments listed in Table II. Polyester SiO2 Si Sapphire Ti Ge Cr Cu Au ρ ½g cm−3� 1.06 2.23 2.33 3.98 4.51 5.32 7.14 8.96 19.32 ρ ½106 · keV4� 4.75 9.99 1.04 1.78 2.02 2.38 3.20 4.01 8.66 BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-8 where ρglass mN ¼ 1010 eV3 is the number density of nucleons in the glass, ViðrÞ is the potential between the neutron and one nucleon at a distance r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 þ z02 p . The double integral in the expression of the potential can be simplified to a single integral: Vi;zðzÞ ¼ 2π ρglass mN Z ∞ z rðr − zÞViðrÞdr: ð4:16Þ In the case of the potentials Va and Vb, the integrals cannot be calculated analytically. However, we found suitable analytical approximations having the correct asymptotic behavior at zero and infinite height, Va;zðzÞ ¼ − ρglass mN 1 32π2Λ2 Z ∞ 2mz u − 2mz u K1ðuÞdu ≈ − ρglass mN 1 32π2Λ2 K0ð2mzÞ 1þ 2mz ; ð4:17Þ and Vb;zðzÞ ¼ ρglass mN m2 4π2Λ4 Z ∞ 2mz u − 2mz u2 K2ðuÞdu ≈ ρglass mN m2 4π2Λ4 K1ð2mzÞ 2mzð3þ 2mzÞ : ð4:18Þ The approximate expressions have a relative precision of better than 50% for Va;z and better than 3% in the case of Vb;z, for all values of z. The case of Vc;z remains to be done. Using the approximate expressions, we have computed the shift in the energy levels of the neutron quantum bouncer using first order perturbation theory: δEk ¼ Z ∞ 0 jψkðzÞj2VzðzÞdz: ð4:19Þ These predictions are UV-insensitive. The bounds on the extra forces Va and Vb as a function of the mediator massm are obtained from the experimental constraint (4.14). They are reported in Figs. 2 and 3. E. Moon perihelion precession The existence of a fifth force at astrophysical scales would imply a slight modification of planetary motions. Any such fifth force can be treated perturbatively whenever it is small with respect to gravity at the distance between the two bodies. The modification of the equation of motion implies, among other effects, an anomalous precession of the perihelion of the orbit. In the case of the Moon, this precession is experimentally measured to high precision by lunar laser ranging experiments [69]. The fundamental Casimir-Polder forces of Eq. (2.2) are between two nucleons. For macroscopic bodies, the poten- tials are given by m1m2 m2 N Vi. Let us calculate the planetary motion in the presence of these new forces. We follow the formalism of Ref. [1]. The radial component of the Casimir-Polder forces between Earth and Moon is given by FiðrÞ ¼ − m☾m⨁ m2 N ∂rViðrÞ. Introducing u ¼ 1 r, the Earth- Moon orbital equation reads d2u dθ2 þ u ¼ m2 ☾ L2u2 ðm☾m⨁Gu2 − Fið1=uÞÞ; ð4:20Þ where L≡m☾r2 dθ dt is the conserved angular momentum and the first term in the parenthesis is the gravitational force. The solution of the unperturbed equation reads Eöt–Wash Stanford Bouncing neutrons Lamoreaux IUPUI μ–oscillators AFM σ opt/σ tot (Neutron scattering) H2 HD+ ddμ+ 10–1310–1210–1110–10 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 10–7 10–6 10–5 10–4 10–3 10–2 10–1 100 101 102 107 106 105 104 103 102 101 100 10–1 10–2 10–3 10–4 10–5 10–6 c/m [meters] 1/ [G eV –1 ] m [eV] FIG. 2. Bounds on a scalar coupled to nucleons via the Oa interaction. The yellow region is excluded at 95% CL. See Sec. IV for details on exclusion regions. Stanford Bouncing neutrons Lamoreaux IUPUI μ–oscillators AFM σ opt/σ tot (Neutron scattering) σ opt/σ Bragg HD+ H 2 ddμ+ 10–13 10–12 10–11 10–10 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 100 101 102 103 104 105 106 107 108 107 106 105 104 103 102 101 100 10–1 10–2 10–3 10–4 10–5 c/m [meters] 1/ [G eV –1 ] m [eV] FIG. 3. Bounds on a scalar coupled to nucleons via the Ob interaction. Same conventions as Fig. 2. BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-9 uðθÞ¼u☾ð1þϵcosðθ−θ0ÞÞ; u☾¼ m3 ☾m⨁G L2 ; ð4:21Þ where ϵ is the orbital eccentricity (ϵ ¼ 0.0549 for the Moon), θ0 indicates the perihelion of the ellipse, and the major semiaxis a☾ is given by a−1☾ ¼ u☾ð1 − ϵ2Þ. At first order in perturbation theory, the extra force is just as a constant, Fið1=a☾Þ, which only modifies u☾, the overall size of the orbit. At second order in perturbation theory, one has Fið1=uÞ¼Fið1=a☾Þþ � u− 1 a☾ �∂Fið1=uÞ ∂u ���� u¼1=a☾ : ð4:22Þ The term linear in u modifies the frequency of the orbit on the left-hand side of the equation of motion. The motion is now given by uðθÞ ¼ u☾ð1þ ϵ cosωðθ − θ0ÞÞð1þ…Þ; ω2 ¼ 1þ u☾a4☾ Gm2 N ∂2 rViðrÞjr¼a☾ ; ð4:23Þ where the ellipsis denotes irrelevant corrections to the overall magnitude of the orbit. Havingω ≠ 1 implies a precession of the perihelion, which can be seen using cosωðθ − θ0Þ ¼ cosωðθ − θ0 þ 2πn ω Þ. The precession angle between two rotations is finally given by δθi ¼ −π a3☾ Gm2 Nð1 − ϵ2Þ ∂ 2 rViðrÞjr¼a☾ : ð4:24Þ We apply this general formula to the Casimir-Polder potentials of Eq. (2.2). Interestingly, the Va and Vc poten- tials, which are attractive, induce an advance of the peri- helion while Vb, which is repulsive, induces a delay of the perihelion. The Moon precession angle is constrained by lunar laser ranging experiments. Other well-understood perturbations induce Moon’s orbit precession: the quadrupole field of the Earth, other bodies of the solar system, and general relativity. Once all these effects are taken into account, one obtains a bound on an extra, anomalous precession angle. Following Ref. [2], an experimental limit from lunar laser ranging is given as δθi < 2π × 1.6 × 10−11: ð4:25Þ V. BOUNDS ON FORCES FROM DARK SCALARS Let us apply the experimental bounds obtained in Sec. IV to the Casimir-Polder forces from a dark scalar given in Eq. (2.2). It is instructive first to understand qualitatively the landscape of exclusion regions on the Casimir-Polder forces. Let us consider the exclusion regions for the Yukawa force (see e.g., [3]). Starting from large scales, the reach of the experiment starts to decrease very steeply below the scale of the Eöt-Wash experiment, at roughly λ < Oð10−4 mÞ down to atomic scales. In this region of λ, the bound on the strength of the Yukawa force α scales very roughly as α < 10−22ð1mλ Þ5, demonstrating the increasing difficulties in measuring forces at small distances. The Casimir-Polder forces behave as 1=rn with n ≥ 3 at short distance. This has the crucial implication that the con- straints from short distance experiments will gain impor- tance and those from long distance will lose importance compared to the exclusion regions on the Yukawa-like force. In particular, one can expect the Eöt-Wash bound to dominate over the bounds from all experiments at larger scale, to the possible exception of lunar laser ranging. Moreover, when n ¼ 7, the decrease of sensitivity in λ−5 is expected to be overwhelmed by the increase of the force in r−7, implying that bounds from the experiments at the smallest scales (from neutron scattering and molecular spectroscopy) dominate over all the bounds from higher distances. The exclusion regions for the Va, Vb, Vc Casimir-Polder potentials are respectively presented in Figs. 2, 3, 4. For the Va potential, we obtain that the Eöt-Wash bound (deduced from [70], Sec. IX B) is the dominant one for λ > 10−3 m. For both Vb and Vc potentials, we obtain indeed an inversion in the hierarchy of bounds. The two leading bounds turn out to be from the ddμþ molecular ion and from the neutron scattering bound combining optical and total cross sections. This fact can be taken as an incentive to pursue and develop such small scale experiments. We remind that, as explained in Sec. IV B, for the Vb and Vc potential, the ddμþ observable has some dependence in StanfordLamoreaux IUPUI μ–oscillators AFM σ opt/σ tot (Neutron scattering) σ opt/σ Bragg H 2, HD+ ddμ+ 10–13 10–12 10–11 10–10 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 100 101 102 103 104 105 106 107 108 107 106 105 104 103 102 101 100 10–1 10–2 10–3 10–4 10–5 c/m [meters] 1/ [G eV –1 ] m [eV] FIG. 4. Bounds on a scalar coupled to nucleons via the Oc interaction. Same conventions as Fig. 2. BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-10 rUV. In our computations, as is customarily done, rUV is taken to be the radius of the deuteron. Using the calculation given in 4.5, we obtain that limits from lunar laser ranging are indeed subleading. At zero mass, the bounds on Λ for the Va, Vb, Vc potentials are found to be respectively Λ > 2 GeV, 6 × 10−5 eV, 2 × 10−8 eV. All these bounds are overwhelmed by stronger ones from shorter distance experiments. VI. CONCLUSIONS There are many motivations—including dark matter and dark energy—for speculating on the existence of a dark sector containing particles with a bilinear coupling to the Standard Model particles. Whenever one of the dark particles is light enough and couples to nucleons in a spin- independent way, it induces forces of the Casimir-Polder type, that are potentially accessible by fifth force experi- ments across many scales. The short- and long-range behaviors of these forces as well as their sign can all be understood and predicted using dimensional analysis and the optical theorem. We provide a comprehensive (re)interpretation of bounds from neutron scattering to the Moon perihelion precession, applicable to any kind of potential. We then focus on the case of a scalar with a variety of couplings to nucleons, generating forces with 1=r3, 1=r5, 1=r7 short-distance behaviors. It turns out that forces in 1=r5, 1=r7 are best constrained by neutron scattering and molecular spectroscopy, which provides extra motivation to pursue these kind of low-scale experi- ments. Implications for dark matter searches have been discussed in Ref. [6]. ACKNOWLEDGMENTS This work is supported in part by the EU Horizon 2020 research and innovation program under the Marie- Sklodowska Grant No. 690575. S. F. thanks V. Korobov for important clarifications on molecular wave functions. This article is based upon work related to the COST Action CA15117 (CANTATA) supported by COST (European Cooperation in Science and Technology). S. F. was supported by the São Paulo Research Foundation (FAPESP) under Grants No. 2011/11973 and No. 2014/21477-2. APPENDIX A: UV COMPLETIONS In this Appendix, we consider models which lead to the operators O0 a;b;c at low energy. We first show a simple model giving rise to O0 b, then we readily present a more general setting, which can be described in five dimensions using the AdS=CFT correspondence, and which can provide viable UV completions to any kind of dark particle and low-energy interaction we have considered in this paper and in [6]. 1. Minimal scenario for O0 b: A light leptophobic Z0 For operator O0 b, let us assume that the quarks and ϕ are charged under an extra Uð1Þ, whose gauge boson Z0 has a mass mZ0 . Then, we have L ⊃ gBQqZμJ μ q þ gBQϕZμJ μ ϕ: ðA1Þ At low energy, the quarks go into nucleons, and the theory takes the form L ⊃ gBQNZμJ μ N þ gBQϕZμJ μ ϕ: ðA2Þ For E < mZ0, one integrates out the Z0 and the low-energy effective theory takes the form L ⊃ − 1 2m2 Z0 ðgBQNJ μ N þ gBQϕJ μ ϕÞ2: ðA3Þ This gives 1=Λ2 ¼ −gBQNgϕQϕm−2 Z0 ðA4Þ and also a four-nucleon interaction that could be tested by neutron scattering. A light-enough Z0 is inaccessible at the LHC [71]. Other bounds have been discussed in [6,72–76]. The Z0 is assumed to be leptophobic; hence, the theory has gauge anomalies, which should be canceled by new chiral fermions which cannot be arbitrarily heavy. Recent devel- opment on meson decays via Z0 [77,78] seem also to put challenging bounds on Λ (see [77]), in which case the fifth force bounds would need to be improved by several order of magnitudes to be competitive on this minimal Z0 scenario. However, this scenario can be embedded in the framework of next section, in which case experimental constraints from mesons, colliders are relaxed. 2. UV completion from a sub-GeV warped extra dimension There is a scenario in which the dark particles auto- matically decouple from the SM sector above a scale which can be chosen arbitrarily low. This scenario is fully general in the sense that it applies to any kind of dark particle and of operators considered in this paper. This decoupling property was first noted in [15] in a different context. This automatic decoupling between the SM and the dark sector happens when the latter is a strongly interacting theory which is conformal in the UV and which develops bound states in the IR. Although the mechanism may apply in other cases, we focus here on large Nc and large t’Hooft coupling, in which case the AdS=CFT correspondence applies and quantitative features can thus be easily obtained. BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-11 The CFT bound states identified as the dark particles appear below a scale μ—which will typically be below ΛQCD in our context. All the SM fields are assumed to be elementary, and the CFT bound states will form the dark sector. The key feature of this scenario is that above the scale of conformal breaking μ, the CFT bound states become invisible to elementary SM fields. This scenario has a number of attractive features which do not need to be discussed here. In this Appendix, we lay down only the aspects relevant in order to UV-complete the operators studied in this paper, and the five-dimensional computa- tions are not detailed. Let us use a five-dimensional model to illustrate con- cretely what happens. We consider a slice of AdS5 with curvature k ∼MPl where the nucleons—and the other SM fields remaining below ΛQCD— are on the UV brane, at z0 ¼ 1=k. We assume that the ϕ field is purely composite and lies thus on the IR brane at z1 ¼ 1=μ. The mediator X is a bulk field, chosen here to be a scalar Φ with Neumann boundary conditions, and which induces interactions between the fields of two branes i.e., between the SM and the dark sector. The Lagrangian including brane interactions is S ¼ Z d4xμdz ffiffiffiffiffiffi −g p � 1 2 ∇MΦ∇MΦ − 1 2 m2 ΦΦ2 þ λδðz − z0ÞN̄NΦþ κδðz − z1Þϕ2Φ � : ðA5Þ The typical values of the parameters are λ ¼ 1= ffiffiffi k p , κ ¼ μ= ffiffiffi k p The bulk mass has to satisfy the Breitenlohner- Freedman bound, m2 Φ ≥ −4k2 to prevent tachyonic insta- bilities in AdS5. For further convenience the bulk mass will be para- metrized as m2 Φ ¼ ð−3 − 2ϵþ ϵ2Þk2. If the boundary con- ditions were tuned it could have a zero mode, but here such an assumption is unnecessary, our interest rather lies in the Kaluza-Klein modes. The KK mode wave functions have values on the UV brane which are the largest for ϵ ¼ 0, and decrease exponentially for jϵj > 0. For con- creteness we consider the case with a nonzero IR brane mass δðz − z1ÞbIRkΦ2=2 and a zero UV brane mass. To illustrate our point, we consider the Nϕ → Nϕ amplitude for q > μ, where q is the center of mass energy of the process. For ϵ slightly larger than zero (so that ðμ=kÞϵ ≪ 1), one finds iM ∝ iλκ ffiffiffi π p ΓðϵÞ 1ffiffiffiffiffi kμ p � 2k q � 3=2−ϵ e− q μ: ðA6Þ We see that an exponential suppression of the amplitude occurs. This shows how the CFT states become invisible to the elementary fields above μ. Let us now consider q ≪ μ and compute the nucleon- dark particle interaction induced by the KK modes, i.e., by the CFT bound states in the dual picture. All the KK modes are integrated out, and for ϵ slightly larger than 0, we find the nucleon–dark particle interaction to be L4D ⊃ − � 1 2 λ2 kð2þ 2ϵ − bIRÞ μ2bIRð2þ 2ϵÞk � μ k � 2ϵ ðN̄NÞ2 þ λκ k bIRμ2 � μ k � ϵ N̄Nϕ2 þ 1 2 κ2 k bIRμ2 ϕ4 � ðA7Þ ∼ − � 1 2μ2 � μ k � 2ϵ ðN̄NÞ2 þ 1 μ � μ k � ϵ N̄Nϕ2 þ 1 2 ϕ4 � : ðA8Þ where the O0 a contact operator between SM and dark particle appears, with effective scale Λ ¼ μ � k μ � ϵ : ðA9Þ We can see a hierarchy between the contact interactions, simply because the mediators couple strongly to the DS and weakly to the SM. The SM self interaction is suppressed with respect to the SM-dark particle interaction, and the dark particle is strongly self-interacting. A very similar analysis can be done for any kind of IR brane operator and bulk mediator. For example a derivative coupling of the form δðz − z1ÞΦð∂μϕÞ2 would occur if ϕ is a Nambu-Goldstone boson. This generates theO0 c operator. In our setup, there is no source of explicit breaking of the global symmetry; hence, the GB is massless. However, a mass term parametrizing a source of explicit symmetry breaking can always be introduced—in which case brane operators of the form δðz − z1Þm2ϕ2Φ should also taken into account, which will contribute to O0 a. Finally, we note that the CFT sector can be approximately supersymmetric, as first noticed in [15], letting a scalar dark particle be arbitrarily lighter than μ. One can verify using [6] that there are no cancellations in the Casimir-Polder force in presence of superpartners. APPENDIX B: CALCULATION OF THE POTENTIALS This Appendix contains details of the computation for the potentials in Eq. (2.2) and those given in Ref. [6]. The full set of operators considered is BRAX, FICHET, and PIGNOL PHYS. REV. D 97, 115034 (2018) 115034-12 O0 a ¼ 1 Λ N̄Njϕj2; O0 b ¼ 1 Λ2 N̄γμNϕ�i∂↔μϕ; O0 c ¼ 1 Λ3 N̄N∂μϕ�∂μϕ; O½ a ¼ 1 Λ2 N̄Nχ̄χ; O½ b ¼ 1 Λ2 N̄γμNχ̄γμχ; O½ c ¼ 1 Λ2 N̄γμNχ̄γμγ5χ; O1 a ¼ 1 Λ3 N̄NjmXμ þ ∂μπj2; O1 b ¼ 1 Λ2 2N̄γμNImðXμνXν� þ ∂νðXνX� μÞ þ ∂μc̄c�Þ; O1 c ¼ 1 Λ3 N̄NjXμνj2; O1 d ¼ 1 Λ3 N̄NXμνX̃� μν: ðB1Þ A dark particle of spin 0, 1=2, 1 is denoted by ϕ, χ, X. π and c, c̄ are respectively the Goldstone bosons and ghosts accompanying X. At that point the dark particle can be self- conjugate (real scalar or vector, Majorana fermion) or not (complex scalar or vector, Dirac fermion). When X is complex, so are π, c and c̄. We will give the results for all cases. We introduce η ¼ � 0 if self-conjugate 1 otherwise: ðB2Þ We calculate the loop diagram of Fig. 1 induced by each of these operators using dimensional regularization. The matching of the effective theory with the UV theory being done at the scale Λ, we can readily identify the divergent integrals as (see [79,80])6 Z d4l ð2πÞ4 1 ðl2 − ΔÞ2 → −i ð4πÞ2 logðΔ=Λ 2Þ; ðB3Þ Z d4l ð2πÞ4 l2 ðl2 − ΔÞ2 → −2i ð4πÞ2Δ logðΔ=Λ2Þ; ðB4Þ Z d4l ð2πÞ4 ðl2Þ2 ðl2 − ΔÞ2 → −3i ð4πÞ2Δ 2 logðΔ=Λ2Þ: ðB5Þ From these amplitudes, the discontinuities in the non- relativistic scattering potential Ṽ are given by Eq. (2.4) and are found to be ½Ṽ0 a�¼2η ½f0� 32π2Λ2 ½Ṽ0 b�¼η m2½f0�−λ2½f1� 8π2Λ4 ½Ṽ0 c�¼2η ð6m4þm2λ2Þ½f0�−ð24m2λ2þλ4Þ½f1�þ20λ4½f2� 64π2Λ6 ½Ṽ1=2 a �¼2η 3ðλ2½f1�−m2½f0�Þ 8π2Λ4 ½Ṽ1=2 b �¼η −λ2½f1� 2π2Λ4 ½Ṽ1=2 c �¼2η m2½f0�−λ2½f1� 4π2Λ4 ½Ṽ1 a�¼2η ð6m4−m2λ2Þ½f0�−ð12m2λ2þλ4Þ½f1�þ20λ4½f2� 64π2Λ6 ½Ṽ1 b�¼η ð8m2þ5λ2Þ½f0�−10λ2½f1� 16π2Λ4 ½Ṽ1 c�¼2η ð9m4þ3m2λ2Þ½f0�−ð36m2λ2þ3λ4Þ½f1�þ30λ4½f2� 8π2Λ6 ½Ṽ1 d�¼2η 3ðλ4½f1�−λ2m2½f0�Þ 8π2Λ6 ðB6Þ where the discontinuities of f0;1;2 are given in Eq. (2.3). These loop functions are explicitly given by f0ðm2; q2;ΛÞ ¼ 2L � 4m2 q2 � þ log � m2 Λ2 � ðB7Þ f1ðm2; q2;ΛÞ ¼ 2m2 þ q2 3q2 L � 4m2 q2 � þ 1 18 þ 1 6 log � m2 Λ2 � ðB8Þ f2ðm2; q2;ΛÞ ¼ 6m4 þ 2m2q2 þ q4 15q4 L � 4m2 q2 � þ 13 900 þ m2 30q2 þ 1 30 log � m2 Λ2 � ðB9Þ with LðxÞ ¼ 8< : ffiffiffiffiffiffiffiffiffiffiffi x − 1 p arctan � 1ffiffiffiffiffiffi x−1 p � − 1 if x > 1 ffiffiffiffiffiffiffiffiffiffiffi 1 − x p � iπ þ 1 2 log � 1þ ffiffiffiffiffiffi 1−x p 1− ffiffiffiffiffiffi 1−x p �� − 1 if x < 1. ðB10Þ The ½fn� discontinuities can be obtained by noticing that lnΔ ¼ lnðx − xþÞðx − x−Þ where x� ¼ 1 2 � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 − 4m2 p 2q ðB11Þ6The running of the Wilson coefficients is taken into account at leading-log order with this method. BOUNDING QUANTUM DARK FORCES PHYS. REV. D 97, 115034 (2018) 115034-13 has a branch cut between x− and xþ and a discontinuity of 2πi. This leads to ½fn� ¼ 2πi Z xþ x− ðxð1 − xÞÞndx ðB12Þ Finally, the spatial potential is given by Eq. (2.6). The integrals over λ needed in the last step of the calculation are Z ∞ 2m dλ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 − 4m2 p e−λr ¼ 2m r K1ð2mrÞ ðB13Þ Z ∞ 2m dλλ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2−4m2 p e−λr¼8m3 r K1ð2mrÞþ12m2 r2 K2ð2mrÞ ðB14Þ Z ∞ 2m dλλ4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ2 − 4m2 p e−λr ¼ 32m4 r2 K2ð2mrÞ þ � 120m3 r3 þ 32m5 r � K3ð2mrÞ: ðB15Þ APPENDIX C: AMPLITUDES The one-loop amplitudes induced by the operators Oa, Ob, Oc are iMa ¼ 1 2Λ2 ūðp1Þuðp2Þūðp0 1Þuðp0 2Þ × Z d4k ð2πÞ4 1 ðk2 −m2Þððkþ qÞ2 −m2Þ ðC1Þ iMb ¼ 1 Λ4 ūðp1Þγμuðp2Þūðp0 1Þγνuðp0 2Þ × Z d4k ð2πÞ4 ðqþ 2kÞμðqþ 2kÞν ðk2 −m2Þððkþ qÞ2 −m2Þ ðC2Þ iMc ¼ 1 2Λ6 ūðp1Þuðp2Þūðp0 1Þuðp0 2Þ × Z d4k ð2πÞ4 ðq:ðqþ kÞÞ2 ðk2 −m2Þððkþ qÞ2 −m2Þ ðC3Þ with q ¼ p1 − p2. These integrals can be reduced to the basis shown in Eqs. 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