# The shadow of dark matter as a shadow of string theory

## Abstract

We point out that string theory can solve the conundrum to explain the emergence of an electroweak dipole moment from electroweak singlets through induction of those dipole moments through a Kalb–Ramond dipole coupling. This can generate a \(U_Y(1)\) portal to dark matter and entails the possibility that the \(U_Y(1)\) gauge field is related to a fundamental vector field for open string interactions. The requirement to explain the observed dark matter abundance relates the coupling scale *M* in the corresponding low-energy effective \(U_Y(1)\) portal to the dark matter mass \(m_\chi \). The corresponding electron recoil cross sections for a single dipole coupled dark matter species are generically below the current limits from XENON, SuperCDMS and SENSEI, except in the GeV mass range if the electric dipole coupling becomes stronger than the magnetic coupling, \(a_e^2\ge a_m^2\). Furthermore, the recoil cross sections are above the neutrino floor, and the \(U_Y(1)\) portal can be tested with longer exposure or larger detectors. Discovery of electroweak dipole dark matter would therefore open an interesting window into string phenomenology.

## 1 Introduction

*Higgs portal*[2] for dark matter models where the interaction is mediated by Higgs exchange [3, 4, 5, 6, 7, 8, 9, 10] (see [11, 12] for recent reviews), the notion of “portals” for the non-gravitational interaction between dark matter and the Standard Model has been widely adopted, including neutrino portals and vector portals. These standard options for non-gravitational dark matter couplings usually do not include a photon portal, as the optical darkness of the dominant matter component in large scale astronomical structures is usually assumed to be a consequence of the absence of direct photon couplings. However, Sigurdson et al. had pointed out that dipole couplings of MeV or GeV scale dark matter to photons comply with the darkness requirement if the coupling is sufficiently suppressed [13], see also [14, 15, 16, 17, 18, 19], and Profumo and Sigurdson coined the notion of a “shadow of dark matter” for this scenario [20]. Possible dipole couplings to photons involve dark fermions in the form

Couplings of the form (1) were also used in [14, 21, 22, 23, 24] in proposals to explain the DAMA annual modulation signal in nuclear recoils. More recently, Conlon et al. pointed out that the direct photon coupling proposed by Profumo and Sigurdson can reconcile the 3.5 keV data from the Hitomi, XMM-Newton and Chandra observations of the Perseus cluster through X-ray absorption and resubmission [25].

Of course, a mass-suppressed photon portal per se to electroweak singlet dark matter breaks the electroweak symmetry of the Standard Model. Therefore it makes sense to replace the mass suppressed photon portal with a mass suppressed \(B_{\mu \nu }\) portal (or \(U_Y(1)\) portal) where \(B_{\mu \nu }=\partial _\mu B_\nu -\partial _\nu B_\mu \) is the field strength of the electroweak \(U_Y(1)\) symmetry. This then automatically entails the photon portal through electroweak mixing into mass eigenstates, \(B_\mu =A_\mu \cos \theta -Z_\mu \sin \theta \) and leaves the electroweak symmetries unbroken. Indeed, it was noticed already by Cline et al. [18] that couplings of the form (1) should also entail corresponding couplings to the *Z* boson.

We wish to draw attention to the fact that the Kalb–Ramond field of string theory can help to generate dipole couplings of the form (1). The Kalb-Ramond field is an anti-symmetric tensor field \(C=C_{\mu \nu }dx^\mu \wedge dx^\nu /2\) which does not need to be closed, \(dC\ne 0\), and therefore cannot simply be considered as the field strength of a hidden *U*(1) symmetry.

It has recently been pointed out that the strongest constraints for low mass dipole coupled dark matter should arise from direct searches in electron recoils [26]. Therefore we also discuss the corresponding electron recoil constraints under the assumption of generation from thermal freeze-out.

The natural emergence of couplings of the Kalb–Ramond field to *U*(1) gauge fields is reviewed in Sect. 2. The ensuing possibility that the Kalb–Ramond field can induce electroweak dipole couplings for electroweak singlets is introduced in Sect. 3. Abundance constraints on the magnetic dipole coupling scale \(M^{-1}=a_m/M_d\) for a single dipole coupled dark matter species \(\chi \) and the resulting constraints from direct dark matter searches in electron recoils are discussed in Sects. 4 and 5, respectively. Section 6 summarizes our conclusions.

## 2 A shadow of string theory

Closed strings contain anti-symmetric tensor excitations in their low-energy sector through the anti-symmetric Lorentz-irreducible component of the oscillation states \((a_{+,1}^\mu )^+ (a_{-,1}^\nu )^+|0\rangle \) [27, 28]. Anti-symmetric tensor fields can also mediate gauge interactions between string world sheets [29], and these fields also participate in brane interactions [30, 31, 32, 33, 34, 35].

There are two ways how the Kalb–Ramond field can couple to *U*(1) gauge bosons, and both of them are related to the gauge symmetries of string–string interactions. We therefore need to review the string couplings of the Kalb–Ramond field and how they necessitate a coupling to *U*(1) gauge bosons in the presence of open strings. Kalb and Ramond had generalized the work of Feynman and Wheeler for action at a distance in electrodynamics in their seminal work, but with the wisdom of hindsight it is easier to start with the Lagrangian formulation of the pertinent string couplings. This formulation also shows how to generalize the construction for couplings to several *U*(1) gauge fields, and demonstrates that we can keep the *U*(1) gauge fields for the boundary charges of open strings massless.

*T*is the string tension, \(\mu _s\) is a string coupling constant (or

*string charge*) with the dimension of mass, \(\tau _a\) and \(\sigma _a\) are timelike and spacelike coordinates on the world sheet of the

*a*-th string, respectively, and \(x_a\equiv x(\tau _a,\sigma _a)\) describes the embedding of the string world sheet into spacetime. The world sheet string interaction term can be written in the form \(\mu _s\int C\), just like the electromagnetic interaction term in particle physics for particles of charge

*q*can be written as a world line integral \(q\int A\). The dimensionless charge

*g*appears only on the endpoints of open strings.

*KR gauge symmetry*

*U*(1) gauge transformation \({\mathcal {B}}_\mu \rightarrow {\mathcal {B}}_\mu +\partial _\mu f\).

*C*and the accompanying vector field \({\mathcal {B}}\), and the action should be amended with kinetic terms for those fields. The KR gauge symmetry (7) is preserved through the kinetic term

*U*(1) currents of a charge

*g*at \(\sigma _a=\ell \) and a charge \(-g\) at \(\sigma _a=0\). Up to boundary terms at \(\tau _a\rightarrow \pm \infty \) (which also appear in the currents of charged particles in electrodynamics), the currents satisfy \(\partial _\mu j_a^{\mu \nu }(x)=(\mu _s/2g)j_a^\nu (x)\) and \(\partial _\mu j_a^{\mu }(x)=0\) [29].

*U*(1) gauge fields \({\mathcal {B}}_\mu \) in the form \(C_{\mu \nu }{\mathcal {B}}^{\mu \nu }\). Indeed, we can easily generalize the construction to the case of different boundary charges \(g_I\) for different types of open strings with corresponding gauge fields \({\mathcal {B}}_{I,\mu }\). We can simply replace the boundary term in Eq. (3) according to

*a*-th string. The boundary equation (6) for the

*a*-th string becomes

*U*(1) gauge fields transform under KR symmetry according to \({\mathcal {B}}_{I,\mu }\rightarrow {\mathcal {B}}_{I,\mu }+(\mu _s/g_I)f_\mu \), and the KR gauge kinetic term becomes

*I*in the \(g_I\)-dependent terms on the left hand side, and Eq. (11) generalizes to

*a*-th string satisfy \(\partial _\mu j_a^{\mu \nu }(x)=(\mu _s/2g_{I(a)})j_a^\nu (x)\).

## 3 Electroweak dipoles induced by the Kalb–Ramond field

*H*-type fluxes [37, 38, 39, 40, 41, 42, 43, 51] through the internal components of the gauge invariant 3-form \(C_3=C_{KLM}dx^K\wedge dx^L\wedge dx^M/6\). These fluxes would imply

*U*(1) dipoles, and therefore dipole interaction terms and KR gauge symmetry breaking would appear unavoidable in any low energy field theory formulation of the theory which would be based on renormalizable terms.

*U*(1) field strengths [56], and integrating this out for massive Kalb–Ramond fields can also generate gauge invariant low-energy effective dipole couplings. Elimination of \(C_{\mu \nu }\) from the Lagrangian

## 4 Dark matter abundance constraints on the \(U_Y(1)\) portal

*hZ*and

*hhZ*final states are not accessible in the non-relativistic regime and their contributions to the thermally averaged cross section at thermal freeze-out are therefore negligible, but we also report the corresponding annihilation cross section into

*hZ*for completeness. This cross section is with \(s\ge (m_h+m_Z)^2\)

*s*in the thermal averaging), and the corresponding cross section into the

*hhZ*final state (which cannot be integrated analytically) becomes only relevant for masses \(m_\chi > rsim 160\) GeV.

*M*through the

*M*-dependence of \(\sigma (s)\),

*M*as a function of dark matter mass \(m_\chi \). For \(m_\chi \le 10\) GeV,

*M*decreases with increasing \(m_\chi \) with values \(M\simeq 23\) EeV for \(m_\chi =1\) MeV and \(M\simeq 3.7\) TeV for \(m_\chi =10\) GeV, see Fig. 1.

Since *M* is related to the mass \(m_C\) of a possible Kalb-Ramond field through \(m_C=g_{BC}g_{C\chi }M\), coupling scales *M* in the few TeV to thousands of TeV range could indicate a Kalb–Ramond mass in the hundreds of GeV to hundreds of TeV range if we assume weak strength couplings of the Kalb–Ramond field.

## 5 Electron recoil cross section

As explained in the white paper [26] on new ideas in dark matter research, electron recoils are a primary possible signal for light dark matter particles with electromagnetic dipole moments.

The recoil cross sections from a magnetic dipole coupling comply with the current direct exclusion limits throughout the considered mass range. On the other hand, the case \(a_e=\pm a_m\), which corresponds to dipole moments from purely right-handed or left-handed fermions, is excluded for masses in the GeV range. We also note that the recoil cross sections are above the neutrino floor [65] and may be detectable with longer exposures or larger detectors.

## 6 Conclusions

Gauge invariant interactions of open strings require the Kalb-Ramond field to couple to the field strength tensors of *U*(1) gauge fields, whereas a theory with only closed strings permits Cremmer–Scherk couplings to *U*(1) field strength tensors. We found that these couplings can induce dipole couplings of electroweak singlet dark matter to the \(U_Y(1)\) gauge field, thus contributing to the formation of a \(U_Y(1)\) portal both to dark matter and to string theory. We analyzed in particular the case of a single dark matter component and found that the MeV–GeV mass range for dipole coupled dark matter remains viable under recent constraints from direct searches in electron recoils if \(a_e^2<a_m^2\), while the case of dipole coupling only to right-handed or left-handed dark fermions \(a_e=\pm a_m\) is excluded in the GeV mass rang but still allowed in the MeV mass range. Dipole coupled dark matter has a high discovery potential due to yielding recoil cross sections above the neutrino floor. The discovery of a dipole coupled \(U_Y(1)\) portal to dark matter would therefore be very interesting from the perspective of a bottom-up approach to string phenomenology.

The model discussed here does not touch upon the important question of moduli stabilization, except for the observation that dilaton stabilization and an internal magnetic Kalb–Ramond flux decouple the Kalb-Ramond coupling in four dimensions from the mass term. We do assume that moduli are stabilized and that compactification yields the Standard Model at low energies. Our point is that under these circumstances the Kalb–Ramond field provides a natural candidate for inducing a dipole coupled \(U_Y(1)\) portal to electroweak singlet dark matter. The discovery of \(U_Y(1)\) dipole coupled dark matter would therefore provide an important low-energy indication for the existence of the anti-symmetric tensor fields of string theory.

## Notes

### Acknowledgements

AD was supported by a MITACS Globalink internship. The research of RD is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a subatomic physics grant.

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