The shadow of dark matter as a shadow of string theory

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 in the corresponding low-energy effective U_Y(1) portal to the dark matter mass. 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. 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.


I. INTRODUCTION
After the confirmation of the Higgs particle as the capstone of the Standard Model, solving the persistent enigma of dark matter has become a major objective of modern particle physics. The completion of the Standard Model (SM) is a great example of successful collaboration between theory and experiment, and theoretical models are needed in the "beyond SM" era to assist with the search for dark matter in particle physics experiments and to provide ideas on what to look for.
It is an intriguing possibility that dark matter might provide a window into low energy string phenomenology, in particular for models with low string scales [1][2][3][4][5][6]. It has been pointed out recently [7] that electroweak dipole dark matter [8,9] may open a window into low energy string phenomenology through the Kalb-Ramond field. The question of the relevant string scale was not addressed in Ref. [7] because no direct link between the dark matter coupling and mass to the string scale was established. However, it turns out that the Kalb-Ramond field can couple to the fermionic Ramond fields in such a way that a low energy coupling of the Kalb-Ramond field to dark fermions can be traced back to string theory. This then allows for an estimate of the maximal string scale for which perturbatively coupled dipole dark matter could be induced from string theory.
Kalb and Ramond had noticed that gauge interactions between strings can be described in analogy to electromagnetic interactions if the basic Nambu-Goto action is amended with a coupling term to an anti-symmetric tensor field [10]. We follow the conventions of Ref. [7] and denote the Kalb-Ramond field with C µν = − C νµ to avoid confusion with the electroweak U Y (1) field strength tensor B µν . The action for strings coupling to the Kalb- * rainer.dick@usask.ca

Ramond field is
Here T s is the string tension and µ s is a string charge with the dimension of mass. The world sheet measure is The string equations of motion which follow from (1) are invariant under the KR gauge symmetry (2) and under the U (1) gauge transformation B µ → B µ + ∂ µ f , and these gauge symmetries are preserved through addition of a field theory action for the gauge fields, where are the components of the 3-form field strength C 3 = dC of the Kalb-Ramond field. This motivated the proposal in [7] to induce a U Y (1) portal to dark matter through the Lagrangian Elimination of C µν for energies much smaller than m C generates electroweak dipole moments for the dark matter field χ, ×χS µν (a m + ia e γ 5 )χ.
However, while the bosonic terms in (5) could be motivated through the couplings of string world sheets to Kalb-Ramond fields, the coupling of C µν to the dark fermions had to be postulated. The purpose of the present paper is to point out that the Kalb-Ramond field can couple to the Ramond [11] and Neveu-Schwarz [12] fields of superstrings in a way that does induce magnetic dipole couplings at low energies, thus lending more credibility to the proposal that direct search for electroweak dipole dark matter can provide a window into low energy string phenomenology. The possible string origin of the coupling of the Kalb-Ramond field to dark dipole moments is introduced in Sec. II and conclusions are formulated in Sec. III. The mapping between world-sheet spinors and halfdifferentials is reviewed in an Appendix, since it helps to understand the proposed coupling of the Kalb-Ramond field to the half-differentials Ψ µ √ ± .

II. STRING ORIGIN OF MAGNETIC DIPOLE TERMS
We wish to provide a string explanation for the coupling of the Kalb-Ramond field to the dark fermion dipole moments in Eq. (5). This will require a coupling to the Ramond fields ψ µ (τ, σ) on the world sheet, which we write as components Ψ µ √ ± (τ, σ) of half-differentials. The mapping between 2D spinors and half-differentials is reviewed in the Appendix.
The mapping should respect world-sheet and target space symmetries under coordinate and Lorentz transformations. Since 2D spinors are completely equivalent to half-differentials, and since we can only use C µν or C µνρ for the coupling, the unique solution to this problem is with a dimensionless coupling constant g s . The minus sign is required by parity invariance of the world-sheet Lagrangian.
The coupling term (7) is in world-sheet spinor notation (with metric determinant g ≡ g(τ, σ) and zweibein components e α a ≡ e α a (τ, σ)) given by The couplings (1,7) of the Kalb-Ramond field break the world-sheet supersymmetry of the free string theory. However, they do preserve the GSO projection [13], and that is what we really need.
If we evaluate the expression (7) on the low-energy states of the Ramond sector, we find in temporal gauge τ = t, dσ = ℓ = k 0 /T s , This describes the interaction of the Kalb-Ramond field with a classical fermion. The corresponding quantum mechanical action is The non-relativistic limit of Eq. (12) corresponds to a coupling where we invoked the standard assumption that the low energy string states in the a priori massless sector acquire masses through symmetry breaking. Elimination of C ij (x) from the action with the terms (5) yields a magnetic dipole coupling with coupling constant However, the coupling scale M = M d /a m can be determined as a function of dark matter mass m χ through the requirement of thermal dark matter creation in the early universe. This led to M ≃ 23 PeV for m χ = 1 MeV and M ≃ 3.7 TeV for m χ = 10 GeV, or roughly m χ M ≃ 3 × 10 4 GeV 2 [14], see Fig. 1 in Ref. [7]. However, Eq. (14) relates this to the string scale, String-induced electroweak dipole dark matter with perturbative couplings would therefore require a low string scale √ 8πT s 120 TeV.

III. CONCLUSION
Inclusion of the missing piece (7) in the string derivation of electroweak dipole dark matter relates the string tension to dark matter parameters. Assuming perturbative couplings and that dark matter is created from thermal freeze-out then leads to the requirement M s 120 TeV. Since string theory appears to be unique in providing a mechanism to induce electroweak dipole moments in dark matter, discovery of electroweak dipole dark matter with a mass and recoil cross section which agrees with the predictions from thermal dark matter creation would be a strong hint for low-scale string theory.
It is intriguing that the geometry of string ground states allows for the construction of phenomenologically viable models with fundamental string scales all the way down to the TeV scale [1][2][3][4][5][6]. Furthermore, it is also conceivable that there may exist different incarnations of fundamental strings with different tensions T s . In that case the Kalb-Ramond model (1) and its extension (7) could find a natural "pure string" explanation within a hierarchy of string scales T s ≪ T P = M 2 Planck /8π. The Kalb-Ramond field C µν (x) would then correspond to the low energy description of the anti-symmetric tensor fields of Planck scale superstrings with tension T P , and the couplings (1,7) would arise from world-sheet couplings between strings with different tensions. If the low scale string sector contains only Ramond sector states, such a scenario would not need to invoke any further assumption about the nature of the string ground state to explain why M Planck ≫ M s . Either way, the possibility to induce electroweak dipole dark matter from string theory adds to the anticipation that low scale string theory should become a leading paradigm for particle physics searches beyond the Standard Model, both in direct dark matter search experiments and at future colliders.

APPENDIX: HALF-DIFFERENTIALS AND FERMIONS ON THE WORLD SHEET
Here we review the mapping between spinors and halfdifferentials on world sheets in the conformal gauge g τ τ + g σσ =Ẋ 2 + X ′2 = 0, g τ σ =Ẋ · X ′ = 0. (16) We denote the remaining degree of freedom in the metric by φ(τ, σ), The metric in the corresponding two-dimensional lightcone coordinates σ ± = σ ± τ , corresponds to zweibein components e α a (note η +− = 1/2), The set of coordinate transformations is restricted by the conformal gauge to [15] On the other hand, the symmetry group SO(1, 1) of tangent plane Lorentz boosts also factorizes in the light-cone basis of tangent vectors. A boost with parameter u = artanh(β) = 1 2 in the tangent plane yields in the light-cone basis with components This is the exact square of the transformation of the spinor components in Weyl representation. Every Weyl representation of 2D γ matrices yields either iS 10 or − iS 10 as the spinor representation of the boost generator, where The corresponding Lorentz boost on the spinors is i.e. the matrix elements of the spinor representation relate to the matrix elements of the vector representation through This motivates the assignments of indices to the components of the 2D Dirac spinor in the Weyl basis: (ψ √ + ) 2 and (ψ √ − ) 2 transform like the components of a tangent vector in the light-cone basis.
To be specific we use in the following calculations the real Weyl representation of 2D γ matrices However, the mapping between world-sheet spinors and half-differentials, and the fermion action in terms of halfdifferentials, follow in the same form for every Weyl representation.
The tangent plane gamma matrices in the light cone basis follow as and the transformation of the integration measure to the light-cone coordinates on the world sheet is The curved space generalization of the kinetic terms in the Dirac action is However, the spin-connection in two dimensions, anti-commutes both with γ 0 and γ 1 , {Ω α , γ a } = 0. Therefore, splitting the derivative symmetrically between ψ and ψ cancels the spin connection from the Dirac action in two dimensions. The resulting action combining the Fubini-Veneziano fields and Dirac spinors on the world sheet is Substituting the spinor components ψ = (ψ Combining the spinor and zweibein components yields the half-differentials and the action The 2D metric has completely disappeared, in the scalar sector due to the conformal gauge and in the spinor sector due to absorption into the half-differentials. The phases of the anti-commuting half-differentials decouple in the action (36), in agreement with the real transformation behavior (26) of the spinors in Weyl representation. The standard phase convention is The superconformal transformation parameters ǫ √ ± (σ ± ) are then real anti-commuting (−1/2)-differentials [16], which yield the transformations