Monochromatic Gamma Rays from Dark Matter Annihilation to Leptons

We investigate the relation between the annihilation of dark matter (DM) particles into lepton pairs and into 2-body final states including one or two photons. We parametrize the DM interactions with leptons in terms of contact interactions, and calculate the loop-level annihilation into monochromatic gamma rays, specifically computing the ratio of the DM annihilation cross sections into two gamma rays versus lepton pairs. While the loop-level processes are generically suppressed in comparison with the tree-level annihilation into leptons, we find that some choices for the mediator spin and coupling structure lead to large branching fractions into gamma-ray lines. This result has implications for a dark matter contribution to the AMS-02 positron excess. We also explore the possibility of mediators which are charged under a dark symmetry and find that, for these loop-level processes, an effective field theory description is accurate for DM masses up to about half the mediator mass.


I. INTRODUCTION
While there is very strong evidence for the existence of dark matter (DM) on the scales of galaxies, galaxy clusters and cosmology [1,2], determining its specific particle properties remains one of the most important open questions in modern physics [3]. There are many elementary particle candidates for the dark matter in the universe (see e.g. [4] for a review of particle DM candidates), but one of the most compelling class of models is based on a striking coincidence: a stable particle with approximately weak-scale mass and couplings (a so-called weakly-interacting massive particle, or WIMP) will, assuming a standard thermal history of the universe, possess a thermal relic density today that roughly corresponds to that of DM. This coincidence is known as "WIMP miracle".
Much effort is under way in searching for WIMPs. The most direct probe of WIMPs as DM candidates is the search for scattering of Galactic WIMPs with ordinary matter. Many direct detection experiments have searched for this signal, with no definitive detections as of yet [5,6].
The bounds from such searches have begun to have significant implications for our understanding of viable WIMP DM candidates, ruling out the most naïve realizations of the WIMP paradigm.
Of course, not all DM interactions can be effectively probed by these experiments.
A second technique to search for WIMPs is to attempt to produce them in high-energy particle collisions. Once produced, a WIMP would escape undetected from the experiment, leading to a missing energy signature [7]. Searches for missing energy have a very long history, and have been designed for a number of purposes, besides models featuring a stable WIMP. Recently, a focus on comparing the bounds on interactions of WIMPs with other particles from these searches to indirect and direct detection experiments has emerged. This exercise started with a parametrization of DM interactions by contact interactions [8][9][10], and has evolved to include various simplified models which feature a collider-accessible mediator particle as well [11]. Many collider analyses are now being designed explicitly with the purpose of searching for DM [12].
A third technique, known as indirect detection, searches for the visible particles that result from annihilations of WIMPs in the Galaxy or in outer systems [5]. These annihilations are rare enough that they do not lead to any significant depletion of the DM density, but they can potentially produce measurable signals. Many experiments have sensitivities good enough to probe annihilation rates that correspond to the rate needed in the early universe in order for the WIMP Miracle to occur [13]. We must keep in mind, however, that the kinematics of annihilation at the present time are very different from those needed in the early universe, and thus the annihilation rate may be significantly different as well [14].
A variety of target Standard Model (SM) particles can be employed for the purposes of indirect searches. Neutrino telescopes have been used to search for the annihilation of WIMPs captured in the Sun, photons from radio through gamma rays have been used to place bounds on DM annihilation, and cosmic rays can also be employed to search for a WIMP signal. The most promising cosmic rays for such purpose are antimatter particles, as we believe that ordinary astrophysical processes do not produce antimatter in great abundances, and that therefore the only background should be secondary, i.e. arising from cosmic ray interactions with interstellar material. Such secondaries are generally much less abundant and have much less energy than primary cosmic rays, so that WIMP annihilation could potentially yield a detectable primary contribution.
In 2008, the PAMELA experiment precisely measured the positron fraction in the 1.5-100 GeV range [15]. The positron fraction was found to sharply increase around 5 GeV in a manner completely inconsistent with secondary sources in essentially all predictive cosmic-ray models [16,17]. The Fermi Large Area Telescope (LAT) later confirmed that the positron faction continues to rise in the 100-200 GeV range [18]. The International Space Station-based experiment Alpha Magnetic Spectrometer (AMS-02) extended measurements to 350 GeV [19]. Their 6.8×10 6 electron and positron events confirm the PAMELA excess with high precision. The slope of the AMS-02 spectrum decreases by an order of magnitude between 20 and 250 GeV and has no measurable fine structure or anisotropy in the arrival direction. The second AMS-02 data release last fall extended the analysis to 500 GeV, and found that above ∼ 275 GeV the positron fraction ceases to increase [20].
These experiments imply that there must be another significant source of high energy positrons besides secondary processes. While a Galactic WIMP annihilation origin is still possible [21], nearby pulsars have been proposed as another possible explanation for the positron excess [22,23].
WIMP annihilation into charged states also generically produces photons via loop-level processes or from final-state radiation. In the latter case, the secondary photons have an energy spectrum peaked towards low energies. In the former case, however, when DM annihilates at close to zero relative velocity into γX, conservation of energy gives a photon with a fixed (monochromatic) energy of where m χ is the WIMP mass. A gamma-ray line from these processes could stand out from the continuous astrophysical background for high enough values of m χ and be detected by HESS or have been detected [25].
In this paper we study the monochromatic gamma-ray yield stemming from DM models where the DM particle annihilates into a charged lepton-antilepton pair. In particular, we carry out a model-independent study based on contact interactions in the context of an effective field theory (EFT) description of the DM-lepton interaction vertex. For a given contact interaction type, we essentially simply "close the fermion loop" as shown in Figure 1 and calculate the (σv) χχ→γγ cross section for a given annihilation cross section into leptons (σv) χχ→ff . As a result, we calculate in a model-independent way the monochromatic gamma-ray yield for e.g. theories that purportedly explain the measured positron excess; vice versa, our results allow to compute (σv) χχ→ff from a given measured gamma-ray line and model to explain it.
The remainder of this paper is organized as follows: In Section II, we describe the EFTs structure under consideration. We calculate the ratio (σv) χχ→γγ /(σv) χχ→ff in Section III and use it to relate the rate of annihilation to positrons to that into monochromatic gamma rays. We comment on the applicability of the EFT assumption in the case of mediators charged under the dark symmetry in section IV, and we conclude with a discussion of our results in the context of our assumptions as well as their general implications for WIMPs in Section V.

II. MODEL SETUP
We consider a model of WIMP DM where the WIMP is a SM gauge singlet Dirac fermion χ, and we assume that the new particle mediating the interactions between the WIMP and SM Name Operator fields is heavy enough to be "integrated out". The natural way of parametrizing these interactions, then, is to have non-renormalizable contact operators suppressed by a mass scale that is related to the mediator's mass and couplings. In this work we focus on interactions between DM and a SM being the usual helicity projectors. This is obviously an over-complete basis of operators, as the vector and axial interactions can be constructed out of the chiral interactions or vice versa. However, due to the chiral nature of the SM gauge interactions the chiral operators may well be of interest, despite their linear dependence on the others. The full list of EFT interactions we consider are shown in Table I. Our naming scheme is simple. For scalar operators, S and P stand for scalar (1) and pseudoscalar (γ 5 ) couplings, respectively. V , A, R and L identify vector (γ µ ), axial (γ µ γ 5 ), right-handed (γ µ P R ) and left-handed (γ µ P L ) couplings.
The quantity of interest here is the cross section ratio 1 where X is another SM vector boson. As discussed before, (σv) χχ→ff and (σv) χχ→γX are computed by evaluating the diagrams shown in Figure 1. We take X = γ for the scalar-type EFTs to maximize photon production. For operators of the form [χΓ µ χ][f Σ µ f ], X = Z unless the WIMP and SM pieces both contain a factor of γ 5 , in which case we also take X = γ. See Appendix B for details behind these choices, which are related to Yang's theorem.
It is important to note that, while we use the language of EFT interactions, any s-channel mediator with appropriate couplings will in fact give an identical cross section ratio to our EFTbased approach, as the s-channel propagator in both diagrams is identical and the mediator cannot carry the loop momentum. Thus, our resulting ratios are applicable for these theories beyond the regime where the EFT has captured all DM physics of interest. After presenting our EFT results we will discuss the case of t-channel mediators in some detail, determining where the EFT analysis holds for R X,f and where it breaks down and the full set of box diagrams must be calculated explicitly.
When computing the relevant cross sections for each of our theories in the WIMP center of mass frame, the values obtained for R X,f depend on v, the speeds of each of the incoming WIMPs. While v is often taken to be zero in calculations, that cannot be done consistently here as (σv) χχ→ff is p-wave suppressed for theories such as AV. We found that the 3-body processes χχ → ff γ and χχ → ff Z did not significantly alleviate this suppression, nor the helicity suppression found for (σv) χχ→ff in the AA theory, demonstrating that gamma ray lines are the natural signals for our EFTs. We performed all our calculations with v/c = 10 −3 , a choice motivated by the typical Galactic velocity dispersion at the Sun's position [26]. The velocity dependence of our results will be discussed below. We collect our analytic results for the relevant tree-level cross sections in Appendix A 1.

III. RESULTS
Motivated by analyses of AMS-02 data [27][28][29] finding that TeV-scale WIMPs annihilating to τ τ provide the best explanation of the positron excess in terms of DM annihilating to SM states, we focus on this specific lepton pair final state. This has not changed with AMS-02's latest data release [30]. To make quantitative predictions for the gamma-ray line yield, we use for reference the allowed regions in the (m χ , (σv) χχ→τ τ ) plane from [27]. We calculate the ratio R X,τ in each of the EFTs in Table I using the Mathematica packages FeynArts [31], FormCalc and LoopTools [32]. The spin-averaged cross sections are computed with WIMP velocity v/c = 10 −3 in the center of mass frame. The tree-level and loop-level analytic amplitudes for the SP and AA interactions found in Appendix A 2 and A 3 were found to agree with the numerical results. Notice that while our EFT interactions are non-renormalizable, the one-loop amplitudes are finite.
The cross section ratios R X,τ for a representative set of the EFTs are shown in Figure 2 with WIMP masses ranging from ∼ 100 GeV to ∼ 10 TeV. Ratio curves are omitted for some EFTs as they are related simply to others shown in the plot: • In the case of the scalar theories, the WIMP and fermion (gamma) parts of the tree-level (loop-level) cross sections decouple. Since the WIMP components are the same for the tree and loop-level cross sections, the SS and PS results are the same, as are the PS and PP ones. Moreover, the loop parts of the amplitudes differ by a factor of order 1 for scalar and pseudoscalar coupling to f . We therefore omit the PS, SP and PP results, as they are numerically very similar to the SS one.
• Averaging over initial WIMP spins allows us to omit LA, LV, LL and LR from our plot, since they give the same results as the RA, RV, RL and RR theories.
• The AR, AL, RR and RL theories allow annihilations of WIMPs to two photons. Since photons couplings to f are helicity independent, we also left AL and RL out of our plot.
• The R Z,τ ratios for the VR and VL theories differ by a factor Most of the EFTs give R X,τ ∼ 10 −10 − 10 −5 . However, for AA, AL, AR, R X,τ ∼ 10 −2 − 10 and for AV, R Z,τ ∼ 1. While surprising at first that a loop-level to tree-level diagram give such close cross sections, this is easily understood: Looking at the results in Appendix A 1 for the AV theory shows that the tree level amplitude is p-wave suppressed. Since we are considering the f = τ case, for the AA, AL and AR theories, the helicity suppressed term is larger than the p-wave suppressed term. In these theories, all of these suppression factors that are present at tree-level are alleviated in the loop diagrams, leading to large cross section ratios.

A. Predicting the gamma ray flux
Suppose now we postulated WIMP annihilation to explain the rising positron fraction: our calculated ratios enable us to predict the gamma-ray monochromatic line signal induced through the loop diagram for a given contact interaction. By rescaling the 99% CL allowed regions in the (M χ , (σv) χχ→τ τ ) plane given in [27] by R X,τ and putting in a factor of 1/2 for theories with X = Z, we calculated the regions in the (M χ , (σv) χχ→γγ ) plane preferred by a WIMP annihilation explanation to the positron fraction anomaly. Figure 3 shows the resulting regions for the AR EFT along with recent upper limits on (σv) χχ→γγ from HESS [33] and Fermi LAT [34] line searches.
The predicted allowed regions for AA, AL and AV lie above those for AR while the regions for all other theories fall well below current observational constraints, by at least 3 orders of magnitude.
HESS-II, GAMMA-400 and CTA line searches are only expected to tighten these constraints by an order of magnitude [35], and so will not be able to probe line signals from AMS-fitting points in these theories.  Figure 4 shows the velocity dependence of the results for the AR theory. The differences between the two panels stem from changes in the relative strengths of the helicity and p-wave suppressed parts of the tree-level cross section. While increasing or decreasing v by an order of magnitude changes the shape of the limit regions, it does not affect whether they fall above or below the line search constraints for any of our theories. We therefore find that AA, AR, AL and AV contact interactions cannot explain the positron excess as they would violate bounds from gamma-ray line searches. All other contact interactions remain viable.

IV. EFT FOR t-CHANNEL MEDIATORS
The loop diagrams we consider require care as arbitrarily large loop momenta run through our effective operators. As mentioned before, if the contact operator comes from an s-channel mediator, the mediator is not contained in the loop and the EFT gives the correct result for the ratio of cross sections. In the case of a charged t-channel mediator, we must consider the box diagram shown in Figure 5 (along with the crossed ones). In this case, the mediator's propagator could lead to significant corrections to the cross section from the EFT prediction at high loop momenta.
To explore this potential issue, we considered a theory in which the WIMP-SM interaction is of the form where χ is the WIMP (taken to be Majorana for simplicity) with mass m χ andτ + is a charged scalar with mass mτ . This is a simplified model of a SUSY theory in which one tau slepton is the dominant contributor to DM annihilations and the DM neutralino's gauge eigenstate content is unknown. We consider the CP-conserving case, which requires that λ S , λ P ∈ R.
To lowest order, the effective Lagrangian for this theory will contain a linear combination of four-fermion operators from Table I matrix element for χχ → τ + τ − , giving Note that in the case of a Dirac WIMP, there would be additional terms involving its tensor and vector currents. In the chiral case |λ S | = |λ P |, only the AA and AV terms are present, implying that R γ,τ will be large for this theory. In what follows we will call the EFT and exact cross sections (σv) EFT χχ→γγ and (σv) exact χχ→γγ . The latter has been calculated in many papers (see e.g. [36][37][38][39][40]) in the v = 0 limit. Here we use the result from [39] which is corrected in the footnote on page 8 of [40].
In Figure 6 we plot (σv) exact χχ→γγ and (σv) EFT χχ→γγ as a function of m χ for mτ = 10 TeV, λ S = 0 and λ P = 1. The cross sections agree to better than 1% all the way up to m χ ≈ 1 2 mτ . Near m χ ≈ mτ the EFT underestimates the cross section when the slepton can be put on shell. For m χ > ∼ mτ the EFT result is larger than the exact result since the EFT loop only contains three propagators. The coupling constants' values have no effect on this agreement. It is surprising that the EFT is valid for such high values of m χ , and furthermore that it works so well for our SUSY-like theory, as this is the classic internal bremsstrahlung case in which EFT is usually expected to be inaccurate.
Finally, we consider whether our calculation of R γ,τ changes in this t-channel theory. The cross section (σv) χχ→ff gets a new contribution from the internal bremsstrahlung diagram for χχ → ff γ. This diagram only contributes significantly to (σv) χχ→ff if the helicity suppression in the lowest-order process χχ → ff is very large. Since we are interested in the final state f = τ , this is not the case [37], and we can safely ignore the internal bremsstrahlung diagram for the range of WIMP masses considered above; for the largest WIMP masses considered the helicity suppression is comparable to the one-loop suppression, leading to a change of only a factor of a few in the ratio. Our type of EFT analysis of R γ,τ from above thus applies for m χ < ∼ 1 2 mτ , although of course having to consider a linear combination of effective operators in equation (4) adds additional complexity. For a lighter final state fermion or much heavier WIMP, a more careful analysis of (σv) χχ→ff including the internal bremsstrahlung diagram would have to be performed, but note that the WIMP mass for a simplified theory of this type has been shown to have an upper bound comparable to the masses considered here [41].

V. DISCUSSION AND CONCLUSIONS
In this study we have calculated the cross section for Dirac WIMPs annihilating to monochromatic gamma rays stemming from an annihilation mode to a charged lepton pair. As a result, we could predict the cross section for a monochromatic gamma-ray line for models where WIMP annihilation explains the observed anomalous rising positron fraction through WIMP pair-annihilation to τ s. To relate (σv) χχ→γγ and (σv) χχ→ff , we used an over-complete set of dimension 6 operators.
The ratio R X,τ varies by many orders of magnitude depending on the theory under consideration, since helicity and p-wave suppressions in the tree-level amplitude are sometimes alleviated at one loop.
We find that for a set of contact operators involving an axial WIMP bilinear (namely the AA, AR, AL and AV cases) cross sections that would explain the positron excess would yield a monochromatic gamma-ray line orders of magnitude brighter than current observational limits from Fermi LAT and HESS. This conclusion is robust to changes in the relative WIMP velocity v.
We also investigate the validity of the EFT description for a specific SUSY-inspired case (t-channel stau-mediated annihilations into tau pairs) and demonstrated that the EFT accurately describes theories where a charged boson mediates SM-WIMP interactions. Since the EFT gives the same value for R X,τ as an s-channel mediator, this shows that our conclusions hold for a wide range of theories whose UV completion involves a mediator heavier than the WIMP.