# Dark Side of Higgs Diphoton Decays and Muon

###### Abstract

We propose that the LHC hints for a Higgs diphoton excess and the muon () discrepancy between theory and experiment may be related by vector-like “leptons” charged under both hypercharge and a “dark” . Quantum loops of such leptons can enhance the Higgs diphoton rate and also generically lead to - kinetic mixing. The induced coupling of a light gauge boson to electric charge can naturally explain the measured . We update mass and coupling constraints based on comparison of the electron experiment and theory, and find that explaining while satisfying other constraints requires to have a mass . We predict new Higgs decay channels and , with rates below the diphoton mode but potentially observable. The boosted in these decays would mimic a promptly converted photon and could provide a fraction of the apparent diphoton excess. More statistics or a closer inspection of extant data may reveal such events.

The discovery of a new Higgs-like state, at a mass of about , by the ATLAS ATLAS0712 () and CMS CMS0712 () collaborations at the Large Hadron Collider (LHC) represents a historical breakthrough in particle physics which is likely to provide a major step toward understanding electroweak symmetry breaking (EWSB). It remains to be seen whether this new state is the long-sought Standard Model (SM) Higgs or some variant of it. While addressing this important question requires more data, the current results by both ATLAS and CMS hint, at a roughly level ATLAS0712 (); CMS0712 (), that the diphoton branching fraction of the new state, which we will henceforth refer to as the Higgs boson , seems to be a factor of larger than the SM prediction Ellis:1975ap (); Shifman:1979eb (). If that diphoton excess is confirmed with more statistics it would be an important clue for physics beyond the SM.

Besides the recent hints from the Higgs data, the current discrepancy between the SM prediction and the measured value of the muon Bennett:2006fi (); PDG (), denoted by , is another potential clue pointing to new physics Czarnecki:2001pv (). In this work, we propose that the Higgs diphoton excess and the discrepancy can be naturally related through the introduction of heavy new vector-like leptons charged under both as well as a new gauge symmetry with a relatively light boson SUSYframework (). (The lower bound of will be discussed later.) We will refer to the new quantum number as “dark charge” since does not have direct couplings to the ordinary “visible” SM sector. However, our study needs not assume a specific connection with dark matter (DM) physics, although it is a possibility. This issue will be briefly addressed.

To set the stage for our discussion, we first address the main problem with trying to explain an excessive Higgs diphoton, , decay rate. The SM prediction arises from destructive interference of loops (the dominant contribution) and a smaller (relatively negative) top quark loop. Adding new heavy charged chiral fermions leads to additional negative loop contributions which further reduce the diphoton Higgs decay amplitude. (We do not consider cases where many fermions are added which change the overall sign and magnitude of the amplitude.) However, if the new fermions are vector-like doublets and singlets, the existence of heavy gauge invariant masses combined with mixing induced by the Higgs-singlet-doublet Yukawa couplings can change the sign of the new fermion loop contribution and actually enhance the Higgs diphoton branching ratio. Variants of that possibility have been suggested by a number of authors Dawson:2012di (); Carena:2012xa (); Bonne:2012im (); An:2012vp (); Joglekar:2012hb (); ArkaniHamed:2012kq (); Almeida:2012bq (); Kearney:2012zi (); Ajaib:2012eb () who have discussed such scenarios in detail, including experimental constraints on properties of new vector-like fermions SUSYdiphot ().

Here, we assume the above solution to the diphoton excess as our starting point, but endow the new fermions with an additional gauge symmetry with a light gauge boson, of mass . To avoid changing the Higgs production rate through gluon fusion, thereby affecting rates for other final states, these fermions should not carry color quantum numbers. Hence, we focus on new “charged leptons.”

One-loop diagrams involving the new vector-like leptons can also induce, via kinetic mixing, a suppressed coupling of to ordinarily charged particles. Such a gauge boson has recently been invoked in generic discussions of DM particles and their potential phenomenology DMzprime (); Davoudiasl:2010am (). As we shall see, the provides a natural viable solution to the discrepancy for a narrow range of and leads to predictions for new Higgs decay modes, and at the LHC, which may occur at observable rates or if not seen, used to constrain such models.

Vector-like heavy fermions can be found in various new physics models including composite Higgs Kaplan:1983sm () and supersymmetric models Langacker:2008yv () motivated to address the -problem Kim:1983dt (). To avoid problems with new stable charged particles, we consider charges which are multiples of the SM particle charges. This would typically imply that the new particles carry hypercharge (for the sake of minimality, we do not consider triplet fermions as they would require additional Higgs content).

Let us denote the ratio of the enhanced rate for compared to that in the SM by

(1) |

Using the results of Ref. ArkaniHamed:2012kq (), the contribution of a new fermion of electric charge and mass (see Fig. 1) to the above ratio is given by

(2) |

where stems from the SM amplitude for the decay, GeV, and GeV is the Higgs mass. Eq. (2) implies that for we need , that is, the contribution of EWSB () to the mass of must be negative.

We next describe a simple scenario which explains the diphoton excess. We extend one of the examples proposed in Ref. ArkaniHamed:2012kq () to include dark interactions. The extended model contains vector-like “leptons”, and with charge assignments

(3) |

where upper (lower) signs are for (), etc. Here, electric charge is related to hypercharge and the third component of isospin by . The mass matrix is obtained from the following Lagrangian:

(4) |

After EWSB, we get two Dirac fermions and with electric charge and masses and , where , and one neutral Dirac fermion of mass with . All the new heavy leptons have pure vector couplings to gauge bosons; hence, no gauge anomalies are present.

One can show that the above field content results in an enhancement of the diphoton rate, which to a good approximation is given by ArkaniHamed:2012kq ()

(5) |

which is valid for , with

(6) |

The enhancement can be achieved for and , assuming .

Without further assumptions, the model in Eqs. (3) and (4) will lead to stable charged particles since the lightest new vector-like lepton will be charged. To avoid this unphysical situation, following Ref. ArkaniHamed:2012kq (), we mention two approaches:

(I) One possibility is to allow very small mass mixing between the new charged heavy leptons (), and ordinary SM charged leptons () which gives rise to decays. For us, this could be simply achieved if we introduce a “dark” Higgs field, , with which allows small Yukawa interactions where is a general Yukawa coupling matrix which connects the new heavy charged leptons with ordinary charged SM leptons. The left-right and right-left couplings will in general be different. The is well motivated since it can also provide a mechanism for spontaneous symmetry breaking and lead to if ().

The Yukawa mixing interaction will induce potentially dangerous flavor changing weak neutral current interactions for the , and bosons. Here, we focus on the couplings. They are important because the small leads to interesting enhancements and are specific to the model we are considering. The induced non-diagonal interaction appears to be highly suppressed by the factor. However, for the longitudinal component (or Goldstone mode ), that factor is cancelled Davoudiasl:2012ag () and one finds the coupling as required by the Goldstone boson equivalence theorem with different for left and right handed . To avoid generating large chiral changing loop effects in quantities such as the electron and muon anomalous magnetic moments, lepton number flavor violating amplitudes for , , light lepton loop induced masses, etc, some of the (particularly those involving or ) have to be quite small . However, even with such small couplings, the decay rates are likely to provide very prompt signals, , at the LHC, where the light , which subsequently decays into , can mimic a converted high energy photon. (One expects pairs at colliders actually giving rise to di- events.) We note that by assuming small mixing parameters (), we can avoid conflict with precision measurements of -- universality such as those discussed in Ref. Batell:2012mj (). Of course, one might use lepton mixing effects to make further predictions or to accommodate all or part of the discrepancy. The constraints on this model and its phenomenology are potentially rich and interesting, but beyond the scope of this paper. Here, we only suggest it as a means to avoid stable heavy charged leptons.

(II) Another possibility is to avoid mixing with the SM leptons and instead add the fields (carrying only ), with the new expanded neutral fermion mass terms

(7) |

which together with Eq. (4) result in two neutral Dirac fermions, and , with masses and potentially . Given the mixing terms in Eq. (7), the new charged particle would decay into (possibly virtual) and .

It may be tempting to think of the lightest new neutral state, as a stable relic DM candidate. However, without further assumptions, this turns out to be not phenomenologically viable. We will address this question and discuss potentially viable DM alternatives in the appendix.

We now turn to the problem which has persisted over the last several years. The discrepancy between the measured value and the SM prediction is about Bennett:2006fi (); PDG ():

(8) |

where . A simple explanation Zdgm2 () of this difference postulates a new hidden symmetry which kinetically mixes with by

(9) |

where parametrizes the mixing, is the weak mixing angle, and , , is a field strength tensor. Upon kinetic diagonalization, one finds that the massive dark boson obtains an induced coupling , where is the the electromagnetic current darkZ (). This light boson is a target of active and planned searches at JLAB and MAMI in Mainz Bjorken:2009mm (); Freytsis:2009bh (); McKeown:2011yj (); Merkel:2011ze (); Abrahamyan:2011gv (); DarkLight (); Russell:2012zz (). Early results from those experiments are illustrated as constraints in Fig. 2.

In Fig. 2, we give the current exclusion bounds on (adopted from Refs. McKeown:2011yj (); Davoudiasl:2012qa ()). There, we have updated the bounds coming from the recently improved electron anomalous magnetic moment comparison between experiment Hanneke:2008tm () and SM theory Bouchendira:2010es (); Aoyama:2012wj ():

(11) |

Because of the small momentum transfer in Rydberg measurements , the effect of a light on the determination of in Ref. Bouchendira:2010es () is expected to be negligible for the mass range considered. That constraint implies at the level (with obtained from Eq. (10) with ) which rules out a significant region which would otherwise provide a viable solution (i.e. , . Hence, one finds that the discrepancy in Eq. (8) can be explained for

(12) |

and

(13) |

without conflict with current experimental bounds on . We note that the exclusion region due to is somewhat enhanced because of the sign of Eq. (11) which is opposite to that expected from .

Our new constraint rules out the (previously allowed) region of the “” band in Fig. 2. That explicit part of the band is the focus of a proposed direct search at VEPP-3 Wojtsekhowski:2012zq (); however, the experiment will also explore smaller .

In a simple framework, is an arbitrary renormalized parameter set by experiment. Normally, we expect which for gives . That expectation would be natural, if either of the symmetries at low energy descend from a non-abelian group in the ultraviolet sector, such that the high energy (bare) value of is zero. In that case, finite kinetic mixing of the type in Eq. (9) can be naturally induced by loops of fermions which are charged under both and Holdom:1985ag (). The typical value of the kinetic mixing at low energies can then be estimated from a 1-loop diagram (see Fig. 1), which is roughly in the range indicated by Eq. (13). In fact, given an appropriate assignment of fermion charges, one can calculate a finite 1-loop result Holdom:1985ag (). For example, a 2-generation extension of the reference model in Eq. (3) with opposite sign dark charges results in a finite and computable value

(14) |

where correspond to the masses from the second generation of charged vector-like leptons. We see that for typical values of parameters, in Eq. (14) has the size needed to address the anomaly. For example, if the logarithm involving the masses is order unity, for and we find . So, the model introduced in Eq. (3) to explain a excess can also accommodate the discrepancy.

Our model leads to the interesting prediction that the Higgs has new decay channels and (Fig. 3) with rates somewhat smaller than that of the mode. To see this, note that Eqs. (13) and (14) imply we need to explain the measured . Hence, we expect that the new decay modes in Fig. 3 will have a similar amplitude as the extra contribution to (Fig. 1).

To connect the Higgs diphoton excess and , as proposed here, it is sufficient to have a single fermion which carries both and charges. This minimal setup can effectively emerge in our reference model in Eq. (3) if there is a modest hierarchy of masses and the new Higgs decay amplitudes (Figs. 1 and 3) are dominated by the lightest charged state. Under such a simplifying assumption, a rough estimate for the rate of the compared to the observed rate of (not the SM expectation) can be given in terms of by

(15) |

where the factor of 2 accounts for the nonidentical final state particles, and the first set of parentheses factors out the new lepton contribution to the decay. Similarly, the rate of the compared to is

(16) |

Based on our preceding discussion, let us take as a typical value for our scenario. This would imply and for (see Fig. 4). We see that the rates for these new channels are expected to be well below that of but potentially within the reach of the LHC experiments. Note that for in the range of Eq. (12), the will mainly decay promptly into Davoudiasl:2012ag (), i.e. within the beam pipe region. However, since , the decay products will be boosted and highly collimated. In general, we expect that the in the final state will mimic a promptly converted photon () but with a small nonzero mass and production vertex near the beam rather than in the tracker. Such properties could be used as a signal for these new decays.

For the range, the opening angle of order is well below the roughly required for separating the ; so, except for the effect of the magnetic field in the detector, the pair would be indistinguishable from a photon (see Ref. Draper:2012xt () for some related discussions). However, the magnetic field will lead to separated tracks for and in the tracker. Under the assumption of a converted photon, these tracks would normally be fitted assuming zero invariant mass. For our purposes, it would be useful to modify the fitting program to allow masses . A careful examination of the available diphoton data, which is outside the scope of this work and more appropriate for experimental scrutiny, could reveal or constrain the presence of or even events.

The above rough estimates of the relationships between , and decay rates can change if the underlying model deviates from the simplifying assumptions used to derive Eqs. (15) and (16). For example, in a 2-generation extension of the model, the charged leptons carrying opposite dark charges are likely to cancel partially and somewhat reduce the rate of (Fig. 3). In this circumstance, our estimate of the should be viewed as an approximate upper bound. In the case of the amplitude (Fig. 3), if the model is enlarged to include the interactions in Eq. (7), the resulting neutral leptons could either increase or reduce the rate for . In addition, a light may be only a small part of the discrepancy, consistent with a much smaller below the “ explained” band in Fig. 2. In that case, the and rates could be much reduced.

Overall, our estimates for the new decay rates are meant to be suggestive and to stimulate experimental searches for those decays. Definitive predictions require more detailed studies using specific models and parameters. We note that the search for and with resembling a promptly converted photon in the LHC experiments would be largely complementary to light “dark boson” searches such as those at JLAB and in Mainz Bjorken:2009mm (); Freytsis:2009bh (); McKeown:2011yj (); Merkel:2011ze (); Abrahamyan:2011gv (); DarkLight (); Russell:2012zz (). Those programs will not only directly probe the explained region in Fig. 2 for boson, but will also explore significant parts of parameter space outside that band.

“Dark” decay modes of the Higgs may also arise through other mechanisms Gopalakrishna:2008dv (); Baumgart:2009tn (); Davoudiasl:2012ag (). For example, it is possible to have a mode from Higgs () and dark Higgs () mixing Gopalakrishna:2008dv (), and in the presence of mass mixing between and , as studied in Ref. Davoudiasl:2012ag (), a new Higgs decay mode would also be possible. The latter channel could mimic with a promptly converted photon if is sufficiently light. However, the predictions in those cases are more arbitrary; whereas the connection between the Higgs diphoton rate and in our model allows us to make more quantitative estimates for the dark decay rates of the Higgs. We also note that our loop induced and decays involve primarily transverse bosons while the and decays in Refs. Gopalakrishna:2008dv (); Davoudiasl:2012ag () are dominated by longitudinal .

In this paper, we have discussed a possible link between the reported excess of the Higgs to diphoton decay at the LHC experiments and via heavy new vector-like leptons and a light dark gauge boson. A gauge boson of mass with small induced coupling to the SM particles is well motivated as a rather simple explanation of the deviation of from the SM. The required coupling of the to the SM fermions is naturally obtained when it arises from loops of charged extra fermions that couple to both the SM and a dark sector with similar size couplings. This scenario yields an additional contribution to through a loop of the charged extra fermions, which is consistent with the recent level deviation at the LHC experiments. The Higgs boson can also decay into and , with the light bosons looking like promptly converted photons in the ATLAS and CMS detectors. Such a connection implies a few definite predictions in high energy experiments at the LHC and complementary low energy searches at JLAB and in Mainz.

Acknowledgments: This work was supported in part by the United States Department of Energy under Grant No. DE-AC02-98CH10886. WM acknowledges partial support as a Fellow in the Gutenberg Research College. We are grateful to I. Lewis for useful discussions.

Note Added: After this paper was posted and submitted for publication, a preprint Endo:2012hp () appeared which reached similar conclusions regarding the updated exclusion region due to new results in Refs. Bouchendira:2010es (); Aoyama:2012wj (). The authors of Ref. Endo:2012hp () also gave a detailed analysis of the effect of a on explicitly showing that for the range considered, it has a negligible effect.

## Appendix A Relation to Dark Matter

As mentioned before, the lightest neutral Dirac lepton in our simple framework, as presented above, is not a good DM candidate. Starting from our basic model in Eq. (3), let us examine solutions (I) and (II), discussed earlier, for avoiding a stable charged state . In case (I), a small degree of mass mixing with the SM leptons would not alter the spectrum of the model significantly, and we still expect the neutral state to be more massive than and thus allow the decay . Hence, cannot be a long-lived DM candidate.

In case (II), the neutral particle is the lightest new state and could be stable. However, it carries dark charge and can hence elastically scatter from protons, through light exchanges, at too high a rate. For the parameter space of interest here, we would typically expect the scattering cross section of from a nucleon to be cm (see, for example, the last paper in Ref. DMzprime ()) which, for masses , is ruled out by about 11 orders of magnitude from direct DM search constraints :2012nq ().

One could arrange for the early universe annihilation cross section of to be large, so that its relic density is very suppressed and it is not a primary component of DM. Alternatively, one could instead take a positive approach and extend our framework to allow for the presence of a DM candidate. For example, let us postulate the breaking scalar , with MeV, which was invoked earlier in our discussion of case (I). We also add a singlet vector-like lepton without any gauge charges and assume that all of the new fermions are odd under a parity to forbid DM decay. With in the spectrum, we can write down an interaction , with . If , the above interaction would lead to which will be prompt, and all particles will decay into and . The scalar will eventually decay into SM states. However, would be a viable, stable DM candidate. Note that due to the very small mixing induced by , the interactions of with could be quite suppressed, and hence one would avoid the stringent bounds from direct detection.

The relic density of is set by , through -channel exchange of neutral states, which for weak scale masses could be of the correct thermal relic size. The mixing is of order , assuming GeV, typical of the particles considered here. Hence, the direct detection cross section via exchange with protons could be suppressed by , which is just below the current sensitivities :2012nq ().

An alternative possibility entails adding a lepton number violating interaction to our model which splits the neutral Dirac lepton and its antiparticle partner into two nondegenerate Majorana states. Such states have zero dark charge and will not scatter elastically (at least at leading order) off ordinary matter via exchange TuckerSmith:2004jv (). There will be off-diagonal couplings between the two Majorana states which allow inelastic scattering, but for the lighter state, that can be kinematically suppressed. A detailed evaluation of this scenario is beyond the scope of our study.

## References

- (1) G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214 [hep-ex]].
- (2) S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716, 30 (2012) [arXiv:1207.7235 [hep-ex]].
- (3) J. R. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B 106, 292 (1976).
- (4) M. A. Shifman, A. I. Vainshtein, M. B. Voloshin and V. I. Zakharov, Sov. J. Nucl. Phys. 30, 711 (1979) [Yad. Fiz. 30, 1368 (1979)].
- (5) G. W. Bennett et al. [Muon g-2 Collaboration], Phys. Rev. D 73, 072003 (2006) [hep-ex/0602035].
- (6) J. Beringer et al. [Particle Data Group], Phys. Rev. D 86, 010001 (2012).
- (7) A. Czarnecki and W. J. Marciano, Phys. Rev. D 64, 013014 (2001) [hep-ph/0102122].
- (8) For discussions of the relation between the Higgs diphoton excess and anomaly in a supersymmetric framework, see J. -J. Cao, Z. -X. Heng, J. M. Yang, Y. -M. Zhang and J. -Y. Zhu, JHEP 1203, 086 (2012) [arXiv:1202.5821 [hep-ph]]; G. F. Giudice, P. Paradisi and A. Strumia, arXiv:1207.6393 [hep-ph]; R. Sato, K. Tobioka and N. Yokozaki, Phys. Lett. B 716, 441 (2012) [arXiv:1208.2630 [hep-ph]].
- (9) S. Dawson and E. Furlan, Phys. Rev. D 86, 015021 (2012) [arXiv:1205.4733 [hep-ph]].
- (10) M. Carena, I. Low and C. E. M. Wagner, JHEP 1208, 060 (2012) [arXiv:1206.1082 [hep-ph]].
- (11) N. Bonne and G. Moreau, arXiv:1206.3360 [hep-ph].
- (12) H. An, T. Liu and L. -T. Wang, arXiv:1207.2473 [hep-ph].
- (13) A. Joglekar, P. Schwaller and C. E. M. Wagner, arXiv:1207.4235 [hep-ph].
- (14) N. Arkani-Hamed, K. Blum, R. T. D’Agnolo and J. Fan, arXiv:1207.4482 [hep-ph].
- (15) L. G. Almeida, E. Bertuzzo, P. A. N. Machado and R. Z. Funchal, arXiv:1207.5254 [hep-ph].
- (16) J. Kearney, A. Pierce and N. Weiner, arXiv:1207.7062 [hep-ph].
- (17) M. A. Ajaib, I. Gogoladze and Q. Shafi, arXiv:1207.7068 [hep-ph].
- (18) For discussions of the diphoton excess in supersymmetric models, see, for example: M. Carena, S. Gori, N. R. Shah and C. E. M. Wagner, JHEP 1203, 014 (2012) [arXiv:1112.3336 [hep-ph]]; M. Carena, S. Gori, N. R. Shah, C. E. M. Wagner and L. -T. Wang, JHEP 1207, 175 (2012) [arXiv:1205.5842 [hep-ph]]; V. Barger, M. Ishida and W. -Y. Keung, arXiv:1207.0779 [hep-ph]; K. Schmidt-Hoberg and F. Staub, arXiv:1208.1683 [hep-ph].
- (19) P. Fayet, Phys. Rev. D 70, 023514 (2004) [hep-ph/0403226]; D. P. Finkbeiner and N. Weiner, Phys. Rev. D 76, 083519 (2007) [astro-ph/0702587]; N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. D 79, 015014 (2009) [arXiv:0810.0713 [hep-ph]].
- (20) H. Davoudiasl, D. E. Morrissey, K. Sigurdson and S. Tulin, Phys. Rev. Lett. 105, 211304 (2010) [arXiv:1008.2399 [hep-ph]].
- (21) D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. B 136, 187 (1984).
- (22) P. Langacker, Rev. Mod. Phys. 81, 1199 (2009) [arXiv:0801.1345 [hep-ph]].
- (23) J. E. Kim and H. P. Nilles, Phys. Lett. B 138, 150 (1984).
- (24) H. Davoudiasl, H. S. Lee and W. J. Marciano, Phys. Rev. D 85, 115019 (2012) [arXiv:1203.2947 [hep-ph]].
- (25) B. Batell, D. McKeen and M. Pospelov, arXiv:1207.6252 [hep-ph].
- (26) P. Fayet, Phys. Rev. D 75, 115017 (2007) [hep-ph/0702176]; M. Pospelov, Phys. Rev. D 80, 095002 (2009) [arXiv:0811.1030 [hep-ph]].
- (27) See Refs. Davoudiasl:2012ag (); Davoudiasl:2012qa () for a variant, dark , which couples to the neutral current as well.
- (28) H. Davoudiasl, H. S. Lee and W. J. Marciano, Phys. Rev. Lett. 109, 031802 (2012) [arXiv:1205.2709 [hep-ph]].
- (29) J. D. Bjorken, R. Essig, P. Schuster and N. Toro, Phys. Rev. D 80, 075018 (2009) [arXiv:0906.0580 [hep-ph]].
- (30) M. Freytsis, G. Ovanesyan and J. Thaler, JHEP 1001, 111 (2010) [arXiv:0909.2862 [hep-ph]].
- (31) R. D. McKeown, AIP Conf. Proc. 1423, 289 (2012) [arXiv:1109.4855 [hep-ex]].
- (32) H. Merkel et al. [A1 Collaboration], Phys. Rev. Lett. 106, 251802 (2011) [arXiv:1101.4091 [nucl-ex]].
- (33) S. Abrahamyan et al. [APEX Collaboration], Phys. Rev. Lett. 107, 191804 (2011) [arXiv:1108.2750 [hep-ex]].
- (34) P. Balakrishnan et al. [DarkLight Collaboration], JLab PAC37 proposal (http://dmtpc.mit.edu/twiki/bin/view/Projects/DarkLight).
- (35) R. Russell [DarkLight Collaboration], AIP Conf. Proc. 1441, 497 (2012).
- (36) B. E. Lautrup, A. Peterman and E. de Rafael, Phys. Rept. 3, 193 (1972); J. P. Leveille, Nucl. Phys. B 137, 63 (1978).
- (37) D. Hanneke, S. Fogwell and G. Gabrielse, Phys. Rev. Lett. 100, 120801 (2008) [arXiv:0801.1134 [physics.atom-ph]].
- (38) R. Bouchendira, P. Clade, S. Guellati-Khelifa, F. Nez and F. Biraben, Phys. Rev. Lett. 106, 080801 (2011) [arXiv:1012.3627 [physics.atom-ph]].
- (39) T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. 109, 111807 (2012) [arXiv:1205.5368 [hep-ph]].
- (40) B. Wojtsekhowski, D. Nikolenko and I. Rachek, arXiv:1207.5089 [hep-ex].
- (41) B. Holdom, Phys. Lett. B 166, 196 (1986).
- (42) P. Draper and D. McKeen, Phys. Rev. D 85, 115023 (2012) [arXiv:1204.1061 [hep-ph]].
- (43) S. Gopalakrishna, S. Jung and J. D. Wells, Phys. Rev. D 78, 055002 (2008) [arXiv:0801.3456 [hep-ph]].
- (44) M. Baumgart, C. Cheung, J. T. Ruderman, L. -T. Wang and I. Yavin, JHEP 0904, 014 (2009) [arXiv:0901.0283 [hep-ph]].
- (45) E. Aprile et al. [XENON100 Collaboration], arXiv:1207.5988 [astro-ph.CO].
- (46) D. Tucker-Smith and N. Weiner, Phys. Rev. D 72, 063509 (2005) [hep-ph/0402065].
- (47) M. Endo, K. Hamaguchi and G. Mishima, arXiv:1209.2558 [hep-ph].