Revised constraints on hidden photons produced in nuclear reactors
New light vector particles – hidden photons – are present in many extensions of the Standard Model of particle physics. They can be produced in nuclear reactors and registered by neutrino detectors. New limits on the models with hidden photons have been obtained in Ref. Park (2017) from analysis of published results of the NEOS and TEXONO neutrino experiments. We criticize that paper from both theoretical and experimental points of view. Accounting for oscillations between visible and hidden photons, we find that the neutrino experiments are generally insensitive to the hidden photons lighter than eV, and present revised limits for heavier hidden photons from results of the TEXONO experiment.
A number of extensions of the Standard Model of particle physics (SM) introduce massive vectors, singlet with respect to the SM gauge group. These hypothetical particles are called hidden photons, dark photons, paraphotons, etc Okun (1982). They can serve as dark matter particles or messengers between the visible sector (SM) and hidden sector(s), which dynamics solves (some of) SM phenomenological and theoretical problems (neutrino oscillations, matter-antimatter asymmetry of the Universe, etc.) or suggests some phenomena which impacts on physics and Universe are not recognizable at present (see e.g. Pospelov et al. (2008); Bjorken et al. (2009); Ilten et al. (2018)).
Hidden photon can couple to the SM via vector portal interaction. The corresponding coupling constant is dimensionless and hence low- and high-energy experiments exhibit similar sensitivity to this type of new physics, if the hidden photon is sufficiently light. In particular, hidden photon of mass can mix with SM photon , the relevant lagrangian Holdom (1986) reads
Here , ; is dimensionless parameter of visible-hidden mixing and is electromagnetic current.
The mixing term in (1) can be responsible for the hidden photon production in a nuclear reactor, if kinematically allowed, i.e. MeV. Recent paper Park (2017) suggests searches for these particles with help of neutrino detectors, and from results of the NEOS Ko et al. (2017) and TEXONO Deniz et al. (2010) experiments places upper limits on the mixing parameter for a wide range of the hidden photon mass from 1 MeV to eV. In this letter we criticize that study, which wrongly describes production, propagation and detection of the hidden photons and incorrectly treats the NEOS and TEXONO experimental results to constrain the hidden photon parameters. Below we explain all the relevant issues and present correct upper limits on the mixing at various .
It is argued in Ref. Park (2017) that for the NEOS and TEXONO setups the hidden photons are produced mostly by photons of energy MeV via Compton scattering off free electrons in a nuclear reactor. However, we observe that the mixing in (1) actually converts photons into hidden photons via oscillations similar to the well-studied neutrino oscillations. The visible-to-hidden photon oscillations are fully described e.g. in Ref. Redondo (2015) devoted to the hidden photon production and propagation in the Sun.
Let us note that Ref. Park (2017) defines photon and hidden photon states not as electromagnetic interaction eigenstates but as free Hamiltonian or propagation eigenstates. In these terms the hidden photon couples to electromagnetic current. Therefore, this state is produced not only via Compton scattering off electrons but also in radiative nuclear transitions coherently along with usual photon. This contribution in particular was not taken into account in Ref. Park (2017). Given the above analogy with neutrino oscillations, we consider the interaction eigenstates to be more convenient for the estimate of hidden photon yield. Typically, one replaces (see, e.g. Redondo (2008)) the hidden photon in (1) as , with being sterile with respect to the electromagnetic interaction. The kinetic term of (1) becomes diagonal in terms of the new variables, while the mass term gains off-diagonal components. Similar to the case of neutrinos, the difference in eigenvalues of the mass squared matrix, , defines the oscillation length,
Photons interact in matter and gain effective non-zero mass saturated for the reactor case by the plasma frequency Lifshitz and Pitaevski (1981), , where is fine-structure constant, is electron mass and is density of free111For the reactor photon energies all electrons in matter can be considered as free. electrons inside the reactor. Numerically, for the reactor material one finds that the photon mass varies within
With effectively massive photon the mass squared difference entering (2) reads
It never falls below , except for the resonance region, . Hence, the oscillation length (2) is much smaller than the size of the nuclear reactor core of order meters.
The oscillation approximation to the hidden photon production implies coherence between mass eigenstates of the system, which in particular can be lost due to fast separation of travelling wave packets. The relevant coherence length in vacuum can be estimated as where is size of the photon wave packet at production and is difference in velocities of the mass eigenstates. Using typical half-lifes of prompt -decays of fission fragments about s (see e.g. Johansson (1964); Albinsson (1971)), one obtains cm. Corresponding coherence length reads
and always exceeds the oscillation length (2). This statement remains true even in the media where the coherence length decreases due to interactions of fission products and after rescattering of photons off electrons. So the oscillation approximation is justified in our case.
The oscillation terminates by absorption of the photon in the reactor material. The photon absorption length in a nuclear reactor, , varies from a few to few tens cm for the relevant energy range MeV. It is much shorter than reactor size of about several meters, but it is typically longer than the oscillation length (2) except for the resonance region . In the following numerical estimates we take cm. Then most photons produced in the core get absorbed in the reactor material (mostly in water and steel) unless they oscillate into hidden photon with probability Redondo (2015)
The term in denominator of (6) responsible for the photon absorption dominates only in the resonance region,
where with accuracy of a few percent.
In the non-resonance case the condition (7) is invalid, and the probability depends on the relation between and . For heavier hidden photons, i.e. when eV, the probability turns to simple law, while in the opposite case one obtains
where we use eV for the estimate. Here we find a huge suppression with respect to Compton-based estimate used in Park (2017).
The hidden photon production rate is obtained by convolution of the probability (6) with photon flux in the reactor, which we take as in Ref. Park (2017). The hidden photons leave the reactor and can be observed at some distance in a detector designed to measure the reactor antineutrino flux. The hidden photons oscillate and produce photons which can be registered at any part of the detector by Compton scattering off electrons if the detector is sufficiently large. Each hidden photon then produces the Compton-like signature with probability (6) where ,
and in (4) should be calculated for material along the hidden photon path inside the detector. Even neglecting (that is for ) one obtains an estimate , which deviates from that in Ref. Park (2017), where a numerical factor was introduced to account for difference in numbers of polarization states between photon (two) and massive photon (three). We find this factor irrelevant since only two transverse polarizations of the hidden photon are produced via oscillations of the massless photon. The production of the longitudinal component of the massive photon is suppressed for the interesting here ranges of masses and photon energies, , see e.g. An et al. (2013); Redondo and Raffelt (2013).
However, the neutrino detectors are made of dense material, so the effective photon mass is certainly not smaller than that in water. For the numerical estimates below we chose eV inside neutrino detector as well. Hence, the total probability in general non-resonance case is just a product of (6) and (9), which implies a huge suppression factor for the light hidden photons, , with respect to the estimates of Ref. Park (2017).
In the resonance case, , the mass difference (4) becomes equal to
and the oscillation length (2) is
Similar enhancement one expects for the registration, however it generally occurs for another resonance mass , since reactor and detector materials are different, and so the corresponding photon masses.
Therefore, in the two very narrow mass ranges the number of signal events in the detector gets amplified up to hundred thousand times with respect to the Compton-based result, . Since the estimate (10) is valid only in the very narrow mass ranges defined by the photon effective mass in the matter, its applicability requires a good knowledge of the nuclear reactor core structure and the detector, which is unavailable for us. However, it can be applied by the NEOS and TEXONO collaborations. Note, that in the realistic case of inhomogeneous materials the probability formula (6) gets modified, see Ref. Redondo (2015). The amplification one observes in (10) may be partially reduced then.
The number of signal events scales with mixing as . To summarize our critics of the theoretical part of Ref. Park (2017) we find that at eV its results for the event numbers are underestimated by factor 3/2, except the resonance regions where they may be (a special study is needed) underestimated by a factor of . For lighter hidden photons, eV, the number of signal events are overestimated by factor . Since Ref. Park (2017) claims the mass-independent upper limit of about , our observation suggests, instead, that the neutrino experiments are absolutely insensitive to the hidden photons lighter than about 0.05 eV.
Now we turn to the analysis of the experimental data of the NEOS and TEXONO performed in Ref. Park (2017). Both estimates are wrong and overoptimistic with respect to the sensitivity of the neutrino experiments to the model with hidden photons. We start with the TEXONO. In the analysis the collaboration applies a special anti-Compton selection, which reduces the background by a factor of 6 in the energy range 3-8 MeV utilized for the searches. The signature of a hidden photon is very similar to the Compton scattering process. Therefore, the anti-Compton selection decreases the efficiency of the hidden photon detection and consequently the upper limit derived in Ref. Park (2017) should be corrected for this efficiency. The efficiency of the anti-Compton selection for single photons is not given by the TEXONO collaboration. It can be smaller than the efficiency for the background suppression. However, for the upper limit estimates we can assume them to be equal and hence to reduce the expected number of hidden photons in Park (2017) by a factor of 6.
In Fig. 1 we present revised 95% CL upper limit on the parameters and of the model from TEXONO data where in recalculation of the results from Park (2017) we take into account both theoretical and experimental issues discussed above.
At large masses of the hidden photon, i.e. eV, the overall correction to the signal estimate claimed in Park (2017) is a factor of 4; given dependence the corresponding limit on is only slightly weaker than that presented in Park (2017). At the same time we see that the sensitivity of the reactor experiment to the hidden photon model is drastically decreased for eV.
Analysis of the NEOS data is wrong at several points. Since Ref.  from Park (2017) is not available for us, we naturally assume that the conclusion ’all of the reactor-on event candidates are due to background’ is based on the approximate equality . Here () and () are the number of -events and data taking time during the reactor on (off) period. Then absence of the signal events associated with the hidden photons implies
Its statistical uncertainty
determines the upper limit on the hidden-visible mixing, 1.96 would correspond to 95% CL. Numerically for the NEOS on- and off- time intervals we obtain instead of adopted in Ref. Park (2017).
Even more important is that the estimation of the upper limit ignores possible systematic errors. The upper limit corresponds to of the total number of -events. This small number implicitly assumes the absence of time variations of the detector efficiency and background contribution at a similar very low relative level. The NEOS experiment Ko et al. (2017) has not presented any evidence of such a challenging stability.
Indeed, signatures of the hidden photon interactions in the NEOS detector are practically indistinguishable from signatures of positrons in the electron antineutrino induced Inverse Beta Decay (IBD) reactions. The NEOS experiment detected 339.1 thousand IBD events whose signature includes apart from a prompt positron signal also a delayed event resulting from -Gd capture. At least 292.7 thousand of them (1626 events per day for data taking of 180 days) had the total positron energy (prompt energy) in the range 1-5 MeV, at which the estimation of the expected number of hidden photons was performed. This number is 5.5 times larger than the 95% CL upper limit on number of observed hidden photon events presented in the paper Park (2017). Here we again point out that the hidden photon event candidates would look like single prompt events and thus their number can not be smaller than the number of fully reconstructed IBD events divided by the neutron detection efficiency and other IBD process selection efficiencies. These efficiencies are not given in the NEOS paper but most probably are not larger than 80%. This observation again explicitly demonstrates that the estimation Park (2017) of the upper limit on the number of detected hidden photons in the NEOS experiment can not be correct. It is at least a factor of 6 too optimistic. For a typical stability of the detector efficiency and background level of about 1% the limit on the number of the detected hidden photons in the NEOS experiment derived in Park (2017) is 2 orders of magnitude too optimistic. Therefore, we ignore the NEOS data in our recalculations of the upper limit on the mixing parameter shown in Fig. 1.
To conclude, we present theoretical description of the dark photon signal in reactor neutrino experiments and obtain corrected upper limits on the hidden photon mass and the mixing parameter from the TEXONO data, see Fig. 1. There are two comments in order. First, we disregard Compton rescattering of the photons during their propagation in the reactor core. The secondary photons may contribute to the hidden photon production. In this respect our limits are conservative. Second, in our estimates we take minimum value of the effective photon mass equal eV for the reactor and detector. The sensitivity to lighter hidden photons, , are strongly reduced. The chosen value refers to water and grows in denser materials, thus suppressing the sensitivity for correspondingly heavier hidden photons. Therefore, we expect, that limits in Fig. 1 may be improved, but only mildly, with dedicated analyses performed by the NEOS and TEXONO collaborations. The only promising case is the resonance, , where the amplified production of the hidden photons may significantly improve the limit on mixing .
We thank S. Gninenko, S. Troitsky and I. Tkachev for valuable discussions. The theoretical analysis of hidden photon production and detection within the oscillation approximation was supported by the RSF grant 17-12-01547. The analysis of the experimental data adopted to limit the model parameters was supported by the Grant of the Russian Federation Government, Agreement #14.W03.31.0026 from 15.02.2018.
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