Coherent scattering and macroscopic coherence: Implications for neutrino and dark matter detection
We study the question of whether coherent neutrino scattering can occur on macroscopic scales, leading to a significant increase of the detection cross section. We concentrate on radiative neutrino scattering on atomic electrons (or on free electrons in a conductor). Such processes can be coherent provided that the net electron recoil momentum, i.e. the momentum transfer from the neutrino minus the momentum of the emitted photon, is sufficiently small. The radiative processes is an attractive possibility as the energy of the emitted photons can be as large as the momentum transfer to the electron system and therefore the problem of detecting extremely low energy recoils can be avoided. The requirement of macroscopic coherence severely constrains the phase space available for the scattered particle and the emitted photon. We show that in the case of the scattering mediated by the usual weak neutral current and charged current interactions this leads to a strong suppression of the elementary cross sections and therefore the requirement of macroscopic coherence results in a reduction rather than an increase of the total detection cross section. However, for the scattering mediated by neutrino magnetic or electric dipole moments coherence effects can actually increase the detection rates. Effects of macroscopic coherence can also allow detection of neutrinos in 100 eV – a few keV energy range, which is currently not accessible to the experiment. A similar coherent enhancement mechanism can work for relativistic particles in the dark sector, but not for the conventionally considered non-relativistic dark matter.
- 1 Introduction
- 2 Radiative neutrino scattering on electrons
- 3 DM detection through radiative coherent scattering?
- 4 Effects of atomic binding
- 5 Coherent effects and the cross sections
- 6 Summary and discussion
Recently, the COHERENT collaboration has reported the first observation of coherent elastic neutrino–nucleus scattering [1, 2], a process predicted over forty years ago [3, 4]. This observation completed the standard-model picture of neutrino interactions with nucleons and nuclei and opened up a new window to probe physics beyond the standard model and nuclear structure; it also has important implications for astrophysics. Very recently, the CONUS collaboration has reported the first experimental indication of coherent elastic neutrino-nucleus scattering with reactor antineutrinos . Coherence of the process implies that the total cross section is proportional to the squared number of the target particles rather than to their number; as a result, for the first time it became possible to observe neutrinos with a hand-held detector rather than with ton- or kiloton-scale ones – a spectacular achievement indeed. One then naturally wonders if it is possible to achieve coherence of neutrino detection on scales that are larger than the nuclear scale, such as atomic or even macroscopic scales, leading to a further significant increase of the detection cross sections. This would also be of great interest for detecting Dark Matter (DM) particles which are currently being actively looked for.
Coherent neutrino scattering on atoms [6, 7, 8] has a two-fold advantage. First, the scattering would occur not just on nucleons inside the nucleus but also on atomic electrons, and the increased number of scatterers would mean additional enhancement of the detection cross section. Second, within the standard model, scattering proceeds through both charged-current (CC) and neutral-current (NC) weak interactions, whereas the scattering is mediated only by neutral currents. Therefore, coherent neutrino–atom scattering would be sensitive to neutrino flavour and thus could potentially be used for studying neutrino oscillations. This is in contrast with the already observed coherent elastic neutrino–nucleus scattering proceeding only through neutral-current interactions which are flavour blind.
The problem with coherent neutrino-atom scattering is that the atomic recoil energies would be very small and extremely difficult to measure. Indeed, coherence requires the momentum transfer to the scatterer to be smaller than or at most of the order of the inverse radius of the scatterer. It is only under this condition that it will be impossible to find out on which constituent of the target particle has the neutrino scattered, and the neutrino waves scattered from the different constituents will be in phase with each other, which are the necessary conditions for coherent scattering. For neutrino–atom scattering, this would imply
where is the Bohr radius. For an atom with the atomic number the recoil energy would then be
about eight orders of magnitude below the currently achieved sensitivity. Measuring such small recoil energies presents a formidable experimental challenge and, if possible at all, would probably require new technologies.
1.1 Macroscopic coherence?
How about scattering with coherence on macroscopic scales? Clearly, this would require measuring even much smaller recoil energies and so does not look practical. It is interesting, however, to inquire what could be the increase of the detection cross sections if such measurements were possible, leaving for the moment the detection problem aside. For an estimate, we will be assuming coherence on the target length scale of cm and the target mass . The total cross section of the elementary neutrino elastic scattering process (i.e. of the scattering on a single target particle) with non-relativistic target particle recoil is , where is the Fermi constant and is the energy of the incident neutrino. To achieve macroscopic coherence, we need momentum transfers satisfying .222Note that by we mean the maximum allowed value of and not the time component of the 4-vector . However, the energies of neutrinos we normally deal with are many orders of magnitude larger than this value, and so are the typical momentum transfers. From the kinematics of elastic scattering it follows that , where is the neutrino scattering angle; therefore, to achieve macroscopic coherence one has to restrict neutrino scattering to nearly forward directions:
This means a severe restriction of the phase space accessible to the final-state neutrino, which, in turn, leads to a strong suppression of the corresponding elementary cross section:
However, in order to find the cross section per one target particle one has to multiply the elementary cross section (4) by the number of particles that contribute coherently to the scattering process, i.e. by the number of particles in the coherent volume . As a result, the cross section per target particle will be proportional to , where is the number of scatterers in the target and we have assumed that the coherent volume is comparable with the total volume of the target. Thus, by going to smaller one could increase the detection cross section. 333Obviously, one cannot go to the limit , as the above estimates are only valid for coherence volumes not exceeding the total volume of the detector. The total cross section obtained by summing over all the scatterers in the target will then scale as , i.e. the cross section increase due to the coherence effects is . While this is much smaller than an extra factor of one could naively expect, it still would mean a very strong enhancement of the detection cross section.
The problem is, of course, that the recoil energies are too small to be detected. For eV and the total mass of particles in the coherent volume , one finds , the quantity which is not going to be ever measured. To give just one reason for that, in order to measure recoil energy of this magnitude, one needs an energy resolution of at least the same order of magnitude, . By time-energy uncertainty relation, the duration of the measurement process should then exceed , which is about 10 orders of magnitude larger than the age of the Universe. To summarize, macroscopic coherence holds and the cross section becomes very large only for neutrino scattering in a very narrow forward cone, which corresponds to unmeasurably small recoil energies of the target particles.
As is seen from the above discussion, one reason why it is difficult to achieve macroscopic coherence in neutrino scattering processes is that one usually measures the recoil energy of the target particles, which for small recoils is suppressed compared to the recoil momentum by a very small factor . The same applies, of course, to experiments on direct DM detection.444 Macroscopic coherence is, however, readily achieved in experiments on light, -ray or neutron scattering from macroscopic targets because what is detected are the scattered particles and not the recoil of the target.
1.2 Weber’s approach and structure factors
Weber suggested to detect neutrinos through their coherent scattering on crystals in torsion balance experiments. This approach combines two interesting ideas. First, as the force coincides with momentum transfer per unit time, the force neutrinos impinge on a crystal is directly related to the momentum transfer to the target rather than to the recoil energy. As discussed above, this is a very desirable feature. Second, when the expected recoil energy of the individual atoms is below the Debye temperature of the crystal , the recoil momentum is with high probability given to the crystal as a whole rather than to the individual atoms, similarly to what happens in the Mössbauer effect. Indeed, the recoil-free fraction is approximately given by 
where is the crystal temperature.555 This formula is valid for harmonic crystals in the limit . For a general , the second term in the round brackets in the exponent should be replaced by . For the quantity , i.e. the momentum is transferred to the crystal as a whole with probability close to 1. For typical Mössbauer crystals keV, and the condition is easily satisfied even for neutrinos in the 10 MeV energy range. Weber asserted that, since in this case it is impossible to find out on exactly which atom the neutrino had scattered, the contributions of different scatterers should add up coherently, leading to macroscopic coherence and a very strong enhancement of the detection cross section.
He developed a theoretical approach to describe neutrino coherent scattering on crystals and obtained encouraging results. He then performed experiments with solar neutrinos, reactor antineutrinos and a radioactive neutrino source and in all three cases reported positive results, in reasonable agreement with his theoretical expectations.
These results were met with scepticism, and were strongly criticized by a number of authors. It was pointed out that the same ideas applied to the -ray  and neutron  scattering on crystals would lead to unrealistically large cross sections in direct contradiction with experiment. In Refs. [15, 16, 17, 18, 19, 20] the theoretical approach of [9, 10, 11] was criticized. It was concluded that the effect had been overestimated by about 24 orders of magnitude. Finally, subsequent torsion balance experiments on neutrino-crystal scattering with sensitivities much higher than the sensitivity of the Weber’s device have reported null result [21, 22].
So, what went wrong with Weber’s ideas? The absence of recoil of the individual atoms, which was the main ingredient of his approach, is necessary for macroscopic coherence, but is not sufficient. It is also necessary that the neutrino waves scattered from different centers be in phase with each other. The amplitudes of particle scattering on a group of scattering centers rather than on a single center should contain the relevant structure factors, which describe the relative phases of the amplitudes corresponding to different scatterers. For elastic neutrino scattering the structure factor is given by
where and are the momenta of the incident and scattered neutrinos, is the coordinate of the th scatterer and is the total number of scatterers in the target. Introducing the number density of scatterers , one can rewrite the structure factor (6) in the familiar form-factor form
where is the momentum transfer to the target.
The structure factors are crucial to the issue of coherence of the scattering process, i.e. to the question of whether the amplitudes of neutrino scattering on different target particles should be added coherently. While the exact form of these factors depend on the specific target utilized in the experiment, the fully coherent and completely incoherent regimes can be studied in a rather general way. Indeed, the squared modulus of the transition amplitude contains the factor
If the momentum transfer satisfies the condition
(where is a linear size of the target), one can replace all the phase factors under the sum in eq. (8) by unity, which gives .666As discussed above, in the cross section this dependence reduces to if neutrino scattering has to be restricted to nearly forward directions in order to achieve sufficiently small momentum transfers. In this case neutrinos scattered from different constituents of the target are in phase with each other. In the opposite limit only the diagonal () terms in the sum survive, and one finds , i.e. we obtain the usual dependence of the total cross section on the number of the target particles. This corresponds to incoherent neutrino scattering.
For scattering on crystals, yet another possibility of having macroscopically coherent effects exists, namely, when the phase differences in eq. (8) are integer multiples of . This leads to the well known Bragg condition for diffraction on crystals,
where is the interplanar distance in the crystal, is the angle between the neutrino momentum and the atomic plane (the scattering angle being ), and is an integer. Just like for -ray diffraction on crystals, the intensity of the scattered neutrino wave in the directions of the Bragg maxima is . It is noticeably different from zero in narrow cones around the Bragg directions, with the corresponding solid angles , and is practically zero outside these cones. Thus, the intensity of the scattered neutrino wave around each Bragg maximum is . Since the scattered neutrinos are not detected, the quantity that is in principle measurable is the crystal recoil momentum, or the force impinged on the crystal. For a given direction of the momenta of the incident neutrinos with respect to the crystal atomic planes and , eq. (10) selects the neutrino energy that satisfies the Bragg condition.777For the Bragg condition is satisfied for all neutrino energies. However, it corresponds to forward scattering in which there is no momentum transfer from neutrinos to the crystal. As the Bragg maxima have finite widths, neutrinos in finite energy intervals will actually experience Bragg diffraction; these intervals are, however, very small and scale as . As a result, the overall momentum transfer to the crystal scales as , just like for the scattering on amorphous bodies .
Thus, scattering on crystals unfortunately does not give any advantage for neutrino detection, and one is back to consider the condition in eq. (9). Since it was not satisfied in Weber’s experiments, macroscopic coherence could not be achieved.888 Note that Weber actually did consider the structure factors, but evaluated them incorrectly . 999 When the preliminary results of this work were presented at CERN neutrino platform week in January 2018, we were informed by P. Huber that some of the considerations presented in Sections 1.1 and 1.2 had appeared earlier in the unpublished (but not classified) Jason Report by Callan, Dashen and Treiman . We thank Patrick Huber for this comment and for sending us a scanned copy of .
1.3 Our approach: Radiative neutrino scattering on electrons
In the present paper we consider a different realization of the idea of employing the momentum transfer to the target rather than the recoil energy of the target particle – radiative neutrino scattering on atomic electrons or on free electrons in a conductor:
In this case the emitted photon rather than the recoil electron is detected, and the photon energy can be as large as the neutrino momentum transfer . Most importantly, the momentum transfer itself (and so also ) need not be small in order to ensure macroscopic coherence of the process. What has to be small ( where is the macroscopic length scale of the coherent volume) is the net recoil momentum of the target particle, which is the difference between the momentum transfer from the neutrinos and the momentum carried away by the photon. This can happen even when and are both large compared to . The above points directly follow from the expression for the structure factor in the case of the process (11) (cf. eq. (6)),
The condition for the scatterers within a volume to contribute coherently is .
Note that in both radiative and elastic scattering cases, momentum conservation implies that the argument of the structure factor coincides with the momentum of the recoil electron. This has a simple physical interpretation. As was discussed above, coherence requires . Since the uncertainty of the magnitude of momentum cannot be much larger than the momentum itself, we also have in this case . The Heisenberg uncertainty relation then means that the coordinate uncertainty of the recoiling electron exceeds the size of the target, that is, one cannot identify which electron the neutrino was scattered off. The requirement also ensures that the neutrino waves scattered from all the electrons within the volume are in phase with each other. These are precisely the conditions of coherence of the contributions of different individual electrons to the amplitude of the process.
1.3.1 Previous studies
The radiative neutrino scattering on electrons (11) was first considered by Lee and Sirlin back in 1964  and since then has been studied by many authors (see, e.g., [26, 27, 28, 29, 30, 31, 32]). To the best of our knowledge, only two studies [28, 29] concern the issue of macroscopic coherence of the process. In  it was suggested to use radiative neutrino scattering on free electrons in a conductor in order to detect cosmic background neutrinos. It was argued that macroscopic coherence of the process can be achieved, leading to measurable photon production cross sections. These results have been criticized in , where a crucial flaw of  was pointed out. It was demonstrated that, as neutrino impact pushes the conduction electrons deeper inside the target, the excess positive ion charge on its surface creates a restoring force which pulls the electrons back. As a result, the cross section of coherent radiative neutrino scattering gets suppressed by a factor , where eV is the plasma frequency. For cosmic background neutrinos , which makes the process completely unobservable.
It is actually not difficult to understand the reason for this drastic suppression of the photon production cross section. Photon radiation in process (11) is due to the time dependent dipole (and in general higher multipole) moments induced by the neutrino scattering on the electrons of the target. In the very long wavelength limit, when the energy transfer to the system (and so also the frequency of the induced radiation) is small compared to the characteristic frequencies of the system, the induced moments are small and the photon radiation is strongly suppressed. This situation is very similar to the one encountered when comparing the cross section of the Rayleigh scattering (photon scattering on bound electrons in atoms) to that of the Thomson scattering (scattering of photons on free electrons). In the classical limit the two cross sections are related by  101010The accurate quantum mechanical formula is more complicated and depends sensitively on the atomic structure, see Section 4 below. However, the limits of large and small are reproduced by the classical formula (13) correctly.
where is a characteristic atomic frequency. In the limit the two cross sections coincide, i.e. the photon scattering on atomic electrons proceeds as if the electrons were free. In the opposite limit one finds . This is the famous law which is responsible for the blue color of the sky. The suppression of the cross section of radiative scattering of cosmic background neutrinos on conduction electrons found in  is of exactly the same nature.111111In Ref.  this suppression was incorrectly interpreted as being due to the electric neutrality of the target. For neutrino scattering on a charged conductor the restoring force on the electrons accelerated by the neutrino impact would still be there, and would be due to both the pull from the positive ions and push from the excess electrons. As a result, the suppression would still be present. This is quite analogous to the situation with photon scattering on atomic systems, where the scattering on charged ions exhibits at the same suppression as the scattering on neutral atoms [34, 35].
1.3.2 Radiative scattering with and phase space constraints
In the present paper we shall consider neutrino detection through coherent radiative neutrino scattering on atomic electrons or on free electrons in a conductor. We will be assuming the energies of the incident neutrinos to be higher than the corresponding characteristic atomic frequencies or plasma frequencies . This will allow the momenta of the emitted photons to exceed and , thus avoiding the suppression of the cross sections discussed above. We will concentrate on the situations when the momentum carried away by the emitted photon nearly compensates the momentum transfer to electrons from neutrinos, leading to very small net recoil momenta of the target electrons. As discussed at the beginning of Section 1.3, this will result in macroscopic coherence of the detection process, while completely avoiding the problem of measuring extremely small recoil energies of the target.
There is a price to pay, however. The requirement puts a stringent constraint on the phase space volume accessible to the final-state particles, and in general also on the amplitude of the process. This should lead to a suppression of the cross section of the individual process, just like in the case of elastic neutrino scattering discussed in Section 1.1 (see eq. (4)). It has to be seen if the enhancement of the cross section due to macroscopic coherence can overcome this suppression, as it is the case for the elastic neutrino scattering (which, however, is unobservable because of the vanishingly small recoil energies). In the present paper we study this issue in detail.
We find that for radiative neutrino scattering mediated by the standard NC and CC weak interactions macroscopic coherence can occur, but only at the expense of severe restriction of the kinematics of the process, resulting in the net suppression rather than enhancement of the total cross section. In contrast to this, for radiative neutrino scattering mediated by neutrino magnetic or electric dipole moments the net effect is an enhancement of the cross section per target electron compared to that for the elastic scattering, though only for the kinetic energies of electron recoil in the elastic process exceeding keV. In addition, coherent radiative scattering due to neutrino magnetic or electric dipole moments could potentially allow detection of neutrinos of very low energies, which are currently not accessible to the experiment. The mechanism we consider here is, unfortunately, not operative for conventional (non-relativistic) DM particle candidates; however, it could work for relativistic particles that may exist in the dark sector.
1.4 The structure of this paper
The paper is organized as follows. In Section 2 we consider the radiative neutrino scattering on free non-relativistic electrons, both without any additional kinematic constraints and assuming that the electron recoil momentum is limited from above by a small value , allowing for macroscopic coherence of the process. In Section 2.1 we discuss the radiative neutrino scattering on free electrons mediated by the usual NC and CC weak interactions, whereas in Section 2.2 we study the case when the scattering is mediated by the neutrino magnetic or electric dipole moments. In Section 3 we briefly discuss the question of whether macroscopic coherence could be realized and the same enhancement mechanism could work for direct DM detection and conclude that for conventional DM this is not possible (mainly for kinematic reasons). In Section 4 we discuss atomic binding effects in the case when radiative scattering takes place on electrons in an atom rather than on free electrons. We demonstrate that these effects can be neglected in the cases of interest to us. In Section 5 we use the cross sections obtained in Section 2 to consider the effects of macroscopic coherence on radiative neutrino scattering and the question of whether it can increase the detection cross sections. We summarize and discuss our results in Section 6. In Appendix A the kinematics of radiative scattering is considered, whereas Appendix B contains some technical details of calculations of the integrals over the 3-body phase space.
2 Radiative neutrino scattering on electrons
We shall consider the process
in the rest frame of the initial electron. Here
are the 4-momenta of the incident neutrino, initial-state electron, scattered neutrino, final-state electron and emitted photon, respectively. In this section we consider radiative neutrino scattering on free non-relativistic electrons; possible effects of atomic binding will be discussed in Section 4. Eventually, we will be interested in coherent radiative neutrino scattering on a macroscopic lump of electrons, which we will assume to be unpolarised, i.e. to have zero total spin. This allows us to simplify the problem by neglecting the electron spin, i.e. to consider neutrino scattering on a “spinless electron” – a particle with the electron’s charge and mass but zero spin. Neutrinos are assumed to be ultra-relativistic, so that the neutrino mass can be neglected both in the kinematics of the process and in calculating transition matrix elements.
We shall consider neutrino-electron scattering mediated either by the usual NC and CC weak interactions or by neutrino magnetic (or electric) dipole moments. In each case we calculate the cross section first allowing all the final-state momenta to span the full ranges allowed by 4-momentum conservation,121212 Except that for -integrated cross sections an infrared cutoff will be introduced for the photon energies, see below. and then restricting the net recoil momentum of the electron to satisfy , where is small compared to the maximum value of allowed by the kinematics of process. We will need the cross sections with such a kinematic restriction when considering macroscopic coherence effects in Section 5.
2.1 Weak interactions induced radiative process
In calculating the cross section of radiative neutrino scattering (14) on “spinless electron” we take into account only the vector current part of the weak NC and CC interactions of electrons since the axial-vector current does not contribute to neutrino scattering on zero-spin targets.
The amplitude of the weak interaction induced radiative neutrino scattering on a “spinless electron” can be written as
Here is the vector weak coupling constant, is the electron charge, is the polarization vector of the produced photon and
is the matrix element of the neutrino current. To leading order in electroweak interaction the tensor is given by
The subscript at stands for neutrino-electron scattering due to the weak interactions; for and scattering the interaction is mediated by the weak neutral current, whereas for the scattering both neutral and charged currents contribute. For NC induced radiative scattering, the leading order amplitude is described by the diagrams of Fig. 1. The three terms in correspond to the three diagrams shown there. For CC induced radiative scattering of on a “spinless electron”, one cannot directly draw diagrams similar to those in Fig. 1, as the vertex connecting spin 1, 0 and 1/2 fields does not exist. Instead, one should consider the scattering on the “standard” spin 1/2 electron described by the left and middle diagrams of Fig. 1 with the boson line replaced by the boson one and the electron and neutrino lines in the final state interchanged. For unpolarised target electrons, in the limit of non-relativistic electron recoils the electron spin becomes relatively unimportant, and the corresponding CC amplitude again has the form (16) with given by eq. (18).131313To arrive at this result one has to make use of the Fierz transformation and consider unpolarised electrons in the limit when their recoil energy is non-relativistic in the rest frame of the initial-state electrons. Note that, as we are interested in coherent effects, the summation over the electron spin states should be done at the amplitude level. Note that satisfies the gauge invariance conditions
Thus, expressions (16) and (18) adequately describe the amplitude of process (11) for non-relativistic electrons, with both NC and CC contributions properly taken into account. The coupling constant is given by
We now proceed to calculate the cross sections, first without constraining . For the double and single differential cross sections one finds
Here is the electron mass and is the angle between the momentum of the emitted photon and that of the incident neutrino. Because of the usual infrared divergence, in order to calculate the -integrated cross section one has to introduce a lower cutoff for the energy of the emitted photon . In our case a natural choice of follows from the requirement that the photon energy exceed the characteristic frequency of the target system, for scattering on atomic electrons or eV for scattering on free electrons in a conductor. As discussed in Section 1.3, this will allow one to avoid the suppression of the cross section.
Next, we constrain the momentum of the final-state electron by requiring , where is small compared to allowed by 4-momentum conservation. The kinematics of the process in this case is considered in Appendix B. As shown there, for a given the photon emission angle is now constrained by
The smallness of implies that the photons are emitted in nearly forward direction. From the kinematics of the process it follows that the same is true for the scattered neutrino. In the leading order in we obtain
Here the integration over was performed in its allowed range given in eq. (26).
The cross section for the emission of photons with energies reads
For one can retain only the first two terms in the curly brackets.
The cross sections in eqs. (27) and (28) scale as the fourth power of ; a factor is expected from the phase space volume of the process with the electron recoil momentum constrained by (see Appendix B), and one more power of comes from the squared modulus of the transition matrix element of the process.
2.2 Radiative scattering and the neutrino magnetic dipole moment
Let us now consider the radiative neutrino scattering process (11) in the case when the neutrino-electron scattering is mediated by the photon exchange due to neutrino magnetic or electric dipole moments. In what follows we will for definiteness discuss the case of neutrino magnetic dipole moment . We will comment on the general case at the end of this subsection.
The amplitude of the process corresponds to the diagrams in Fig. 1 in which the intermediate vector boson connecting the neutrino and electron lines is the photon.141414We ignore the possibility that the final-state photon is emitted from the neutrino line, as this would be a process of higher order in the very small neutrino magnetic moment . In this case one can expect some kinematic enhancement compared to the usual weak NC and CC induced processes considered in the previous subsection. Indeed, we are interested in the kinematic region in which the momentum carried away by the photon nearly coincides with the momentum transfer from the neutrino, . In the regime of small net recoil momentum of the electron, the same is true for the corresponding energies: . This means that the 4-momentum of the virtual photon nearly coincides with that of the final-state photon, . As the produced photon is on the mass shell, the virtual photon is nearly on the mass shell, and its propagator should lead to an enhancement of the amplitude of the process.
The transition matrix element of the neutrino magnetic moment induced radiative scattering process on a “spinless electron” is
where we have used the Gordon identity and took into account that neutrinos are treated as massless particles. As before, the kinematic regime of non-relativistic electron recoil is considered. Without constraining , for the double and single differential cross sections we find
The cross section for the emission of photons with energies is
For this equation gives
Next, we again constrain the momentum of the final-state electron by requiring . Integrating over the in the allowed range given in eq. (26), we find, to leading order in ,
For this gives
Interestingly, in this approximation the cross section is essentially independent of the incident neutrino energy , except that should satisfy .
The cross sections (35)-(37) increase with decreasing minimum photon energy . Recall, however, that the photon energy cannot be too small: It should exceed the characteristic frequency ( for neutrino scattering and for scattering on free electrons in a conductor) in order to avoid the suppression.
As discussed at the beginning of this subsection, the cross section of the neutrino magnetic moment induced process exhibits for small a kinematic enhancement due to the propagator of the virtual photon being close to its pole. The enhancement, however, turns out to be relatively mild: the cross sections (35)-(37) scale as , which is to be compared with the dependence found in Section 2.1.
We have considered here radiative neutrino scattered process (11) induced by the neutrino magnetic dipole moment . In general, neutrinos may have both the magnetic and electric dipole moments, which, in addition, are matrices in flavour space. One can take this into account by replacing in the expression for the transition amplitude the quantity by , where is the matrix of neutrino electric dipole moments. Such an amplitude will then describe the transition of to a neutrino which may be of the same or different flavour. As the final-state neutrino is not detected, in calculating the cross section of the process one has to sum over . For the ultra-relativistic neutrinos we confine ourselves to, this amounts to replacing in the expressions for the cross sections .
3 DM detection through radiative coherent scattering?
It would be interesting to extend the above considerations to detection of other particles, such as DM. Unfortunately, the mechanism of enhancement of the detection cross section through macroscopic coherence considered here for neutrinos would not work for non-relativistic projectiles. The reason is actually mostly kinematic. Macroscopic coherence requires tiny net recoil momenta of the target electrons. It is easy to see that for non-relativistic projectiles vanishing is excluded by energy-momentum conservation (see Appendix A). Small non-zero values of are allowed, but only for extremely soft emitted photons, (or in the case of moderately relativistic projectiles). As discussed above, the cross sections of radiative scattering on electrons get a very strong suppression in this case.
As the conventionally discussed DM particles are supposed to be non-relativistic, the detection enhancement mechanism considered here will not be operative for them.151515In Ref.  it was suggested to use the radiative coherent scattering on nuclei to detect DM particles, but the issue of macroscopic coherence has not been addressed there. It may, however, work for detection of relativistic particles that may exist in the dark sector.
4 Effects of atomic binding
In Section 2 we considered radiative neutrino scattering on free electrons. This is suitable for conduction electrons in metals; however, for scattering on atomic electrons in dielectrics the effects of atomic binding should in general be taken into account. We shall show now that for the kinematic regime of interest to us, when the net recoil momentum of the electron is small and at same time the neutrino satisfies , the atomic effects can be neglected and the results found in Section 2 apply.
To demonstrate this, let us first note that for radiative scattering on free electrons, in the regime of small the contribution of the first two terms in the expression for (18) is small, and the main contribution comes from the third term, corresponding to the right diagram of Fig. 1.161616Indeed, for small electron recoil momenta the terms in proportional to and to nearly cancel each other, whereas the terms are subleading in the case of non-relativistic electrons and weak NC and CC mediated neutrino-electron scattering (their contributions vanish exactly for neutrino magnetic moment mediated scattering). The terms do not contribute by gauge invariance. Note that for calculations in the Coulomb gauge in the rest frame of the initial-state electron the whole second term in does not contribute to the amplitude, and the contribution of the first term is small because of . The same holds true when atomic effects are taken into account: in the kinematic region of interest to us the analogues of the first two terms in are small, and the main contribution comes form the analogue of the third term, which is largely insensitive to the effects of atomic structure (see below). This is fully analogous to what happens for elastic scattering of photons on atoms when the photon energy is much higher than the characteristic atomic frequencies . As discussed in Section 1.3, in this limit the cross section essentially coincides with that of photon scattering on free electrons. This can be readily seen from the expression for the amplitude of elastic photon-atom scattering. For non-relativistic electrons, the leading order amplitude in the Coulomb gauge is proportional to [37, 38]
Here is the 3-momentum operator, and are the momenta of the incident and scattered photons, and are their polarizations vectors, and the sum is over the intermediate atomic states. In eq. (38) we have taken into account that for elastic scattering on a heavy system coincides with . For all three terms in (38) are of the same order of magnitude and nearly cancel each other, leading to the suppression mentioned in Section 1.3; however, in the regime that is of interest to us, the first two terms in (38) are small compared to the third term and to a good accuracy can be neglected. Moreover, for spherically symmetric atomic states they tend to cancel each other.171717Indeed, using the closure property of the atomic states and commuting the factors , with the momentum operator, for the sum of the first two terms in (38) one finds in this limit , which vanishes for spherically symmetric states . As we are interested in coherent scattering on a group of atoms, by one should actually understand the ground state of such a system. The cancellation then happens also in the case when this state is spherically symmetric (i.e. has zero total angular momentum), even if the ground states of the individual atoms are not. The remaining term, , in general depends on the electron charge distribution in the state . For it is actually independent of the atomic structure, and for photon scattering on a single atom reduces to , where is the total number of the atomic electrons. This corresponds to coherent elastic photon-atom scattering. If the more stringent condition is satisfied where is the linear size of the target, the scattering on all electrons in the target is coherent. Otherwise, one would need to take into account structure factors describing electron distribution in the target, as discussed in Section 1.2.
Similar arguments apply to radiative neutrino scattering on atoms. Note that in this case one has to replace in eq. (38)
where is the 4-momentum operator and is the relevant matrix element of the neutrino current. The condition is then replaced by , which we always assume to be satisfied with a large margin when discussing macroscopically coherent effects.
It should be noted that for inelastic scattering with ionization or excitation of atoms typically dominates, while the processes in which the atom remains in its initial state are only important for nearly forward scattering. This is, however, exactly the case we are interested in. The fact that the probability of the radiative scattering without excitation or ionization of the target atoms is small is already taken into account by the suppression of the individual cross sections which we found upon constraining the electron recoil momentum by eV.
5 Coherent effects and the cross sections
Let us now assess the effects of macroscopic coherence on the cross sections of neutrino detection process (14).
As discussed in Sections 1.2 and 1.3, in order to take possible macroscopic coherence effects into account one has to multiply the elementary amplitude of the process by the relevant structure factor (such as (6) or (12)). The structure factor depends on the target used in the experiment, and the calculated cross section will therefore also be target-dependent. However, simple estimates of the effects of macroscopic coherence can be obtained in a rather general way as follows.
Assume that all the scatterers contained in some volume of a linear size within the target contribute to the cross section coherently; for this to occur, the net recoil momentum of the scatterer must satisfy . The coherent volume can in principle range from just the volume per one scatterer (no coherence) to the total volume of the target (complete coherence). To assess the coherence effects one can first calculate the elementary cross section of the process with the constraint imposed. In calculating such constrained elementary cross sections the corresponding structure factors can be replaced by unity. To find the cross section per target particle with coherence effects taken into account one would then have to multiply the constrained elementary cross section by the number of scatterers in the coherent volume .181818 Let the number of scatterers within one coherent volume be , and the number of coherent volumes in the target be . The total number of scatterers in the target is . If is the elementary cross section of the process, the cross section corresponding to scattering on all the target particles contained within one coherent volume is . The total cross section is . The cross section per one target particle is then , as stated. In the fully coherent case () and completely incoherent case () the total cross sections are and , respectively, and the corresponding cross sections per target particle are and . The choice of the recoil momentum cutoff (i.e. of the linear size of of the coherent volume) would then have to be optimized, within the range allowed by the kinematics of the process and the geometry of the experiment, by maximizing the resulting cross section.
In doing this, one should not forget the issue of observability of the process, which may be fully coherent but completely unobservable. For example, as discussed in Section 1.1, for elastic neutrino scattering the optimization requires to choose for the maximum recoil momentum (denoted there) the smallest possible value , but the scattering will then be unobservable due to the vanishingly small recoil energy of the target particles (see the discussion around eqs. (3) and (4)). No such problems arise for radiative processes discussed in the present paper.
We shall now estimate the effects of possible macroscopic coherence on radiative neutrino scattering on electrons. The corresponding cross sections with the net electron recoil momentum constrained by with a small cutoff were found in Section 2. Consider first radiative neutrino-electron scattering mediated by the usual NC and CC weak interactions. The constrained differential and integrated elementary cross sections are given in eqs. (27) and (28), and are proportional to . To find the cross section per one target electron one has to multipy these cross sections by the number of electrons in the coherent volume,
where is the electron number density in the target. As a result, the cross sections per one target electron turn out to be proportional to and are maximized for maximal possible value of , which corresponds to the absence of macrosopic coherence. What actually happens in this case is that macroscopic coherence can be achieved, but it requires such a stringent constraint on the value of (and so on the phase space available to the final-state particles) that the resulting cross sections are much smaller than those in the incoherent case. That is, macroscopic coherence is possible, but it leads to a reduction of the cross section rather than to its increase.
The situation is different for neutrino magnetic (or electric) dipole moment mediated radiative neutrino scattering. As discussed in Section 2.2, for small the cross sections get an enhancement due to the propagator of the virtual photon being close to its mass-shell pole. The enhacement is, however, rather modest: the constrained elementary cross sections (35)-(37) are proportional to rather than to , as it was in the case of weak NC and CC mediated radiative process. As before, to obtain the cross sections per target electron we have to multiply the constrained elementary cross sections by given by eq. (40). The factor in the cross sections (35)-(37) then gets canceled by from eq. (40), i.e. to leading order in the small the resulting cross sections per target electron are -independent.191919 As follows from the derivation of eqs. (35)-(37), this is correct only when satisfies . From eqs. (35) and (37) we then find
Here the lines over are to denote the cross sections per one target electron with coherence effects taken into account, and we have assumed in eq. (42).
The simplified approach we have adopted to evaluate the coherence effects, namely, to introduce the cutoff on the electron recoil momentum, replace the structure factors within the coherence volume by unity and then multiply the obtained elementary cross sections by the number electrons in the coherent volume, actually proves to be rather accurate. As we shall see, it just slightly overestimates the numerical factors in the cross sections (41) and (42). A more accurate estimate is obtained if one notes that for a macroscopically large number of electrons of the target contributing coherently to the cross section of the process, the summation in the expression for the structure factor in eq. (12) can be replaced by integration. This yields
where is the total electron number in the target and is the target’s volume. Mutiplying the squared matrix element of the elementary process by
performing the integration over the momenta of the scattered neutrino and the recoil electron as well as over the directions of the photon emission and dividing by , for the differential cross section per one target electron we obtain
which has the same structure as (41), but is smaller by a factor of . The -integrated cross section in the limit will be smaller than the expression in eq. (42) by the same factor. Note that the same approach applied to the radiative neutrino-electron scattering mediated by the NC and CC weak interactions would yield vanishing cross sections of coherent scattering per target electron. This corresponds to the already discussed fact that in this case macroscopic coherence, though possible for sufficiently small electron recoils, would lead to vanishingly small cross sections.
Can the coherent enhancement of the neutrino magnetic moment mediated radiative neutrino scattering help us to increase the experimental sensitivity to the neutrino magnetic dipole moments or even to detect them? The best laboratory limits on neutrino magnetic moments come from the experiments on elastic scattering at reactors, where one looks for possible deviations of the measured differential cross section from the usual one mediated by the weak CC and NC processes. The cross section due to the -induced elastic scattering is
where is the kinetic energy of the recoil electron and in the last (approximate) equality it is assumed that . This can be compared with the differential cross section (45) of the radiative neutrino-electron scattering, where a forward photon rather than the recoil electron is detected. For a numerical estimate, we set in eq. (45)
where is the Avogadro constant, is the density of the target material, is the number of electrons per nucleon in the target, and in the last (approximate) equality we have set . In the regime this gives
Taking for an estimate eV, which is about the smallest value that would allow to avoid the suppression of the radiative cross section, g/cm, and comparing eqs. (46) and (48), we find that even in the most optimistic case the cross section of coherently enhanced radiative neutrino-electron scattering exceeds that of the incoherent elastic scattering only for the electron recoil energies satisfying keV. At the same time, reactor experiments are currently probing scattering in the sub-keV region of the recoil energies , where the cross section of the incoherent elastic scattering dominates. Still, it should be noted that experimentally detecting – 100 eV photons may be easier than detecting electron recoil energies in the same range.
A potentially important advantage of the coherent radiative -mediated scattering is that it could in principle allow detection of very low energy neutrinos. Consider, e.g., neutrinos of energy eV. For the elastic scattering the electron recoil energies would then be