The cross sections for the processes $\gamma \nu \to \gamma \gamma \nu$ and effective Lagrangian derived from the Standard model. These cross sections are shown to be much larger than the elastic cross section $\sigma(\gamma\nu implications are discussed

Dicus, D A

Repko, W W

arXiv

Duane A. Dicus

Wayne W. Repko

Crossref

Physical Review Letters

Photon-Neutrino Interactions

The cross sections for the processes {gamma}{nu} {yields} {gamma}{gamma} {yields} {nu}{anti {nu}} are calculated with the aid of an effective Lagrangian derived from the Standard model. These cross sections are shown to be much larger than the elastic cross section {sigma}({gamma}{nu} {yields} {gamma}{nu}) for photon energies {omega} {ge} 1 keV. Possible astrophysical implications are discussed

Dicus, D.A.

Repko, W.W.

UNT (University of North Texas) Digital Library

MSUHEP-70205 Phot on-neutrino interactions Duane A. Dicus Center for Particle Physics and Department of Physics University of Texas, Austin, Texas 78712 Wayne W. Repko Department of Physics and Astronomy Michigan State University, East Lansing, Michigan 48824 Draft(February 15, 1997) Abstract The cross sections for the processes yv -+ yyv and yy -+ yvii are calculated with the aid of an effective Lagrangian derived from the Standard model. These cross sections are shown to be much larger than the elastic cross section'cr(yv -+ 7 v )  for photon energies w 2 1 keV. Possible astrophysical implications are discussed. 1. Elastic processes Photon-neutrino scattering is of potential interest in astrophysical processes. The various yv reaction cross sections, which are expected to be small due to the weak interaction, are further suppressed because of the vector-axial vector nature of the weak coupling [l, 21. For the channel ufi -+ yy, Yang's theorem [3] implies a vanishing decay amplitude to order GF when the neutrinos are massless. This means that the amplitude in any channel contains additional factors of w / m w ,  where w is the photon energy and m w  is the W mass. If the neutrinos have a mass m,, the amplitude is reduced by factors of m,/mw [4]. In the case of massless neutrinos, the amplitude for yv -, yv in the Standard model is [5], to leading order in ms2, where A and A' are the photon helicities, s = 4u2, t = -2w2(1 - cos 0 )  and B is the scattering angle. Notice that there are no inverse powers of me in this expression and hence the scale is set by m w .  As a consequence, all 2 + 2 yv channels can be calculated using the effective Lagrangian where F,, is the photon field tensor and $I is the neutrino field. The resulting cross section for elastic scattering is Despite the enhancement of the ln2 (mk/m,2) factor, this cross section is exceedingly small and likely to be of little practical importance in astrophysics. 2. Inelastic processes The source of the large suppression in the 7 v  elastic amplitude is the Yang theorem prohibition of a two photon coupling to a J = 1 state. There is no similar restriction on the coupling of three photons. This suggests an examination of the inelastic process yv --+ yyv to determine if the scale of the loop integrals resulting from the set of diagrams represented by Fig. 1 is set by the electron mass rather than m w .  This turns out to be the case, and one can obtain an effective Lagrangian of the form where NPv is N P V  = 8, (47V(1 + 7 5 ) 4  - 8” ( 4 7 P ( l +  7 5 ) + )  f (5) The numerical factors 5/180 and -14/180 are familiar from the Euler-Heisenberg [6] expan- sion of the photon-photon scattering amplitude. This occurs because after replacing the W propagator by m$ and performing a Fierz transformation, the vV -+ yyy amplitude is the product of a neutrino current and the photon to  three photon amplitude. Using Eq. (4), the cross section for yv + yyv is This cross section is illustrated in Fig. 2 for 1 keV 5 w 5 me. Further details of the scattering process are presented in Fig.3. By retaining the polar- ization vector of one of the final state photons, it is possible to obtain the cross section for this photon to be produced with either positive or negative helicity. This is illustrated in the left panel of Fig.3 by the dashed and dot-dashed lines. The solid line in this panel is the angular distribution for unpolarized scattering and 6 is the angle between the outgoing photon and the incident photon. The difference between the positive and negative helicity cross sections, which are fifth order polynomials in cos8, results in the polarization P(6)  illustrated in the right panel of Fig. 3. The existence of a net circular polarization is possible because the weak interaction violates parity. 2 It is also of interest to examine the annihilation channel yy yvV, which provides an energy loss mechanism for stellar processes. In this case, the double differential cross section d 2a/ sin f3dBdw‘ is 1 G:a3 sin BdBdw‘ 3,110,400 ?r4mz - w3wI3(w - w’)[2224w2 - ~ ( 5 9 2 ~  + 520w‘) sin2f3 d ’c +139w‘ ’ sin4B] , (7) where 8 is the scattering angle of the final photon in the center of mass, w’ is its energy and w is the initial photon energy. When integrated, Eq. (7) gives the total cross section 67 G&a3 8 c(y7 + y v v )  = 7 O2 ‘ 1,275,750 Unlike yv -+ yyv, the final photon in the annihilation channel does not acquire any circular polarization. 3. Discussion and conclusions A direct comparison of the elastic and inelastic cross sections, Eqs. ( 3 ) ,  (6) and ( 8 ) ,  is given in Table 1. The w10 behavior of the inelastic cross sections versus the w6 behavior of the elastic cross section is evident as is the 12 orders of magnitude difference at w = 2m,. Certainly, the effective Lagrangian Eq. (4) provides an adequate description of the inelastic processes for w < me. The application of Eq. (7) to stellar energy loss is therefore completely justified. At some point beyond w = me, the cross section ceases to grow as the tenth power, begins a transition to a ‘soft’ behavior and eventually decreases. The precise range of applicability of the power law is somewhat subjective in the sense that numerical factors resulting from the loop integrals which define Lefi are often included in the definition of the scale factor. Including the denominator factors of Eq. (6), the effective scale is - 47rm,. Table 1: a(w)  in cm2 for w in keV 100 1.46 x 10-64 2.39 x 10-64 1022 1.81 x 10-54 2.96 x 10-54 511 1.77 x 3.30 x uyY4-f” 2.05 x 10-84 2.05 x 10-72 3.65 x 2.34 x As a rough indication of the importance of the 7 v  --+ yyv process in cosmology, consider the mean number of collisions in an expansion time t .  Assuming that there is an effective scale greater than me, Eq. (6) can be written ( 9 )  “(7” --+ yyv) = 3.5 x 10- 55 TI,  10 cm2,  3 where 2-10 is the photon energy in units of lOl0K and w - T. The mean number of collisions in this time [7], a(7v --+ 77v)n,d, where n, denotes the neutrino density, is For this to be of order 1, 2'10 - 13.6 or T - 20772,. Strictly speaking, this result is beyond the scale - 47rm,, and it is probably necessary to treat the transition region more carefully in order to determine the decoupling temperature. If this temperature were shown to be low enough, the process yv ---f yyv might be of some importance in cosmological considerations. There is the possibility, albeit remote, that the detection of circular polarization, which only occurs in parity violating processes, could provide evidence for relic neutrino interactions. Acknowledgements We would like to thank V. Teplite and M. Einhorn for helpful conversations. This research was supported in part by the U.S. Department of Energy under Contract No. DE-FGO13- 93ER40757 and in part by the National Science Foundation under Grant No. PHY-93-07980. [l] V. K. Cung and M. Yoshimura, Nuovo Cimento 29A, 557 (1975). [2] J. Liu, Phys. Rev. D 44, 2879 (1991). [3] C. N. Yang, Phys. Rev. 77, 242 (1950). [4] M. Gell-Mann, Phys. Rev. Lett. 6, 70 (1961). [5] D. A. Dicus and W. W. Repko, Phys. Rev. D 48, 5106 (1993). [6] H. Euler, Ann. Phys. 26, 398 (1936); W. Heisenberg and H. Euler, Zeit. Phys. 98, 714 (1936). [7] P. J. E. Peebles, PrincipEes of PhysicaE Cosmology, Princeton University Press, Prince- ton, New Jersey (1993). 4 Figures I I ‘7 I \ e /  \ a Figure 1: A typical diagram for the process yv + yyv is shown. The complete set is obtained by permuting the photons. w (MeV) o (MeV) Figure 2: The yv t yyv cross section is shown for 1 keV 5 w 5 me. 5 7” + 77v 6 (radians) Figure 3: The differential cross section dc /  sin BdB and polarization P(B) are shown. In the left panel, the dashed line denotes the cross section for the production of a positive helicity photon, the dot-dashed line the cross section for the production of a negative helicity photon and the solid line is the sum of the two. DISCLAIMER This report was prepared as an accuunt of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, rccom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 6 

Photon-neutrino interactions

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Photon-neutrino interactions

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