A general decomposition of the amplitudes for the $2\to 2$ processes gauge invariance and Bose symmetry. The restrictions implied by this decomposition are investigated for the reaction $\gamma\gamma\to\nu\bar{\nu}$ by computing the one-loop helicity amplitudes in the standard model. In the center of mass, where $\sqrt{s} = 2\omega$, the cross section grows roughly as Astrophysical implications of very high energy photon-neutrino interactions are discussed

Abbasabadi, A

Devoto, A

Dicus, D A

Repko, W W

Physical Review D

arXiv

Ali Abbasabadi

Alberto Devoto

Duane A. Dicus

Wayne W. Repko

Crossref

High energy photon-neutrino interactions

A general decomposition of the amplitudes for the 2 {r_arrow} 2 processes {gamma}{nu} {r_arrow} {gamma}{nu} and {gamma}{gamma} {r_arrow} {nu}{bar {nu}} is obtained using gauge invariance and Bose symmetry. The restrictions implied by this decomposition are investigated for the reaction {gamma}{gamma} {r_arrow} {nu}{bar {nu}} by computing the one-loop helicity amplitudes in the standard model. In the center of mass, where {radical}s = 2{omega}, the cross section grows roughly as {omega}{sup 6} up to the threshold for W-boson production, {radical}s = 2m{sub W}. Astrophysical implications of very high energy photon-neutrino interactions are discussed

Abbasabadi, A.

Devoto, A.

Dicus, D.A.

Repko, W.W.

UNT (University of North Texas) Digital Library

I- / D 0 4 h 4 0 7 5 7 -  ! 11 3 UTEXAS-HEP-98-10 MSUHEP-80715 High energy photon-neutrino interactions Ali Abbasabadil, Alberto Devoto2, Duane A. Dicus3 and Wayne W. Repko4 Department of Physical Sciences, Femk State University, Big Rapids, Michigan 49307 Dipartmento di Fisica, Universita' di Cagliari and I.N.F.N., Sezione di Cagliae Center for Particle Physics and Department of Physics University of Texas, Austin, Texas 78712 Department of Physics and Astronomy, Michigan State University East Lansing, Michigan 48824 (July 14, 1998) Abstract R:ECEIVED A general decomposition of the amplitudes for the 2 3 2 processes 7v --+ 7 v  and 77 + UP is obtained using gauge invariance and Bose symmetry. The restrictions implied by this decomposition are investigated for the reaction 77 + UB by computing the one-loop helicity amplitudes in the standard model. In the center of mass, where 4 = 2w, the cross section grows roughly as w6 up to the threshold for W-boson production, f i  = 2mw. Astrophysical implications of very high energy photon-neutrino interactions are discussed. 13.15.+g, 14.60.Lm,14.70.Bh, 95.30.Cq Typeset using R E V W  1 DISCLAIMER This report was prepared as an account 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 responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise docs not necessarily constitute or imply its endorsement, m o m -  mcnhtion, 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. DISCLAIMER Portions of this document may be illegible electronic image products. Images are produced from the best available original document. I. INTRODUCTION Investigations of the astrophysical importance of the 2 + 2 processes 7v + 7u, 77 + UG and UP + 77 have a long history. These reactions have been examined using the four-Fermi interaction [ l ] ,  charged vector boson theories [2,3], the standard model [4,5] and model- independent parameterizations [6]. Because of the vector-axial vector nature of the weak coupling, the cross sections are much smaller than a simple counting of powers of the weak coupling, GF, and the fine structure constant, a, in the diagrams of Fig. 1 would suggest [7,8]. For massless neutrinos, the cross sections vanish to order G$a2u2 [8 ] ,  and are known to be of order G$w6 when w < me [3,5]. This is due to the fact that the scale of the loop integrals for the diagrams in Fig. 1 is set by the W-boson mass, m W ,  rather than the electron mass. As a consequence, for center of mass energies 2w between 1 keV and 1 MeV, the cross sections for 2 + 3 processes such as 7 u  + 77" are larger than the 2 + 2 cross sections [9]. Although of little practical importance in stellar processes where the energy scale is 5 1  MeV, the 2 -7, 2 processes could be important in very high energy reactions such as u3 77 [ l o ] ,  particularly if the w6 behavior persists for w > me. The next Section contains the development of a general decomposition of the elastic amplitude A(s,t,u) for 7 u  4 7u. In SectionIII, we apply this decomposition to the crossed channel 77 --+ uG and obtain numerical results for the complete one-loop helicity amplitudes. This is followed by a discussion of our results. 11. GENERAL DECOMPOSITION OF THE 7~ ELASTIC AMPLITUDE The amplitude for the process 7v + 7v can be expressed as where hipa@ is a sum of gauge invariant, Bose symmetric tensors constructed from the neutrino momenta p l ,  p2 and the photon momenta kl, kz. Construction of these tensors is aided by the fact that, for massless neutrinos, we have This suggests the use of the combinations P =  P l + P 2 ,  .IC+ = ki + k2 , k- = kl - k2 . From the above, of pa,pp andp,, only pa andpp wil l  result in non-vanishing contributions. Thus, the vector-vector and axial vector-axial vector contributions to  Mpa@ will consist of combinations of pa ,  pp,  (k+),,, ( l ~ 2 ) ~  and ( k l ) p  as well as Spa, bpp and Sap. The gauge invariant combinations of pa and pp are 2 and we note that k + . p  = 2kl.p = 2k2-p and kt = 2kl-k2. Gauge invariant second rank tensors involving Spa and Spp are @l.k24m - ( k l )p (h )a )  and (kl.k.244 - (k2)&1)p) , (kk24Yp - (kZ)a(kl)p) * (9) Gauge invariant third rank tensors can then be constructed from products of (k+)p, the vectors in Eq. ( 6 ) ,  and the second rank tensors in Eqs. (7-9). In doing so, it should be noted that combinations such as (10) k+ ' P  k+ (kl*klS,, - (kl)p(k2)a)(PP - 7 ( k l ) P )  9 are not linearly independent since k+ 'P -(lo) - (12) = 2(11). kl.k2 As a consequence, the tensors of Eq.(8) can be omitted when constructing a complete set of gauge invariant third rank tensors. The requirement of Bose symmetry can easily be included if one constructs gauge in- variant tensors that have a definite symmetry under the exchanges kl + -k2, IC2 + -kl and a t) p. The vector k+ --+ -k+ under these exchanges. It is possible to construct four linearly independent, gauge invariant third rank tensors with definite symmetry under the exchange of the photons. They can be chosen to be 3 It can be seen that TPaa (1) is symmetric under the exchange of photons, while TPap, (2) T$$ and TPap (4) are antisymmetric. To obtain tensors which correspond to the vector-axial vector interference terms, one can use a set of independent momenta together with e,pxp to form gauge invariant tensors with definite symmetry under the exchange of the photons. When this is done, one finds two tensors with the appropriate properties. They have the form T::b = (k+),GY,AP(k+)x(k-)P 7 (18) k,€apxp + kaq3xpp + kpExppa + kxeppap + kp+apx = 0 , (20) which holds for any vector k .  Using this identity, the factor (, = ii(p2)7,(l + c(5)u(p1) in Eq.(l) can always be rewritten so that it is contracted with ePap instead of with (k+) , .  The relation then enables us to express any tensor containing an epvap as a sum of scalar products. These are simply combinations of the Tf$ of Eqs. (14-17). Explicitly, we find [ll] Consequently, the TPap, (4 i = 1, - -. ,4 ,  are sufficient to parameterize the 7 u elastic amplitude. The tensor Mpap can thus be written where Bose symmetry requires Ml(s, t ,  21) = Mi(u, t ,  8 )  , Mj(s,t,u) = -Mj(u,t,s), j = 2,3,4. (25) (26) Using the center of mass frame, and the explicit forms of all the vectors, we can obtain the following helicity basis 4 where 0 is the angle between the incoming and outgoing neutrinos, A1 is the helicity of the initial photon, X 2  is the helicity of the final photon and it is unnecessary to specify the neutrino helicities since they have a definite handedness. Notice that the first, third and fourth of these amplitudes are symmetric under the exchange of A1 and A 2 ,  while the second is antisymmetric under this exchange. Since time reversal invariance implies that the amplitude should be symmetric under the exchange of initial and final helicities, the second amplitude is T-violating. There are thus three helicity amplitudes if we require time reversal symmetry. This is consistent with simple helicity counting arguments. 111. THE CHANNEL yy .--) v0 The results of Section I1 can be expressed in the crossed channel 77 --+ uv using the relations pl  --+ -pl and k2 --+ -k2, which imply the interchange s f+ t .  The combinations k+ and p become k+ = kl + 3 k1- k2 k- (314 ( 3 W  p =  Pl +P2 3 -P1 +p2 = -P These transformations allow us to write the cross channel counterparts of Eqs. (14-17) as The amplitude can be expanded as M p a d  = Mi(t ,  S, u)p;S + Mz(t, s , u ) ~ ; $  + M3(t, S, u)P;$ + M4\', s,u)P;:L, (36) where Ml( t ,  s , u )  = Ml(% 0) , Mj(t ,  S, U) = -Mj(u, S, t ) ,  j = 2,3,4 (37) (38) In this case, the center of mass helicity basis is 5 FLtb = 4tu sin ~ X J ~ .  Here, tp = i i ( ~ ~ ) 7 ~ ( l  + 75)~(p2) and 0 is the angle between photon 1, which is in the z direction, and - the outgoing neutrino. Using the helicity basis, we can express the helicity amplitudes AA1A2(s, z), where z = cos 8, as A++(s, z )  = sin 8 [s(t - u)Ml(t ,  s, v) + s2M3(t, s,u)  + 4tuMq(t, s , u ) ]  , A+-(s,z) = sin8 [-2stMl(t,s,u) - 4tuMq(t,s,u)] , L + ( s ,  z )  = sin 8 [2suMl(t, s , u )  - 4tuMq(t, s,u)] , d_-(s, z )  = sin 0 [s(t - u)Ml(t,  s, u) + s2M3(t, s,u)  + 4tuM4(t, s, u)] , (43) (44) (45) (46) where we have assumed time reversal symmetry and omitted M2(t,s,u).  From Eqs.(43) and (46), is is clear that &+(s, z )  = c&-(s, z). Moreover, since t = -is( 1 - z )  and u = -Ls(l+ 2 z), the symmetries of Eqs. (37) and (38) imply Similarly, we find that the helicity flip amplitudes satisfy 2) = - L + ( s ,  -2 )  To explore these general results, we calculated the standard model diagrams of Fig. 1 using a nonlinear Be gauge such that the coupling between the photon, the W-boson and the Goldstone boson vanishes [12,13,5]. These diagrams can be decomposed in terms of scalar two-point, three-point and four-point functions, which are expressible in terms of dilogarithms [14]. The expressions for the helicity amplitudes in terms of dilogarithms were evaluated numerically and the results checked against FORTRAN codes developed for one- loop integrals [15,16]. In doing so, we assumed that s >> m;, where ml is the lepton mass, and showed that the dependence on h(mz) or h2(mz) from individual diagrams actually cancels. The results for the helicity amplitudes are shown in Figs.(2) and (3), where the cross sections for various helicities are plotted for f i  = 20 GeV and f i  = 200 GeV using These figures clearly illustrate the symmetries of Eqs.(47) and (48), particularly the van- ishing of the non-flip amplitudes at z = 0. The total cross section for unpolarized photons is given by 6 and is plotted in Fig.(4). This figure illustrates the roughly s3 behavior of the total cross section up to the threshold for W pair production. Also shown in dashes is the helicity non- flip contribution to the cross section, which can be seen to be a factor - smaller than the helicity flip contribution. This feature does not appear to be related to any symmetry, but it is reminiscent of the low energy case, where the non-fip amplitudes vanish [5]. IV. DISCUSSION AND CONCLUSIONS As mentioned in the Introduction, the low energy 2 4 2 photon-neutrino cross sections are much smaller than the corresponding 2 + 3 cross sections for center of mass energies between 1 keV and 1 MeV. Recently, the 2 + 3 cross sections have been computed for a range of center of mass energies from well below 2me, where the cross sections vary as w1', to well above 2me, but less than mw [17]. For me << w << mw, the only scale in the 2 + 3 processes is mw or, equivalently, GF. From dimensional considerations, the center of mass cross section must behave as Gg w2 in this range of energy. Explicitly, it is found that can be expressed as For the 2 + 2 process 77 + uij, we expect an w6 behavior, and a fit to the data in Fig. (4) yields These expressions are equal for w = 1.94 x 104me or 4 - 20GeV. Thus, for sufficiently high energies, the 2 + 2 process dominates the 2 + 3 process. To assess the importance of very high energy photon-neutrino interactions in processes of interest in cosmology, consider the scattering of high energy neutrinos from the cosmic neutrino background. This process has recently been studied by assuming that the neutrino collision produces a 2 at resonance whose decay chain contains photons and protons which could account for the flux of > 1020eV cosmic rays [lo]. For the processes considered here, the uti collision would directly produce high energy photons. The cross section for uti + 77 can be obtained from Eq. (52) by supplying a factor of 2. Assuming the relic neutrinos have a small mass [10,18], the cross section in the rest frame of the target is 3 = 5.62 x (fi,E,,) cm2, (53) where n?, is m,/l eV and 8, is &/lo2' eV. The interaction rate on the cosmic neutrino background is ~,+,Trn,c, where n, is the relic neutrino density. If this is multiplied by t o ,  the age of the universe, the condition that at least one interaction occur is 7 or - 5.62 x 1013m-2/3 ni,E, = (n,do)1/3 (55) Taking nu = 5 6 ~ m - ~  and t o  = 15 x lo9 years, niV& = 6.07 x lo3, which translates into f i  = 3.5TeV. Even if there were sources of such high energy neutrinos (which seems unlikely), this is well above the region for which the s3 behavior is valid. In fact, the relation (n,cto)-l = 1.26 x cm2 together with Fig.(4) show that ~ ~ f i + - , ~  is not large enough for any value of s to attenuate a high energy neutrino ffux given the present values of n, and the age of the universe. The temperature at which the reaction vij --+ 77 ceases to occur can be determined from the reaction rate per unit volume where $1 and $2 are the neutrino and antineutrino momenta, El and E2 their energies, I.'[ is the flux and T the temperature. Using the invariance of uE~E~IGI, the relationship between ala1 in the center of mass frame and any other frame is If the angle between $1 and & is 6'12, Eq. (57) gives The integration is straightforward and Eq. (56) gives where ((z) is the Riemann zeta function. The interaction rate dividing P , ~ + ~  by the neutrino density n, = 3C(3)T3/47r2, and we find can be obtained by (60) 24 9 -1 T1Os RVfi.+-,T = 7.32 x 10- with TI0 denoting T/lO1° K. Expressing the age of the universe as t = 2TG2 s, the condition for at least one interaction, RuD+-,-,t - 1 gives T - 1.58 GeV, which is within the region of validity of the s3 behavior. ACKNOWLEDGMENTS One of us (A. D.) wishes to thank the Department of Physics and Astronomy of Michigan State University for its support and hospitality. This research was supported in part by the U.S. Department of Energy under Contract No. DE-FG013-93ER40757 and in part by the National Science Foundation under Grants No. PHY-93-07980 and PHY-98-02439. 8 REFERENCES [l] H.-Y. Chiu and P. Morrison, Phys. Rev. Lett. 5 ,  573 (1960). [2] M. J. Levine, Nuovo Cimento 48A, 67 (1967). [3] L. F. Landovitz and W. M. Schreiber, Nuovo Cimento 2A, 359 (1971). [4] V. K. Cung and M. Yoshimura, Nuovo Cimento 29A, 557 (1975). [5] D. A. Dicus and W. W. Repko, Phys. Rev. D 48, 5106 (1993). [6] J. Liu, Phys. Rev. D 44,2879 (1991). [7] C. N. Yang, Phys. Rev. 77, 242 (1950); L. D. Landau, Sov. Phys. Doklady 60, 207 [8] M. Gell-Mann, Phys. Rev. Lett. 6, 70 (1961). [9] D. A. Dicus and W. W. Repko, Phys. Rev. Lett. 79,569 (1997). The channel77 + vfi7 is discussed in: N. Van Hieu and E. P. Shabalin, Sov. Phys. JETP 17,681 (1963). See also: A. Abada, J. Matias and R. Pittau, hep-ph/9806383 (1998). (1948). [lo] E. Waxman, astro-ph/9804023 (1998). [ l l ]  This is most easily shown using Eqs. (27-30) and the corresponding expressions for TPap (5) and Tj$. M. Bace and N. D. Hari Dass, Ann. of Phys. 94, 349 (1975). [12] M. B. Gavela, G. Girardi, C. Malleville, and P. Sorba, Nucl. Phys. B193, 257 (1981); [13] J. F. Nieves, P. B. Pal, and D. G. Unger, Phys. Rev. D28, 908 (1983). [14] G. Passarino and M. Veltman, Nucl. Phys. B160, 151 (1979). [15] C. Kao and D. A. Dicus, LOOP, a FORTRAN program for evaluating loop integrals based on the results in: G. 't Hooft and M. Veltman, Nucl. Phys. B153, 365 (1979), and Ref. [14]. [16] G. J. van Oldenborgh, NIKHEF-H/90-15, (1990). [17] D. A. Dicus, C. Kao and W. W. Repko, hep-ph/9806499 (1998) (submitted to Phys. [18] The corrections to the s3 behavior of Eq. (52) due to a small neutrino mass are of order Rev.). m:/s, which is negligible. 9 FIGURES V FIG. 1. Diagrams for the process yy --+ ve&. For each of (a), (b), (c) there is also a diagram with the photons interchanged. h e 2 2 Y P 1 0 4  104 104 10-7 10-8 10-9 10-10 -1 0 FIG. 2. The helicity dependent differential cross sections for y7 vij are shown for 4 = 20 GeV. The solid line is du+-/dz, the dashed line is da-+/ds and the dot-dashed line is da++/dz. 10 n e Y b loo - 10-1 e N 10-2 a lo-= W s 10-4 10-5 / 0 / / f I I I -1 0 1 FIG. 3. Same as Fig. (2) with 4 = 200 GeV. n e Y b 1. . . . . I  . , loa * (GaV) FIG. 4. The solid line is the total cross section Q ~ + , F  and the dashed line is the contribution fiom the helicity non-flip amplitudes. 11 

High energy photon-neutrino interactions

Abstract

Similar works

Full text

Available Versions

Crossref

UNT (University of North Texas) Digital Library

Archivio istituzionale della ricerca - Università di Cagliari