In a first part, we justify the feasibility of substituting a photon leg by a
neutrino current in the Euler-Heisenberg Lagrangian to obtain an effective
Lagrangian for the process $\gamma\nu\to\gamma\gamma \nu$ and its crossed
reactions.
  We establish the link between these processes and the four-photon scattering
in both the Standard Model and the effective theory. As an application, we
compute in this effective theory the processes $\gamma\nu\to\gamma\gamma\nu$
and $\gamma\gamma\to\gamma{\nu}\bar\nu$ and show how to use the
$\gamma\gamma\to\gamma\gamma$ results as a check. We settle the question of the
disagreement between two computations in the literature concerning the reaction
$\gamma\gamma\to \gamma \nu\bar\nu$. In the second part, we present results of
the direct computation of the photon-neutrino five-leg processes in the
Standard Model, discuss possible astrophysical implications of our results, and
provide simple fits to the exact expressions.Comment: 6 pages, axodraw, ltwol2e, 5 figures, contributed paper to the 29th
  International Conference on High Energy Physics (Vancouver

Abada, A.

Matias, J.

Pittau, R.

English

arXiv

In a first part, we justify the feasibility of substituting a photon leg by a neutrino current in the Euler-Heisenberg Lagrangian to obtain an effective Lagrangian for the process $\gamma\nu\to\gamma\gamma \nu$ and its crossed reactions. We establish the link between these processes and the four-photon scattering in both the Standard Model and the effective theory. As an application, we compute in this effective theory the processes $\gamma\nu\to\gamma\gamma\nu$ and $\gamma\gamma\to\gamma{\nu}\bar\nu$ and show how to use the disagreement between two computations in the literature concerning the reaction the direct computation of the photon-neutrino five-leg processes in the Standard Model, discuss possible astrophysical implications of our results, and provide simple fits to the exact expressions.In a first part, we justify the feasibility of substituting a photon leg by a neutrino current in the Euler-Heisenberg Lagrangian to obtain an effective Lagrangian for the process $\gamma\nu\to\gamma\gamma \nu$ and its crossed reactions. We establish the link between these processes and the four-photon scattering in both the Standard Model and the effective theory. As an application, we compute in this effective theory the processes $\gamma\nu\to\gamma\gamma\nu$ and $\gamma\gamma\to\gamma{\nu}\bar\nu$ and show how to use the $\gamma\gamma\to\gamma\gamma$ results as a check. We settle the question of the disagreement between two computations in the literature concerning the reaction $\gamma\gamma\to \gamma \nu\bar\nu$. In the second part, we present results of the direct computation of the photon-neutrino five-leg processes in the Standard Model, discuss possible astrophysical implications of our results, and provide simple fits to the exact expressions

CERN Document Server

CERN-TH/98-305LPTHE Orsay-98/59hep-ph/yymmdddFive-leg photon{neutrino interactionsA. Abada, J. Matias and R. PittauTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandAbstract:In a rst part, we justify the feasibility of substituting a photon leg bya neutrino current in the Euler{Heisenberg Lagrangian to obtain an ef-fective Lagrangian for the process γ ! γγ and its crossed reactions.We establish the link between these processes and the four-photon scat-tering in both the Standard Model and the eective theory. As an ap-plication, we compute in this eective theory the processes γ ! γγand γγ ! γ and show how to use the γγ ! γγ results as a check.We settle the question of the disagreement between two computations inthe literature concerning the reaction γγ ! γ. In the second part, wepresent results of the direct computation of the photon{neutrino ve-legprocesses in the Standard Model, discuss possible astrophysical implica-tions of our results, and provide simple ts to the exact expressions.Contributed paper toXXIXth International Conference on High Energy PhysicsVancouver, B.C., Canada, 23{29 July 1998abstract 1025CERN-TH/98-305September 1998FIVE-LEG PHOTON{NEUTRINO INTERACTIONSA. Abada, J. Matias, R. PittauTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandE-mail: abada@mail.cern.ch, matias@mail.cern.ch, pittau@mail.cern.chIn a rst part, we justify the feasibility of substituting a photon leg by a neutrino current in the Euler{Heisenberg Lagrangian toobtain an eective Lagrangian for the process γ ! γγ and its crossed reactions. We establish the link between these processesand the four-photon scattering in both the Standard Model and the eective theory. As an application, we compute in this eectivetheory the processes γ ! γγ and γγ ! γ and show how to use the γγ ! γγ results as a check. We settle the question ofthe disagreement between two computations in the literature concerning the reaction γγ ! γ. In the second part, we presentresults of the direct computation of the photon{neutrino ve-leg processes in the Standard Model, discuss possible astrophysicalimplications of our results, and provide simple ts to the exact expressions.1 IntroductionProcesses involving photons and neutrinos are potentiallyof interest in astrophysics and cosmology. However, itwas realized some time ago in1 that 4-leg processes (γ !γ, γγ !  and  ! γγ) are too strongly suppressedto be of relevance. The reason for this suppression is theprohibition of a two-photon coupling to a J = 1 state.This is because of Yang’s theorem 2. On the other hand,this theorem does not apply to 5-leg processes involvingtwo neutrinos and three photons, such asγ ! γγγγ ! γ ! γγγ ; (1)The extra  in the cross section is compensated by an in-terchange of the 1=MW suppression by an 1=me enhance-ment 3;4. The relative enhancement of the 5-leg processversus the 4-leg one happens to be of several orders ofmagnitude, depending on the energy.In3, Dicus and Repko derived an eective Lagrangianfor the above ve-leg photon{neutrino interactions by us-ing the Euler{Heisenberg Lagrangian that describes thephoton{photon scattering 5. Moreover, in the literaturethere already existed a computation by Shabalin andHieu 6 of the second process in (1) whose result disagreeswith the one given in 3.To settle this question, in a recent work 4 we com-puted the rst and the second processes (1), in theframework of the eective theory, conrming the resultsreported in ref. 3. In section 2, we justify the feasibil-ity of this approach 4 and give another derivation of the5-leg eective vertex starting from the Euler{HeisenbergLagrangian 5.The eective approach gives reliable results for ener-gies below the threshold for e+e− pair production, whileits extrapolation to energies above 1 MeV, interestingto study, for example, supernova dynamics, is suspect.Therefore, an exact calculation of the processes (1) isimportant to see their role in astrophysics and the rangeof validity of the eective theory. This calculation 7 issummarized in section 3, with the main results.Very recently, parallel work in the same direction hasbeen carried out by Dicus, Kao and Repko 8, so that wehad a chance to compare our numerical results, ndingcomplete agreement between the two independent calcu-lations.We end up with a discussion on some of the impli-cations of the exact results of these 5-leg processes inastrophysics and cosmology, and we give our conclusions.2 Eective theoryThe starting point is the leading term of the Euler{Heisenberg Lagrangian 5, which describes the photon{photon scattering of Fig. 1a:LE−H =2180m4eh5 (FF)2 − 14FFFFi(2)where F is the photon eld-strength tensor and  theQED coupling constant.γγγ γp1; 1 p3; 3p2; 2p4; 4(a)γeγγγp1; 1 p3; 3p2; 2p4; 4(b)Figure. 1: Four-photon interaction (a) in the eectiveLagrangian and (b) in the SM.In what follows we will show the relation between theeective Lagrangian of eq. (2) and the one describingprocesses (1), which is given byLe =C180h5~FF(FF− 14~FFFFi(3)1whereC =g5s3W (1 + ve)322m4eM2W=2GF3=2(1 + ve)p2m4e; (4)and ~F stands for the eld strength of the new \gaugeeld" ~A   γ(1 − γ5) = 2Γ , with Γ the neutrinocurrentΓ = v+(5)γw−u−(4) :We use the notation w = (1  γ5)=2 and ve = −1=2 +2s2W , where sW is the sine of the Weinberg angle.p4   p5Zγγ γfig fkgfjge(a)p4   p5Wγγ γfig fkgfjge(b)Figure 2. SM leading diagrams contributing to ve-legphoton{neutrino processes.The keypoint of the proof is to demonstrate that theSM amplitudes of diagrams (a) and (b) of Fig. 2, whichwe called Aijk and Bijk, respectively, can be rewritten interms of diagram (b) of Fig. 1.The total amplitude M(; ; ) of the 5-leg processreadsM(; ; ) = [(A123 +A321) + (B123 +B321)+ (A132 +A231) + (B132 +B231) ++ (A213 +A312) + (B213 +B312)] ; (5)where we denote the photon polarization by ,  and (see 4 for the explicit form of Aijk and Bijk).The reason why the terms in eq. (5) are collected inpairs is because, when adding the two terms in each pair,and using the reversing invariance of the γ-matrix traces,changing q to −q and me to −me, the γ5 piece of the Zeevertex cancels 4. For instance, the coupleA123 +A321 = (~P1; )(~P2; )γ(~P3; )(Aγ123 +Aγ321 );whereAγ123 +Aγ321 = −2(2)4 (gsW )3g2cW2veΓ1ZLγ1andLγ1 =Zdqn Trγ1Q−23γγ1Q−2γ1Q−0γ1Q−1;cancels completely its axial part. We are using the nota-tions of Refs. 4;7:Qi = /Qi me ; Q(ij) = /Q(ij) me ;Qi = Q0  pi ; Q(ij) = Q0  pi  pj ; Q0 = q ;Di = Q+i Q−i ; D(ij) = Q+(ij) Q−(ij) : (6)and 1=Z = 1=(p4 + p5)2 −M2Z −1=M2Z.The same trick can be applied to the couples of B’s.Let us take, for instance,Bγ123 +Bγ321 = −4(2)4 (gsW )3g2p22ΓLγ2 (7)Lγ2 =Zdqn Trγ1Q−23γγ1Q−2γ1Q−0γ1Q−11W (q)where W (q) = (q + p2 + p3 + p5)2 −M2W .The second step of the proof is to shrink diagram(b) of Fig. 2 to diagram (b) of Fig. 1 with the photonleg substituted by the neutrino current. This is done byexpanding the W propagator inside the loop, using1W (q)=1q2 −M2W−k2 + 2q  k(q2 −M2W )((q + k)2 −M2W )(8)where W (q)  (q+k)2−M2W and k = p2+p3+p5. Onceintroduced in eq. (7), the rst term in the r.h.s. of eq.(8) gives rise to an L1 type of integral plus a contributionof next order in 1=M2W . On the other hand, the secondterm on the r.h.s. of eq. (8) is of order 1=M4W , and cantherefore be neglected.In conclusion, at leading order in 1=M2W , the set offour diagrams (from a total of 12) is always proportionalto L1:Aγ123 +Aγ321 +Bγ123 +Bγ321 = −g5s3W2(1 + ve)Zc2WΓLγ1We have shown that in the SM the 5-leg processes (1)at leading order correspond up to a global factor to the4-photon scattering process; both theories should thus bedescribed at this order by the same eective theory, aftersubstitution of one polarization vector by the neutrinocurrent(−!P4; 4)!12Γ (9)and xing the overall constant C. This constant C (eq.(4)) is xed unambiguously by considering the followingratios of amplitudes in the large-me limitlimlarge meASM4γASMP=Ae4γAeP; (10)where P stands for our 5-leg processes.2.1 Cross sectionsUsing the obtained eective Lagrangian eq. (3) that co-incides with the one used in3 we have evaluated the crosssection for the rst two processes:e(γ ! γγ) =262127575G2Fa234!me8!2 :2e(γγ ! γ) =2144637875G2Fa234!me8!2 : (11)In 4 we have also computed the unpolarized dierentialcross sections and the polarized cross sections, using asa check for each polarized cross section of the secondprocess the well-known polarized cross sections of the4-photon scattering process. The results in 4 coincidefor the dierential and total cross sections with the onesobtained in3 and are, therefore, in disagreement with theones obtained by Shabalin et al. in 6.3 Direct computation of the 5-leg processesThe range of energy in which the processes (1) are rel-evant is well below the W mass, so that we treated theneutrino{electron coupling as a four-Fermi interaction.We also assumed massless neutrinos. The total ampli-tude readsM(; ; )=−2(1 + ve) Γ03Xi=1Ii (p1; p2; p3; ; ; )(12)I2 and I3 are obtained from I1 by the replacementsp2 $ p3 and (p2; ) $ (p3; ) for I2 , and p2 $ p1and (p2; )$ (p1; ) for I3 . The factor Γ0 isΓ0=(gsW )3g2cW21Z1(2)4 v+(5)γu−(4): (13)A detailed description of the method used for the compu-tation of the total amplitude eq. (12) can be found in 7.The reduction of M(; ; ) to scalar one-loop integralsis performed with the help of the technique describedin 9. The general philosophy of such a method is usingthe γ algebra in the traces to reconstruct the denomina-tors appearing in the loop integrals, rather than makinga more standard tensorial decomposition 10. The algo-rithm can be iterated in such a way that only scalar andrank-one functions appear at the end of the reduction,at worst together with higher-rank two-point tensors. Inthat way, all the results are expressed only in terms ofscalar functions with 3 and 4 denominators, rank-1 inte-grals with 3 and 4 denominators, rank-2 integrals with3 denominators, and rank-3 functions with 3 denomina-tors. This already provides an important simplicationwith respect to the standard decomposition, in that thecomputation of tensors such asT ;; =Zdnqqq ; qqq; qqqqD0D−1D2D(23)(14)is completely avoided. A suitable choice of the polariza-tion vectors11 as explained in7 is the key ingredient in thecase at hand. To obtain compact expressions, we made alarge use of the Kahane{Chisholm manipulations over γmatrices 12. Such identities are strictly four-dimensional,while we are, at the same time, using dimensional reg-ularization. Our solution is splitting, before any tracemanipulation, the n-dimensional integration momentumappearing in the traces as 9 q ! q + ~q, where q and ~qare the four-dimensional and -dimensional components( = n− 4), respectively, so that q  ~q = 0. The γ algebracan then be safely performed in four dimensions, at theprice of having additional terms. In fact, the splittingq ! q+ ~q is equivalent to redening m2e ! m2e − ~q2 fromthe beginning. The net eects are then the extra inte-grals containing powers of ~q2 in the numerator, wheneverm2e is present in the formulae. The computation of suchintegrals in the limit ! 0 is straightforward9. Finally, astandard Passarino{Veltman decomposition10 of the sim-ple remaining tensorial structures in terms of scalar loopfunctions, concludes our calculation. We implementedthe outcoming formulae in a Fortran code, performingthe phase-space integration by Monte Carlo. Numericalresults are reported in the next section.3.1 ResultsOur formulae remain valid also when including all neu-trino species. In this case, only the rst diagram in Fig.2a contributes, at leading order in !=me, because the sec-ond one is suppressed by powers of !=m; . Therefore,the inclusion of all neutrinos can be achieved by simplyreplacing (1 + ve) with (1 + 3 ve) in eq. (12). However,we only considered e in our numerical results. Eectiveand exact computations are compared in Figs. 3, 4 and5, for the three processes. Furthermore, Table 1 showsthe ratio between exact cross section and e for severalvalues of !=me, then exhibiting the range of validity ofthe eective theory.From the above gures and numbers it is clear that,for all three processes, the eective theory is valid onlywhen, roughly, !=me  2, as expected. At larger !, theexact computation predicts a softer energy dependencewith respect to the (!=me)10behaviour given by the ef-fective Lagrangian.All these results are in full agreement with those re-ported in8. In order that the results be useful, the only aoption is to t the curves in Figs. 3, 4 and 5. The resultsof the ts are(γ ! γγ)e(γ ! γγ)= r−2:76046  exp[2:13317−2:12629log2(r) + 0:406718log3(r) − 0:029852log4(r)];aA large-me expansion has been performed. The rst term does notgive numerical predictions that are useful to extend the eectivetheory beyond ! = me.3!=me γ ! γγ γγ ! γ  ! γγγ0.3 0.969(8) 1.09(1) 1.20(1)0.4 0.923(6) 1.17(1) 1.37(1)0.5 0.888(6) 1.28(1) 1.68(1)0.6 0.852(4) 1.47(1) 2.20(1)0.7 0.826(5) 1.80(1) 3.17(1)0.8 0.811(5) 2.41(1) 5.31(2)0.9 0.819(6) 3.95(2) 11.88(3)1.0 0.880(7) 23.1(1) 176.3(3)1.1 1.19(1) 18.2(1) 94.7(2)1.3 1.71(2) 8.31(5) 31.6(1)1.5 1.44(1) 3.37(2) 11.9(1)1.7 0.996(8) 1.40(1) 4.96(3)1.9 0.635(4) 0.622(3) 2.23(1)2.0 0.503(3) 0.424(2) 1.54(1)Table 1: Ratio between exact and eective results for the threecross sections. The error on the last digit comes from the phase-space Monte Carlo integration.(γγ ! γ)e(γγ ! γ)= r−7:85491  exp[4:42122 +0:343516log2(r) − 0:114058log3(r) + 0:0103219log4(r)];( ! γγγ)e( ! γγγ)= r−6:57374  exp[5:27548−0:689808log2(r) + 0:15014log3(r) − 0:0123385log4(r)](15)where the eective cross sections e are given in eq. (11)and ref. 3 and r = !=me. All the above ts are valid inthe energy range 1:7 < r < 100.0.1 1.0 10.0 100.010-2610-2210-1810-1410-1010-610-2ExactEffectiveFigure 3: γ ! γγ cross section in fb as a function of!=me.0.1 1.0 10.0 100.010-2610-2210-1810-1410-1010-610-2ExactEffectiveFigure 4: γγ ! γ cross sec-tion in fb as a function of !=me.0.1 1.0 10.0 100.010-2610-2210-1810-1410-1010-610-2ExactEffectiveFigure 5:  ! γγγ cross section in fb as a function of!=me.4 Astrophysical and cosmological possibleimplicationsThe 5-leg photon{neutrino processes are of interest inastrophysics. The processes γ ! γγ and  ! γγγcan aect the mean free path of neutrinos inside the su-pernova core, while the process γγ ! γ is a possiblecooling mechanism for hot objects 13.Basing their results on the assumption(γ ! γγ) = 0 !1 MeVγ; 0 = 10−52 cm2 ;(16)and on the data collected from supernova 1987 A, theauthors of 13 tted the exponent γ in eq. (16) to be lessthan 8:4, for ! of the order of a few MeV. The physicalrequirement behind this is that neutrinos of a few MeVshould immediately leave the supernova, so that theirmean free path is constrained to be larger than 1011 cm.4The eective theory predicts γ = 10, while, usingthe curves from the exact calculation, a softer energydependence is observed in the region of interest (see Fig.3). A t to the exact curve gives γ  3 for 1 MeV < ! <10 MeV, thus conrming the expectations of 13.A second interesting quantity is the range of param-eters for which the neutrino mean free path for such re-actions is inside the supernova core (of 10 km typicalsize), therefore aecting its dynamics. Always in 13 itwas found, with the help of Monte Carlo simulations,that for several choices of temperature and chemical po-tential, and assuming the validity of the eective theory(γ = 10), this happens when !  5 MeV. Since the exactresults are now available, it would be of extreme interestto see how the above prediction is aected. More in gen-eral, we think that the reactions in (1) should be includedin supernova codes.On the basis of the eective theory results, the au-thors of 3, suggested that these processes could also havesome relevance in cosmology. Consider, in fact, the meannumber N of neutrino collisions, via the rst of processes(1), in an expansion time t equal to the age of the Uni-verse 14:N = (γ ! γγ)n c t ; n = neutrino number density:By writing n and t in terms of the photon energy atthermal equilibrium (!  kT ), expressed in units of1010 K and denoted by T10, we getn = 1:6 1031T 310 cm−3 ; t = 2T−210 s: (17)When N is large, the neutrinos are in thermal contactwith matter and radiation, while, for N  1 (namely  10−42T−110 cm2), the neutrinos decouple. Applyingthe eective formula in eq. (11), the resulting decouplingtemperature is T10  9:5, namely !  8:2 MeV, there-fore outside the range of validity of the eective theory.By repeating the same analysis with the exact result,we found instead that N becomes of the order of 1 at!  1 GeV, and at these energies, other processes enterthe game. In conclusion, the ve-leg reactions in eq. (1)are unlikely to be important for a study of the neutrinodecoupling temperature, contrary to what the eectivetheory seemed to suggest.AcknowledgementsWe thank the authors of 8 for having informed us abouttheir recent computation.We also wish to thank M.B. Gavela, G.F. Giudice andO. Pene for helpful remarks.References1. D. A. Dicus and W. W. Repko, Phys. Rev. D48(1993) 5106.2. C. N. Yang, Phys. Rev. 77 (1950) 242;M. Gell-Mann, Phys. Rev. Lett. 6 (1961) 70.3. D. A. Dicus and W. W. Repko, Phys. Rev. Lett.79 (1997) 569.4. A. Abada, J. Matias and R. Pittau, hep-ph/9806383.5. H. Euler, Ann. Phys. 26 (1936) 398; W. Heisen-berg and H. Euler, Z. Phys. 98 (1936)714.6. N. Van Hieu and E. P. Shabalin, Sov. Phys. JETP17 (1963) 681.7. A. Abada, J. Matias and R. Pittau, hep-ph/9808294.8. D. A. Dicus, C. Kao and W. W. Repko, hep-ph/9806499.9. R. Pittau, Comput. Phys. Commun. 104 (1997)23 and 111 (1998) 48.10. G. Passarino and M. Veltman, Nucl. Phys. B160(1979) 151.11. F. A. Berends et al., Phys. Lett. B103 (1981) 124;P. De Causmaecker at al., Nucl. Phys. B206(1982) 53;R. Kleiss and W. J. Stirling, Nucl. Phys. B262(1985) 235;J. F. Gunion and Z. Kunszt, Phys. Lett. B161(1985) 333;Z. Xu, D. -H. Zhang and L. Chang, Nucl. Phys.B291 (1987) 392.12. J. Kahane, J. Math. Phys. 9 (1968) 1732;J. S. R. Chisholm, Comput. Phys. Commun. 4(1972) 205;E. R. Caianiello and S. Fubini, Nuovo Cimento 9(1952) 1218.13. M. Harris, J. Wang and V. L. Teplitz, astro-ph/9707113.14. P. J. E. Peebles, Principles of Physical Cosmology,Princeton University Press (1993).5

Five-leg photon-neutrino interactions

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