arXiv:1409.4339v3 [hep-ph] 30 Sep 2014 Anomalous radiative transitions Kenzo Ishikawa1, Toshiki Tajima2,3, and Yutaka Tobita1 October 1, 2014 1 Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan 2 Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA 3 KEK High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan Abstract Anomalous transitions involving photons derived by many-body interaction of the form, ∂µ Gµ , in the standard model are studied. This does not affect the equation of motion in the bulk, but makes wave functions modified, and causes the unusual transition characterized by the time-independent probability. In the transition probability at a time-interval T expressed generally in the form P = T Γ0 + P (d) , now with P (d) 6= 0. The diffractive term P (d) has the origin in the overlap of waves of the initial and final states, and reveals the characteristics of waves. In particular, the processes of the neutrino-photon interaction ordinarily forbidden by Landau-Yang’s theorem (Γ0 = 0) manifests itself through the boundary interaction. The new term leads to physical processes over a wide energy range to have finite probabilities. New methods of detecting neutrinos using laser are proposed that are based on this difractive term, which enhance the detectability of neutrinos by many orders of magnitude. 1 Matter wave and S[T] 1 In modern science and technology, quantum mechanics plays fundamental roles. Despite the fact that stationary phenomena and method have been well developed, those of non-stationary phenomena have not. In the former, the de Broglie wave length p~ determines a typical length and is of a smicroscopic size, and scatterings or reactions in macroscopic scale are considered independent, and successive reactions have been treated under the independent scattering hypothesis. The probability of the event that they occur are computed by the incoherent sum of each value. In the latter, time and space variables vary simultaneously, and a new scale, which can be much larger than the de Broglie wave length, emerges. They appear in overlapping regions of the initial and final waves, and show unique properties of intriguing quantum mechanical waves. A transition rate computed with a method for stationary waves with initial and final states defined at the infinite time-interval T = ∞ is independent of the details of the wave functions. They hold characteristics of particles and preserve symmetry of the system. Transitions occurring at a finite T , however, reveal characteristics of waves, the dependence on the boundary conditions [1, 2] 1 , and the probability, P = T Γ0 + P (d) , (1) where P (d) is the diffractive term which has often escaped attention by researchers. The rate Γ0 is computed with Fermi’s golden rule [6, 7, 8, 9, 10], and preserves the internal and space-time symmetries, including the kineticenergy conservation. Γ0 holds the characteristic properties of particles and the hypothesis of independent scatterings is valid. For a particle of small mass, ms , Γ0 (pi , ms ) behaves Γ0 (pi , ms ) ≈ Γ0 (pi , 0), (2) because the characteristic length, the de Broglie wave length, is determined with pi . The region where P (d) is ignorable is called the particle-zone. Overlapping waves in the initial and final states have the finite interaction energy, and reveal unique properties of waves [1, 2]. Because the interaction 1 It was pointed out by Sakurai [3], Peierls [4], and Greiner [5] that the probability at finite T would be different from that at T = ∞. Hereafter we compute the difference for the detected particle that has the mean free path lmf p of lmf p ≫ cT . 2 energy is part of the total energy, sharing with the kinetic energy, the conservation law of the kinetic-energy is violated. Consequently, the state becomes non-uniform in time and the transition probability has a new component P (d) , (d) showing characteristics of waves. The term PT was shown to behave with a new scale of length, ( m~s c ) · ( mEsic2 ). Accordingly the correction is proportional to the ratio of two small quantities 1/T P (d) ), = f( 2 3 T ms c /~Ei (3) and does not follow Eq. (2). P (d) reflects the non-stationary waves and is not computed with the stationary waves. In the region where P (d) is important, the hypothesis of independent scattering is invalid, and interference unique to waves manifests. This region is called the wave-zone, and extends to a large area for light particles. P (d) has been ignored, but gives important contributions to the probability in various processes. Especially P (d) is inevitable for the process of Γ0 = 0 and P (d) 6= 0, which often appears. Furthermore, P (d) can be enhanced drastically, if the overlap of the waves is constructive in wide area. This happens for small ms , large Ei , even for large T , and reveals macroscopic quantum phenomena. Processes of large P (d) involving photon and neutrino in the standard model are studied in the present paper. An example of showing Γ0 = 0, P (d) 6= 0 is a system of fields described by a free part L0 and interaction part Lint of total derivative, L = L0 + Lint , Lint = d G, dt (4) where G is a polynomial of fields φl (x). φl (x) follows free equation, ∂L0 ∂ ∂L0 = 0. − ∂φl (x) ∂t ∂ ∂φl (x) ∂t (5) Lint decouples from the equation and does not modify the equation of motion in classical and quantum mechanics. Nevertheless, a wave function |Ψ(t)i follows a Schrödinger equation in the interaction picture, i~ ∂ ∂ |Ψiint = ( Gint (t))|Ψiint, ∂t ∂t 3 (6) where the free part, H0 , and the interaction part, Hint , are derived from the previous Lagrangian, and Gint stands for G of the interaction picture. A solution at t, |Ψ(t)iint = e Gint (t)−Gint (0) i~ |Ψ(0)iint, (7) is expressed with G(t), and the state at t > 0 is modified by the interaction. The initial state |Ψ(0)iint prepared at t = 0 is transformed to the other state of t-independent weight. Hence, Eq. (7), is like stationary, and Γ0 = 0 , and P (d) 6= 0. Physical observables are expressed by the probability of the events, which are specified by the initial and final states. For those at finite T , normal S-matrix, S[∞], which satisfies the boundary condition at T = ∞ instead of those at finite T , is useless. S[T ] that satisfies the boundary condition at T [1, 2] is necessary and was constructed. S[T ] is applied to the system described by Eq. (4). S[T ] is constructed with the Møller operators at a finite T , Ω± (T ), as S[T ] = Ω†− (T )Ω+ (T ). Ω± (T ) are expressed by a free Hamiltonian H0 and a total Hamiltonian H by Ω± (T ) = lim eiHt e−iH0 t . From this expression, t→∓T /2 S[T ] is unitary and satisfies [S[T ], H0 ] 6= 0, (8) hence a matrix element of S[T ] between two eigenstates of H0 , |αi and |βi of eigenvalues Eα and Eβ , is decomposed into two components hβ|S[T ]|αi = hβ|S (n)[T ]|αi + hβ|S (d) [T ]|αi, (9) where hβ|S (n) |αi and hβ|S (d) |αi get contributions from cases Eβ = Eα and Eβ 6= Eα , and give Γ0 T and P (d) respectively. The deviation of the kineticenergies, Eβ − Eα , in the latter is due to the interaction energy of the overlapping waves, which depends on the coordinate system. Therefore, it is understood that Hint is not Lorentz invariant. Thus the kinetic-energy nonconserving term, which was mentioned by Pierls and Landau [4] so as to give negligibly small correction, yields P (d) [5] 2 . Because H0 is an generator of the Poincare group, Eq.(8) shows that S (d) [T] and P (d) violate the Poincare invariance. In the system described by Eq. (4), the first term disappears but the second term does not, Γ0 = 0, and P (d) 6= 0. 2 Unusual enhancement observed in laser Compton experiment [11] may be connected. 4 S[T ] is expressed with the boundary conditions for the scalar field φ(x) [12, 13], lim hα|φf |βi = hα|φfin |βi, (10) lim hφf |0i = hα|φfout|βi, (11) t→−T /2 t→+T /2 where φin (x) and φout (x) satisfy the free wave equation, and φf , φfin , and φfout are the expansion coefficient of φ(x), φin (x), and φout (x), with the normalized wave functions f (x) of the form Z ← → f φ (t) = i d3 xf ∗ (~x, t) ∂0 φ(~x, t). (12) The function f (x) indicates the wave function that the out-going wave interacts in a successive reaction of the process. The out-going photon studied in the following section interacts with atom or nucleus and their wave functions are used for f (x). Consequently, S (d) [T ] depends on f (x), and is appropriate to write as S (d) [T ; f ]. Accordingly the probability of the events is expressed by this normalized wave function, called wavepacket. Wavepackets that satisfy free wave equations and are localized in space are important for rigorously defining scattering amplitude [12, 13]. S (d) [T ; f ] expresses the wave nature due to the states of continuous kinetic-energy. S (d) [T ; f ] does not preserve Poincaré invariance defined by L0 . The state |βi of Eβ is orthogonal to |αi of Eα 6= Eβ and P (d) approaches constant at T = ∞. The wavepackets [12, 13, 8, 9, 10] can be replaced with plane waves for a practical computation for S[∞] [18, 19, 20, 21, 22, 23, 24, 25], but can not be done so for P (d) [14, 15, 16]. P (d) is derived from S (d) [T ; f ], and depends on f (x). Photon is massless in vacuum and has a small effective mass determined by the plasma frequency in matter, and neutrino is nearly massless. Thus they have the large wave-zone of revealing wave phenomena caused by P (d) . These small masses make P (d) appear in a macroscopic scale and significantly affect physical reactions. In this small (or zero) mass region the effects of diffractive term P (d) are pronounced. This is our interest in the present paper. The produced photon interacts with matter with the electromagnetic interaction, which leads to macroscopic observables. The term P (d) of the processes of Γ0 = 0 such as 2γ decays of 1+ meson and γ and ν reactions are 5 shown to be relevant to many physical processes, including possible experimental observation of relic neutrino. The enhancement of the probability for light particles with intense photons based on the normal component Γ0 was proposed in Refs. [27, 28], and the collective interaction between electrons and neutrino derived from the normal component Γ0 was considered in Ref. [29]. Our theory is based on the probability P (d) , hence differs from the previous ones in many respects. This paper is organized in the following manner. In Section 2, the couplings of two photon with 1+ state through triangle diagram is obtained. In section 3 and 4, positronium and heavy quarkonium are studied and their P (d) are computed. Based on these studies we go on to investigate the interaction of photons and neutrinos. In section 5, neutrino–photon interaction of the order αGF , and various implications to high energy neutrino phenomena are presented. In Section 6 we explore the implication of the photon-neutrino coupling on experimental settings. Summary is given in section 7. 2 Coupling of 1+ meson with two photons The coupling of γγ with axial vector states, 1+ meson composed of e− e+ , QQ̄, and ν ν̄ are studied using an effective Lagrangian expressed by local fields. From symmetry considerations, an effective interaction of the 1+ state φµ1 with two photons has the form, Z Sint = g d4 x∂µ (φµ1 (x)F̃αβ (x)F αβ (x)), (13) ∂µ φµ1 (x) = 0, F̃αβ (x) = ǫαβγν F γν (x), where Fαβ is the electromagnetic field, and the coupling strength g is computed later. In a transition of plane waves in infinite-time interval, the spacetime boundary is at the infinity, and the transition amplitude is computed with the plane waves, in the form, M = (pi − pf )µ (2π)4 δ(pi − pf )M̃ µ (14) and vanishes, where pi and pf are four-dimensional momenta of the initial and final states and M̃ is the invariant amplitude. This shows that the amplitude proportional to δ 4 (pi − pf ) and the transition rate Γ0 vanish. The rate of 6 1+ → γγ decay vanishes in general systems, because the state of two photons of momenta (~p, −~p) does not couple with a massive 1+ particle. Hence + →γγ Γ10 = 0, (15) which is known as Landau-Yang’s theorem [30, 31]. The term Sint is written as a surface term in four-dimensional space-time, Z Sint = dSµ g(φµ1 (x)F̃αβ (x)F αβ (x)), (16) surf ace which is determined by the wave functions of the initial and final states. Accordingly, the transition amplitude derived from this surface action is not proportional to T , but has a weaker T dependence. Thus P (d) comes from the surface term, and does not have the delta function of kinetic-energy conservation. Kinetic energy of the final states deviates from that of the initial state due to the finite interaction energy between them. The deviation becomes larger and P (d) is expected to increase with larger overlap. We find P (d) in the following. 2.1 Triangle diagram The interaction of the form Eq. (13) is generated by one loop effect in the standard model. The scalar and axial vector current J(0) = ¯l(0)l(0), (17) J 5,µ = ¯l(0)γ5γ µ l(0) (18) in QED, we have 1 L = L0 + Lint , L0 = ¯l(x)(γ · p − ml )l(x) − Fµν (x)F µν (x), 4 µ ¯ Lint = eJµ A (x), Jµ (x) = l(x)γµ l(x), (19) where Aµ (x) is the photon field and l(x) is the electron field, coupled through two photons in the bulk through the triangle diagram Fig. 1. The matrix elements are e2 (20) Γ0 = h0|J(0)|k1, k2i = 2 ǫµ (k1 )ǫν (k2 )m[k2,µ k1ν − gµν k1 · k2 ]f0 , 4π e2 Γ5,α = h0|J5,α(0)|k1 , k2i = −i 2 2f1 ǫµ1 (k1 )ǫµ2 (k2 ) 4π × [(k1,µ2 ǫµ1 ν1 ν2 α − k2,µ1 ǫµ2 ν1 ν2 α )k1ν1 k2ν2 + (k1 · k2 )ǫµ1 µ2 να (k1 − k2 )ν ], (21) 7 ν pν,2 k1 ν γ pν,2 e k2 e e pν,1 γ e e pν,1 k2 e γ ν k1 γ ν Figure 1: Triangle diagrams of the electron loop which give contributions to 1+ (l¯l) → γγ, ν + γ → ν + γ and ν + ν̄ → γ + γ. where ǫµ (k) is the polarization vector for the photon. The triangle diagram for the axial vector current Eq. (21) has been studied in connection with axial anomaly and π 0 → γγ decay [32, 33, 34, 35, 36] and now is applied to P (d) for two photon transitions of the axial vector meson and neutrino. The triangle diagram Fig. 1 shows that the interaction occurs localy in space and time, but the transition amplitude is the integral over the coordinates and receives the large diffractive contribution if the neutrino and photon are spatially spread waves. In Fig. 1, the in-coming and out-going waves are expressed by lines, but they are in fact the spread waves, which is obvious in the figures in Ref. [37] and in Fig. 2. Γ5,α is expressed also with f1 in the form αem f1 (k1 + k2 )α (F̃1,ρλ F2ρλ + F̃2,ρλ F1ρλ ), 2π 1 = k1,ρ ǫ1,λ − k1,λ ǫ2,ρ , F̃ρλ = ǫρλξη F ξη . 2 Γ5,α = F1,ρλ (22) The coefficients f0 and f1 are given by the integral over the Feynman parameters, Z 1 Z 1−x 1 f0 = , (23) dx dy 2 ml − 2xyk1 · k2 − iǫ 0 0 Z 1 Z 1−x xy f1 = dx dy 2 , (24) ml − 2xyk1 · k2 − iǫ 0 0 8 ν ν γ (detected) γ Figure 2: Diagram of the neutrino photon scattering, ν + γ → ν + γ of spatially spread waves. where m2l 1 + I1 , (25) 4k1 · k2 4(k1 · k2 )2 Z 1 Z 1−x 1 = 2k1 · k2 dx dy 2 ml − 2xyk1 · k2 − iǫ 0 0   q k1 ·k2   , if k1 · k2 < 2m2l , 2Sin−1  2m2l  r r 2m2 2m2 l = (26) 1+ 1− 1− k ·kl 1+ 2 k1 ·k2 2 1 π 1 2 2  r r − log + iπ log , if k · k ≥ 2m .  1 2 l 2 2  2 2m 2m 2 l l 1− 1− 1− 1− f1 = − I1 k1 ·k2 f1 in various kinematical regions is  −1 m2   4k1 ·k2 + 4(k1 ·k2 )2 π(1 − f1 = 4k−1 ; k1 · k2 ≫ m2l , 1 ·k2   1 ; k1 · k2 ≪ m2l . 2m2 k1 ·k2 2δE ); ml l 9 k1 · k2 = 2m2l − ml δE, (27) 3 Positronium The bound states of a positronium with the orbital angular momentum L = 1, S = 1 have total angular momentum J = 2, 1, 0. These states at rest of P~ = 0 are expressed with non-relativistic wave functions and creation operators of l+ and l− of momentum ~p, spin ±1/2, and P-wave wave functions pi F (p) as are given in Appendix A, where Z |1/2, 1/2; −1i = d~p b†+1/2 (~p )d†+1/2 (−~p )(px − ipy )F (p)|0i. (28) Others are defined in the same manner. These bound states couple with the lepton pairs with effective local interactions which are expressed as ¯ Lint = g0 φ0 (x)¯l(x)l(x) + g1 φµ1 (x)¯l(x)γ 5 γµ l(x) + g2 φµν 2 (x)l(x)γµ ∂ν l(x), (29) where the coupling strengths are computed from g0 = h0|¯l(0)l(0)|φ0 ; ~p = 0i, g1 ǫµ (~p = 0) = h0|¯l(0)γ 5 γ µ l(0)|φ1 ; ~p = 0i, g2 ǫµν (~p = 0) = h0|¯l(0)γ µ ∂ ν l(0)|φ2; p~ = 0i, (30) (31) (32) where the ǫµ , ǫµν are polarization vector or tensor for the massive vector and tensor mesons. We have g0 = gN0 , g1 = gN1 , g2 = gN2 , Z 2 g = −2π dp p4 F (p) , |E| + m N0 = 1, N1 = 0.862, N2 = 1.33. (33) The decay rates for 0+ and 2+ were studied in [38, 39], so that here we concentrate on 1+ and 0+ as a reference for 1+ . Fields φ0 (x), φ1 (x), and φ2 (x) couple with two photons through the triangle diagrams. Their interactions with two photons are summarized in the following effective Lagrangian, α α α Lint = g0 f0 φ0 Fµν F µν + g1 f1 ∂µ (φµ1 ǫνρστ F νρ F στ ) + g2 T µν Fµρ F ρν . π π π (34) 10 3.1 Axial vector positronium Here we study the two photon decay of axial vector positronium, which is governed by the second term of the right-hand side of Eq. (34). The matrix element of the axial current between the vacuum and two photon state was computed by Refs. [32, 33, 34, 35, 36]. Because Γ0 of two photon decay of axial vector meson vanishes due to Landau-Yang’s theorem, but P (d) does not, we give the detailed derivation of P (d) . From the effective interaction, Eq. (34), the probability amplitude of the event that one of the photons of ~kγ from the decay of cµ of p~c is detected at ~ γ , Tγ ) is (X Z α ∂ ~ γ , Tγ ), k1 |ǫνρστ F νρ F στ |0i], (35) M = −g1 d4 x [f1 h0|cµ (x)|~pc ih(~kγ , X π ∂xµ where ~ γ , Tγ |Aµ (x)|0i h~kγ , X Z σγ ~ ~ 2 ~ ~ ~ = Nγ d~k2 ργ (~k2 )e− 2 (k2 −kγ ) +i(E(k2 )(t−Tγ )−k2 (~x−Xγ )) ǫµ (~k2 ), ~ ~ h~k1 |Aµ (x)|0i = ργ (~k1 )ǫµ (~k2 )e+i(E(k1 )t−k1 ~x) , ~ (36) ~ h0|cµ (x)|~pc i = (2π)3/2 ρc (~pc )ǫµ (~pc )e−i(E(kc )t−kc ~x) . 3 The initial state is normalized, and the coupling in Eq. (33) has (2π) 2 for the initial state, and kµ ǫµ (k) = 0, (pc )µ ǫµ (pc ) = 0,   12 1 . ρ(~k) = 2E(k)(2π)3 (37) ~ γ , Tγ | is normalized and, The state h~kγ , X  σ  34 γ . (38) Nγ = π Integration over ~k2 is made prior to the integration over x, in order for M to satisfy the boundary condition of S[T ], and we have,  1 3 2 (2π) µ ~ γ , Tγ |A (x)|0i = θ(λ) h~kγ , X ρ(~kγ + δ~k)ǫµ (~kγ + δ~k) σγ σT2 ~ ~ ~ ×ei(E(kγ )(t−Tγ )−kγ (~x−Xγ ))−χ(xµ ) , 11 (39) where ~ γ )2 , λ = (t − Tγ )2 − (~x − X 1 ~ γ )2 , ~ γ )l − vγ (x0 − Tγ ))2 + 1 (~x − X χ(x) = ((~x − X T 2σ γ 2σT i ~ γ − ~vγ (t − Tγ )), (δ~k(x))i = − i δ~x, δ~x = (~x − X σγ (40) (41) σγl = σγ ; i = longitudinal, σγT = σT ; i = transverse. The wavepacket expands in the transverse direction, and σT in large x0 − Tγ is given by σT = σγ − i (x0 − Tγ ). E (42) We later use X spin ǫµ (~kγ + δ~k)ǫν (~kγ + δ~k) = −gµν , δk 0 (x) = (43) i 0 ~ γ − ~vγ (t − Tγ )). δx , δx0 = ~v(~x − X σγ The stationary phase for large x0 − Tγ exists in the time-like region λ ≥ 0 [14], and the function and its derivative is proportional to θ(λ). Thus the integration in Eq. (35) is made over the region λ ≥ 0, which has the boundary, λ = 0. Consequently the transition amplitude does not vanish if the integrand at the boundary is finite. It is shown that this is the case in fact. σγ is the size of the nucleus or atomic wave function that the photon interacts with and is estimated later. For the sake of simplicity, we use the Gaussian form for the main part in this paper. Substituting Eq. (21), we have the amplitude Z α ∂ i(pc −k1 )x µ M = −ig1 N d4 x [e M̃ ], (44) π ∂xµ λ≥0 ~ ~ M̃µ = f1 {k1 · (kγ + δk(x))}ei(E(kγ )(t−Tγ )−kγ (~x−Xγ ))−χ(x) T µ , T µ = ǫµ (pc )ǫν (k1 )∗ ǫρλξν k1ξ (kγ + δkγ (x))ρ ǫλ (kγ + δkγ (x)),  1 (2π)3 2 3/2 ~ ~ , N = (2π) ρc (~pc )ργ (kγ )ρ(k1 )Nγ σγ σT2 12 (45) ~ γ ) of the final state which depends on the momenta and coordinates (Tγ , X and Tm . Although M is written as the integral over the surface, λ = 0, the expression Eq. (44) is useful and applied for computing the probability per particle Z X ~γ 1 dX ~kγ d~k1 P = |M|2, (46) d V (2π)3 s ,s 1 2 where V is a normalization volume for the initial state, and the momentum of the non-observed final state is integrated over the whole positive energy region and the position of the observed particle are integrated inside the detector. Following the method of the previous works [1, 2] and Appendix B, we write the probability with a correlation function. In the integral  2 Z Z Z gαN ∂2 2 2 ~ ~ ~ dk1|M| = (47) d k 1 ρ (k 1 ) d4 x1 d4 x2 µ1 µ2 F, πρ(k1 ) ∂1 ∂2 λ1 ≥0 F =f1 {k1 (kγ + δkγ (x1 ))f1∗ (k1 (kγ + δkγ (x2 )))}T µ1 (x1 )T µ2 ,∗ (x2 ) × e−i(pc −k1 −kγ )(x1 −x2 )−χ(x1 )−χ we have X ∗ (x 2) , T µ1 (x1 )T µ2 ,∗ (x2 ) = spin 2(−g µ1 µ2 + pµc 1 pµc 2 /M 2 )(k1 (kγ + δkγ (x2 )∗ ))(k1 (kγ + δk(x1 ))). (48) With variables xµ1 + xµ2 − 2X µ , X0 = Tγ , 2 δxµ = xµ1 − xµ2 , xµ+ = (49) (50) the integral is written as   Z Z ∂2 1 ∂2 ∂2 4 4 4 4 d x1 d x2 µ F = d x+ d δx − F ∂x1 ∂x2,µ 4 ∂x+,µ 2 ∂δxµ 2 λi ≥0 λ+ ≥0 Z Z 1 3 µ ∂ 4 d S+ F, (51) = d δx ∂x+,µ λ+ =0 4 (52) 13 where λ+ = x2+ and x+ is integrated over the region λ+ ≥ 0, and the integral is computed with the value at the boundary, λ+ = 0. δx are integrated over the whole region, and the second term in the second line vanishes. Using the formulas [1, 2], Z 1 d~k1 −i(pc −k1 )·δx 1 e = −i δ(λ− )ǫ(δt)θ(phase-space) + regular, 3 (2π) 2E(k1 ) 4π 2 θ(phase-space) = θ(M − 2pc pγ ), (53) Z iφ̄c (δt) 1 σT e 2 1 dδ~xeipγ δ~x− 4σ (δ~x−~vδt) δ(λ) = , (54) 4π 2 δt m2γ , φ̄c (δt) = ωγ δt, ωγ = 2Eγ and the integrals given in Appendix E, we compute the probability. It is worthwhile to note that the right-cone singularity exists in the kinematical region θ(phase-space) and the probability becomes finite in this region [1, 2]. The natural unit, c = ~ = 1, is taken in the majority of places, but c and ~ are written explicitly when it is necessary, and MKSA unit is taken in later parts. After tedious calculations, we have the probability ! Z ~ X 1 dXγ d~pγ 1 P = i Ii ∆1+,γ θ(M 2 − 2pc · pγ ), (55) N2 3 V (2π)3 Eγ 4π i where Ii is given in Appendix E, and 1 π 3/2 2 ( ) , N2 = (g α)2 π 2Ec σγ (56) ∆1+,γ = |f1 ((pc − kγ ) · kγ )|2 2((pc − kγ ) · kγ )2 = (π − 2)2 ; (δE → 0), (57) 32 and 1 π3 2 = ρs (x01 − x02 ), σγ3 x0 − x02 ρs (x01 − x02 ) = 1 + i 1 Eγ σγ  π 3 σγ σT2 (1)σT2 (2)∗  21  were substituted. 14 (58) ~ γ , we obtain the total volume, Integrating over the gamma’s coordinate X −1 which is canceled by the factor V from the normalization of the initial state. The total probability, for the high-energy gamma rays, is then expressed as Z  1 2 d 3 pγ 1 σl 2 P = π σl log(ωT ) + 2 N2 θ(M 2 − 2pc · pγ )∆1+,γ , (59) 3 mγ (2π)3 Eγ 2 where L = cT is the length of the decay region. The kinematical region of the final states is expressed by the step function [2], which is different from the on-mass shell condition. Γ0 vanishes and P is composed of log T and constant terms. The constant is inversely proportional to m2γ , and becomes large for the small mγ . 3.2 Transition probability A photon γ is massless in vacuum and has an effective mass in matter, which is given by plasma frequency in matter [40] as s r ne e2 4παne ~c =~ , (60) mef f = ~ me ǫ0 me where ne is the density of electron, and for the value, n0e = 1 (10−10 )3 (m)−3 = 1030 (m)−3 . (61) Substituting α= 1 , me = 0.5 MeV/c2 , 137 (62) we have, mef f c2 = 30 r ne (0) ne eV. (63) In the air, ne = 3 × 1025 /m3 , the mass agrees with mef f c2 = 4 × 10−3 eV, 15 (64) which is comparable to the neutrino mass. At a macroscopic T , ωT = T /T0 , T0−1 = m2ef f c4 ne 1 = 900 (0) sec−1 , −16 × 106 ~E 6 × 10 ne (65) ωT is much larger than 1. We have the probability at the system of pc = (Ec , 0, 0, pc ),   M2 Ec M2 M2 dP ; = C (1 − )pγ + ≤ pγ ≤ , (66) dpγ pc 2pc 2(Ec + pc ) 2(Ec − pc )   π2 T 1 σγ N2 2 C= 3σγ log + ∆1+.γ , 3 T0 4 m2γ 2 and the total probability Ptotal 4 (π − 2)2 1 √ = 1536 π √ 1 1 T √ 3 σ γ log + T0 4 m2γ σ γ ! (gα)2 Ec + 2pc . (67) Ec (Ec + pc ) Heavy quarkonium The decay of axial vector mesons composed of heavy quarks exhibits the same phenomena. The heavy quark mesons composed of charm and bottom quarks are observed, and show rich decay properties in the two photon decay, radiative transitions, and two gluon decays. Because quarks interact via electromagnetic and strong interactions, non-perturbative effects are not negligible, but the symmetry consideration is valid. Moreover, the nonrelativistic representations are good for these bound states, because quark masses are much greater than the confinement scale. Furthermore they have small spatial sizes. Accordingly we represent them with local fields and find their interactions using the coupling strengths of Eq. (33) and the values of the triangle diagrams, Eqs. (20), and (21). 4.1 q q̄ → γ + γ Up and down quarks have charge 2e/3 or e/3, and color triplet. Hence the probabilities of two photon decays are obtained by the expression Eq. (67) with charges of quarks 2e/3, e/3, and color factor. It is highly desired to obtain the experimental value for 1+ . 16 4.2 q q̄ → gluon + gluon A meson composed of heavy quarks decays to light hadrons through gluons. Color singlet two gluon states are equivalent to two photon states. Accordingly the two gluon decays are calculated in the equivalent way to that of two photon decays as far as the perturbative calculations are concerned. The transition rates for 0+ and 2+ may be calculated in this manner. The total rates for L = 0 charmonium, J/Ψ and ηc agree with the values obtained by the perturbative calculations. J/Ψ is C = −1 and decays to three gluons and the latter is C = +1 and decays to two gluons. The former rate is of αs3 and the latter rate is of αs2 , where αs is the coupling strength of gluon. Their widths are Γ = 93 keV or Γ = 26.7 MeV, and are consistent with small coupling strength αs ≈ 0.2. Now L = 1 states have C = +1. Hence a meson of J = 2 and that of J = 0 decay to two gluons, whereas the decay rate of a J = 1 meson vanishes by Landau-Yang’s theorem. A cc̄ meson of the quantum number 1+ is slightly different from that of positronium, because a gluon hadronizes by a non-perturbative effect, which is peculiar to the gluon. The hadronization length is not rigorously known, but it would be reasonable to assume that the length is on the order of the size of pion. The time interval T is then a microscopic value. P (d) for this time T is estimated in the following. A gluon also hadronizes in the interval of the lightest hadron size, T = Rhadron c~ . = c mπ The gluon plasma frequency is estimated from quark density, s nq e2strong , mef f = mq ǫ0 (68) (69) and nq = 1 , αs = 0.2, mq c2 = 2 MeV, (fm)3 (70) Then we have m2ef f c4 T c4 αs2 (c~)3 = = αs2 3 3 E (fm) mq mπ E mq mπ E(fm) 3 (197MeV) = × 4 × 10−2 = 0.9. 2 × 130 × 1500(MeV)3 ωT = 17 (71) At small ωT , g̃(ωT ) varies with T as is shown in Fig. 2 of Ref. [2] and g̃(ωT )|ωT =1 = 2.5. (72) Higher order corrections also modify the rates for q q̄. The values to light hadrons are estimated by [41, 42, 43, 44, 45]. The total values for light hadrons are expressed with a singlet and octet components H1 and H8 as Γ(χ0 → light hadrons) = 6.6αs2H1 + 3.96H8 αs2 , Γ(χ1 → light hadrons) = 0 × αs2 H1 + 3.96H8 αs2 , Γ(χ2 → light hadrons) = 0.682αs2H1 + 3.96H8 αs2 . (73) (74) (75) Using values for χ0 and χ2 from [49] Γ(χ0 → light hadrons) = 1.8 MeV, Γ(χ2 → light hadrons) = 0.278 MeV, (76) (77) we have the rate for χ1 from H8 , Γ(8) (χ1 → light hadrons) = 0.056 MeV. (78) The experimental value for χ1 is Γ(χ1 → light hadrons ) = 0.086 MeV (world average), Γ(χ1 → light hadrons) = 0.139 MeV (BESS II). (79) (80) The large discrepancy among experiments for Γχ1 →light hadrons may suggest that the value depends on the experimental situation, which may be a feature of P (d) . We have ∆Γ = Γ(χ1 → light hadrons ) − Γ(8) (χ1 → light hadrons), ∆Γ(world average) = 0.030 MeV , ∆Γ(BESSII ) = 0.083 MeV, which are attributed to P (d) . 18 (81) (82) 4.3 E1 transition: Ψ′ → φ1 + γ, φ1 → J/Ψ + γ φ1 is produced in the radiative decays of Ψ′ through the E1 transition Ψ′ → φ1 + γ, which is expressed by the effective interaction Z ′ d4 xO µν Fµ,ν , O µν = ǫµνρσ Ψ′µ φ1,ν Sint = e (83) (84) in the local limit. The action Eq. (84) does not take the form of the total derivative, but is written as Z Z 4 µν ′ ′ d4 x(∂ν O µν )Aµ . (85) d x∂ν (O Aµ ) − e Sint = e The second term shows the interaction of the local electromagnetic coupling of the current j µ = ∂ν O µν and the first term shows the surface term. This leads to the constant probability at a finite T , P (d) . The induced P (d) for muon decay was computed in [1], and large probability from P (d) compared with T Γ0 was found in the region cT = 1 m. The radiative transitions of heavy quarkonium are deeply connected with other radiative transitions and the detailed analysis will be presented elsewhere. Spin 0 and 2 mesons , φ0 and φ2 , show the same E1 transitions and photons show the same behavior from P (d) [46, 47, 48, 49]. A pair of photons of the continuous energy spectrum are produced in the wave zone, and are correlated. On the other hand, a pair of photons of the discrete energy spectrum are produced in the particle zones and are not correlated. Thus the photons in the continuous spectrum are different from the simple background, and it would be possible to confirm the correlation by measuring the time coincident of the two photons. 5 Neutrino-photon interaction The neutrino photon interaction of the strength αGF induced from higher order effects vanishes due to the Landau-Yang-Gell-Mann theorem. But P (d) is free from the theorem and gives observable effects. Moreover, although the strength seems much weaker than the normal weak process, that is enhanced 19 drastically if the photon’s effective mass is extremely small. Because P (d) does not preserve Lorentz invariance, a careful treatment is required. The triangle diagram of Fig. 1 is expressed in terms of the action Z 1 GF ∂ Sνγ = αem √ (JµA (x)F̃αβ F αβ ), (86) d4 xf1 2π ∂x 2 µ JµA (x) = ν̄(x)(1 − γ5 )γµ ν(x), where the axial vector meson in Eq. (34) was replaced by the neutrino current. The mass of the neutrino is extremely small, and was neglected. The Sνγ leads to the neutrino gamma reactions with Γ0 = 0, P (d) 6= 0 on the order αGF . 5.1 ν +γ →ν +γ The rates Γ0 of the events ν + ν̄ → γ + γ ν+γ →ν +γ (87) vanishes on the order αGF [50] due to Landau-Yang’s theorem. The higher order effects were also shown to be extremely small [34] and these processes have been ignored. The theorem is derived from the rigorous conservation law of the kinetic-energy and angular momentum. However these do not hold in P (d) , due to the interaction energy caused by the overlap of the initial and final wave functions. Consequently, neither P (d) nor the transition probability vanish. These processes are reconsidered with P (d) . From Eq. (86), we have the probability amplitude of the event in which ~ γ as one of the photon of ~kγ interacts with another object or is detected at X Z GF αem ∂ ~ γ , Tγ )|F̃αβ F αβ |k1 i]. (88) M=− √ d4 x µ [hpν,2 |JµA (x)|pν,1 ih(~kγ , X ∂x 2π 2 The amplitude is expressed in the same manner as Eq.(44), Z GF 2 ∂ M = −i √ αN (89) d4 x µ [e−i(pν1 −pν2 +k1 )x M̃µ ], ∂x 2π λ≥0 2 ~2 ~ M̃µ = f (k1 · kγ2 )ei(E(kγ )(t−Tγ )−kγ (~x−Xγ ))−χ(x) T µ , T µ = ν̄(pν2 )(1 − γ5 )γ µ ν(pν1 )ǫαβδζ ǫα (k 1 )ǫβ (kγ2 + δkγ (x))k1δ (k2 + δkγ (x))ζ ,  12 1   3 m m 1 (2π)3 2 ν ν , N= Nγ (2π) 2 ργ (k 1 )ργ (kγ2 + δk) 3 σγ σT2 (2π) 2 Eν1 Eν2 20 where the corrections due to higher order in δ~k are ignored in the following calculations as in Section (3.1). The probability averaged over the initial spin per unit of particles of initial state, is Z X ~γ 1 dX ~k 2 d~k 2 P = |M|2 d 2 V (2π)3 γ ν s ,s 1 2 Z ~ dkγ,2 = N 2 (i)(I0 p0ν1 (pν1 + pγ,1 − pγ2 )0 + Ii piν1 (pν1 + pγ1 − pγ2 )i ) (2π)3 2Eγ,2 0 × (4k1 · kγ f1 )2 θ(Phase-space), (90) θ(Phase-space) = θ((pν1 + kγ,1 )2 − 2(pν1 + kγ,1 )kγ,2 ),  2  3 1 GF 2α 1 1 1 1 2 2 √ N0 = , |Nγ | 2 4π σγ 2Eν1 2Eγ1 2 π where V is the normalization volume of the initial state, and ( 1 ; high energy, f1 (k1 · kγ ) = 4k11 ·kγ ; low energy, 2m2 (91) e where I0 and Ii are given in Appendix E. The probability P is not Lorentz invariant and the values in the CM frame of the initial neutrino and photon and those of the general frame do not agree generally. 5.1.1 Center of mass frame The phase space integral over the momentum in the CM frame p~ν1 + ~pγ1 = 0, p = |~pν1 | (92) is d~kγ,2 (p(2p − pγ2 ))0 (4f1 k1 · kγ )2 θ(phase-space) (2π)3 2Eγ,2 1 = 2 p4 ; high-energy, 6π Z d~kγ,2 1 (p(2p − pγ2 ))0 (4f1 k1 · kγ )2 2 θ(phase-space) 3 (2π) 2Eγ,2 kγ,2  4 p 1 p2 ; low-energy. = 2 12π me Z 21 (93) (94) Thus we have the probability  2  2  1 GF 2α 1 p 1 √ √ P = ; high-energy, (95) σγ log(ω̄T ) + √ 5 2 2 2mγ σγ 12 2 π 4π 2  2  4 1 GF 2α 1 p 1 − 21 1 √ = ; low-energy, (96) 5 5 σγ 3 2 ǫ me 2 π 4π 2 4 6π 2 where ω̄ in the log term is the average ω, and ǫ is the deviation of the index of refraction from unity given in Appendix C. The log term was ignored in the right-hand side of the low energy region. We found that in the majority √ 1 becomes much larger than σγ log(ωT ). Here we assume of region, ωE √ σγ that the medium is not ionized. 5.1.2 Moving frame At the frame ~pν1 = (0, 0, pν1 ), ~pγ1 = (0, 0, −pγ1 ), pν1 > pγ1 the probability in high energy region is P =N02 Z d~kγ,2 θ((p1 + k1 )2 − 2(p1 + k1 )kγ,2 ) (2π)3 2Eγ,2  σγ ) × (p0ν1 (pν1 + pγ1 − pγ2 )0 + plν1 (pν1 + pγ1 − pγ2 )l )l (σγ2 log ωT + 4ωE  i i 2 + pν1 (pν1 + pγ1 − pγ2 ) (σγ log ωT )   1 √ p2ν1 ; pν1 ≫ pγ1 . (97) =C0 σγ log ωT + √ σγ m2γ In the low energy region, the second term of P is given in the form,  4 1 pν 1 ′ P = C0 √ , σγ ǫ me (98) and the first term proportional to p6ν1 was ignored. Numerical constants )2 . The photon effective mass at high C0 and C0′ are proportional to ( G√F2 2α π energy region, mγ , and the deviation of the refraction constant from unity at low energy region, ǫ, are extremely small in the dilute gas, and P (d) becomes large in these situations. 22 5.2 Neutrino interaction with uniform magnetic field ν +B →ν +γ The action Eq. (86) leads to the coherent interactions of neutrinos with the macroscopic electric or magnetic fields. These fields are expressed in MKSA unit. Accordingly we express the Lagrangian in MKSA unit, which is summarized inAppendix D, and compute the probabilities. The magnetic field B in the z-direction is expressed by the field strength Fµν (x) = ǫ3,µν B, (99) and we have the action Sνγ (B) = gB Z d4 x ∂ (J A (x)F0,z ), ∂xµ µ (100) where gB is given in Appendix D. Because Sνγ (B) is reduced to the surface term, the rate vanishes, Γ0 = 0, but P (d) 6= 0. Furthermore, Sνγ (B) is not Lorentz invariant, and P (d) for νi → νj + γ becomes proportional to m2ν of much larger magnitude than the naive expectation. The amplitude is Z ∂ M = −iNgB d4 x µ [e−i(pν1 −pν2 )x M̃µ ], (101) ∂x λ≥0 0 ~ ~ M̃µ = ei(k0 (kγ )(x0 −Xγ )−kγ (~x−Xγ ))−χ(x) T µ , T µ = ν̄(kν2 )(1 − γ5 )γ µ ν(kν1 )(ǫ0 (kγ )kγz − ǫz (kγ )kγ0 ),  1 1  0 0  12  ω̃ν1 ω̃ν2 (2π)3 2 ~ 2 1 1 N= , Nγ ργ (kγ ) , ργ (kγ ) = 3 σγ σT2 (2π)3 2kγ0 ǫ0 (2π) 2 kν01 kν02 where X (T µ1 T µ2 ∗ ) = spin 8 2ω̃ν01 2ω̃ν02 2 (kνµ11 kνµ22 − g µ1 µ2 kν1 kν2 + kνµ12 kνµ21 )(kγ0 − kγz 2 ). (102) We have the probability from Eq. (46), Z 1 1 d~kγ 2 2 2 2 3 P = Ñ0 gB (2π) 16 (kγ0 − kγ3 ) 3 2Eν1 ǫ0 (2π) 2Eγ 0 0 l ×(i)(I0 kν1 (kν1 − kγ ) + Il kν1 (kν1 − kγ )l )θ(phase-space), θ(phase-space) = θ(δων212 − 2kν1 kγ ), 23 (103) 3 2 where Ñ02 = (πσγ )− 2 , δω12 = (ω̃ν01 )2 − (ω̃ν02 )2 , I0,l and IT,i are given in Appendix E. The convergence condition on the light-cone singularity is satisfied in the kinematical region, θ(phase-space). Thus the momentum satisfies 2(kν01 kγ0 − kν1 kγ cos θ) ≤ δω12 2 . Solving kγ , we have the condition for fraction x = (104) kγ , k ν1 α− ≤ x ≤ α+ , q 2 4 − 4(ω̃ν01 )2 m2γ δω12 ± δω12 , α± = 2(ω̃ν01 )2 (105) (106) where α± = O(1). The process νi → νj + γ occurs with the probability Eq. (103) if m2γ ≤ 2 2 (δω12 ) , 4(ω̃ν01 )2 (107) which is satisfied in dilute gas. If the inequality Eq. (107) is not satisfied, this probability vanishes. The probablity P reflects the large overlap of initial and final states and is not Lorentz invariant. Consequently, although the integration region in Eq. (103) is narrow in phase space determined by θ(phase-space), which is 2 proportional to the mass-squared difference, δω12 , the integrand is as large 3 as pν . P becomes much larger than the value obtained from Fermi’s golden rule. 5.2.1 High energy neutrino At high energy, Eq. (A.49) are substituted. For the case that p~ν1 is non~ we have parallel to B, dP 1 = Ñ02 gB2 (2π)3 16 (1 − cos2 ζ)(i)I0 kν31 C(m, x) (108) dx ǫ0 2 2 C(m, x) = m2γ x4 − (δω12 + m2γ )x3 + (ω̃ν1 )2 − ω12 )x2 − (ω̃ν01 )2 x, ~ where ζ is the angle between ~pν1 and B, ~ · p~ν1 = Bpν1 cos ζ. B 24 (109) The integral I0 is almost independent of kγ . Ignoring the dependence, we integrate the photon’s momentum, and have 1 P = Ñ02 gB2 (2π)3 16 (1 − cos2 ζ)(i)I0 kν31 C(m) (110) ǫ0 1 1 5 5 2 4 4 − α− ) − (δω12 + m2γ )(α+ − α− ) C(m) = m2γ (α+ 5 4 1 1 2 3 3 2 2 + ((ω̃ν01 )2 − ω12 )(α+ − α− ) − (ω̃ν01 )2 (α+ − α− ). 3 2 The probability P of Eq. (110) is proportional to (ω̃ν0 )2 kν31 , which is very different from the rate of the normal neutrino radiative decay, Γ = G2F m5ν (mν1 /Eν1 )× (numerial factor), especially for high energy neutrino. Moreover, I0 = m12 , γ can be extremely large in dilute gas, thus P is enormously enhanced. If the momentum of initial neutrino is parallel to the magnetic field, ζ = 0, we have 1 (111) P = Ñ02 gB2 (2π)3 16 (i)I0 kν1 D(m), ǫ0 2 2 2 α 2 − α− 4 α+ − α− 2 D(m) = δω12 − δω12 (ω̃ν01 )2 (α+ − α− − 2 + ) 2 2 α+ + (ω̃ν01 )2 (log − α+ + α− ). α− P in Eq. (111) is proportional to (ω̃ν0)4 kν1 , and is negligibly small compared to that of Eq. (110). 5.2.2 Low energy neutrino At low energy, Eq. (A.52) are substituted. I0 is inversely proportional to kγ2 and we have dP 1 1 1 2 = Ñ02 gB2 4πEν1 (1 − cos2 ζ) (1 − x)[δω12 − xm2γ − m2γ ], dx ǫ0 ǫ x and 1 1 (112) P = Ñ02 gB2 4π (1 − cos2 ζ)kν1 Clow (m), ǫ0 ǫ 2 2 3 α 2 − α− α 2 − α− α 3 − α− 2 Clow (m) = δω12 (α+ − α− − + ) − m2γ ( + − + ) 2 2 3 α+ + (ω̃ν01 )2 (α+ − α− ) − (ω̃ν01 )2 log . α− Using α± , C(m) and D(m) are computed easily. 25 5.3 Neutrino interaction with uniform electric field ν + E →ν +γ For a uniform electric field in the z-direction, Fµν (x) = E 03µν ǫ , c (113) we have T µ = ν̄(kν2 )(1 − γ5 )γ µ ν(kν1 )(ǫx (kγ )kγy − ǫy (kγ )kγx ), (114) X 1 (T µ1 T µ2 ∗ ) = 8 0 0 (kνµ11 kνµ22 − g µ1 µ2 kν1 kν2 + kνµ12 kνµ21 ) 2ων1 2ων2 spin ×(kγx 2 + kγy 2 ). (115) νi → νj + γ in the electric field is almost the same as that in the magnetic field. 5.3.1 High energy neutrino We have the probability in the high energy region, 1 P = Ñ02 gE2 (2π)3 16 2 (1 − cos2 ζ)(i)I0 kν31 C(m), c ǫ0 1 1 5 5 2 4 4 − α− ) − (δω12 + m2γ )(α+ − α− ) C(m) = m2γ (α+ 5 4 1 1 2 3 3 2 2 + ((ω̃ν01 )2 − ω12 )(α+ − α− ) − (ω̃ν01 )2 (α+ − α− ), 3 2 where ~ ~kν1 = Ekν1 cos ζ. E 5.3.2 (116) (117) Low energy neutrino The probability in the low energy region is 1 1 (118) P = Ñ02 gE2 4π 2 (1 − cos2 ζ)kν1 Clow (m), c ǫ0 ǫ     2 2 2 2 3 3 α+ − α− α+ − α− α+ − α− 2 2 Clow (m) = δω12 α+ − α− − − mγ − 2 2 3 α + + (ω̃ν01 )2 (α+ − α− ) − (ω̃ν01 )2 log . α− 26 5.4 Neutrino interaction with nucleus electric field ν + Enuc → ν + γ In space-time near a nucleus, there is the Coulombic electric field Enucl due to the nucleus, and one of Fµν in the action is replaced with Enucl . The rate estimated in Ref. [34] was much smaller by a factor 10−4 or more than the value of the normal process due to charged current interaction. Here we estimate P (d) for the same process. The action becomes Z 1 GF ∂ µν Sνγ (Enucl ) = αem √ (JµA (x)Fµν Fnucl ), (119) d4 xf1 2π ∂x 2 µ which causes the unusual radiative interaction of neutrino in matter. The probability P (d) of the high energy neutrino, where 2k1 · k2 ≫ m2e , and we µν due to a bound nucleus is shortsubstitute the value f1 = 8m1 2 . Since Fnucl l range, the probability is not enhanced. 5.5 Neutrino interaction with laser wave ν + Elaser → ν+γ In the scattering of neutrino with a classical electromagnetic wave due to laser, one of Fµν in the action is replaced with the electromagnetic field Elaser of laser of the form µν Flaser = E i ǫ0iµν eik1 x . The probability P (d) of this process is computed with the action Z 1 GF ∂ µν Sνγ (Enucl ) = αem √ (J A (x)Fµν Flaser ), d4 xf1 2π ∂xµ µ 2 (120) (121) where 2k1 · k2 ≈ 0, and we substitute the value f1 = 8m1 2 . The amplitude and l probability are almost equivalent to those of the uniform electric field. 6 Implications to neutrino reactions in matter and fields An initial neutrino is transformed to another neutrino and a photon following the probability P (d) . The photon in the final state interacts with a microscopic object in matter with the electromagnetic interaction, and loses the 27 energy. Thus the size σγ in P (d) is determined by its wave function, and the probability P (d) × σγA , where σγA is the cross section of the photon and is much larger than that of weak reactions, determines the effective cross section of the whole process. Hence the effective cross section can be as large as that of the normal weak process caused by the charged current interaction. 6.1 Effective cross section The probability of the event that the photon reacts on another object is expressed by P (d) in Eqs. (96), (97), and (98), and that of the final photon. If a system initially has photons of density nγ (Eγ ), the number of photons are multiplied, and the probability of the event that the initial neutrino is transformed is given by P (d) × nγ . In the system of electric or magnetic fields, the initial neutrino is transformed to the final neutrino and photon. Hence P (d) ×nγ for the former case, and P (d) for the latter case are important parameters to be compared with the experiments. The effective cross section, for the process where the photon in the final state interacts with atoms A of the cross section σγA , is (d) σγA = P (d) (γ)nγ × σγA , (122) for the former case, and (d) σγA = P (d) (γ) × σγA . (123) for the latter cases. The cross sections Eqs. (122) and (123) are compared with that of the charged current weak process weak σνA = G2F Eν MA . 2 (124) (d) weak Since σγA is much larger than σνA , by a factor 1014 or more, σγA in Eqs. (122) and (123) can be as large Eq. (124), if P (d) is around 10−14 . Accordingly, 10−14 or 10−15 is the critical value for the photon neutrino process to be relevant and important. If the value is larger, then the reaction that is dectated to vanish due to Landau-Yang’s theorem manifests with a sizable probability. 28 6.2 ν +γ →ν +γ The probability P (d) is of the order of αGF and almost independent of time. The probability in this order has been considered vanishing, and this process has not been studied. If the magnitude is sizable, these neutrino processes should be included in astronomy and others. The process ν + ν̄ → γ + γ is almost equivalent to ν + γ → ν + γ, and we do not study in this paper. A system of high temperature has many photons, and a neutrino makes a transition through its collision with the photons. The probability is determined by the product between the number of photons nγ and each probability P (d) (γ)nγ . (125) In a thermal equilibrium of higher temperature, the density is about nγ = (kT )3 , (126) and we have the product for a head-on collision P nγ = (GF α)2 6.2.1 2 p2ν (kT )3 8π 9/2 m2γ σ 1/2 12 The sun In the core of the sun, R = 109 meters, kT ≈ 2 keV, (127) and the solar neutrino has the energy around 1 − 10 MeV. The photon’s energy distribution is given by the Planck distribution, and the mean free path for the head-on collision is l= 1 P (d) (γ)nγ σγA = 5 × 1015 meters, (128) where σγA = 10−24 cm2 , mγ = 1 eV and nA = 1029 /cm3 are used. The value is much longer than the sun’s radius. For the neutrino of higher energy, we use Eq. (97), and have  0 2 1 pν 9 meters, (129) l = (d) = 6.2 × 10 P (γ)nγ nA pν p0ν = 10 GeV, 29 thus the length exceeds the sun’s radius for pν > 25 GeV. The high energy neutrino does not escape from the core if the energy is higher than around 25 GeV. 6.2.2 Supernova In supernova, the temperature is as high as 10 MeV and the probability becomes much higher. We have l= 1 P (d) (γ)nγ nA = 1m, (130) pγ = 107 , nA = 1025 /cm3 mγ or l = 104 m, pγ = 105 , nA = 1024 /cm3 . mγ (131) The mean free path becomes smaller in the lower matter density region. Thus in the region of small photon’s effective mass, the neutrino does not escape but loses its majority of energy. This is totally different from the standard behavior of the supernova neutrino. 6.2.3 Neutron star If magnetic field is as high as 109 [Tesla], then the probability becomes large. The energy of the neutrino is transfered to the photon’s energy. 6.2.4 Low energy reaction In low energy region, P (d) behaves as Eq.(96) and −3.5 σγA → CEphoton , Ephoton → 0. (132) The effective transition probability P (d) nγ and the cross section depend on the photon density. 30 6.3 ν + (E, B) → ν′ + γ The radiative transition of one neutrino to another lighter neutrino and photon in the electromagnetic field occurs with the probability P (d) . Because P (d) is not proportional to T but almost constant, the number of parents decreases fast at small T , and remains the same afterward without decreasing. Now the photon in the final state reacts with matter with sizable magnitude and the probability of whole process is expressed with the effective cross section. 6.3.1 High energy neutrinos The transition probability of the high energy neutrino in the magnetic field B [Tesla] and electric field E [V/meter] are PB = 4α  ecB me c2 PE = 4α  eE me c2 2 2 1 G2F −3/2 1 3 π √ pν C(mν ) 2 , 2 2 4mγ σγ me (133) G2F −3/2 1 1 3 π √ pν C(mν ) 2 . 2 2 4mγ σγ me (134) √ (135) For the parameters, mγ c2 = 10−9 eV, σγ = 10−13 meters, they become in the magnetic field PB = 6.4 × 10−27  B B0 2 pν (0) pν !3 , (136) p(0) ν = 10 MeV, B0 = 1 Tesla. and in the electric field PE = 2.8 × 10−20  E E0 2 pν (0) pν !3 , (137) 3 p(0) ν = 10 MeV, E0 = 10 GV/m, Neutrino energy dependences are shown in Figs. 3 (ν + B) and 4 (ν + E). 31 Probability 10 -15 10 -18 10 -22 0 5 10 Neutrino Energy [GeV] Figure 3: Eν dependence of the probability Eq. (136) is shown. B = 10 Tesla is used for the calculation. Probability 10 -12 10 -15 10 -19 0 5 10 Neutrino Energy [GeV] Figure 4: Eν dependence of the probability Eq. (137) is shown. E = 100 GV/m is used for the calculation. 32 6.3.2 Low energy neutrinos The transition probability of the low energy neutrino are  2 2 ecB GF −3/2 1 1 PB = 4α π √ pν C(mν ) 2 2 me c 2 ǫ σγ me  2 2 eE 1 GF −3/2 1 PE = 4α π √ pν C(mν ) 2 . 2 me c 2 ǫ σγ me They become in the typical situations √ ǫ = 10−20 , σγ = 10−13 m, pν = 1 eV, E0 = 103 GV/m,  2 E −38 . PE = 6.4 × 10 E0 (138) (139) (140) The probabilities Eqs. (136), (137), and (140) are caused by the overlap of waves on the parent and daughters. The phases of waves become cancelled at small mγ or ǫ, and more waves are added constructively then. Thus the effect become larger as they become smaller. The probability shows this and is inversely proportional to m2γ or ǫ. They become extremely large as mγ → 0 or ǫ → 0. 6.4 Table of processes The P (d) could be tested in various neutrino processes of wide energy regions. Neutrino source Solar neutrino Reactor neutrino Cumulonimbus cloud Accelerator neutrino Cosmic neutrino Relic neutrino Energy 1–10 MeV 3–8 MeV 1–10 MeV 0.5–50 GeV ∼ 106 GeV 1 meV Flux 1015 /m2 s 1020 /s 1015 /m2 s(solar ν) 1020 10−9 GeV/cm2 s sr 1020 /year B 1–5 Tesla 1–5 Tesla 10−5 Tesla 1–5 Tesla 10−5 Tesla – E – – 3 10 GV/m – – 5 10 GV/m The diffractive term P (d) becomes substantial in magnitude at high energy or fields. So this process may be relevant to the neutrino of high energy or at high fields, and may give new insights into or measurability of the following processes. 33 1. The accelerator neutrino has high energy and total intensity of the order 1020 . At B = 10 Tesla and Pν = 1(10) GeV, we have P ≈ 10−18 (10−15 ), and using the laser of E0 = 103 GV/m at Pν = 1(10) GeV, we have P ≈ 10−16 (10−13 ). The neutrino from the accelerator may be probed by detecting the final gamma rays. For example, at the beam dump of LHC we may set up laser to detect the neutrino interaction. 2. For the reactor neutrino, of the flux ≈ 1020 /s per reactor, a detector of high magnetic field of the order 10 Tesla, or of high intensity laser, may be able to detect the neutrino. 3. Direct observations of solar neutrinos, which have the energy in 0.5 MeV to 10 MeV and flux around 1015 /m2 sec, using the detector with strong magnetic field similar to that for axion search [51] would be possible. 4. In Supernova or neutron star, the neutrino photon reaction would give a new important process, because the final photon interacts with matter strongly. As to the detectability of neutrinos from the sun, 10 GeV is the threshold from the sun, while they from SN interact with 10 MeV photons. 5. We find that P (d) becomes maximum at ζ = π2 from Eq.(118), and may apply this effect to enhancing the neutrino flux. The term also has the momentum dependence, which may be exploites for “optical” effect of neutrino through the photon interaction. 6. The photon neutrino reaction may be useful for the relic neutrino detection. The reaction rate may be enhanced with such method as the neutrinos mirrors that collect them.[52] We may take advantage of the above effect. 7. The probability becomes huge at extremely high energy. So, this process may be relevant to ultra-high energy neutrino process. [53, 54, 55]), 8. Neutrino may interact with electromagnetic fields in Cumulonimbus cloud. 8-1. Lightening has total energy of the order 900 MJ ≈ 109 CV and current 106 A in a short period. cB at a radius r = 1 cm is 1.5 × 109 N/C. Assuming E = cB = 1.5 × 109 N/C, and mγ c2 = 10−11 eV, Pν = 10 MeV we have PE+B ≈ 10−15 . The neutrino inevitably loses its energy, and the photons of the continuous spectrum are emitted. This may be related with the upper-atmospheric lightening [56] . 8-2. The gamma rays observed in Cumulonimbus cloud, [57], may be connected with the diffractive component. 9. Primordial magnetic fluctuations with zero frequency [58] may have interacted with neutrinos before neutrino detachment from the hot neutrino 34 plasma(>GeV temperature) during the Big Bang. Such signatures may be carried by neutrinos (which is now relic neutrinos). 7 Summary and the future prospect We have found that the photon interaction expressed by the total derivative Eqs. (13) or (86), which are derived from the triangle diagram in the standard model, causes the unusual transitions characterized by the time-independent probability. The interaction Lagrangian of this form does not give rise to any physical effect in classical physics, because the equation of motion is not affected. In quantum mechanics, this assertion is correct for the transition rate Γ0 . However, this does not apply to the diffractive term P (d) , which manifests the wave characteristics of the initial and final waves. Our results show that P (d) is relevant to experiments and important in understanding many phenomena in nature. The neutral particles do not interact with the photon in classical mechanics. In quantum field theory, the vacuum fluctuation expressed by triangle diagram gives the effective interaction to the neutral particles such as 1+ meson → γγ and ν +γ(B, E) → ν +γ. However, they have vanishing rates due to Landau-Yang-Gell-Mann’s theorem. P (d) does not vanish, nevertheless, and holds unusual properties such as the violation of the kinetic-energy conservation and that of Lorentz invariance. Furthermore the magnitudes become comparable to or even larger than the normal weak processes. Accordingly, the two photon or two gluon decays of the neutral axial vector mesons composed of a pair of electrons or quarks come to have finite decay probabilities. They will be tested in experiments. The neutrino photon processes, which have been ignored, also have finite probabilities from P (d) . It will be interesting to observe the neutrino photon processes directly using electric or magnetic fields, or laser and neutrino beams in various energy regions. The diffractive probability P (d) would be also important for understanding the wide neutrino processes in earth, star, astronomy, and cosmology. The diffractive probability P (d) is caused by the overlap of wave functions of the parent and decay products, which makes the interaction energy finite and the kinetic energy vary. Consequently the final state has continuous spectrum of the kinetic energy and possesses a wave nature unique to the waves. The unique feature of P (d) , i.e., independence of on the time-interval T , shows that the number of parents, which decrease like e−Γt in the normal 35 decay, is constant now. The state of parent and daughter is expressed by the quasi-stationary states, which is expressed by the superposition of different −iEt energies and different from the normal stationary state of the form e ~ ψ(xi ). The probability of the events that the neutrinos or photons are detected is computed with S[T ] that satisfies the boundary condition of the physical processes. Applying S[T ] we obtained the results that can be compared with experiments. The pattern of the probability is determined by the difference of angular velocities, ω = ω E − ω dB , where ω E = Eν /~ and ω dB = c|~pν |/~. The quantity ω takes the extremely small value m2 c4 /(2E~) for light particles such as neutrinos or photons [49] in matter. Consequently, the diffraction term becomes finite in the macroscopic spatial region of r ≤ 2πE~c . m2 c4 This allows us to introduce a new class of experimental measurement possibilities of the deployment of photons to detect weakly interacting particles such as neutrinos. Because the modern technology on the electric and magnetic fields and laser, a large number of coherent photons are possible and the effects we have derived may have important implications in detecting and enhancing the measurement of neutrinos with photons. We see a variety of detectability opportunities that have eluded attention till now. These happen either with high energy neutrinos such as from cosmic rays and accelerators; or in high fields (such as intense laser and strong magnetic fields ). In the latter examples, neutrinos from reactors, accelerators, the sun, supernovae, thunder clouds and even polar ice may be detected with enhanced probabilities using intense lasers. We also mention the probability estimate for the primordial relic neutrinos and embedded information in them. Acknowledgments This work was partially supported by a Grant-in-Aid for Scientific Research (Grant No. 24340043). 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Hosono, J.Geophys.Res.107,4324(2002); H.Tsuchiya, et. al, Phys.Rev.Letters. 99165002(2007) [58] T.Tajima, S. Cable, K.Shibata, and R. M. Kulsrud, Ap. J. 390, 309 (1992). [59] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. Photons and atoms: Introduction to Quantum Elctrodynamics (John Wiley & Sons, Inc. New York, 1989). Appendix A Wave functions including CG coefficients are 1 1 φ0 : |J = 0, P~ = 0i = ( √ |1/2, 1/2; −1i − √ |1/2, −1/2; −1i 3 6 1 + √ | − 1/2, 1/2; −1i + | − 1/2, −1/2; −1i, (A.1) 6 1 1 φ1 : |J = 1, P~ = 0i = ( √ |1/2, 1/2; 0i − | + 1/2, −1/2; −1i 2 2 1 (A.2) − | − 1/2, 1/2; 1i, 2 φ2 : |J = 2P~ = 0i = |1/2, 1/2; 1i. (A.3) Appendix B The integration over a semi-infinite region of satisfying the causality in two dimensional variables, x1 (t1 , x1 ), x2 (t2 , x2 ) Z ∂ ∂ I= d2 x1 d2 x2 f (x1 − x2 )g(x1 + x2 ), (A.4) ∂t1 ∂t2 λ1 ,λ2 ≥0 40 where the integrands satisfy g(t+ , x+ = ±∞) = 0, f (t− , x− = ±∞) = 0 (A.5) is made with the change of variables. Due to Eq. (A.5), I would vanish if the integration region were from −∞ to +∞. We write I with variables 2 x+ = x1 +x , x− = x1 − x2 , and have 2 I = I1 − I2 , Z ∂2 I1 = d2 x− f (x− )d2 x+ ( 2 g(x+ )), ∂t+ λ+ ≥0 Z ∂2 I2 = d2 x+ g(x+ )d2 x− ( 2 f (x− )), ∂t− λ+ ≥0 (A.6) where λ+ = x2+ . It is noted that x+ is integrated in the restricted region but x− is integrated in whole region. Thus Z ∂ g(x+ ))|x+ =x+,min , (A.7) I1 = − d2 x− f (x− )dt+ v 2 ( ∂x+ I2 = 0, where the functional form g(x+ ) = g(x+ − vt+ ) was used. I is computed with the slope of g(x+ ) at the boundary x+ = t+ . We apply this method for computing four dimensional integrals. Appendix C In the case of the low energy region the photon has no effective mass, but is expressed by the index of refraction very close to unity in dilute gas, n = 1 + ǫ. (A.8) Accordingly, the angular velocity in φ̄c (δt) is given by ω = (1 + ǫ)Pγ − Pγ = ǫPγ . (A.9) ǫ = 0.000292, 0C, 1 atomspheric pressure, (A.10) ǫ in air is and ǫ in dilute gas becomes extremely small of the order 10−14 − 10−15 . Consequently the integrand in P (d) is proportional to P2γ ǫ . 41 Appendix D Notations :MKSA unit It is convenient to express the Lagrangian with MKSA unit to study quantum phenomena caused by macroscopic electric and magnetic field [59]. The Maxwell equations for electric and magnetic fields are expressed with the dielectric constant and magnetic permeability of the vacuum, ǫ0 , µ0 which are related with the speed of light, c, c= √ 1 ǫ0 µ0 (A.11) The zeroth component x0 in 4-dimensional coordinates xµ , is x0 = ct; t = sec. (A.12) The Maxwell equations in vacuum are, ~ ·E ~ = 1 ρ(x), ∇ ǫ0 ~ ·B ~ = 0, ∇ ~ ~ = − ∂B , ~ × ( 1 E) ∇ c ∂x0 1~ ~ ×B ~ = µ0~j(x) + ∂( c E) , ∇ ∂x0 (A.13) where the charge density and electric current, ρ(x), ~j(x), satisfy c ∂ ~ ~j(x) = 0. ρ(x) + ∇ ∂x0 (A.14) Using the vector potential ~ ), Aµ = (A0 , −A (A.15) we write ∂ ~ 1~ ~ 0 (x), E(x) = − A(x) − ∇A c ∂x0 ~ =∇ ~ × A(x), ~ B 42 (A.16) and the Lagrangian density of electric and magnetic fields LEM = − 1 1 1 ~2 ǫ0 ~ 2 − B . Fµν F µν = E 4 µ0 2 2µ0 The Lagrangian density of electronic fields is,   ∂ ∂ 2 Le = ψ̄(x) γ0 ic~ − γl ic~ − mc ψ, ∂x0 ∂xl (A.17) (A.18) and that of QED is LQED = Le + LEM + +ecA0 ψ̄γ0 ψ − ecAl ψ̄γl ψ. (A.19) Canonical momenta and commutation relations from Eqs. (A.17) and (A.19) are ∂ L = i~ψ † (x), ∂ ψ̇(x) ∂ 1 Ei , Πl (x)(x) = L= µ 0 c2 ∂ Ȧi (x) {ψ(x1 ), ψ † (x2 )}δ(t1 − t2 ) = δ(~x1 − ~x2 )δ(t1 − t2 ), 1 [Ai (x1 ), Ȧj (x2 )]δ(t1 − t2 ) = i~ δij δ(~x1 − ~x2 )δ(t1 − t2 ), ǫ0 πψ (x) = (A.20) (A.21) where the gauge dependent term was ignored in the last equation. Thus the commutation relations for electron fields do not have ~, and those of electromagnetic fields have ~ and the dielectric constant ǫ0 . ǫ~0 shows the unit size in phase space. Accordingly the number of states per unit area and the strength of the light-cone singularity are proportional to 1/ǫ0 . The fields are expanded with the wave vectors as ω̃0 1/2 ~ d~k ( ) (u(k, s)b(~k, s)e−ikx + v(~k, s)d† (~k, s)eikx ), 3/2 0 ~ (2π) k (k) s (A.22) Z X ~ d~k ( )1/2 (ǫi (~k, s)a(~k, s)e−ikx + ǫ∗ (~k, s)a† (~k, s)eikx ), Ai (x) = 3 1/2 ((2π) 2k0 ) ǫ0 s (A.23) ψ(x) = XZ 43 where kx = k0 x0 − ~k~x, q mc k0 (~k) = ~k 2 + ω̃02 , ω̃0 = . ~ (A.24) The creation and annihilation operators satisfy {b(~k1 , s1 ), b† (~k2 , s2 )} = δ(~k1 − ~k2 )δs1 s2 , [a(~k1 , s1 ), a† (~k2 , s2 )] = δ(~k1 − ~k2 )δs s . 1 2 (A.25) (A.26) The spinor is normalized as X s γ · k + ω̃0 , u(~k, s)ū(~k, s) = 2ω̃0 (A.27) and the light-cone singularity is ∆+ (x1 − x2 ) = Z d~k 1 −ik·(x1 −x2 ) e = 2i δ(λ)ǫ(δx0 ) + · · · , (A.28) (2π)3 2k0 4π where the less-singular and regular terms are in · · · . The action Z S = dx(LQED + ψ̄(x)γµ (1 − γ5 )ψ(x)J µ (x)), (A.29) GF J µ (x) = √ ν̄(x)γ µ (1 − γ5 )ν(x). 2 governs the dynamics of electron, photon and neutrino. Integrating ψ(x) and ψ̄(x), we have Det(D + J µ γµ (1 − γ5 )) and the effective action between neutrino and photon   Z ∂ β ∂ σ ∂ e2 ν (A.30) J (x)ǫαβρσ A A , Sef f = 2 2 f (0) dx 8π c~ ∂xν ∂xα ∂xρ where f (0) = 1 . 2ω̃02 44 (A.31) The magnetic field and electric field in the 3-rd direction, and laser field expressed by the external fields, µν µν Fext = ǫ03µν B, Fext = ǫ12µν E , Aµlaser (x) c (A.32) are substituted to the action Eq .(A.30), and we have the actions Z ∂ ∂ σ Sef f,B = gB ǫ03ρσ dx [ν̄(x)γ ν (1 − γ5 )ν(x) A (x)], (A.33) ∂xν ∂xρ Z ∂ σ ∂ [ν̄(x)γ ν (1 − γ5 )ν(x) A (x)], (A.34) Sef f,E = gE ǫ12ρσ dx ∂xν ∂xρ Z ∂ ∂ σ µν Sef f,laser = gl ǫµνρσ dx [ν̄(x)γ ν (1 − γ5 )ν(x)Flaser (x) A (x)], (A.35) ∂xν ∂xρ where the coupling strengths are e2 GF f (0)B √ , 2 2 8π c~ 2 2 E GF e gE = 2 2 f (0) √ , 8π c~ c 2 2 e GF gl = 2 2 f (0) √ . 8π c~ 2 gB = (A.36) (A.37) (A.38) The wave vectors are connected with the energy and momentum E(~k) = ~ck0 (~k), ~p = ~c~k. Appendix E (A.39) Integration formulae The integrals Z ∂ ∗ I0 = d4 x1 d4 x2 ρs (x0− )( 0 )2 e−χ(x1 )−χ(x2 ) δ((λ− )2 )eip·x− , ∂x+ λ =0 Z + ∂ ∗ Ii = d4 x1 d4 x2 ρs (x0− )( i )2 e−χ(x1 )−χ(x2 ) δ((λ− )2 )eip·x− , ∂x+ λ+ =0 x0 − x02 , ρs (x0− ) = 1 + i 1 Eσγ 45 (A.40) (A.41) where the spreading in the transverse direction in Eq. (58) leads to ρs (x0− ) in the right hand sides. Changing the variables to x+ and x− , we have χ(x1 ) + χ(x2 )∗ = χ(+) (x+ ) + χ(−) (x− ) + χ(+−) (x+ , x− ), 1 1 χ(+) (x+ ) = ((x+,l − v(x+,0 ))2 + ((~x+ )2T ), σγ 2σT (+) 1 1 χ(−) (x− ) = ((~x− )2T ), (x−,l − v(x−,0 ))2 + 4σ γ 8σT (+) 1 ((~x+ )T (~x− )T ), χ(+−) (x+ , x− ) = σT (−) (A.42) where the wavepacket sizes in the transverse direction are 1 1 1 = + , 2σT (+) 2σT (1) 2σT (2)∗ 1 1 1 = − . 2σT (−) 2σT (1) 2σT (2)∗ (A.43) The wavepacket expands in the transverse direction and the size σT (+) is given by σT (+) = σγ i − (x0 ) + O(x2− , x2+ ). 2 4E − (A.44) The off-diagonal term χ(x+ , x− ) gives small corrections and is ignored. We have 1 I0 (+)I(−), (A.45) σγ Z Z 2 1 x+ + 2 + −χ(x+ ) (~ T) = dxl+ π(2σT (+))2 l , I0 (+) = dxl d ~xT e + x+ xl x0 =|~ x+ | Z + Z 4σT (+) iωx0− I(−) = d4 x− ρ(x0− )e−χ(x− ) δ(λ2− )eipx− = dx0− ρ(x0− ) e , x0− I0 = and I0 = π Z 0 T dxl+ 1 xl+ Z +xl+ −xl+ = −i2π 2 (σγ2 log(ωT ) + 16σT (+) dx0− ρ(x0− ) σl σγ ). 4ωE 46 3 0 eiωx− x0− (A.46) Eq. (A.46) is composed of the log T term and constant. At ω ≈ 0, the latter is important and at a larger ω, the former is important. Similarly Ii = Ii (+)I(−) (A.47) Z i q i x ′ Ii (+) = = ± (x0+ )2 − (xl+ )2 − (xi+ )2 dx0+ dxl+ d~xi+ + e−χ(x+ ) σT (+) x+ Z 2σT (+) = π dx0+ , x0+ and Ii = π Z 0 T dx0+ 1 x0+ Z x0+ x0+ = −i2π 2 (σγ2 log(ωT )). In high energy regions, ω = m2γ , 2E iωx0− 0 0 2e dx− ρ(x− )8σT (+) x0− (A.48) and I0 = −i2π 2 (σγ2 log(ωT ) + σγ ), 4m2γ IT,i = −i2π 2 (σγ2 log(ωT )), Il = I0 . (A.49) (A.50) (A.51) In low energy regions, ω = ǫp, and I0 = −i2π 2 (σγ2 log(ωT ) + Ii = −i2π 2 (σγ2 log(ωT )), Il = I0 . 47 σγ ), 4ǫp2 (A.52) (A.53) (A.54)