1 Multi messenger Astrophysics

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1 1 Multi messenger Astrophysics Definition for the field of research discussed here is multi messenger astrophysics, that is the study of the sources and environments in the cosmos with the (simultaneous) observations of different particles that act as messengers, or carriers of information. Four cosmic messengers exist: photons, neutrinos, cosmic rays and gravitational waves. Photons The Photon of course has been the fundamental Nuncius Sidereus ( Messenger from the stars ) since the prehistory of humankind, and nearly everything we know about the universe has been obtained from observations of light of different wavelengths. This is reflected in the fact that in common language the words observation and seeing describe decoding of the information contained in the light we detect with out eyes. Crucial steps have been the development of the optical telescope at the beginning of the 17th century, and then the opening of different windows in photon wavelength (or equivalently energy), observed with always more sensitive instruments. The recent development of γ ray telescopes for observation of very high energy photons represents a recent important chapter in this history. Cosmic Rays Cosmic rays are relativistic charged particles, that arrive at the Earth with an approximately isotropic flux, because their trajectories are bent by (galactic and extragalactic) magnetic fields. They have a very broad energy range that extends more than ten orders of magnitude, from particles that are only moderately relativistic, to particles with an energy of order ev. The discovery of cosmic rays approximately hundred years ago played a crucial role in the early development of Particle Physics when these particles were the only source of very high energy particles before the development of particle accelerators. This discovery revealed the existence of a hidden High Energy Universe, an ensemble of astrophysical objects, environments and mechanisms capable of producing particles of extraordinarily high energy. Neutrinos The idea of neutrino astronomy, that is the use of neutrinos as cosmic messengers from astrophysical sources, is as natural as profoundly fascinating, and was immediately formulated as soon as the neutrino was discovered. Neutrinos, as photons, travel along straight lines, and their detection allows a direct imaging of their sources. Because of their very small interaction cross section, neutrinos can emerge from deep inside astrophysical objects, and carry information that is very different from what can be inferred from the photons emitted by the same source. The dream of neutrino astronomy is now a reality. Three sources have been identified: The Sun, Supernova 1987A and the Earth (with the detection of geo (anti) neutrinos generated by the radioactive decays of unstable nuclei inside the Earth). More recently (in 2013), at much higher energy (E ν 30 TeV) the IceCube detector at the South Pole has detected, a diffuse flux of extraterrestrial neutrinos generated by astrophysical sources, 1

2 that have not yet been identified. Gravitational Waves The existence of gravitational waves, ripples in curvature of space time that propagate at the speed of light, is a key prediction of General Relativity. Large mass astrophysical bodies undergoing large accelerations should emit waves that are in principle directly observable by detector at the Earth. The existing detectors have now the sensitivity to observe the first and their results are eagerly awaited. The study of Gravitational Waves will not be discussed here, What is important to stress is that the sources of gravitational waves are very likely also copious emitters of high energy particles in the form of photons, neutrinos and cosmic rays, and therefore these fields are intimately related. One example of several possible multi messenger studies is the search for the simultaneous emission of gravitational waves from Gamma Ray Bursts events (very energetic explosive events observed in other galaxies that are also predicted to be observable as high energy neutrino emitters). 2 Low energy neutrino detection The detection of neutrinos with energy between MeV (that here will be considered as low ) have given results that are extraordinarily interesting. The most remarkable results are: The successful observations of solar neutrinos (E ν MeV) The observations of supernova neutrinos (E ν MeV). The observations of geophysical neutrinos (E ν MeV, the endpoint of the spectrum of decay of radioactive nuclei). The importance of the solar and supernova observations has been recognized by the Nobel prizes of Raymond Davis and Masatoshi Koshiba in 2002, and the Nobel prize of Arthur McDonald and Takaaki Kajita in The observations of solar neutrinos, has also resulted in the discovery of neutrino flavor oscillations. The field of solar neutrino detection is now mature, but far from exhausted, and several studies remain interesting. Perhaps the most interesting challenge is the measurement of the neutrinos created in the CNO-cycle [a subdominant (in the Sun) set of reactions for the fusion of hydrogen into helium]. The handful of neutrinos events from SuperNova SN 1987A observed by Kamiokande and IMB (and perhaps also Baksan) [One should also remember the puzzling and controversial results of the LSD detector in the Mont Blanc Tunnel] has given us an extraordinary amount of information on the physics of gravitational collapse. The existing (and even more, the future) detectors for SN neutrinos (will) have much higher sensitivity and therefore will yield a much more detailed and valuable information. The obvious problem is that one has to wait for such an explosion to occur, but the results will be so interesting that there are no doubts that one has to be well prepared. Also of the highest interest has been the detection (performed by KamLAND in Japan, and BOREXINO in the Gran Sasso laboratory) of geo neutrinos, the ν e generated by 2

3 radioactive decays of unstable nuclei inside the Earth. This study has the fascinating potential to give us important information about the internal structure of the Earth. Solar, SuperNova and geo neutrinos have energies that are approximately in the same energy range, that also overlaps with the energy range of the anti neutrinos generated in nuclear reactors Reactor neutrinos have played a crucial role in the study of neutrino flavor oscillations, and have a very important potential for future studies. This has allowed several detectors (Kamiokande, Super Kamiokande, BOREXINO) to obtain important results for different problems, including measurements of higher energy atmospheric and accelerator neutrinos. The possibility of developing multi purpose detectors is clearly a fundamental consideration for the design of future instruments. 3 The High Energy Universe Three particle messengers : cosmic rays, gamma rays, and neutrino can give us information can give us information about the High Energy Universe. In most cases the astrophysical sources directly accelerate to very high energy charged particles (protons, nuclei and electrons). These particles can then generate photons and neutrinos when they interact with matter or radiation fields inside or near the accelerators. The observations of the three particle messengers give complementary information about the nature and structure of the astrophysical sources and the mechanisms that operate in them. [...] 3

4 4 Hadronic mechanism of production Hadronic mechanism for the production of high energy gamma rays and neutrinos: (i) protons (or ionized nuclei) are accelerated to relativistic energies by some electromagnetic mechanism. (ii) These relativistic particles interact with some target (gas of radiation fields) creating hadrons (such as charged and neutral pions, and kaons kaons). (iiia) The decays of neutral pions generate gamma rays vias π γγ. Additional photons are created in the decay of other mesons (such as η and η ). (iiib) The Weak decays of the hadrons generate neutrinos directly (as in π + ν µ µ + ) or indirectly in the subsequent decay of muons produced in the same hadronic decays (µ + ν µ ν e e + and charge conjugate channel). Any environment where relativistic hadrons are accelerated or stored by magnetic fields is therefore a high energy neutrino source. 5 Two body decay of a particle Let is consider the two body decay π γγ Decay of π in center of mass system (isotropic decay) Energy in the c.m. Isotropic distribution in c.m. frame: E γ = m π 0 2 (1) Normalization: 2 particles in the final state +1 1 γ d cos θ = 1 (2) d cos θ γ d cos θ = 2 (3) Calculation of the spectrum of the photons in the laboratory frame (where the pion has energy E π ). The expression of the energy of the photon in th lab system can be calculated with a Lorentz boost: E γ = γ [E γ + β π E γ cos θ ] (4) (The angle θ is with respect to the boost direction). The Lorentz parameter γ = E π /m π, and E = m π /2): E γ = E π [1 + β π cos θ ] m π m π 2 E γ = E π 2 [1 + β π cos θ ] (6) The limits of the photon energy can be set putting cos θ = 1: E min,max γ (5) = E π 2 [1 β π] (7) 4

5 In the limit of an ultrarelativistic pion one finds that the limit are β π = 1 m2 π E 2 π 1 (8) E min γ = 0, E max γ = E π. (9) The spectrum can be calculated simply with a change of variable: γ = [ ] 1 γ deγ de γ d cos θ d cos θ (10) with the result: γ de γ = 2 E π β π (11) That is the spectrum is flat, between the kinematical limits of Eq. (??). It is simple to check that the Normalization: de γ γ de γ = 2 (12) de γ E γ γ de γ = E π (13) Ultrarelativistic limit: between limits : Useful to introduce a scaling variable γ de γ = 2 E π (14) E min γ = 0 (15) E min γ = E π (16) y = E γ E π (17) γ dy = 2 β π (18) Ultrarelativistic limit: Note: scaling function: y min = 1 β π 2 y max = 1 + β π 2 (19) (20) γ dy = 2 (21) y min = 0 (22) y max = 1 (23) 5

6 6 Charged pion decay. Branching ratios The charged pion has two modes of decay possible: π + µ + ν µ π + e + ν e The decay happens nearly entireli into muons, with the electron mode havinbg a branching ratio of order The expression for the rate of decay into the mode π + l + ν l is: Γ π µν = G2 F V ud 2 8π f 2 π m π m 2 l ( ) 2 1 m2 l m 2 (24) π Note the factor m 2 l that is the consequence of the (V-A) stricture of the Weak interactions. The strong suppression of the electron mode is an outstanding manifestation of the V A nature of the charged current weak interactions, and a clear illustration of the difference between the chirality and the helicity. Let us consider (see fig.??) the decay of pions at rest: π + l + + ν l. S S e e + p e e + + e + pe Figure 1: Charged pion decay in the rest frame of the pion. The V-A structure of weak interactions requires the emitted ν l to be of left handed chirality. For m ν 0 this also means that it has the left handed (or negative) helicity (spin antiparallel to its momentum). Conservation of total angular momentum then requires l + to have negative helicity. However, the l + is an antiparticle, and again for the V-A structure of weak interactions it must be produced in a state of right handed chirality. Therefore the amplitude of the process must be proportional to the admixture of left handed (negative) helicity for a right handed chirality charged lepton, that is proportional to its mass: A(π + l + + ν l ) m l. Including phase space effects (m 2 π m 2 l ) one has the expectation: R π Γ(π+ e + + ν e ) Γ(π + µ + + ν µ ) = ( me m µ ) 2 ( ) m 2 2 π m e = (25) m 2 π m 2 µ In agreement (after including a 4% radiative correction) with the experimental value: R π = (1.230 ± 0.004) Charged pion decay. Kinematics General case of a two body decay: x a + b 6

7 Energy in the c.m. frame: E a = (m2 x + m 2 a m 2 b ) 2 m x (26) E b = (m2 x + m 2 b + m2 a) 2 m x (27) It is easy to check that energy and momentum are conserved (the two particles are produced in the c.m. frame with equal and opposite momenta). Ea + Eb = m x (28) E a m 2 a = Eb m2 b (29) Appplying this general rules to the pion decay (and using the approximation m ν 0) one finds: ( ) E µ = m π m2 µ m 2 π = m π 2 (1 + r π ) (30) and E ν = p = m π 2 ( 1 m2 µ m 2 π ) = m π 2 (1 r π ) (31) The factor r π is: r π = m2 µ m 2 = (32) π The important point is that the energy of the pion mass is not shared equally between the two particles in the final state. The muon, that has a mass close to the pion, takes away a fraction (1 + r π )/2 0.79, while the neutrino carries away a fraction (1 r π )/ To compute the spectrum in the laboratory frame (where the pion has energy E π ) one can perform a Lorentz transformation: whith γ = E π /m π and β = p π /E π : E lab = γ (E + β p cos θ ) (33) The decay is isotropic in the rest frame (because the pion is a scalar particle of spin zero), and accordingly the distribution of the angle cos θ is flat: d cos θ = 1 2 The distribution in E lab can be calculated as: (34) de lab = d cos θ d cos θ. (35) de lab One obtains the result: That is the distribution is flat. de lab = γ β π p (36) 7

8 We still have to compute the kinematical limits that is the minimum and maximum energy of the particles in the lab. frame. The limits are obviously obtained for cos θ = 1: E min,max µ = γ ( E µ β π p ) = E π m π m π 2 [(1 + r π ) β π (1 r π )] Similarly: = E π 2 [(1 + r π) β π (1 r π )] E min,max ν = γ (p β π p ) = E π m π m π 2 [(1 r π ) β π (1 r π )] = E π 2 [(1 r π) β π (1 r π )] It is again possible to study the spectra in term of the fractional energy y ν,µ = E ν,µ /E π. In the limit of an ultrarelativistic pion, one has that the kinamtically allowed range of the y ν,µ are: y ν [0, (1 r π )] (37) y µ [r π, 1] (38) The spectra of neutrinos and muons created in pion decay are shown in Fig.??. One can see how the spectra take an asymptotic form for E π m π. 8

9 E Π 0.2, 0.5, GeV 3 dy Figure 2: Spectra of muon and neutrinos in the laboratory in terms of the scaling variable y 9

10 8 Muon decay Decay of unpolarized muons Matrix element: µ ν µ + e + ν e (39) To compute the distribution of energy (and direction) of the particles in the final state, one has to take into account the dynamics of the decay, that is controlled by the matrix element. Note that the muon has spin 1/2. One can consider the decay of a polarized muon (with spin described by the 4 vector s µ ). Summing over the different polarizations of the final state particles, the matrix element can be expressed in terms of the 4 momenta of the particles and the 4 vector of the muon spin. For the decay of a polarized muon with spin 4-vector s µ : M 2 (p νµ p e ) (p µ p νe ) (40) M 2 (p νµ p e ) [(p µ p νe ) m µ (s µ p νe )] (41) The differential rate for muon decay can be calculated in the standard way: dγ = 1 2 m µ M 2 dφ (42) In this equation M is the matrix element for the decay, and dφ is the differential phase space element for three bodies in the final state, that written explicitely is: [ d 3 ] [ p 1 d 3 ] [ p 2 d 3 ] p 3 dφ = (2π) 3 2E 1 (2π) 3 2E 2 (2π) 3 2E 3 (2π) 4 δ (4) [p µ p 1 p 2 p 3 ] (43) Integrating over the entire phase space one obtains the total decay rate: Numerically: Γ µ = 1 τ µ = G2 F m5 µ 192 π 3 (44) τ µ s (45) ( ) E µ Eµ l µ = c τ µ β µ 6.23 km (46) m µ GeV The inclusive spectra of the neutrinos can be calculated obtaining dγ/de j after integration in all other kinematical variables. Note: the phase space for a final state of three particles is described by 5 variables (one has three 3 momenta, that is 9 variables, and 4 constraints that come from the conservation of energy and momentum). For an inclusive spectrum, one choose the variable in the rest frame of the parent particle (the muon) the variables: {E, cos θ, ϕ}, that is the energy of the particle, he angle θ with respect to the spin direction, and an azimuth angle, that by symmetry is clearly 10

11 dynamically irrelevant. To obtain dγ/(de d cos θ) one needs to integrate over the rest of the phase space. An outline of the integration follows here: One can perform the integration over d 3 p 3 using the delta function (obtaining p 3 = P p 1 p 2 ) dφ 3 body = = 1 (2π) 5 1 d 3 p 1 d 3 p 2 δ[e E 1 E 2 E 3 ] 8 E 1 E 2 E 3 [ ] E p p 21 + m m22 p m2 (47) 3 1 (2π) E 1 E 2 E 3 d 3 p 1 d 3 p 2 δ writing d 3 p 1 = dp 1 p 2 1 d cos θ 1 dϕ 1 d 3 p 2 = dp 2 p 2 2 d cos θ 12 dϕ 12 One can perform the integral over d cos θ 12 using the delta over energy with the result: d cos θ 12 [...] δ[e p p 21 + m m22 p p p 1p 2 cos θ 12 + m 2 3 ] = E 3 p 1 p 2 (48) Let us consider the decay of a muon (a µ ) at rest, and polarized with the spin along the +z direction. The spectra of the final state particles will depend on the direction of emission (with respect to the axis of the muon spin). One can define the variable x: x = 2E ν (49) m µ where E is energy in the rest frame of the muon. The variable x can vary in the interval: (0 x 1). Neglecting the neutrino and the electron mass one has the distributions: dx d cos θ (x, cos θ) = x 2 (3 2x) + cos θ x 2 (1 2x) (50) µ ν µ dx d cos θ (x, cos θ) = 6x 2 (1 x) + cos θ 6 x 2 (1 x) (51) µ ν e Note that in the approximation of neglecting the electron mass, the spectrum of the electron is identical to the spectrum of the ν µ. This can be understood simply inspecting the structure of the matrix element (that is symmetric for the exchange p e p νµ ). Integrating in angle, the cos θ term gives zero, and one finds the same result as in the decay of an upolarized muon: dx dx These two spectra are shown in Fig.??. (x) = 2x 2 (3 2x) (52) µ ν µ (x) = 12x 2 (1 x) (53) µ ν e 11

12 The averages of these distributions are: x νµ = 7 10 (54) x νe = 6 10 (55) Note that more energy goes into the ν µ (and less energy goes into the ν e ). Summing over all particles (and using the fact that x e = x νµ ) one obtains: x νµ + x νe + x e = 2 (56) (so all energy is accounted for). Integrating in energy (that is in x) one finds the results: d cos θ (x, cos θ) = 1 µ ν µ 2 cos θ 6 d cos θ (x, cos θ) = 1 µ ν e 2 + cos θ 2 These angular distributions have average: cos θ νµ = 1 9 (57) (58) (59) cos θ νe = (60) This indicated that the ν µ is emitted preferentially in the direction opposite to the spin of the muon, while the ν e is emitted in the direction of the spin of the muon. [Note that in the decay of of a µ +, one has the opposite, the ν µ is emitted in the direction of the spin of the µ + and the ν e is emitted preferentially in the opposite direction. This can be immediately understood on the basis of CP symmetry] The qualitative results are the following The ν µ carries more energy than the ν e The ν µ emitted preferentially in the direction of the µ spin. The ν e is emitted preferentially in the direction opposite to the µ spin. These results are easy to understand qualitatively in the basis of simple considerations about the the spins of the different particles in the final state In the decay of µ ν µ + ν e + e, the ν µ and the e have helicity 1, while the ν e has helicity +1. Figure?? wants to illustrate qualitatively why in µ decay the ν µ has a harder spectrum that the ν e (and similarly in µ + decay the ν µ has a harder spectrum that the ν e ). One can inspect a configuration where, in the c.m. frame, one particle (the ν µ in the left panel and the ν e in the right panel) is emitted with the maximum kinematically allowed energy (E = m µ /2). The other two particles must then be antiparallel, and the 12

13 1.5 dx Figure 3: Spectra of neutrinos in muon decay. The spectra are shown in the muon rest frame. The solid line is for the spectrum of ν µ in µ decay, The dashed line is for the spectrum of ν e in µ decay. sum of their energy must also be equal to m µ /2. Because of the (V A) structure of the Weak interaction, one has that the helicity of the e and the ν e are negative, and the helicity of the ν e is positive. One can see that when the particle with the maximum energy m µ /2 is the ν µ (left panel), the spins of the three particles combine to give a total angular momentum of 1/2 (in units of ). This is of course allowed. However when the particle with the maximum energy is the ν e (right panel), the spins of the three particles combine to give a total angular momentum of 3/2, and this is forbidden (because the angular momentum of the initial state is 1/2. It follows that the emission of ν e at high energy in the c.m. frame is more suppressed than the production of ν µ, and the spectrum is softer. Note that this argument remains valid also in the case of the decay of µ +, because the helicity of all particles in the final state is reversed (and the spectrum of the ν e is softer than the spectrum of the ν µ ). Figure?? illustrates qualitatively why in the decay µ ν µ e ν e, in the rest frame of the µ, the ν e is emitted prevalently in the direction of the muon spin, and the ν µ is mostly emitted in the direction that is opposite to the muon spin. Note that the effect is inverted in the decay of µ +, where the ν µ is emitted mostly parallel to the spin, and the ν e is emitted mostly anti parallel to the spin. The qualitative argument is simply that, because of the V A structure of Weak decays, the neutrinos are emitted with negative helicity, (spin anti-paralle to 3-momentum vector) and the anti neutrinos have positive helicity (spin parallel to 3-momentum vector). Conservation of the angular momentum favors emission with the spin of the final state 13

14 Μ decay Ν Μ Μ decay Ν e Ν e Ν Μ e Allowed e Forbidden Figure 4: Qualitative discussion of the neutrino spectra in muon decay. particle parallel to the spin of the parent particle. Since the spin direction and the 3 momentum direction are perfectly correlated, the angular distribution of the emission is distorted. 14

15 Μ decay Ν e Μ decay Ν Μ Figure 5: Qualitative discussion of the neutrino spectra in muon decay. 15

16 8.1 Laboratory frame distributions To compute the spectrum in the laboratory frame, one can start from the expression of the energy of the a particle in that frame: E ν = γ (1 + β µ cos θ ) E ν (61) In the ultrarelativistic limit: β µ 1, and using γ = E µ /m µ, E = m µ x/2, and with the scaling variable y = E ν /E µ, one has: y = x 2 (1 + cos θ ) (62) To obtain the energy distribution of the neutrinos in the laboratory frame one has to integrate over x and cos θ (with the appropriate constraint) { } dy = dx d cos θ [ µ ν dx d cos θ (x, cos θ ) δ y x ] 2 (1 + cos θ ) (63) 0 1 One can perform one integration using the delta function. Integrating in x: 1 { } dx dy = µ ν x dx d cos θ (x, cos θ = 2y/x 1) y Note that in this integration θ is the angle with respect to the boost direction, however we have calculated the energy angle distribution of the neutrino with the angle θ with respect to the spin of the muon. It is however straightforward to compute the two special cases where the muon is in an eigenstate of helicity h = ±1, because in these cases (64) cos θ spin = ± cos θ boost (65) and therefore one can use equations (??) and (??) to insert them in the integration of equation (??). The result for the general case of a muon that has helicity h (with range of possible values h [ 1, 1]) One obtains the distributions: dy (y) = 5 µ ν µ 3 3 y2 + 4 [ 1 3 y3 + helicity 3 3 y2 + 8 ] 3 y3 (66) dy (y) = y y 3 helicity [ 2 (1 4y) (1 y) 2] (67) µ ν e The results can be summarized as follows: In the decay of a µ ± the muon (anti) neutrino and the electron neutrinos carry away an fractional energy: y νµ = 7 20 helicity 1 20 (68) y νe = helicity 1 (69) 10 In the case of unpolarized parent muons, the muon neutrino (or anti neutrino) carries a larger energy than the electron anti neutrino (or neutrino). The sharing of energy between the two neutrino types depends however on the helicity state of the parent muon. 16

17 dy Figure 6: Spectra of neutrinos in the decay of unpolarized µ. The spectra are shown in a frame where the muon is ultrarelativistic. The solid line is for ν ν, The dashed line is for ν e.. 17

18 dy Figure 7: Spectra of ν µ in µ decay. The spectra are shown in a frame where the muon is ultrarealtivistic. The solid line is for an unpolarized muon. The dashed line is for helicity +1. The dot dashed line is for helicity 1. 18

19 3 dy Figure 8: Spectra of ν e in µ decay. The spectra are shown in a frame where the muon is ultrarealtivistic. The solid line is for an unpolarized muon. The dashed line is for helicity +1. The dot dashed line is for helicity 1. 19

20 8.2 Chain decay of pions As discussed in the previous subsection, the spectrum of the neutrinos created in the decay of a muon depends on its polarization state. Muons that are created in the decay of charged pions are polarized, and therefore in calculating the spectrum of neutrinos generated in the chain decay πµν one has to take into account this fact. The reason why the muon created in the decay of a charged pion is polarized has been already discussed. The muon polarization is a simple consequence of fact of the conservation of angular momentum and that the neutrino is left handed. The problem of calculating the muon polarization is very simple in the rest frame of the pion. In this frame the muon and the neutrino are emitted back to back. In the decay π µ ν µ, the ν is right handed, and to conserve angular momentum, also the µ must be right handed (so that the total angular momentum of the final state is zero). In a general frame, the calculation of the polarization of the muon is more complicated, and the helicity of the muon depends on its energy. The problem is illustrated in fig.??. Inspecting the figure one can understand that in a frame where the parent pion is relativistic, the µ with highest energy have helicity +1, and the muons with the lowest energy have helicity 1. [Note that in the case of the charged conjugate decay π + µ + ν µ, all spins must be reversed. The µ + is left handed in the rest frame. In a frame where the π + is relativistic, high energy µ + have helicity 1 and low energy muons have helicity +1]. 20

21 px p px p px p px p Figure 9: Muon polarization in charged pion decay (For the π µ ν µ. In all panels the ellipse shows the kinematically allowed values of p and p z for the muon in the final state. The red arrows shows the direction of the muon spin. In the first panel the decay is seen in the pion rest frame, and the µ has always helicity +1. In the other panels the decay is seen in frames where the muon has different energies (0.143, and GeV). In general the muon helicity depends on the angle of emission of the muon (or equivalently on the muon momentum). Note that when the pion has energy E π = , the particles has β , that is equal to the velocity of the muon is the pion rest frame, and a muon produced backward in the pion rest frame is at rest in the lab. frame. 21

22 The spin 4 momentum for the muon s, is the 4 vector that in the rest frame of the muon is purely spatial s = {0, ˆn} (70) with ˆn is the versor of the spin direction. Note that one has: The spin 4 vector is orthogonal to the 4 momentum p: If a particle has 4 momentum s s = 1 (71) s p = 0 (72) p = m γ {1, β ˆn} then the spin 4 vector in the case of helicity +1 is: s = γ {β, ˆn} Let us now consider the decay of a pion (π µ ν µ ) in the pion rest frame, where the muon is emitted in the direction cos θ (and azimuth angle 0, so that the 3 momentum is in the (x, z) plane). In the rest frame the µ is polarized 100% with helicity +1. The momentum and spin 4-vectors are: p µ = m π 2 The spin 4-vector is then: {(1 + r), (1 r) sin θ, 0, (1 r) cos θ } (73) s µ = m π 2m µ {(1 r), (1 + r) sin θ, 0, (1 + r) cos θ } (74) Transforming in the laboratory frame, where the pion has energy E π (and momentum in the +z direction), one finds that the muon energy, and the 0 component of the spin 4 vector are: Eµ lab = E π 2 [(1 + r) + β π (1 r) cos θ ] (75) s lab 0 = E π 2 m µ [(1 r) + β π (1 + r) cos θ ] (76) To compute the polarization, we can now rotate the coordinates, so that the 4 momentum of the muon has form: The spin 4 vectors has components: p lab,rot µ = {E, 0, 0, E β µ } (77) s lab,rot µ = {s 0, s x, 0, s z } = {s 0, s x, s 0 /β µ } (78) The last equality is obtained requiring that the 4 vectors s and p are orthogonal. 22

23 We can now perform a boost along the new z axis that brings the muon at rest. The z component of the spin 4 vectors becomes: s r.f. z = γ µ (s z β µ s 0 ) = γ µ ( s0 β µ β µ s 0 ) = γ µ ( ) s 0 1 β 2 β µ µ = s 0 γ µ β µ (79) The quantity s r.f. z is in fact the helicity of the muon. This is because is simply equal to cos θ with θ the angle with respect to the muon momentum. The helicity h of a particle is equal to h = P + P (80) where P + and P are the probability of having spin +1/2 and spin 1/2 in the direction of the momentum. If the spin is pointing in the direction θ with respect to the momentum, then P + = cos 2 θ/2 P = sin 2 θ/2 and h = cos 2 θ 2 sin2 θ 2 = cos θ (81) Helicity = 1 β µ (1 r) + (1 + r) cos θ β π (1 + r) + (1 r) cos θ β π (82) In the ultarelativistic limit one has β π 1, β µ 1, and the helicity of the muon becomes (1 r) + (1 + r) cos θ Helicity = (1 + r) + (1 r) cos θ (83) this can be also expressed a function of the quantity x = E µ /E π using the fact that: x = 1 2 [(1 + r) + (1 r) cos θ ] (84) Helicity = The result can therefore be written as: This function is shown in Fig.??. Helicity = (1 r) + (1 + r) cos θ (1 + r) + (1 r) cos θ (85) x r (2 x) x(1 r) (86) 23

24 Helicity Μ Helicity Μ Figure 10: Helicity of the muon generated in the decay of a pion. The helicity is calculated in a frame where the particles are ultrarelativistic, and is plotted as a function of the ratio E µ/e π. 24

25 Helicity Μ Helicity Μ Figure 11: Helicity of the muon generated in the decay of a pion. The helicity is calculated in a frame where the pion has energy E π and is plotted as a function of the ratio E µ/(e µ,max E µ,min). The pion energy are: E π = GeV, E π = E π = (m 2 π + m 2 µ)/(2m π), E π = 0.2 GeV, E π = 0.3 GeV and E π. 25

26 The final step of the calculation (to obtain the spectrum of neutrinos in the chain decay πµ ν requires the convolution of the spectrum of muons in pion decay, and then the decay of the ploarized muon. Defining: x = E µ /E π, y = E ν /E µ and z = E ν /E π One obviously has z = x y, and dz (z) = π µ ν 1 r dx 1 1 r With the appropriate helicity The spectra are shown in the following figures. The average energies are: dy µ mu (y; h[x]) δ[z x y] (87) dy yνµ = r π (88) y νe = 2 + r π (89) 10 Note that 2 y νµ + yνe = 1 + r π (90) 2 that is the fraction of the pion energy carried by the muon (conservation of energy is satisfied) dy Figure 12: Spectra of neutrinos in the chain decay of charged pions. The spectra are shown in a frame where the pion is ultrarelativistic. The solid (dashed) line is the spectrum of ν µ (ν e) in the chain decay π µ ν µ. It is easy to understand qualitatively these results. In the decay of π µ + ν µ, the µ is created in a state of helicity +1. This is because the anti neutrino is right handed, and one has to conserve the angular momentum. 26

27 dy Figure 13: Spectra of neutrinos in the chain decay of charged pions. The spectra are shown in a frame where the pion is ultrarealtivistic. The solid line is the spectrum of ν µ in the chain decay π µ ν µ. The dashed line is the same spectrum calculated neglecting the muon polarization effects dy Figure 14: Spectra of neutrinos in the chain decay of charged pions. The spectra are shown in a frame where the pion is ultrarealtivistic. The solid line is the spectrum of ν e in the chain decay π µ ν e. The dashed line is the same spectrum calculated neglecting the muon polarization effects. 27

28 8.3 Decay of Kaons In most circumstances (the only exception is interaction in very dense media) the dominant particles for neutrino production are the charged pions π ± that, together with their neutral isospin partner, are the particles most abundantly produced in hadronic interactions, accounting for over one half of the total energy in the final state. Charged pions decay with approximately 100% branching ratio in the mode π + µ + + ν µ (91) (and charge conjugate mode). In most astrophysical environments the medium where the muons propagates has a density sufficiently low, so that the µ ± decay before loosing an appreciably amount of energy in the well known mode µ + e + + ν e + ν µ (92) Therefore the chain decay of a π + results in the emission of three neutrinos with flavors (ν µ, ν µ, ν e ), and the decay of a π in the charge conjugate triplet (ν µ, ν µ, ν e ). This, together with a knowledge of the π + /π production ratio essentially determines the flavor composition before propagation for most neutrino sources. The neutral isospin partner of the charged pions decays electromagnetically in the mode: π γ +γ generating photons. The dynamics of production of π +, π and π are connected by isospin symmetry, and this is the origin of a strong correlation between the neutrino and photon fluxes. The second source of neutrinos in order of importance is the decay of kaons. The important decay modes for charged kaons are: K + µ + + ν µ (BR = 0.634) (93) K + π + e + + ν e (BR = ) (94) K + π + µ + + ν µ (BR = ) (95) (and charge conjugate channels). For neutral kaons, only the K L has significant semi leptonic branching ratios in the modes: K L π + e ± + ν e (ν e ) (BR = 0.194) (96) K L π + µ ± + ν µ (ν µ ) (BR = 0.135) (97) The amount of energy that goes into kaon production is only approximately 10% of the energy that goes into producing pions, however in the case of atmospheric neutrinos the shorter lifetime of kaons results in a larger decay probability and an enhanced contribution of kaons. Additional sources of neutrinos are the decay of hadrons containing charm (and heavier quarks). 28

29 9 Decay spectra in the ultrarelativistic limit The inclusive energy spectra of a secondary particle f generated in the decay of a primary particle i, calculated in a frame where the parent particle is ultrarelativistic has a scaling form: In general the spectrum is a function of two variables: (the energies of the initial and final particles) de f = f i f (E f ; E i ) (98) i f but in the limit of large E i, then the spectrum takes the scaling form: de f = 1 ( ) Ei F i f i f E i E f [Note that the inclusive spectrum is normalized so that after integration over all energies one obtains the average number of particles of type f created in a decay] Equation (??) can be also rewritten in the form: i f = F i f (y) (100) dy where we have introduced the fractional energy (99) y = E i E f. (101) The inclusive spectrum of particle f in the rest frame can be described in the direction of emission cos θ in terms of the variable x: The variable x i, in general can only vary in the interval x = 2 E m i (102) 0 x 1 (103) (depending on the decay mode, the limits on the range of x will be in general more restricted with x min > 0 and x max < 1.) The quantities E lab and E (the energy of particle f in the laboratory frame and in the c.m. frame) are related by a Lorentz transformation: E f,lab = γ i [E + β i p cos θ ] = γ i [1 + β i β f cos θ ] E = E i m i [1 + β i β f = E i x 2 = E i x 2 = E i x 2 cos θ ] m i x 2 [1 + β i β f cos θ ] [1 + β i β f cos θ ] [1 + cos θ ] (104) 29

30 The bottom line is that asymptotically one has the simple relation y = E f E i x 2 [1 + cos θ ] (105) Note that the energy E i of the parent particle is not present in the asymptotic expression. The spectrum /dy can be obtained summing (integrating) on all combinations of x and cos θ that yield y: dy = i f 1 0 dx +1 1 d cos θ [ dx d cos θ (x, cos θ ) δ y x ] 2 (1 + cos θ ) (106) One can perform one integration using the delta function. For example one can integrate in x: 1 dx dy = i f x dx d cos θ (x, cos θ = 2y/x 1) (107) y This is explicitely independent from the energy of the parent particles 30

31 10 Decay of a power law spectrum of parent particles Let us consider the decay of an ensemble of primary particles that have a power law spectrum. For example an ensemble of charged pions with spectrum: N π (E π ) = K π E α π. (108) After the complete decay of all of these particles, one finds a spectrum of neutrinos that can be calculted performing the integral: N ν (E ν ) = de π N π (E π ) E ν (E ν ; E π ) (109) π ν Using the scaling relation E ν E ν (E ν ; E π ) = 1 F π ν π ν E π ( Eν E π ) (110) and the power law form of the pion spectrum one can express the integration (??) as: N ν (E ν ) = de π N π (E π ) E ν E ν (E ν ; E π ) π ν ( ) = de π K π Eπ α 1 Eν F π ν (111) E π E ν One can now change variable, passing from the integration in E π to the integration in the variable y = E ν /E π. Using: or: One can rewrite Eq. (??) as: N ν (E ν ) = = E π = E ν y dy = E ν E 2 π E ν 1 0 E π (112) = y de π E π (113) dy y = de π E π (114) de π K π E α π dy y K π = K π E α ν 1 where we have defined the so called Z factor: 0 ( Eν y 1 E π F π ν ) α F π ν(y) dy y α 1 F π ν (y) ( Eν = K π Z π ν (α) E α ν (115) Z π ν (α) = 1 The (α 1) momentum of the (scaling) decay distribution. 0 E π dy y α 1 F π ν (y) (116) 31 )

32 11 Z factors for Weak decays The Z factors for Weak decays can be calculated from their definition: Z a b (p) = For the two body decay of charged pions one has for the neutrino: 1 0 dx x p 1 F a b (x) (117) and for the muon: For the decay of muons one has: Z µ ν µ (p) = Z π + ν µ (p) = (1 r π) p 1 Z π + µ +(p) = (1 rp π) p (1 r π ) p 2(p + 5) p (p 2 + 5p + 6) + h 2 2p p 3 + 5p 2 + 6p (118) (119) (120) and: Z µ ν e (p) = 12 p 3 + 5p 2 + 6p + h 12(p 1) p (p 3 + 6p p + 6) (121) (where h is the helicity of the muon in the laboratory frame. The Z factors for the decay of µ + are given by the same expressions but inverting the sign of the helicity (that is with the replacement h h). For the chain decay π + µ + ν one has: Z π + µ + ν µ (p) = 4 ((p(r 1) + 2r 3)rp 2r + 3) p 2 (p + 2)(p + 3)(r 1) 2 (122) and Z π + µ + ν e (p) = 24 (r (rp 1) + p( r) + p) p 2 (p + 1)(p + 2)(p + 3)(r 1) 2 (123) 32

33 0.4 Ν Μ Π Μ Ν Μ e Ν e Ν Μ Ν Μ Z Α Ν Μ Νe Figure 15: The figure shows the Z factors for the decays π + ν µ, π + µ + ν µ and π + µ + ν e as a function of the exponent α Ν Μ Ν e K Μ Ν Μ e Ν e Ν Μ Ν Μ Z Α 0.2 Ν Μ Figure 16: The figure shows the Z factors for the decays K + ν µ, K + µ + ν µ and K + µ + ν e as a function of the exponent α (considering only the two body mode K + µ + ν µ). 33

34 Relative Neutrino production Π Μ Ν Μ e Ν e Ν Μ Ν Μ Ν Μ Ν e Ν Μ Figure 17: Relative size of the Z factors for the decays π + ν µ, π + µ + ν µ and π + µ + ν e as a function of the exponent α. One can see that the ν µ (created in the first decay) has the smallest Z factor. The difference between the three neutrino types grows with the exponent α. 34

35 Relative neutrino production K Μ Ν Μ e Ν e Ν Μ Ν Μ Ν e Ν Μ Ν Μ Figure 18: Relative size of the Z factors for the decays K + ν µ, K + µ + ν µ and K + µ + ν e as a function of the exponent α (only the two body decay K + µ + ν µ is taken into account). 35

36 Zpolarization Α Z Α Π Μ e Ν e Ν Μ Ν e Ν Μ Figure 19: This figure shows the effect of taking into account the polarization of the muon in the calculation of the neutrino spectra for the chain decay of charged pions (π + µ + + ν µ (e + ν eν µ) + ν µ). The figure shows the quantity Z/Z pol = (Z pol Z 0)/Z pol, where Z pol and Z 0 are the momenta of the netrino calculated including and neglecting the effects of the muon polarization. The polarization effect suppresses (enhances) the flux for the muon (electron) anti neutrino produced in the muon decay. 36

37 Zpolarization Α Z Α K Μ e Ν e Ν Μ Ν e Ν Μ Figure 20: Same in fig.??, but for the chain decay of charged kaons. Note that the effect of polarization is larger. 37

38 Neutrinos from pion decay Π Π Π 0 Γ Ν Figure 21: This figure shows the quantity R π νγ(α) = 2(Z π + ν µ + Z π + µ + ν e + Z π + µ + ν µ )/Z π 0 γ. This quantity is an approximation of the ratio φ ν/φ γ if the source of neutrinos and the photons is a power law of pions with equal numbers for the three charge states (π ± and π ). 38