NNeutrino Sources and Properties
Francesco Vissani ∗ INFN, Gran Sasso Science Institute & Laboratori Nazionali del Gran SassoE-mail: [email protected]
In this lecture, prepared for PhD students, basic considerations on neutrino interactions, proper-ties and sites of production are overviewed. The detailed content is as follows:
Sect. 1, Weakinteractions and neutrinos:
Fermi coupling; definition of neutrinos; global numbers.
Sect. 2,A list of neutrino sources:
Explanatory note and examples (solar pp- and supernova-neutrinos).
Sect. 3, Neutrinos oscillations:
Basic formalism (Pontecorvo); matter effect (Mikheev, Smirnov,Wolfenstein); status of neutrino masses and mixings.
Sect. 4, Modifying the standard model toinclude neutrinos masses:
The fermions of the standard model; one additional operator in thestandard model (Weinberg); implications. One summary table and several exercises offer thestudents occasions to check, consolidate and extend their understanding; the brief reference listincludes historical and review papers and some entry points to active research in neutrino physics.
Gran Sasso Summer Institute 2014 Hands-On Experimental Underground Physics at LNGS - GSSI14,22 September - 03 October 2014INFN - Laboratori Nazionali del Gran Sasso, Assergi, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - ph ] M a y eutrino Sources and Properties Francesco Vissani
1. Weak interactions and neutrinos
We assume that the reader has some acquaintance with the standard model of electroweakinteractions. We use this assumption to introduce the topics of interest as effectively as possible.
The hamiltonian that causes the weak charged-currents transitions is, H cc = G F √ (cid:90) d x ( J µ cc ) † J cc , µ where we sum over µ = , , , √ G F , thus any decay width or cross section is proportional to itssquare. (We discuss later some physical quantities that are linear in G F .) The numerical value is, G F = . × − cm MeV (1.2)where we have used ¯ hc ∼
200 MeV fm, with 1 fm=10 − cm. From the point of view of thestandard model, the above interaction derives from the (tree-level) exchange of a W boson in thelow energy limit: Thus, we get G F √ = g M W (1.3)where the value of the W mass is M W ∼
80 GeV and its coupling to the fermions is g ∼ . The weak charged current J µ cc decreases the electric charge of the fermionic state by one unit.It contains two parts, one leptonic and one hadronic (or in fact ‘quarkonic’). The first one is, J µ cc, lept / = ¯ e γ µ P L ν e + ¯ µγ µ P L ν µ + ¯ τγ µ P L ν τ (1.4)where e , µ , τ and ν e , ν µ , ν τ are relativistic quantum fields, and where the chiral projector P L =( − γ ) / definition ofneutrino ‘species’ (or ‘type’, or ‘flavor’): the electronic neutrino field is the one associated toelectronic field, and similarly for the other ones. Thus, by definition, the neutrino emitted in thepion decay π + → µ + + ν µ is a muonic neutrino, whereas the one emitted in the beta decay ofthe neutron n → p + e − + ¯ ν e is an electronic antineutrino. In the standard model neutrinos aremassless, that agrees with the observation that neutrinos have negative helicity and antineutrinospositive helicity. Experience shows that, in any known reaction, the number of leptons of any type does notchange. This leads to the conclusion that the electronic lepton number N e , the muonic leptonnumber N µ etc, are conserved, just as the total number of leptons, N L = N e + N µ + N τ (1.5)2 eutrino Sources and Properties Francesco Vissani
Figure 1:
The baryon and the lepton numbers are conserved in theclassical theory: their currents obey dJ µ / dx µ = . Instead, in quantumfield theory the divergence is non-zero when we consider loop diagramswith W bosons in external states (‘anomaly’). This leads to transitionsthat change B and L, when W are in thermal equilibrium. or the total number of baryons N B . This observational fact is neatly accounted for in the standardmodel, since there are global symmetries associated to conserved currents, e.g. dJ µ e / dx µ =
0, wherethe time-component of the electronic current is J e = e † e + ν † e P L ν e and N e = (cid:82) J e ( t ,(cid:126) x ) d x .For completeness, note that the last statement is true only neglecting quantum fluctuations.Indeed, strictly speaking N L and N B (the baryon number) are not respected in the standard model,since the divergence of the leptonic and baryonic currents are not zero. E.g. the diagram of Fig. 1yields the contribution, dJ µ L dx µ = dJ µ B dx µ = g π Tr [ F µν (cid:101) F µν ] (1.6)Thus the divergence is non-zero and the global numbers are violated. But these effects becomeconspicuous only when W -bosons are copiously produced, e.g. at high temperatures that occurin the early Universe. In the following, we will not develop these remarks further and focus onphenomena that are directly observable in terrestrial laboratories. Exercises of Sect. 1
1) Check the dimensions of all formulae in these notes, by using the rules prescribed by so-calledsystem of natural units (i.e. the system used in particle physics where ¯ h = c = ) namely:length=time=1/mass=1/momentum=1/energy.2) Write down the (leptonic and hadronic) currents that appear in electromagnetic, charged currentand neutral current interactions. Discuss their similarities and differences.3) Find/guess which are combinations of global numbers that are conserved in the standard modeleven accounting for quantum fluctuations. Can they be promoted to new gauge symmetries?
2. A list of neutrino sources
In table 1, that appears at the end of these note, several neutrino sources are considered. Foreach of them, we list various features, including the most important one: the number of observed(or potentially observable) events. This is given byevents = N × T × Φ × σ (2.1)where the 4 terms in the r.h.s. are the number of targets N , the time of data taking T , the flux Φ and the cross section of the relevant interaction σ . Few experiments are mentioned in table 1, butmaking reference only to the number of relevant targets and to the time of data taking.3 eutrino Sources and Properties Francesco Vissani
This table is useful for a first orientation. We fixed the value of the total cross section atsome relevant energy, and then checked that with a suitable average value of the flux, this gives thecorrect (measured or expected) number of events.Often, the most relevant quantity is the number of events. Note that in the scientific literaturethe time T is sometimes included in the fluence F , i.e. the time integrated flux F = (cid:90) Φ ( t ) dt = Φ × T (2.2)where Φ in the r.h.s. is the time-averaged flux. Alternatively, the time is included in the exposure ,namely the product N × T . Another combination that is commonly used is the effective area givenby N × σ . We do not take into account explicitly any efficiency factor, that can be attached to theeffective cross section or to the exposure or to the effective area. Note finally that in the table andin our simplified estimations, the flux (and the fluence) are always integrated in the relevant energyrange. Let us repeat that the estimations in the table are not supposed to be precise, they shouldonly convey the correct order of magnitude of the number of events.If we know the average distance of production of the flux, D , and when the emission isisotropic–to some degree of approximation–we can connect the observed flux with the intensity of emission I (i.e., number of neutrinos per second) namely, Φ = I π D (2.3)In this case, the total power radiated in neutrinos–in astrophysical parlance, the luminosity –will be L = I × (cid:104) E (cid:105) (2.4)where we introduced the average energy of the neutrinos, (cid:104) E (cid:105) . If the emission is not isotropic, wehave to replace 4 π with Ω , the solid angle of emission. We proceed by illustrating with elementaryconsiderations a couple of entries of table 1. The measured solar luminosity (of photons!) is L (cid:12) = × erg/s (2.5)this is in ultimate analysis due to the fusion of 4 hydrogen nuclei into helium, by a series of reactionsthat leads to 4 p → He + e + + ν e (2.6)and liberates Q = . ∼ N ≡ L (cid:12) / Q = /s = I / Φ ≈ × electron neutrinos per cm per second.4 eutrino Sources and Properties Francesco Vissani
Figure 2:
By counting the number of neutrons produced inthe dissociation of deuterium, due to neutrino-induced neutral-current reactions, we can measure the flux of solar neutrinos ofany type. The SNO collaboration performed successfully thistype of experiment and found a result in agreement with the the-oretical predictions.
The reaction used in Borexino for their detection is the elastic scattering ν + e → ν + e (2.8)Its cross section can be estimated as σ ES ∼ G F E ν ∼ − cm (2.9)It is a direct prediction of the standard model and it depends upon the flavor of the impingingneutrino; see e.g. [6]. Note that when the neutrino energy E ν is much larger than the electron mass m e = . σ ES with the energy changes from E ν → m e E ν . In order to allow the formation of a neutron star, the energy that should be radiated is E = G N M ns R ns ∼ const. × erg (2.10)where we have the Newton constant G N = × − erg cm/g , the mass of the star M ns = . M (cid:12) = × g, the radius of the star R ns =
15 km; then, the number ‘const.’ in previous estimation is4, a somewhat more accurate value used in the following is 3. This energy is carried away by thesix types of neutrinos; if they have an average energy of 12 MeV, this will mean about few 10 electron antineutrinos. Thus, we can equate it to the average power over the time of emission: E = T L (2.11)where L is the power emitted per type of neutrino, T ∼
10 s is the time of emission, and 6 are thetypes of neutrinos and antineutrinos. If they are emitted at a typical Galactic distance of 10 kpc,the expected fluence per type of neutrino is F = × cm − (2.12)The cross section of detection is the inverse beta decay , entailing electron antineutrino interactionson hydrogen nuclei, ¯ ν e + p → e + + n (2.13)A precise expression of the cross section is in [7]; it can estimated as σ IBD ∼ G F p e E e with E e + = E ν − ( m n − m p ) (2.14)5 eutrino Sources and Properties Francesco Vissani
Figure 3:
A ray of light propagating in a crys-tal changes its polarization by ◦ if the phasevelocities of the horizontal and vertical compo-nents are different and the size is chosen appro-priately [9]. The description in terms of elemen-tary particles is more dramatic: upon propaga-tion, each photon changes its nature (i.e. eachstate is transformed into its orthogonal state).Free neutrinos are subject to similar transfor-mations acting on flavor space (Sect. 3.1). Insuitable situations, the propagation in matterproduces additional effects (Sect. 3.2). In pure scintillators such as Borexino, the positron and the neutron are both observable. The energyreleased from positron annihilation, 2 m e , adds to the kinetic energy of the positron E e + − m e . Exercises of Sect. 2
4) Compare σ ES and σ IBD cross sections. Calculate them in the standard model.5) Choose one of the (or more) lines of table 1 and check the calculations in some detail.6) Use the values of the solar luminosity and of the solar temperature, about 6000 K, to estimatethe solar radius by thermodynamics. Repeat the same steps to predict the inner radius of thesupernova from where neutrinos but not light quanta can escape freely ( neutrino-sphere ).7) Estimate the number of neutrinos and antineutrinos from a supernova and compare these num-bers with the number of electrons originally present.
3. Neutrinos oscillations
One of the first things one should know (and that, plausibly, most readers know already) aboutneutrino observations is that, in many cases, there is a severe disagreement between the measuredand the predicted neutrino fluxes. An important example is the one of solar neutrinos, that havebeen predicted by Bahcall in sixties; in fact, all observations, beginning with those of the Homes-take experiment, found a flux of electron neutrinos that is systematically smaller. A related crucialobservation was made in 2002. The SNO experiment [8] used neutral current reactions to see solarneutrinos, thereby counting all types of neutrinos and not only the electronic ones: see Fig. 2. Sincetheir observation agrees with the predictions–i.e. there is no shortage of events–one concludes thatneutrinos do not disappear, but rather, they partially change type along their trip to the Earth.This can be explained by the occurrence of the phenomenon known as neutrino oscillation(aka flavor transformation) that is analogous to well-known phenomena, as the rotation of thepolarization of the light in certain crystals, that is illustrated in Fig. 3. Below, we provide the readerwith a basic description of this phenomenon; for a more complete account see [6]. Recall that the leader of Homestake, Ray Davis Jr. and the one of Kamiokande, Masatoshi Koshiba, were awardedthe Nobel prize in 2002 for the detection of cosmic neutrinos, not for the discovery of oscillations. Note also that thedisagreement is today evident in the Borexino experiment, able to observe solar neutrinos of all possible energies. eutrino Sources and Properties Francesco Vissani
Figure 4:
Illustration of the massspectra compatible with the datafrom neutrino oscillations; left,normal hierarchy; right, invertedhierarchy. The minimum mass isnot probed by oscillations.
This first idea was proposed in 1957 by B. Pontecorvo. He postulated the possibility thatneutrinos could become antineutrinos, in analogy with what happens in the particle physics world. Ten years later, before solar neutrino results were known, Pontecorvo refined this proposal andsuggested that this conjecture was valid for neutrinos of different species. The starting point is theformalism elaborated in 1962 by Maki, Nakagawa and Sakata in Nagoya, that allows us to writethe known neutrino states as a superposition of mass eigenstates. Let us consider the simple casewith two states, (cid:40) | ν e (cid:105) = + cos θ | ν (cid:105) + sin θ | ν (cid:105)| ν µ (cid:105) = − sin θ | ν (cid:105) + cos θ | ν (cid:105) (3.1)Suppose to produce a state | ν e (cid:105) , that, in the moment of the production is orthogonal to the state | ν µ (cid:105) . Now, suppose to have a distribution over the momentum (=a wavepacket) and consider thetwo mass components that have the same momentum. Their propagation (=de Broglie’s) phases,exp (cid:20) i (cid:126) p (cid:126) x − E i t ¯ h (cid:21) with E i = (cid:113) (cid:126) p + m i (3.2)will become different in the course of the propagation since E (cid:54) = E . The difference of phase is, E − E = E − E E + E = m − m E + E ≈ m − m E (3.3)where in the last step, we used the assumption of ultra-relativistic neutrinos, E ≈ | (cid:126) p | (cid:29) m i andwe set c = |(cid:104) ν e , | ν e , t (cid:105)| (cid:54) = t , or in other words, the electronneutrinos will not survive as such; conversely, there will be a finite probability that it will be turninto a muon neutrino; and vice versa. It is not difficult to derive the following formulae, P ν e → ν e ≡ |(cid:104) ν e , | ν e , t (cid:105)| = − sin θ sin (cid:104) m − m E L (cid:105) [disappearance/survival probability] P ν e → ν µ ≡ |(cid:104) ν µ , | ν e , t (cid:105)| = sin θ sin (cid:104) m − m E L (cid:105) [appearance probability] (3.4)where L is the distance of propagation L ≈ ct and the energy in this formulae is just the kinetic one E ≈ | (cid:126) p | (and where we use ¯ h = c = The story is beautifully recounted in the review paper [10]. In 1955 Gell-Mann and Pais undestood that weak interactions mix K and ¯ K , removing the mass degeneracy. If eutrino Sources and Properties Francesco Vissani
Figure 5:
Matter effect for normal mass hierarchy. Left panel: The probability of oscillation P ν µ → ν e for adistance of 730 (black), 2500 (green), 6371 (red) km. Right panel: the same but for P ¯ ν µ → ¯ ν e . The case ofinverted hierarchy corresponds, in good approximation, simply to a swap of the two panels. Vacuum oscillations occur only when the phases between the mass eigenstates depart fromthem. But there are other phases that modify the transformation of neutrinos. In fact, the abundantpresence of electrons in ordinary matter provides the electron neutrinos with a peculiar phase ofscattering. This corresponds to the forward scattering of neutrinos due to weak interaction hamil-tonian, H = G F √ (cid:90) d x ν e γ µ ( − γ ) ν e (cid:104) ¯ e γ µ ( − γ ) e (cid:105) (3.5)where the average is on the electron state. The only non-zero term for ordinary matter–i.e. un-polarized matter at rest–is (cid:104) ¯ e γ e (cid:105) = (cid:104) e † e (cid:105) ≡ n e , namely, the density of electrons (measured e.g. inelectrons per cm ) that equates to the molar density ρ e times the Avogadro number. The phase ofscattering of ν e in a sample of matter of size dx is then given by √ G F n e dx . This term leads tomatter effect on neutrino oscillations–in short, matter effect or MSW effect after Wolfenstein [11],Mikheev and Smirnov [12]. The size of the new phase can be compared with the vacuum phase asfollows, ε ≡ √ G F n e ∆ m / ( E ν ) ≈ (cid:16) . × − eV ∆ m (cid:17) (cid:0) E ν MeV (cid:1) (cid:16) ρ e mol/cm (cid:17)(cid:16) . × − eV ∆ m (cid:17) (cid:0) E ν GeV (cid:1) (cid:16) ρ e mol/cm (cid:17) (3.6)where ∆ m stands for a difference of squared masses. E.g., let us focus on the solar neutrinos (forwhich ∆ m = . × − eV and that are produced where ρ e ∼
100 mol/cm ). When the energy E ν (cid:28) ρ e ∼ and ∆ m = . × − eV ) but this still to beobserved. In fact, it depends on the arrangement of the neutrino spectrum, that is unknown to date: one of them is produced at rest, after some time t its two components with masses m (cid:54) = m will acquire different phasesexp ( − im i c t / ¯ h ) causing a partial transformation into the other particle. This was observed in 1956. Note that: the two words ‘virtual transmutation’ in the English translation of the Nagoya paper suggest that theyhad some insight of the phenomenon we are discussing; the hypothesis of ‘mixed neutrinos’ is anticipated by a team inTokyo, Katayama et al, 1962; the circulation of ideas was much less efficient than today. eutrino Sources and Properties Francesco Vissani
Figure 6:
The surfaces of the circles represent the size of the mixing elements. From top to bottom, fromleft to right: Left panel, quark mixing (CKM) elements | V ud | , V us | , V ub | , | V cd | , ... ; Right panel, lepton mixing(PMNS) elements | U e | , U e | , U e | , | U µ | , ... in both panels, the mixing matrices are supposed to be unitary.The hierarchical structure of quark mixing elements contrasts with the one of lepton mixing elements. The two possibilities are shown in Fig. 4. The relevant analytical formulae, obtained assumingconstant matter density, give a reasonable approximation of the result, e.g. P ν µ → ν e = sin θ (cid:18) sin 2 θ Ξ (cid:19) sin (cid:20) ∆ m L E Ξ (cid:21) with Ξ = (cid:113) sin θ + ( cos 2 θ − ε ) (3.7)Here we used the sign of ε for the case of normal hierarchy; this has to be flipped for invertedhierarchy (left plot of Fig. 4). Note that when ε =
0, we have Ξ =
1, and this formula remembersclosely the vacuum formulae in Eqs. 3.4. The probabilities of oscillations can be calculated at theweb site [13] that runs the code described in [14]; a sample output is in Fig. 5. For more discussionon matter effect see also [6] and the appendix of [15].
Starting from the standard model, the most natural hypothesis is to assume that we have justthree light neutrinos. We can take into account the evidences of oscillations by postulating thatthey have mass and mix among them. The two types of mass arrangements (aka hierarchies orordering or spectra) compatible with the present data are illustrated in Fig. 4. This figure shows thateach of the neutrino mass eigenstates ν i is a superposition of the flavor states ν (cid:96) . The connection isgiven by the leptonic (or PMNS) mixing matrix, that links these two kinds of field, ν (cid:96) = ∑ i = U (cid:96) i ν i with (cid:96) = e , µ , τ (3.8) Actually, it coincides with the vacuum formula | U e U µ × [ exp ( ∆ m L / E ) − ] | , written with the conventionalchoice of the mixing elements U e = sin θ and U µ = cos θ sin θ , see e.g. [6]. This is supported by the width of the Z , that measures the number of weakly interacting neutrinos, but also bymeasurements based on the big-bang cosmology, namely: the primordial nucleosynthesis of light elements and thedistribution of the inhomogeneites of the cosmic microwave background. eutrino Sources and Properties Francesco Vissani
A quantitative information is given in Fig. 6, where we compare the sizes of the elements of theleptonic and of the quark mixing matrices, | U (cid:96) i | and | V ud | . Recall that the physical parameters are 3mixing angles and one phase that describes CP violating phenomena. All 3 leptonic mixing anglesare known in good approximation and we have the first hints that the CP violating phase is differentfrom zero; but, let us repeat, the mass hierarchy is unknown to date.Can we determine the mass hierarchy by means of oscillations? Considering 3 flavor oscil-lations, the survival probability P ¯ ν e → ¯ ν e of reactor antineutrinos is not the same for the two masshierarchies shown in Fig 4. The effect is maximum at about 50 km, but it is not large and re-quires very precise measurements; this is the target of JUNO. The study of Earth matter effect via P ν µ → ν µ , P ν µ → ν e , P ν e → ν µ , P ν e → ν e and connected antineutrino channels (see Fig. 5) offers other possi-bilities, that can be pursued already with atmospheric neutrinos (PINGU, ORCA, INO, Super- andHyper-Kamiokande). Also long-baseline experiments such as NO ν A, T2K (possibly with Hyper-Kamiokande) can study the mass hierarchy and the CP violating phase.Finally, we would like to discuss neutrino masses. A quantitative illustration of the spectrum,assuming normal mass hierarchy, is in Fig. 7. Note that oscillations fix only the squared massdifferences without providing us information on lightest neutrino mass. Other experiments can giveus information on some combination of neutrino mass. These include cosmological measurementsof the distribution of the matter, the search of mass effects in the (endpoint of) beta spectrum,the search for the lepton number violating transition named neutrinoless double beta decay thatreceives a contribution from Majorana neutrino masses (see Sect. 4.3). The present 95% CL boundon the sum of masses from cosmology is 140 meV [16]. This bound allows us to deduce otherstringent bounds, theory beta spectrum neutrinoless β β parameter m lightest m β m ββ ( max ) NORMAL HIERARCHY
38 meV 39 meV 39 meV
INVERTED HIERARCHY
28 meV 56 meV 55 meVAll these quantities are below the sensitivity of near future experiments. If the bound is correct, apositive detection in laboratory experiments ( β or β β ) would point to physics beyond the minimalscenario; e.g. new sources of lepton number violation beyond light neutrino masses contributingto neutrinoless double beta decay. Note that if we treat the cosmological bound as Gaussian, wefind at 1 sigma that the sum of mass is below 71 meV; thus, the normal mass hierarchy is slightlyfavored by the interpretation of cosmological observations. Exercises of Sect. 3
8) Derive the formula of the survival and of the appearance probabilities; note that in order toobtain it, a crucial hypothesis is to consider the case of ultra-relativistic neutrinos . [This is the onlycase that applies in practice, though it could be interesting for theoretical–or academic–purposesto consider the other one.]9) Including ¯ h and c factors in Eqs. 3.4, prove the important numerical formula ϕ ≡ ∆ m L E = . ∆ m eV L km GeV E (3.9) eutrino Sources and Properties Francesco Vissani
Figure 7:
Masses of the known elementary spin 1/2 fermions. Red, up quarks; yellow, down quarks; orange,charged leptons; green, neutrinos. In the last case, we assume normal mass hierarchy and a mass ratio of1/10 between the two lightest neutrinos for illustration purposes. that allows to estimate easily the cases when oscillations are absent ( ϕ (cid:28) ) when they exist inproper sense ( ϕ ∼ ) and when they are described by a constant factor instead ( ϕ (cid:29) ).10) Check that the maximum of the red curve in the left panel of Fig. 5, due to matter effect,corresponds to the region where the phase of scattering due to matter is comparable to the phasedue to vacuum oscillations.11) Knowing that the mass measured in cosmology is Σ = m + m + m and using the values ofthe mixing elements and of the differences of mass squared, derive the bounds on the massesdefined as m β = ∑ i | U ei | m i and m ββ = | ∑ i U ei m i | (the last one implying m ββ ( max ) = ∑ i | U ei | m i ).
4. Modifying the standard model to include neutrinos masses
In this last section, we discuss a modification of the standard model of elementary particles,that allows us to take into account neutrinos masses. In this manner, we can explain oscillationsand discuss on firmer bases new phenomena.
Let us examine the meaning of neutrino masses from the point of view of the standard model.The latter is based on the gauge group G SM = SU ( ) color × SU ( ) left × U ( ) hypercharge (4.1)and it includes 3 families of fermions, each one with 2 quarks (coming in 3 colors) and 2 leptons,see Fig. 8. It allows the transitions between the 3 different families of quarks but it forbids thosebetween different families of leptons. However, the observation of neutrino oscillations/flavor con-version shows that this possibility does occur, just as it occurs for quarks. The most reasonableexplanation, suggested in the previous pages, is that neutrinos have mass and mix among them.Thus, we ask the question of how to introduce neutrino masses in a suitable extension of the stan-dard model.The minimal modification (by definition) is to require that we do not add any new light particle.In this case, if we want to build some sort of neutrino mass, we have no other choice but to deal Applying the Cauchy-Schwarz inequality |(cid:104) (cid:126) x ,(cid:126) y (cid:105)| ≤ || (cid:126) x || || (cid:126) y || to the vectors with components ( (cid:126) x ) i = U ∗ ei and ( (cid:126) y ) i = U ei m i we have m ββ ≤ m β , since || (cid:126) x || =
1. Note that m lightest = m (= m ) in normal (inverted) hierarchy, see Fig. 4. eutrino Sources and Properties Francesco Vissani
Figure 8:
In the standard model, each of the threefamilies contain 15 spin-half chiral (Weyl) states, in-cluding one neutrino with left chirality. The corre-sponding antiparticles are automatically included bythe general principle of relativistic quantum field the-ory. In the figure, we show the states of the first family,that corresponds to the following mass eigenstates:u and d quarks (charged and colored), the lepton e(charged) and neutrino ν e (neutral and massless). with the (left) leptonic doublet L (cid:96) a = (cid:32) ν (cid:96) a (cid:96) a (cid:33) where (cid:96) = e , µ , τ and a = , , , SU ( ) left . The standard model quantum numbers ofthis doublet are ( , , − / ) meaning that: it is a singlet under the color group SU ( ) color (of course!it is a lepton); it is a doublet under SU ( ) left (even more evident) and it has Y = diag ( − / , − / ) ,and thus the electric charges are Q = σ / + Y = diag ( , − ) , as it should be. The Higgs doubletinstead has the quantum numbers ( , , + / ) , thus it can be written H = ( H + , H ) . We can form a gauge singlet by taking the product, Hi σ L (cid:96) a = − ν (cid:96) a H + (cid:96) a H + but this bilinearcombination is not Lorentz invariant, since it has one free spinorial index a . When the Higgsfield takes vacuum expectation value, we get a term proportional to the neutrino field, Hi σ L (cid:96) a = −(cid:104) H (cid:105) ν (cid:96) a + ... It is sufficient to contract two of these terms to obtain an invariant term, that can beadded to the standard model hamiltonian density, namely [17] H σ L (cid:96) a C − ab H σ L (cid:96) (cid:48) b / ( M ) (4.3)where repeated indices are contracted. Here, M is a constant with dimensions of mass, that isincluded to ensure that the hamiltonian density has the correct dimensions, while C is the chargeconjugation matrix, needed to form an invariant quantity out of two spinors. When the Higgs fieldtakes vacuum expectation value we get a bilinear term built with the (left handed) neutrino fieldsonly. This is, m ν (cid:96) C − ν (cid:96) (cid:48) + hermitian conjugate, where m = (cid:104) H (cid:105) M (4.4)Note that the mass parameter m is inversely proportional to the mass scale of the Weinberg operator, M . We can generalize this position to include similar terms for all types of neutrinos by replacing m → m (cid:96)(cid:96) (cid:48) where (cid:96), (cid:96) (cid:48) = e , µ , τ . Finally, we define the following real (or Majorana) spinor: χ (cid:96) = ν (cid:96) + C ν (cid:96) t such that χ (cid:96) = C χ (cid:96) t (4.5)If m (cid:96)(cid:96) (cid:48) is real, we can rewrite the above bilinear term in a manner that looks familiar, namely m (cid:96)(cid:96) (cid:48) ν (cid:96) C − ν (cid:96) (cid:48) + hermitian conjugate = − m (cid:96)(cid:96) (cid:48) χ (cid:96) χ (cid:96) (cid:48) (4.6)We conclude that the new operator produces a mass term for the neutrinos, called Majorana mass .12 eutrino Sources and Properties
Francesco Vissani
Evidently, the above possibility is particularly interesting, since (apart from the technicalities)it is compatible with the standard model gauge symmetries. Moreover, it gives rise to lepton numberviolating phenomena, such as the neutrinoless double beta decay –namely the nuclear transition ( A , Z ) → ( A , Z + ) + e − (4.7)that is forbidden in the standard model. Thus there is some connection between neutrino oscilla-tions and other phenomena–simply because they both result from neutrino masses.However, ‘there is no such thing as a free lunch’. What is the price that we have to pay if wecorrect for the shortcomings of the standard model by adding the new operator of Eq. 4.3? Thefact is that it has canonical (mass) dimension 5, thus it breaks the renormalizability of the resultingquantum field theory. This is not so dramatic as it might look, though; the same happens with Fermiinteractions. In fact, G F has dimensions of inverse mass squared (recall Eq. 1.3) but this is far frommeaning that Fermi interactions have no practical applications in nuclear and particle physics!To summarize, we can hypothesize that neutrino masses are due to the operator shown inEq. 4.3, but we are not yet ready to discuss its origin ; we have to postpone the question of howit emerges from a renormalizable theory (that extends the standard model) to a future and morecomplete model of the world of elementary particles, that will be hopefully based on new data andinformation. Unfortunately, it is not clear whether the scale of new physics will allow direct accessby terrestrial accelerators–see the last exercise. Exercises of Sect. 4
12) Assuming that the spinor ν has left chirality, ν = P L ν , check that the spinor C ν t has right chiral-ity.13) Prove that the spinor χ has to be a trivial representation of any U ( ) group, and in particular,it cannot transform under the U ( ) e.m. of the electromagnetism–it cannot be a charged field. Whatabout U ( ) hypercharge ?14) Show that the two mass terms written in Eq. 4.6 are identical. Prove that, in general, thematrix m (cid:96)(cid:96) (cid:48) is symmetric. Write the mass term using the Majorana field χ (cid:96) , assuming that m (cid:96)(cid:96) (cid:48) = Re ( m (cid:96)(cid:96) (cid:48) ) + i Im ( m (cid:96)(cid:96) (cid:48) ) is complex.15) Show by direct calculation that the combination H σ L is a gauge singlet of the standard modelgauge group, and in particular it is invariant under SU ( ) left transformations. What are the transfor-mation properties of H σ (cid:126) σ L ? Combine the SU ( ) left spinors L and H in other manners and showthat the result is always proportional to the Weinberg operator. (Recall that ⊗ = ⊕ .)16) Discuss which are the values of masses M in Eq. 4.3 that are compatible with the massesof neutrinos observed by mean of oscillations. Discuss the meaning on M by elaborating on thecorrespondence between Eqs. 1.3 and 4.3 possibly by envisaging a specific renormalizable modelto account for this operator. Acknowledgments
I would like to thank A. Ianni and the Organizers for the invitation, and P. Sapienza for carefulreading of the text. An appeal to potential readers: please feel free to contact me at to discuss thisnote, the exercises, or to get more information but do not worry too much if your results do notagree with those in table 1; they are only meant to allow first orientation.13 eutrino Sources and Properties
Francesco Vissani
References [1] G. Bellini et al. [Borexino Collaboration], “Neutrinos from the primary proton-proton fusion processin the Sun,”
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Phys. Rev. Lett. (2002) 011301[nucl-ex/0204008].[9] http://en.wikipedia.org/wiki/Waveplate [10] B. M. Pontecorvo, “Pages In The Development Of Neutrino Physics,” Sov. Phys. Usp. (1983) 1087[Usp. Fiz. Nauk (1983) 675].[11] L. Wolfenstein, “Neutrino Oscillations in Matter,” Phys. Rev. D 17 (1978) 2369.[12] S.P. Mikheev and A.Yu. Smirnov, “Resonance Amplification of Oscillations in Matter andSpectroscopy of Solar Neutrinos,” , Sov. J. Nucl. Phys. 42 (1985) 913.[13] http://pcbat1.mi.infn.it/ ∼ battist/cgi-bin/oscil/index.r [14] G. Battistoni, A. Ferrari, C. Rubbia, P. R. Sala and F. Vissani, “Atmospheric neutrinos in a largeliquid argon detector,” hep-ph/0604182.[15] C. Lujan-Peschard, G. Pagliaroli and F. Vissani, “Counting muons to probe the neutrino massspectrum,” Eur. Phys. J. C (2013) 2439 [arXiv:1301.4577].[16] N. Palanque-Delabrouille et al. , “Constraint on neutrino masses from SDSS-III/BOSS Ly α forest andother cosmological probes,” JCAP 1502 (2015) 045 [arXiv:1410.7244].[17] S. Weinberg, “Baryon and Lepton Non-conserving Processes,”
Phys. Rev. Lett. (1979) 1566. eutrino Sources and Properties Francesco Vissani N e u t r i no F l ux A v e r a g e T a r g e t T i m e D i s t a n ce C r o ss R e m a r k s & E v e n t s ou r ce [ / c m s ] e n e r gy & t yp e [ c m ] s ec ti on [ c m ]r e f e r e n ce S un ( pp ) × . M e V × e a y1 . × k m − ( E S ) B X [ ] E a r t h3 × M e V p4y r ? × − (I B D ) B X [ ] R eac t o r × M e V p4y r m − (I B D ) B X [ ] R e li c S N . M e V p1y r c × − (I B D ) S K [ ] . ? G a l ac ti c S N M e V p10 s c × − (I B D ) S K [ ] A t m o s ph e r e G e V N r m − ( Q EL + D I S ) S K ( a ll ν ) A cce l e r a t o r s ν µ . G e V × N r m − ( D I S ) S K A cce l e r a t o r s ν τ . G e V N r m − ( D I S ) O p e r a G a l ac ti c s ou r ce − T e V × N r c − ( D I S ) I ce ? H E n e u t r i no s × − T e V × N r ? − ( D I S ) I ce [ ] T a b l e : A bb r e v i a ti on s : ( ) B X = B o r e x i no ; S K = S up e r- K a m i ok a nd e ( . t on ) ; I ce = I ce C U B E . ( ) E S = e l a s ti c s ca tt e r i ng ; I B D = i nv e r s e b e t a d eca y ; Q EL = qu a s i e l a s ti c nu c l e on i n t e r ac ti on ; D I S = d ee p i n e l a s ti c s ca tt e r i ng . ( ) e = e l ec t r on ; p = p r o t on ; N = nu c l e on . c = × c m . Q u e s ti on m a r k s d e no t e pu r e l y t h e o r e ti ca l p r e d i c ti on s ..