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(fm )
3000
167
1000
0 0
4
sm
8
12
Fig. 3. – Quadrupole correlations in the ground state of 82 Kr as a function of the amount of quadrupole-quadrupole interaction λQ · Q added to the Hamiltonian and of the maximum seniority sm permitted in the wave functions.
The operator Q represents the mass quadrupole. No effective “nuclear” charges are included. Using 82 Kr as our test bench, we proceed to compute Q2 , first with our standing effective interaction and different seniority truncations (sm means the maximum seniority allowed in the wave functions of parent and grand-daughter nuclei). The results are drawn in fig. 3 as the black circles labeled λ = 0. We can see that at sm = 4 —roughly, the implicit level of seniority truncation in the spherical QRPA— some 70% of the full quadrupole correlations are incorporated in the wave function. We would like to know how this behavior evolves when more correlations are enforced in the system. For this we recalculate the ground state of 82 Kr with a new Hamiltonian that consists of the standing one plus a quadrupole-quadrupole term λQ · Q, whose effect will be gauged by its influence in the sum rule. To have an idea of the relevant range of values of Q2 in this nucleus and valence space, we have gone to the limit of pure quadrupole-quadrupole interaction with degenerate single-particle energies, getting Q2 ≈ 4500 fm4 . The results for λ = 1 and λ = 2 are also shown in fig. 3 (λ = 1 corresponds to λqq = 0.025 in figs. 6-9 and λqq = 1 in fig. 10 to λqq = 0.1). It is evident in the figure that, as we try to increase the correlations, the sm = 4 truncation becomes more and more ineffective. For λ = 1, only 57% of the full correlations are present, and for λ = 2 only 50%. The situation is different for 82 Se: while for λ = 0 the values of Q2 as a function of seniority are similar, albeit a bit smaller than the 82 Kr ones, for λ = 1 and λ = 2 there is scarcely any increase of the ground-state correlations. This means also that, as we increase λ, the “deformation” of 82 Kr grows, whereas that of 82 Se remains constant. This behavior offers us the opportunity of exploring the effect of the difference in deformation between parent and grand-daughter in the 0νββ NME’s. To this goal, we have computed the Gamow-Teller matrix element for different values of λ —the amount of extra quadrupole-quadrupole interaction— and sm —the maximum seniority allowed in the wave function. The results are gathered in fig. 4. For sm = 0, we observe that the
168
´ndez, A. Poves, E. Caurier and F. Nowacki J. Mene 8
sm=0 sm=4 sm=8 sm=12
MGT(0ν)
6
4
2
0
0
1
λ
2
Fig. 4. – 82 Se → 82 Kr Gamow-Teller matrix element, M GT , as a function of the maximum seniority of the wave functions, for different values of the strength of the extra quadrupolequadrupole interaction.
Gamow-Teller matrix element grows as a function of λ. This may seem paradoxical, but is not, because at this seniority truncation, the only effect of adding more quadrupolequadrupole interaction is to augment the pairing content of the wave functions, thus increasing M GT . At sm = 4, M GT remains constant as a function of λ, meaning that the minor increase of the correlations of 82 Kr, that we have shown in fig. 3, is barely enough to compensate the increase of M GT at sm = 0. On the contrary, the full space results are sensitive to the difference in deformation —or, to be more precise, to the difference in the level of quadrupole correlations in the ground state— between parent and grand-daughter. The effect goes in the direction of reducing the value of M GT . In the A = 82 decay, doubling the quadrupole correlations in 82 Kr, roughly halves M GT . We can go much further in the exploration of the deformation effects, calculating the NME of the decay for initial and final states computed with different amounts of supplementary quadrupole-quadrupole interaction. As before, we measure the quadrupole correlations by means of the mass quadrupole sum rule. The results of this search are plotted in fig. 5. We observe that the NME decreases almost linearly as the difference of the sum rules for the final and initial states increases. In fact, the maximum values of the NME are reached when this difference is close to zero. On the contrary for large differences the NME can be extremely quenched. Notice that the values in the figure range between 0.07 and 2.7, a factor of 40 span. The λ = 0 value is 2.18, indicating that the difference in quadrupole correlations between 82 Kr and 82 Se is not very large. 4. – 0ν (unphysical) mirror decays: a case study We have also studied the transitions between mirror nuclei in order to have a clearer view of the role of deformations in the NME’s. These transitions have the peculiarity that the wave functions of the initial and final nuclei are identical (provided Coulomb
Deformation and the nuclear matrix elements of the neutrinoless ββ decay
169
4
(Kr) -(Se) (in fm )
3000
2000
2
2
2
1000
0
-1000 0
1860=(Kr) 2480 2945 3160 3300 3620 1010 550 0.5
1
1.5
2
2.5
0ν
M
Fig. 5. – 82 Se → 82 Kr NME, M 0ν , as a function of the difference between the mass quadrupole sum rule between 82 Kr and 82 Se for a large number of different values of the strength of the added quadrupole-quadrupole interaction.
effects are neglected) and consequently the interplay of the 0νββ operator and of the nuclear wave functions in the NME may be easier to understand. We have studied four parent nuclei with six valence protons and four valence neutrons, 26 Mg, 50 Cr, 66 Ge and 110 Xe, decaying into the four grand-daughter nuclei with four valence protons and six valence neutrons, 26 Si, 50 Fe, 66 Se and 110 Ba. The valence spaces considered are the sd-shell, the pf -shell, r3 g (1p3/2 , 1p1/2 , 0f5/2 , 0g9/2 ) and r4 h (0g7/2 , 1d5/2 , 1d3/2 , 2s1/2 , 0h11/2 ). The starting interactions are USD [27], KB3 [28], GCN28.50 and GCN50.82 [29]. The deformation of the nuclei is modified as in the previous section, and it is quantified also in the same way. The results for A = 66 in the case of equally deformed initial and final nuclei are shown in fig. 6. There we see that, as the nuclei become more deformed, the NME and the pairing content of the wave function get smaller, while the quadrupole sum rule grows. All these changes are nearly linear for reasonable deformations and then the saturation is approached more smoothly. Note that the purely quadrupole interaction (λqq → ∞ limit) gives a NME which is about a half of the value obtaind with no additional quadrupole. Figure 7 shows the same quantities as fig. 6 but now only the final nucleus has been artificially deformed by adding an extra quadrupole-quadupole term. In addition, the overlap between initial and final wave functions has been included. We see that now, the reduction of the NME is more pronounced and, what is more interesting, that it follows closely the overlap between wave functions. This means that, if we write the final wave function as |Ψ = a|Ψ0 + b|Ψqq , the 0νββ operator only connects the parts of the wave functions that have the same deformation among themselves. The behavior of the NME’s with respect to the difference of deformation between parent and grand-daughter is common to all the other transitions between mirror nuclei that we have studied. Therefore we can submit that this is a robust result. However, when we consider the transitions between equally deformed nuclei, the evolution of the
170
´ndez, A. Poves, E. Caurier and F. Nowacki J. Mene 5 1 4.5 0.9
NME
0.8 3.5 0.7
,<pair>
4
3 0.6 NME<pair>
2.5
0.5
2 0
0.05
0.1
0.15
λqq
0.2
0.6
1
1.4
Fig. 6. – 66 Ge → 66 Se NME, M 0ν , as a function of the strength of the added quadrupole-quadrupole interaction. Equally deformed case; the same amount of extra QQ interaction is added to 66 Ge and 66 Se. On the right-hand y axis the pairing and quadrupole sum rules are represented, normalized so that their maximum value is 1. Note the change of scale in the x-axis at λ = 0.2.
NME’s with the deformation that we have found in A = 66 is only shared by the A = 110 case. When the valence space is a full major oscillator shell —A = 26 and A = 50— the situation is quite different. Indeed, what is observed is that the NME does not decrease for moderate values of λ but remains rather constant until a point —with large deformation— where its value increases significantly —up to 50%. This is due to the 5
4
0.8
3 NME
0.6
2
0.4 NMEnondiag<pair>
1
,<pair>,
1
0.2
0
0 0
0.05
0.1
0.15
λqq
0.2
0.6
1
1.4
Fig. 7. – Same as the previous figure, but now the additional quadrupole interaction is only added to 66 Se. The normalized overlap between the initial and final states is also included.
Deformation and the nuclear matrix elements of the neutrinoless ββ decay
171
1 0.9
0.25
0.8 0.7 0.6 0.15
0.5
0.1
2ν NME
0.05
<pair>
0.4
,<pair>
2ν NME
0.2
0.3 0.2 0.1
0
0 0
Fig. 8. – Equally deformed λ = 0.2.
66
0.05
0.1
Ge →
66
0.15
λqq
0.2
0.6
1
1.4
Se 2νNME. Note the change of scale in the x axis at
fact that, at this point, the major contribution to the NME ceases to come only form the decay of pairs coupled to J = 0, since other values like J = 2, 4, 6, which usually have a contribution to the NME contrary to that of J = 0, reverse sign and grow until being comparable with this contribution, thus resulting in this notorious rise of the NME. In the sd-shell and pf -shell cases, the λqq → ∞ limit is equivalent to Elliott’s SU (3) limit, and the fact that both the initial and final nuclei should belong to the same irrep of SU (3) may be the reason of the increase of the NME, but, for the moment we have not found a formal explanation. 5. – 2ν (unphysical) mirror decays: a case study Very similar conclusions may be reached for the effect of deformation in 2νββ decay. For instance, the mirror nuclei diagonal and non-diagonal NME’s are represented as in the 0νββ case in figs. 8 and 9 for the A = 66 transition. We see that the figures resemble very much that of the previous section, with the only exception that, in the equally deformed case, the lowering of the NME due to deformation is more pronounced. If we look to a non-mirror —but again fictitious— transition, for instance that of 48 Ti → 48 Cr, we get the results shown in fig. 10. If we compare with those of ref. [26], which are again the equivalent ones for the 0νββ transition, we see that there are not substantial changes. In this sense, deformation seems to affect similarly 0νββ and 2νββ decays. 6. – Summary After a brief discussion of the “state of the art” results for the nuclear matrix elements of the neutrinoless double beta decay in the context of the Interacting Shell Model and
172
´ndez, A. Poves, E. Caurier and F. Nowacki J. Mene 1
0.8
2ν NME
0.2 0.6 0.15
0.1
2ν NMEnondiag
0.05
<pair>
0.4
,<pair>,
0.25
0.2
0
0 0
0.05
0.1
0.15
λqq
0.2
0.6
1
1.4
Fig. 9. – The same as the previous figure, but now the only nucleus calculated with additional quadrupole interaction is the final one. The normalized overlap between initial and final states is also included. 0.4
λ qq(Cr)=-2 λ qq(Cr)=0 λ qq(Cr)=+2
MGT
2ν
0.3
0.2
0.1 -2
-1
Fig. 10. – Influence of deformation in the
48
0
λ qq(Ti)
1
2
Ti → 48 Cr decay.
of the Quasiparticle Random Phase Approximation, we have analyzed the role of the pairing correlations and the deformation in the NME’s, concluding that seniority truncations are less reliable when the quadrupole correlations are large. Since the NME’s are reduced when the deformation of parent and grand-daughter is different, a bad treatment of the quadrupole correlations can lead to an artificial enhancement of the NME’s in the transitions among nuclei with unequal deformations, or in the cases of nuclei with different —and large— amounts of quadrupole correlations.
Deformation and the nuclear matrix elements of the neutrinoless ββ decay
173
∗ ∗ ∗ This work has been supported by a grant of the Spanish Ministry of Education and Science, FPA2007-66069, by the IN2P3-CICyT collaboration agreements, by the Spanish Consolider-Ingenio 2010 Program CPAN (CSD2007-00042), and by the Comunidad de Madrid (Spain), project HEPHACOS P-ESP-00346. REFERENCES [1] Fukuda Y., Hayakawa T., Ichihara E., Inoue K., Ishihara K., Ishino H., Itow Y., Kajita T., Kameda J., Kasuga S., Kobayashi K., Kobayashi Y., Koshio Y., Miura M., Nakahata M., Nakayama S., Okada A., Okumura K., Sakurai N., Shiozawa M., Suzuki Y., Takeuchi Y., Totsuka Y., Yamada S., Earl M., Habig A. and Kearns E., Phys. Rev. Lett., 81 (1998) 1562. [2] Ahmad Q. R., Allen R. C., Andersen T. C., D. Anglin J., Barton J. C., Beier E. W., Bercovitch M., Bigu J., Biller S. D., Black R. A., Blevis I., Boardman R. J., Boger J., Bonvin E., Boulay M. G., Bowler M. G., Bowles T. J., Brice ¨hler G., Cameron J., Chan Y. D., Chen S. J., Browne M. C., Bullard T. V., B u H. H., Chen M., Chen X. and Cleveland B. T., Phys. Rev. Lett., 89 (2002) 011301. [3] Eguchi K., Enomoto S., Furuno K., Goldman J., Hanada H., Ikeda H., Ikeda K., Inoue K., Ishihara K., Itoh W., Iwamoto T., Kawaguchi T., Kawashima T., Kinoshita H., Kishimoto Y., Koga M., Koseki Y., Maeda T., Mitsui T., Motoki M., Nakajima K., Nakajima M., Nakajima T., Ogawa H., Owada K., Sakabe T. and Shimizu I., Phys. Rev. Lett., 90 (2003) 021802. [4] Klapdor-Kleingrothaus H. V., Dietz A., Harney H. L. and Krivosheina I. V., Mod. Phys. Lett. A, 16 (2001) 2409. [5] Klapdor-Kleingrothaus H. V., Krivosheina I. V., Dietz A. and Chkvorets O., Phys. Lett. B, 586 (2004) 198. [6] Arnold R., Augier C., Baker J., Barabash A., Broudin G., Brudanin V., Caffrey A. J., Caurier E., Egorov V., Errahmane K., Etienvre A. I., Guyonnet J. L., Hubert F., Hubert P., Jollet C., Jullian S., Kochetov O., Kovalenko V., Konovalov S., Lalanne D., Leccia F., Longuemare C., Lutter G., Marquet C., Mauger F., Nowacki F., Ohsumi H., Piquemal F., Reyss J. L., Saakyan R., Sarazin X., Simard L., Simkovic F., Shitov Y., Smolnikov A., Stekl L., Suhonen J., Sutton C. S., Szklarz G., Thomas J., Timkin V., Tretyak V., Umatov V., Vala L., Vanushin I., Vasilyev V., Vorobel V. and Vylov T., Phys. Rev. Lett., 95 (2005) 182302. [7] Arnaboldi C., Artusa D. R., Avignone F. T., Balata M., Bandac I., Barucci M., Beeman J. W., Brofferio C., Bucci C., Capelli S., Carbone L., Cebrian S., Cremonesi O., Creswick R. J., de Waard A., Farach H. A., Fiorini E., Frossati G., Guardincerri E., Giuliani A., Gorla P., Haller E. E., McDonald R. J., Morales A., Norman E. B., Nucciotti A., Olivieri E., Pallavicini M., Palmieri E., Pasca E., Pavan M., Pedretti M., Pessina G., Pirro S., Previtali E., Risegari L., Rosenfeld C., Sangiorgio S., Sisti M., Smith A. R., Torres L. and Ventura G., Phys. Rev. Lett., 95 (2005) 142501. [8] Bloxham T., Boston A., Dawson J., Dobos D., Fox S. P., Freer M., Fulton ¨ ssling C., Harrison P. F., Junker M., Kiel H., McGrath J., Morgan B. R., Go ¨nstermann D., Nolan P., Oehl S., Ramachers Y., Reeve C., Stewart D., B., Mu Wadsworth R., Wilson J. R. and Zuber K., Phys. Rev. C, 76 (2007) 025501.
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[9] [10] [11] [12] [13]
Avignone F. T., Elliott S. R. and Engel J., Rev. Mod. Phys., 80 (2008) 481. Suhonen J. and Civitarese O., Phys. Rept., 300 (1998) 123. Rodin V. A., Faessler A., Simkovic F. and Vogel P., Nucl. Phys. A, 766 (2006) 107. Rodin V. A., Faessler A., Simkovic F. and Vogel P., Nucl. Phys. A, 793 (2007) 213. Caurier E., Martinez-Pinedo G., Nowacki F., Poves A. and Zuker A. P., Rev. Mod. Phys., 77 (2005) 427. Retamosa J., Caurier E. and Nowacki F., Phys. Rev. C, 51 (1995) 371. Caurier E., Nowacki F., Poves A. and Retamosa J., Phys. Rev. Lett., 77 (1996) 1954. Simkovic F., Pantis G., Vergados J. D. and Faessler A., Phys. Rev. C, 60 (1999) 055502. Kortelainen M. and Suhonen J., Phys. Rev. C, 75 (2007) 051303. Kortelainen M. and Suhonen J., Phys. Rev. C, 76 (2007) 024315. Caurier E., Menendez J., Nowacki F. and Poves A., Phys. Rev. Lett., 100 (2008) 052503. Feldmeier H., Neff T., Roth R. and Schnack J., Nucl. Phys. A, 632 (1998) 61. Doi M., Kotani T., Nishiura H., Okuda K. and Takasugi E., Phys. Lett. B, 103 (1981) 219. Doi M., Kotani T. and Takasugi E., Prog. Theor. Phys. Suppl., 83 (1985) 1. Simkovic F., Faessler A., Rodin V. A., Vogel P. and Engel J., Phys. Rev. C, 77 (2008) 045503. Suhonen J. and Kortelainen M., Int. J. Mod. Phys. E, 17 (2008) 1. Roth R., Hergert H., Papakonstantinou P., Neff T. and Feldmeier H., Phys. Rev. C, 72 (2005) 034002. Caurier E., Nowacki F. and Poves A., Eur. Phys. J. A, 36 (2008) 195. Wildenthal B. H., Prog. Part. Nucl. Phys., 11 (1984) 5. Poves A. and Zuker A., Phys. Rep., 70 (1981) 235. Gniady A., Caurier E. and Nowacki F., to be published.
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
DOI 10.3254/978-1-60750-038-4-175
Charge-exchange reactions and nuclear matrix elements for ββ decay D. Frekers Institut f¨ ur Kernphysik, Westf¨ alische Wilhelms-Universit¨ at - D-48149 M¨ unster, Germany
Summary. — Charge-exchange reactions of (n, p) and (p, n) type at intermediate energies are a powerful tool for the study of nuclear matrix element in ββ decay. The present paper reviews some of the most recent experiments in this context. Here, the (n, p) type reactions are realized through (d, 2 He), where 2 He refers to two protons in a singlet 1 S0 state and where both of these are momentum analyzed and detected by the same spectrometer and detector. These reactions have been developed and performed exclusively at KVI, Groningen (NL), using an incident deuteron energy of 183 MeV. Final-state resolutions of about 100 keV have routinely been available. On the other hand, the (3 He, t) reaction is of (p, n) type and was developed at the RCNP facility in Osaka (JP). Measurements with an unprecedented high resolution of 30 keV at incident energies of 420 MeV are now readily possible. Using both reaction types, one can extract the Gamow-Teller transition strengths B(GT+ ) and B(GT− ), which define the two “legs” of the ββ decay matrix elements for the 2νββ decay. The high-resolution available in both reactions allows a detailed insight into the excitations of the intermediate odd-odd nuclei and, as will be shown, some unexpected features are being unveiled.
1. – Charge-exchange reactions Charge-exchange reactions of (p, n) and (n, p) type at intermediate energies and at forward angles, i.e., low momentum transfers (qtr ∼ 0 and ΔL = 0), selectively excite c Societ` a Italiana di Fisica
175
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D. Frekers
GT transitions owing to the dominance of the Vστ component of the effective interaction [1-4]. However, experiments which employ the elementary (p, n) and (n, p) reactions have rather limited resolution and alternatives to them have now successfully been established through the (n, p)-type (d, 2 He) or (t, 3 He) reactions and the (p, n)-type (3 He, t) reaction. These have been performed at the Kernfysisch Versneller Instituut (KVI), Groningen [5, 6], at the National Superconducting Cyclotron Laboratory at Michigan State University [7], and at the Research Center for Nuclear Physics (RCNP), Osaka [8,9]. Resolutions on the order of 100 keV in the case of (d, 2 He), 190 keV for (t, 3 He) and 30 keV for (3 He, t) have routinely been achieved. The high resolution of 30 keV using a 420 MeV 3 He beam for the (3 He, t) reaction at the RCNP and of 100 keV using a 180 MeV d beam for the (d, 2 He) reaction at the KVI is unprecedented and unique in the world. It allows one to obtain detailed and much warranted information about the nuclear structure that is relevant for the 2νββ decay as well as (though to a lesser extent) for the 0νββ decay. This has recently been exemplified in the cases of 48 Ca 116 Cd and 64 Zn [10-13]. A more recent experiment at RCNP centered around 76 Ge, 82 Se, 96 Zr and 100 Mo has produced results which give even more importance to the high resolution. In the present context it may first be instructive to recall the connection between the 2νββ-decay rate Γ(β2ν− β − ) and the GT transition strength B(GT): (2ν) 2 Γ(β2ν− β − ) = G2ν (Q, Z) MDGT .
(1)
G2ν (Q, Z) is a phase-space factor depending on the Q-value of the reaction and the Z-value of the decaying nucleus. It further contains the weak interaction coupling constant. The 2νββ decay matrix element can be deduced by combining GT+ and GT− distributions in the following way: (2)
(2ν) MDGT
=
) − + − (i) + 0(f g.s. || k σk τk ||1m 1m || k σk τk ||0g.s. m
=
m
(3)
B(GT± ) =
1/ Q (0(f ) ) 2 ββ g.s. +
+ Ex (1+ m ) − E0 −
GT GT Mm · Mm 1/ Q (0(f ) ) 2 ββ g.s.
+ Ex (1+ m ) − E0
± 1 |M GT |2 . 2Ji + 1
1+ state and Here, E(1+ m ) − E0 is the energy difference between the m-th intermediate the initial ground state, Qββ the Q-value of the ββ decay, and the sum k runs over all the neutrons of the decaying nucleus. In this formula the only unknowns are the GamowTeller transition strength values B(GT), which are the quantities to be extracted from the charge-exchange reactions. Of course, the 0νββ decay is potentially the most interesting one, because in weak-interaction gauge theories it requires the neutrino to be a massive Majorana particle
Charge-exchange reactions and nuclear matrix elements etc.
177
irrespective of the mechanism which drives the decay [14]. Its decay rate is given as (4)
Γ(β0ν− β − )
=G
0ν
2 (0ν) gV (0ν) (Q, Z) MDGT − M mνe 2 . gA DF
G0ν (Q, Z) is in general a more favorable phase-space factor than the one in 2νββ mode, (0ν) (0ν) although it only scales with Q5 . The quantities MDGT and MDF are generalized GamowTeller and Fermi matrix elements for 0νββ decay, and mνe is the effective Majorana neutrino mass given as (5) mνe = Uei2 mi . i
The Uei are the elements of the mixing matrix containing two mixing angles θ12 and θ13 as well as two CP phases φ12 and φ13 , and mi are the three corresponding mass eigenvalues. In order to extract the neutrino mass from an observed decay rate, the nuclear matrix elements need to be known with some reasonable reliability. Whereas the matrix elements in the 2νββ decay have a rather simple structure, the ones for the 0νββ decay are significantly more complex, since the neutrino enters into the description as a virtual particle. Usually, the generalized matrix elements are expressed in terms of a neutrino potential operator (cf. refs. [15-18] and references therein): (0ν)
(6)
MDGT = f |
(7)
MDF = f |
σl σk τl− τk− HGT (rlk , Ea )|i
lk
(0ν)
τl− τk− HF (rlk , Ea )|i,
lk
where rlk is the proton-neutron distance in the nucleus, and Ea is an energy parameter related to the excitation energy. (Note that short-range effects become important here.) As the distance rlk is of order the size of the nucleus, the momentum transfers involved can be large, typically of order 0.5 fm−1 , which then allows excitation of many intermediate states and many multipoles up to considerably high energies (∼ 50 MeV). An experimental determination of these matrix elements is an almost impossible task. Further, because of the different operator active for exciting high-J states in hadronic reactions, one cannot easily relate the strength observed in the charge-exchange reaction with the one in the weak-interaction process of ββ decay anymore. Instead, one is forced to resort to theoretical models, but in order for them to be reliable, they must at least be confronted with the experimental matrix elements for the 2νββ decay. 2. – The case of
48
Ca and
64
Zn
In this section we briefly review the experiments, which were performed on the mass A = 48 (48 Ca and 48 Ti) and the mass A = 64 system (64 Zn and 64 Ni). Both experiments have been published in refs. [10,11,13,19], and we refer to these as many of the intricacies of the experiments and the analyses can be found there.
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D. Frekers
Fig. 1. – Excitation spectra for the 48 Ca(3 He, t)48 Sc (upper left panel, from ref. [11]) and 48 Ti(d, 2 He)48 Sc experiments (lower left panel, from ref. [10]). On the right side are the excitation spectra for the 64 Zn(d, 2 He)64 Cu (upper right panel, from ref. [13]) and 64 Ni(3 He, t)64 Cu experiments (lower right panel, from ref. [19]). Levels excited in both spectra are connected by the vertical lines.
The (d, 2 He) experiment on 48 Ti at the KVI Groningen [10] and the (3 He, t) experiment on 48 Ca at the RCNP [11] yielded a resolution of about 100 keV and 40 keV, respectively. The results of the two experiments are shown in fig. 1. Most surprising in the case of 48 Ca was the strong anti-correlation of the GT+ strength with its GT− counterpart, i.e. intermediate states, which were strongly excited via the (p, n)-type reaction were only weakly excited via the (n, p) reaction and vice versa. This anti-correlation has an immediate bearing on the size of the ββ matrix element and, therefore, on the half-life 2ν of 48 Ca. The extracted ββ matrix element was MDGT = 0.063 ± 0.016 MeV−1 and the 48 2ν half-life of the ββ decay of Ca was in the range of 2.4 × 1019 y < T1/2 < 12.9 × 1019 y. The approach of quoting two extremes as the likely interval for the true 48 Ca ββ-decay half-life is due to fact that charge-exchange reactions do not provide information about the sign of the matrix elements [11]. Shell model calculations make some predictions about phases, and in the above paper [11] we have tried to accommodate this additional information, which then widens the interval for the extracted half-life. The result is in agreement with the recently communicated counting result from the NEMO-III experi2ν ment T1/2 = 3.9 × 1019 y [20, 21]. One should note, however, that the present studies are not primarily geared towards a precise determination of the half-life, but rather to bring the details of the underlying nuclear structure to light.
Charge-exchange reactions and nuclear matrix elements etc.
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The anti-correlation of the GT+ and GT− strength was a surprising effect and it was speculated that this was due to the difference of the intrinsic deformation of the mother nucleus 48 Ca and the grand-daughter nucleus 48 Ti. The difference of deformation could, in fact, be a new aspect in the theoretical treatment of ββ decay, which has so far been dealt with only in a rather coarse way. We will see this effect re-appearing in a much more pronounced way in the case of 76 Ge. Figure 1 also shows results from a similar study in the mass A = 64 system, i.e. 64 Zn and 64 Ni. Here, 64 Zn is a candidate for ββ decay into the β + β + direction. We performed a 64 Zn(d, 2 He)64 Cu measurement at the KVI Groningen [13] and compared that to a 64 Ni(3 He, t)64 Cu measurement performed at the RCNP [19]. Through these two measurements the β + β + decay matrix elements were constructed. This marks the first time that experimental ββ decay nuclear matrix elements have been measured for a nucleus, which decays in the β + β + direction. As β + β + decaying nuclei feature a comparatively low neutron excess (e.g., N –Z = 4 for 64 Zn), the GT+ suppression owing to the Pauli blocking of occupied levels in the daughter nucleus was expected to be less severe than for typical β − β − decaying nuclei. This is what was observed by the relatively large B(GT+ ) strength extracted from the 64 Cu daughter at low excitation energies. These large B(GT+ ) values then yielded a comparatively large 2νββ decay matrix element, which in the present case was about an order of magnitude larger than that of 48 Ca. This larger matrix element of 64 Zn accelerates the 2νββ decay by about 2 orders of magnitude, which indicates the general importance of the nuclear structure entering into the dynamics of the ββ decay. Unfortunately, this rather advantageous factor does not offset the large extra suppression, which enters through the β + β + phase space factor compared to the β − β − case, let alone the experimental difficulties associated with the detection of the decay. Using the phase space factors for 2νβ + EC and 2νECEC decay given in ref. [22], one then arrives at half-lives for the 64 Zn decay T1/2 (2νβ + EC) = (4.7 ± 0.9) · 1031 y, T1/2 (2νECEC) = (1.2 ± 0.2) · 1025 y. Here, the rather large value for the 2νβ + EC decay is due to the fact that the Q-value of the 64 Zn decay is only 1.096 MeV leaving for the 2 neutrinos and the positron a mere 74 keV to share (Q − 2m0 c2 ). 3. – The case of
76
Ge
The nucleus 76 Ge is considered one of the most important ββ decaying nuclei. This is the only nucleus, for which a signature for 0νββ decay has so far been reported [23, 24]. The positive report has prompted two new efforts, GERDA and MAJORANA, which will put this observation to a serious test [25, 26]. Both experiments expect to increase the sensitivity level by about 2 orders of magnitude compared to the previous experiment. This constitutes an enormous challenge and both experiments will have to be staged over
2.22 MeV (1+)
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D. Frekers 0.20 76Se(d,2He)76As Θc.m. ~ 0.4° ∆E = 120 keV KVI 2005
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76Ge(3He,t)76As Θc.m. ~ 0.3° ∆E = 30 keV RCNP 2007
60
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76Ge(3He,t)76As Θc.m. ~ 0.3° ∆E = 30 keV
B(GT-)
0
1
2
3
4
5 Ex[MeV]
Fig. 2. – The top left shows the excitation spectra for the 76 Se(d, 2 He)76 As reaction (from ref. [27]), below the 76 Ge(3 He, t)76 As reaction. Levels marked in red are identified as J π = 1+ . An anti-correlation of the transition strength is observed as indicated in the extracted B(GT) values in the figure on the right. Red bars show levels that can be correlated, whereas full black bars show levels that do not have a clearly visible partner.
several phases. Clearly, if a positive result is found, one wishes to extract the mass of the Majorana neutrino with as little theoretical uncertainty as possible. We performed the 76 Se(d, 2 He)76 As experiment at the KVI [27], which yielded a resolution of 120 keV and the 76 Ge(3 He, t)76 As experiment at RCNP with an energy resolution of about 30 keV (see fig. 2). In this A = 76 system we find again a surprisingly strong anti-correlation between the GT− and GT+ transition strengths, which is even more strongly pronounced than in the case of 48 Ca. Many 1+ states are observed in the direction of (3 He, t) and only a few clearly visible states in the direction of (d, 2 He). Moreover, those that do show up strongly in one of the two directions are not significantly correlated with the other. In the right part of fig. 2 we show the experimental distribution of GT strength for the two directions covering the low-energy region up to about 5 MeV. The significant lack of correlation indicates a significant lack of overlap between the wave functions involved and thereby a significant lack of connectivity between the two paths. 2ν In fig. 3 (left) we show the running sum of the matrix element MDGT evaluated from the B(GT) values of the two directions. Assuming an undifferentiated and evenly distributed general background of 0.2 units in the B(GT+ ) distribution up to about 5 MeV (as was identified from the (d, 2 He) reaction, see ref. [27]), then correlating this with all relevant transitions appearing in the (3 He, t) direction and adding this on top of those transitions that can be uniquely connected from the two reaction sides, one arrives at almost exactly the matrix element one would extract from the known half-life. The extracted half-life of 2ν the 2νββ decay of 76 Ge is T1/2 = (1.2±0.2)×1021 y, which agrees with the recommended
Charge-exchange reactions and nuclear matrix elements etc.
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2ν for the 2νββ decay of 76 Ge. The broFig. 3. – The left side shows the running sum of the MDGT ken line corresponds to a running sum from correlated and clearly visible transitions from both directions, whereas the full line includes the contribution from the undifferentiated background, 2ν ≈ 0.16 MeV−1 . The half-life calculated from this value corresponds which then adds to MDGT to the reported half-life from the counting experiments. The right side shows the overlap of BCS wave functions in a deformed QRPA calculation as a function of the difference of intrinsic deformation of the two nuclei 76 Ge and 76 Se (from ref. [28]). This overlap can be taken as a rough measure of the relative size of the matrix elements.
2ν average experimental value of T1/2 = (1.5 ± 0.1) × 1021 y [29]. ˇ In fig. 3 (right) we also show the result of a recent calculation by Simkovic et al. [28], who have calculated the BCS wave function overlap between the two nuclei 76 Ge and 76 Se as a function of the difference between their intrinsic deformation. The shaded area indicates the variation of the results as function of the absolute size of the deformation (here from β = −0.2 to β = +0.2). The idea here is that one can assume that the BCS overlap of the two wave functions is a rough indicator for the relative size of the ββ matrix element. It is generally accepted that the two nuclei 76 Ge and 76 Se have different intrinsic deformation, and we may therefore speculate that the present observation of the mismatch between the GT transition from the different paths could be a result of this wave function mismatch. At present, the theoretical calculations do not give guidance to what the effect of the intrinsic deformation could be on the more interesting 0νββ decay. Two important conclusions may be drawn from these experimental findings:
– The low excitation energy region seems to exhaust almost all the relevant contri2ν butions to the 2νββ decay matrix element MDGT , which is an observation already 48 64 made in the cases of Ca and Zn. – The effect of deformation seems to manifest itself in a mismatch between the different transitions rather than in an overall suppression of B(GT) strength. Further, taking theoretical models as a guide, it seems that the difference of the intrinsic de-
0.04
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182
50
1
2
3
4 5 Ex [MeV]
0
1
2
3
4 5 Ex [MeV]
Fig. 4. – Excitation energy spectrum of the 96 Mo(d, 2 He)96 Nb experiment [30] (left spectrum) performed at KVI and the 96 Zr(3 He, t)96 Nb experiment (right spectrum) performed at RCNP. The entire low-energy GT+ and GT− strength is concentrated in one single state only.
formation between mother and grand-daughter nucleus and not their absolute size has the largest bearing on the overall reduction of the ββ-decay matrix element. 4. – The case of
96
Zr
The GT distribution in the charge-exchange reaction on the mass A = 96 system is completely contrary to what was observed in the A = 76 system. In the (d, 2 He) reaction [30] only one strong GT transition was found at 0.69 MeV with B(GT) = 0.3 units (see left part of fig. 4). This is rather remarkable, since in the simple shell model the conversion of any of the two g9/2 valence protons into a g7/2 neutron can proceed largely unhindered and should, in fact, give rise to several comparatively strong GT transitions. On the other hand, comparing this to the (3 He, t) reaction, we find that the only strongly excited 1+ state is again the one at 0.69 MeV. Almost all of the low-energy GT− strength is concentrated in this state as shown in the right part of fig. 4. This is a rather unique situation of the single state dominance (SSD), which is a term given by Ejiri et al. [31]. A subsequent evaluation of the 2νββ matrix element shows that the entire transition path for the 2νββ decay indeed seems to proceed through this state only. In fact, the extracted half-life from the charge-exchange experiments is T1/2 = (2.4 ± 0.3) × 1019 y, in perfect agreement with the half-life measured by the NEMO experiment [32] given as 2ν 19 19 T1/2 = (2.1+0.8 −0.4 ) × 10 y and recently updated to T1/2 = (2.3 ± 0.3) × 10 y [33]. 5. – The case of
100
Mo
The nucleus 100 Mo is interesting not only from the aspect of ββ decay, but also and foremost, because in a sufficiently large experiment like MOON [34] or SUPER-NEMO
183
yield/10keV
Charge-exchange reactions and nuclear matrix elements etc.
500 100Mo(3He,t)100Tc Θc.m. ~ 0.3° ΔE = 29 keV
400
300 g.s. 200
100
0
-1
0
1
2
3 Ex [MeV]
Fig. 5. – Excitation energy spectrum of the 100 Mo(3 He, t)100 Tc experiment performed at RCNP. The entire GT− strength is concentrated in one single state only, similar to the case of 96 Zr(3 He, t).
it can be used as a neutrino detector for solar neutrinos or even for neutrinos from a supernova explosion. The measured 100 Mo(3 He, t) spectrum leading to the intermediate nucleus 100 Tc is shown in fig. 5. Similar to the case of 96 Zr(3 He, t)96 Nb, we observe that the entire low-energy GT− strength up to at least 5 MeV is concentrated in one single transition only, which is even the one to the ground state. Again, we are faced here with another example of a near perfect SSD situation. Apart from the relevance to ββ decay, this feature has important and significant advantages for the detection of neutrinos, in particular for those originating from a nearby (i.e. in our Galaxy) supernova explosion. These neutrinos, whose mean energy would be about 10 MeV, will almost exclusively populate the ground state through the charged current 100 Mo(ν, e− )100 Tc reaction, thereby leaving their full energy to the electron, which subsequently can be tagged by the 15 s delayed β − decay (Emax = 3.202 MeV) of 100 Tc. This constitutes a rather unique way to measure the neutrino temperature spectrum and possible distortions imprinted onto it due to collective flavor transitions mediated through the self-interactions and matter potentials, as is discussed in ref. [35]. 6. – Conclusion Double-beta decay is presently at the forefront of experimental research in sub-atomic physics. This is because a mere observation of the 0νββ-decay mode would imply the neutrino to be a Majorana particle and thereby signal physics beyond the Standard Model. However, the rate of the 0ν decay critically depends on two unknowns, the mass
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of the neutrino on the one side and the underlying nuclear physics embedded in the nuclear matrix elements on the other. This is a rather uncomfortable situation, as it makes solid experimental planning a challenge. We have shown that charge-exchange reactions like (d, 2 He) and (3 He, t) can probe the ββ decay matrix elements at least for the 2ν variant in a rather detailed way. This was exemplified by new results for the masses A = 76, 96, and 100. Good experimental resolution was one of the important pre-requisites. We have also shown that some new aspects seem to emerge, one of them being the influence of deformation on the size of the matrix elements. A suppression of the matrix elements can be expected if the mother and grand-daughter nuclei exhibit rather different intrinsic deformations. However, this suppression is not caused by a general suppression of the GT strength, but rather by an overall mismatch of the strength in the GT+ and GT− direction. On the other hand, the nuclear structure in the region of A 100 exhibits rather simple features, which theoretical models should be able to deal with. Clearly, in view of the various upcoming initiatives for measuring ββ decay in different systems, a concerted theoretical and experimental effort is needed to address the important issue of the ββ decay nuclear matrix elements. REFERENCES [1] Goodman C. D. et al., Phys. Rev. Lett., 44 (1980) 1755. [2] Taddeucci T. N. et al., Nucl. Phys. A, 469 (1987) 125. [3] Alford W. P. and Jackson K. P., Proceedings of the Workshop on Isovector Excitations in Nuclei, Can. J. Phys., 65 (1987). [4] Jackson K. P. et al., Phys. Lett. B, 201 (1988) 25. [5] Rakers S. et al., Nucl. Instrum. Methods Phys. Res. B, 481 (2002) 253. [6] Rakers S. et al., Phys. Rev. C, 65 (2002) 044323. [7] Hitt G. W. et al., Nucl. Instrum. Methods Phys. Res. A, 566 (2006) 264. [8] Adachi T. et al., Phys. Rev. C, 73 (2006) 024311. [9] Fujita Y. et al., Phys. Rev. Lett., 95 (2005) 212501. [10] Rakers S. et al., Phys. Rev. C, 70 (2004) 054302. [11] Grewe E.-W. et al., Phys. Rev. C, 76 (2007) 054307. [12] Rakers S. et al., Phys. Rev. C, 71 (2005) 054313. [13] Grewe E.-W. et al., Phys. Rev. C, 77 (2008) 064303. [14] Schlechter J. and Valle J. W. F., Phys. Rev. D, 25 (1982) 2951. [15] Suhonen J. and Civitarese O., Phys. Rep., 300 (1998) 123. [16] Suhonen J., Khadkikar S. B. and Faessler A., Phys. Lett. B, 237 (1990) 8. [17] Suhonen J., Khadkikar S. B. and Faessler A., Nucl. Phys. A, 529 (1991) 727. [18] Vergados J. D., Nucl. Phys. A, 506 (1990) 482. [19] Popescu L., Ph.D. thesis, Gent University (2006); Popescu L. et al., to be published (2009). [20] Arnold R. et al., Nucl. Instrum. Methods Phys. Res. A, 536 (2005) 79. [21] Vasiliev V., preliminary value communicated during Physics of Massive Neutrinos, Blaubeuren, Germany, July 2007, Joint Annual Meeting of EU project ILIAS, N4 IDEA, ¨ssler A., Giuliani A., Lalanne D. and Rodin and N6/WP1 ENTApP, organized by Fa V. [22] Kim C. W. and Kubodera K., Phys. Rev. D, 27 (1983) 2765.
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[23] Klapdor-Kleingrothaus H.-V. et al., Mod. Phys. Lett. A, 16 (2001) 2409. [24] Klapdor-Kleingrothaus H.-V. et al., Phys. Lett. B, 586 (2004) 198. KlapdorKleingrothaus H.-V. et al., Nucl. Instrum. Methods Phys. Res. A, 522 (2004) 371. [25] Simgen H. et al., Nucl. Phys. B, 143 (2005) 567; Abt I. et al., hep-ex/0404039, GERDA proposal submitted to Gran Sasso Scientific Committee. [26] Aalseth C. E. et al., Nucl. Phys. B. Proc. Suppl., 138 (2005) 217; Aalseth C. E. et al., Phys. At. Nuclei, 67 (2004) 2002. [27] Grewe E.-W. et al., Phys. Rev. C, 78 (2008) 044301. ˇ [28] Simkovic F. et al., arXiv:nucl-th/0308037 (2003). [29] Barabash A. S., Czech. J. Phys., 56 (2006) 437. [30] Dohmann H. et al., Phys. Rev. C, 78 (2008) 041602. [31] Ejiri H. et al., Phys. Lett. B, 258 (1991) 17. [32] Arnold R. et al., Nucl. Phys. A, 658 (1999) 299. [33] Barabash A., arXiv:0807.2336v2 [nucl-ex] (2008). [34] Ejiri H. et al., Phys. Rev. Lett., 85 (2000) 2917; Ejiri H., Phys. Rep., 338 (2000) 265; Ejiri H. et al., Nucl. Phys. B. Proc. Suppl., 110 (2002) 375; Ejiri H. et al., Phys. Lett. B, 530 (27) 2002; Doe P. et al., Nucl. Phys. A, 721 (2003) 517c; Hazama R. et al., Nucl. Phys. B. Proc. Suppl., 138 (2005) 102; Nomachi H. et al., Nucl. Phys. B. Proc. Suppl., 138 (2005) 221. [35] Fogli G. L. et al., Phys. Rev. D, 78 (2008) 097301; Fogli G. L. et al., arXiv: 0812.3031[hep-ph].
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DOI 10.3254/978-1-60750-038-4-187
Cosmological probes of neutrino masses S. Pastor Institut de F´ısica Corpuscular (CSIC-Universitat de Val` encia) Ed. Instituts d’Investigaci´ o, Ap. correus 22085, 46071 Val` encia, Spain
Summary. — Neutrinos can play an important role in the evolution of the Universe, modifying some of the cosmological observables. In this contribution we summarize the main aspects of cosmological relic neutrinos and we describe how the precision of present cosmological data can be used to learn about neutrino properties, in particular their mass, providing complementary information with respect to beta decay and neutrinoless double-beta decay experiments. We show how the analysis of current cosmological observations, such as the anisotropies of the cosmic microwave background or the distribution of large-scale structure, provides an upper bound on the sum of neutrino masses, with very good perspectives from future cosmological measurements which are expected to be sensitive to neutrino masses well into the sub-eV range.
1. – Introduction In this contribution I summarize the topics discussed in my lectures at the CLXX Course of the International School of Physics “Enrico Fermi”. The subject of my presentations was the role of neutrinos in Cosmology, one of the best examples of the very close ties that have developed between nuclear physics, particle physics, astrophysics and cosmology. I tried to present the most interesting aspects, but many others that were left out can be found in the review by A. D. Dolgov [1]. c Societ` a Italiana di Fisica
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S. Pastor
We begin with a description of the properties and evolution of the background of relic neutrinos that fills the Universe. Then we review the influence of neutrinos on Primordial Nucleosynthesis and the possible effects of neutrino oscillations on Cosmology. The largest part of this contribution is devoted to the impact of massive neutrinos on cosmological observables, that can be used to extract bounds on neutrino masses from present data. Finally we discuss the sensitivities on neutrino masses from future cosmological experiments. Note that light massive neutrinos could also play a role in the generation of the baryon asymmetry of the Universe from a previously created lepton asymmetry. In these leptogenesis scenarios, one can also obtain quite restrictive bounds on light neutrino masses, which are, however, strongly model-dependent. We do not discuss this subject here, as was covered by other lecturer at the School [2]. For further details, the reader is referred to the short reviews on neutrino cosmology [3] or [4], while more information can be found in [5] and in particular in [6]. A more general review on the connection between particle physics and cosmology can be found in [7]. 2. – The cosmic neutrino background The existence of a relic sea of neutrinos is a generic feature of the standard hot Big-Bang model, in number only slightly below that of relic photons that constitute the cosmic microwave background (CMB). This cosmic neutrino background (CNB) has not been detected yet, but its presence is indirectly established by the accurate agreement between the calculated and observed primordial abundances of light elements, as well as from the analysis of the power spectrum of CMB anisotropies. In this section we will summarize the evolution and main properties of the CNB. . 2 1. Relic neutrino production and decoupling. – Produced at large temperatures by frequent weak interactions, cosmic neutrinos of any flavour (νe , νμ , ντ ) were kept in equilibrium until these processes became ineffective in the course of the expansion of the early Universe. While coupled to the rest of the primeval plasma (relativistic particles such as electrons, positrons and photons), neutrinos had a momentum spectrum with an equilibrium Fermi-Dirac form with temperature T , −1 p − μν feq (p, T ) = exp +1 , T
(1)
which is just one example of the general case of particles in equilibrium (fermions or bosons, relativistic or non-relativistic), as shown, e.g., in [8]. In the previous equation we have included a neutrino chemical potential μν that would exist in the presence of . a neutrino-antineutrino asymmetry, but we will see later in subsect. 5 1 that even if it exists, its contribution can be safely ignored. As the Universe cools, the weak interaction rate Γν falls below the expansion rate and one says that neutrinos decouple from the rest of the plasma. An estimate of the
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decoupling temperature Tdec can be found by equating the thermally averaged value of the weak interaction rate (2)
Γν = σν nν ,
where σν ∝ G2F is the cross section of the electron-neutrino processes with GF the Fermi constant and nν is the neutrino number density, with the expansion rate given by the Hubble parameter H (3)
H2 =
8πρ , 3MP2
where ρ ∝ T 4 is the total energy density, dominated by relativistic particles, and MP = 1/G1/2 is the Planck mass. If we approximate the numerical factors to unity, with Γν ≈ G2F T 5 and H ≈ T 2 /MP , we obtain the rough estimate Tdec ≈ 1 MeV. More accurate calculations give slightly higher values of Tdec which are flavour-dependent since electron neutrinos and antineutrinos are in closer contact with electrons and positrons, as shown, e.g., in [1]. Although neutrino decoupling is not described by a unique Tdec , it can be approximated as an instantaneous process. The standard picture of instantaneous neutrino decoupling is very simple (see, e.g., [8] or [9]) and reasonably accurate. In this approximation, the spectrum in eq. (1) is preserved after decoupling, since both neutrino momenta and temperature redshift identically with the expansion of the Universe. In other words, the number density of non-interacting neutrinos remains constant in a comoving volume since the decoupling epoch. We will see later that active neutrinos cannot possess masses much larger than 1 eV, so they were ultra-relativistic at decoupling. This is the reason why the momentum distribution in eq. (1) does not depend on the neutrino masses, even after decoupling, i.e. there is no neutrino energy in the exponential of feq (p). When calculating quantities related to relic neutrinos, one must consider the various possible degrees of freedom per flavour. If neutrinos are massless or Majorana particles, there are two degrees of freedom for each flavour, one for neutrinos (one negative helicity state) and one for antineutrinos (one positive helicity state). Instead, for Dirac neutrinos there are in principle twice more degrees of freedom, corresponding to the two helicity states. However, the extra degrees of freedom should be included in the computation only if they are populated and brought into equilibrium before the time of neutrino decoupling. In practice, the Dirac neutrinos with the “wrong-helicity” states do not interact with the plasma at temperatures of the MeV order and have a vanishingly small density with respect to the usual left-handed neutrinos (unless neutrinos have masses close to the keV range, as explained in sect. 6.4 of [1], but such a large mass is excluded for active neutrinos). Thus the relic density of active neutrinos does not depend on their nature, either Dirac or Majorana particles. Shortly after neutrino decoupling the temperature drops below the electron mass, favouring e± annihilations that heat the photons. If one assumes that this entropy
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1.5
10 TJ = 1.401 TQ
1.4
1 Ti (MeV)
1.3 TJ / TQ
neutrinos photons
e+e- oJJ
1.2
0.1 1.1 TJ = TQ
1 10
1
0.01 0.1
0.1 TJ (MeV)
1 a/a(1 MeV)
10
Fig. 1. – Photon and neutrino temperatures during the process of e± annihilations: evolution of their ratio (left) and their decrease with the expansion of the Universe (right).
transfer did not affect the neutrinos because they were already completely decoupled, it is easy to calculate the change in the photon temperature before any e± annihilation and after the electron-positron pairs disappear by assuming entropy conservation of the electromagnetic plasma. The result is (4)
Tγafter = Tγbefore
11 4
1/3
1.40102,
which is also the ratio between the temperatures of relic photons and neutrinos Tγ /Tν = (11/4)1/3 . The evolution of this ratio during the process of e± annihilations is shown in the left panel of fig. 1, while one can see in the right panel how in this epoch the photon temperature decreases with the expansion less than the inverse of the scale factor a. Instead the temperature of the decoupled neutrinos always falls as 1/a. It turns out that the standard picture of neutrino decoupling described above is slightly modified: the processes of neutrino decoupling and e± annihilations are sufficiently close in time so that some relic interactions between e± and neutrinos exist. These relic processes are more efficient for larger neutrino energies, leading to non-thermal distortions in the neutrino spectra at the per cent level and a slightly smaller increase of the comoving photon temperature, as noted in a series of works (see the full list given in the review [1]). A proper calculation of the process of non-instantaneous neutrino decoupling demands solving the momentum-dependent Boltzmann equations for the neutrino spectra, a set of integro-differential kinetic equations that are difficult to solve numerically. The most recent analysis [10] of this problem has included the effect of flavour neutrino oscillations on the neutrino decoupling process. One finds an increase in the neutrino energy densities with respect to the instantaneous decoupling approximation (0.73% and
Cosmological probes of neutrino masses
191
0.52% for νe ’s and νμ,τ ’s, respectively) and a value of the comoving photon temperature after e± annihilations which is a factor 1.3978 larger, instead of 1.40102. These changes modify the contribution of relativistic relic neutrinos to the total energy density which is taken into account using Neff 3.046, as defined later in eq. (12). In practice, the distortions calculated in [10] only have small consequences on the evolution of cosmological perturbations, and for many purposes they can be safely neglected. Any quantity related to relic neutrinos can be calculated after decoupling with the spectrum in eq. (1) and Tν . For instance, the number density per flavour is fixed by the temperature, (5)
nν =
3 6ζ(3) 3 nγ = T , 11 11π 2 γ
which leads to a present value of 113 neutrinos and antineutrinos of each flavour per cm3 . Instead, the energy density for massive neutrinos should in principle be calculated numerically, with two well-defined analytical limits, (6a) (6b)
4/3 7π 2 4 Tγ4 , 120 11 ρν (mν Tν ) = mν nν .
ρν (mν Tν ) =
. 2 2. Background evolution. – Let us discuss the evolution of the CNB after decoupling in the expanding Universe, which is described by the Friedmann-Robertson-Walker metric [9] (7)
ds2 = dt2 − a(t)2 δij dxi dxj ,
where we assumed negligible spatial curvature. Here a(t) is the scale factor usually normalized to unity now (a(t0 ) = 1) and related to the redshift z as a = 1/(1 + z). General relativity tells us the relation between the metric and the matter and energy in the Universe via the Einstein equations, whose time-time component is the Friedmann equation (8)
2 8πG ρ a˙ ρ = H02 0 , = H2 = a 3 ρc
that gives the Hubble rate in terms of the total energy density ρ. At any time, the critical density ρc is defined as ρc = 3H 2 /8πG, and the current value H0 of the Hubble parameter gives the critical density today (9)
ρ0c = 1.8788 × 10−29 h2 g cm−3 ,
where h ≡ H0 /(100 km s−1 Mpc−1 ). The different contributions to the total energy
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S. Pastor
TQ (eV) 10
Ui1/4 (eV)
10
10
6
3
TQ (K)
1
10
-3
1
cdm b J / Q3 Q2 Q1
3
1
10
9
10
6
10
3
1.95
0.1
:i
10
6
0.01
-3
BBN
10
J dec.
0.001
10-9
10-6 a/a0
10-3
1e-04 1
10-9
10-6 a/a0
10-3
1
Fig. 2. – Evolution of the background energy densities (left) and density fractions Ωi (right) from the time when Tν = 1 MeV until now, for each component of a flat Universe with h = 0.7 and current density fractions ΩΛ = 0.70, Ωb = 0.05, Ων = 0.0013 and Ωcdm = 1 − ΩΛ − Ωb − Ων . The three neutrino masses are m1 = 0, m2 = 0.009 eV and m3 = 0.05 eV.
density are (10)
ρ = ργ + ρcdm + ρb + ρν + ρΛ ,
and the evolution of each component is given by the energy conservation law in an expanding Universe, ρ˙ = −3H(ρ + p), where p is the pressure. Thus the homogeneous density of photons ργ scales like a−4 , that of non-relativistic matter (ρcdm for cold dark matter and ρb for baryons) like a−3 , and the cosmological constant density ρΛ is of course time-independent. Instead, the energy density of neutrinos contributes to the radiation density at early times but behaves as matter after the non-relativistic transition. The evolution of all densities is shown on the left plot of fig. 2, starting at MeV temperatures until now. We also display the characteristic times for the end of Primordial Nucleosynthesis and for photon decoupling or recombination. The evolution of the density fractions Ωi ≡ ρi /ρc is shown on the right panel, where it is easier to see which of the Universe components is dominant, fixing its expansion rate: first radiation in the form of photons and neutrinos (Radiation Domination or RD), then matter which can be CDM, baryons and massive neutrinos at late times (Matter Domination or MD) and finally the cosmological constant density takes over at low redshift (typically z < 0.5). Massive neutrinos are the only particles that present the transition from radiation to matter, when their density is clearly enhanced (upper solid lines in the right panel of fig. 2). Obviously the contribution of massive neutrinos to the energy density in the nonrelativistic limit is a function of the mass (or the sum of all masses for which mi Tν ), and the present value Ων could be of order unity for eV masses (see sect. 6).
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3. – Neutrinos and primordial nucleosynthesis In the course of its expansion, when the early Universe was only less than a second old, the conditions of temperature and density of its nucleon component were such that light nuclei could be created via nuclear reactions (for a recent review, see [11]). During this epoch, known as Primordial or Big-Bang Nucleosynthesis (BBN), the primordial abundances of light elements were produced: mostly 4 He but also smaller quantities of less stable nuclei such as D, 3 He and 7 Li. Heavier elements could not be produced because of the rapid evolution of the Universe and its small nucleon content, related to the small value of the baryon asymmetry which normalized to the photon density, ηb ≡ (nb −nb¯ )/nγ , was about a few times 10−10 . Measuring these primordial abundances today is a very difficult task, because stellar process may have altered the chemical compositions. Still, data on the primordial abundances of 4 He, D and 7 Li exist and can be compared with the theoretical predictions to learn about the conditions of the Universe at such an early period. Thus BBN can be used as a cosmological test of any non-standard physics or cosmology [12]. The physics of BBN is well understood, since in principle only involves the Standard Model of particle physics and the time evolution of the expansion rate as given by the Friedmann equation (3). In the first phase of BBN the weak processes that had kept the neutrons and protons in equilibrium,
(11)
n + νe ↔ p + e− ,
n + e+ ↔ p + ν¯e ,
freeze and the neutron-to-proton ratio becomes a constant (later diminished due to neutron decays, n → p + e− + ν¯e ). This ratio largely fixes the produced 4 He abundance. Later all the primordial abundances of light elements are produced and their value depend on the competition between the nuclear reaction rates and the expansion rate of the Universe. These values can be quite precisely calculated with a BBN numerical code (see, e.g., [13]). At present there exists a nice agreement with the observed abundance of D for a value of the baryon asymmetry η = 6.1 ± 0.6 [11], which also agrees with the region determined by CMB and large-scale structure data (LSS). Instead, the predicted primordial abundance of 4 He tends to be a bit larger than the observed value. However, it is difficult to consider this as a serious discrepancy, because the accuracy of the observations of 4 He is limited by systematic uncertainties. There are two main effects of relic neutrinos at BBN. The first one is that they contribute to the relativistic energy density of the universe (if mν Tν ), thus fixing the expansion rate. This is why BBN gave the first allowed range of the number of neutrino species before accelerators (see the next section). On the other hand, BBN is the last period of the Universe sensitive to neutrino flavour, since electron neutrinos and antineutrinos play a direct role in the processes in eq. (11). We will see some examples of BBN bounds on neutrinos (effective number or oscillations) in the following sections.
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4. – Extra radiation and the effective number of neutrinos Together with photons, in the standard case neutrinos fix the expansion rate during the cosmological era when the Universe is dominated by radiation. Their contribution to the total radiation content can be parametrized in terms of the effective number of neutrinos Neff , through the relation
(12)
7 ρr = ργ + ρν = 1 + 8
4 11
4/3 Neff
ργ ,
where we have normalized to the photon energy density because its value today is known from the measurement of the CMB temperature. This equation is valid when neutrino decoupling is complete and holds as long as all neutrinos are relativistic. We know that the number of light neutrinos sensitive to weak interactions (flavour or active neutrinos) equals three from the analysis of the invisible Z-boson width at LEP, Nν = 2.994 ± 0.012 [14], and we saw in a previous section from the analysis of neutrino decoupling that they contribute as Neff 3.046. Any departure of Neff from this last value would be due to non-standard neutrino features or to the contribution of other relativistic relics. For instance, the energy density of a hypothetical scalar particle φ in equilibrium with the same temperature as neutrinos would be ρφ = (π/30) Tν4 , leading to a departure of Neff from the standard value of 4/7. A detailed discussion of cosmological scenarios where Neff is not fixed to three can be found in [12] or [1]. In the previous section we saw that the expansion rate during BBN fixes the produced abundances of light elements, and in particular that of 4 He. Thus the value of Neff can be constrained at the BBN epoch from the comparison of theoretical predictions and experimental data on the primordial abundances of light elements. In addition, a value of Neff different from the standard one would modify the transition epoch from a radiation-dominated to a matter-dominated Universe, which has some consequences on some cosmological observables such as the power spectrum of CMB anisotropies, leading to independent bounds on the radiation content. These are two complementary ways of constraining Neff at very different epochs. Here, as an example, we will only describe the results of a recent analysis [15] (see the references therein for a list of recent works), who considered both BBN and CMB/LSS data. The allowed regions on the plane defined by Neff and the baryon contribution to the present energy density Ωb h2 are shown in fig. 3. The BBN contours were calculated with the 4 He and D abundances, leading to the allowed range Neff = 3.1+1.4 −1.2 (95% CL). Instead, the filled contours correspond to the regions found from the analysis of the most recent cosmological data on CMB temperature anisotropies and polarization, Large Scale galaxy clustering from SDSS and 2dF and luminosity distances of type-Ia Supernovae (we will describe these measurements later in sect. 8). The bounds on the effective number of neutrinos for this case are Neff = 5.2+2.7 −2.2 , while in a less conservative analysis data on the Lyman-α absorption clouds (Ly-α) and the Baryonic Acoustic Oscillations (BAO) from SDSS were added, leading to Neff = 4.6+1.6 −1.5 .
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Fig. 3. – Allowed regions on the (ωb , Nνeff )-plane at 68% and 95% CL from the analysis of BBN (dotted lines) and cosmological data from other observables as shown (filled contours). The horizontal lines correspond to the standard prediction of Neff 3.046. From [15].
These ranges are in reasonable agreement with the standard prediction of Neff 3.046, shown as a horizontal line on the plots in fig. 3. Moreover, they show that there exists an allowed region of Neff values that is common at early (BBN) and more recent epochs, although some tension remains particularly when adding Lyman-α and BAO data. However, the reader should be cautious in the interpretation of any of these allowed regions as an indication for relativistic degrees of freedom beyond the contribution of flavour neutrinos. Similar results for Neff with more updated cosmological data can be found in [16-18]. 5. – Neutrino oscillations in the Early Universe Nowadays there exist compelling evidences for flavour neutrino oscillations from a variety of experimental data on solar, atmospheric, reactor and accelerator neutrinos. These are very important results, because the existence of flavour change implies that neutrinos mix and have non-zero masses, which in turn requires particle physics beyond the Standard Model. Thus, it is interesting to check whether neutrino oscillations can modify any of the cosmological observables. More on neutrino oscillations and their implications can be found in [19] or any of the existing reviews such as [20-22], to which we refer the reader for more details. It turns out that in the standard cosmological picture all flavour neutrinos were pro. duced with the same energy spectrum, as we saw in subsect. 2 1, so we do not expect any effect from the oscillations among these three states. This is true up to the small spectral distortion caused by the heating of neutrinos from e+ e− annihilations [10], as we described before. In this section we will briefly consider two cases where neutrino oscillations could have cosmological consequences: flavour oscillations with non-zero relic neutrino asymmetries and active-sterile neutrino oscillations.
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. 5 1. Active-active neutrino oscillations: relic neutrino asymmetries. – A non-zero relic neutrino asymmetry exists when the number densities of neutrinos and antineutrinos of a given flavour are different. Such a putative asymmetry, that was produced by some mechanism in the very early Universe well before the thermal decoupling epoch, can be quantified assuming that a given flavour is characterized by a Fermi-Dirac distribution as in eq. (1) with a non-zero chemical potential μν or equivalently with the dimensionless degeneracy parameter ξν ≡ μν /T (for antineutrinos, ξν¯ = −ξν ). In such a case, sometimes one says that the relic neutrinos are degenerate (but not in the sense of equal masses). Degenerate electron neutrinos have a direct effect on BBN: we saw in sect. 3 that any change in the νe /¯ νe spectra modifies the primordial neutron-to-proton ratio, which in this case is n/p ∝ exp[−ξνe ]. Therefore, a positive ξνe decreases the primordial 4 He mass fraction, while a negative ξνe increases it, leading to an allowed range −0.01 < ξνe < 0.07,
(13)
compatible with ξνe = 0 and very restrictive for negative values. In addition a non-zero relic neutrino asymmetry always enhances the contribution of the CNB to the relativistic energy density, since for any ξν one has a departure from the standard value of the effective number of neutrinos Neff given by (14)
ΔNeff
4 2 15 ξν ξν 2 . = + 7 π π
We have seen that this increased radiation modifies the outcome of BBN and that bounds on Neff can be obtained. In addition, another consequence of the extra radiation density is that it postpones the epoch of matter-radiation equality, producing observable effects on the spectrum of CMB anisotropies and the distribution of cosmic large-scale structures (LSS). Both independent bounds on the radiation content can be translated into flavourindependent limits on ξν . Altogether these cosmological limits on the neutrino chemical potentials or relic neutrino asymmetries are not very restrictive, since at least for BBN their effect in the νμ or ντ sector can be compensated by a positive ξνe . For example, an analysis of the combined effect of a non-zero neutrino asymmetry on BBN and CMB/LSS yields the allowed regions [23] (15)
−0.01 < ξνe < 0.22,
|ξνμ,τ | < 2.6,
in agreement with similar but more updated bounds as cited in [6]. These limits allow for a very significant radiation contribution of degenerate neutrinos, leading many authors to discuss the implications of a large neutrino asymmetry in different physical situations (see, e.g., [1]).
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It is obvious that the limits in eq. (15) would be modified if neutrino flavour oscillations were effective before BBN, equalizing the neutrino chemical potentials. Actually, it was shown in [24] that this is the case for the neutrino mixing parameters in the region favoured by present data. This result is obtained only after the proper inclusion of the refractive terms produced by the background neutrinos, which synchronize the oscillations of neutrinos with different momenta (which would evolve independently without them). In summary, since flavour equilibrium is reached before BBN, the restrictive limits on ξνe in eq. (13) apply to all flavours. The current bounds on the common value of the neutrino degeneracy parameter ξν ≡ μν /T are −0.05 < ξν < 0.07 at 2σ [25]. Thus the contribution of a relic neutrino asymmetry can be safely ignored, in turn implying that the cosmic neutrino radiation density is close to its standard value. . 5 2. Active-sterile neutrino oscillations. – In addition to the flavour or active neutrinos (three species as we saw from accelerator data), there could also exist extra massive neutrino states that are sterile, i.e. singlets of the Standard Model gauge group and thus insensitive to weak interactions. These sterile states νs were either not present in the early Universe or severely suppressed with respect to the active ones, but they could be populated through the effect of active-sterile oscillations if non-zero mixing exists, additional to that among the flavour states. This “thermalization” of the sterile neutrinos is a well-known phenomenon that is very difficult to avoid unless the cosmological scenario is drastically modified. For instance, the suppression of active-sterile oscillations may require very large pre-existing neutrino asymmetries (see, e.g., [26] and references therein). There are two possible regimes for the active-sterile oscillations in cosmology, depending if they are effective before or after the decoupling of the flavour states. If oscillations occur before thermal decoupling, the abundance of sterile neutrinos grows via the conversion of the active states, whose spectrum is kept in equilibrium by the frequent weak interactions. In such a case, the contribution of sterile neutrinos to the radiation energy density is additional to that of the active states, leading to an extra Neff between 0 and 1 (for only one sterile species, it would be larger for more states) depending on how effective was the thermalization of the sterile neutrinos. Instead, for active-sterile oscillations effective after T < 1 MeV, i.e. after decoupling, the total number of neutrinos is conserved and the main feature is that large distortions may appear on the spectra of both the sterile and the active states, with its consequences on BBN when the active neutrinos are of the electron flavour (see, e.g. [27], also for the case of non-zero initial νs abundance). The detailed evolution of active-sterile oscillations in the early Universe is a difficult task, that requires solving the corresponding kinetic equations taking into account the medium effects. More details can be found, for instance, in the works [28] and [29] where detailed numerical calculations were carried out also including the unavoidable mixing between the flavour neutrinos (a 4 × 4 neutrino case). In general, one can obtain bounds on the region of active-sterile mixing parameters from the comparison with BBN and CMB/LSS data, as can be seen in the figures of the previous references or the summary in [1].
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Albeit the current data on neutrino oscillations does not seem to favour the existence of mixing with sterile states, the cosmological bounds provide complementary information valid also for some regions of parameters beyond the sensitivity of the laboratory experiments. Note, however, that in order to explain the results of the Liquid Scintillator Neutrino Detector (LSND) [30], an experiment that measured the appearance of electron antineutrinos in a muon antineutrino beam, a fourth sterile neutrino would be required with mass of O(eV). Recently, the MiniBoone experiment [31] has excluded almost completely two-neutrino oscillations as an explanation of the LSND results. In any case, the existence of light sterile neutrinos would have profound consequences in cosmology, first of all because it is very difficult to avoid its full thermalization via active-sterile oscillations. 6. – Massive neutrinos as Dark Matter Nowadays the existence of Dark Matter (DM), the dominant non-baryonic component of the matter density in the Universe, is well established. A priori, massive neutrinos are excellent DM candidates, in particular because we are certain that they exist, in contrast with other candidate particles. Together with CMB photons, relic neutrinos can be found anywhere in the Universe with a number density given by the present value of eq. (5) of 339 neutrinos and antineutrinos per cm3 , and their energy density in units of the critical value of the energy density (see eq. (9)) is ρν i mi . Ων = 0 = ρc 93.14 h2 eV
(16)
Here i mi includes all masses of the neutrino states which are non-relativistic today. It is also useful to define the neutrino density fraction fν with respect to the total matter density fν ≡
(17)
Ων ρν = . (ρcdm + ρb + ρν ) Ωm
In order to check whether relic neutrinos can have a contribution of order unity to the present values of Ων or fν , we should consider which neutrino masses are allowed by non-cosmological data. Oscillation experiments measure the differences of squared neutrino masses Δm221 = m22 − m21 and Δm231 = m23 − m21 , the relevant ones for solar and atmospheric neutrinos, respectively [19]. As a reference, we take the following 3σ ranges of mixing parameters from an update of [20], (18)
−5 Δm221 = (7.6+0.7 eV2 , −0.5 ) × 10
s212 = 0.32+0.08 −0.06 ,
|Δm231 | = (2.4 ± 0.4) × 10−3 eV2 ,
s223 = 0.50+0.17 −0.16 ,
s213 ≤ 0.05.
Here sij = sin θij , where θij (ij = 12, 23 or 13) are the three mixing angles. Unfortunately oscillation experiments are insensitive to the absolute scale of neutrino masses, since the
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1
3 Inverted Normal 1 3
6mi (eV)
mi (eV)
0.1
1, 2 2
0.3
0.01 0.1 1
0.001 0.05
3
0.1
Inverted Normal 0.3 6mi (eV)
0.5
1
0.03 0.001
0.01 0.1 lightest mQ (eV)
1
Fig. 4. – Expected values of neutrino masses according to the values in eq. (18). Left: individual neutrino masses as a function of the total mass for the best-fit values of the Δm2 . Right: ranges of total neutrino mass as a function of the lightest state within the 3σ regions (thick lines) and for a future determination at the 5% level (thin lines).
knowledge of Δm221 > 0 and |Δm231 | leads to the two possible schemes shown in fig. 1 of [6], but leaves one neutrino mass unconstrained. These two schemes are known as normal (NH) and inverted (IH) hierarchies, characterized by the sign of Δm231 , positive and negative, respectively. For small values of the lightest neutrino mass m0 , i.e. m1 (m3 ) for NH (IH), the mass states follow a hierarchical scenario, while for masses much larger than the differences all neutrinos share in practice the same mass and then we say that they are degenerate. In general, the relation between the individual masses and the total neutrino mass can be found numerically, as shown in fig. 4. There are two types of laboratory experiments searching for the absolute scale of neutrino masses, a crucial piece of information for constructing models of neutrino masses and mixings [32]. The neutrinoless double-beta decay (Z, A) → (Z + 2, A) + 2e− (in short 0ν2β) is a rare nuclear processes where lepton number is violated and whose observation would mean that neutrinos are Majorana particles. If the 0ν2β process is mediated by a light neutrino, the results from neutrinoless double-beta decay experiments are converted into an upper bound or a measurement of the effective mass mββ (19)
mββ = |c212 c213 m1 + s212 c213 m2 eiφ2 + s213 m3 eiφ3 |,
where φ1,2 are the two Majorana phases that appear in lepton-number–violating processes. An important issue for 0ν2β results is related to the uncertainties on the corresponding nuclear matrix elements, that were discussed in detail by other speakers in the School [33-35]. For more details and the current experimental results, see [36]. Beta decay experiments, which involve only the kinematics of electrons, are in principle the best strategy for measuring directly the neutrino mass [37,38]. The current limits
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from tritium beta decay apply only to the range of degenerate neutrino masses, so that mβ m0 , where (20)
mβ = (c212 c213 m21 + s212 c213 m22 + s213 m23 )1/2 ,
is the relevant parameter for beta decay experiments. The bound at 95% CL is m0 < 2.05–2.3 eV from the Troitsk and Mainz experiments, respectively. This value is expected to be improved by the KATRIN project to reach a discovery potential for 0.3–0.35 eV masses (or a sensitivity of 0.2 eV at 90% CL). Taking into account this upper bound and the minimal values of the total neutrino mass in the normal (inverted) hierarchy, the sum of neutrino masses is restricted to the approximate range (21) 0.06 (0.1) eV mi 6 eV. i
As we discuss in the next sections, cosmology is at first order sensitive to the total neutrino mass i mi if all states have the same number density, providing information on m0 but blind to neutrino mixing angles or possible CP violating phases. Thus cosmological results are complementary to terrestrial experiments. The interested reader can find the allowed regions in the parameter space defined by any pair of parameters ( i mi , mββ , mβ ) in [21, 39]. Now we can find the possible present values of Ων in agreement with the three neutrino masses shown in fig. 4 and the approximate bounds of eq. (21). Note that even if the three neutrinos are non-degenerate in mass, eq. (16) can be safely applied, because we know from neutrino oscillation data that at least two of the neutrino states are nonrelativistic today, since both (Δm231 )1/2 0.05 eV and (Δm221 )1/2 0.009 eV are larger than the temperature Tν 1.96 K 1.7 × 10−4 eV. If the third neutrino state is very light and still relativistic, its relative contribution to Ων is negligible and eq. (16) remains an excellent approximation of the total density. One finds that Ων is restricted to the approximate range (22)
0.0013 (0.0022) Ων 0.13,
where we already included that h ≈ 0.7. This applies only to the standard case of three light active neutrinos, while in general a cosmological upper bound on Ων has been used since the 1970s to constrain the possible values of neutrino masses. For instance, if we demand that neutrinos should not be heavy enough to overclose the Universe (Ων < 1), we obtain an upper bound i mi 45 eV (again fixing h = 0.7). Moreover, since from present analyses of cosmological data we know that the approximate contribution of matter is Ωm 0.3, the neutrino masses should obey the stronger bound i mi 15 eV. We see that with this simple argument one obtains a bound which is roughly only a factor 2 worse than the bound from tritium beta decay, but of course with the caveats that apply to any cosmological analysis. In the three-neutrino case, these bounds should be understood in terms of m0 = i mi /3.
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Dark-matter particles with a large velocity dispersion such as that of neutrinos are called hot dark matter (HDM). The role of neutrinos as HDM particles has been widely discussed since the 1970s, and the reader can find a historical review in [40]. It was realized in the mid-1980s that HDM affects the evolution of cosmological perturbations in a particular way: it erases the density contrasts on wavelengths smaller than a massdependent free-streaming scale. In a universe dominated by HDM, this suppression is in contradiction with various observations. For instance, large objects such as superclusters of galaxies form first, while smaller structures like clusters and galaxies form via a fragmentation process. This top-down scenario is at odds with the fact that galaxies seem older than clusters. Given the failure of HDM-dominated scenarios, the attention then turned to cold dark matter (CDM) candidates, i.e. particles which were non-relativistic at the epoch when the universe became matter-dominated, which provided a better agreement with observations. Still in the mid-1990s it appeared that a small mixture of HDM in a universe dominated by CDM fitted better the observational data on density fluctuations at small scales than a pure CDM model. However, within the presently favoured ΛCDM model dominated at late times by a cosmological constant (or some form of dark energy) there is no need for a significant contribution of HDM. Instead, one can use the available cosmological data to find how large the neutrino contribution can be, as we will see later. Before concluding this section, we would like to mention the case of a sterile neutrino with a mass of the order of a few keV’s and a very small mixing with the flavour neutrinos. Such “heavy” neutrinos could be produced by active-sterile oscillations but not fully thermalized, so that they could play the role of dark matter and replace the usual CDM component. But due to their large thermal velocity (slightly smaller than that of active neutrinos), they would behave as Warm Dark Matter and erase small-scale cosmological structures. Their mass can be bounded from below using Lyman-α forest data from quasar spectra, and from above using X-ray observations. The viability of this scenario is currently under careful examination (see, e.g., [41, 42] for recent analyses and a list of references). 7. – Effects of neutrino masses on cosmology In this section we will briefly describe the main cosmological observables and the effects that neutrino masses cause on them. A more detailed discussion of the effects of massive neutrinos on the evolution of cosmological perturbations can be found in sects. 4.5 and 4.6 of [6]. . 7 1. Brief description of cosmological observables. – Although there exist many different types of cosmological measurements, here we will restrict the discussion to those that are at present the more important for obtaining an upper bound or eventually a measurement of neutrino masses. First of all, we have the CMB temperature anisotropy power spectrum [43], defined as the angular two-point correlation function of CMB maps δT /T¯(ˆ n) (ˆ n being a direction
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in the sky). This function is usually expanded in Legendre multipoles (23)
∞ δT (2l + 1) δT Cl Pl (ˆ ) = n·n ˆ ), (ˆ n ) (ˆ n ¯ ¯ 4π T T l=0
where Pl (x) are the Legendre polynomials. For Gaussian fluctuations, all the information is encoded in the multipoles Cl which probe correlations on angular scales θ = π/l. We have seen that each neutrino family can only have a mass of the order of 1 eV, so that the transition of relic neutrinos to the non-relativistic regime is expected to take place after the time of recombination between electrons and nucleons, i.e. after photon decoupling. Since the shape of the CMB spectrum is related mainly to the physical evolution before recombination, it will be only marginally affected by the neutrino mass, except for an indirect effect through the modified background evolution. There exists interesting complementary information to the temperature power spectrum if the CMB polarization is measured, and currently we have some less precise data on the temperature × E-polarization (TE) correlation function and the E-polarization self-correlation spectrum (EE). The current Large Scale Structure (LSS) of the Universe is probed by the matter power spectrum, observed with various techniques described in the next section (directly or indirectly, today or in the near past at redshift z). It is defined as the two-point correlation function of non-relativistic matter fluctuations in Fourier space (24)
P (k, z) = |δm (k, z)|2 ,
ρm . Usually P (k) refers to the matter power spectrum evaluated where δm = δρm /¯ today (at z = 0). In the case of several fluids (e.g., CDM, baryons and non-relativistic neutrinos), the total matter perturbation can be expanded as ρ¯i δi (25) δm = i . ¯i i ρ Since the energy density is related to the mass density of non-relativistic matter through E = mc2 , δm represents indifferently the energy or mass power spectrum. The shape of the matter power spectrum is affected in a scale-dependent way by the free streaming caused by small neutrino masses of O(eV) and thus it is the key observable for constraining mν with cosmological methods. We will show later in fig. 5 the typical shape of both the CMB temperature anisotropy spectrum Cl and the matter power spectrum P (k). . 7 2. Neutrino free streaming. – After thermal decoupling, relic neutrinos constitute a collisionless fluid, where the individual particles free-stream with a characteristic velocity that, in average, is the thermal velocity vth . It is possible to define a horizon as the typical distance on which particles travel between time ti and t. When the Universe was dominated by radiation or matter t ti , this horizon is, as usual, asymptotically equal
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no Q’s fQ=0
5000
fQ=0.1
10
4
10
3
10
2
3
P(k) (Mpc/h)
l(l+1) Cl / 2S (PK)
2
6000
4000 3000 2000
no Q’s fQ=0
1000
fQ=0.1 0 2
200
400
600
800 l
1000
1200
1400
10
-3
-2
10 k (h/Mpc)
10
-1
Fig. 5. – CMB temperature anisotropy spectrum ClT and matter power spectrum P (k) for three models: the neutrinoless ΛCDM model, a more realistic ΛCDM model with three massless neutrinos (fν 0), and finally a ΛMDM model with three massive degenerate neutrinos and a total density fraction fν = 0.1. In all models, the values of the cosmological parameters (ωb = Ωb h2 , ωm = Ωm h2 , ΩΛ , As , n, τ ) have been kept fixed. From [6].
to vth /H, up to a numerical factor of order one. Similar to the definition of the Jeans length (see sect. 4.4 in [6]), we can define the neutrino free-streaming wave number and length as 1/2 4πG¯ ρ(t)a2 (t) kFS (t) = , 2 (t) vth ) a(t) 2 vth (t) λFS (t) = 2π = 2π . kFS (t) 3 H(t)
(26a)
(26b)
As long as neutrinos are relativistic, they travel at the speed of light and their freestreaming length is simply equal to the Hubble radius. When they become nonrelativistic, their thermal velocity decays like (27)
vth
3Tν 3Tν0 a0 1 eV p
=
150(1 + z) km s−1 , ≡ m m m a m
where we used for the present neutrino temperature Tν0 (4/11)1/3 Tγ0 and Tγ0 2.726 K. This gives for the free-streaming wavelength and wave number during matter or Λ domination
(28a) (28b)
ΩΛ + Ωm (1 + z)3 m h Mpc−1 , (1 + z)2 1 eV 1 eV 1+z h−1 Mpc , λFS (t) = 7.7 3 m ΩΛ + Ωm (1 + z) kFS (t) = 0.82
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where ΩΛ and Ωm are the cosmological constant and matter density fractions, respectively, evaluated today. After the non-relativistic transition and during matter domination, the free-streaming length continues to increase, but only like (aH)−1 ∝ t1/3 , i.e. more slowly than the scale factor a ∝ t2/3 . Therefore, the comoving free-streaming length λFS /a actually decreases like (a2 H)−1 ∝ t−1/3 . As a consequence, for neutrinos becoming non-relativistic during matter domination, the comoving free-streaming wave number passes through a minimum knr at the time of the transition, i.e. when m = p = 3Tν and a0 /a = (1 + z) = 2.0 × 103 (m/1 eV). This minimum value is found to be (29)
knr 0.018 Ω1/2 m
m 1/2 h Mpc−1 . 1 eV
The physical effect of free streaming is to damp small-scale neutrino density fluctuations: neutrinos cannot be confined into (or kept outside of) regions smaller than the free-streaming length, for obvious kinematic reasons. There exists a gravitational backreaction effect that also damps the metric perturbations on those scales. Instead, on scales much larger than the free-streaming scale the neutrino velocity can be effectively considered as vanishing and after the non-relativistic transition the neutrino perturbations behave like CDM perturbations. In particular, modes with k < knr are never affected by free-streaming and evolve like in a pure ΛCDM model. . 7 3. Impact of massive neutrinos on the matter power spectrum. – The small initial cosmological perturbations in the early Universe evolve, under the linear regime at any scale at early times and on the largest scales more recently, and produce the structures we see today. We will not review here all the details (see [6] and references therein), but we will emphasize the main effects caused by massive neutrinos in the framework of the standard cosmological scenario: a Λ Mixed Dark Matter (ΛMDM) model, where Mixed refers to the inclusion of some HDM component. First let us describe the changes in the background evolution of the Universe. We have seen that massless neutrinos are always part of the radiation content, so in this case the present value of the matter contribution Ω0m is equal to the contribution of CDM and baryons. Instead, massive neutrinos contribute to radiation at early times but to matter after becoming non-relativistic. Thus with respect to the massless neutrino case, massive neutrinos also contribute to Ω0m , reducing the values of Ω0CDM and Ω0b . As a result, if these massive neutrinos have not yet become non-relativistic at the time of radiation/matter equality (the epoch of the Universe when its starts to be dominated by matter and the contribution of radiation becomes subdominant), then this transition is delayed. The consequence of a late equality for the LSS matter power spectrum is the following: since on sub-Hubble scales the matter density contrast δm grows more efficiently during MD than during RD, the matter power spectrum is suppressed on small scales relatively to large scales. At the perturbation level, we also saw that free-streaming damps small-scale neutrino density fluctuations. This produces a direct effect on the matter power spectrum
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(see sect. 4.5 of [6]) that depends on the value k with respect to knr in eq. (29), (30)
P (k) = = =
δρcdm + δρb + δρν ρcdm + ρb + ρν
2
Ωcdm δcdm + Ωb δb + Ων δν Ωcdm + Ωb + Ων
2 δcdm ,
2 , [1 − Ων /Ωm ]2 δcdm
2
for k < knr , for k knr ,
with Ωm ≡ Ωcdm + Ωb + Ων . Thus the role of the neutrino masses would be simply to cut the power spectrum by a factor [1 − Ων /Ωm ]2 for k knr . However, it turns out that the presence of neutrinos actually modifies the evolution of the CDM and baryon density contrasts in such way that the suppression factor is greatly enhanced, more or less by a factor four. In conclusion, the combined effect of the shift in the time of equality and of the reduced CDM fluctuation growth during matter domination produces an attenuation of small-scale perturbations for k > knr . It can be shown that for small values of fν this effect can be approximated in the large k limit by the well-known linear expression [44] (31)
P (k)fν
1 − 8 fν . P (k)fν =0
For the comparison with the data, one could use instead some better analytical approximations to the full MDM or ΛMDM matter power spectrum, valid for arbitrary scales and redshifts, as listed in [6]. However, nowadays the analyses are performed using the matter power spectra calculated by Boltzmann codes such as cmbfast [45] or camb [46], that solve numerically the evolution of the cosmological perturbations. An example of P (k) with and without massive neutrinos is shown in fig. 5, where the effect of mν at large k’s can be clearly visible. Such suppression is probably better seen in fig. 6, where we plot the ratio of the matter power spectrum for ΛMDM over that of ΛCDM, for different values of fν and three degenerate massive neutrinos, but for fixed parameters (ωm , ΩΛ ). For large k’s, eq. (31) is a reasonable first-order approximation for 0 < fν < 0.07. Is it possible to mimic the effect of massive neutrinos on the matter power spectrum with some combination of other cosmological parameters? If so, one would say that a parameter degeneracy exists, reducing the sensitivity to neutrino masses. This possibility depends on the interval [kmin , kmax ] in which the P (k) can be accurately measured. Ideally, if we could have kmin ≤ 10−2 h Mpc−1 and kmax ≥ 1 h Mpc−1 , the effect of the neutrino mass would be non-degenerate, because of its very characteristic step-like effect. In contrast, other cosmological parameters like the scalar tilt or the tilt running change the spectrum slope on all scales. The problem is that usually the matter power spectrum can only be accurately measured in the intermediate region where the mass
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0.8 0.6
f
P(k) ν / P(k) ν
f =0
1
0.4 0.2 0 10-4
10-3
10-2 10-1 k (h/Mpc)
1
Fig. 6. – Ratio of the matter power spectrum including three degenerate massive neutrinos with density fraction fν to that with three massless neutrinos. The parameters (ωm , ΩΛ ) = (0.147, 0.70) are kept fixed, and from top to bottom the curves correspond to fν = 0.01, 0.02, 0.03, . . . , 0.10. The individual masses mν range from 0.046 to 0.46 eV, and the scale knr from 2.1 × 10−3 h Mpc−1 to 6.7 × 10−3 h Mpc−1 as shown on the top of the figure. From [6].
effect is neither null nor maximal: in other words, many experiments only have access to the transition region in the step-like transfer function. In this region, the neutrino mass affects the slope of the matter power spectrum in a way which can be easily confused with the effect of other cosmological parameters. Because of these parameter degeneracies, the LSS data alone cannot provide significant constraints on the neutrino mass, and it is necessary to combine them with other cosmological data, in particular the CMB anisotropy spectrum, which could lift most of the degeneracies. Still, for exotic models with, e.g., extra relativistic degrees of freedom, a constant equation-of-state parameter of the dark energy different from −1 or a non-power-law primordial spectrum, the neutrino mass bound can become significantly weaker. . 7 4. Impact of massive neutrinos on the CMB anisotropy spectrum. – For neutrino masses of the order of 1 eV (about fν ≤ 0.1) the three neutrino species are still relativistic at the time of photon decoupling, and the direct effect of free-streaming neutrinos on the evolution of the baryon-photon acoustic oscillations is the same in the ΛCDM and ΛMDM cases. Therefore, the effect of the mass is indirect, appearing only at the level of the background evolution: the fact that the neutrinos account today for a fraction Ων of the critical density implies some change either in the present value of the spatial curvature, or in the relative density of other species. If neutrinos were heavier than a few eV, they would already be non-relativistic at decoupling. This case would have more complicated consequences for the CMB, as described in [47]. However, we will see later that this situation is disfavoured by current upper bounds on the neutrino mass.
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Let us describe one example: we choose to maintain a flat Universe (the sum of all Ωi is one) with fixed (ωb = Ωb h2 , ωm = Ωm h2 , ΩΛ ). Thus, while Ωb and ΩΛ are constant, Ωcdm is constrained to decrease as Ων increases. The main effect on the CMB anisotropy spectrum results from a change in the time of equality. Since neutrinos are still relativistic at decoupling, they should be counted as radiation instead of matter around the time of equality, which is found by solving ρb + ρcdm = ργ + ρν . This gives aeq = Ωr /(Ωb +Ωcdm ), where Ωr stands for the radiation density extrapolated until today assuming that all neutrinos would remain massless, given by eq. (12) with Neff 3.04. So, when fν increases, aeq increases proportionally to [1 − fν ]−1 : equality is postponed. This produces an enhancement of small-scale perturbations, especially near the first acoustic peak. Also, postponing the time of equality increases slightly the size of the sound horizon at recombination. These two features explain why in fig. 5 the acoustic peaks are slightly enhanced and shifted to the left in the ΛMDM case. Since the effect of the neutrino mass on CMB fluctuations is indirect and appears only at the background level, one could think that by changing the value of other cosmological parameters it would be possible to cancel exactly this effect (i.e. a parameter degeneracy). It can be actually shown that in the simplest ΛMDM model, with only seven cosmological parameters, one cannot vary the neutrino mass while keeping fixed aeq and all other quantities governing the CMB spectrum. Therefore, it is possible to constrain the neutrino mass using CMB experiments alone [48, 6], although neutrinos are still relativistic at decoupling. This conclusion can be altered in more complicated models with extra cosmological parameters. For instance, allowing for an open Universe or varying the number of relativistic degrees of freedom. In such extended models the CMB alone is not sufficient for constraining the mass, but fortunately the LSS power spectrum can lift the degeneracy. 8. – Current bounds on neutrino masses In this section we review how the available cosmological data is used to get information on the absolute scale of neutrino masses, complementary to laboratory experiments. Note that the bounds in the next subsections are all based on the Bayesian inference method, and the upper bounds on the sum of neutrino masses are given at 95% CL after marginalization over all free cosmological parameters. We refer the reader to sect. 5.1 of [6] for a detailed discussion on this statistical method, as well as for most of the references for the experimental data or parameter analysis. . 8 1. CMB anisotropies. – The experimental situation of the measurement of the CMB anisotropies is dominated by the five-year release of WMAP data (WMAP5), which improved the already precise TT and TE angular power spectra of the previous data releases (WMAP3 and WMAP1), and adds a detection of the E-polarization self-correlation spectrum (EE). On similar or smaller angular scales than WMAP, we have results from experiments that are either ground-based (ACBAR, VSA, CBI, DASI, . . . ) or balloonborne (ARCHEOPS, BOOMERANG, MAXIMA, . . . ). When using data from different
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CMB experiments, one should take into account that they overlap in some multipole region, and not all data are uncorrelated. More details can by found in [43]. We saw in the previous section that the signature on the CMB spectrum of a neutrino mass smaller than 0.5 eV is small but does not vanish due to a background effect, proportional to Ων , which changes some characteristic times and scales in the evolution of the Universe, and affects mainly the amplitude of the first acoustic peak as well as the location of all the peaks. Therefore, it is possible to constrain neutrino masses using CMB experiments only, down to the level at which this background effect is masked by instrumental noise, or by cosmic variance, or by parameter degeneracies in the case of some cosmological models beyond the minimal Λ Mixed Dark matter framework. Here it is assumed that the total neutrino mass was the only additional parameter with respect to a flat ΛCDM cosmological model characterized by 6 parameters (this will be the case for the bounds reviewed in this section, unless specified otherwise). In this framework, many analyses support the conclusion that a sensible bound on neutrino masses exists using CMB data only, of order of 2–3 eV for the total mass Mν ≡ i mi (depending on the data included in addition to WMAP data). The most recent analyses found an upper limit of Mν < 1.2–1.3 eV [49, 50, 39]. This is an important result, since it does not depend on the uncertainties from LSS data discussed next. . 8 2. Galaxy redshift surveys. – We have seen that free streaming of massive neutrinos produces a direct effect on the formation of cosmological structures. As shown in fig. 6, the presence of neutrino masses leads to an attenuation of the linear matter power spectrum on small scales. In a seminal paper [44] it was shown that an efficient way to probe neutrino masses of order eV was to use data from large redshift surveys, which measure the distance to a large number of galaxies, giving us a three-dimensional picture of the universe. At present, we have data from two large projects: the 2 degree Field (2dF) galaxy redshift survey, whose final results were obtained from more than 220000 galaxy redshifts, and the Sloan Digital Sky Survey (SDSS), which will be completed soon with data from one million galaxies. One of the main goals of galaxy redshift surveys is to reconstruct the power spectrum of matter fluctuations on very large scales, whose cosmological evolution is described entirely by linear perturbation theory. However, the linear power spectrum must be reconstructed from individual galaxies which underwent a strongly non-linear evolution. A simple analytic model of structure formation suggests that on large scales, the galaxygalaxy correlation function should be, not equal, but proportional to the linear matter density power spectrum, up to a constant factor that is called the light-to-mass bias (b). This parameter can be obtained from independent methods, which tend to confirm that the linear biasing assumption is correct, at least in first approximation. A conservative way to use the measurements of galaxy-galaxy correlations in an analysis of cosmological data is to take the bias as a free parameter, i.e. to consider only the shape of the matter power spectrum at the corresponding scales and not its amplitude (denoted as galaxy clustering data). An upper limit on Mν between 0.8 and 1.7 eV is found from the analysis of galaxy clustering data (SDSS and/or 2dF, leaving the bias
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209
as a free parameter) added to CMB data. These values improve those found with CMB data only. In general, one obtains weaker bounds on neutrino masses using preliminary SDSS results instead of 2dF data, but this conclusion could change after the next SDSS releases. The bounds on neutrino masses are more stringent when the amplitude of the matter power spectrum is fixed with a measurement of the bias, instead of leaving it as a free parameter. The upper limits on Mν are reduced to values of order 0.5–0.9 eV although some analyses also add Lyman-α data (see next subsection). Finally, a galaxy redshift survey performed in a large volume can also be sensitive to the imprint created by the baryon acoustic oscillations (BAO) at large scales on the power spectrum of non-relativistic matter. Since baryons are only a subdominant component of the non-relativistic matter, the BAO feature is manifested as a small single peak in the galaxy correlation function in real space that was recently detected from the analysis of the SDSS luminous red galaxy (LRG) sample. The observed position of this baryon oscillation peak provides a way to measure the angular diameter distance out to the typical LRG redshift of z = 0.35, which in turn can be used to constrain the parameters of the underlying cosmological model. The SDSS measurement was included by [51] to get a bound of 0.44 eV on the total neutrino mass Mν , while an upper limit of Mν < 0.61 eV was recently found [50] including BAO but without data on the shape of the galaxy power spectra. . 8 3. Lyman-α forest. – The matter power spectrum on small scales can also be inferred from data on the so-called Lyman-α forest. This corresponds to the Lyman-α absorption of photons traveling from distant quasars (z ∼ 2–3) by the neutral hydrogen in the intergalactic medium. As an effect of the Universe expansion, photons are continuously red-shifted along the line of sight, and can be absorbed when they reach a wavelength of 1216 ˚ A in the rest frame of the intervening medium. Therefore, the quasar spectrum contains a series of absorption lines, whose amplitude as a function of wavelength traces back the density and temperature fluctuations of neutral hydrogen along the line of sight. It is then possible to infer the matter density fluctuations in the linear or quasi-linear regime. In order to use the Lyman-α forest data, one needs to recover the matter power spectrum from the spectrum of the transmitted flux, a task that requires the use of hydro-dynamical simulations for the corresponding cosmological model. This is a difficult procedure, and given the various systematics involved in the analysis, the robustness of Lyman-α forest data is still a subject of intense discussion between experts. In any case, the recovered matter power spectrum is again sensitive to the suppression of growth of mass fluctuations caused by massive neutrinos, and in many cosmological analyses the Lyman-α data is added to CMB and other LSS data. For a free bias, one finds that Lyman-α data help to reduce the upper bounds on the total neutrino mass to the level Mν < 0.5–0.7 eV. But those analyses that include Lyman-α data and a measurement of the bias do not always lead to a lower limit, ranging from 0.4 to 0.7 eV. Finally, [51] found the upper bound Mν < 0.30 eV adding simultaneously Lyα and BAO data, both from SDSS.
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3
CMB only
Total neutrino mass MQ (eV)
+ 2dF/SDSS-gal
1
+ bias and/or Ly-D and/or SDSS-BAO
0.3
0.1
Inverted Normal 0.03 0.001
0.01 0.1 lightest mQ (eV)
1
Fig. 7. – Current upper bounds (95%CL) from cosmological data on the sum of neutrino masses, compared to the values in agreement at a 3σ level with neutrino oscillation data in eq. (18). From [6].
. 8 4. Summary and discussion of current bounds. – The upper bounds on Mν from the previous subsections are representative of an important fact: a single cosmological bound on neutrino masses does not exist. A graphical summary is presented in fig. 7, where the cosmological bounds correspond to the bands, which were grouped according to the included set of data and whose thickness roughly describes the spread of values obtained from similar cosmological data: 1.3–3 eV for CMB only, 0.9–1.7 eV for CMB and 2dF/SDSS-gal or 0.2–0.9 eV with the inclusion of a measurement of the bias and/or Lyman-α forest data and/or the SDSS measurement of the baryon oscillation peak. For a recent discussion on the dependence of the upper limits on the considered set of cosmological data, see, e.g., [39]. Note that it is usual to add other measurements of cosmological parameters to CMB and LSS data. Probably the most important case is the measurement of the present value of the Hubble parameter by the Key Project of the Hubble Space Telescope [52], giving h = 0.72 ± 0.08 (1σ), which excludes low values of h and leads to a stronger upper bound on the total neutrino mass. In addition, one can include the constraints on the current density of the dark-energy component deduced from the redshift dependence of type-Ia supernovae (SNIa) luminosity, which measures the late evolution of the expansion rate of the Universe [53]. For a flat Universe with a cosmological constant, these constraints can be translated into bounds for the matter density Ωm . One can see from fig. 7 that current cosmological data probe the region of neutrino masses where the 3 neutrino states are degenerate, with a mass Mν /3. This mass region is conservatively bounded to values below approximately 1 eV from CMB results
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combined only with galaxy clustering data from 2dF and/or SDSS (i.e. the shape of the matter power spectrum for the relevant scales). The addition of further data leads to an improvement of the bounds, which reach the lowest values when data from Lyman-α and/or the SDSS measurement of the baryon oscillation peak are included or the bias is fixed. In such cases the contribution of a total neutrino mass of the order 0.2–0.6 eV seems already disfavoured. It is interesting to compare these bounds with those coming from tritium beta decay and neutrinoless double beta decay, as recently done in [21, 39]. Finally, we remind the reader that the impressive cosmological bounds on neutrino masses shown in fig. 7 may change if additional cosmological parameters, beyond those included in the minimal ΛCDM, are allowed. This could be the case whenever a new parameter degeneracy with the neutrino masses arises. For instance, in the presence of extra radiation parametrized by a larger Neff the bound on Mν gets less stringent [54]. Another possible parameter degeneracy exists between neutrino masses and the parameter w, that characterizes the equation of state of the dark-energy component X (pX = wρX ). This degeneracy can be broken adding data on baryon acoustic oscillations [51]. Finally, the cosmological implications of neutrino masses could be very different if the spectrum or evolution of the cosmic neutrino background was non-standard (see the discussion in sect. 5.7 of [6]). 9. – Future sensitivities on neutrino masses from cosmology In the near future we will have more precise data on cosmological observables from various experimental techniques and experiments. If the characteristics of these future experiments are known with some precision, it is possible to assume a “fiducial model”, i.e. a cosmological model that would yield the best fit to future data, and to estimate the error bar on a particular parameter that will be obtained after marginalizing the hypothetical likelihood distribution over all the other free parameters. Technically, the simplest way to forecast this error is to compute a Fisher matrix, a technique has been widely used in the literature, for many different models and hypothetical datasets, now complemented by Monte Carlo methods. Here we will focus on the results for σ(Mν ), the forecast 68% CL error on the total neutrino mass, assuming various combinations of future observations: CMB anisotropies measured with ground-based experiments or satellites such as Planck, galaxy redshift surveys, galaxy cluster surveys, . . . . In particular, the potentiality for measuring small neutrino masses of weak lensing experiments has been recently emphasized, which will look for the lensing effect caused by the large-scale structure of the neighboring universe, either on the CMB signal [55, 56] or on the apparent shape of galaxies as measured by cosmic shear surveys, see, e.g., [57, 58]. For these two observables we refer the reader to sect. 6 of [6] for further details. We give a graphical summary of the forecast sensitivities to neutrino masses of different cosmological data in fig. 8, compared to the allowed values of neutrino masses in the two possible 3-neutrino schemes. One can see from this figure that there are very good prospects for testing neutrino masses in the degenerate and quasi-degenerate mass regions above 0.2 eV or so. A detection at a significant level of the minimal value of the
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3
3
PLANCK
1
CMBpol or PLANCK+SDSS
0.3
2V sensitivities including CMB lensing / cosmic shear surveys
Inverted Normal
6mi (eV)
6mi (eV)
1
2V sensitivities without weak lensing
CMBpol+SDSS
0.1
0.3
PLANCK lensing + S300/S1000
0.1 Inverted Normal
0.03 0.001
0.01 0.1 lightest mQ (eV)
1
PLANCK lensing
0.03 0.001
CMBpol lensing CMBpol lensing + G2S/G4S
0.01 0.1 lightest mQ (eV)
1
Fig. 8. – Forecast 2σ sensitivities to the total neutrino mass from future cosmological experiments compared to the values in agreement with present neutrino oscillation data in eq. (18) (assuming a future determination at the 5% level). Left: sensitivities expected for future CMB experiments (without lensing extraction), alone and combined with the completed SDSS galaxy redshift survey. Right: sensitivities expected for future CMB experiments including lensing information, alone and combined with future cosmic shear surveys. Here CMBpol refers to a hypothetical CMB experiment roughly corresponding to the Inflation Probe mission. From [6].
total neutrino mass in the inverted hierarchy scheme will demand the combination of future data from CMB lensing and cosmic shear surveys, whose more ambitious projects will provide a 2σ sensitivity to the minimal value in the case of normal hierarchy (of order 0.05 eV). The combination of CMB observations with future galaxy cluster surveys [59], derived from the same weak lensing observations, as well as X-ray and Sunyaev-Zel’dovich surveys, should yield a similar sensitivity. Finally, before concluding we would like to briefly comment on a new cosmological observable that can potentially probe small-scale modifications of the matter power spectrum at intermediate redshifts (20 < z < 6). It has been shown [60, 61] that the study of fluctuations in the 21 cm line emitted by neutral H could provide stronger constraints on neutrino masses than even very large galaxy surveys, thanks in part to the fact that at higher redshifts the problem of non-linearities becomes less important. The authors of [61] conclude that future low-frequency radio observations could enhance the sensitivity to neutrino masses down to the 0.01 eV scale. 10. – Conclusions Neutrinos, despite the weakness of their interactions and their small masses, can play an important role in Cosmology that we have reviewed in this contribution. In addition,
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cosmological data can be used to constrain neutrino properties, providing information on these elusive particles that complements the efforts of laboratory experiments. In particular, the data on cosmological observables have been used to bound the effective number of neutrinos (including a potential extra contribution from other relativistic particles). But probably the most important contribution of Cosmology to our knowledge of neutrino properties is the information it can provide on the absolute scale of neutrino masses. We have seen that the analysis of cosmological data can lead to either a bound or a measurement of the sum of neutrino masses, an important result complementary to terrestrial experiments such as tritium beta decay and neutrinoless double-beta decay experiments. In the next future, thanks to the data from new cosmological experiments, we could even hope to test the minimal values of neutrino masses guaranteed by the present evidences for flavour neutrino oscillations. For this and many other reasons, we expect that neutrino cosmology will remain an active research field in the next years. ∗ ∗ ∗ I thank the organizers of the International School of Physics “Enrico Fermi” for their invitation and hospitality. Many of the topics discussed here were developed in an enjoyable collaboration with J. Lesgourgues. This work was supported by the European Union (contracts No. RII3-CT-2004-506222 and MRTN-CT-2004-503369) and the Spanish grant FPA2005-01269, as well as by a Ram´ on y Cajal contract of MEC.
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DOI 10.3254/978-1-60750-038-4-215
Direct determination of neutrino mass from 3 H β-spectrum C. Weinheimer(∗ ) Institut f¨ ur Kernphysik, Westf¨ alische Wilhelms-Universit¨ at M¨ unster D-48149 M¨ unster, Germany
Summary. — The investigation of the endpoint region of the tritium β decay spectrum is still the most sensitive direct method to determine the neutrino mass scale. In the nineties and the beginning of this century the tritium β decay experiments at Mainz and Troitsk reached a sensitivity on the neutrino mass of 2 eV/c2 . They were using a new type of high-resolution spectrometer with large sensitivity, the MACE-Filter, and were studying the systematics in detail. Currently, the KATRIN experiment is being set up at Forschungszentrum Karlsruhe, Germany. KATRIN will improve the neutrino mass sensitivity by one order of magnitude down to 0.2 eV/c2 , sufficient to cover the degenerate neutrino mass scenarios and the cosmologically relevant neutrino mass range.
1. – Introduction We know from neutrino oscillation experiments that the different neutrino flavors mix and can oscillate during flight from one flavor state into another. The analysis of all neutrino oscillation experiments yields the mixing angles and the differences of squared neutrino mass eigenstates [1]. Clearly these findings prove that neutrinos have (∗ ) E-mail: [email protected] c Societ` a Italiana di Fisica
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Fig. 1. – Neutrino mass eigenvalues m(νi ) (solid P lines) and one third of the cosmologically relevant sum of the three neutrino mass eigenvalues m(νi )/3 (dashed line) as a function of the smallest neutrino mass eigenvalue mmin for normal hierarchy m(ν3 ) > m(ν2 ) > m(ν1 ) (left) and inverted hierarchy m(ν2 ) > m(ν1 ) > m(ν3 ) (right). The upper limit from the tritium β decay experiments at Mainz and Troitsk on P m(νe ) (solid line), which holds in the degenerate neutrino mass region for each m(νi ), and for m(νi )/3 (dashed line) is also marked. P The hot dark-matter m(νi )/3 is indicated contribution Ων of the universe relating to the average neutrino mass by the right scale in the normal hierarchy plot and compared to all other known matter/energy contributions in the universe (middle). With the relic neutrino density of 336/cm3 the laboratory neutrino mass limit from tritium β decay m(νe ) < 2 eV/c2 corresponds to a maximum allowed neutrino matter contribution in the universe of Ων < 0.12.
non-zero masses, but such “interference experiments” cannot probe the absolute mass scale. We have to parametrize our ignorance by a free parameter mmin , the mass of the smallest neutrino mass eigenstate (see fig. 1). The huge abundance of neutrinos left over in the universe from the Big Bang (336/cm3 ) and their contribution to structure formation [2] as well as the key role of neutrino masses in finding the new Standard Model of particle physics [3, 4] make the absolute value of the neutrino mass one of the most urgent questions of astroparticle physics and cosmology as well as of nuclear and particle physics. There exist 3 different approaches to the absolute neutrino mass scale: – Cosmology Essentially the size of fluctuations is observed at different scales by using cosmic microwave background and large scale structure data. Since the light neutrinos would have smeared out fluctuations at small scales, the power spectrum at small scales is sensitive to the neutrino mass. Up to now, only limits on the sum of the 3 neutrino masses have been obtained around m(νi ) < 0.61 eV/c2 [5], which are to some extent model and analysis dependent [2, 6].
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– Neutrinoless double β decay (0νββ) A neutrinoless double β decay (two β decays in the same nucleus at the same time with emission of two β electrons (positrons) while the (anti)neutrino emitted at one vertex is absorbed at the other vertex as a neutrino (antineutrino)) is forbidden in the Standard Model of particle physics. It could exist, if the neutrino is its own antiparticle (“Majorana-neutrino” in contrast to “Dirac-neutrino”). Furthermore, a finite neutrino mass is required in order to produce in the chirality-selective interaction a neutrino with a small component of opposite handedness on which this neutrino exchange subsists. The decay rate would scale with the absolute square of the so-called effective neutrino mass, which takes into account the neutrino mixing matrix U : 2 (1) Γ0νββ ∝ Uei2 m(νi ) := mee 2 . Here mee represents the coherent sum of the m(νi ) components of the 0νββ decay amplitudes and hence carries their relative phases (the usual CP -violating phase of a unitary 3 × 3 mixing matrix plus two so-called Majorana phases). A significant additional uncertainty which enters the relation of mee and the decay rate is the nuclear matrix element of the neutrinoless double β decay. There is one claim for evidence at mee ≈ 0.4 eV/c2 by part of the Heidelberg-Moscow collaboration [7] and limits from different experiments in the 1 eV/c2 range [8]. – Direct neutrino mass determination The direct neutrino mass determination is based purely on relativistic kinematics without further assumptions. Therefore it is sensitive to the neutrino mass squared m2 (ν). In principle there are two methods: time-of-flight measurements and precision investigations of weak decays. The former requires very long baselines and therefore very strong sources, which only cataclysmic cosmological events like a core-collapse supernova could provide. The non-observation of the dependence of the arrival time on the energy of supernova neutrinos from SN1987a gave an upper limit on the neutrino mass of 5.7 eV/c2 [9]. Unfortunately nearby supernova explosions are too rare and too little understood to allow an improvement into the sub-eV range. Therefore, aiming for this sensitivity, the investigation of the kinematics of weak decays and more explicitly the investigation of the endpoint region of a β decay spectrum is still the most sensitive model-independent and direct method to determine the neutrino mass. Here the neutrino is not observed but the charged decay products are precisely measured. Using energy and momentum conservation the neutrino mass can be obtained. In the case of the investigation of a β spectrum usually the “average electron neutrino mass” m(νe ) is determined (see sect. 2 below): |Uei2 |m(νi )2 . (2) m(νe )2 := This incoherent sum is not sensitive to the phases of the neutrino mixing matrix.
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Fig. 2. – Observables of neutrinoless double β decay mee (open band) and of direct neutrino end of the mee mass determination by single β decay m(νe ) (thin gray area sitting at the upper P band) versus the cosmologically relevant sum of neutrino mass eigenvalues m(νi ) for the case of normal hierarchy (left) and of inverted hierarchy (right). The width of the bands/areas is caused by the experimental uncertainties of the neutrino mixing angles [9] and in the case of mee also by the completely unknown Majorana and CP phases. Uncertainties of the nuclear matrix elements, which enter mee , are not considered.
Figure 2 demonstrates that the different methods are complementary to each other and compares them. These lecture notes are structured as follows: In sect. 2 the neutrino mass determination from the kinematics of tritium β decay is described. Section 3 presents the previous tritium β decay experiments, especially the experiments at Mainz and Troitsk. In sect. 4 an overview of the present KATRIN experiment is given. This paper closes with a conclusion in sect. 5. For a more detailed and complete overview on this subject we would like to refer to the reviews [10-13], with [14] being the most recent one. 2. – Neutrino mass from the tritium β decay spectrum According Fermi’s Golden Rule the decay rate for a β decay is given by the square of the transition matrix element M summed and integrated over all possible discrete and continuous final states f (from here on we use the convention h ¯ = 1 = c for simplicity): (3)
Γ = 2π
|M 2 |df.
Let us first calculate the density of the final states. The number of different final states dn of outgoing particles inside a normalization volume V into the solid angle dΩ with momenta between p and p + dp, or, respectively, with energies in the corresponding
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interval around the total energy Etot , is (4)
dn =
V · p2 · dp · dΩ V · p2 · dp · dΩ V · p · Etot · dEtot · dΩ = = . 3 3 h (2π) (2π)3
This gives a state density per energy interval and solid angle of V · p · Etot dn = . dEtot dΩ (2π)3
(5)
Since the mass of the nucleus is —especially in our case— much larger than the energies of the two emitted leptons, we can use the following simplification: The nucleus takes nearly no energy but balances all momenta. The recoil energy of the daughter nucleus of mass mdaught is bound within the following limits: (6)
0 ≤ Erec =
( p + pν )2 p2max E 2 + 2E0 m ≤ Erec,max = = 0 . 2mdaught 2mdaught 2mdaught
Here we denote without indices the quantities of the electron and with index ν the ones of the neutrinos. Since we will reserve the notation E for the kinetic energy of the β electron, we denote its total energy with Ee . E0 is the maximum kinetic energy the electron can obtain. Therefore we need to count the state density of the electron and the neutrino only (7)
dne dnν V 2 · pe · Ee · pν · Eν · = dEe dΩe dEν dΩν (2π)6 V 2 · Ee2 − m2 · Ee · Eν2 − m2 (νe ) · Eν = . (2π)6
ρ(Ee , Eν , dΩe , dΩν ) =
The transition matrix element M can be divided into a leptonic part, Mlep , and a hadronic one, Mnucl . Usually the coupling is written separately and expressed in terms of Fermi’s coupling constant GF and the Cabibbo angle ΘC : (8)
M = GF · cos ΘC · Mlep · Mnucl .
2 | essenFor an allowed or superallowed decay like that in tritium, the leptonic part |Mlep tially results in the probability of the two leptons to be found at the nucleus, which is 1/V for the neutrino and 1/V · F (E, Z + 1) for the electron, yielding
(9)
2 |Mlep |=
1 · F (E, Z + 1). V2
The Fermi function F (E, Z+1) takes into account the final electromagnetic interaction of the emitted β electron with the daughter nucleus of nuclear charge (Z + 1). The Fermi
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function is approximately given by [11] (10)
F (E, Z + 1) =
2πη , 1 − exp(−2πη)
with the Sommerfeld parameter η = α(Z + 1)/β. The spin structure and coupling to the nuclear spin, as well as its (β, ν) angular correlation, is usually contracted into the nuclear matrix element. For an allowed or super-allowed transition the hadronic matrix element is independent of the kinetic energy of the electron. Generally it can be divided into a vector current or Fermi part (ΔInucl = 0) and into an axial current or Gamov-Teller part (ΔInucl = ±1). In the former case, the spins of electron and neutrino couple to S = 0, in the latter case to S = 1. Summing over spin states and averaging over the (β, ν) angular correlation factor 1 + a · (β · βν )
(11)
(with the electron velocity β = v/c and the neutrino velocity βν = vν /c), the hadronic matrix element for tritium is [10] 2 |Mnucl (tritium)| = 5.55.
(12)
The phase space density (7) is distributed over a surface in the two-particle phase space which is defined by a δ-function conserving the decay energy. With this prescription, we can integrate (3) over the continuum states and get the partial decay rate into a single channel; for instance, the ground state of the daughter system with probability P0
(13)
2 G2F · cos2 ΘC · |Mnucl |· · F (E, Z + 1) 5 (2π) Etot ,Etot,ν ,Ω,Ων ·β ν ) · Ee2 − m2 · Ee · Eν2 − m2 (νe ) · Eν · 1 + a · (β
Γ 0 = P0 ·
· δ(Q + m − Etot − Etot,ν − Erec ) dEtot dEtot,ν dΩ dΩν . A correct integration over the unobserved neutrino variables in (13) has to respect the (β, ν) angular correlation factor (11), which enters the recoil energy (6). The variation of Erec near the endpoint is tiny [15]. Even for the most sensitive tritium β decay experiment, the upcoming KATRIN experiment, the variation of Erec over the energy interval of investigation (the last 25 eV below the endpoint) can be neglected and replaced with a constant value of Erec = 1.72 eV, yielding a fixed endpoint [16] (14)
E0 = Q − Erec .
We can then integrate over Etot,ν simply by fixing it through the δ-function to the missing energy Etot,ν = (E0 − E): the difference between endpoint energy E0 and kinetic
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energy E of the β electron. Further integration over the angles yields through (11) an averaged nuclear matrix element, as mentioned above. Besides integrating over the (β, ν)-continuum, we have to sum over all other final states. It is a double sum, one over the 3 neutrino mass eigenstates m(νi ) with probabilities |Uei2 |, the other over all of the electronic final states of the daughter system with probabilities Pj and excitation energies Vj . The latter are caused by the sudden change of the nuclear charge and the different nuclear charge of the daughter atom/molecule. They give rise to shifted endpoint energies. Introducing the definition ε := (E0 − E),
(15)
the total neutrino energy now amounts to Etot,ν = ε − Vj . Rather than in the total decay rate, we are interested in its energy spectrum γ = dΓ/dE, which we can read directly from (13) without performing the second integration over the β energy. Written in terms of ε and summed up over the final states it reads (16)
γ=
G2F · cos2 ΘC 2 · |Mnucl | · F (E, Z + 1) 2π 3 · (E0 + m − ε) · (E0 + m − ε)2 − m2 2 · |Uei | · Pj · (ε − Vj ) · (ε − Vj )2 − m2 (νi ) · Θ(ε − Vj − m(νi )). i,j
We directly see the validity of the definition of the average electron neutrino mass squared m2 (νe ) by (2), if the different neutrino mass states cannot be resolved experimentally. The Θ-function confines the spectral components to the physical sector ε−Vj −m(νi ) > 0. This causes a technical difficulty in fitting mass values smaller than the sensitivity limit of the data, as statistical fluctuations of the measured spectrum might occur which can no longer be fitted within the allowed physical parameter space. Therefore, one has to define a reasonable mathematical continuation of the spectrum into the region which leads to χ2 -parabolas around m2 (νi ) ≈ 0 (see, e.g., [15]). But one may equally well use formulas describing a physical model with the signature of a spectrum stretching beyond E0 like tachyonic neutrinos [17] (with the caution, of course, that one should not jump to spectacular conclusions from significant fit values m2 (νi ) < 0 instead of carefully searching for systematic errors in the data). Furthermore, one may apply radiative corrections to the spectrum [18, 19]. However, they are quite small and would influence the result on m2 (νe ) only by few a percent of its present systematic uncertainty. One may also raise the point of whether possible contributions from right-handed currents might lead to measurable spectral anomalies [20, 21]. It has been checked that the present limits on the corresponding right-handed boson mass [9, 22] rule out a sizeable contribution within present experimental uncertainties. Even the forthcoming KATRIN experiment will hardly be sensitive to this problem [23]. The β electrons are leaving the nucleus on a time scale much shorter than the typical Bohr velocities of the shell electrons of the mother isotope. Therefore, the excitation
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probabilies of electronic states —and of vibrational-rotational excitations in the case of molecules— can be calculated in the so-called sudden approximation from the overlap of the primary electron wave function Ψ0 with the wave functions of the daughter ion Ψf,j Vj = |Ψ0 |Ψf,j |2 .
(17)
We will calculate the first excited electronic states for the case of a decaying tritium atom. The tritium (i.e., hydrogen) wave function for the electronic ground state is Ψ0 = ΨZ=1 100 (r, ϑ, φ) =
(18)
1 πa30
· e−r/a0 ,
with Bohr radius a0 = 4πε0 /me2 . The final daughter atom is a 3 He+ ion. Therefore its wave functions Ψf,j = ΨZ=2 nlm are hydrogen-like functions with nuclear charge Z = 2. Due to the orthogonality of the spherical harmonics Ylm (ϑ, φ) the overlap intergral (17) can only be non-zero for excited final states Ψf,j = ΨZ=2 n00 . For Z = 2 the first 3 hydrogen-like wave functions are : 8 ΨZ=2 (19) · e−2r/a0 , 100 = πa30 (20)
ΨZ=2 200 =
(21)
ΨZ=2 300
: =
1 πa30
· (1 − r/a0 ) · e−r/a0 ,
4r 8r2 8 · e−2r/3a0 . · 1 − + 27πa30 3a0 27a20
We can compute the overlap integral (17) using the following relation: (22)
∞
rn exp (−r/μ) dr = n! μn+1 .
0
The transition probability P0 to the 3 He+ ground state (n = 1) amounts to
(23)
P0 =
=
=
=
2 : 8 1 −r/a0 −2r/a0 2 · e · · e · r sin θdθdφ dr 3 % &' ( πa0 πa30 =4π 2 √ 8 2 3 · e−3r/a0 r2 dr a0 √ 2 8 2 2a3 3 · 0 a0 27 √ 2 16 2 512 = 0.702. = 27 729
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Fig. 3. – Excitation spectrum of the daugther (3 HeT)+ in β decay of molecular tritium [24].
The transition probability P1 to the first excited state of the 3 He+ ion (n = 2) is
(24)
2 1 1 −r/a0 −r/a0 2 P1 = ·e · 3 · (1 − r/a0 ) · e · r sin θdθdφ dr 3 % &' ( πa0 πa0 =4π 2 4π = 3 · (1 − r/a0 ) · e−2r/a0 · r2 dr πa0 2 3 4 3a3 a = 3 · 2 0 − 0 a0 8 8 2 −1 = 0.25. = 2
Therefore, the first two electronic final states P0 + P1 comprise already more than 95% of all final states. In addition to the excited states of the 3 He+ ion, there are also continuum states, which are more difficult to compute. The excitation energies of the excited electronic states of the 3 He+ ion are (25)
Z=2 Vn−1 = E(ΨZ=2 n00 ) − E(Ψ100 ) =
m(αZ)2 2
1 1 1 − 2 = 1 − 2 · 54.4 eV. n n
Thus, the excitation energy to the first excited level is V2 = 40.8 eV. In practice, all tritium sources so far have been using molecular tritium sources, containing the molecule T2 . The wave functions of the tritium molecule are much more complicated, since in addition to two identical electrons they comprise also the description of rotational and vibrational states, which will be excited during the β decay as well. Figure 3 shows a recent numerical calculation of the final states of the T2 molecule.
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C. Weinheimer
2 const. offset ∼ m ( νe ) 2 := Σ |U ei| m 2( ν ) i i
m ν = 0 eV ∼2 ∗10 −13 m ν = 1 eV
Fig. 4. – Expanded β spectrum around its endpoint E0 for m(νe ) = 0 (dashed line) and for an arbitrarily chosen neutrino mass of 1 eV (solid line). In the case of tritium, the gray-shaded area corresponds to a fraction of 2 · 10−13 of all tritium β decays.
The transition to the electronic ground state of the 3 HeT+ daughter ion is not a single state, but broadened due to rotational-vibrational excitation with a Gaussian standard deviation of σ = 0.42 eV. Secondly the first group of excitated states starts at around Vj = 25 eV. More recent calculations agree to these results [25]. The neutrino mass influences the β spectrum only at the upper end below E0 , where the neutrino is non-relativistic and can exhibit its massive character. The relative influence decreases in proportion to m2 (νe )/ε2 (see fig. 4) leading far below the endpoint to a small constant offset proportional to −m2 (νe ). Figure 4 defines the requirements of a direct neutrino mass experiment which investigates a β spectrum: The task is to resolve the tiny change of the spectral shape due to the neutrino mass in the region just below the endpoint E0 , where the count rate is going to vanish. Therefore, high energy resolution is required combined with large source strength and acceptance as well as low background rate. Now we should firstly discuss what is the best β emitter for such a task. Figure 5 shows the total count rate of a super-allowed β emitter as a function of the endpoint energy. Of course, the total count rate rises strongly with E0 , while the relative fraction in the last 10 eV below E0 decreases. Interestingly, the total count rate in the last 10 eV below E0 , which we can take as our energy region of interest for determining the neutrino mass, is rather stable with regard to E0 . From fig. 5 one might argue that the endpoint energy does not play a significant role in selecting the right β isotope, but we have to consider the fact that we need a certain energy resolution ΔE to determine the neutrino mass. Experimentally it makes a huge difference, whether we have to achieve a certain
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Fig. 5. – Dependence on the endpoint energy E0 of total count rate (left), relative fraction in the last 10 eV below the endpoint (middle) and total count rate in the last 10 eV of a β emitter (right). These numbers have been calculated for a super-allowed β decay using (16) for m(νe ) = 0 and neglecting possible final states as well as the Fermi function F .
ΔE at a low energy E0 or at a higher one. Secondly, the β electrons of no interest with regard to the neutrino mass could cause experimental problems (e.g. as background or pile-up) and again this arguement favors a low E0 . Therefore, tritium is the standard isotope for this kind of study due to its low endpoint of 18.6 keV, its rather short half-life of 12.3 y, its super-allowed shape of the β spectrum, and its simple electronic structure. Tritium β decay experiments using a tritium source and a separated electron spectrometer have been performed in search for the neutrino mass for more than 50 years. 187 Re is a second isotope suited to determine the neutrino mass. Due to the complicated electronic structure of 187 Re and its primordial half-life of 4.3·1010 y, the advantage of the 7 times lower endpoint energy E0 = 2.47 keV of 187 Re with respect to tritium can only be exploited if the β spectrometer measures the entire released energy, except that of the neutrino. This situation can be realized by using a cryogenic bolometer as the β spectrometer, which at the same time contains the β emitter 187 Re [26]. 3. – Previous tritium neutrino mass experiments The majority of the published direct laboratory results on m(νe ) originates from the investigation of tritium β decay, while only two results from 187 Re have been reported very recently (there are also results from investigations of electron capture [27] and bound state β decay [28], which are about 2 orders of magnitude less stringent on the neutrino mass). In the long history of tritium β decay experiments, about a dozen results have been reported starting with the experiment of Curran in the late forties yielding m2 (νe ) < 1 keV [29]. In the beginning of the eighties a group from the Institute of Theoretical and Experimental Physics (ITEP) at Moscow [30] claimed the discovery of a non-zero neutrino mass of around 30 eV/c2 . The ITEP group used as β source a thin film of tritiated valine combined with a new type of magnetic “Tretyakov” spectrometer. The first results
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Fig. 6. – Recent results of tritium β decay experiments on the observable m2 (νe ). The experiments at Los Alamos, Z¨ urich, Tokyo, Beijing and Livermore [36-40] used magnetic spectrometers, the tritium experiments at Mainz and Troitsk [41-44] are using electrostatic spectrometers of the MAC-E-Filter type (see text).
testing the ITEP claim came from the experiments at the University of Z¨ urich [31] and the Los Alamos National Laboratory (LANL) [32]. Both groups used similar Tretyakovtype spectrometers, but more advanced tritium sources with respect to the ITEP group. The Z¨ urich group used a solid source of tritium implanted into carbon and later a selfassembling film of tritiated hydrocarbon chains. The LANL group developed a gaseous molecular tritium source avoiding solid-state corrections. Both experiments disproved the ITEP result. The reason for the “mass signal” at ITEP was twofold: the energy loss correction was probably overestimated, and a 3 He-T mass difference measurement [33] confirming the endpoint energy of the ITEP result, turned out only later to be significantly wrong [34, 35]. Also in the nineties tritium β decay experiments yielded controversially discussed results: Figure 6 shows the final results of the experiments at LANL and Z¨ urich together with the results from other more recent measurements with magnetic spectrometers at University of Tokyo, Lawrence Livermore National Laboratory and Beijing. The sensitivity on the neutrino mass have improved a lot but the values for the observable m2 (νe ) populated the unphysical negative m2 (νe ) region. In 1991 and 1994 two new experiments started data taking at Mainz and at Troitsk, which used a new type of electrostatic spectrometer, so-called MAC-E-Filters, which were superior in energy resolution and luminosity with respect to the previous magnetic spectrometers. However, even their early data were confirming the large negative m2 (νe ) values of the LANL and Livermore experiments when being analyzed over the last 500 eV of the β spectrum
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below the endpoint E0 . But the large negative values of m2 (νe ) disappeared when analyzing only small intervals below the endpoint E0 . This effect, which could only be investigated by the high-luminosity MAC-E-Filters, pointed towards an underestimated or missing energy loss process, seemingly to be present in all experiments. The only common feature of the various experiment seemed to be the calculations of the electronic excitation energies and excitation probabilities of the daughter ions. Different theory groups checked these calculations in detail. The expansion was calculated to one order further and new interesting insight into this problem was obtained, but no significant changes were found [24, 25]. Then the Mainz group found the origin of the missing energy loss process for its experiment. The Mainz experiment used as tritium source a film of molecular tritium quench-condensed onto aluminum or graphite substrates. Although the film was prepared as a homogenous thin film with flat surface, detailed studies showed [45] that the film undergoes a temperature-activated roughening transition into an inhomogeneous film by formation of microcrystals leading to unexpected large inelastic scattering probabilities. The Troitsk experiment on the other hand used a windowless gaseous molecular tritium source, similar to the LANL apparatus. Here, the influence of large-angle scattering of electrons magnetically trapped in the tritium source was not considered in the first analysis. After correcting for this effect the negative values for m2 (νe ) disappeared. The fact that more experimental results of the early nineties populate the region of negative m2 (νe ) values (see fig. 6) can be understood by the following consideration [10]: For ε m(νe ), eq. (16) can be expanded into (26)
dN ∝ ε2 − m2 (νe )/2 . dE
On the other hand the convolution of a β spectrum (16) with a Gaussian of width σ leads to (27)
dN ∝ ε2 + σ 2 . dE
Therefore, in the presence of a missed experimental broadening with Gaussian width σ one expects a shift of the result on m2 (νe ) of (28)
Δm2 (νe ) ≈ −2 · σ 2 ,
which gives rise to a negative value of m2 (νe ) [10]. . 3 1. MAC-E-Filter. – The significant improvement in the neutrino mass sensitivity by the Troitsk and the Mainz experiments are due to MAC-E-Filters (Magnetic Adiabatic Collimation with an Electrostatic Filter). This new type of spectrometer —based on early work by Kruit [46]— was developed for the application to the tritium β decay at Mainz and Troitsk independently [47, 48]. The MAC-E-Filter combines high luminosity
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Fig. 7. – Principle of the MAC-E-Filter. Top: experimental setup; bottom: momentum transformation due to adiabatic invariance of the orbital magnetic momentum μ in the inhomogeneous magnetic field.
at low background and a high energy resolution, which are essential features to measure the neutrino mass from the endpoint region of a β decay spectrum. The main features of the MAC-E-Filter are illustrated in fig. 7: two superconducting solenoids are producing a magnetic guiding field. The β electrons, starting from the tritium source in the left solenoid into the forward hemisphere, are guided magnetically on a cyclotron motion along the magnetic field lines into the spectrometer, thus resulting in an accepted solid angle of nearly 2π. On their way into the center of the spectrometer the magnetic field B drops adiabatically by several orders of magnitude keeping the magnetic orbital moment μ invariant (equation given in non-relativistic approximation): (29)
μ=
E⊥ = const. B
Therefore nearly all the cyclotron energy E⊥ is transformed into longitudinal motion (see fig. 7 bottom) giving rise to a broad beam of electrons flying almost parallel to the magnetic field lines. This parallel beam of electrons is energetically analyzed by applying an electrostatic barrier made up by a system of one or more cylindrical electrodes. The relative sharpness of this energy high-pass filter is only given by the ratio of the minimum magnetic
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Fig. 8. – Transmission function of the KATRIN experiment as a function of the surplus energy E −qU . The KATRIN design settings [60] were used: Bmin = 3·10−4 T, Bmax = 6 T, BS = 3.6 T. The upper horizontal axis illustrates the dependence of the maximum starting angle, which is transmitted at a given surplus energy. Clearly, electrons with larger starting angles reach the transmission condition later, since they still have a significant amount of cyclotron energy in the analysing plane at Bmin .
field Bmin reached at the electrostatic barrier in the so-called analyzing plane and the maximum magnetic field between β electron source and spectrometer Bmax , Bmin ΔE = . E Bmax
(30)
The exact shape of the transmission function can be calculated analytically. For an isotropically emitting source of particles with charge q, it reads
(31)
⎧ 0, ⎪ ⎪ ⎨ 1− 1− T (E, U ) = ⎪ ⎪ ⎩ 1− 1−
for E−qU E BS Bmax
,
·
BS Bmin
, for for
E ≤ qU, qU < E < qU + ΔE, E ≥ qU + ΔE.
We assume the electron source to be placed in a magnetic field BS and that the retarding voltage of the spectrometer is U . Figure 8 shows the transmission functions for the settings of the KATRIN experiment (see sect. 4). The β electrons are spiralling around the guiding magnetic field lines in zeroth approximation. Additionally, in non-homogeneous electrical and magnetic fields they feel a small drift u, which reads in first order [47] (c = 1) (32)
u =
×B (E⊥ + 2E|| ) E . − (B × ∇⊥ B) B2 e · B3
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C. Weinheimer
Fig. 9. – The upgraded Mainz setup shown schematically. The outer diameter amounts to 1 m, the distance from source to detector is 6 m.
The two recent tritium β decay experiments at Mainz and at Troitsk use similar MAC-E-Filters with an energy resolution of 4.8 eV (3.5 eV) at Mainz (Troitsk). The spectrometers differ slightly in size: the diameter and length of the Mainz (Troitsk) spectrometer are 1 m (1.5 m) and 4 m (7 m). The major differences between the two setups are the tritium sources: Mainz uses as tritium source a thin film of molecular tritium quench-condensed on a cold graphite substrate, whereas Troitsk has chosen a windowless gaseous molecular tritium source. After the upgrade of the Mainz experiment in 1995-1997 both experiments ran with similar signal and similar background rates. . 3 2. The Mainz neutrino mass experiment. – The Mainz setup was upgraded in 1995-1997 (see fig. 9), including the installation of a new tilted pair of superconducting solenoids between the tritium source and the spectrometer and the use of a new cryostat providing tritium film temperatures of below 2 K. The first measure eliminated sourcecorrelated background and allowed the source strength to be increased significantly. The second measure avoids the roughening transition of the homogeneously condensed tritium films with time [45], which previously gave rise to negative values of m2 (νe ) when the data analysis used large intervals of the β spectrum below the endpoint E0 . The upgrade was completed by the application of HF pulses on one of the electrodes between measurements every 20 s, and a full automation of the apparatus and remote control. This former improvement lowers and stabilizes the background, the latter one allows long-term measurements. Figure 10 shows the endpoint region of the Mainz 1998, 1999 and 2001 data in comparison with the former Mainz 1994 data. An improvement of the signal-to-background ratio by a factor 10 by the upgrade of the Mainz experiment as well as a significant enhancement of the statistical quality of the data by long-term measurements are clearly visible. The main systematic uncertainties of the Mainz experiment are the inelastic scattering of β electrons within the tritium film, the excitation of neighbor molecules due to sudden change of the nuclear charge during β decay, and the self-charging of the tritium film as a consequence of its radioactivity. As a result of detailed investigations in Mainz [49-51, 42] —mostly by dedicated experiments— the systematic corrections
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Fig. 10. – Averaged count rate of the Mainz 1998/1999 data (filled squares) with fit for m(νe ) = 0 (line) and of the 2001 data (open squares) with fit for m(νe ) = 0 (line) in comparison with previous Mainz data from 1994 (open circles) as a function of the retarding energy near the endpoint E0 and effective endpoint E0,eff (taking into account the width of the response function of the setup and the mean rotation-vibration excitation energy of the electronic ground state of the 3 HeT+ daughter molecule).
became much better understood and their uncertainties were reduced significantly. The high-statistics Mainz data from 1998 to 2001 allowed the first determination of the probability of the neighbor excitation to occur in (5.0 ± 1.6 ± 2.2)% of all β decays [42] in good agreement with the theoretical expectation [52]. The analysis of the last 70 eV below the endpoint of the 1998, 1999 and 2001 data, resulted in [42] (33)
m2 (νe ) = (−0.6 ± 2.2 ± 2.1) eV2 /c4 ,
which corresponds to an upper limit of (34)
m(νe ) < 2.3 eV/c2
(95% C.L.)
This is the lowest model-independent upper limit of the neutrino mass obtained thus far. . 3 3. The Troitsk neutrino mass experiment. – The windowless gaseous tritium source of the Troitsk experiment [44] is essentially a tube of 5 cm diameter filled with T2 resulting in a column density of 1017 molecules/cm2 . The source is connected to the ultrahigh vacuum of the spectrometer by a series a differential pumping stations.
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From their first measurement in 1994 onwards the Troitsk group has reported the observation of a small, but significant anomaly in its experimental spectra starting a few eV below the β endpoint E0 . This anomaly appears as a sharp step of the count rate [43]. Because of the integrating property of the MAC-E-Filter, this step should correspond to a narrow line in the primary spectrum with a relative intensity of about 10−10 of the total decay rate. In 1998 the Troitsk group reported that the position of this line oscillates with a frequency of 0.5 years between 5 eV and 15 eV below E0 [44]. By 2000 the anomaly did no longer follow the 0.5 year periodicity, but still existed in most data sets. The reason for such an anomaly with these features is not clear. In Mainz a similar behavior has been found only in one run taken under unfavorable conditions [42]. In dedicated measurements at Mainz, synchronously taken with the Troitsk experiment, the anomaly was seen at Troitsk, but not at Mainz. After some experimental inprovements the first two runs of 2001 at Troitsk either gave no indication for an anomaly or only showed a small effect with 2.5 mHz amplitude compared to the previous ones with amplitudes between 2.5 mHz and 13 mHz. These findings as well as the Mainz data clearly support the assumption that the Troitsk anomaly is due to an still unknown experimental artifact. In the presence of this problem, the Troitsk experiment is correcting for this anomaly by fitting an additional line to the β spectrum run by run. Combining the 2001 results with the previous ones since 1994 gives [53] (35)
m2 (νe ) = (−2.3 ± 2.5 ± 2.0) eV2 /c4 ,
from which the Troitsk group deduces an upper limit of (36)
m(νe ) < 2.05 eV/c2
(95% C.L.)
The values of eqs. (35) and (36) do not include the systematic uncertainty which is needed to be taken into account when the timely varying anomalous excess count rate at Troitsk is described run by run by an additional line. 4. – The Karlsruhe tritium neutrino experiment KATRIN The previous Mainz and Troitsk experiments have reached their sensitivity limit on the neutrino mass with 2 eV/c2 . Concerning a further and significant improvement on neutrino mass sensitivity, the following lessons can be learned from these experiments: – The MAC-E-Filter is a superior instrument to measure the endpoint region of the tritium β spectrum with utmost sensitivity. Special care has to be taken of the background rate originating in the spectrometer. – The quench-condensed tritium source of the Mainz experiment is very well understood with small systematic uncertainties with regard to the Mainz sensitivity. All uncertainties could be improved except the self-charging of the tritium film, which
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causes an energy spread of 20 meV/monolayer of tritium. If this effect cannot be reduced or completely avoided, a significant further improvement is not possible using such a source. – The windowless gaseous tritium source at Troitsk which is based on the pioneer work at Los Alamos is complicated but served as a rather reliable source for a long time. Special care has to be taken to stabilize the source to allow a longterm running and to avoid particle trapping. On the other hand such a windowless gaseous tritium source exhibits the smallest systematic uncertainties and would allow a significant improvement of the sensitivity on the neutrino mass. To improve the luminosity and stability of such a windowless gaseous tritium source for a nextgeneration of tritium β decay experiment, great effort has to be made. However, this seems to be achievable based on the strong expertise in tritium handling and purification, which is available in fusion technology. Preliminary ideas for next-generation experiments on tritium β decay in search for the absolute neutrino mass were presented by the Troitsk [54] and Mainz [50] groups at a meeting in Erice in 1997. More details on the latter have been published by Bonn et al. [55]. With the discovery of neutrino oscillations in 1998 [56] the discussion gained momentum. Motivated by a long record in neutrino physics through the GALLEX and KARMEN experiments [57, 58] and backed by the presence of a dedicated tritium laboratory on site, the Forschungszentrum Karlsruhe decided to get involved in the plans for a new neutrino mass experiment. It was named KATRIN standing for “KArlsruhe TRItium Neutrino experiment”. From a workshop in Bad Liebenzell in 2001 a letter of intent for the KATRIN experiment [59] emerged from close collaboration of group members from the earlier neutrino mass experiments at Los Alamos (now at University of Washington, Seattle and at University of North Carolina, Chapel Hill), Mainz, and Troitsk with Forschungszentrum Karlsruhe. A design report [60] was approved in 2004. Construction of the experiment is under way and expected to be completed in 2011/12. The experiment aims for an improvement of the sensitivity limit by an order of magnitude down to to check the cosmologically relevant neutrino mass range and to distinguish degenerate neutrino mass scenarios from hierarchical ones. Furthermore, Majorana neutrinos sufficiently massive to cause the neutrinoless double β decay rate of 76 Ge which part of the Heidelberg Moscow collaboration claims to have observed [7] would be observable in the KATRIN experiment in a model-independent way. The true challenge becomes clear by drawing attention to the experimental observable whose uncertainties have then to be lowered by two orders of magnitude. Improving tritium β-spectroscopy by a factor of 100 evidently requires brute force, based on proven experimental concepts. It was decided, therefore, to build a MAC-EFilter with a diameter of 10 m, corresponding to a 100 times larger analyzing plane as compared to the pilot instruments at Mainz and Troitsk. Accordingly one gains a factor of 100 in quality factor which we may define as the product of accepted cross-section of the source (“luminosity”) times the resolving power E/ΔE for the emitted β-particles.
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Fig. 11. – The KATRIN main spectrometer passes through the village of Leopoldshafen on its way from the river Rhine to the Forschungszentrum Karlsruhe on November 25, 2006 (photo: FZ Karlsruhe).
Figure 11 shows the spectrometer tank of KATRIN on its way to Forschungszentrum Karlsruhe, fig. 12 depicts a schematic plan of the whole 70 m long setup. Meanwhile the spectrometer has been set up and has reached its designed outgassing rate in the range of 10−12 mbar l/(s cm2 ). A decay rate of the order of 1011 Bq is aimed for in a source with a diameter of 9 cm. For the reason given above the KATRIN collaboration decided to build a windowless gaseous tritium source (WGTS) in spite of its extraordinary demands in terms of size and cryo-techniques, which would be required to handle the flux of 1019 T2 molecules/s safely. T2 is injected at the midpoint of a 10 m long source tube kept at a temperature of 27 K by a 2-phase liquid neon bath. The integral column density of the source of 5 · 1017 molecules/cm2 has to be stabilized within 0.1%. Owing to background considerations, the T2 flux entering the spectrometer should not exceed 105 T2 molecules/s. This will be achieved by differential pumping sections (DPS), followed by cryo-pumping sections (CPS) which trap residual T2 on argon frost at 4 K [62]. Each system reduces the throughput by 107 , which has been demonstrated for the cryo-pumping section by a dedicated experiment at Forschungszentrum Karlsruhe. The T2 gas collected by the DPS pumps will be purified and recycled. A pre-spectrometer will transmit only the uppermost part of the β spectrum into the main spectrometer in order to reduce the rate of background-producing ionization events therein. The entire pre- and main spectrometer vessels will each be put on their
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Fig. 12. – Schematic view of the 70 m long KATRIN experiment consisting of calibration and monitor rear system, windowless gaseous T 2 source, differential pumping and cryo-trapping section, small pre-spectrometer and large main spectrometer, segmented PIN-diode detector and separate monitor spectrometer.
respective analyzing potentials, which are shifted within the vacuum tank by about −200 V, however, due to the installation of a background-reducing inner screen grid system (fig. 13). A ratio of the maximum magnetic field in the pinch magnet over the minimum magnetic field in the central analyzing plane of the main spectrometer of 20000 provides an energy resolution of ΔE = 0.93 eV near the tritium endpoint E0 . The residual inhomogeneities of the electric retarding potential and the magnetic fields in the analyzing plane will be corrected by the spatial information from a 148 pixel PIN diode detector. Active and passive shields will minimize the background rate at the detector. Additional post-acceleration will reduce the background rate within the energy window of interest. Special care has to be taken to stabilize and to measure the retarding voltage. Therefore, the spectrometer of the former Mainz neutrino mass experiment will be operated at KATRIN as a high-voltage monitor spectrometer which continuously measures the position of the 83m Kr-K32 conversion electron line at 17.8 keV, in parallel to the retarding energy of the main spectrometer. To that end its energy resolution has been refined to ΔE = 1 eV. The β electrons will be guided from the source through the spectrometer to the detector within a magnetic flux tube of 191 T cm2 , which is provided by a series of superconducting solenoids. This tight transverse confinement by the Lorentz force applies also to the 1011 daughter ions per second, emerging from β decay in the source tube, as well as to the 1012 electron-ion pairs per second produced therein by the β electron-flux through ionization of T2 molecules. The strong magnetic field of 3.5 T within the source is confining this plasma strictly in the transverse direction such that charged particles cannot diffuse to the conducting wall of the source tube for getting neutralized. The question
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Fig. 13. – Prototype of one of the 248 modules of the double-layer wire electrode system for the KATRIN main spectrometer. Wires with a diameter of 300 μm (200 μm) are used for the outer (inner) layer. The wires are mounted via precision ceramic holders onto a frame consisting of “combs” and C-profiles and keep their relative distance along their length within a few tenths of a mm. Materials are chosen to be non-magnetic and bakable at 350 ◦ C in order to reach the required low outgassing rate of 10−12 mbar l/(s cm2 ) [69].
how the plasma in the source becomes then neutralized or to which potential it might charge up eventually, has been raised and dealt with only recently [61]. The salient point is, however, that the longitudinal mobility is not influenced by the magnetic field. Hence the resulting high longitudinal conductance of the plasma will stabilize the potential along a magnetic field line to that value which this field line meets at the point where it crosses a rear wall. This provides a lever to control the plasma potential. Meanwhile the Troitsk group has performed a first experiment on the problem [63]. They have mixed 83m Kr into their gaseous T2 and searched for a broadening of the LIII 32-conversion line at 30.47 keV which might be due to an inhomogeneous source potential. Their data fit is compatible with a possible broadening of 0.2 eV, which would not affect their results but suggests further investigation at KATRIN. The sensitivity limit of KATRIN has been simulated (see below) on the basis of a background rate of 10−2 cts/s, observed at Mainz and Troitsk under optimal conditions. Whether this small number can also be reached at the so much larger KATRIN instrument —or even be lowered— has yet to be proven. On the one side, the large dimensions of the main spectrometer are helpful, as they improve straight adiabatic motion due to reduced field gradients. On the other hand, the central flux tube faces a 100 times larger electrode surface at the analyzing potential from which secondary electrons might sneak in. Measurements at Mainz demonstrated that a large number of slow electrons at full potential emerge from the surface of the large central electrodes which are hit by
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Fig. 14. – One of the final double-layer wire electrode modules on the 3-axis measurement table for quality assurance. The fixing of the wires inside the ceramics holders (see inserted smaller photos on the top right) with the connecting wires is checked with a high-resolution camera, whereas wire position and wire tension are monitored by a specially developed 2-dimensional laser sensor [70].
cosmic muons and local radioactivity. But they are born outside the magnetic flux tube which crosses the detector; hence they are guided adiabatically past the detector. This decisive magnetic shielding effect was investigated at Mainz with an external γ-source, as well as by coincidence with traversing cosmic muons; a magnetic shielding factor of around 105 was measured [64]. Furthermore similar checks at Troitsk pointed to 10 times better shielding [65], which probably results from the better adiabaticity conditions of this larger instrument. In case the axial symmetry of the electromagnetic field configuration is broken (e.g. by stray fields) the drift u (32) develops a radial component, which will be all the faster the weaker the guiding field. This drift can transport slow electrons from the surface into the inner sensitive flux tube within which they are accelerated onto the detector. The effect is probably present at Mainz [66]. After finishing tritium measurements in 2001, electrostatic solutions were developed at Mainz, which strengthened shielding of surface electrons by an additional factor of ≈ 10. This was achieved by covering the electrodes with negatively biased grids built from thin wires [67, 68]. Such grids are now under construction for the KATRIN spectrometer (see figs. 13, 14). This measure (in addition to improved adiabaticity) will contribute decisively to keep-
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Fig. 15. – View into the KATRIN main spectrometer with scaffold installed. This scaffold was built by FZ Karlsruhe experts completely for the clean-room conditions of the KATRIN main spectrometer (residual gas design pressure after out-baking: 10−11 mbar) [71].
ing the background rate from this much larger instrument down to the design level of 10−2 cts/s [60]. The installation of wire electrode modules inside the KATRIN main spectrometer is a very challenging engineering task (see fig. 15). A simulated spectrum covering 3 years of data taking at KATRIN is shown in fig. 16; a spectrum for the typical measurement conditions at Mainz is added for comparison. Due to the gain in the signal-to-background ratio, the region of optimal mass sensitivity around E0 has moved much closer to the endpoint and one already notices at first glance a marked mass effect for m(νe ) = 0.5 eV. One also notices that the typical third-power rise of the integral spectrum below E0 is delayed. This is mainly due to rotational-vibrational excitations of the daughter molecule which centre at 1.72 eV and stretch up to more than 4 eV with a width of σro-vib = 0.42 eV (see fig. 3). This width diminishes the mass sensitivity as compared to an atomic source with a sharp endpoint. At KATRIN this effect will be felt for the first time, but still amounts to only 5.5% sensitivity loss on m2 (νe ), according to a simulation with standard KATRIN-parameters. Figure 17 shows simulations of the statistical uncertainty of the observable and corresponding upper mass limits (without systematic uncertainties) which are expected from the KATRIN experiment after 3 years of data taking at background rates of 10−2 cts/s and 10−3 cts/s, respectively. They are plotted as a function of the width of the spectral interval, as measured with equidistant or optimized distribution of settings for analyzing potential as well as for measuring time. The dependence on the interval length is rather flat, in particular assuming a lower background. For the reference value one expects to reach a total uncertainty Δm2 (νe )stat somewhat below 0.02 eV2 .
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Fig. 16. – Top: simulated β-spectra (assuming mν = 0 and E0 = 18.575 keV) resulting from 3 years of KATRIN running under KATRIN standard conditions (filled circles) and from phase 2 of the Mainz experiment for comparison (open squares). Middle: difference of data and fit normalized to the statistical uncertainty for m(νe ) fixed in the fit to 0 eV/c2 (filled circles), 0.35 eV/c2 (open circles) and 0.5 eV/c2 (open squares). Bottom: distribution of measuring points, optimized in position and measuring time.
Fortunately, the improved signal-to-noise ratio is very helpful with regard to the systematic uncertainties, as it allows to shorten the spectral interval under investigation below E0 : Some of the systematic uncertaines decrease, others even vanish completely as soon as the measurement interval drops below energy thresholds of inelastic processes, like the first electronic excitation of the (3 HeT)+ -ion at around 25 eV (see fig. 3) and the minimum energy loss of inelastic scattering on T2 molecules of about 10 eV [49]. From fig. 17 it is clear that KATRIN aims at measuring intervals of about 25 eV below E0 , for which the following systematic uncertainties and the corresponding countermeasures play a role:
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Fig. 17. – Simulations of statistical neutrino mass-squared uncertainty expected at KATRIN after 3 years of running, calculated in dependence on the fit interval under the following conditions. Spectrometer diameter = 7 m as originally proposed [59]: (a); final 10 m design [60]: (b, c, d); background = 10−2 counts/s: (a, b, c); background = 10−3 counts/s: (d); equidistant measuring point distribution: (a, b); measuring point distribution optimized according to local mass sensitivity: (c, d) (reprinted from ref. [60]).
– Uncertainty of the energy-dependent cross-section of inelastic scattering of β electrons on T2 in the windowless gaseous tritium source. Countermeasures: energy loss measurements with an electron gun as done in Troitsk [49] analyzed by special deconvolution methods [72]. – Fluctuations of the T2 column density in the windowless gaseous tritium source. Countermeasures: temperature and pressure control of the tritium source to the 10−3 level, laser Raman spectroscopy to monitor the T2 concentration compared to HT, DT, H2 , D2 and HD [74]. – Spatial inhomogeneity of the transmission function by inhomogeneities of electric retarding potential and the magnetic field in the analyzing plane of the main spectrometer. Countermeasures: spatially resolved measurements with an electron gun or, alternatively, with an 83m Kr conversion electron source. – Stability of retardation voltage [76]. Countermeasures: a) measurement of HV with ppm precision by a HV-divider [73] and a voltage standard; b) applying the retarding voltage also to the monitor spectrometer, which continuously measures 83m Kr conversion electron lines [76-79]. – Electric potential inhomogeneities in the WGTS due to plasma effects.
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Countermeasures: potential-defining plate at the rear exit of the WGTS; monitoring of the potential within WGTS possible by special runs with 83m Kr/T2 mixtures. Each systematic uncertainty contributes to the uncertainty of m2 (νe ) with less than 0.0075 eV2 /c4 , resulting in a total systematic uncertainty of Δm2 (νe )sys = 0.017 eV2 /c4 . The improvement on the observable m2 (νe ) will be two orders of magnitude compared to previous experiments at Mainz and Troitsk. The total uncertainty will allow a sensitivity on m(νe ) of 0.2 eV/c2 to be reached. If no neutrino mass is observed, this sensitivity corresponds to an upper limit on m(νe ) of 0.2 eV/c2 at 90% C.L, or, otherwise, to evidence for (discovery of) a non-zero neutrino mass value at m(νe ) = 0.3 eV/c2 (0.35 eV/c2 ) with 3σ (5σ) significance. For more details, we refer to the KATRIN Design Report [60]. 5. – Conclusion Among various ways to address the absolute neutrino mass scale, the investigation of the shape of β decay spectra around the endpoint is the only model-independent method. This direct method is complementary to the search for the neutrinoless double β decay and to the information from astrophysics and cosmology. The investigation of the endpoint spectrum of the tritium β decay is still the most sensitive direct method. The tritium β decay experiments at Mainz and Troitsk have ended yielding upper limits of about 2 eV/c2 . The new KATRIN experiment is being set up at the Forschungszentrum Karlsruhe by an international collaboration. To measure the tritium β spectrum near the endpoint with lowest systematic uncertainties and highest count rate, the KATRIN collaboration is setting up a) a windowless gaseous tritium source with a factor 100 more count rate than previous experiments and b) a doublet of two spectrometers of MAC-E-Filter type, which is connected to the windowless gaseous tritium source via a complex tritium elimination and electron transport chain. KATRIN’s large main spectrometer has a 100 times larger cross-section and a 5 times higher energy resolution compared to the previous tritium β spectrometers. The background design value is based on active background reduction methods at the spectrometer (double layer screening wire electrode system) and at the electron detector (active and passive shielding, low activity materials). The systematic uncertainties of KATRIN will be well under control by many calibration and monitoring activities, as well as by virtue of the small energy interval of interest below the endpoint reducing the influence of inelastic processes. KATRIN will enhance the model-independent sensitivity on the neutrino mass further by one order of magnitude down to 0.2 eV. ∗ ∗ ∗ The author would like to thank all colleagues and friends from the KATRIN, Mainz and Troitsk collaborations for fruitful discussions. Among them he would like especially ¨ck and E. Otten; as well as K. Valerius to name J. Bonn, G. Drexlin, F. Gl u for carefully reading and correcting these lecture notes. The work by the author for the KATRIN experiment is supported by the German Bundesministerium f¨ ur Bildung und Forschung, the Deutsche Forschungsgemeinschaft and the University of M¨ unster.
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DOI 10.3254/978-1-60750-038-4-245
Precision measurements of the Cosmic Microwave Background P. de Bernardis and S. Masi Dipartimento di Fisica, Sapienza Universit` a di Roma and INFN Sezione di Roma 1 P.le A. Moro 2, 00185 Roma, Italy
Summary. — In this paper we give a tutorial introduction to the Cosmic Microwave Background and its measurements, focusing on the current efforts to obtain the full detailed picture of all its observables: the spectrum, the anisotropy, the polarization.
1. – Introduction We live in an expanding evolving universe, coming from an initial hot and dense phase. Neutrinos played a key role in the early evolution of the universe. Neutrino masses can be efficiently constrained from cosmology, and in particular from precise observations of the Cosmic Microwave Background (CMB) and of other cosmological observables. Moreover, some of the methods developed to measure CMB photons can be modified and used to measure neutrino masses (bolometers/micro-calorimeters). For these reasons we believe that precision measurements of the CMB are of interest in a school devoted to neutrinos. 2. – Modern cosmology and the CMB Cosmology is the description of the Universe at large scales, and of its evolution. The current Cosmological Model is the Hot Big Bang model, based on the Cosmological c Societ` a Italiana di Fisica
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Principle, on general relativity, on particle, nuclear, atomic physics. The observational evidences are: 1) The expansion of the Universe, 2) The measured abundance of light elements, 3) The Cosmic Microwave Background. If we believe that we do not occupy a special position in the Universe, i.e. that the Universe at large scales is the same everywhere, and that the correct description of gravity is general relativity, then we get the Friedmann equation, describing the evolution of the Friedmann-Robertson-Walker metric (see Pastor’s paper in these proceedings) ⎡ ⎤ 2 dχ (1) ds2 = c2 dt2 + a(t)2 ⎣ + (χdθ)2 + (χ sin θdφ)2 ⎦ , 1 − χ2 where a(t) is the scale factor, describing the common variation with time of the physical lengths in the Universe, and normalized to be unity today (a(t0 ) = 1). Under this metric, the Friedmann equation can be derived from Einstein’s equations (2)
2 ΩM 0 (1 − Ω0 ) a˙ 2 ΩR0 = H0 + 3 + + ΩΛ . a a4 a a2
˙ 0 )/a(t0 ); Ωi0 are the density parameters for the H0 is the Hubble constant, H0 = a(t different energy-density components of the universe today (3)
Ωi0 =
ρi0 8πGρi0 = . ρc0 3H02
Here i = R, M, Λ for radiation, matter and cosmological constant respectively, and Ω0 = ΩM 0 + ΩR0 + ΩΛ . The solution for the scale factor a(t) depends on the different kinds of energy density relevant at the considered epoch. The first fundamental result of this equation is that the universe is not static: a = a(t). From observations, we know that the Universe expands isotropically today: a(t ˙ 0 ) > 0. In an expanding Universe the wavelengths of photons expand in the same way as all other lengths: this is the cosmological redshift, a direct consequence of general relativity. Consider a source at distance R(t) = a(t)χ1 (comoving coordinate χ1 ). Photons emitted from the source propagate radially towards us along the coordinate χ, occupying sequentially all coordinates between χ1 and 0. From the FRW metric, since ds = 0 for photons and the propagation is radial, we get cdt dχ = . a(t) 1 − χ2
(4)
Consider a first crest of the EM wave emitted at time t1 and received at time t0 ; next crest is emitted at t1 + λ1 /c and received at t0 + λ0 /c. Since χ1 is constant, we have that (5)
t0
c t1
dt = a(t)
0
χ1
dχ 1−
χ2
t0 +λ0 /c
=c t1 +λ1 /c
dt . a(t)
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The equality between the two integrals over time can be rewritten as (6)
t0 +λ0 /c
c t0
dt =c a(t)
t1 +λ1 /c t1
dt . a(t)
The times λ0 /c and λ1 /c are both H0−1 , the typical variation time of a(t). So we can consider a(t) as constant in the integrals, and we get λ1 λ0 = . a(t0 ) a(t1 )
(7)
The wavelengths of photons elongate in the same way as all other cosmological distances, following the same scale factor a(t). This phenomenon is called cosmological redshift, in our expanding universe. Light coming from distant galaxies is shifted towards longer wavelengths, in a way approximately proportional to distance: the farther the source, the longer is the travel time of photons, the larger is the growth of the scale factor between emission “em” and detection “det”, the larger the elongation of the wavelength. This is the Hubble law, measured since ∼ 1930 for distant galaxies, and valid for Δλ/λ = (λdet − λem )/λem 1: (8)
c
Δλ = H0 D, λ
where D is the distance of the galaxy. We define the redshift z from the relationship (9)
z=
Δλ λdet − λem , = λem λ
or (10)
1+z =
λdet adet = . λem aem
The Hubble law, valid for z 1 is rewritten as (11)
cz = H0 D.
As evident from the previous equations, the cosmological redshift has nothing to do with the Doppler effect, and has enormous observational consequences. Measurements using standard candles (Cepheids, SN1a, . . . ) to estimate the distance of galaxies, have now established that H0 = (75 ± 8) km/s/Mpc and that the expansion of the universe is indeed approximately isotropic. If the Universe is expanding, it was denser and hotter in the past. In the Early Universe, the temperature was high enough that nuclear reactions produced light elements starting from a plasma of particles (the primeval fireball). The observed primordial abundance of light elements can be produced only if an abundant background of photons
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is present (∼ 109 γ/b). This background is observed today as the Cosmic Microwave Background. According to modern cosmology the CMB was generated in the very early universe, less than 4 μs after the Big Bang, from a small matter-antimatter asymmetry. The CMB was then thermalized in the primeval fireball (in the first 380000 years after the big bang) by repeated scattering against free electrons. When the temperature dropped below 3000 K, electrons and protons combined into neutral hydrogen atoms (recombination), radiation decoupled from matter, and the universe became transparent. The CMB was then diluted and redshifted to microwave frequencies (zCMB = 1100) in the subsequent 14 Gy of expansion of the Universe. The result is a very isotropic blackbody with T = 2.725 K, as observed with extreme accuracy by the COBE-FIRAS experiment [1]. A long-standing need of observational cosmology is Dark Matter, i.e. a form of mass which interacts gravitationally but does not interact electromagnetically. The presence of dark matter is required at different scales in the Universe, from the rotation curves of galaxies to the dispersion of galaxy velocities in clusters, to the formation of structures in the Universe, to the details of the anisotropy of the Cosmic Microwave Background. It is widely believed that the large-scale structure of the Universe observed today (see, e.g., [4]) derives from the growth of initial density seeds, already visible as small anisotropies in the maps of the Cosmic Microwave Background. This scenario works only if dark matter is already clumped at the epoch of CMB decoupling, gravitationally inducing anisotropy in the Cosmic Microwave Background. There are three physical processes converting the density perturbations present at recombination into observable CMB temperature fluctuations ΔT /T . They are: photon density fluctuations δγ , which can be related to the matter density fluctuations Δρ once a specific class of perturbations is specified; the gravitational redshift of photons scattered in an over-density or an under-density with gravitational potential difference φr ; the Doppler effect produced by the proper motion with velocity v of the electrons scattering the CMB photons. In formulas (12)
ΔT 1 1 φr vr (n ) ≈ δγr + − n , T 4 3 c2 c
where n is the line of sight vector and the subscript r labels quantities at recombination. Our description of fluctuations with respect to the FRW isotropic and homogeneous metric is totally statistical. So we are not able to forecast the map ΔT /T (θ, φ), but we are able to predict its statistical properties. If the fluctuations are random and Gaussian, all the information encoded in the image is contained in the angular power spectrum of the map, detailing the contributions of the different angular scales to the fluctuations in the map. In other words, the power spectrum of the image of the CMB details the relative abundance of the spots with different angular scales. If we expand the temperature of the CMB in spherical harmonics (13)
ΔT = a,m Ym (θ, φ), T
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the power spectrum of the CMB is defined as (14)
c = a2,m ,
with no dependence on m since there are not preferred directions in the image. Since we have only a statistical description of the observable, the precision with which the theory can be compared to measurement is limited both by experimental errors and by the statistical uncertainty in the theory itself. Each observable has an associated cosmic and sampling variance, which depends on how many independent samples can be observed in the sky. In the case of the c ’s, their distribution is a χ2 with 2 + 1 degrees of freedom, which means that low multipoles have a larger intrinsic variance than high multipoles (see, e.g., [7]). Detailed models and codes are available to compute the angular power spectrum of the CMB image given a cosmological model for the generation of density fluctuations in the Universe, and a set of parameters describing the background cosmology (see, e.g., [8, 9]). The power spectrum of CMB anisotropy is now measured quite well (see, e.g., [10-19]); an adiabatic inflationary model with cold dark matter and a cosmological constant fits extremely well the measured data (see, e.g., [20-31, 14, 32-36]). Due to the phenomenon of free streaming, neutrinos cannot be responsible for the spectrum of perturbations implied by CMB anisotropy measurements: cold dark matter is much more consistent. However, massive neutrinos can be a minor component of the dark matter in the universe, and detailed observations of the c can constrain their properties (see, e.g., [37-39]). 3. – CMB observables How can we measure the CMB anisotropy with the needed detail? The spectrum of the CMB was measured with high accuracy by COBE-FIRAS: a Martin-Puplett FourierTransform Spectrometer with bolometric detectors, placed in a 400 km orbit. This nullinstrument compared the specific sky brightness to the brightness of an internal cryogenic blackbody reference. The output was precisely nulled (within detector noise) for Tref = 2.725 K. This implies that the brightness of the empty sky is a blackbody at the same temperature, and that the early Universe was in thermal equilibrium at high temperature. The brightness of a 2.725 K blackbody is relatively large (compared to the typical noise of mm-wave detectors). However, it should be stressed that everything at room temperature emits microwaves in the same frequency range: the instrument itself, the surrounding environment, the Earth atmosphere. A room temperature blackbody produces orders of magnitude more power than the CMB. The main difficulty in the measurement of the brightness of the CMB is thus the extremely efficient rejection of local radiation required. Low emissivity, reflective surfaces must be used to shield the instrument, which needs to be cooled to cryogenic temperatures. Also, to avoid a very wide dynamic range, a cryogenic reference should be used in the comparison. All this justifies the design of the COBE-FIRAS instrument [1]. The FIRAS one can be considered a definitive
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measurement of the spectrum of the CMB in the mm range: the deviations from a pure blackbody are less than 0.01% in the peak region, low enough to be fully convincing about the thermal nature of the CMB. However, there are regions of the spectrum where small deviations from a pure blackbody could be expected. The ARCADE experiment [2], for example, focused on the low-frequency end of the spectrum, looking for cm-wave deviations. Processes like reionization due to the first stars, and particle decays in the early Universe, would heat the diffuse matter, which in turn would cool, injecting the excess heat in the CMB (see, e.g., [3]). Sub-percent deviations of the CMB spectrum in the cm range could still be undetected. At a much lower level we expect the CMB to be anisotropic and weakly polarized. The wavelength spectrum of the anisotropy and of the polarization is a modified blackbody (15)
ΔB = B(ν, TCMB )
xex ΔT , −1 T
ex
peaking at 220 GHz. However, since ΔT /T 1 (eq. (12)), the amplitude of the anisotropy signal is very small, and differential instruments have been developed to measure it. Also, a real-world instrument will be sensitive to a limited range of angular scales, depending on its angular resolution (FWHM), on the size of the observed region θ, and on the beam-switch amplitude α if it is a differential instrument. This translates into a range of multipoles in eq. (13) contributing to√the measured signal. For example an instrument with a Gaussian beam (FWHM = 2σ 2 ln 2) and covering the whole sky will respond to multipole through the beam function B = exp[− ( + 1/2)σ 2 ] as plotted in fig. 1, where we compare it to the “standard” power spectrum of CMB anisotropy and EE polarization. This means that to see the “acoustic peaks” due to oscillations of the photon-baryon fluid before recombination, an angular resolution of the instrument better than 1◦ is needed. The other problem evident from fig. 2 is that CMB anisotropy is extremely weak; as a rule of thumb the r.m.s. anisotropy is in the order of 30–100 μK. The polarization signal generated by scalar (density) perturbations at recombination is of the order of a few μK. The polarization signal generated by tensor perturbations during the inflation phase (if inflation really happened) depends on the energy scale of inflation, but is even weaker, in the range of 100 nK or lower (see, e.g., [5, 6]). Measuring the weak anisotropy of the CMB at the 1% level seemed science fiction 20 years ago. Now it is reality, and even the scalar polarization signal is currently being measured. This gives us confidence that the weaker B-mode polarization measurements can be measured in the future. To do this, however, observation methods need to be significantly improved. 4. – CMB observation techniques Observing the CMB is not an easy task. The spectrum of the CMB, a 2.725 K blackbody, is very weak with respect to the emission of the local environment. For this reason, only a space mission like COBE-FIRAS has been able to measure it precisely. The anisotropy and polarization of the CMB offer a more difficult challenge. Their level
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Fig. 1. – Top: angular power spectrum for CMB anisotropy (T T ) and for EE polarization. The latter has been amplified 20 times to make it visible in the same plot of T T . The angular scale γ corresponding to multipole is approximately γ(◦ ) = 180/. Bottom: filter functions of absolute instruments with different angular resolutions. A FWHM smaller than 1◦ is needed to be sensitive to the “acoustic peaks” due to photon-baryon oscillations in the early universe. The curves are labeled with the beam FWHM. Differential instruments will not be sensitive to multipoles < 180/α(◦ ), where α is the angular separation of the beam switch; experiments scanning a limited sky region with angular size θ cannot be sensitive to multipoles with < 180/θ(◦ ).
is 10−5 of the absolute intensity: extremely sensitive detectors, excellent sites, and differential methods are needed to measure it. In the following we give some insight on each of these issues. . 4 1. CMB observation sites. – The Earth atmosphere represents a formidable microwave absorber and emitter in the mm range, due to the presence of water vapor, oxygen and ozone molecules and their rotational transitions. It is also a turbulent medium, featuring time-varying anisotropy in its parameters. A few “windows” feature lower opacity and emission in the mm range. These are located at ν 50 GHz; 70 GHz ν 110 GHz; 125 GHz ν 175 GHz; 190 GHz ν 300 GHz; 320 GHz ν 360 GHz and 390 GHz ν 420 GHz. The atmospheric emissivity in the windows can be as low as 0.01, improving in high mountain cold and dry locations. The problem of extracting
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Fig. 2. – Specific brightness of the CMB and of its anisotropy, compared to the brightness of the atmosphere at a mountain site and on a stratospheric balloon.
CMB signals from ground-based observations is illustrated in fig. 2, where we compare the brightness of a 250 K greybody (with emissivities of 1, 0.1, 0.01), the brightness of a good mountain site (0.5 mm precipitable water vapor, 4000 m o.s.l.), the atmospheric brightness at 41 km of altitude (stratospheric balloon) to the spectrum of the CMB and of its anisotropy. It is evident that CMB anisotropy measurements can be carried out only if atmospheric emission is very isotropic and stable: stability is even more important than the absolute value of the emission. The advantage of a balloon platform is also evident: balloon-borne missions can access a much wider frequency range, and have a better chance to observe spectral features in the CMB, if present. All this explains the huge effort by CMB experimentalists, mounting their instrument in high, cold and dry mountain sites (like White Mountain in California, Testa Grigia in the Alps, Tenerife, Chajnantor in Chile (5100 m), Dome C and South Pole in Antarctica, to quote some of the best locations). Even more challenging is the use of stratospheric balloons, which balances the huge advantage of a lower atmospheric background with a shorter duration of the mission and a more complex pointing of the telescope. Historically, coherent detectors operating at low frequencies have been used mostly from ground-based locations, while broad-band sensitive bolometers have been used on stratospheric balloon platforms.
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Of course the best way to avoid atmospheric emission is to carry the instrument above the atmosphere. The extremely successful WMAP mission of NASA has been operated at 1.5 million km of altitude, in the Lagrangian point L2 of the Earth-Sun system. The forthcoming Planck mission of ESA will also operate from the same advantage point. . 4 2. Detectors. – The peak brightness of the CMB is at a wavelength of 2 mm. This wavelength range has been for a long time shorter than the shortest wavelength efficiently measurable with coherent radiometers, and longer than the longest wavelength accessible with far IR photoconductors. Only the development of sensitive microwave amplifiers based on high electron mobility transistors, on one side, and of cryogenic spider-web bolometers, on the other side, has recently allowed experimentalists to perform detailed measurements of the CMB. Today we have three kinds of mm-wave detectors for CMB studies: – Coherent detectors, where the incoming radiation field is converted into current by an antenna, and the current is amplified by suitable ultra-fast amplifiers, or downconverted beating with a local oscillator, and then amplified in a radio-frequency system (see, e.g., [40]). – Thermal bolometric detectors (see, e.g., [41]), where the integrated effect of many CMB photons heats a radiation absorber, and the resulting temperature increase is measured by an ultra-sensitive thermistor, either a current-biased semiconductor (see, e.g., [42, 43]) or a voltage-biased superconductor (see, e.g., [44, 45]). – Direct detectors, where CMB photons break Cooper pairs in a superconducting film (KIDs, see, e.g., [46]) and the resulting change in the kinetic impedance of the film is detected. Noise in bolometers has Johnson, phonon and photon origin (see, e.g., [47]). Bolometric detectors used in space conditions are already limited by the quantum noise of the CMB itself (like in the HFI instrument of the Planck mission [48]). For this reason, it would be useless for CMB measurements to reduce the noise equivalent power of the detectors. Progress in the field can be obtained only by increasing the mapping speed of the instruments, by means of large arrays of detectors, to be accommodated in telescopes with large corrected focal planes. In CMB photon-noise–limited conditions, the CMB anisotropy signal is given by (16)
ΔB =
ΔT AΩ T
x2
(x)B(ν, TCMB ) x1
xex dx, ex − 1
where AΩ is the throughput of the instrument, x is the adimensional frequency x = hν/kT , (x) is the spectral response of the instrument, including the effects of the detectors, of the filters, of the atmosphere. In the same conditions the variance of the
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Fig. 3. – Signal-to-noise ratio for CMB anisotropy measurements limited by CMB photon noise. A case with ΔT /T ∼ 10−5 is considered. Center frequency, bandwidth and throughput are specified in the labels of the lines.
measured brightness after an integration time T is given by (17)
σ 2 (ΔB) =
4k 5 T 5 AΩ c2 h3
x2
(x) x1
x4 (ex − 1 + (x)) 1 . dx (ex − 1)2 T
From these two equations it is possible to compute the signal-to-noise ratio ΔB/σ(ΔB) for CMB anisotropy measurements, as a function of integration time. This is shown in fig. 3. It is evident that, in order to reach the few μK level necessary for precise anisotropy measurements, long integration times are needed for each observed pixel. The situation is even worse for CMB polarization measurements, where the expected signal is much lower. Since a large number of pixels must be observed to fight cosmic variance, the only solution is the use of large format arrays, so that the mapping speed of the measurement is boosted. In this respect, coherent detectors have progressed with the introduction of MMIC radiometers (see, e.g., [49]), while fully litographed TES bolometers have been replicated in large arrays (see, e.g., [50, 51]) and are already working in the focal plane of large telescopes (see, e.g., [52-54]). Both these techniques involve massive electronics for the amplifiers themselves in the first case, and for the multiplexing readout in the second
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case. In this respect, although less developed, KIDs promise a large simplification in the readout system. Moreover, lumped-elements KIDs are being developed, which will further simplify the absorption of microwaves in the array [55]. . 4 3. Optics. – CMB anisotropy measurements require large microwave telescopes to achieve sufficient angular resolution. In the case of an off-axis telescope, the angular response is approximated by the Airy function (18)
2 dI 2J1 (akθ) (θ) = I0 , dΩ akθ
where a is the radius of the input aperture (normally the primary mirror of the telescope), k = 1/λ, θ is the off-axis angle of the incoming radiation whose amplitude is I0 , and J1 is the Bessel function of order 1. From the equation above we see that at 2 mm of wavelength (150 GHz), the diameter of the aperture of the telescope required to obtain a 10 FWHM beam (needed to study the features in the power spectrum, see fig. 1) is of the order of D 1.22λ/θ ∼ 0.8 m. Moreover, the equation above shows that there are important sidelobes in the angular response of the telescope. This means that the rejection of ground spillover is an important issue. In addition to what comes from the main beam, pointed to the sky to measure the CMB, the detector will receive power from all the surrounding sources, weighted by the angular response of eq. (18), as follows: (19)
W =A
B(θ, φ) 4π
dI (θ, φ)dΩ, dΩ
where B(θ, φ) is the brightness from direction θ, φ. Beyond the main beam (θ λ/D) the envelope of the angular response in eq. (18) scales as θ−3 . For a ground-based experiment, where the ground emission fills about 2π srad of the environment surrounding the instrument, the “nuisance” signal of ground emission can be comparable to or even larger than the CMB signal from the main beam. This can be estimated as follows: dI dI (20) W =A (θ, φ)dΩ + (θ, φ)dΩ , B(θ, φ) B(θ, φ) dΩ dΩ M S where M stands for main lobe and S stands for side lobes. Equation (20) can be approximated as dI dI (21) W A Bsky (θ, φ) ΩM + Bground (θ, φ) ΩS = A[IM + IS ], dΩ M dΩ S where M,S represent the averages of the angular response over the main lobe ( 1) and over the sidelobes ( 1). The ratio IM /IS depends on the main lobe FWHM and on the dI average response in the sidelobes dΩ S . In the case of a 2.725 K sky emission and of a dI 250 K ground emission, for example, in order to have IS IM we need dΩ S 4×10−5
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Fig. 4. – Picture of the BOOMERanG telescope, where the Sun shield and the ground shield surrounding the off-axis telescope are evident. The primary mirror is shown in the top left inset, while in the bottom left inset we plot a schematics of the off-axis optical scheme.
dI for a 10◦ FWHM experiment, and dΩ S 1×10−8 for a 10 FWHM experiment. Hence the necessity of additional shields surrounding the telescope, to increase the number of diffractions that radiation from the ground must undergo before reaching the detectors. As an example we show in fig. 4 a picture of the BOOMERanG balloon-borne experiment, where the ground and Sun shields are evident. The situation is even worse in the case of anisotropy measurements, where the interesting signal is of a few μK. Here a differential instrument is needed, which helps in reducing the sidelobes. The last resource is to send the instrument far from the Earth, so that the solid angle occupied by the ground emission is 2π. This is the case for the WMAP and Planck space missions devoted to CMB anisotropy measurements. They both operate from the Lagrange point L2 of the Sun-Earth system, where the solid angle occupied by the Earth is only 2 × 10−4 srad, with an improvement of a factor ∼ 30000 with respect to ground-based or balloon-borne experiments. The telescope and shields configuration is optimized using numerical methods (see, e.g., [56]), normally based on the geometrical theory of diffraction [57] to speed up the computations.
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The measurement of small signals embedded in a large common mode signal requires efficient modulation techniques. This is the case of anisotropy and polarization measurements. CMB anisotropy has been measured in the past using beam switching techniques, introducing a movable element in the optical system to alternate on the detectors the target and a neighbor reference region (see, e.g., [58]). To first order, the CMB emission, the instrument and telescope emission and the atmospheric emission are the same in the target and in the reference regions. Using a synchronous demodulator (a lock-in amplifier) the small anisotropy signal is efficiently extracted from an overwhelming common mode. This technique, however, has a very slow mapping speed. With the availability of lower noise detectors, the beam-switch technique has been gradually replaced with the fast scan technique, where the entire telescope scans the sky, and only the AC component of the signal is sampled and measured. In this way each pixel of the sky is referenced to the average surrounding emission, so that the common mode signal is rejected. Scanning telescopes have been BOOMERanG [59, 60], Archeops [61], WMAP [62, 63], and will be ˙ so that each Planck [64]. In these instruments the telescope scans the sky at a speed φ, multipole is modulated by the sky scan at a different frequency. Assuming a scan at constant zenith angle Θ, the temperature fluctuations of the CMB along the scan can be expressed as a Fourier series T (Θ, φ) = m αm eimφ . The m-th component of the CMB ∗ signal (which has a mean square amplitude Γm = αm αm ) will produce in the detector a ˙ signal at the electrical frequency f = φm/2π. So a frequency analysis of the detector signal can be performed to get m-space spectroscopy of the CMB anisotropy. Moreover, the 1-D m-space spectrum Γm is related in a simple way [65] to the 2-D -space power spec∞ 2 trum c of the CMB: Γm = =|m| c B2 Pm (Θ). This relationship is valid for scans along full circles; shorter scans on circle sections have a lower -space resolution. All the spherical harmonics components of the CMB with > m contribute to the detector signal at ˙ frequency mφ/2π. If min is the lowest spherical harmonic of interest, the experimentalist will set up the scan in such a way that noise in the system is confined at frequencies lower ˙ than fmin = min φ/2π. This condition is stricter for the HEMT-based receivers, which feature higher 1/f noise knee. On the other hand, if the highest spherical harmonics of interest is max (which depends on the beam size of the experiment), the scan speed should ˙ be adjusted so that the frequency fmax = max φ/2π is lower than the high-frequency cutoff of the experiment. This condition is quite strict for bolometric receivers, which feature 10 ms thermal time constants. In practice, with a scanning speed of ∼ 1 rpm = 6◦ /s, even with a 50 ms time constant the maximum multipole accessible is max ∼ 2400: this figure is appropriate for the slowest bolometers in the forthcoming Planck HFI. In the case of polarization measurements, the small linear polarization degree must be extracted from the larger temperature anisotropy by means of a polarization modulator. Most of the detections of CMB polarization to date have been made using coherent detectors at frequencies 100 GHz [66-72]. In these instruments the detector is intrinsically sensitive to one polarization of the incoming signal, and two orthogonal polarizations can be switched on the detector by means of a ferrite modulator in a circular waveguide, either in a direct or in a correlation receiver. Other instruments, instead, have used thermal detectors [73-78], using either polarization sensitive bolometers [79] or a rotating
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waveplate modulator [80]. In the first case, the signal from two different bolometers, each sensitive to one of the two orthogonal polarizations, is compared, while the telescope scans the sky. This approach requires a good matching of the characteristics of the two detectors. In fact, selected thermistors are mounted on two independent orthogonal wire grids, which are placed inside the same groove of a corrugated waveguide. This approach offers the same mapping speed of anisotropy measurements, but is prone to significant cross-polarization and poor common mode rejection. These have to be properly characterized and calibrated (see, e.g., [73]). This is the methodology selected for the HFI instrument on Planck, after validation on the BOOMERanG-03 balloon flight. The other approach modulates on the same bolometer the two orthogonal polarizations of the incoming radiation, by means of a rotating waveplate. The method is a dynamic implementation of the original measurement method by Stokes, which analyzes the polarization of the incoming radiation by means of a waveplate followed by a polarizer. With a half-wave plate a modulated signal at 4 times the rotation frequency is produced. Suitable materials for efficient waveplates at mm wavelengths are quartz or sapphire, and using a stack of waveplates a wide bandwidth can be covered with the same modulator [81, 82]; moreover, metal-mesh waveplates are also being developed [83]. The main problem with this approach is the necessity of a uniform waveplate, with its optical axis perfectly aligned to the spin axis. Moreover, the waveplate must spin at cryogenic temperature, without introducing vibrations and microphonics in the detectors. The mapping speed of this method is limited by the necessity of integrating on each sky pixel for several rotations of the waveplate. To solve this problem, fast spinning levitating waveplates are being developed [84]: this technology is required for space-borne missions devoted to CMB polarization currently under study. 5. – CMB anisotropy: current status and open issues Precision cosmology is becoming a reality, and this is largely due to the measurements of CMB anisotropy carried out in the last 10 years. The emerging scenario, however, has opened three enigmas: Inflation, Dark Matter, Dark Energy. In the following we show how forthcoming measurements of the CMB can shed light on these issues. The all-sky maps produced by COBE have been confirmed and improved by WMAP: the large-scale structure of the last scattering surface is thus now well measured. The power spectrum at large angular scales (multipoles lower than 20) is well consistent with a Sachs-Wolfe plateau, with one noticeable exception: the quadrupole component is somewhat low. This has been confirmed by the WMAP 3 years data. However, the detailed likelihood function of the quadrupole is non-trivial, and the probability that this is a chance result is not negligible (see, e.g., [85]). In addition, there is some degree of alignment of the lowest multipoles [86, 87], and there is an evident galactic North-South anomaly in the CMB map of WMAP: the distribution is smoother in the North than in the South (see, e.g., [88-90]). These deviations are all at most at 3-σ level. However, they have triggered a lively debate, and several explanations have been attempted (see, e.g., [91] and references therein).
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At intermediate angular scales there is evidence for localized non-Gaussian spots in the maps (see, e.g., [92, 93]). The “cold-spot” evident in the southern sky CMB image of WMAP seems to be also devoid of radio-sources [94, 95], thus suggesting a postrecombination origin, but also non-trivial homogeneity properties of our Universe or the presence of defects [96]. All this seems enough to call for an independent measurement of CMB anisotropy at large angular scales, with a wider frequency coverage to better monitor the foregrounds, and with the highest possible sensitivity, to make it easier to detect instrumental systematics. The Planck mission of the European Space Agency, covering the frequency range 30–850 GHz with an angular resolution of 30 –5 and with unprecedented sensitivity (ΔT /T ∼ 2 × 10−6 ) will assess all these issues. Planck promises a high control of systematics, due to the redundancy of the detectors, to the scanning strategy and to the location, in the Lagrange point L2 of the Sun-Earth system. The launch will be in 2009. The payload module consists of two instruments and one telescope. The Low-Frequency Instrument (LFI) uses HEMT amplifiers, while the High Frequency Instrument (HFI) uses cryogenic bolometers. The telescope is an off-axis Gregorian with a 1.50 × 1.89 m ellipsoid primary. The two instruments have been described in [97, 98], and the science case is described in detail in [99]. To date, there is no full-sky map of the mm-wave sky available at the frequencies covered by HFI! HFI is a unique tool to measure the full sky and to separate the different components contributing to diffuse emission. Photon noise of the CMB itself will be the major limitation to the sensitivity of the CMB channels in HFI. Higher-frequency channels will measure galactic foregrounds. The two instruments have been calibrated, and their performance is consistent with or exceeds the specifications. So we can expect a precisely calibrated instrument, operating in the best possible space environment, producing by 2010 maps covering the full wavelength range and angular resolution of primary CMB anisotropy. At small angular scales, the third peak and the damping tail of the angular power spectrum of the CMB are not measured as well as multipoles < 500. There is evidence for excess anisotropy at multipoles > 2000 [100, 101, 17]. Is this due to unresolved clusters of galaxies, via the Sunyaev-Zeldovich effect? Planck will assess these issues very well. Surveys of Sunyaev-Zeldovich effect in a large number of clusters of galaxies can provide fundamental cosmological information. Planck will contribute with a shallow survey of many thousand clusters; very powerful machines, based on larger telescopes and larger arrays of bolometric detectors (see, e.g., [52]) will provide huge data sets, allowing an independent measurement of the Hubble constant, and investigation of Dark Energy via cluster counts (see, e.g., [102]). 6. – Testing inflation There are basically two CMB observables related to inflation: anisotropy and polarization. The general slope of the power spectrum of CMB anisotropy is related to the spectral index of the power spectrum of primordial density perturbations. In the inflationary
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paradigm, primordial density perturbations derive from quantum fluctuations present in the pre-inflation era, and their power spectrum can be computed quite accurately (see, e.g., [103]). A power law with a spectral index ns 1 is expected. The power spectrum of CMB anisotropy is very sensitive to this parameter, when measured on a wide multipoles range to break degeneracies with other parameters. Current CMB measurements constrain ns to be 0.958 ± 0.016 (WMAP only [35]) or 0.947 ± 0.015 (WMAP plus other anisotropy experiments [35]), fully consistent with the prediction of inflation. Currently we have a 2-σ detection of ns < 1 from WMAP. It has been shown [99] that the power spectrum measurements of Planck will be so precise that a spectral index around 0.99 will be immediately detected, improving the current indication to the level of a definitive measurement. . 6 1. CMB polarization measurements. – CMB photons are Thomson-scattered by electrons at recombination. Linear polarization in the scattered radiation is obtained if there is a quadrupole anisotropy in the incoming, unpolarized photons [104]. This can be produced in two ways. The first, unavoidable source of quadrupole anisotropy of the photon field is the density fluctuation field present at recombination. Density fluctuations induce peculiar velocities in the primeval plasma. For this reason a given electron receives redshifted or blueshifted radiation from the surrounding electrons. Where there is a velocity gradient, a quadrupole anisotropy in the radiation is generated, and the scattered radiation is polarized. So the measurement of the polarization of the CMB probes the velocity field present at recombination. This effect, due to scalar fluctuations, produces a non-rotational polarization field (called E-modes of CMB polarization). The E-mode polarization spectrum (and its correlation with the anisotropy, T E) can be computed very accurately from the density fluctuations inferred by the measured anisotropy spectra. A signal of the order of a few μK r.m.s. is expected for an experiment with angular resolution better than 1◦ . The power spectrum of this signal has maxima where there are minima in the anisotropy power spectrum, just because in a density oscillation there is maximum velocity when the density is equal to the average, while there is zero velocity when the density fluctuation is maximum. The second source of quadrupole anisotropy at recombination is the presence of long-wavelength gravitational waves. If, as expected in the inflationary scenario, a stochastic background of gravitational waves is generated in the very early Universe, then we should be able to see an additional component of the polarization field. The amplitude of this component is very small (100 nK or lower, depending on the energy scale of inflation), but this has also a curl component (Bmode). Using proper analysis methods, the B-mode inflationary component can thus be disentangled from the dominant E-mode, opening a unique window to probe the very early Universe and the physics of extremely high energies (around 1016 GeV) (see, e.g., [105]). In addition, polarization measurements will provide essential information on the reionization process. Considerable effort is being spent by CMB experimentalists to prepare high-accuracy and -sensitivity experiments to detect the rotational component of CMB polarization.
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There are three big issues to be solved for this measurement: – Detector noise: the signal we are looking for is in the 1–100 nK range. Successful polarization surveys to date have reported measurements of signals of the order of 2 μK r.m.s. Since cryogenic bolometers are already close to be limited by photon noise of the CMB itself, the 100× improvement in sensitivity can only be obtained increasing the number of detectors, so that the mapping speed is improved. The likelihood to detect the primordial gravitational waves for different levels of noise has been estimated in several papers (see, e.g., [106]). However, the real difficulty is to solve simultaneously the other issues listed below. – Systematic effects: the signal to be detected is smaller than detector noise that it is very difficult to detect low-level systematics. Current polarization experiments have managed to keep the systematic effects below about 0.1 μK. Here we want to improve this by a factor around 100. The only way to improve here is to experiment. New polarization modulators and optical components must be characterized with unprecedented accuracy. Enough to keep the experimentalists busy for years with laboratory developments and ground-based and balloon-borne experiments. – Polarized foregrounds: our Galaxy emits polarized radiation: elongated interstellar grains, aligned by the Galactic magnetic field, produce thermal emission with a few percent polarization degree [107]; radiation emitted by the electrons of the interstellar medium spiraling in the Galactic magnetic field produces highly polarized synchrotron radiation. A first survey of this emission has been carried out by the WMAP satellite [71]. The level of total polarized emission is up to 100 times larger than the goal polarization signal from primordial gravitational waves. It is thus necessary to characterize the polarized foreground at the 1% level. The EBEX balloon-borne instrument [108] aims at detecting the B-modes by means of a high resolution (< 8 ) telescope and large-format arrays of bolometers working at 150, 250, 350, 450 GHz. The polarization is modulated by an achromatic rotating half-wave plate: a technique pioneered by the MAXIPOL experiment [80]. The instrument is to be flown on a long-duration balloon in Antarctica. SPIDER [109] is devoted to large-scale measurements of CMB polarization, obtained with cryogenic bolometers covering the spectral range from 80 to 270 GHz. A large fraction of the sky is observed during an ultra-long duration balloon flight. The instrument is a precursor of the EPIC CMB Polarization satellite, studied in the framework of the NASA Beyond Einstein program [110]. On the European side, in the framework of the Cosmic Vision ESA program, the BPol satellite has been proposed [111, 112]. This is based on a set of refractive telescopes (one per band, from 45 to 350 GHz), each with its own wave-plate polarization modulator and its bolometric detector array. Although this experiment was not selected for the first round of Cosmic Vision 2015-2025 missions, the community of B-Pol is still extremely active in developing the needed technology and test it in pathfinder observations: detectors
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and modulators, for example, will be tested with the ground-based ClOVER [113] and BRAIN [114] experiments, operating with sub-degree resolution, and with the balloon B-B-Pol, operating at large angular scales. These missions will allow us to validate the involved technologies, and, most important, to detect systematic effects related to the measurement, and identify mitigation strategies. In the case of the large-scale polarization mission B-B-Pol (funded by ASI), the idea is to scan the sky with a balloon-borne telescope spinning in azimuth (like Archeops) during a circumpolar flight in the polar night. The payload design strategy is based on the use of sensitive (and many, a few hundred) detectors directly fed by corrugated feedhorns. The focal plane of the telescope is kept small to avoid off-axis positions: to accommodate all the detectors we prefer to multiply the number of telescopes. A separate cryogenic polarization modulator (rotating waveplate) is used for each telescope, so that it can be optimized for relatively narrow frequency bands. 7. – High-resolution CMB observations Our knowledge of the distribution of visible matter in the Universe has improved significantly, thanks to the 3D galaxy surveys like 2DF and SDSS. Galaxy filaments form a sort of “cosmic web” with clusters and voids. From X-rays images of the clusters we have evidence that the potential wells of clusters of galaxies are full of hot (around 10 keV), ionized and diluted gas, which is bright in the X-rays. Can CMB observations help us in understanding the formation and evolution of structures? . 7 1. Sunyaev-Zeldovich effect. – The photons of the Cosmic Microwave Background can interact with the hot gas, receiving a small boost in energy from the electrons in the gas (by inverse Compton): this is the so called Sunyaev-Zeldovich (S-Z) effect. A first-order calculation of the Inverse Compton Effect for CMB photons against charged particles in the hot gas of clusters can be done as follows: the cluster optical depth is τ nσ , where is a few Mpc, n < 10−3 cm−3 and σ = 6.65 × 10−25 cm2 . So τ 0.01: there is a 1% likelihood that a CMB photon crossing the cluster is scattered by an electron of the hot gas. Since Ee Eγ , the electron transfers part of his energy to the photon. To first order, the energy gain of the photon and the resulting CMB temperature anisotropy are (22)
Δν kTe Δν ΔT
τ
10−4 .
0.01 → 2 ν me c T ν
This is not a small signal: maps of the CMB with sensitivity of 10−5 of the background per pixel are now routinely obtained by CMB anisotropy experiments. Since all photons get a positive boost in energy and the number of photons is conserved, there is a shift of the spectrum of the CMB anisotropy in the direction of the cluster, which means a decrement of the brightness at frequencies below 217 GHz, where the CMB anisotropy spectrum is increasing, and an increment at frequencies above 217 GHz. This spectrum is very peculiar and can be measured by comparing the signal from the cluster to the
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Fig. 5. – The differential spectrum of the Sunyaev-Zeldovich effect (S-Z) in a cluster of galaxies, for different electron temperatures. The S-Z is compared to the spectra of competing effects: primordial CMB anisotropy (CMB) and local emission from synchrotron (S), free-free (F) and interstellar dust (D). Due to the peculiar positive-negative spectrum, the S-Z effect can be easily separated by local effects if a multiband instrument (like OLIMPO) or a spectrometer (like SAGACE) are used. The bands explored by the SAGACE spectrometer are also shown as dashed regions.
signal from a reference region outside the cluster. In fig. 5 we plot the typical spectrum of the S-Z effect. The S-Z effect is one of the three main sources of anisotropy in the microwave sky at high galactic latitudes and millimetre wavelengths. The primary anisotropy of the CMB, and the anisotropy of the Extragalactic Far Infrared Background (FIRB) are the other main contributors. The “cosmological window” where these components are dominant extends roughly from 90 to 600 GHz: at lower frequencies interstellar emission of spinning dust grains, free-free and synchrotron dominate over the cosmological background; at higher frequencies the clumpy foreground from cirrus dust dominates the sky brightness even at high galactic latitudes. The intensity of the S-Z effect is proportional to the density of the intergalactic electrons (n), while the X-ray brightness of the same cluster is proportional to n2 . So
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the S-Z measurements are sensitive to the intracluster gas in the peripheral regions of the cluster, while X-ray measurements are not. Moreover, combining measurements of the two quantities it is possible to derive the angular diameter distance of the clusters of galaxies [115-117]. Observations of many clusters would allow to build an Hubble diagram, and from this to measure the Hubble constant. These measurements are being carried out (see, e.g., [118]), but the error in the determination of H0 is still quite large. To improve these measurements, we need to collect a larger sample of clusters (and forthcoming experiments like SPT [52] and ACT [119] will do a wonderful job in this respect), and also to improve our knowledge of the details of the S-Z effect (cooling flows, inhomogeneities, relativistic corrections and so on): a survey of nearby clusters with excellent inter-channel calibration and wide frequency coverage will be instrumental in this. The S-Z effect depends on the optical depth, but it does not depend on the distance of the clusters (it is like an opacity effect). So we can see clusters that are too faint to be visible in the optical or in the X-rays bands. The number of clusters seen at different distances is a strong function of the Dark Energy density: clusters can in principle be used as probes of the history of Dark Energy [120]. Observing selected clusters where dark matter is separated from baryons it is possible to study the S-Z effect generated by annihilation products of the Dark Matter, thus testing the nature of Dark Matter [121]. This requires ∼ arcmin resolution to resolve the clouds of dark and baryonic matter. The best frequency to operate is around 220 GHz, where the baryonic S-Z signal is nearly null, and only the DM signal and the kinetic S-Z are present. This high frequency represents a challenge for ground-based telescopes. OLIMPO [122], a 2.6 m stratospheric balloon-borne telescope, will carry out its survey in four frequency bands centered at 140, 220, 410 and 540 GHz, in order to be optimally sensitive to the S-Z effect and to efficiently reject competing sources of emission. A satellite version of this telescope has been recently proposed to the Italian Space Agency. The payload, named SAGACE (Spectroscopic Active Galaxies And Clusters Explorer) is able to perform spectral measurements in the range 100 to 760 GHz, using an imaging FTS in the focus of an OLIMPO-like telescope. Four medium size arrays of bolometers improve the mapping speed of the instrument, which can produce maps of diffuse mm emission with a resolution of ∼ 1 . The result will be a catalog of spectra of a few thousand AGNs and Clusters. The high S/N of the measurement allows a very detailed study of the thermal, non-thermal and kinematic S-Z effect, the Hubble diagram, the study of TCMB (z) and the identification of the physical mechanisms of emission in AGNs. In addition, the high-frequency band (720–260 GHz) will provide tomography of early galaxies using the C + line in the “redshift desert”, thus complementing optical surveys of galaxies. . 7 2. Wavelength spectrum of CMB anisotropy. – The CMB is differentially absorbed/scattered by atoms. In particular, it is believed that, during reionization, the first metals are produced by population-III stars, enriching the IGM. In principle, a tomography of the abundance of particular metals can be carried out by precisely calibrated, high-resolution, multi-band CMB anisotropy experiments [123]. This kind of
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observations requires very high angular resolution, very accurate spectroscopy, and the absence of atmospheric contamination of the measurements. A large space telescope mission coupled to FTS analyzers or other forms of advanced spectrometers is probably the only way to carry out such demanding observations (SAGACE could be a perfect pathfinder for this mission). In turn, these observations can provide essential information on reionization, complementary to other probes like Ly-α emitters, 21 cm and IR surveys, CMB polarization. . 7 3. CMB anisotropy and large-scale structure. – The path of the photons, from the last scattering surface to the observer, traverses the large-scale structure of the Universe: these density inhomogeneities produce mainly two effects. Gravitational lensing by intervening mass distorts the anisotropy and the polarization of the CMB. Photons are deflected typically by a few arcmin, with a coherence scale of a few degrees [124]. The main effect is a smoothing of the acoustic peaks of anisotropy and polarization power spectra, and the production of B-mode polarization at high multipoles. The former will be detected by the high-accuracy anisotropy measurement of Planck. The latter will be measured by current polarization experiments like QUAD [77] and ClOVER [113]. These measurements have the potential to measure the growth of structure and the properties of dark matter, including massive neutrinos [125]. And, of course, are needed to remove this contaminant in searches of primordial B-modes. Gravitational redshift produced by large-scale structures also produces CMB anisotropy, via the Integrated Sachs-Wolfe (ISW) effect [126, 127], observable at large angular scales because at small scales the effect is averaged out on the line of sight through many structures. Since density perturbations stop their growth when dark energy starts to dominate the expansion, the ISW is very sensitive to dark energy and its equation of state. However, being a large-scale effect, cosmic variance prevents an accurate measurement. A possible solution is the correlation of the ISW with lensing maps of the CMB: this requires extremely accurate anisotropy and polarization measurements. 8. – Conclusions The outstanding enigmas of current cosmology (Inflation, Dark Matter, Dark Energy) can be tested with ambitious CMB experiments. The technology and methodology is now mature for precise measurements, which are carried out by large international collaborations. These activities require significant resources, but promise fundamental results for our knowledge of the Universe and of fundamental physics. REFERENCES [1] [2] [3] [4]
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DOI 10.3254/978-1-60750-038-4-269
Theory of neutrino masses and mixings ´ G. Senjanovic ICTP, Strada Costiera 11, 34014 Trieste, Italy
Summary. — The tiny neutrino masses and the associated large lepton mixings provide an interesting puzzle and a likely window to the physics beyond the standard model. This is certainly true if neutrinos are Majorana particles, since unlike in the Dirac case, the standard model is not a complete theory. The Majorana case leads to lepton number violation manifested through a neutrinoless double-beta decay and same-sign dileptons possibly produced at colliders such as LHC. I discuss in these lectures possible theories of neutrino mass whose predictions are dictated by their structure only and this points strongly to grand unification. I cover in detail both SU (5) and SO(10) grand unified theories, and study the predictions of their minimal versions. I argue that the theory allows for a (moderate) optimism of probing the origin of neutrino mass in near future.
1. – Foreword The theory of neutrino masses and mixings is a rich subject, with a continuos flow of papers as you are reading these lecture notes. There is no way I could do justice to this vast field in such a short time and space and so I chose to concentrate on what my taste dictated. In order to be as complete and as pedagogical as possible on the issues chosen to be discussed, I have completely omitted a popular field of horizontal symmetries which are used in order to make statements on neutrino masses and mixings, and I apologize to the workers in the field. My decision is prompted by my lack of belief in this approach which to me amounts often to a change of language. c Societ` a Italiana di Fisica
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Instead of accepting the values of these parameters, one typically choses some textures of fermion mass matrices (this is done by the use of symmetries, often discrete ones) which then leads to definite values of masses and/or mixings. The problem that I have with this approach is that this is like saying that the proton is stable because of baryon number symmetry or that the photon is massless because of gauge invariance. The symmetries we assume need not to be exact, and the departures from these symmetries will give departures from the values that follow consequently. It does not make sense to me to say that proton and neutron should have the same mass because of SU (2) isospin invariance, and here I am sure that the reader will agree with me. The small mass difference between the proton and the neutron only says that the isospin symmetry is quite good, albeit approximate symmetry. In searching for the origin of neutrino mass, I have opted here to theories whose inner structure leads to neutrino mass and whose predictions depend only on the same inner structure. Two such examples, the very ones that lead originally to the understanding of the smallness of neutrino mass through the so-called see-saw mechanism, are provided by left-right symmetric theories and the SO(10) grand unified theory. They provide the core of my lectures, and I have included one of the Appendices (D) on the group theory of SO(2N ) to in order to facilitate the reader’s job. Grand unified theories are particularly interesting since they typically fix their own scale. For this reason, I make an exception and discuss in detail also an SU (5) grand unified theory, although in its minimal form it was tailor fit for massless neutrinos, just as the minimal standard model. However, a minimal extension needed to account for neutrino masses and mixings leads to exciting predictions of new particles and interactions likely to be tested at LHC. Furthermore, an understanding of SO(10) becomes much easier after one masters a simple, minimal SU (5) theory, which will always remain as a laboratory of the theory of grand unification and thus a large portion of these notes is devoted to it, including a short Appendix C. The readers familiar with SU (5) can go directly to the last subsection relevant for neutrino mass. Since my lectures are far from being complete, I suggest here to complement them with these two pedagogical expos´es on the subject of neutrino masses and mixings. At the end of the lectures, I include some references for further reading. 1) Mohapatra, Pal [1]. An excellent book, with a detailed analysis of Majorana neutrinos, left-right symmetry, see-saw mechanism and SO(10) grand unification, which provides the core of my lectures. 2) Strumia, Vissani review [2]. Highly recommended especially for the phenomenology of neutrino masses and mixings. Extremely well written, continuously updated, concise, clear and surprisingly complete study of neutrino oscillations and related topics. 2. – Introduction Today we know for fact that at least two neutrinos are massive and by analogy with quarks we need the leptonic mixing matrix (see the lectures by Strumia in these Proceedings).
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We start by reviewing what the Standard Model (SM) says about neutrino masses and mixings. . 2 1. Standard Model review . – The minimal Standard Model (MSM) is an SU (3) × SU (2) × U (1) gauge theory with the following fermionic assignment [3]: u ; d ν ; L ≡ e
qL ≡
(1)
(uc )L , (dc )L , (ec )L ,
where we have omitted the color index for quarks and we work here with left-handed anti-fermions instead of right-handed fermions (see Appendix A, formula (A.11))) T (ψ C )L ≡ C ψ¯R .
(2)
Actually, we will sometimes work with right-handed fermions too (as in sect. 4 on L-R symmetry), and it is important to be familiar and at ease with both notations. The maximal parity violation in the usual charged weak interactions is characterized by the maximal asymmetry between left and right: only left-handed fermions interact with W ± gauge bosons. On top of that, the quark-lepton symmetry is broken by the minimality assumption: NO right-handed neutrinos. Hence a clear prediction: neutrinos are massless. In order to see that, recall that fermionic masses in the MSM stem from the Yukawa interactions with a Higgs doublet Φ T T LY = yu qlT Ciσ2 ΦucL + yd qL CΦ∗ dcL + yl lL CΦ∗ ecL + h.c.,
(3)
where the generation index is suppressed for simplicity. An equivalent expression involves right-handed particles instead of left-handed anti-particles LY = yu q¯l iσ2 Φ∗ uR + yd q¯L ΦdR + yl ¯lL ΦeR + h.c.
(4)
From the charge formula Q = T3 + Y /2,
(5)
The usual charges are reproduced with (6)
Yq =
1 , 3
Y = −1,
YuR =
4 , 3
2 YdR = − , 3
YeR = −2,
YΦ = 1.
Notice the physical interpretation for the hypercharge of the left-handed particles (7)
YL = B − L,
whereas YR has no physical interpretation and needs to be memorized.
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The B-L symmetry of the MSM is selected out: it is an anomaly free combination of accidental global symmetries B and L. In other words, B-L can be gauged. We will come back often to this important and suggestive fact. The minimality of (1), the broken symmetry between quarks and leptons is thus responsible for the only failure of this, otherwise extremely successful, theory. As it is, the MSM must be augmented in order to account for neutrino mass. If you insist, though, on the MSM degrees of freedom in (1), the Yukawa interactions that could lead to neutrino mass must clearly be higher dimensional LY (d = 5) = yν
(8)
( TL Ciσ2 Φ)(φT iσ2 L ) , M
where the new scale M signifies some new physics. Exercise: Show that there are only three possible d = 5, SU (2) × U (1) invariant operators of type (8). Show then that they are all equivalent. When the Higgs doublet gets a nonvanishing vacuum expectation value (vev) (9)
0 , Φ = v
the charged fermions get the usual Dirac mass (10)
mf f¯ f ≡ mf (f¯L fR + f¯R fL ),
with mf = yf v. In the same manner, from (8) the neutrino gets a Majorana mass (11)
mν νLT CνL ,
with (12)
mν = yν
v2 . M
If M θ, neutrinos are automatically lighter than the charged fermions; however if M v (or even M v), small mν may result from yν 1. Since this is an effective theory, we can say nothing about mν . In short, the absence of new light degrees of freedom indicates Majorana neutrino masses and the violation of the lepton number at the new scale M . From (8) and (11), one has ΔL = 2 which allows for the neutrinoless double-beta decay ββ 0 as shown in fig. 1 (13)
n + n → p + p + e + e¯.
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n
p W
e
xm n
e
W p
Fig. 1. – Neutrinoless double-β decay through a Majorana mass mM which breaks a neutrino fermionic line.
It is often argued that ββ 0 probes mM , however, the situation is more complex. Namely, the MSM with neutrino Majorana mass is not a complete theory —it must be completed through a d = 5 operator (8) and a new physics at M . We will see that the predictions for ββ 0 depend on the completion, to which we now turn to. The effective operator (8) is useful in discussing the qualitative nature of neutrino mass, but if we wish to probe the origin of neutrino masses we need a renormalizable theory beyond the MSM. There are three different possibilities of completing the MSM which all lead to the d = 5 operator upon integrating out the new physics; these are three different see-saw mechanisms. 3. – The see-saw mechanism We discuss here different realizations of the see-saw mechanism, in order of their popularity which coincides with the historic development. The idea is a renormalizable completion of the MSM that will lead to small neutrino mass. The end result must be a d = 5 operator discussed above, since that is dictated by the MSM gauge symmetry, as long as new physics is at a scale above MW . Fortunately, there are only three different possibilities and therefore we can and will discuss all of them in what follows. . 3 1. Right-handed neutrinos: type-I see-saw . – The most suggestive completion of the MSM is the introduction of νR (per family of fermions), a gauge singlet chiral fermion. This leads to new renormalizable Yukawa couplings (written here for one generation case only) (14)
ΔL = yD ¯L σ2 Φ∗ νR +
MR T ν CνR + h.c. 2 R
Introduce (15)
ν ≡ νL + C ν¯LT , T N ≡ νR + C ν¯R ,
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N X
YD
N YD
Fig. 2. – Diagrammatic representation of the type-I see-saw.
which gives the mass matrix for ν and N (see Appendix B) (16)
0 mTD
mD . MR
If MR mD , neutrinos would be predominantly Dirac particles. For MR mD , we have a messy combination of Majorana and Dirac, whereas for mD MR we would have a predominantly Majorana case (this case is rather interesting, since the gauge invariant scale MR is expected to be above MW : MR > MW ). In this case the approximate eigenstates are N with mass MN ≡ MR and ν with a tiny mass (17)
Mν = −mTD
1 mD . MN
This is the original see-saw formula [7] today called type I. As we know from (8), with heavy νR , neutrino mass must be of the type (11), confirmed here. Exercise: Prove explicitly (16) in the case of two generations. Hint: work with mD diagonal. It is clear from (16) that the number of νR ’s determines the number of massive light neutrinos: for each νR , only one νL gets a mass. In other words, we need at least two νR ’s in order to account for both solar and atmospheric neutrino mass differences. It is suggestive, though, to have a νR per family, in which case an accidental anomaly free global symmetry of the MSM can be gauged. A neutrino per generation is needed to cancel the U (1)3B-L anomaly. The diagrammatic representation of the see-saw in fig. 2 may be even clearer; it is easy to see that the heavy neutrino propagator gives the see-saw result. . 3 2. Y = 2, SU (2)L triplet Higgs: type-II see-saw. – Instead of νR , a Y = 2 triplet L · σ can play the same role [8]. From the new Yukawas ΔL ≡ Δ (18)
ij T i CΔL j + h.c., ΔL(Δ) = yΔ
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Y Fig. 3. – Diagrammatic representation of the type-II see-saw.
where i, j = 1, . . . , N counts the generations, neutrinos get a mass when ΔL gets a vev (19)
Mν = yΔ Δ.
The vev Δ results form the cubic scalar interaction (20)
2 Tr Δ†L ΔL + . . . , ΔV = μΦT σ2 Δ∗L Φ + MΔ
with (21)
Δ
μv 2 2 , MΔ
where one expects μ of order MΔ . If MΔ v, neutrinos are naturally light. Notice that (19) and (21) reproduce again the formula (11) as it must be: for large scales of new physics, neutrino mass must come from d = 5 operator in (8). Again, the diagrammatic representation may be even clearer, see fig. 3. . 3 3. Y = 0, SU (2)L triplet fermion: type-III see-saw. – The Yukawa interaction in (14) for new singlet fermions carries on straightforwardly to SU (2) triplets too, written now in the Majorana notation (where for simplicity the generation index is suppressed and also an index counting the number of triplet —recall that at least two are needed in order to provide two massive light neutrinos) (22)
ΔL(TF ) = yT T Cσ2σ · TF Φ + MT TFT C TF .
In exactly the same manner as before in Type I, one gets a type-III see-saw for MT v, (23)
Mν = −yTT
1 yT v 2 MT
Again, as in the Type-I case, one would need at least two such triplets to account for the solar and atmospheric neutrino oscillations (or a triplet and a singlet). And, as before, (23) simply reproduces (11) for large MT , and SU (2) × U (1) symmetry dictates.
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It can easily be shown that these three types of see-saw exhaust all the possibilities of reproducing (8) and (11). Exercise: Show that the three possible different operators of the type (8) correspond to the three different types of see-saw. Since (8) and (11) describe effectively neutrino Majorana masses in the MSM, the question is wether we gain anything by going to the renormalizable see-saw scenarios. If the new scales MR , MΔ and MT are huge and not accessible to experiment, then arguably (16), or (19) and (21), or (23), are the (8) or (11). In a sense, they are only a change of language, but not a useful language. We have traded the couplings yν between physical, observable particles, to the unknown yD (or yΔ or yT ) couplings and the unknown masses of the heavy particles that we integrate out. The issue, in any case, is not so much to explain the smallness of the neutrino mass, but to relate it to some other, new, physical phenomenon. After all, small fermion masses are controlled by small Yukawa couplings. This is reminiscent of the Fermi theory of weak interactions. At low energies E
MW , the concept of a massive gauge boson W was not useful and for many years one kept working on the Fermi theory instead. For otherwise, one would be trading the interactions between light physical states for the unknown coupling with W and unknown MW . There are two cases when one is better off talking of W , though: 1) When one can reach the energy E MW and thus make W experimentally accessible. 2) Even when E MW , but one has a dynamical theory of W interactions as in the MSM. The SU (2) × U (1) gauge symmetry of MSM made clear predictions at low energies by correlating charged and neutral current processes. Ideally, we would like both 1 and 2. By complete analogy, we need then either MR , MΔ or MT close to MW in order to be accessible at LHC, or we need a theory of new interactions. The nice example for the latter is Grand Unification: through q- symmetry it, in principle, correlates quark and lepton masses and mixings. A particularly appealing GUT is SO(10), since it unifies a family of fermions and has L-R symmetry as a finite gauge transformation in the form of Dirac’s charge conjugation. I will be discussing it at length later; for the moment suffice it to say that it predicts both Type-I and Type-II see-saw, but in minimal predictive versions their scale is very large, much above MW —and hopeless to detect directly. In summary, the main message of this chapter should be that the Majorana neutrino mass is rather suggestive from the theoretical point of view. As such, it provides a window to new physics at scale M of (8). The crucial prediction of this picture is the ΔL = 2 lepton number violation in processes such as ββ 0 . However, ββ 0 depends in general on the new physics at scale M , and it is desirable to have a direct probe of lepton number violation. In 1983, Keung and myself [36] suggested ΔL = 2 production of same-sign
Theory of neutrino masses and mixings
277
dileptons at colliders, accompanied by jets, as a direct probe of the origin of neutrino mass. We will discuss lepton number violation at length in sect. 7. What happens if the neutrino has a pure Dirac mass? In this case, mν = yD v and the smallness of mν simply requires the smallness of yD . The smallness of mν remains a puzzle controlled by small yD , as much as the smallness of me is controlled by a small electron Yukawa coupling. The MSM with Dirac couplings is a complete theory and needs no theory beyond it. The diversity of fermion masses and mixings encourages many workers in the field to look for flavor symmetries at high energies, precisely since the MSM is complete one has no sense of direction and the possibilities are infinite. The danger here is to be caught in semantics rather in physics, for one often trades the known masses and mixings of the physical states for the unmeasurable properties of the new heavy particles and/or textures of mass matrices that often cannot be probed. This is a generic problem of large-scale theories ad in order to verify them we would need to correlate the neutrino masses and mixings with some new physics. A nice example is proton decay in GUTs, to which we will come later. 4. – Left-right symmetry and neutrino mass This section is devoted to the left-right symmetric extension of the standard model and the issue of the origin of the breaking of parity. This theory played an important historic role in leading automatically to nonzero neutrino masses and the see-saw mechanism. There are two different possible left-right symmetries: parity and charge conjugation. The latter is the finite gauge transformation in SO(10), and is thus rather suggestive. Still, parity is normally identified with L-R symmetry, so I discuss next parity. The write-up here is rather simple and pedagogical, without too many technicalities. . 4 1. Parity as L-R symmetry. – Parity is the fundamental symmetry between left and right and its breaking, I believe, should be understood. In the standard model P is broken explicitly and clearly, in order to break P spontaneously, we must enlarge the gauge group. The minimal model is based on the gauge group [9-12] GLR = SU (2)L × SU (2)R × U (1)Y with the quarks and leptons completely symmetric under L ↔ R (24)
u u P ←→ QR = , QL = d L d R ν ν P ←→ R = . L = e L e R
Notice that the requirement of left-right symmetry leads to the existence of the righthanded neutrino and now the neutrino mass becomes a dynamical issue, related to the pattern of symmetry breaking. In the Standard Model, where νR is absent, mν = 0; here
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instead we shall need to explain why neutrinos are so much lighter than the corresponding charged leptons. In this theory, the formula for the electromagnetic charge becomes Qem = I3L + I3R +
(25)
B−L . 2
This is in sharp contrast with the Standard Model, where the hypercharge Y was completely devoid of any physical meaning. So L-R symmetry is deeply connected with B-L symmetry; the existence of right-handed neutrinos implied by L-R symmetry is necessary in order to cancel anomalies when gauging B-L. Namely, the B-L symmetry is a global anomaly free symmetry of the SM, but without νR the gauged version would have (B-L)3 anomaly. Our primary task is to break L-R symmetry, i.e. to account for the fact that MWR MWL , WR and WL denoting right-handed and left-handed gauge bosons, respectively. In order to do so, we need a set of left-handed and right-handed Higgs scalars whose quantum numbers we will specify later. Imagine for the moment two scalars ϕL and ϕR with P
ϕL ←→ ϕR .
(26)
Assume no terms linear in the fields (since ϕL and ϕR should carry quantum numbers under SU (2)L and SU (2)R ), we can write down the left-right symmetric potential (27)
V =−
μ2 2 λ λ (ϕL + ϕ2R ) + (ϕ4L + ϕ4R ) + ϕ2L ϕ2R , 2 4 2
where λ > 0 in order for V to be bounded from below, and we choose μ2 > 0 in order to achieve symmetry breaking in the usual manner. We rewrite the potential as (28)
V =−
μ2 2 λ λ − λ 2 2 (ϕL + ϕ2R ) + (ϕ2L + ϕ2R )2 + ϕL ϕR , 2 4 2
which tells us that the pattern of symmetry breaking depends crucially on the sign of λ − λ, since the first two terms do not depend on the direction of symmetry breaking (of course μ2 > 0 guarantees that ϕL = ϕR = 0 is a maximum and not a minimum of the potential). Exercise: Show that if 1) λ − λ > 0, in order to minimize V , we have either ϕL = 0, ϕR = 0, or vice versa. Due to the symmetry of V both solutions are equally probable. 2) λ − λ < 0, we need ϕL = 0 = ϕR and L-R symmetry implies ϕL = ϕR . Obviously we choose 1), which implies that P is broken in nature [11, 12].
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. 4 2. Left-Right symmetry and massive neutrinos. – What fields should we choose for the role of ϕL and ϕR ? From the neutrino mass point of view, the ideal candidates should be triplets, i.e. ΔL (¯ 3L , 1R , 2),
(29)
ΔR (¯1L , 3R , 2),
where the quantum numbers denote SU (2)L , SU (2)R and B-L transformation properties. Simply speaking, ΔL and ΔR are SU (2)L and SU (2)R triplets, respectively, with B-L numbers equal to two. Writing ΔL,R = ΔiL,R τi /2 (τi being the Pauli matrices) as is usual for the adjoint representations, we find Yukawa couplings LΔ = hΔ ( TL C iτ2 ΔL L + L → R) + h.c.
(30)
To check the invariance of (30) under the Lorentz group and the gauge symmetry SU (2)L × SU (2)R × U (1)B-L , recall that T – ψL CψL is a Lorentz invariant quantity for a chiral Weyl spinor ψL (and similarly for ψR ).
– under the gauge symmetry SU (2)L (31)
L −→ UL L ,
ΔL −→ UL ΔL UL† ULT (iτ2 ) = (iτ2 )UL†
and similarly for SU (2)R – the B-L number of the ΔL,R fields is two. This proves the invariance of (30) under all the relevant symmetries. Now, from their definition, the fields ΔL,R have the following decomposition under the charge eigenstates: √ Δ+ / 2 = Δ0
(32)
ΔL,R
Δ++√ −Δ+ / 2
, L,R
where we use the fact that Tr ΔL,R = 0 and the charge is computed from Q = I3L + I3R + (B-L)/2. Notice an interesting consequence of doubly charged physical Higgs scalars in this theory. From the general analysis of the spontaneous L-R symmetry breaking, we know that for a range of parameters of the potential the minimum of the theory can be chosen as
(33)
ΔL = 0,
0 ΔR = vR
0 . 0
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From (30), we the obtain the mass for the right-handed neutrino νR (34)
† T ∗ Lm = hΔ vR (νR C νR + νR C † νR ).
Thus the right-handed neutrino gets a large mass MR = hΔ vR , which corresponds to the scale of breaking of parity. At the same time, the original gauge symmetry is broken down to the Standard Model one (35)
ΔR
SU (2)L × SU (2)R × U (1)B-L −→ SU (2)L × U (1)Y .
This can be checked by computing the gauge boson mass matrix. By defining the right-handed charged gauge boson (36)
WR± =
A1R ∓ iA2R √ , 2
we get (37)
2 2 2 MW = gR vR , R
(38)
2 2 MZ2 R = 2(g 2 + gB-L ) vR ,
where (39)
ZR =
gB-L A3R + gR AB-L 2 g 2 + gB-L
is the new massive neutral gauge field, and gR and gB-L gauge couplings correspond to SU (2)R and B-L 2 , respectively. Thus the scale of parity breaking is related to the mass of the right-handed charged gauge bosons WR± . The predominant V -A nature of the weak interactions puts a lower limit on MWR , but the limit depends on the details of the model. In general the left and right mixings between quarks (and leptons too) are not correlated and the MWR can be quite low. If the L and R mixings are the same (or approximately the same) as in some minimal versions of the theory, the best limit comes from the KL − KS mass difference (40)
MWR > 2 TeV.
To complete the theory, one needs a Higgs bidoublet which contains the SM Higgs, so that one can give masses to quarks and leptons. In the process we get the Dirac neutrino mass between νL and νR and in turn we end up with the type-I see-saw mechanism for light neutrino masses.
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Type-I see-saw. From the Dirac Yukawas (41)
L = hΦ L Φ R + h.c.,
after the symmetry breaking the neutrino Dirac mass term is mD = hΦ Φ. The neutrino mass terms become (42)
mD L R + MR TR C R + h.c.
and the neutrino mass matrix takes clearly the see-saw form. The important point here is that the mass of νR is determined by the scale of parity breaking and the smallness of the neutrino mass is a reflection of the predominant V -A structure of the weak interaction and provides a probe of parity restoration at high energies E > MWR . Type-II see-saw. The gauge symmetry of the Left-Right model allows also for the following term in the potential that we have ignored before for simplicity: (43)
ΔV = αΔ†L ΦΔR Φ† ,
which implies that ΔL cannot vanish. Exercise: Show that (44)
ΔL α
2 MW M2 ΔR
α W , MΔ L MR
which leads to type-II see-saw. The predictions for neutrino mass depend crucially on MWR , but the L-R symmetric model by itself cannot give us its value. This is cured in SO(10) grand unified theory, where we will see that this scale is very large which fits perfectly with observed neutrino masses. . 4 3. Charge conjugation as L-R symmetry. – Since charge conjugation (see Appendix A) (45)
T (ψ C )L ≡ C ψ¯R
is also a transformation between left and right, one can as well use C as a L-R symmetry of this theory. In the limit of CP invariance, these symmetries are equivalent; the difference lies only in the tiny breaking of CP . The above discussion goes almost unchanged and we leave it as an exercise for a reader to go through. Exercise: Rewrite the above left-right symmetric theory, both gauge and Yukawa couplings with L-R symmetry as C instead of P .
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We will see that in SO(10) this symmetry introduced here ad hoc is an automatic finite gauge transformation. It would be natural to go directly to SO(10) now, but it will be helpful to master first the minimal grand unified theory based on SU (5) symmetry, the minimal gauge group that embed the SM symmetry. In order to be as pedagogical as possible, I have included Appendices C and D on SU (N ) and SO(2N ) groups, respectively. In particular, Appendix D deals with the spinorial representations of SO(2N ), a possibly new topic for most of the readers. There are a number of exercises that should help you know whether you have a mastery of the necessary group theory. 5. – SU (5): A prototype GUT The minimal group that can unify the Standard Model (SM) is SU (5), a group of rank four. It is actually the minimal group that can unify the SU (2)L and SU (3)c of the SM, the U (1) comes for free. It is natural that we should try to put the electro-weak doublet Φ and the new color triplet hα in the 5-dimensional fundamental representation ⎫ ⎛ r⎞ h ⎬ ⎜ hg ⎟ SU (3)c ⎜ b⎟ ⎭ ⎟ , (46) 5H = Φ = ⎜ h ⎜ ⎟ " ⎝φ+ ⎠ SU (2)L φ0 where in the obvious notation the SU (3)c symmetry is acting on the first 3 components and the SU (2)L on the last two. . 5 1. Structure. . 5 1.1. Fermions. We have 15 Weyl fields in each generation and it is natural to try to put them in a 15-dimensional symmetric representation of SU (5). Now (47)
5 ⊗ 5 = 15s + 10as
Since 5 = (3c , 1L ) + (1c , 2L ) (in an obvious notation), since (3c ⊗ 3c )s = 6c , and since quarks come only in color triplets, we must abandon the idea of 15S . It is not anomaly free anyway, it could not have worked. What about 5 and 10as ? The quantum numbers of 5 from (46) imply uniquely ⎛
(48)
(recall that (f C )R ≡ C f¯L ).
⎞ dr ⎜ dg ⎟ ⎜ b ⎟ ⎟ 5F ≡ ψ = ⎜ ⎜ d ⎟ ⎝ e+ ⎠ −ν C R
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Now, from ψ −→ U ψ under SU (5), the 10-dimensional representation χ must transform as χ −→ U χ U T .
(49)
This is enough to give the quantum numbers of the particles in 10 ⎡
(50)
0 ⎢−uC 1 ⎢ Cb χ= √ ⎢ ug 2⎢ ⎣ ur dr
uC b 0 −uC r ug dg
−uC g uC r 0 ub db
−ur −ug −ub 0 −e+
⎤ −dr −dg ⎥ ⎥ −db ⎥ ⎥ . + ⎦ e 0
L +
e Notice that in (48), a minus sign convention for the ν C field is to ensure that ( −ν C )R e and ( ν )L transform identically, and in (50) the signs are the property of χ being antisymmetric. We will work in the future with 10F and ¯5F (instead of 5F ). We can see furthermore that a unified theory such as SU (5) explains charge quantization, i.e. it relates quark and lepton charges. From (48)
(51)
1 1 Q(dC ) = − Q(e) = 3 3
and then from (50) we see that Q(u) = Q(d) + 1 = 2/3. . 5 1.2. Interactions. The interactions of fermions with gauge bosons are (52)
¯ μ Dμ ψ − i Tr χγ Lf = iψγ ¯ μ Dμ χ,
where (53)
Dμ χ = ∂μ χ − ig(Aμ χ + χATμ ).
There are of course the old QCD and SU (2)L × U (1) interactions with gs = gW = g, and sin2 θW = 3/8, the couplings at the unification scale where full SU (5) is operative. Furthermore, there are new X and Y bosons who carry both color and flavor with charges 4/3 and 1/3, respectively. Their interactions are (54)
g ¯α ¯ μ + μ + ¯ L(X, Y ) = √ X ¯cγ μ dαR γ eR + dαL γ eL + αβγ u L γμ uβL 2 g C + √ Y¯μα −d¯αR γ μ νR +u ¯αL γ μ e+ ¯cγ L γμ dβL + h.c. L + αβγ u 2
As expected, due to the nontrivial color and flavor characteristics of the quarks, the X and Y couple to the quark-quark and quark-lepton states. It is clear that B and L are violated, although for some magic reason B-L is conserved (more about it later). This
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leads to the decay of the proton. By analogy with the usual weak decay n → p + e + ν¯, μ → e + ν¯e + νμ the proton decay rate can be estimated as Γp
(55)
g4 5 4 mp . MX
From (τp )exp > 1033 y we get MX > 1015.5 GeV; later we will show that we can actually compute MX . . 5 2. Symmetry breaking. – The first stage of symmetry breaking down to the SM is achieved by the adjoint Higgs Σ = 24H . Assume, only for the sake of simplicity, the discrete symmetry Σ → −Σ. Then the most general renormalizable potential for Σ is given by (56)
V (Σ) = −
μ2 1 1 Tr Σ2 + a (Tr Σ2 )2 + b Tr Σ4 . 2 4 2
Now, since Σ is a Hermitian matrix, it can be diagonalized by an SU (5) rotation. Assume now that it is in the same direction as the hypercharge: Σ ∝ Y = vX diag(1, 1, 1, −3/2, −3/2). 2 From (56) you get then μ2 = 21 (15a+7b) vX , which, for μ2 > 0, implies (15a+7b) > 0. In order to check that this is a local minimum, we must show that all the second derivatives are positive. Since Σ has exactly the same form as the gauge boson matrix, we can write ⎛ ⎞ ¯X ¯Y Σ Σ Σ8 + 35 − 32 1c Σ0 ⎜ ⎟ ⎜ ⎟ 1 (57) Σ = Σ + ⎜ ⎟, ΣX Σ3 + 35 Σ0 Σ+ 2 ⎝ ⎠ ΣY Σ− − 21 Σ3 + 35 Σ0 where Σ8 are the analogs of gluons, ΣX and ΣY the analogs of X and Y , Σ3 , Σ+ , Σ− and Σ0 the analogs of W 3 , W + , W − and B, respectively. The masses of the particle masses in Σ are (58)
5 2 bv , 4 X 2 m2 (Σ3 ) = m2 (Σ± ) = 5b vX , 15a + 7b 2 vX , m2 (Σ0 ) = 2 m2 (ΣX ) = m2 (ΣY ) = 0. m2 (Σ8 ) =
Thus for 15a + 7b > 0, b > 0 the extremum is a local minimum of the theory. Notice that ΣX and ΣY are would-be Goldstone bosons of the theory; they get “eaten” by the X and Y gauge fields, i.e. they become their longitudinal components.
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Finally, one can show that the vev of Σ is actually a global minimum. In fact, other extrema can be shown to be at best saddle points. Exercise: HARD. Prove that the above minimum is in fact global. Thus SU (5) can be successfully broken down to the standard model, since as we said Y commutes with both the SU (3)c and SU (2)L × U (1)Y generators. This will be even more evident from the study of the gauge bosons mass matrix. Since Σ is in the adjoint representation, Dμ Σ = ∂μ Σ − ig[Aμ , Σ], and one has (59)
1 25 2 2 ¯ a μ ¯ a μ (Dμ Σ)† (Dμ Σ) = g vX Xμ Xa + Yμ Ya , 2 8
where a, as usual, is the color index, a = r, g, b. As expected, the gluons and the electro-weak gauge bosons remain massless, but X and Y get equal masses (60)
2 m2X = m2Y ≡ MX =
25 2 2 g vX , 8
as a consequence of both SU (3)c and SU (2)L remaining unbroken. The original SU (5) symmetry is broken down to SU (3)c × SU (2)L × U (1)Y . The rest of the breaking is completed by a 5-dimensional Higgs multiplet Φ5 which contains the Standard Model doublet. Let us study this in some detail including the full SU (5) invariant potential. We can write (61)
1 1 μ2Σ Tr Σ2 + a(Tr Σ2 )2 + b Tr Σ4 2 4 2 μ2Φ † λ † 2 − Φ Φ + (Φ Φ) 2 4 +αΦ† Φ Tr Σ2 − βΦ† Σ2 Φ,
V (Σ, Φ) = −
with a > 0, λ > 0, 15a + 7b > 0 and β > 0. Since both SU (3)c and SU (2)L are unbroken at this point, we can always rotate Φ into the form ΦT = (vc , 0, 0, 0, vW ). It is only 2 2 the β term that is sensitive to the direction of Φ and it gives −βvX (vc2 +9/4vW ), which, for β > 0, forms the solution vW = 0, vc = 0 in order to minimize the energy. It is an easy exercise to compute the mass of the colored triplet scalar ha in Φ, it is 2 m2h = 52 βvX , which justifies the choice β > 0. It is also easy to show that (62)
2 = MW
2 g2 8MX 9 μ2Φ + β) . (−15α + 4λ 25g 2 2
But MX > 1015 GeV, which implies an extraordinary fine tuning in the above equation of at least 26 orders of magnitude. The number on the right-hand side of (62) is naturally
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2 of order MX > 1030 GeV2 ; instead it ends up being (100 GeV)2 . This is known as the hierarchy problem. In the next subsection we will see that the colored triplet ha mediates proton decay and thus it must be very heavy: mh > 1012 GeV, implying that β cannot be taken arbitrarily small. On the other hand, its partner η weighs < 1 TeV, and this aspect of the hierarchy problem is known as the doublet-triplet splitting problem. Before we close this subsection, let us say a few words more on the hierarchy problem. The problem is that the mass term for the Higgs scalars cannot be made small (or zero) by any symmetry, unlike the case of fermions. There the limit mf = 0 corresponds to the chiral symmetry f → γ5 f , and thus the higher-order corrections must also vanish if mf = 0 at the tree level. In other words, the higher-order corrections are necessarily proportional to mf (tree), and so only logarithmically divergent. In the case of scalars the divergence is quadratic and thus in the context of grand unified theories (GUTs) such as SU (5) the natural value for MW is of order MX .
. 5 3. Yukawa couplings and fermion mass relations. – In the Standard Model the lefthanded fermions are doublets and the right-handed fermions are singlets, and so their chiral property is more than manifest. In SU (5) the V -A structure of a family of fermions is left-intact and here also there are no direct mass terms for fermions. In the minimal SU (5) theory the fermion masses originate through the Yukawa couplings of fermions with the light Higgs Φ (63)
1 LY = fd ψ¯R χ Φ† + fu χT C χ Φ + h.c., 2
where C is the Dirac conjugation matrix, and fu is clearly a symmetric matrix. The symbolic notation of (63) should read in the SU (5) notation as (64)
ψ¯R χ Φ† = ψ¯R i χij Φ†j , χT C χ Φ = ijklm (χT )ij C χkl Φm .
With ΦT = (0 0 0 0 vW ), we get for fermionic masses (65)
+ c T Lm = fd vW (d¯R dL + e¯+ R eL ) − fu vW (u )L C uL + h.c. ¯ + e¯e) − fu vW u = −[fd vW (dd ¯u].
In other words, just as in the Standard Model mf = hf vW , furthermore charged lepton and down-quark masses are equal. Exercise: Explain why this happens. Unfortunately, this works bad even for the third family, since at MX one finds mb = 0.6mτ . This means that one must include higher-dimensional operators in the Yukawa
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sector, up to now neglected. Alternatively, you can include other Higgs representations that can contribute to the fermionic masses; for example, you can add 45H . Now, besides the usual Yukawa structure of the Higgs doublet in the SM, one has new interactions of the color triplet hα . From (63) and (64) it is easy to compute its couplings to fermions T ij kl α Lh = fd ψ¯R i χi α h+ α + fu ijklα (χ ) Cχ h ,
(66) which gives (67)
αβγ c @ α ? + + ¯α c h . Lh = fd αβγ u ¯cL β dγR + u ¯α u ¯R β dγL + u ¯α L eR + dL νR + fu R eL
Notice that the structure of the above couplings (not the strength, though) is dictated by the SU (3)C × SU (2)L × U (1)Y gauge invariance only. This becomes clearer if we write u¯cL dR = uTR CdR and u¯cR dL = uTL CdL . It is clear that the interactions of H break B and L, just like those of X and Y . Notice, though, that B-L is again conserved. In complete analogy with the situation encountered before for the X and Y bosons, we have the possible exchanges of hα which leads to the proton decay. Of course, the amplitude is proportional to small Yukawa couplings and the corresponding limit on its mass is somewhat less strict: mh ≥ 1012 GeV. . 5 3.1. Generations and their mixings. We know that in the standard model the neutral current interactions are flavor diagonal and that the charged current processes lead to flavor mixing and CP violation. How is this feature incorporated in the SU (5) theory and what about new superweak interactions of the X and Y bosons? The analysis is straightforward and it proceeds along the same lines as in the SU (2)L ×U (1)Y theory [5]. I should stress that the predictions we will obtain are of course not realistic since in this minimal theory neutrinos are massless and the down quark and charged lepton mass relations come out wrong. The minimal model discussed here should be viewed only as a prototype of the what predictive theory should be like. We diagonalize as usual fermion mass matrices by bi-unitary transformations (68)
† ULf Mf URf = Df ,
where Df is diagonal, with its elements being real, positive numbers. Furthermore, since Mu is symmetric (69)
∗ URu = ULu K ∗,
where
(70)
⎛ iφu e ⎜ K=⎜ ⎝
⎞ eiφe iφt
e
⎟ ⎟ ⎠ ...
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is the matrix of phases needed to ensure that the elements of Du are real and positive. The above statements are equivalent to the redefinition of our original fermionic fields in the Lagrangian † fL,R → UL,R fL,R ,
(71) +
d e with UL,R = UL,R . Since, on the other hand, the neutrinos are massless, we can rotate c d c them any way we wish and so we chose νR → UR νR . Thus we can write for the 5d d dimensional representation ψR → UR ψR , which means that UR disappears since it is just an overall factor. Suppressing the color index, we can write
⎡
ULu Kuc
(72)
χ→⎣
−ULu u
−ULd d
⎡
−ULd e+ UCKM Kuc
= ULd ⎣
−UCKM u
⎤ ⎦ L
−d −e
⎤ ⎦ ,
+ L
† where UCKM = ULd ULu . Again ULd is just an overall factor and so it will disappear. We are left with the Cabibbo-Kobayashi-Maskawa unitary matrix and the phase matrix K only. Thus the leptonic interactions are flavor conserving (since neutrinos are massless), and the weak quark interactions involve UCKM only, as it must be. Finally, the X and Y boson interactions involve no new flavor mixings besides UCKM , however there will be new phases hidden in K. In the physical basis we get
(73)
g ¯ ¯ μ + ¯ μ + L(X, Y ) = √ X ¯cL γ μ K ∗ uL μ dR γ eR + dL γ eL + u 2 g † † c + √ Y¯μ −d¯R γ μ νR +u ¯L γ μ UCKM e+ ¯cL γ μ UCKM dL + h.c. L +u 2
From U11 ∝ cos θc , U12 ∝ sin θc we would expect (74)
Γ(p → π 0 μ+ ) ∝ sin2 θc . Γ(p → π 0 e+ )
Of course, this minimal SU (5) model is not realistic, for down and strange quark masses are not equal to their leptonic counterparts at the unification scale. It is only an illustration how proton decay partial rates are connected to the fermion masses and mixings. The true test can only be possible in a completely realistic theory of fermion masses and mixings (for a review and references, see [6]). In any case, the minimal SU (5) theory fails to explain neutrino masses; it is custom fit for massless neutrinos. While nonminimal models can lead to nonvanishing neutrino masses, by itself, SU (5) just like the standard model cannot relate neutrino masses to
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charged fermion masses nor relate quark and lepton mixing angles. This is treated beautifully in the SO(10) theory which requires the existence of right-handed neutrinos and leads to small, nonvanishing neutrino masses through the see-saw mechanism. The main ingredients are the left-right and quark-lepton symmetry inbuilt in SO(10) automatically. However, SU (5) offers an interesting possibility of neutrino Yukawa couplings of being probed at LHC and before moving to SO(10) in sect. 6 we will discuss a simple and predictive SU (5) theory with an adjoint fermionic representation added to the minimal model discussed above. We will show that the theory is completely realistic and testable at colliders. . 5 4. Low-energy predictions. . 5 4.1. Ordinary SU (5). As is well known, the couplings run logarithmically with energy. We have (75)
1 1 1 MX = bG ln − αG (MW ) αU 2π MW
for the gauge group G; MX is the energy where we imagine the unification to take place, and αU is the value of the unified coupling at MX . One has a generic formula for the running coefficient (76)
bG =
2 1 11 TGB − TF − TH , 3 3 3
where the Casimir TR for the representation R is defined by TR δij = Tr Ti Tj
(77)
and Ti are the Hermitian traceless generators of the group in question. For the fundamental representation of SU (N ) the convention is the one of SU (2): Tfund = 12 , which implies for the adjoint representation (relevant for gauge bosons) in SU (N ): Ta dj = TGB = N . This gives for the SU (3)C , SU (2)L and U (1), respectively (78)
33 4 − ng , 3 3 22 4 1 − ng − nH , b2 = 3 3 6 3 4 1 b1 = bY = − ng − nH , 5 3 10 b3 =
where Ng is the number of generations, nH is the number of Higgs doublets (nH = 1 in the minimal standard model). We are now fully armed to check the evolution of these couplings above MW . From above (79)
1 1 b j − bi MX − = ln . αi (MW ) αj (MW ) 2π MW
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In the above we have used α1 (MX ) = α2 (MX ) = α3 (MX ) = αU . sin2 θW α2 = cos2 θW αY and αY = 3/5α1 , we get easily (80)
From αem =
1 22 + nH MX 1 − = ln , α2 (MW ) α3 (MW ) 12π MW 3 110 − nH MX αem (MW ) ln . sin2 θW (MW ) = − 8 48π MW
Notice the prediction sin2 θW = 38 at MX which we discussed before. Now, for nH = 1 1 and by taking a α3 (MW ) 0.12, α2 (MW ) 30 , we find MX 1016 GeV, but sin2 θW (MW ) 0.2.
(81)
The minimal SU (5) theory thus fails to meet the experiment. . 5 4.2. Supersymmetric SU (5). Supersymmetry, i.e. symmetry between bosons and fermions, guarantees the cancellation of quadratic divergences for the Higgs mass and thus can make MW insensitive to MX . That is, we do not know why MW /MX is small, but it is not a problem, since it will stay small in perturbation theory as long as the scale of supersymmetry breaking is small, ΛSS MW . The point is that the Higgs mass term is invariant under the internal symmetries and thus is normally not protected from high scales as manifested by quadratic divergences. The fermion masses, on the other hand, are protected by chiral symmetry and thus insensitive to large scales as manifested by “small” logarithmic divergences. In supersymmetry scalars and fermions are not distinguishable and thus Higgs mass is under control too. Then for every particle of the standard model there is a supersymmetric partner of the opposite statistics: fermions (quarks, leptons) s = 1/2
⇐⇒
sfermions (squarks, sleptons) s=0
gauge bosons (W ± , Z, γ, gluons) s=1
⇐⇒
gauginos (Wino, Zino, photino, gluinos) s = 1/2
Higgs scalar s=0
⇐⇒
Higgsino s = 1/2
It is easy to see that the formulas for the running of the gauge couplings will be affected by the presence of the new particles. From (76) we get (82)
bSS G
=
11 2 − 3 3
TGB −
2 1 + 3 3
TF −
1 2 + 3 3
TH ,
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or (83)
bG = 3TGB − TF − TH ,
where the added contributions in (82) are due to the superpartners. From (83) we get for the individual gauge couplings (84)
bSS 3 = 9 − 2ng , 1 bSS 2 = 6 − 2ng − nH , 2 3 nH , bSS 1 = −2ng − 10
where nH is again the number of Higgs doublets. In exactly the same way as before, assuming the unification of couplings at MX , we find (85)
1 6 + nH MX 1 − = ln , α2 (MW ) α3 (MW ) 4π MW 3 30 − nH MX αem (MW ) ln . sin2 θW (MW ) = − 8 16π MW
In the minimal supersymmetric standard model (MSSM) nH = 2, and we find (86)
MX 1016 GeV
and (87)
sin2 θW (MW ) =
7 αem (MW ) 1 +
0.23. 5 15 α3 (MW )
MSSM agrees perfectly well with the experiment and with the above value for MX we predict the proton lifetime (88)
τp 1036 y
which is above the experimental bound (89)
(τp )exp ≥ 6 · 1033 y.
Now, if we are to take supersymmetry seriously, all the way up to the scale MX , we expect ˜ Y˜ , associated with the superheavy bosons X and Y of SU (5); of course new gauginos X, ˜ α from 5 of SU (5). The exchange of the heavy Higgsinos and also heavy Higgsinos h
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leads to proton decay, suppressed only linearly by the GUT scale. More precisely, the exchange of heavy Higgsinos gives the effective operator of the type (90)
1 ˜ Q, ˜ QLQ MX
˜ stands for squarks. In turn the where Q and L stand for quarks and leptons and Q squarks are changed into quarks through the exchange of gauginos and one obtains an operator of the form QQQL of the proton decay. While it depends on the Yukawa sector and the sfermion masses and mixings, and thus not easy to predict precisely, proton lifetime is typically very close (or below) the experimental limit. . 5 5. SU (5) and neutrino mass. – The minimal theory of Georgi and Glashow fails in two crucial ways: a) it predicts massless neutrinos, b) gauge couplings do not unify. We need a minimal extension that cures both problems. It does not suffice to add right-handed neutrinos for they are gauge singlets and do not contribute to the running of gauge couplings and thus cannot help the unification. In other words, type-I see-saw fails in minimal SU (5). One could try type II, which requires a 15-dimensional Higgs representation, but instead I wish to discuss here a particularly simple and predictive theory [13], since it only requires adding the adjoint fermions 24F to the existing minimal model with three generations of quarks and leptons, and 24H and 5H Higgs fields. This automatically leads to the hybrid scenario of both type-I and type-III see-saw, since 24F has also a SM singlet fermion, i.e. the right-handed neutrino. This should be clear to the alert student. After all, the 24F is completely analogous to the 24H or even better the adjoint gauge boson representation, which we studied at length. The fermionic triplet simply corresponds to the SU (2) gauge boson triplet, whereas the singlet corresponds to the U (1) gauge boson. This singlet can be interpreted as a right-handed neutrino, for it is a SM neutral particle with Yukawa couplings to the light neutrinos. The triplet fermion on the other hand has the quantum numbers of the winos, the supersymmetric partners of the SU (2) charged and neutral gauge bosons. The main prediction of this theory is the lightness of the fermionic triplet. For a conventional value of MGU T ≈ 1016 GeV, the unification constraints strongly suggest its mass below TeV, relevant for the future colliders such as LHC. The triplet fermion decays predominantly into W (or Z) and leptons, with lifetimes shorter that about 10−12 s. Equally important, the decays of the triplet are dictated by the same Yukawa couplings that lead to neutrino masses and thus one has an example of predicted low-energy see-saw directly testable at colliders and likely already at LHC. The minimal implementation of the type-III see-saw in nonsupersymmetric SU (5) requires a fermionic adjoint 24F in addition to the usual field content 24H , 5H and three generations of fermionic 10F and 5F . The consistency of the charged fermion masses requires higher-dimensional operators in the usual Yukawa sector [14]. One must add
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the new Yukawa interactions (91)
5iF 24F 5H LY ν = y0i ¯ 1 ¯i i i i + 5 F y1 24F 24H + y2 24H 24F + y3 Tr (24F 24H ) 5H + h.c. Λ
After the SU (5) breaking, one obtains the following physical relevant Yukawa interactions for neutrino with the triplet TF ≡ TF · σ and singlet SF fermions (together with mass terms for TF and SF : (92)
mS mT S F SF + TF TF + h.c., LY ν = Li yTi TF + ySi SF H + 2 2
where yTi , ySi are two different linear combinations of y0i and yai vGU T /Λ (a = 1, 2, 3), Li are the lepton doublets and H is the Higgs doublet. It is clear from the above formula that, besides the new appearance of the triplet fermion, the singlet fermion in 24F acts precisely as the right-handed neutrino; it should not come out as a surprise, as it has the right SM quantum numbers. After the SU (2) × U (1) symmetry breaking (H = v ≈ 174 GeV), one obtains in the usual manner the light neutrino mass matrix upon integrating out SF and TF ySi ySj yTi yTj ij 2 , + (93) mν = v mT mS with mT ≤ 1 TeV (see below) and mS undetermined. From the above formula, one important prediction emerges immediately: only two light neutrinos get mass, while the third one remains massless. This is understood readily. First, the Yukawas here are vectors, and for example the vector coupling corresponding to the triplet can be rotated in the, say, 3rd direction. Thus only one light neutrino effectively coupled to the triplet, i.e. only one neutrino gets the mass through this coupling. Obviously, the same could have been said about the singlet and thus only two massive light neutrinos. This is of course independent of the nature of the heavy states, and the number of light massive neutrinos is in direct proportion to the number of heavy fermions, be they singlets or triplets. The mass of the fermionic triplet is found by performing the renormalization group analysis as before. From [13] one has (94)
(95)
exp 30π α1−1 − α2−1 (MZ ) = 20 84 F 4 B 5 m 3 m3 MGU T MGU T MGU T , MZ MZ5 mT mF (3,2) exp 20π α1−1 − α3−1 (MZ ) = 20 86 F 4 B 5 −1 m 8 m8 MGU T MGU T MGU T , MZ MZ5 mT mF (3,2)
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where mF,B , mF,B , mF 3 8 (3,2) and mT are the masses of weak triplets, color octets (only fermionic) leptoquarks and (only bosonic) color triplets, respectively. We discussed at length the well-known problem in the standard model of the low meeting scale of α1 and α2 . It is clear that the SU (2) triplet fermions are ideal from this point of view since they slow down the running of α2 , while leaving α1 intact (other particles have nonvanishing hypercharge and thus make α1 grow faster as to meet α2 even before). They should clearly be as light as possible while the color triplet as heavy B as possible. In order to illustrate the point, take mF 3 = m3 = MZ and mT = MGU T . −1 −1 −1 This gives (α1 (MZ ) = 59, α2 (MZ ) = 29.57, α3 (MZ ) = 8.55) MGU T ≈ 1015.5 GeV. Increasing the triplet masses mF,B reduces MGU T dangerously, making proton decay 3 too fast. Finally, one can ask, where must the octets be. Since the triplets slowed down the running of α2 , the meting point of α2 and α3 would become too large, unless α3 gets slowed down too. Thus the octets must lie much below MGU T , but since they contribute to the running more than the triplets, they should be also much above the weak scale, and one gets m8 = 107 –108 GeV For a more detailed discussion of unification constraints and especially the phenomenology of the triplet relevant for LHC, see [15]. The bottom line is a prediction of the light weak fermion triplet (96)
mT < TeV.
Its decays proceed via its Yukawa couplings yT and thus probe the neutrino mass. One can parametrize yT through the lepton mixing matrix. In normal hierarchy (NH) i.e. mν1 = 0, √ vyTi∗ = i mT Ui2 mν2 cos z ± Ui3 mν3 sin z ,
(97)
while in inverted hierarchy (IH) i.e. mν3 = 0, √ vyTi∗ = i mT Ui1 mν1 cos z ± Ui2 mν2 sin z ,
(98)
where, z is a complex parameter. You can readily show that in NH the neutrino masses are (99)
mν1 = 0,
mν2 =
Δm2S ,
mν3 =
Δm2A + Δm2S ,
while in the IH case (100)
mν1 =
Δm2A − Δm2S ,
mν2 =
Δm2A ,
mν3 = 0.
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The predominant decay modes of the triplets [15] are T → W (Z)+ light lepton whose strength is dictated by the neutral Dirac Yukawa couplings. (101) (102)
2 m2Z m2Z mT k 2 Γ(T → y 1− 2 1+2 2 , = 32π T m mT T 2 m2W m2W mT k 2 − − 1− 2 1+2 2 , yT Γ(T → W νk ) = 16π mT mT −
Ze− k)
k
(103)
(104)
k
0 − + Γ(T 0 → W + e− k ) = Γ(T → W ek ) 2 m2W m2W mT k 2 y 1− 2 1+2 2 , = 32π T m mT T 2 2 k 2 m2Z mZ mT 0 1+2 2 , 1− 2 yT Γ(T → Zνk ) = 32π mT mT k
k
where we averaged over initial polarizations and summed over final ones. From (103) one sees that the decays of T 0 , just as those of right-handed neutrinos, violate lepton number. In a machine such as LHC one would typically produce a pair T + T 0 (or T − T 0 ), whose decays then allow for interesting ΔL = 2 signatures of same-sign dileptons and 4 jets. This fairly SM background-free signature is characteristic of any theory with right-handed neutrinos as discussed in [36]. The main point here is that these triplets are really predicted to be light, unlike in the case of right-handed neutrinos. We discuss this further in sect. 7 on lepton number violation. 6. – SO(10): family unified The minimal gauge group that unifies the gauge interactions of the standard model was seen in the previous subsection to be based on SU (5) and studied at length. It is tailor fit for massless neutrinos just as the SM, for in the minimal version of the theory neutrinos get neither Dirac nor Majorana mass terms. Furthermore, the ordinary, nonsupersymmetric theory fails to unify gauge couplings. We found that the simple extension with the adjoint fermion representation provides a minimal and remarkably predictive theory with light fermionic triplet expected at LHC and whose decay rates probe the Dirac Yukawa couplings of neutrinos. We have a theory that works and furthermore gives serious hope for an old dream of verifying see-saw mechanism at colliders. So why should one ever wish to go beyond SU (5)? We can think of at least two reasons. First, if one is to worry about the Higgs mass naturalness, one may wish to include supersymmetry. While SU (5) with the low-energy supersymmetry has a rather appealing feature of providing automatically (as predicted many years ago) a gauge coupling unification, it is not an interesting theory of fermion masses and mixings. First of all, it offers no explanation for the smallness of R-parity violation in nature, and at the same time it requires a certain amount of arbitrary and unpredicted R-parity violation in order to provide neutrino masses. One can also include the type-II see-saw into the theory through the
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15H supermultiplet, and even attribute to it a mediation of supersymmetry breaking, but one ends up without any direct low-energy probes or interesting quark-lepton mass and mixings relations. This is where SO(10) fits ideally, for it also unifies matter besides the interactions. It works nicely without supersymmetry too, for it provides a natural unification of gauge couplings through the intermediate scale of L-R symmetry breaking. The general case SO(2N ) is presented in Appendix D. The one important representation of SO(10) is a 16-dimensional spinor, which can be decomposed under SU (5) as 16 = 10+ ¯ 5+1. It unifies a family of fermions with an addition of a right-handed neutrino per family. This minimal grand unified theory that unifies matter on top of interactions suggests naturally small neutrino masses through the see-saw mechanism. Furthermore, it relates neutrino masses and mixings to the ones of charged fermions, and is predictive in its minimal version. In this section I discuss some salient features in this theory while focusing on its minimal realizations. The crucial representation is a self-dual five-index anti-symmetric one responsible for right-handed neutrino masses and is a must, whether being elementary or composed at the loop level or through the higher-dimensional operators. A number of different minimal realizations of SO(10) depends on this construction, and what follows summarizes a few of them. There are a number of features that make SO(10) special: 1) A family of fermions is unified in a 16-dimensional spinorial representation; this in turn predicts the existence of right-handed neutrinos. 2) L-R symmetry is a finite gauge transformation in the form of charge conjugation. This is a consequence of both left-handed fermions fL and its charged-conjugated T counterparts (f c )L ≡ Cf R residing in the same representation 16F . 3) In the supersymmetric version, matter parity M = (−1)3(B-L) , equivalent to the R-parity R = M (−1)2S , is a gauge transformation [16], a part of the center Z4 of SO(10). It simply reads 16 → −16, 10 → 10. Its fate depends then on the pattern of symmetry breaking (or the choice of Higgs fields); it turns out that in the renormalizable version of the theory R-parity remains exact at all energies [17, 18]. The lightest supersymmetric partner (LSP) is then stable and is a natural candidate for the dark matter of the universe. 4) Its other maximal subgroup, besides SU (5) × U (1), is SO(4) × SO(6) = SU (2)L × SU (2)R × SU (4)c symmetry of Pati and Salam. It explains immediately the somewhat mysterious relations md = me (or md = 1/3me ) of SU (5). 5) The unification of gauge couplings can be achieved with or without supersymmetry. 6) The minimal renormalizable version (with no higher-dimensional 1/MP l terms) offers a simple and deep connection between b−τ unification and a large atmospheric mixing angle in the context of the type-II see-saw [19]. In order to understand some of these results, and in order to address the issue of construction of the theory, we turn now to the Yukawa sector.
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Theory of neutrino masses and mixings
. 6 1. Yukawa sector . – Fermions belong to the spinor representation 16F [4]. From 16 × 16 = 10 + 120 + 126,
(105)
the most general Yukawa sector in general contains 10H , 120H and 126H , respectively the fundamental vector representation, the three-index antisymmetric representation and the five-index antisymmetric and anti-self-dual representation. This can be seen by analogy with the Yukawa couplings of SO(6) (see Appendix D), Ly = y10 ΨT BΓi ΨΦi + y120 ΨT BΓi Γj Γk ΨΦ[ijk]
(106)
+y126 ΨT BΓi Γj Γk Γl Γm ΨΦ− [ijklm] . 126H is necessarily complex, supersymmetric or not; 10H and 126H Yukawa matrices are symmetric in generation space, while the 120H one is antisymmetric. Understanding fermion masses is easier in the Pati-Salam language of one of the two maximal subgroups of SO(10), GP S = SU (4)c × SU (2)L × SU (2)R (the other being SU (5) × U (1)). Let us decompose the relevant representations under GP S 16 = (4, 2, 1) + (¯ 4, 1, 2),
(107)
10 = (1, 2, 2) + (6, 1, 1), 120 = (1, 2, 2) + (6, 3, 1) + (6, 1, 3) + (15, 2, 2) + (10, 1, 1) + (10, 1, 1), 126 = (10, 3, 1) + (10, 1, 3) + (15, 2, 2) + (6, 1, 1). I illustrate the decomposition of a spinor representation 16 = Ψ+ (see Appendix D) (108)
Ψ+ ≡ |1 . . . 5 ;
1 . . . 5 = +1.
It contains (109)
1 2 3 = +1;
4 5 = +1
1 2 3 = −1;
4 5 = −1.
and (110)
The first one is 4 of SU (4)C , doublet of SU (2)L and the latter ¯4 of 4 of SU (4)C , doublet of SU (2)R , as can be read off readily from the sections on SO(4) and SO(6) of Appendix D. Exercise: Try to arrive at the rest of the above decomposition using the material in Appendix D. Clearly, the see-saw mechanism, whether type I or II, requires 126: it contains both (10, 1, 3) whose vev gives a mass to νR (type I), and (10, 3, 1), which contains a color
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singlet, B-L = 2 field ΔL , that can give directly a small mass to νL (type II). A reader familiar with the SU (5) language sees this immediately from the decomposition under this group (111)
126 = 1 + 5 + 15 + 45 + 50.
The 1 of SU (5) belongs to the (10, 1, 3) of GP S and gives a mass for νR , while 15 corresponds to the (10, 3, 1) and gives the direct mass to νL . Of course, 126H can be a fundamental field, or a composite of two 16H fields, or can even be induced as a two-loop effective representation built out of a 10H and two gauge 45-dimensional representations. In what follows I shall discuss carefully all three possibilities. Normally the light Higgs is chosen to be the smallest one, 10H . Since 10H = (1, 2, 2)P S is a SU (4)c singlet, md = me follows immediately, independently of the number of 10H you wish to have. Thus we must add either 120H or 126H or both in order to correct the bad mass relations. Both of these fields contain (15, 2, 2)P S , and its vev gives the relation me = −3md . As 126H is needed anyway for the see-saw, it is natural to take this first. The crucial point here is that in general (1, 2, 2) and (15, 2, 2) mix through (10, 1, 3) [20] and thus the light Higgs is a mixture if the two. In other words, (15, 2, 2) in 126H is in general nonvanishing(1 ). It is rather appealing that 10H and 126H may be sufficient for all the fermion masses, with only two sets of symmetric Yukawa coupling matrices. . 6 2. An instructive failure. – Before proceeding, let me emphasize the crucial point of the necessity of 120H or 126H in the charged fermion sector on an instructive failure: a simple and beautiful model by Witten [21]. The model is nonsupersymmetric and the SUSY lovers may place the blame for the failure here. It uses 16H in order to break B-L, and the “light” Higgs is 10H . Witten noticed an ingenious and simple way of generating an effective mass for the right-handed neutrino, through a two-loop effect which gives (112)
MνR yup
α 2 π
MGU T ,
where one takes all the large mass scales, together with 16H , of the order MGU T . Since 10H = (1, 2, 2)P S preserves quark-lepton symmetry, it is easy to see that (113)
Mν ∝ Mu , Me = Md , Mu ∝ Md ,
(1 ) In supersymmetry this is not automatic, but depends on the Higgs superfields needed to break SO(10) at MGU T .
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so that Vlepton = Vquark = 1. The model fails badly. The original motivation of Witten was a desire to know the scale of MνR and increase Mν , at that time neutrino masses were expected to be larger. But the real achievement of this simple, elegant, minimal SO(10) theory is the predictivity of the structure of MνR and thus Mν . It is an example of a good, albeit wrong theory: it fails because it predicts. What is the moral behind the failure? Not easy to answer. The main problem, in my opinion, was to ignore the fact that with only 10H already charged fermion masses fail. One needs to enlarge the Higgs sector, by adding for example a 120H ; the theory still leads to interesting predictions while possible completely realistic. . 6 3. Non-supersymmetric SO(10). – In the last two decades, and especially after its success with gauge coupling unification, grand unification by and large got tied up with low-energy supersymmetry. This is certainly well motivated, since supersymmetry is the only mechanism in field theory which controls the gauge hierarchy. In SO(10), gauge coupling unification needs no supersymmetry whatsoever. It only says that there must be intermediate scales [22], such as Pati-Salam SU (4)c × SU (2)L × SU (2)R or Left-Right SU (3)c × SU (2)L × SU (2)R × U (1)B-L symmetry, between MW and MGU T . An oasis or two in the desert is always welcome. Thus if we accept the fine-tuning, as we seem to be forced in the case of the cosmological constant, we can as well study the ordinary, nonsupersymmetric version of the theory. In this context the idea of the cosmic attractors [23] as the solution to the gauge hierarchy becomes extremely appealing. It needs no supersymmetry whatsoever, and enhances the motivation for ordinary grand unified theories. In what follows I discuss some essential features of a possible minimal such theory with 126H as a necessary ingredient for see-saw. Let us start by analyzing the case with an extra 10H field [35]. The most general Yukawa interaction is (114)
LY = 16F 10H Y10 + 126H Y126 16F + h.c.,
where Y10 and Y126 are symmetric matrices in the generation space. With this one obtains relations for the Dirac fermion masses (115)
M D = M1 + M0 , ME = −3M1 + M0 ,
MU = c1 M1 + c0 M0 , MνD = −3c1 M1 + c0 M0 ,
where we have defined (116)
M1 = 2, 2, 15d126 Y126 ,
M0 = 2, 2, 1d10 Y10
and (117)
c0 =
2, 2, 1u10 , 2, 2, 1d10
c1 =
2, 2, 15u126 . 2, 2, 15d126
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In the physically sensible approximation θq = Vcb = 0, these relations imply c0 =
(118)
mc (mτ − mb ) − mt (mμ − ms ) mt ≈ . ms mτ − m μ mb mb
Exercise: Derive this formula. Notice that this means that 10H cannot be real, since in that case one would have |2, 2, 1u10 | = |2, 2, 1d10 |, implying mt /mb of order one. It is necessary to complexify 10H , just as in a supersymmetric theory. If, taking advantage of this fact one decides to impose a Peccei-Quinn symmetry, thus providing a dark matter candidate, the Yukawa sector in nonsupersymmetric and supersymmetric models is similar. In this case, this model has the interesting feature of automatic connection between b − τ unification and large atmospheric mixing angle in the type-II see-saw. From MνL ∝ Y126 , one has MνL ∝ MD − ME , as shown in [19, 24]. This fact has inspired the careful study of the analogous supersymmetric version where mτ mb at the GUT scale works rather well [29]. In the nonsupersymmetric theory, b − τ unification fails badly, mτ ∼ 2mb [30]. The realistic theory will require a type-I see-saw, or an admixture of both possibilities. Suppose now that we choose instead 120H [35]. Since Y120 is antisymmetric, this means only 3 new complex couplings on top of Y126 . On gets in this case (119)
MD = M1 + M2 ,
MU = c1 M1 + c2 M2 ,
ME = −3M1 + c3 M2 ,
MνD = −3c1 M1 + c4 M2 ,
where M1 and c1 are defined in (116), (117), and (120)
M2 = Y120 2, 2, 1d120 + 2, 2, 15d120 ,
c2 =
2, 2, 1u120 + 2, 2, 15u120 , 2, 2, 1d120 + 2, 2, 15d120
2, 2, 1d120 − 32, 2, 15d120 , 2, 2, 1d120 + 2, 2, 15d120
c4 =
2, 2, 1u120 − 32, 2, 15u120 . 2, 2, 1d120 + 2, 2, 15d120
c3 =
It is easy to see that again there is a need to complexify the Higgs fields, by arguments similar to the case of 10H . In order to obtain algebraic expressions, from which a clearer physical meaning can be extracted, one can restrict the analysis to the second and third generations. Later, numerical studies could include the effects of the first generation as a perturbation. In the basis where M1 is diagonal, real and non-negative, for the two-generation case one gets (121)
M1 ∝
sin2 θ 0
0 cos2 θ
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and the most general charged fermion matrix can be written as Mf = μf
(122)
sin2 θ i(sin θ cos θ + f ) , −i(sin θ cos θ + f ) cos2 θ
where f = D, U, E stands for charged fermions and f vanishes for negligible second-generation masses. In other words |f | ∝ mf2 /mf3 . Furthermore the real parameter μf sets the third generation mass scale. By calculating up to leading order in |f |, we have the following interesting predictions [35]: 1) Type-I and type-II see-saw lead to the same structure I II ∝ MN ∝ M1 , MN
(123)
so that in the selected basis the neutrino mass matrix is diagonal. We see that the angle θ has to be identified with the leptonic (atmospheric) mixing angle θA up to terms of the order of |E | ≈ mμ /mτ . For the neutrino masses we obtain from (121) (124)
cos 2θA m23 − m22 + O(||). = 2 2 m3 + m 2 1 − sin2 2θA /2
Exercise: Derive this formula. This equation points to an intriguing correlation: the degeneracy of neutrino masses is measured by the maximality of the atmospheric mixing angle. 2) The ratio of tau and bottom mass at the GUT scale is given by mτ = 3 + O(||). mb
(125)
This is not correct in principle, the extrapolation in standard model gives mτ ≈ 2mb . However, several effects modify this conclusion, such as, for example, the inclusion of the first generation or the running of Yukawa couplings. We would in any case expect that mb comes out as small as possible. 3) The quark mixing is found to be (126)
|Vcb | = | cos 2θA (D − U )| + O(|2 |).
This equation demonstrates the successful coexistence of small and large mixing angles. In order for it to work quantitatively, | cos 2θA | should be as large as possible, i.e. θA should be as far as possible from the maximal value 45◦ . To make a definite numerical statement, again, the effects from the first generation and the loops have to be included.
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. 6 4. Supersymmetric case. – In supersymmetry 10H is necessarily complex and the bidoublet (1, 2, 2) in 10H contains the two Higgs doublets of the MSSM, with the vevs v u and v d in general different: tan β ≡ v u /v d = 1 in general. In order to study the physics of SO(10), we need to know what the theory is, i.e. its Higgs content. There are two orthogonal approaches to the issue, as we discuss now. Small representations. The idea: take the smallest Higgs fields (least number of fields, not of representations) that can break SO(10) down to the MSSM and give realistic fermion masses and mixings. The following fields are both necessary and sufficient (127)
45H , 16H + 16H , 10H .
It all looks simple and easy to deal with, but the superpotential becomes extremely complicated. First, at the renormalizable level it is too simple. The pure Higgs and the Yukawa superpotential at the renormalizable level take the form (128)
WH = m45 452H + m16 16H 16H + λ1 16H Γ2 16H 45H +m10 102H + λ2 16H Γ16F 10H + λ3 16H Γ16H 10H ,
(129)
Wy = y10 16F Γ16F 10H ,
where Γ stands for the Clifford algebra matrices of SO(10), Γ1 . . . Γ10 , and the products of Γ’s are written in a symbolic notation (both internal and Lorentz charge conjugation are omitted). Clearly, both WH and Wy are insufficient. The fermion mass matrices would be completely unrealistic and the vevs 45H , 6H , 16H would all point in the SU (5) direction. Thus, one adds non-renormalizable operators (130) ΔWH =
(131) ΔWy =
1 2 2 (45H ) + 454H + (16H 16H )2 + (16H Γ2 16H )2 + (16H Γ4 16H )2 MP l +(16H Γ16H )2 + (16H Γ5 16H )2 + {16H → 16H } +16H Γ4 16H 452H + 16H Γ3 16H 45H 10H + {16H → 16H } , 1 16F Γ16F 16H Γ16H + {16H → 16H } MP l +16F Γ3 16F 45H 10H + 16F Γ5 16F 16H Γ5 16H ,
where I take for simplicity all the couplings to be unity; there are simply too many of them. The large number of Yukawa couplings means very little predictivity. The way out is to add flavor symmetries and to play the texture game and thus reduce the number of couplings. This in a sense goes beyond grand unification and appeals to new physics at MP l and/or new symmetries. To me, maybe the least appealing aspect of this approach is the loss of R (matter) parity due to 16H and 16H ; it must be postulated by hand as much as in the MSSM.
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On the positive side, it is an asymptotically free theory and one can work in the perturbative regime all the way up to MP l . While this sounds nice, I am not sure what it means in practice. It would be crucial if you were able to make high precision determination of MGU T or mT , the mass of colored triplets responsible for d = 5 proton decay. The trouble is that the lack of knowledge of the superpotential couplings is sufficient even in the minimal SU (5) theory to prevent this task; in SO(10) it gets even worse. 2 Maybe more relevant is the fact that in this scenario MR MGU T /MP l 13 14 10 –10 GeV, which fits nicely with the neutrino masses via see-saw. Furthermore, see-saw can be considered “clean”, of the pure type I, since the type-II effect is suppressed by 1/MP l . Most important, the mb mτ relation from (129) is maintained due to small 1/MP l effects relevant only for the first two generations. Large representations. The nonrenormalizable operators in reality mean invoking new physics beyond grand unification. This may be necessary, but still, one should be more ambitious and try to use the renormalizable theory only. This means large representations necessarily: at least 126H is needed in order to give the mass to νR (in supersymmetry, one must add 126H ). The consequence is the loss of asymptotic freedom above MGU T , the coupling constants grow large at the scale ΛF 10MGU T . Once we accept large representations, we should minimize their number. The minimal theory contains, on top of 10H , 126H and 126H , also 210H [31-34] with the decomposition (132)
210H = (1, 1, 1)− + (15, 1, 1)+ + (15, 1, 3) + (15, 3, 1) +(6, 2, 2) + (10, 2, 2) + (10, 2, 2),
where the −(+) subscript denotes the properties of the color singlets under charge conjugation. The Higgs superpotential is remarkably simple, (133)
WH = m210 (210H )2 + m126 126H 126H + m10 (10H )2 + λ(210H )3 +η126H 126H 210H + α10H 126H 210H + α10H 126H 210H
and the Yukawa one even simpler (134)
WY = y10 16F Γ16F 10H + y126 16F Γ5 16F 126H .
Remarkably enough, this may be sufficient, without any higher-dimensional operators; however, the situation is not completely clear. There is a small number of parameters: 3 + 6 × 2 = 15 real Yukawa couplings, and 11 real parameters in the Higgs sector. In this sense the theory can be considered as the minimal supersymmetric GUT in general [34]. As usual, I am not counting the parameters associated with the SUSY breaking terms.
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The nicest feature of this program (and the best justification for the use of large representations) is the following. Besides the (10, 1, 3) which gives masses to the νR ’s, also the (15, 2, 2) in 126H gets a vev [32, 20]. Approximately (135)
15, 2, 2126
MP S 1, 2, 2, MGU T
with MP S = 15, 2, 2 being the scale of SU (4)c symmetry breaking. In SUSY, MP S ≤ MGU T and thus one can have correct mass relations for the charged fermions. What is lost, though, is the b − τ unification, i.e. with (15, 2, 2)126 = 0, mb = mτ at MGU T becomes an accident. However, in the case of type-II see-saw, there is a profound connection between b−τ unification and a large atmospheric mixing angle. The fermionic mass matrices are obtained from (134) u u Mu = v10 y10 + v126 y126 , d d Md = v10 y10 + v126 y126 , d d Me = v10 y10 − 3v126 y126 ,
(136)
u u MνD = v10 y10 − 3v126 y126 ,
(137)
MνR = y126 (10, 1, 3),
(138)
MνL = y126 (10, 3, 1),
2 where (10, 3, 1) MW /MGU T provides a direct (type II) see-saw mass for light neutrinos. The form in (136) is readily understandable, if you notice that (1, 2, 2) is a SU (4)c singlet with mq = m , and (15, 2, 2) is a SU (4)c adjoint, with m = −3mq . The vevs of the bidoublets are denoted by v u and v d as usual. Now, suppose that type II dominates, or Mν ∝ y126 ∝ Me − Md , so that
Mν ∝ Me − M d .
(139)
Let us now look at the 2nd and 3rd generations first. In the basis of diagonal Me , and for the small mixing de (140)
Mν ∝
m μ − ms de
de , m τ − mb
obviously, large atmospheric mixing can only be obtained for mb mτ [19]. Exercise: Prove that the above neutrino mass matrix requires b − τ unification in order to lead to a large mixing angle. Use the fact that the second-generation masses are small in comparison with the third-generation ones. Of course, there was no reason whatsoever to assume type-II see-saw. Actually, we should reverse the argument: the experimental fact of mb mτ at MGU T , and large
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θatm seem to favor the type-II see-saw. It can be shown, in the same approximation of 2-3 generations, that type I cannot dominate: it gives a small θatm [24]. This gives hope to disentangle the nature of the see-saw in this theory. As a check, it can be shown that the two types of see-saw are really inequivalent [24]. I wish to stress an important feature of this programme. Since 126 (126) is invariant under matter parity, R parity remains exact at all energies and thus the lightest supersymmetric particle is stable and a natural candidate for the dark matter. Mass scales. In SO(10) we have in principle more than one scale above MW (and ΛSU SY ): the GUT scale, the Pati-Salam scale where SU (4)c is broken, the L-R scale where parity (charge conjugation) is broken, the scales of the breaking of SU (2)R and U (1)B-L . Of course, these may be one and the same scale, as expected with low-energy supersymmetry. This solution is certainly there, since the gauge couplings of the MSSM unify successfully and encourage the single step breaking of SO(10). Is there any room for intermediate mass scales in SUSY SO(10)? It is certainly appealing to have an intermediate see-saw mass scale MR , between 1012 and 1015 GeV or so. In the nonrenormalizable case, with 16H and 16H , this is precisely what happens: 2 13 14 MR cMGU T /MP l c(10 –10 ) GeV. In the renormalizable case, with 126H and 126H , one needs to perform a renormalization group study using unification constraints. While this is in principle possible, in practice it is hard due to the large number of fields. The stage has recently been set, for all the particle masses were computed [25,26], and the preliminary studies show that the situation may be under control [27]. It is interesting that the existence of intermediate mass scales lowers the GUT scale [25, 28], allowing for a possibly observable d = 6 proton decay. Notice that a complete study is basically impossible. In order to perform the running, you need to know particle masses precisely. Now, suppose you stick to the principle of minimal fine tuning. As an example, you fine-tune the mass of the W and Z in the SM, then you know that the Higgs mass and the fermion masses are at the same scale √ (141)
mH =
λ mW , g
mf =
yf mW , g
where λ is a φ4 coupling, and yf an appropriate fermionic Yukawa coupling. Of course, you know the fermion masses in the SM model, and you know mH mW . In an analogous manner, at some large scale mG a group G is broken and there are usually a number of states that lie at mG , with masses (142)
mi = αi mG ,
where αi is an approximate dimensionless coupling. Most renormalization group studies typically argue that αi O(1) is natural, and rely on that heavily. In the SM, you could then take mH mW , mf mW ; while reasonable for the Higgs, it is nonsense for the fermions (except for the top quark).
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In supersymmetry all the couplings are of Yukawa type, i.e. self-renormalizable, and thus taking αi O(1) may be as wrong as taking all yf O(1). While a possibly reasonable approach when trying to get a qualitative idea of a theory, it is clearly unacceptable when a high-precision study of MGU T is called for. 4 Proton decay. As you know, d = 6 proton decay gives τp (d = 6) ∝ MGU T , while (d = 5) 2 gives τp (d = 5) ∝ MGU T . In view of the discussion above, the high-precision determination of τp appears almost impossible in SO(10) (and even in SU (5)). You may wonder if our renormalizable theory makes sense at all. After all, we are ignoring the higher-dimensional operators of order MGU T /MP l 10−2 –10−3 . If they are present with the coefficients of order one, we can forget almost everything we said about the predictions, especially in the Yukawa sector. However, we actually know that the presence of 1/MP l operators is not automatic (at least not with the coefficients of order 1). Operators of the type (in symbolic notation)
O5p =
(143)
c 164 MP l F
are allowed by SO(10) and they give (144)
O5p =
c [(QQQL) + (Qc Qc Qc Lc )] . MP l
These are the well-known d = 5 proton decay operators, and for c O(1) they give τp 1023 y. Agreement with experiment requires (145)
c ≤ 10−6 .
Exercise: Hard. Prove the above result. Use the fact that the supersymmetric oper˜Q ˜ and then use the ator of the type QQQL corresponds to an effective interaction QLQ ˜ ˜ interactions with gauginos to transform QQ into QQ in order to create a proton decay operator QQQL. It happens at the one-loop level. Could this be a signal that 1/MP l operators are small in general? Alternatively, you need to understand why just this one is to be so small. It is appealing to assume that this may be generic; if so, neglecting 1/MP l contributions in the study of fermion masses and mixings is fully justified. 7. – Majorana neutrinos: lepton number violation and the origin of neutrino mass Majorana neutrino mass implies ΔL = 2 processes: 1) neutrinoless double-β decay, 2) same-sign dilepton par production at colliders [36].
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p
n WR
e
x mN
n
WR
e p
Fig. 4. – Neutrinoless double-β decay through WR and N .
. 7 1. Neutrinoless double-β decay. – This is the usual text-book example of ΔL = 2 and is often considered a probe of Majorana mν . However, the Majorana case needs a completion of the SM and ββ 0 depends in general on the completion. A simple and clear example is provided by L-R symmetric theories with low MR scale in which case there are new contributions to ββ 0 . The dominant one is due to the WR exchange and right-handed neutrinos N , as shown in fig. 4. It gives 1 1 (146) (ββ 0 )RR ∝ 4 ee, M W R MN to be compared with the usual WL contribution (147)
(ββ 0 )LL ∝
1 (mν )ee , 4 MW p2 L
where we assume gL gR and p is the momentum exchange p 100 MeV. We have 4 1 MWL p2 (ββ 0 )RR (148) ee.
(ββ 0 )LL MW R (mν )ee MN For MR in the few TeV region and MN TeV, the (RR) contribution tends to dominate over the (LL) one, and clearly right-handed neutrinos should not be too light. Since mν → 0 when yD → 0, you can imagine a situation when neutrino mass is arbitrarily small, but (ββ 0 )RR = 0 due to the N exchange. Srictly speaking, ββ 0 is not a measure of light neutrino masses and it will be hard to disentangle the origin of the see-saw through this process. In particular, we would need to know whether it is due to exchange of ν’s or heavy particles needed to complete the SM in order to have mν = 0 (such as NR ).
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j j
WR N
d
uŦ
WR
l
l Fig. 5. – Production of lepton-number–violating same-sign dileptons at colliders through WR and N .
It is thus crucial to have a direct measure of lepton number violation which can probe the source of neutrino Majorana mass. This is provided by the same-sign dilepton production at colliders as we discuss below. . 7 2. Lepton number violation at colliders. – We have just seen that ββ 0 is obscured by various contributions which are not easy to disentangle. We need some direct tests of the origin of ΔL = 2, i.e. the see-saw mechanism. This comes about from possible direct production of the right-handed neutrinos through a WR production. The crucial point here is the Majorana nature of N : once produced it should decay equally often into leptons and antileptons. This led us [36] to suggest a direct production of the same-sign dileptons at colliders as a manifestation of ΔL = 2. The most promising channel is + 2 jets as seen form fig. 5. One can also imagine a production of N through its couplings to WL (proportional to yD ), but this is a long shot. It would require large yD and large cancellations among them in order to have small mν . This could be achieved in principle by fine-tuning, but is not the see-saw mechanism. The crucial characteristics: 1) no missing energy which helps to fight the background, 2) by measuring energies and momenta of the final states, one can reconstruct both the mas of Wr and of the right-handed neutrino, 3) the process can be amplified by the WR resonance.
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Ŧ
l
Y ++
Ŧ
l
Z
ŦŦ
Y l l Fig. 6. – Production of a pair of double-charged Higgs scalars and subsequent decay into pairs of same-sign dileptons.
The main background comes from b¯b + jets, but can be fought against with the usual cuts of large pt for leptons and jets. Also important is tt¯ + jets, which is less present but more resistent to large pT cuts. Careful and complete studies were performed with encouraging results: one can easily discover WR at the LHC up to MWR 3–4 TeV and mN 100 GeV–TeV. In the L-R symmetric theories one also predicts type-II see-saw as discussed before. Type II can also exist by itself in which case it can lead to rather interesting signatures at the colliders if the scale of the SU (2)L triplet Δ is light enough. In particular, it can lead to the production of doubly charged scalars that decay into same-sign dilepton pairs as in fig. 6. Notice that Δ++ and Δ−− decay through the Yukawas yΔ , these decays thus probe the neutrino mass matrix (149)
Mν = yΔ Δ.
One can derive the sum rules for the flavor structure of fig. 6. Of course, this is valid only when these decays dominate over the decays with W bosons through Δ. The relative strength of Δ−− → and Δ−− → W − W − depends on yΔ . From (150)
Γ(Δ−− → )
2 yΔ MΔ 8π
and (151)
Γ(Δ−− → W − W − )
g 2 Δ2 8πMΔ
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YT TŦ
l
Z j
WŦ j T
0
W+
YT
j l j
Fig. 7. – The same-sign dilepton signature of type-III seesaw through the production of the charged and neutral components of a fermion triplet TF .
for MΔ MW , one gets (152)
B(Δ−− → ) ≡
2 2 yΔ Γ(Δ−− → ) MΔ
. Γ(Δ−− → W − W − ) g 2 Δ2
Thus B(Δ−− → ) ≥ 1 requires that the vev of Δ be small and yΔ large. Ideally, observing both decays would establish the SU (2) gauge triplet property of Δ and could measure the form of the neutrino mass matrix. The widely separated dilepton pairs in the case of B(Δ−− → ) ≥ 1 provide a clean manifestation of the type-II see-saw mechanism and allow for the discovery of Δ++ with MΔ ≤ 800 GeV. In short, both type I and II could lead to exciting ΔL = 2 signatures at LHC, if WR and N and/or Δ are light enough. But, as will be discussed later, in predictive grand unified theories such as minimal SO(10), they are expected to be rather heavy, out of reach for LHC. One can ask the same question in the case of type-III see-saw. As we said, one would need at least the fermionic triplets in order to have at least two massive neutrinos, one could have a hybrid situation of type-I and type-III see-saw, with a heavy fermionic singlet (N ) and triplet (T ). This case is particularly interesting, since it emerges naturally in the SU (5) grand unified theory. Again, the process of interest for LHC is a production of same-sign dileptons (but now with 4 jets) as in fig. 7. The main point here is that in the minimal SU (5) theory augmented by an adjoint fermionic representation 24F the fermion triplet TF is predicted to lie below TeV, and thus the above process is a realistic possibility at colliders such as LHC.
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311
8. – Summary and outlook The smallness of neutrino mass is an intriguing fact that gives hope of being a window into a new physics beyond the standard model. This crucially depends on the nature of neutrino mass, i.e. whether it is Dirac or Majorana. In the former case, the standard model is a complete theory and although the smallness of neutrino mass is attributed to the smallness of Dirac Yukawa couplings. True, this is not explained, but strictly speaking there may be no new physics, the same way that there may be no new physics behind the smallness of electron mass. In the limit of small Yukawas one has more symmetry, and thus small Yukawas are technically natural, protected from high-energy physics. The Dirac case thus gives no clue where to look for a new physics. Of course, one can always search for horizontal symmetries as the explanation of small Yukawas, but here there is the danger of only changing the language. The Majorana case, on the other hand, provides a clear window into new physics for the MSM with Majorana neutrino mass is not a complete theory. At the same time, this case implies a violation of lepton number through a neutrinoless double-beta decay as is well known and the possible production of the same-sign dileptons, less known but becoming a new hot field in itself. The completion of the MSM that produces small neutrino Majorana mass results in the celebrated see-saw mechanism which comes in three different varieties. In order to be predictive, though, the see-saw mechanism needs a theory behind, for otherwise it is simply a linguistic variation on the effective d = 5 operator that we saw necessarily describe neutrino mass after the new states are integrated out. One important theory which leads to both type-I and -II see-saw is based on L-R symmetry, and has been a principal source of neutrino mass and see-saw. If the scale of L-R symmetry breaking were to be in the TeV region, one would have a possibility of seeing both the parity restoration and the origin of the neutrino mass through the production of a right-handed charged boson and right-handed neutrinos. Similarly, one could in principle produce the scalar triplet responsible for the type-II see-saw. The scale of L-R breaking can be predicted only in grand unification and in simple, predictive models it is quite large, far above the TeV scale of colliders. Still, one may be able to connect the values of neutrino masses and mixings with the predictions for the branching ratios of proton decay an thus have a check on the theory, albeit indirect. On the other hand, the type-III see-saw finds its natural realization in SU (5) grand unified theory, when the minimal model of Georgi and Glashow is augmented by an adjoint fermion representation. This allows for the unification of gauge couplings and provides a hybrid type-I and -III see-saw. One predicts one massless neutrino and more important a light weak triplet fermion, with a mass below TeV. The decays of the triplet probe neutrino masses and mixings through the lepton number violating production of same-sign dileptons accompanied by four jets. The hope of finding the origin of neutrino mass becomes feasible at colliders such as LHC. In summary, I tried to argue in these lectures in favor of Majorana masses of neutrinos, and the possibility of seeing its origin through lepton number violation or the connection with proton decays. The lepton number violation will be searched for in the
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new generation of neutrinoless double-beta decay and at LHC. Hopefully, a serious effort will be put in the next generation of proton decay experiments; they could be simultaneously a probe of baryon number violation in nature and an origin of neutrino masses and mixings. Appendix A. Dirac and Majorana masses The irreducible spin 1/2 representations of the Lorentz group are the two-component left- and right-handed chiral fermion Weyl fields uL and uR , which transform under the Lorentz group as uL,R → ΛL,R uL,R ,
(A.1) with
ΛL ≡ eiσ/2(θ+iφ ) ,
(A.2)
ΛR ≡ eiσ/2(θ−iφ ) . denotes the boosts. The spinors ψL The three Euler angles θ stand for rotations, ad φ and ψR transform the same under the rotations, but in an opposite manner under the boosts. It is straightforward to show that the following bilinear combinations are Lorentz invariant: (A.3)
(M )
uTL iσ2 uL
(D)
u†L uR
uTR iσ2 uR
and and
u†R uL
(Majorana type),
(Dirac type).
Historically, the Dirac type came first, but in a sense the Majorana invariant is even more fundamental for it needs only one species of fermions. To bridge the gap with Dirac four-component fermions, we need the Dirac algebra (A.4)
{γ μ , γ ν } = 2g μν ,
g μν = diag(1, −1, −1, −1),
with (A.5)
i
γ =
0 −σ i
σi , 0
0
γ =
(A.6)
γ5 = iγ 1 γ 2 γ 3 γ 0 =
12 0
and the projectors (A.7)
PL.R ≡
1 ± γ5 . 2
0 12
0 −12
12 , 0
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The Dirac charge conjugation, defined through C T γ μ C = −γμT ,
(A.8)
C T = −C
is with my conventions C = iγ2 γ0 .
(A.9)
In other words, the Majorana mass term can be written as T CψL + h.c.) (M ) mM (ψL
(A.10) and the Dirac one as (A.11)
¯ (D) mD (ψ¯L ψR + ψ¯R ψL ) ≡ mD ψψ,
ψ ≡ ψ L + ψR .
It is convenient to work with left-handed antiparticles instead of right-handed particles (A.12)
T , (ψ C )L ≡ C ψ¯R
in which case one can write a mass matrix for ψL and (ψ C )L in the Majorana notation (ψ1T Cψ2 ) (A.13)
mL mD
mD , mR
where mL and mR are the Majorana mass terms of ψL and ψR , respectively. The case of a pure Dirac fermion simply means mL = mR = 0. If neutrino mass is of the Majorana type, on the other hand, it will imply a violation of the lepton number and a new rich physics associated with it. Appendix B. Majorana spinors: Feynman rules Take a two-component spinor with left-handed chirality ψL with the following Lagrangian: m M T ψL CψL + h.c. , (B.1) LM = iψ¯L γ μ ∂μ ΨL − 2 where the subscript M indicates the Majorana nature of the mass term. In order to bridge the gap with the familiar 4-component Dirac case, introduce by analogy (B.2)
T ψM ≡ ψL + C ψ¯L .
From (B.3)
ψ¯M γ μ ∂μ ψM = 2ψ¯L γ μ ∂μ ψL
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and T CψL + h.c., ψ¯M ψM = ψL
(B.4) we get
1 ¯ μ iψM γ ∂μ − mM ψ¯M ψM . 2
LM =
(B.5)
Two important facts emerge 1) mM is the (Majorana) mass of ψM , 2) one can use the usual Dirac case Feynman rules. Appendix C. SU (N ) group theory On a fundamental N -dimensional complex representation Φ, the SU (N ) group acts as (C.1)
U † U = 1,
Φ → U Φ,
det(U ) = 1
and U can be written as (C.2)
U = e−iθa Ta ,
a = 1 . . . N 2 − 1,
where the group generators Ta satisfy (C.3)
Ta = Ta† ,
Tr(Ta ) = 0,
[Ta , Tb ] = ifabc Tc ,
where fabc are the group structure constants. There is also a complex conjugate representation (C.4)
Φ∗ → U ∗ Φ∗
and an (N 2 − 1)-dimensional adjoint representation (C.5)
A → U AU † = A − iθa [Ta , A] + . . . .
In other words, the generators act on A as commutators. One can write A = Aa Ta , so that Aa transforms under a small group rotation as (C.6)
Aa → Aa + fabc θb Ac .
Examples of fields transforming as the adjoint representation are the gauge bosons A of SU (N ) and the heavy scalars Σ employed to break the grand unified symmetry. The
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reason for the latter is the fact that under a unitary transformation Σ → U ΣU † , one can have Σ diagonal, which in turn implies (C.7)
[Σ, Ta ∈ Cartan] = 0.
The adjoint Higgs preserves the rank of the group after the symmetry breaking. This is specially important in SU (5) since it has the same rank (= 4) as the SM gauge group. All other representations are built out of the fundamental Φ (and/or Φ∗ ) by symmetrizing and antisymmetrizing (and subtracting the trace when necessary). For example (C.8)
Φi Φj = Φ[i,j] + Φ{i,j} N (N − 1) N (N + 1) . 2 2
This means that all the charges get summed up (C.9)
Q(Φi Φj ) = Q(Φi ) + Q(Φj ).
Appendix D. SO(2N ) group theory SO(2N ) is the group of real orthogonal transformations, OT O = OOT = 1, with det(O) = 1. It can be generated by N (N − 1)/2 Hermitian antisymmetric matrices (D.1)
O = e−iθij Lij ,
with (D.2)
(Lij )kl = −i(δik δjl − δil δjk ),
so that one has the following commutation relations: (D.3)
[Lij , Lkl ] = i(δik Ljl − δjl Lik ).
The N -dimensional Cartan subalgebra is spanned by (D.4)
Cartan = {L12 , L34 , . . . , L2N −1,2N },
whose eigenvalues are ±1. The fundamental (vector) representation transforms as (D.5)
Φi → Oij Φj
and is generated by Lij in Appendix D.2. One can construct the general N -index irreducible representation by antisymmetrizing or symmetrizing (and subtracting traces)
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N times the vector representation. Rather interesting are the [N ]-index antisymmetric ones, for one can complexify them by introducing Φ± [a1 ...aN ] = Φ[a1 ...aN ] ±
(D.6)
iN a ...a b ...b Φb ...b . N! 1 N 1 N 1 N
We illustrate this on a simple example below in SO(2) where this amounts to just complexifying a fundamental representation. It turns out that such 5 index antisymmetric 126-dimensional representation of SO(10) plays a profound role in the physics of neutrino mass; this is discussed in sect. 6. . D 1. SO(2N ): spinors. – By analogy with the Dirac algebra in Minkowski space, an Euclidean version is based on the Clifford algebra of the Γi matrices (i = 1 . . . 2N ) {Γi , Γj } = 2δij ,
(D.7)
out of which one can construct N (N − 1)/2 generators Σij =
(D.8)
1 [Γi , Γj ], 4i
which satisfy the usual commutation relations of the SO(2N ) generators in Appendix D.3. It is easy to see that the Cartan subalgebra consists of N generators Cartan = {Σ12 , . . . , Σ2N −1,2N },
(D.9)
whose eigenvalues are ±1/2. The appropriate 2N -dimensional complex representation Ψ is called a spinor of SO(2N ). Adding a spinor changes of course a group, just as SO(3) becomes SU (2). One often calls SO(10) with spinors Spin(2N ). The spinors transforms in the following manner: Ψ → e−iθij Σij Ψ.
(D.10)
Again, by analogy with Dirac γ5 matrix, one can introduce ΓFIVE = (−1)N Γ1 . . . Γ2N ,
(D.11) with the properties (D.12)
Γ2FIVE = 1,
[ΓFIVE , Σij ] = 0,
{ΓFIVE , Γi } = 0.
By using the projectors (D.13)
Γ+(−) ≡
1 ± ΓFIVE , 2
one can construct the irreducible 2N −1 dimensional spinors (D.14)
Ψ± ≡ Γ+(−) Ψ,
by analogy with Weyl spinors of the Lorentz group.
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Theory of neutrino masses and mixings
One can also introduce the analogue of the usual charge conjugation by demanding that (D.15)
ΨT BΨ = invariant ⇔ Ψc ≡ BΨ∗ ,
which amounts to (D.16)
ΣT B + BΣ = 0.
There are two possible solutions for B (D.17)
B(1) = Γ1 . . . Γ2N −1 ,
B(2) = Γ2 . . . Γ2N .
. D 2. The ket notation for spinors. – From (D.18)
ΓFIVE = 2Σ12 . . . 2Σ2N +1,2N ,
one can write (D.19)
ΓFIVE = 1 2 . . . N ,
where i are ±1, the eigenvalues of Σ2i−1,2i . Then one can denote the Ψ+ spinors as a ket (D.20)
Ψ+ ≡ |1 . . . N
For example, take the spinors Ψ+ of SO(10) (D.21)
Ψ+ ≡ |1 . . . 5 ;
1 . . . 5 = +1.
The 16-component Ψ+ can be decomposed as ⎧ 1 field | + + + ++ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | + + + −−, | + + − +−, | + + − −+ ⎪ ⎪ ⎨ 10 fields | + − + +−, | + − + −+, | + − − ++ (D.22) Ψ+ = ⎪ | − + + +−, | − + + −+, | − + − ++, | − − + ++ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | + − − −−, | − + − −− ⎪ ⎪ ⎩ 5 fields | − − + −, | − − − +−, | − − − −+ We will see that this can be interpreted as a decomposition under SU (5) (D.23)
16 = 10 + 5 + 1,
in other words, a family of fermions augmented by a right-handed neutrino makes an irreducible spinorial representation of SO(10). The unification of matter, on top of gauge interactions, points strongly towards SO(10). However, in order to appreciate this fact and have fun with SO(10), we first go through some pedagogical exposition of smaller groups.
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. D 3. SO(2): a prototype for SO(4n + 2). – We choose (D.24)
Γ1 = σ1 ,
Γ2 = σ2 ,
so that (D.25)
ΓFIVE = σ3
σ3 , 2
Σ12 =
which illustrates clearly [ΓFIVE , Σi,j ] = 0. The irreducible 1-component spinors transform as (D.26)
Ψ+ → e−iθ/2 Ψ+ ,
Ψ− → e+iθ/2 Ψ− ,
since (D.27)
Ψ≡
Ψ+ Ψ−
−iθσ3 /2
→e
Ψ+ . Ψ−
On the other hand, the two-component vectors transform as φ1 φ1 cos θ sin θ (D.28) → , − sin θ cos θ φ2 φ2 or (D.29)
φ1 ± iφ2 → e±iθ (φ1 ± iφ2 ).
Equations (D.26) and (D.29) simply account for the fact SO(2) U (1). The internal “charge” conjugation B can be chosen as B1 = σ1 , so that (D.30)
ΨT BΨ = Ψ+ Ψ− .
However, only Ψ+ (or Ψ− ) is an irreducible spinor, therefore there is no mass term for an irreducible spinor of SO(2). In other words, the spinors Ψ+ (Ψ− ) are chiral and can represent physical particles such as the fermions of the SM. This is true in any SO(4n + 2) theory. In particular, in SO(10), which means that it offers hope of being realistic. Dual representation. From (D.31)
ij det O = Oik Ojl kl ,
is is easy to see that φi and ij φi transform in the same way. We can introduce the self(anti-self) dual representation (D.32)
1 Φi (±) = √ (φi ± iij φj ), 2
which is nothing but the complex representation of U (1) (D.29). This should make clear the generic concept of self-dual representations in SO(2N ) discussed before.
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Theory of neutrino masses and mixings
Yukawa couplings. We have seen that there is no direct mass term. There are Yukawa couplings, though, of the type (D.33)
LY = ΨT Bσi Ψφi = Ψ+ Ψ+ (φ1 − iφ2 ) + Ψ− Ψ− (φ1 + iφ2 ),
as dictated by U (1) charges. . D 4. SO(4). – One knows that SO(4) is isomorphic to SU (2) × SU (2), and it plays an important role in providing a left-right symmetric subgroup of SO(10). It is a Euclidean analog of the Lorentz group and the Clifford algebra can be generated by
(D.34)
0 Γ1 = σ1 0 Γ3 = σ3
σ1 0 σ2 Γ2 = 0 σ2 0 σ3 0 −i , Γ4 = i 0 0
so that (D.35)
0 , −iσ2
1 0 0 −1
ΓFIVE =
and “charge” conjugation can be taken as (D.36)
B(1) = Γ1 Γ3 =
−iσ2 0
or (D.37)
B(2) = Γ2 Γ4 =
iσ2 0
0 . −iσ2
The mass term (D.38)
ΨT BΨ ∝ ΨT+ iσ2 Ψ+ + . . . ,
where (D.39)
Ψ± =
1 ± Γ5 Ψ± . 2
In other words, the mass term for Ψ+ (or Ψ− ) is invariant, which means that we can have no chiral fermions in SO(4). This is true for all SO(4n) groups. In the ket notation (D.40)
Ψ+ = |1 2 ;
1 2 = 1;
1,2 = ±1,
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or | + + . | − −
(D.41)
Ψ+ =
Introduce the neutral generator of SU (2)L and SU (2)R (D.42)
T3L ≡
1 (Σ12 + Σ34 ), 2
T3R ≡
1 (Σ12 − Σ34 ), 2
and you see that Ψ+ is an SU (2)L doublet, SU (2)R singlet field, an analog of left-handed Weyl spinors of the Lorentz group. Similarly, Ψ− is an SU (2)L singlet, SU (2)R doublet field. . D 5. SO(6). – SO(6) ∼ SU (4)C is the Pati-Salam group of quark-lepton symmetry, with leptons as the fourth color. It deserves a brief description. Start with a six-dimensional vector Φi (i = 1 . . . 6). It is easy to see that the components (φ1 ± φ2 ), (φ3 ± φ4 ), (φ5 ± φ6 ) transform as 3 and 3∗ of its subgroup SU (3) which we identify with the color. The neutral generators are identified as 1 (Σ12 − Σ34 ), 2 1 = (Σ12 + Σ34 − 2Σ56 ). 2
T3C =
(D.43)
T8C
The additional neutral generator of SU (4), identifiable as B-L, can be written as 2 B-L = − (Σ12 + Σ34 + Σ56 ). 3
(D.44)
Regarding spinors, the positive chirality can be written as
(D.45)
Ψ+ =
⎧ ⎨color singlet
| + ++,
⎩color triplet (B-L) = 1/3
| + −−, | − +−, | − −+.
It says simply that the irreducible 4-component spinor of SO(6) is a fundamental of SU (4) with the decomposition under SU (3)C (with B-L) (D.46)
Ψ+ = 4 = 1−1 + 31/3 ,
which is precisely a combination of a lepton and a colored quark. Similarly, Ψi = 4∗ = 1+1 + 3−1/3 stands for an antilepton and antiquark. Exercise: As a check, show that 4 × 4 = 6 + 10. Show that 6 of SO(6) has the quantum number of the 6 (antisymmetric) of SU (4).
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Theory of neutrino masses and mixings
Yukawa couplings in SO(6). We know that the irreducible spinors of SO(6) are fundamental representations of SU (4) and 4 × 4 = 6 + 10. There are then two types of Yukawa couplings (D.47)
LY = y6 ΨT BΓi ΨΦi + y10 ΨT BΓi Γj Γk ΨΦ− [ijk] ,
where it is a simple exercise to show that Φ− [ijk] is an anti-self-dual representation (D.48)
− Φ− ijk = Φ[ijk] =
i ijklmn Φ[lmn] , 3!
and where Φ− [ijk] is the 3-index antisymmetric tensor of SO(6). Exercise: Construct the self-dual and anti-self-dual representation of SO(6) out of the 3-index antisymmetric representation Φ[ijk] . Show that 20 = 10 + 10. Then prove equation (D.47) and show that there are no other couplings. Exercise: Take the Pati-Salam group SO(4) × SO(6) SU (2)L × SU (2)R × SU (4)c . Show that the representations (2, 1, 4) and (1, 2, 4) give a family of quarks and leptons augmented by a right-handed neutrino. Exercise: The chiral anomalies are proportional to Λijk = Tr({Ti , Tj }Tk ). Show that the SO(2N ) groups are anomaly free, except for the SO(6). Comment on why SO(6) must have an anomaly. REFERENCES [1] Mohapatra R. N. and Pal P. B., World Sci. Lect. Notes Phys., 60 (1998) 1 (World Sci. Lect. Notes Phys., 72 (2004) 1). [2] Strumia A. and Vissani F., arXiv:hep-ph/0606054. [3] Glashow S. L., Nucl. Phys., 22 (1961) 579. [4] For useful reviews on spinors in SO(2N ), see Mohapatra R. N. and Sakita B., Phys. Rev. D, 21 (1980) 1062; Wilczek F. and Zee A., Phys. Rev. D, 25 (1982) 553. See also Nath P. and Syed R. M., Nucl. Phys. B, 618 (2001) 138; Aulakh C. S. and Girdhar A., arXiv:hep-ph/0204097. [5] Mohapatra R. N., Phys. Rev. Lett., 43 (1979) 893. [6] Nath P. and Perez P. F., arXiv:hep-ph/0601023. [7] Minkowski P., Phys. Lett. B, 67 (1977) 421; Yanagida T., Proceedings of the Workshop on Unified Theories and Baryon Number in the Universe, Tsukuba, 1979, edited by Sawada A., Sugamoto A. and Glashow S., in Cargese 1979, Proceedings, Quarks and Leptons (1979); Gell-Mann M., Ramond P. and Slansky R., Proceedings of the Supergravity Stony Brook Workshop, New York, 1979, edited by Van Niewenhuizen P., ´ G., Phys. Rev. Lett., 44 (1980) 912. Freeman D., Mohapatra R. and Senjanovic [8] Magg M. and Wetterich C., Phys. Lett. B, 94 (1980) 61; Lazarides G., Shafi Q. and ´ G., Wetterich C., Nucl. Phys. B, 181 (1981) 287; Mohapatra R. N. and Senjanovic Phys. Rev. D, 23 (1981) 165. [9] Pati J. C. and Salam A., Phys. Rev. D, 10 (1974) 275. [10] Mohapatra R. N. and Pati J. C., Phys. Rev. D, 11 (1975) 2558.
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DOI 10.3254/978-1-60750-038-4-323
Calorimetric beta spectroscopy F. Gatti Dipartimento di Fisica, Universit` a di Genova e INFN, Sezione di Genova - Italy
Summary. — In the last decade the energy dispersive spectroscopy of nuclear radiation has made impressive progresses by means of small thermal microcalorimeters operating at about 0.1 K. The present status of this technology, which has achieved 2 eV energy resolution, allows to design a true calorimetric experiment for neutrino mass direct determination with sub-eV sensitivity from the β spectrum of 187 Re and the E.C. spectrum of 163 Ho. The calorimetric method, often indicated as solution for a model-independent measurement, allows to overcome the final states problem of impulse spectroscopy. A further reduction of the systematic uncertatinties might be achived by comparing the finite neutrino mass effect of the two isotopes. Here, the motivations, the principles of operations, the results from the first pilot measurements and the future perspectives are described.
1. – Introduction In 1914 Chadwick made the discovery of the continuous spectra of β-ray of RaB (214 Pb) with a magnetic spectrometer and electron counter [1]. The large spread of the electron kinetic energy from the nuclear decay was in contrast with the quantum theory of the disintegration of the bodies. Actually, at that time, it was already established that α-rays were emitted as mono-energetic lines in transitions from a nucleus to a quantum state of a final nucleus, which could subsequently decay emitting γ lines, definitively demonstrating that the nuclear phenomena were regulated by the quantum c Societ` a Italiana di Fisica
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Fig. 1. – Schematics of the Ellis and Wooster experiment: two identical lead calorimeters are suspended with low thermal conductivity mica supports in an adiabatic box. The RaE source was deposited onto one of the two thin rods, which can be fully inserted in the lead calorimeter. The second rod, without radioactive source, is used for subtracting the systematic heating effect caused by its introduction.
rules. Rutherford [2], taking into consideration that discrete electron lines have been found over the continuous spectrum of RaB, supposed the existence of slowing down processes of the primary mono-energetic electrons in the atom itself, therefore accounting for the observed wide distributions. Similarly, Mitner [3] attributed these secondary processes to the surrounding materials. After the observation of the RaE (210 Bi) β decay, which was showing a pure continuous distribution without lines, more fundamental concerns about the nuclear phenomenology were posed, including the non-conservation of the energy. Bohr [4] supported the explanation that in the nuclear processes the energy conservation was no longer valid. In the paper The Average Energy of Disintegration of Radium E [5], in 1927, Ellis and Wooster described an experiment for testing whether the β-ray bodies emit always the same energy in the disintegration or, as they supposed in previous reports, the electrons were directly emitted from the nucleus with varying energies. They performed the first calorimetric experiment that has been applied to the study of the β decay. The calorimeter, which has been depicted in fig. 1, was made with a cylinder of lead 13 mm long and 3.5 mm in diameter with a central hole of 1 mm in diameter. A second copy of the lead calorimeter was built and mounted close to the first. The thermal conductance was minimized by using supports of thin foils of mica and the temperature measured with thermocouples. The calorimeter was placed in a copper box and the RaE sources were inserted and extracted in and from the lead cylinder using small rods. The same tools without radioactive source were used for the second lead cylinder in order to subtract the systematic heating effect of the introduction. Temperature increases of about 1 mK at room temperature were measured with the source inside the calorimeter,
Calorimetric beta spectroscopy
325
with a precision good enough to establish that the mean energy per disintegrating atom was 1/3 of the maximum energy of the β-ray spectrum, within 15% accuracy. This result excludes the presence of a “slowing-down” mechanism in the atom or the surrounding materials and opens the way to the Pauli conjecture [6] of the emission of a weakly neutral non-interacting particle accompanying the electron, later named neutrino by Fermi [7]. 2. – Towards the calorimetric single-event detection The Ellis and Wooster calorimeter was not able to detect single decay events. An a important step forward in the single-particle thermal detection has been made in 1949 by Andrews, Fowler and Williams [8] in bombarding with α-particles a superconducting strip of NbN, 6 μm thick, that was maintained at the normal-to-superconducting critical temperature of about 15.5 K with a current of 40 mA. They reported a countable electrical pulse for each particle impact, corresponding to an equivalent pulse height of 10−7 V. The readout electronics were made crudely with an audio transformer and a pulse amplifier. This technique is incredibly similar to the present one that is used in the best performing calorimetric detectors. However, the sensitivity of the thermal detectors to single particles has been for a long time unsatisfactory with respect to gas proportional chambers and solid-state silicon or germanium PIN diodes. For many decades the development of thermal detectors has concentrated mainly on bolometers for radiant power measurements or for molecular beam experiments. Single-particle thermal detection experiments have been only sporadic tests of prototypes. The boost in the evolution of these detectors started in the ’80s, thanks to the wider availability of new techniques in cryogenics, cold electronics and superconductivity. First of all, the development of cryogenic techniques in the ’60s and ’70s made easy tools available for working at 0.1 K with the He3 -He4 dilution fridges. The achievements of precise surface implantation doping of silicon for the microelectronics and the high-accuracy neutron bulk doping of germanium have made new sensitive semiconducting thermistors available operating at 0.1 K, very close to the metal-to-insulating transition. Furthermore, the achievements in superconductivity with the invention of the SQUID made today a new wide set of very low-noise readout techniques accessible. After two decades of developments, the present generations of thermal detectors show unsurpassed spectral resolution in the energy-dispersive spectroscopy applications. For reference purpose, the microcalorimeters have achieved 2 eV FWHM at 5.9 keV [9], about 70 times better than Si(Li) detectors, and 50 eV FWHM at 100 keV [10], about 10 times better than Ge(Li) detectors. The cryogenic microcalorimeters are commonly composed of a radiation absorber, a thermal sensor that detects the absorber temperature and a weak thermal link between the detector and the heat sink. The operating principle of the microcalorimeter is in principle very simple: when the radiation releases the energy ER in the absorber, this is fully and instantaneously converted into thermal phonons. The temperature of the detector first rises and then drops to its original value, due to the weak thermal link with the heat sink. The height of the temperature transient is ΔT = ER /C, where C is
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Fig. 2. – TES microcalorimeter of heat capacity C and thermal conductance G biased at voltage V . The current signal flowing in the inductance L is measured by means of a SQUID in current amplifier configuration.
the microcalorimeter heat capacity. The sensor is generally a resistor whose resistance has a strong dependence on the temperature. Such a thermistor can be biased either at constant current or constant voltage, then the temperature transient is readout as voltage or current pulse, respectively. The modern detectors working with a sensor at the normal-to-superconducting transition, as in the previously cited work of Andrews et al., are usually called Transition Edge Sensor (TES) microcalorimenters (see fig. 2). A simple model of a microcalorimeter is explained in the following. For highsensitivity operations, the detector is first of all brought to a steady state at temperature T0 higher than the one of the thermal sink TS , by means of a Joule power PJ0 , so that the power (1)
W (T0 , TS ) = PJ0
flows through the weak link to the sink. If the radiation gives rise to a small impulsive thermal release in the absorber, the corresponding time evolution of the power PI (t) drives the thermal balance equation of the microcalorimeter (2)
C
T (t) + W (T (t), TS ) = PJ (t) + PI (t). dt
Within the limit of small signals, the difference of the two powers, W (T, TS ) and W (T0 , TS ) of eq. (1) and eq. (2), are approximated by the thermal conductancetemperature drop product: G(T0 ) × ΔT (t), in which ΔT = T (t) − T0 . Therefore it can be written as (3)
W (T (t), TS ) − W (T0 , TS ) ≈
dW (T (t), T0 ) dT
× ΔT (t) = G(T0 ) × ΔT (t). T =T0
In the case in which a constant voltage V is applied, and always in the small signals approximation, the difference of the two Joule powers of eq. (1) and eq. (2) can be
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written as (4) PJ (t) − PJ0
V 2 T0 d V2 × δT (t) ≈ 0 ≈ dT R R0 R0
dR dT
× T =T0
ΔT (t) ΔT (t) = PJ0 α(T0 ) , T0 T0
where α = T /R(dR/dT ) is the TES thermometric sensitivity, and R its resistance. Finally, subtracting term by term eq. (1) from eq. (2), and taking the first-order approximation, the pulse regime of the TES microcalorimeter is described by the equation (5)
dΔT 1 + dt C
PJ0 α G+ δT = PR (t), T0
or (6)
dΔT + dt
1+L τ
ΔT = PR (t),
where τ is the decay time constant of the heat pulse and L = (αPJ0 /GT0 ) is the electrothermal feedback (ETF) parameter. In the TES microcalorimeters L can achieve values as high as several hundreds. In this regime, which is called strong electrothermal feedback, the effective temperature transient becomes so small that the detector works almost at constant temperature. This improves linearity, dynamic range and response speed. Indeed, the effective time constant goes inversely proportional to L, for L 1, (7)
τeff =
τ 1+L
and becomes much smaller than the physical time constant τ = C/G. Because the strong electrothermal feedback forces the detector to work at constant temperature, the bias current changes in order to balance the power PR (t) produced by the primary energy release of the radiation. This current is typically measured with a low-noise SQUID galvanometer in series with the TES thermistor (see fig. 2). Actually, most of the approximations that have been done in the previous crude model of the microcalorimeter cannot be applied in practice, because of the complexity of thermal systems at this low temperatures and the strong dependence from the temperature of several thermodynamic quantities. As an example, a simple normal conducting metal film at the sub-kelvin temperatures behaves as two systems, respectively, of heat capacity Ce ∝ T of the conduction electron gas, and Cph ∝ T 3 of the phonons of the crystal lattice, which are connected by a thermal conductance Ge-ph ∝ T 5 . If a heat excess is produced in the electron gas, this flows to the heat sink through the series of Ge-ph , the phononphonon conductance GKapitza ∝ T 3 and Gph ∝ T 3 of a dielectric link. Therefore, a set of non-linear differential equations should be used for modelling the near-equilibrium thermal evolution of the microcalorimeter. Far out-of-equilibrium evolution needs dedicated models, including also Monte Carlo simulations, like in ballistic or quasidiffusive phonon
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Fig. 3. – Left: an example of a first modern microcalorimeter prototype (Genoa, 1995) made with a neutron transmutation doped germanium sensor, (Ge-NTD) 100 × 100 × 250 μm3 , with ultrasonically bonded Al wires at the opposite contact faces. This prototype achieved an energy resolution of about 30 eV FWHM at 5.9 keV. Right: a present (Genoa, 2004) Ir-Au Transition Edge Sensor, TES, 100 × 200 × 0.05 μm3 microcalorimeter onto a suspended membrane of 1 μm thick silicon nitride (Si3 N4 ). This was built with all thin metal film planar techniques. A gold radiation absorber (vertical bar) was grown on top. This prototype achieved an energy resolution of about 4 eV FWHM at 5.9 keV.
transport or in diffusion of high-energy quasiparticles. For a more detailed discussion, see ref. [11]. The cryogenic microcalorimeters are so attractive because the spectral resolutions are much higher than in any other energy-dispersive detector. A simple estimation of the intrinsic limit of the energy resolution is obtained by considering the microcalorimeter at the thermal equilibrium with the heat sink through a weak thermal link. Actually this is not the case, but it can be seen that this is a good approximation. The intrinsic limits in energy resolution are caused by the unavoidable temperature fluctuations in the microcalorimeter. These are determined by the Brownian motion of the phonons between the two bodies. If U is the internal energy, the corresponding average number of phonons is Nph = (U/kT ) = (CT /kT ). Consequently, the r.m.s. energy fluctuations √ are ΔUr.m.s. = kT Nph and then ΔUr.m.s. = kT 2 C. It can be easily calculated that at T = 0.1 K, and C = 10−13 J/K, the r.m.s. energy noise ΔUr.m.s. is as small as 1 eV. A more detailed model that is based on the hypothesis of the steady state and includes the electrothermal feedback effect, gives the similar expression
(8)
√ ΔUr.m.s. = ζ kT 2 C,
with a correction parameter ζ that is a function of L, G, α and is smaller than 1 in conditions of strong electrothermal feedback. Examples of modern microcalorimeters are shown in fig. 3.
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329
3. – The calorimetric measurement of neutrino mass In 1980 Simpson reported the results of a measurement of the 3 H β-energy spectrum for antineutrino mass determination that was made with a Si(Li) X-ray detector into which a dose of tritium was implanted with a Van der Graff accelerator [12]. The motivation of this new approach lies in the necessity to examine with a different method the results on neutrino mass measurements with β magnetic spectrometer. After the claiming of finite neutrino mass that was made by the group of ITEP [13], it was pointed out that an incorrect estimation of the detector response function (see Bergkvist [14]) could simulate finite neutrino mass. The analysis of β spectra with resolution of tens of eV or better with an external source requires special and accurate calculations of the energy losses and its distribution. Energy losses of the β are due to inelastic scattering inside the source, that depends from the geometry and composition, but also to the so-called final-state interactions. The final states available to the decay have energies appropriate to the helium ion, in the 25–50 eV range, so that any precise measurement might take into account the probabilities of the various final states. In the conjecture of Simpson these final states should be automatically accounted for in a full absorption experiment with the source inside the detector. Here we analyze more deeply this experiment. First of all, a dose of about 13 3 10 H atoms/cm3 has been implanted in Si(Li) diode with energy between 8.0 and 9.1 MeV, corresponding to an implantation depth of 0.2 mm, that is large if compared to the range of 18.6 keV β particles and the absorption length of the Si K X-ray line and external bremsstrahlung. Then the probability of energy escaping the active volume of the detector was negligible. The 3 H implanted ions move by channelling inside the crystal and stop as neutral atoms in a well-defined site inside the tetragonal structure of the silicon lattice at about 0.2 ˚ A from the nearest Si atom. This can be assumed as initial state. The final state of the β transition is 3 He+ ion with one electron in 1s or 2s orbital. The transition to the ground state might follow two ways. A second electron fills the 1s orbital, relaxing the atoms in 1S0 ground state with a typical time scale of 10 ns. The 2s state is metastable in free atoms, but because of the mixing of 2s and 2p orbitals due to the screening effect of Si and the Stark effect, which is caused by the crystal electric field, it undergoes fast (≤ ns) decay to 1S with the emission of 58.4 nm photon. Finally it can be concluded that the excited final states decay within the charge collection time of the Si(Li)diode, which is about hundreds of ns. Further, the main de-excitation processes release several tens of eV. As a reference example, the decay from the electron configuration (1s)(2p) to the (1s)(1s) gives about 20 eV. Being the energy for electronhole pair creation w = 1.1 eV, these small-energy releases contribute to the total charge signal of the single β decay. For these reasons the Simpson experiment can be considered a calorimetric energy-dispersive measurement of the β decay, even if a charge mediated detector was used. The first proposals for a true calorimetric β spectroscopy with phonon-mediated detectors for neutrino mass measurement were made independently by McCammon [15]
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and Vitale [16]. In the first proposal, McCammon conceived a tritium-implanted monolithic Si microcalorimeter 0.5 × 0.5 mm2 of a total heat capacity of 10−15 J/K at 0.1 K. The expected energy resolution was calculated to be about 1 eV FWHM. In the proposal of Vitale it has been pointed out that the main disadvantages of the calorimetric technique, i.e. the impossibility of cutting out the non-interesting part of the spectrum, can be balanced by choosing the lowest Q value β isotope that is known in nature: 187 Re, whose Q is about 2.5 keV. This allows gaining a factor 400 in statistical sensitivity with respect to the tritium. The statistical sensitivity is calculated as the ratio of the end-point fraction sensitive to the neutrino mass over the whole spectrum itself: this factor scales as m3ν /Q3 . In Vitale’s proposal, the detector was conceived as a metallic Re cryogenic microcalorimeter with an expected energy resolution of about 2 eV FWHM. As pointed out by Bergkvist [14] the presence of non-defined final states gives rise to a smearing of the spectral shape at the end-point, at least, as wide as the distribution of excited levels. If we call, respectively, i , E0i and E0 the energy of the generic final state i, the corresponding end-point and the ground-to-ground end-point value of the transition, then the β spectrum can be written as follows: (9)
N (E) ∝
(E0i − Ei ) (E0i − Ei )2 − m2ν ,
i
where (10)
E0i = E0 − i .
The term pEFS, that is approximately constant approaching the end-point, has been disregarded. As reference, the calculated excited levels i available for the transition T2 to HeT+ are more than 110 and cover an energy range of about 160 eV [17]. The effect of the sum in eq. (1) is the well-known “banana-like”-shaped Kurie plot at the end-point (see fig. 4, left). In an ideal calorimeter the releases i should be added to the β particle energy in a single event with energy (11)
Ec = Ei + i
and the calorimetric β spectrum becomes (12)
N (Ec ) ∝ (E0 − Ec ) (E0 − Ec )2 − m2ν
In which (13)
E0 = E01 + 1 = E02 + 2 = . . . = E0i + i = . . . .
Thus, each β transition spectrum adds coherently to the others generating a ground-toground state calorimetric energy spectrum, which is shown in fig. 4, right.
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Fig. 4. – Left: simplified Kurie plot for a β transition with 3 excited states: the full spectrum has a typical banana-like feature at the end-point. Right: simplified Kurie plot of the same transition as measured with a calorimetric β spectrometer, in which all the 3 excited state spectra add coherently whith the same end-point.
4. – The case of a Re metal detector for studying
187
Re decay
Here we analyze the case of the cryogenic microcalorimeters with Re metal absorber that is one of the simplest chemical forms in which it can be prepared. The natural isotopic abundance of 187 Re is 63%. Since the beginning of the ’90s the group of the Genoa has undertaken the study of Re metal properties [18-20], showing that single crystals are at the same time suitable radiation absorbers and sources for cryogenic detectors operating around 0.1 K, at which they behave superconducting. The specific heat in the superconducting state approaches the one of dielectric materials, which allows building detectors with absorber masses of the order of mg, otherwise impossible to be achieved in normal conducting state. Another chemical form, the dielectric compound AgReO4 , has been studied by the group of Milan [21, 22], that found them similarly suitable for calorimetric applications. For the sake of simplicity, we focus the attention on the phenomenology of the β decay 187 of Re in Re metal absorber. However, most of the following considerations apply to other kinds of chemical forms. For many years the decay spectrum of 187 Re, a first forbidden unique transition, has been unknown because of the very low energy of the β particles (2.5 keV) in conjunction with the high atomic number Z = 75 and density ρ = 21 g/cm3 . Being the electron range of few hundreds of ˚ A, the escape probability from a solid sample is therefore negligible and any attempt to detect the products of the decay has been vain. Only in 1965 [23] and 1967 [24] two groups, using a gas proportional chamber filled with a Re volatile compound that is stable at 250◦ C, revealed the spectral shape, even if with a poor resolution of about 1 keV FWHM at 3 keV. They found E0 = 2.62 ± 0.09 and 2.65±0.04 keV, respectively. This measurement has been repeated in 1993 with the same method giving E0 = 2.70 ± 0.09 keV [25]. In Re metal microcalorimeter (see an example in fig. 5), the 187 Re decays to the ground level of neutral 187 Os76 atom in the Re crystal. The 187 Re75 end-point energy in
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Fig. 5. – Left: picture of a Re single-crystal microcalorimeter with germanium transmutation doped sensor on top built in Genoa. Centre and right: AgRe3 O4 crystal microcalorimeter built in Milan: front side with wire connections and back side with the crystal.
the crystal is related to one of the isolated isotopes (14)
E0,crystal = E0,isolated − (eΦ + EFermi ) − ΔBlattice ,
where (eΦ+EFermi ) is the electron binding energy in the crystal as a sum of the electronic work function, 5.1 eV, and the Fermi energy, 11.2 eV. ΔBlattice is the change in the potential energy of the isotope in the crystal due to the change of the nuclear charge by one unit. This lattice contribution is negligible: it corresponds to about 2.6% of the total binding energy of rhenium atom in the crystal, which has been evaluated to be 16.9 eV/atom. It is interesting to note that the Os electron cloud is more strongly bound than in Re by ΔBe = −13.77 keV. Therefore the difference of nuclear binding energies M (Os) + me c2 − M (Re) is about +11 keV, preventing the β decay to the continuum in the bare nucleus. The bare 187 Re75+ can only undergo bound state decay to 187 Os75+ , which has been observed in a storage ring having a 32 y half-life and 63 keV Q value [26]. Because the atomic binding energy is the leading term, the decay rate is strongly influenced by the EM fine-structure constant α, making the 187 Re one of the most sensitive probes for the EM coupling over a wide time scale comparable to the age of the universe [27]. It is worth noting that the microcalorimetric detectors are able to measure the relaxation of possible final excited states if their lifetimes are shorter than the heat pulse formation times, which have been measured to be about 10−5 s in fast detectors. However, an assessment of the final excited state energies and probabilities and other causes of energy losses is made in the following. Final excited states. In the course of the 187 Re decay, the β particles pass through the atom, that may not have the time to rearrange the electrons. As can be seen in table I, the atomic binding energy difference from Re to Os+ is very close to the one from Re to the neutral Os. Therefore, being the energy of the Os+ after the decay almost equal to the one of the ground state of Os, high excited final states are very unlikely. A provisional analysis of the possible atomic excited final states shows that, due to the very similar atomic wave functions and levels of Re75 and Os76 , the probability of transitions towards
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Table I. – Re and Os electronic configuration and the total atomic binding energies. 75 Re 76 Os+ 76 Os
Xe 4f 14 5d5 6s2 Xe 4f 14 5d6 6s1 Xe 4f 14 5d6 6s2
Etot = −429402.3 eV Etot = −443164.5 eV Etot = −443172.8 eV
an excited level is very small, being the Os excited eigenstates orthogonal to the Re ground state. An evaluation of this probability gives values lower than 7 × 10−5 . Then, the energy loss via long-living atomic excited state should be negligible. Lattice defects. The possibility of energy losses in the dislocations that are caused by the recoils of the daughters can be excluded. The energy of the recoils is lower than 8 meV, therefore, it is not sufficient to cause crystal defects, but only to create phonons belonging to the elastic branches, which contributes to the heat pulse formation. Recoil-free decay. Another process, that in principle could smear the position of the end-point, is the recoil-free β decay. This effect has not been observed until now, but it can be foreseen on the basis of the extension of the theory of the well-known Mossbauer effect to the β decay. In the case of recoil-free decay the energy of the neutrino-electron pairs is increased by an amount equal to the nucleus recoil. Since this exchanged energy is very small, the effect on the shape of the end-point region is negligible. Shake off. The shake off probability is at 1% level and involves the N and O shells that emit photons or Auger electron with an average energy of 50 eV. These are fully absorbed in much less than a thickness of 1 μm of Re metal. IB and EB. Inner and external bremsstrahlung generate photons with a maximum energy equal to Q, that have a larger penetration depth than the electrons, but however within few μm. Quasiparticle metastable states. Collective excitations in the crystal can also contribute to the generation of long-lived metastable states, which can trap a variable amount of energy for a long time. Because the rhenium crystal is operated well below the normal-to-superconducting transition temperature Tc = 1.69 K, long-living quasiparticle states are predicted by the models that extend the BCS theory to the systems out of the thermal equilibrium [28], as expected in the region of the β tracks. A dedicate set of measurements has shown that the thermalization efficiency, i.e., the fraction of the primary energy converted into heat in a short time, is 100% over a wide range of temperatures and decreases to lower, but non-vanishing values below 80 mK. In order to explain this behaviour, it has been supposed that at so low T /Tc the quasiparticles created in the narrow region of the track re-condense promptly in Cooper pairs before diffusing over the whole crystal volume, providing this unexpected full thermalization of the primary energy till below 0.1 K. Even if these measurements are biased by the specific choice of the sensor type (Ge-NTD) and the absorber geometry (Re crystal and foils), its general trend has been confirmed in other materials (Pb, Sn, Nb, Ti, Al) and indirectly by other groups. At the lowest temperatures, much less than 80 mK, long tails appear on the thermal pulses indicating a possible incomplete thermalization [29].
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In this case the possibility of intrinsic fluctuation caused by incomplete thermalization processes cannot be excluded a priori. 5. – First detector prototypes and pilot experiments Here we describe in detail examples of Re detectors, the measurement methods and the first results. The decay of 187 Re was observed for the first time with a cryogenic microcalorimeter with 660 eV FWHM energy resolution in 1992 by the group of Genoa [20]. After a few years spent in improving the detector [30], a spectrum with 30 eV FWHM resolution and 50 eV threshold has been achieved [31]. The first high-statistics calorimetric measurement has been realized later, in 1998. About 6 × 106 187 Re decays with energy threshold of 420 eV and energy resolution of about 96 eV FWHM have been acquired. The high statistics and good performance of the experiment allowed the measurement of the end-point energy and half-life, with an accuracy never obtained before [32]. In this first experiment a rhenium single crystal of 1.572 mg was used as absorber, with a β activity of about 1.1 Bq. The sensor was a neutron transmutation doped, NTD, germanium thermistor. Sensor and absorber have been connected with a small drop of epoxy. The microcalorimeter was suspended by two ultrasonic-bounded Al wires that provided both the electrical connections and the weak thermal link to the heat sink. During the measurements the refrigerator operated at the base temperature of 60 mK for an uninterrupted period of three months. The detector was thermally and radioactively shielded in a small box of low-activity copper and lead. Only a small hole with a Be window was used for the detector energy calibration with soft X-rays of an external calibration source. This was obtained with the fluorescence of Cl, Ca, Ti, and Va salts that were excited with a 50 μCi 55 Fe X-ray source. The monitor of the detector gain was provided by a weak 55 Fe X-ray at 5898 eV and 6490 eV with a count rate of about 10−2 Hz. Al and Be foils were used to filter out the low-energy emissions of the source itself in the range of interest below 3 keV. The stability of the detector calibration has been proven to be very good over periods of several days, so that such a low-activity source was sufficient for monitoring the energy calibration for periods of about one week during the entire data taking. These measurements led to the discovery of the influence of a crystalline structure on the β decay [33], as shown in fig. 6. This phenomenon, named beta environmental fine structure, BEFS, was hypothesized in 1991 [34]. It appears as an oscillatory modulation of the β spectrum of the order of 1%, whose parameters depend on the crystalline structure. This fine structure is caused by the interference of the outgoing β wave function with the reflected ones from the neighbouring atomic cores. The emitted β, whose wavelength changes from about 2 ˚ A at 10 eV, to about 0.02 ˚ A at 1 keV, moves away from the decaying atoms, expanding over several tens of ˚ A in the crystal. All the backscattered waves from the encountered atoms along their pathway travel back and interfere with each other at the site of the decaying nucleus. Therefore, for a finite coherence time, around the nucleus a complex electronic standing wave pattern is created with maxima and minima that depend on the wavelength. Therefore the amplitude of the electronic
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Fig. 6. – First observation of the fine structure of the beta spectrum (BEFS) in Re metal (Genoa, 1999): data and fit function.
final state of the β decay inside the crystal is modulated by the β energy itself. In general we can state that the general β decay in a solid should be corrected by the fine-structure term (15)
ξ(Ee ) =
Ni fi (π) i
kRi
2 2
sin(2kRi + φi + 2δ0 )e−γRi e−2σi k ,
where the sum is over all neighbouring i-shells containing Ni atoms, fi (π) is the backward scattering amplitude for an electron of energy Ee and wave number k, γ is the inverse of the energy-dependent electron mean free path, Ri is the distance between the central decaying atom and the surrounding i-shell of atoms and σ accounts for the corrections of the thermal lattice motion occurring in the Debye-Waller term. This formula holds for electrons emitted in s-wave. A phase factor is needed to account for p-wave beta emission. The statistics of this first data are not sufficient for determining the phase factor. A first determination of the phase factor and consequently the s- and p-wave mixing has been done by the Milan group in 2005 [35]. In table II the parameters of the selected detectors built in Genoa are presented: improvements of about one order of magnitude of energy resolution have been achieved from Table II. – Parameters of typical Re metal calorimeters built in Genoa from 1998 to 2005. The resolutions are measured at 5.9 keV. Sensor Re (μg) C(sensor) pJ/K C(Re) pJ/K Calculated FWHM resolution Measured FWHM at 6 keV
Ge NTD 1572.3 3.6 3.7 48.3 96.0
Ge NTD 60.1 3.6 0.1 28.4 31.3
Ge NTD 210.3 1.8 0.5 33.2 37.2
Ir/Au TES 215.50 0.02 0.54 5.80 11.0
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Table III. – Parameters of typical AgReO4 calorimeters built in Milan for the experiment in 2002. Detector 1 2 3 4 5 6 7 8
AgReO4 mass (μg)
Toperating (K)
EFWHM (eV) at 1.5 keV
EFWHM (eV) at 2.6 keV
272 259 280 249 284 282 268 278
78.6 70.2 78.5 80.6 67.4 57.3 74.0 76.6
23.7 23.2 28.7 25.4 25.7 29.3 21.3 29.2
26.3 26.6 30.5 28.9 29.2 33.4 24.9 34.5
1998 to 2005. Further efforts are under way to achieve a spectral resolution of 2–5 eV. In 2002–2003 the group of Milan performed a high-statistics measurement with an array of 10 microcalorimeters with AgReO4 dielectric absorber. The AgReO4 absorbers were coupled to doped silicon thermistors. Even if the data from two detectors with poorer resolution were not included in the statistics, the effective total mass of the array was 2.174 mg, for a 187 Re total activity of 1.17 Hz. See table III for the details on the detector parameters. The total lifetime was 210 days, 42 of which have been devoted to the periodic calibrations while 168 days correspond to pure data acquisition. The total efficiency of this run was therefore of 54%. The calibrations were made with X-rays at 1.5, 2.6, 3.7, and 4.5 keV of excited sample of Al, Cl, Ca, and Ti, respectively. The calibrations were repeated periodically every 2 hours. The performance of the detectors was quite stable during the run. The FWHM resolution at 1.5 keV in the single-detector final spectra ranges from 21.3 to 29.3 eV; with an average of 25.5 eV. The 10% to 90% rise time of the 8 detectors was in the range 340–680 μs; with an average value of 492 μs. The FWHM resolution of the entire array extrapolated at the energy of the end-point at 2.46 keV was 28.5 eV [36-38]. In table IV the main results of the two first experiments of calorimetric beta spectroscopy are summarized.
Table IV. – Experiment, energy resolution, end-point energy, half-life, bound to neutrino mass of the pilot experiments in Genoa and Milan. Experiment Resolution-FWHM (eV) E0 (eV) τ1/2 (Gy) mν (eV/c2 )
MANU-1998
MiBeta-2003/04
96 2470 ± 1 ± 4 41.2 ± 0.2 ± 1.1 ≤ 25 95% C.L.
28.5 2465 ± 0.5 ± 1.6 43.2 ± 0.2 ± 0.1 ≤ 15 90% C.L.
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Fig. 7. – Left: an example of 64 microcalorimeters having 2 eV FWHM energy resolution that have been integrated in 8 × 8 array (Courtesy of GSFC-NASA). Rigth: simulated sensitivity on mν of 187 Re experiment (MARE) with 4 ×104 microcalorimeters vs. measurement live time with different detector parameters: energy resolution, time resolution and count rate per detector.
6. – Towards a sub-eV mν calorimetric experiment The sensitivity of a calorimetric experiment to the neutrino mass is related to the total statistics N and energy resolution ΔE (16)
(mν )90%CL 0.9
E03 ΔE N
1/4 ,
where N = Aβ × tM , the β source activity-live time product. The activity and spectral resolution of 187 Re detectors are strictly connected to each other: high counting rates need high absorber masses and conversely high resolutions require instead small heat capacities and consequently small absorbers. In natural Re the ratio activity/mass is roughly 1 Bq/mg. The counting rate might be limited also by the maximum acceptable pile-up rate for achieving the wanted mass sensitivity, once the detector time resolution and the pile-up recognition capability of the analysis have been fixed. The trade off among these parameters can be met by using highly pixelated detectors. Each pixel can arrange a Re single crystal as absorber with mass up to few mg. The conceptual design of an experiment with sub-eV sensitivity has been discussed within the proposal MARE [39]. The present status of the integration technology for microcalorimetric detectors is such that 32× 32 pixels array of area of few cm2 and with spectral resolution of about 2 eV are being produced [40]. An experiment with 4 × 104 detectors that might be built with subarrays of 103 pixels can achieve the same sensitivities as the KATRIN experiment [41]. In fig. 7 are shown a prototype of an array of 8 × 8 detectors with 2 eV FWHM energy resolution and the expected sensitivity-live time plot of the 187 Re microcalorimeters experiment (MARE) with different detector parameters. Researches and developments are under way for fixing the main technological issues of the detector integration and readout. Further improvements of the detector performance
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Fig. 8. – Left: Calculated E.C. spectra for Q = 2150 and 2750 eV. Right: expected normalized deficit vs. mν of 187 Re and 163 Ho with Q = 2150 and 2800 eV.
are foreseen in the next years, thus allowing better sensitivity expectations. The E.C. decaying isotope 163 Ho is another attractive probe for finite neutrino mass, equally to 187 Re. Among several different methods that have been proposed in the past [42-45], the most suitable is the kink search at the end-point of the E.C. spectrum, which is
(17)
dW ΓH /2π , = M 2 (Q − Ec ) (Q − Ec )2 − m2ν φ2H dEc (Ec − EH )2 + Γ2H /4 H
where Ec is the energy released in the electron capture decay, EH is the orbital binding energy, M is the nuclear matrix coefficient, φ2H is the H electron wave function amplitude on the nucleus, Γ is the linewidth. A first attempt of calorimetric measurement of 163 Ho spectrum has been made in 1997 at Genoa [46]. Nevertheless, the results were much better than the previous experiments, the final spectral resolution at the M1 capture line (2047 eV) was about 80 eV FWHM, because of the difficulties in synthesizing a homogenous absorber starting from a water solution of holmium chloride. A novel approach in 163 Ho filled absorber, which can be based on the production by proton activation and subsequent ion implantation in the absorber, might allow building high-spectral-resolution detectors. The main advantages of 163 Ho E.C. calorimetry are that the activity is roughly unconnected from the absorber mass, thus allowing the minimization of heat capacity, and that the spectrum is self-calibrated. A comparison of the merit factors, which are defined as the expected normalized deficit for a finite neutrino mass, is presented in fig. 8. These have been calculated for the minimum and maximum Q values found in the literature: 2150 eV and 2750, respectively. It can be concluded that calorimetric spectroscopy of 163 Ho should have theoretical sensitivities from a factor 2 better to a factor 1.5 poorer than in 187 Re, depending on the effective Q value.
Calorimetric beta spectroscopy
339
7. – Conclusion High-precision calorimetric spectroscopy of 187 Re and 163 Ho can achieve sensitivity in mν deep in the sub-eV region. The present status of cryogenic microcalorimeter technology allows to design an experiment with 0.2–0.3 eV sensitivity. Further developments that are under way might open new perspectives for further improvements in neutrino mass direct searches. REFERENCES [1] [2] [3] [4] [5] [6]
[7]
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Chadwick J., Verh. Deutschen Phys. Ges., 16 (1914) 383. Rutherford E., Philos. Mag., 28 (1914) 305. Mitner L., Z. Phys., 9 (1922) 145. Bohr N., Convegno di Fisica Nucleare (Reale Accademia d’Italia, Roma) 1932, p. 119. Ellis C. D. and Wooster W. A., Proc. R. Soc. London, Ser. A, 117 (1927) 109. Pauli W., letter to L. Mitner and colleagues, Dec. 4th, 1930, in Wis. Brief. mit Bohr, Einstein, Heisenberg (Springer) 1985, p. 39. It is interesting to note that in a letter from Heisenberg to Pauli on Dec. 1st, 1930, there is an explicit reference to a neutral particle emitted in beta decay. These communications were done before the discovery of the neutron by Chadwick in 1932 (Nature, 129 (1932) 312) and the theory of nucleus made up of neutrons and protons by Ivanenko (Nature, 129 (1932) 798), Heisenberg (Z. Phys., 77 (1932) 1), Majorana (Z. Phys., 82 (1933) 137). Fermi E., Convegno di Fisica Nucleare (Reale Accademia d’Italia, Roma) 1932. Fermi referred to the Pauli particle as “neutrino” to distinguish it from the neutron newly discovered by Chadwick. Andrews D. H., Fowler R. D. and Williams M. C., Phys. Rev. A, 76 (1949) 157. Bandler S. R. et al., J. Low Temp. Phys., 151 (2008) 400. Doriese W. B. et al., J. Low Temp. Phys., 151 (2008) 754. Irwin K. D. and Hilton G. C., Cryogenic Particle Detection, Vol. 99 (Springer) 2005, p. 63. Simpson J. J. et al., Phys. Rev. D, 23 (1981) 649. Lubimov V. A. et al., Phys. Lett. B, 94 (1980) 266. Bergkvist K. E. et al., Phys. Lett. B, 154 (1985) 224. McCammon D. et al., 3rd Telemark Conference, Oct. 1984, AIP Conf. Proc. (1985). Vitale S. et al., Erice International School of Physics of Exotic Atoms, 1984, and Blasi A. et al., INFN Report, BE-85/2 (1985). Jonsell K. E. and Monkhorst H. J., Phys. Rev. Lett., 76 (1996) 4476. Gallinaro G., Gatti F. and Vitale S., Europhys. Lett., 14 (1991) 225. Gatti F., Vitale S. and Barabino A., Nucl. Instrum. Methods Phys. Res. A, 315 (1992) 260. Cosulich E. et al., Phys. Lett. B, 295 (1992) 143. Alessandrello A. et al., J. Phys. D., 32 (1999) 3099. Alessandrello A. et al., Phys. Lett. B, 457 (1999) 253. Brodzinski R. L. and Conway D. C., Phys. Rev. B, 138 (1965) 1368. Huster E. and Verbeek H., Z. Phys., 203 (1967) 435. Akshtorab K. et al., Phys. Rev. C., 47 (1993) 2954. Bosch F. et al., Phys. Rev. Lett., 77 (1996) 5190.
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[27] Olive K. A. et al., Phys. Rev. D, 69 (2004) 027701. Low-Q isotopes are the most sensitive to changes of the fine-structure constant α, being the relative variation of the decay rate (Δλ/λ) = 3(ΔQ/Q) large for small Q. If 187 Re were incorporated into a meteorite in the early Solar System, the present abundance of 187 Os in the meteorite is 187 Ospres = 187 Osinit + 187 Repres [exp(−λ187 t) − 1], where λ187 is the time-averaged decay rate for 187 Re and t the meteorite age. Independent determinations of meteorite age allow to measure λ187 over about 5 Gy and then (Δα/α) 5 × 10−5 (Δλ/λ). [28] Kaplan S. B. et al., Phys. Rev. B, 14 (1976) 4854. [29] Cosulich E., Gatti F. and Vitale S., J. Low Temp. Phys., 93 (1993) 263. [30] Cosulich E. et al., Nucl. Phys. A, 529 (1995) 59. [31] Gatti F., in 7th International Workshop on Neutrino Telescopes, edited by M. Baldo Ceolin (1997) 141. [32] Galeazzi M. et al., Phys. Rev. C, 63 (2000) 014302. [33] Gatti F. et al., Nature, 397 (1999) 137. [34] Koonin S. E., Nature, 354 (1991) 468. [35] Arnaboldi C. et al., Phys. Rev. Lett., 96 (2006) 042503. [36] Arnaboldi C. et al., Phys. Rev. Lett., 91 (2003) 161802. [37] Sisti M. et al., Nucl. Instrum. Methods Phys. Res., 520 (2004) 125. [38] Nucciotti A. et al., Nucl. Instrum. Methods Phys. Res., 520 (2004) 148. [39] Gatti F. et al., MARE Proposal, http://www.ge.infn.it/∼numass. [40] Kelley R. et al., GFSC-NASA, private communication. [41] Weinheimer C., this volume, p. 215. [42] De Rujula A. et al., Nucl. Phys. B., 114 (1981) 488. [43] Yasumi S. et al., Phys. Lett. B., 334 (1994) 229. [44] De Rujula A. and Lusignoli M., Nucl. Phys. B., 219 (1983) 277. [45] De Rujula A. and Lusignoli M., Phys. Lett., 9 (1982) 429. [46] Gatti F. et al., Phys. Lett. B, 398 (1997) 415.
DOI 10.3254/978-1-60750-038-4-341
Leptogenesis A. Riotto CERN, Theory Division - Geneva 23, CH-1211, Switzerland
Summary. — We summarize the state of the art of the theory of thermal leptogenesis according to which the baryon asymmetry in the Universe is produced at very high temperatures from the decay of heavy right-handed neutrinos. The same heavy states might be responsible through the see-saw mechanism of the lightness of the left-handed neutrinos. This opens up the possibility that measuring the low-energy neutrino parameters would lead to know something about the primordial Universe.
1. – Introduction The symmetry between particles and antiparticles, firmly established in collider physics, naturally leads to the question of why the observed universe is composed almost entirely of matter with little or no primordial antimatter. Outside of particle accelerators, antimatter can be seen in cosmic rays in the form of a few antiprotons, present at a level of around 10−4 in comparison with the number of protons. However, this proportion is consistent with secondary antiproton production through accelerator-like processes, p + p → 3p + p¯, as the cosmic rays stream towards us. Thus there is no evidence for primordial antimatter in our galaxy. Also, if matter and antimatter galaxies were to coexist in clusters of galaxies, then we would expect there to be a detectable background of γ-radiation from nucleon-antinucleon annihilations within c Societ` a Italiana di Fisica
341
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A. Riotto
the clusters. This background is not observed and so we conclude that there is negligible antimatter on the scale of clusters. More generally, if large domains of matter and antimatter exist, then annihilations would take place at the interfaces between them. If the typical size of such a domain was small enough, then the energy released by these annihilations would result in a diffuse γ-ray background and a distortion of the cosmic microwave radiation, neither of which is observed. A careful numerical analysis of this problem demonstrates that the universe must consist entirely of either matter or antimatter on all scales up to the Hubble size. It therefore seems that the universe is fundamentally matter-antimatter asymmetric. While the above considerations put an experimental upper bound on the amount of antimatter in the universe, strict quantitative estimates of the relative abundances of baryonic matter and antimatter may also be obtained from the standard cosmology. The baryon number density does not remain constant during the evolution of the universe, instead scaling like a−3 , where a is the cosmological scale factor [1]. It is therefore convenient to define the baryon asymmetry of the universe in terms of the quantity (1)
YB ≡
nB , s
where nB = nb − n¯b is the difference between the number of baryons and antibaryons per unit volume and s = (2π 2 /45)g is the entropy density at temperature T when the thermal plasma contained g relativistic degrees of freedom. The observed baryon asymmetry of the Universe is now accurately determined by Cosmic Microwave (CMB) Anisotropy measurements [2] (2)
YBCMB = (8.75 ± 0.23) × 10−11 .
The physics behind the connection behind the determination of YB and the CMB anisotropy would require another set of lectures. It suffices to say that the baryon number plays a crucial role in determining the relative amplitude of the even peaks to the odd ones. Measuring this ratio gives a determination of the baryon asymmetry of the Universe [3]. Primordial nucleosynthesis (for a review see [4]) is also one of the most powerful predictions of the standard cosmological model. The theory allows accurate predictions of the cosmological abundances of all the light elements, H, 3 He, 4 He, D, B and 7 Li, while requiring only the single input parameter YB which has been constant since nucleosynthesis. The range of YB consistent with nucleosynthesis agrees with the one provided by CMB anisotropies, even though they are relevant on various ranges of temperatures, about MeV for nucleosinthesis and a fraction of eV for the CMB anisotropies. To see that the standard cosmological model cannot explain the observed value of YB , suppose that initially we start with YB = 0 [5,6]. We can compute the final number density of nucleons b that are left over after annihilations have frozen out. At temperatures T 1 GeV the equilibrium abundance of nucleons and antinucleons is [1] (3)
m 3/2 mp n¯ nb p
b e− T . nγ nγ T
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Leptogenesis
When the universe cools off, the number of nucleons and antinucleons decreases as long as the annihilation rate Γann nb σA v is larger than the expansion rate of the universe 1/2 T 2 . The thermally averaged annihilation cross-section σA v is of the H 1.66 g M P order of m2π , so at T 20 MeV, Γann H, and annihilations freeze out, nucleons and antinucleons being so rare that they cannot annihilate any longer. Therefore, from (3) we obtain (4)
nb n¯
b 10−18 , nγ nγ
which is much smaller than the value required by nucleosynthesis. In conclusion, in the standard cosmological model there is no explanation for the value of the observed baryon asymmetry, if we start from YB = 0. An initial asymmetry may be imposed by hand as an initial condition, but this would violate any naturalness principle. Rather, the guiding principle behind modern cosmology is to attempt to explain the initial conditions required by the standard cosmology on the basis of quantum field theories of elementary particles in the early universe. In this context, the generation of the observed value of YB is referred to as baryogenesis. The goal of this lecture is to present the thermal leptogenesis scenario. As we shall see, it is strictly related to neutrino physics, the main subject of this School. As such, one would hope, through low-energy neutrino measurements, to open a window on the early Universe dynamics. For more details about the subject, the reader is forwarded to the excellent recent review of the subject [7]. Other useful reviews are [5, 6]. A note about our conventions. Throughout we use a metric with signature +2 and, unless explicitly stated otherwise, we employ units such that = c = k = 1. The review is laid out as follows. In sect. 3 we provide some usful cosmological tools, in sect. 4 we discuss the out-of-equilibroum decay scenario, while sects. 5 and 6 are devoted to leptogenesis. Section 7 contains our conclusions. 2. – The Sakharov criteria A small baryon asymmetry YB may have been produced in the early universe if three necessary conditions are satisfied [8]: i) baryon number (B) violation; ii) violation of C (charge conjugation symmetry) and CP (the composition of parity and C) and iii) departure from thermal equilibrium. The first condition should be clear since, starting from a baryon-symmetric universe with YB = 0, baryon number violation must take place in order to evolve into a universe in which YB does not vanish. The second Sakharov criterion is required because, if C and CP are exact symmetries, then one can prove that the total rate for any process which produces an excess of baryons is equal to the rate of the complementary process which produces an excess of antibaryons and so no net baryon number can be created. That is to say that the thermal average of the baryon number operator B, which is odd under both C and CP , is zero unless those discrete symmetries are violated. CP violation is present either if there are complex phases in the Lagrangian which cannot be reabsorbed by field redifinitions (explicit breaking) or
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if some Higgs scalar field acquires a VEV which is not real (spontaneous breaking). We will discuss this in detail shortly. Finally, to explain the third criterion, one can calculate the equilibrium average of B (5)
BT = Tr(e−βH B) = Tr[(CP T )(CP T )−1 e−βH B] = Tr[e−βH (CP T )−1 B(CP T )] = − Tr(e−βH B),
where we have used that the Hamiltonian H commutes with CP T . Thus BT = 0 in equilibrium and there is no generation of net baryon number. Of the three Sakharov conditions, baryon number violation and C and CP violation may be investigated only within a given particle physics model, while the third condition —the departure from thermal equilibrium— may be discussed in a more general way, as we shall see. Let us discuss the Sakharov criteria in more detail. . 2 1. Baryon number violation. . 2 1.1. B-violation in Grand Unified Theories. Grand Unified Theories (GUTs) describe the fundamental interactions by means of a unique gauge group G which contains the Standard Model (SM) gauge group SU (3)C ⊗ SU (2)L ⊗ U (1)Y . The fundamental idea of GUTs is that at energies higher than a certain energy threshold MGUT , the group symmetry is G and that, at lower energies, the symmetry is broken down to the SM gauge symmetry, possibly through a chain of symmetry breakings. The main motivation for this scenario is that, at least in supersymmetric models, the (running) gauge couplings of the SM unify at the scale MGUT 2 × 1016 GeV, hinting at the presence of a GUT involving a higher symmetry with a single gauge coupling. Baryon number violation seems very natural in GUTs. Indeed, a general property of these theories is that the same representation of G may contain both quarks and leptons, and therefore it is possible for scalar and gauge bosons to mediate gauge interactions among fermions having different baryon number. However, this alone is not sufficient to conclude that baryon number is automatically violated in GUTs, since in some circumstances it is possible to assign a baryonic charge to the gauge bosons in such a way that at each boson-fermion-fermion vertex the baryon number is conserved. In the particular case of the gauge group SU (5), it turns out that among all the scalar and gauge bosons which couple only to the fermions of the SM, five of them may give rise to interactions which violate the baryon number. The fermionic content of SU (5) is the same as that of the SM. Fermions are assigned to the reducible representation 5f ⊕ 10f as 5f = {dcL , L } and 10f = {QL , ucL , ecL }. In addition, there are 24 gauge bosons which belong to the adjoint representation 24V and which may couple to the fermions through the couplings (6)
√ (g/ 2)24V (5f )† (5f ) + (10f )† 10f .
Because of this structure, the gauge bosons XY with SM gauge numbers (3, 2, −5/6) may undergo baryon-number-violating decays. While in the gauge sector of SU (5) the structure is uniquely determined by the gauge group, in the Higgs sector the results
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Leptogenesis
depend upon the choice of the representation. The Higgs fields which couple to the fermions may be in the representation 5H or in the representations 10H , 15H , 45H and 50H . If we consider the minimal choice 5H , we obtain (7)
hU (10f )T (10f )5H + hD (5f )T (10f )5H ,
where hU,D are complex matrices in the flavor space. The representation 5H contains the Higgs doublet of the SM, (1, 2, 1/2) and the triplet H3 = (3, 1, −1/3) which is Bviolating. Note that each of these bosons have the same value of the charge combination (B − L), which means that this symmetry may not be violated in any vertex of SU (5). However, if we extend the fermionic content of the theory beyond that of minimal SU (5), we allow the presence of more heavy bosons which may violate B and even (B − L). In GUTs based on SO(10), for instance, there exists a fermion which is a singlet under the SM gauge group, carries lepton number L = −1, and is identified with the right-handed antineutrino. However, this choice for the lepton number assignment leads to no new gauge boson which violates (B −L). As we mentioned, the generation of a baryon asymmetry requires C violation. Since SO(10) is C-symmetric, the C symmetry must be broken before a baryon asymmetry may be created. If SO(10) breaks down to SU (2)L ⊗ SU (2)R ⊗ SU (4) via the nonzero expectation value of a Higgs field in the 54H representation, the C symmetry is not broken until U (1)B−L is broken at the scale MB−L . In this case, the right-handed neutrino acquires a Majorana mass MN = O(MB−L ) and its out-of-equilibrium decays may generate a nonvanishing (B − L) asymmetry [9]. . 2 2. B-violation in the electroweak theory. – It is well known that the most general Lagrangian invariant under the SM gauge group and containing only color singlet Higgs fields is automatically invariant under global Abelian symmetries which may be identified with the baryonic and leptonic symmetries. These, therefore, are accidental symmetries and as a result it is not possible to violate B and L at tree-level or at any order of perturbation theory. Nevertheless, in many cases the perturbative expansion does not describe all the dynamics of the theory and, indeed, in 1976 ’t Hooft realized that nonperturbative effects (instantons) may give rise to processes which violate the combination (B + L), but not the orthogonal combination (B − L). The probability of these processes occurring today is exponentially suppressed and probably irrelevant. However, in more extreme situations —like the primordial universe at very high temperatures— baryonand lepton-number-violating processes may be fast enough to play a significant role in baryogenesis. Let us have a closer look. At the quantum level, the baryon and the lepton symmetries are anomalous (8)
μ ∂ μ jB = ∂μ jLμ = nf
g2 g 2 a ˜ aμν ˜ μν , W − F W F μν 32π 2 μν 32π 2
where g and g are the gauge couplings of SU (2)L and U (1)Y , respectively, nf is the ˜ μν = (1/2)μναβ Wαβ is the dual of the SU (2)L field strength number of families and W tensor, with an analogous expression holding for F˜ . To understand how the anomaly
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A. Riotto
is closely related to the vacuum structure of the theory, we may compute the change in baryon number from time t = 0 to some arbitrary final time t = tf . For transitions between vacua, the average values of the field strengths are zero at the beginning and the end of the evolution. The change in baryon number may be written as (9)
ΔB = ΔNCS ≡ nf [NCS (tf ) − NCS (0)].
where the Cherns-Simon number is defined to be g2 2 3 ijk d x Tr Ai ∂j Ak + igAi Aj Ak . (10) NCS (t) ≡ 32π 2 3 Although the Chern-Simons number is not gauge invariant, the change ΔNCS is. Thus, changes in Chern-Simons number result in changes in baryon number which are integral multiples of the number of families nf . Gauge transformations U (x) which connect two degenerate vacua of the gauge theory may change the Chern-Simons number by an integer n, the winding number. If the system is able to perform a transition from the (n) (n±1) vacuum Gvac to the closest one Gvac , the Chern-Simons number is changed by unity and ΔB = ΔL = nf . Each transition creates 9 left-handed quarks (3 color states for each generation) and 3 left-handed leptons (one per generation). However, adjacent vacua of the electroweak theory are separated by a ridge of configurations with energies larger than that of the vacuum. The lowest energy point on this ridge is a saddle point solution to the equations of motion with a single negative eigenvalue, and is referred to as the sphaleron. The probability of baryon-number-nonconserving processes at zero temperature has been computed by ’t Hooft and is highly suppressed by a factor exp[−4π/αW ], where αW = g 2 /4π. This factor may be interpreted as the probability of making a transition from one classical vacuum to the closest one by tunneling through an energy barrier of height ∼ 10 TeV corresponding to the sphaleron. On the other hand, one might think that fast baryon-number-violating transitions may be obtained in physical situations which involve a large number of fields. Since the sphaleron may be produced by collective and coherent excitations containing ∼ 1/αW quanta with wavelength of the order of 1/MW , one expects that at temperatures T MW these modes essentially obey statistical mechanics and the transition probability may be computed via classical considerations. Analogously to the case of zero temperature and since the transition which violates the baryon number is sustained by the sphaleron configuration, the thermal rate of baryon number violation in the broken phase is proportional to exp[−S3 /T ], where S3 is the three-diemensional action computed along the sphaleron configuration, (11)
S3 = Esp (T ) ≡ (MW (T )/αW )E,
with the dimensionless parameter E lying in the range 3.1 < E < 5.4 depending on the Higgs mass. The prefactor of the thermal rate reads (12)
Γsp (T ) = μ
MW αW T
3 4 MW
Esp (T ) , exp − T
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Leptogenesis
where μ is a dimensionless constant. Although the Boltzmann suppression in (12) appears large, it is to be expected that, when the electroweak symmetry becomes restored at a temperature of around 100 GeV, there will no longer be an exponential suppression factor. Although calculation of the baryon-number-violating rate in the high-temperature unbroken phase is extremely difficult, a simple estimate is possible. The only important scale in the symmetric phase is the magnetic screening length given by ξ = (αW T )−1 . Thus, on dimensional grounds, we expect the rate per unit volume of sphaleron events to be (13)
Γsp (T ) = κ(αW T )4 ,
with κ another dimensionless constant. The rate of sphaleron processes can be related to the diffusion constant for Chern-Simons number by a fluctuation-dissipation theorem. In almost all numerical calculations of the sphaleron rate, this relationship is used and what is actually evaluated is the diffusion constant. The first attempts to numerically estimate κ in this way yielded κ ∼ 0.1–1, but the approach suffered from limited statistics and large volume systematic errors. Nevertheless, more recent numerical attempts found approximately the same result. However, these approaches employ a poor definition of the Chern-Simons number which compromises their reliability. This simple scaling argument leading to (13) is not correct, however. Damping effects 5 in the plasma suppress the rate by an extra power of αW to give Γsp ∼ αW T 4 . Indeed, 2 since the transition rate involves physics at soft energies g T that are small compared to the typical hard energies ∼ T of the thermal excitations in the plasma, the simplest way of analyzing the problem is to consider an effective theory for the soft modes, where the hard modes have been integrated out, and to keep the dominant contributions, the so-called hard thermal loops. It is the resulting typical frequency ωc of a gauge field configuration immersed in the plasma and with spatial extent (g 2 T )−1 that determines the change of baryon number per unit time and unit volume. This frequency ωc has been estimated to be ∼ g 4 T when taking into account the damping effects of the hard modes. Using the effective dynamics of soft nonAbelian gauge fields at finite temperature, one 5 can find that Γsp ∼ αW T 4 ln(1/αW ). Lattice simulations with hard thermal loops seem 5 4 to indicate the Γsp ∼ 30αW T 4 , which is not far from αW T 4. 3. – Some necessary notions of cosmology and equilibrium thermodynamics . 3 1. Expansion rate, number density, and entropy. – Before launching ourselves into the issue of baryon asymmetry production in the early Universe, let us just remind the reader a few notions about thermodynamics in an expanding Universe that will turn out to be useful in the following [1, 3]. According to general relativity, the space-time evolution is determined via the Einstein equation by the matter content of the Universe, which differs from epoch to epoch depending on what kind of energy dominates the energy density of the Universe at that time. There are three important epochs characterized by different relations between the energy density ρ and the pressure p: 1) vacuum energy
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A. Riotto
dominance with p = −ρ, 2) massless (relativistic) particle dominance with p = ρ/3 and 3) nonrelativistic particle dominance with p ρ. The Einstein equation reads (14)
1 Rμν − gμν R = 8πGN Tμν , 2
where Rμν is the Ricci tensor, R is the Ricci scalar, gμν is the metric, GN = MP−2 = (1.2 × 1019 )−2 GeV−2 is the Newton constant and Tμν is the stress-energy tensor. With the homogeneity and isotropy of the three space, the Einstein equation is much simplified with the Robertson-Walker metric (15)
ds2 = dt2 − a2 (t)x2 ,
where a(t) is the cosmic scale factor and the stress-energy tensor is reduced to Tμν = −pgμν + (p + ρ)uμ uν . Here uμ is the velocity vector which in the rest frame of the plasma reads uμ = (1, 0) and has the property uμ uμ = 1. The (0 − 0) component of eq. (14) becomes the so-called Friedmann equation (16)
H2 +
k 8πGN ρ, = 2 a 3
where k can be chosen to be +1, −1 or 0 for spaces of constant positive, negative or zero spatial curvature, respectively, and we have defined the Hubble parameter H≡
(17)
a˙ , a
which measures how fast the Universe is expanding during the different stages of its evolution. The μ = 0 component of the conservation of the stress-energy tensor (T;νμν = 0) gives the first law of thermodynamics in the familiar form (18)
d(ρa3 ) = −p d(a3 ),
that is, the change in energy in a comiving volume element, d(ρa3 ) is equal to minus the pressure times the change in volume, p d(a3 ). For a simple equation of state p = wρ, where w is independent of time, the energy density evolves like ρ ∝ a−3(1+w) . Examples of interest include radiation (ρ ∝ a−4 ), matter (ρ ∝ a−3 ), vacuum energy (ρ ∝ constant). The time behaviour of the scale factor a(t) then is : (19)
1) a ∝ e
Ht
,
2) a ∝ t1/2 , 3) a ∝ t2/3 .
H=
8πV , 3MP2
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Leptogenesis
The first stage is the inflationary epoch where the constant vacuum energy V gives the exponential growth of the scale factor, which is believed to solve the horizon and the flatness problems of the standard Big-Bang theory of cosmology (for a review, see [10]). Of great importance is the transient stage from inflation to radiation dominance. This epoch is called reheating after inflation and we shall come back to it later in these lectures. What is relevant for us is that the early Universe was to a good approximation in thermal equilibrium at temperature T [1] and we can define the equilibrium number density nEQ X of a generic interacting species X as nEQ X =
(20)
gX (2π)3
fEQ (p, μX ) d3 p,
where gX denotes the number of degrees of freedom of the species X and the phase space occupancy fEQ is given by the familiar Fermi-Dirac or Bose-Einstein distributions fEQ (p, μX ) = [exp[(EX − μX )/T ] ± 1] ,
(21)
where EX = (p2 + m2X )1/2 is the energy, μX is the chemical potential of the species and +1 pertains to the Fermi-Dirac species and −1 to the Bose-Einstein species. In the relativistic regime T mX , μX formula (20) reduces to nEQ X =
(22)
⎧ ⎨(ζ(3)/π 2 )gX T 3
(Bose),
⎩(3/4)(ζ(3)/π 2 )gX T 3
(Fermi),
where ζ(3) 1.2 is the Riemann function of 3. In the nonrelativitic limit, T mX , the number density is the same for Bose and Fermi species and reads nEQ X = gX
(23)
mX T 2π
3/2
e−
mX T
+
μX T
.
It is also important to define the number density of particles minus the number density of antiparticles (24)
nEQ X
−
nEQ X
gX = fEQ (p, μX )d3 p − (μX ↔ −μX ) (2π)3 ⎧ ⎨ gX T23 π 2 μX + μX 3 , 6π T T = ⎩2g (m T /2π)3/2 sinh(μ /T ) exp[−m /T ], X X X X
(T mX ), (T mX ).
Notice that, in the relativistic limit T mX , this difference scales linearly for T μX . This means that detailed balances among particle number asymmetries may be expressed in terms of linear equations in the chemical potentials.
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A. Riotto
We can similarly define the equilibrium energy density ρEQ X of a species X as gX E fEQ (p)d3 p, (25) ρEQ = X (2π)3 which reads in the relativistic limit ⎧ ⎨(π 2 /30)gX T 4 = (26) ρEQ X ⎩(7/8)(π 2 /30)gX T 4
(Bose), (Fermi).
Since the energy density of a nonrelativistic particle species is exponentially smaller than that of a relativisitic species, it is a very convenient approximation to include only relativistic species with energy density ρR in the total energy density ρ of the Universe at temperature T π2 g T 4 , 30
ρ ρR =
(27)
where g counts the total number of effectively massless degrees of freedom of the plasma (28)
g =
gi
i=Bose
4
Ti T
7 + 8
gi
i=Fermi
Ti T
4 .
Here Ti denotes the effective temperature of any species i (which might be decoupled from the thermal bath at temperature T ). In the rest of these lectures we will be always concerned with temperatures higher than about 100 GeV. At these temperatures, all the degrees of freedom of the standard model are in equilibrium and g is at least equal to 106.75. From this expression, we derive that, when the energy density of the Universe was dominated by a gas of relativistic particles, ρ ∝ a−4 ∝ T 4 and, therefore [1] T ∝ a−1 .
(29)
Assuming that during the early radiation-dominated epoch (t 4 × 1010 s), the scale factor scales like tα , where α is a constant, the Hubble parameter scales like t−1 ∝ T 2 ∝ a−2 . This means that the scale factor a(t) scales like t1/2 and we recover 2) of eq. (19). More precisely, the expansion rate of the Universe H is [1] (30)
H=
1/2
8π ρ 3MP2
1/2
1.66 g
T2 . MP
Using the fact that H = (1/2t) and eq. (30), we can easily relate time and temperature as (31)
t 0.301
MP 1/2
g T 2
T MeV
−2 s.
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Leptogenesis
Another quantity that will turn out to be useful in the following is the entropy density. Throughout most of the history of the Universe, local thermal equilibrium is attained and the entropy in a comoving volume element s remains constant. Since it is dominated by the contribution of relativisitic particles, to a very good approximation (32)
s=
2π 2 g∗S T 3 , 45
3
where (33)
g∗S =
i=Bose
gi
Ti T
7 + 8
i=Fermi
gi
Ti T
3 .
For most of the history of the Universe, however, all the particles have the same temperature and we can safely replace g∗S with g . Notice that the conservation of entropy implies that s ∝ a−3 and therefore g∗S T 3 a3 remains a constant as the Universe expands. This means that the number of some species X in a comoving volume NX ≡ a3 nX is proportional to the number density of that species divided by s, NX ∝ nX /s. . 3 2. Local thermal equilibrium and chemical equilibrium. – So far we have been using the fact that, throughout most of the history of the Universe, thermal equilibrium was attained. The characteristic time τX for particles of a species X with respect to the process X + A · · · → C + D + · · · is defined by the rate of change of the number of particles per unit volume nX due to this process: dnX 1 1 (34) =− . τX nX dt X+A···→C+D+··· In the early Universe, if τX is smaller than the characteristic time of the expansion H −1 , then there is enough time for the process to occur and the particles X’s are said to be thermally coupled to the cosmic fluid. By contrast, if τX H −1 , for every process in which the particles X’s are involved, then they are not in thermal equilibrium and they are said to be decoupled. In order to analyze the evolution of the particle populations which constitute the cosmic fluid, it is necessary to compare H −1 with τX at different temperatures. This is done through the Boltzmann equation, which, in an expanding Universe, reads 1 d 3 (a nX ) = πX C[fX ], (35) a3 dt where C is the collision operator. Equation (35) may be rewritten as dnX π f fm · · · (1±fX )(1±fj ) · · · W ( +m+· · · → X +j +· · · ) +3HnX = (36) dt j,,m,···
−fX fj · · · (1 ± f )(1 ± fm ) · · · W (X + j + · · · → + m + · · · ) ,
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A. Riotto
where π ≡ πX πj · · · π πm · · · , πi = (2π)−3 gi (d3 p/2Ei ) is the volume element in the phase space, W is the matrix element of the given process and (+) applies to bosons and (−) to fermions. The second term on the left-hand side of eq. (36) accounts for the nX diluition due to the cosmic expansion and the right-hand side accounts for the nX variations due to any elemenatry process X +j +· · · → +m+· · · in which the X particles are involved. As it stands, eq. (36) is rather formidable and complicated, but some approximations can be made to transform it in a simpler form. Let us consider, for example, a process like X + f → X + f , where the number of X particles does change in the scatterings and let us also suppose that the f particles are light (T mf ) and that the corresponding population is in thermal equilibrium. In the case in which the X distribution function is described by a Maxwell-Boltzmann distribution, i.e. the X particles are in equilibrium at temperatures smaller than mX , it is easy to see that the right-hand side of eq. (36) may be expressed in the form (37)
S, r.h.s. of eq. (36) = − nX − nEQ X
where (38)
S=
πf Ef ffEQ σ(X + f → X + f )
nEQ f σ(X + f → X + f )v.
The notation σv stands for the thermal-average cross-section times the relative velocity −1 = ΓX associated to the elastic process is therefore v. The inverse time scale τX (39)
−1
nEQ ΓX = τX f σ(X + f → X + f )v.
From these very simple considerations, we may conclude that the X degrees of freedom are in thermal equilibrium if (40)
ΓX nEQ f σ(X + f → X + f )v H
(thermal equilibrium is attained).
Departure from thermal equilibrium is expected whenever a rate crucial for mantaining thermal equilibrium becomes smaller than the expansion rate, ΓX H. Another useful concept is that of chemical equilibrium. In general, a species X is in chemical equilibrium if the inelastic scatterings which change the number of X particles in the plasma, X + j → + m, have a rate Γinel larger than the expansion rate of the Universe. In such a case, one is allowed to write down a relation between the different chemical potentials μ’s, (41)
μX + μj = μ + μm ,
of the particles involved in the process. With these simple notions in mind, we may start our voyage towards the country of baryogenesis. According to one of Sakharov’s criteria,
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Leptogenesis
if all the particles in the Universe remained in thermal equilibrium, then no preferred direction for time may be defined and the CP T invariance would prevent the appearance of any baryon excess, making the presence of CP -violating interactions irrelevant. Let us indeed suppose that a certain species X with mass mX is in thermal equilibrium at temperatures T mX . Its number density will be given by (42)
nX gX (mX T )3/2 e−
mX T
+
μX T
,
where μX is the associated chemical potential. As we have mentioned, a species X is in chemical equilibrium if the inelastic scatterings which change the number of X particles in the plasma, X + A → B + C, have a rate Γinel larger than the expansion rate of the Universe. In such a case, one can write down a relation among the different chemical potentials of the particles involved in the process (43)
μX + μA = μB + μC .
In this way the number density in thermal equilibrium of the antiparticle X (mX = mX¯ ) is (44)
nX¯ gX (mX T )3/2 e−
mX T
−
μX T
,
where we have made use of the fact that μX¯ = −μX because of the process (45)
XX → γγ,
and μγ = 0. If the X particle carries baryon number, then B will get a contribution from (46)
B ∝ nX − nX¯ = 2gX (mX T )3/2 e−
mX T
sinh
μ X
T
.
The crucial point is now that, if X and X undergo B-violating reactions, as required by the first Sakharov condition, (47)
XX → XX,
then μX = 0 and the relative contribution of the X particles to the net baryon number vanishes. Only a departure from thermal equilibrium can allow for a finite baryon excess. 4. – The standard out-of-equilibrium decay scenario Out of the three Sakharov’s conditions, the baryon number violation and C and CP violation may be investigated thoroughly only within a given particle physics model, while the third condition —the departure from thermal equilibrium— may be discussed in a more general way. Very roughly speaking, the various models of baryogenesis that
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A. Riotto
have been proposed so far fall into two categories [5]: – models where the out-of-equilibrium condition is attained thanks to the expansion of the Universe and the presence of heavy decaying particles; – models where the departure from thermal equilibrium is attained during the phase transitions which lead to the breaking of some global and/or gauge symmetry. In this lecture we will analyse the first category, the standard out-of-equilibrium decay scenario [1]. . 4 1. The conditions for the out-of-equilibrium decay scenario. – It is obvious that in a static Universe any particle, even very weakly interacting, will attain sooner or later thermodynamical equilibrium with the surroinding plasma. The expansion of the Universe, however, introduces a finite time scale, τU ∼ H −1 . Let suppose that X is a baryon-number-violating superheavy boson field (vector or scalar) which is coupled to 1/2 lighter fermionic degrees of freedom with a strength αX (either a gauge coupling αgauge or a Yukawa coupling αY ). In the case in which the couplings are renormalizable, the decay rate ΓX of the superheavy boson may be easily estimated to be (48)
ΓX ∼ αX MX ,
where MX is the mass of the particle X. In the opposite case in which the boson is a gauge singlet scalar field and it only couples to light matter through gravitational interactions —this is the case of singlets in the hidden sector of supergravity models— the decay rate is from dimensional arguments (49)
ΓX ∼
3 MX . MP2
At very large temperatures T MX , it is assumed that all the particles species are in thermal equilibrium, i.e. nX nX nγ (up to statistical factors) and that B = 0. At T MX the equilibrium abundance of X and X relative to photons is given by (50)
nEQ nEQ X
X nγ nγ
MX T
3/2
e−
MX T
,
where we have neglected the chemical potential μX . For the X and X particles to mantain their equilibrium abundances, they must be able to diminish their number rapidly with respect to the Hubble rate H(T ). The conditions necessary for doing so are easily quantified. The superheavy X and X particles may attain equilibrium through decays with rate ΓX , inverse decays with rate ΓID X ⎧ ⎨1, T MX , (51) ΓID X ΓX ⎩(MX /T )3/2 exp[−MX /T ], T MX ,
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Leptogenesis
and annihilation processes with rate Γann X ∝ nX . The latter, however are “self-quenching” and therefore less important than the decay and inverse decay processes. They will be ignored from now on. Of crucial interest are the B-nonconserving scattering processes 2 ↔ 2 mediated by the X and X particles with rate ΓSX (52)
ΓSX nσ α2 T 3
T2 2 + T 2 )2 , (MX
where α g 2 /4π denotes the coupling strength of the X boson. At high temperatures, the 2 ↔ 2 scatterings cross-section is σ α2 /T 2 , while at low temperatures 4 σ α2 T 2 /MX . For baryogenesis, the most important rate is the decay rate, as decays (and inverse decays) are the mechanisms that regulate the number of X and X particles in the plasma. It is therefore useful to define the following quantity: (53)
K≡
ΓX , H T =MX
which measures the effectiveness of decays at the crucial epoch (T ∼ MX ) when the X and X particles must decrease in number if they are to stay in equilibrium. Note also that for T MX , K determines the effectiveness of inverse decays and 2 ↔ 2 scatterings 3/2 as well: ΓID exp[−MX /T ] K and ΓSX /H α(T /MX )5 K. X /H (MX /T ) Now, if K 1, and therefore (54)
ΓX H|T =MX ,
then the X and X particles will adjust their abundances by decaying to their equilibrium abundances and no baryogenesis can be induced by their decays —this is simply because out-of-equilibrium conditions are not attained. Given the expression (30) for the expansion rate of the Universe, the condition (54) is equivalent to (55)
−1/2
MX g∗
αX MP ,
for strongly coupled scalar bosons, and to (56)
1/2
MX g
MP ,
for gravitationally coupled X particles. Obviously, this last condition is never satisfied for MX MP . However, if the decay rate is such that K 1, and therefore (57)
ΓX H|T =MX ,
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A. Riotto
then the X and X particles cannot decay on the expansion time scale τU and so they remain as abundant as photons for T MX . In other words, at some temperature T > MX , the superheavy bosons X and X are so weakly interacting that they cannot catch up with the expansion of the Universe and they decouple from the thermal bath when still relativistic, nX nX nγ at the time of decoupling. Therefore, at temperature T MX , they will populate the Universe with an abundance which is much larger than the equilibrium one. This overbundance with respect to the equilibrium abundance is precisely the departure from thermal equilibrium needed to produce a final nonvanishing baryon asymmetry. Condition (57) is equivalent to (58)
−1/2
MX g∗
αX MP ,
for strongly coupled scalar bosons, and to (59)
1/2
MX g
MP ,
for gravitationally coupled X particles. It is clear that this last condition is always −1/2 satisfied, whereas condition (58) is based on the smallness of the quantity g∗ αX . In particular, if the X particle is a gauge boson, αX ∼ αgauge can span the range (2.5 × 10−2 –10−1 ), while g is about 102 . In this way we obtain from (58) that the condition of out-of-equilibrium can be satisfied for (60)
MX (10−4 –10−3 ) MP (1015 –1016 ) GeV.
If X is a scalar boson, its coupling αY to fermions f with mass mf is proportional to the squared mass of the fermions (61)
αY ∼
mf mW
2 αgauge ,
where mW is the W -boson mass and αY is typically in the range (10−2 –10−7 ), from which (62)
MX (10−8 –10−3 ) MP (1010 –1016 ) GeV.
Obviously, condition (62) is more easily satisfied than condition (60) and we conclude that baryogenesis is more easily produced through the decay of superheavy scalar bosons. On the other hand, as we have seen above, condition (59) tells us that the out-of-equilibrium condition is automatically satisfied for gravitationally interacting particles. . 4 2. The production of the baryon asymmetry. – Let us now follow the subsequent evolution of the X and X particles. When the Universe becomes as old as the lifetime of
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Leptogenesis
these particles, t ∼ H −1 ∼ Γ−1 X , they start decaying. This takes place at a temperature TD defined by the condition ΓX H|T =TD ,
(63) i.e. at (64)
−1/4
TD g∗
1/2
αX (MX MP )1/2 < MX ,
where the last inequality comes from (58) and is valid for particles with unsuppressed couplings. For particles with only gravitational interactions (65)
−1/4
TD ∼ g
MX
MX MP
1/2 < MX ,
the last inequality coming from (59). At T ∼ TD , X and X particles start to decay and their number decreases. If their decay violates the baryon number, they will generate a net baryon number per decay. Suppose now that the X particle may decay into two channels, let us denote them by a and b, with different baryon numbers Ba and Bb , respectively. Correspondingly, the decay channels of X, a and b, have baryon numbers −Ba and −Bb , respectively. Let r(r) be the branching ratio of the X(X) in channel a(a) and 1 − r(r) the branching ratio of X(X) in channel b(b), (66)
Γ(X → a) , ΓX Γ(X → a) r= , ΓX Γ(X → b) 1−r = , ΓX Γ(X → b) , 1−r = ΓX r=
where we have been using the fact that the total decay rates of X and X are equal because of the CP T theorem plus unitarity. The average net baryon number produced in the X decays is (67)
rBa + (1 − r)Bb ,
and that produced by X decays is (68)
−rBa − (1 − r)Bb .
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A. Riotto
Finally, the mean net baryon number produced in X and X decays is (69)
ΔB = (r − r)Ba + [(1 − r) − (1 − r)] Bb = (r − r)(Ba − Bb ).
Equation (69) may be easily generalized to the case in which X(X) may decay into a set of final states fn (f n ) with baryon number Bn (−Bn ) (70)
ΔB =
1 Bn Γ(X → fn ) − Γ(X → f n ) . ΓX n
At the decay temperature, TD MX , because K 1, both inverse decays and 2 ↔ 2 baryon-violating scatterings are impotent and can be safely ignored and thus the net baryon number produced per decay ΔB is not destroyed by the net baryon number −ΔB produced by the inverse decays and by the baryon-number-violating scatterings. At T TD , nX nX nγ and therefore the net baryon number density produced by the out-of-equilibrium decay is (71)
nB = ΔB nX .
The three Sakharov ingredients for producing a net baryon asymmetry can be easily traced back here: – If B is not violated, then Bn = 0 and ΔB = 0. – If C and CP are not violated, then Γ(X → fn ) = Γ(X → f n ), and also ΔB = 0. – In thermal equilibrium, the inverse processes are not suppressed and the net baryon number produced by decays will be erased by the inverse decays. Since each decay produces a mean net baryon number density nB = ΔBnX ΔBnγ and since the entropy density is s g nγ , the net baryon number produced is (72)
B≡
ΔBnγ ΔB nB
. s g nγ g
Taking g ∼ 102 , we see that only tiny C and CP violations are required to generate ΔB ∼ 10−8 , and thus B ∼ 10−10 . To obtain (72) we have assumed that the entropy release in X decays is negligible. However, sometimes, this is not a good approximation (especially if the X particles decay very late, at TD MX , which is the case of gravitationally interacting particles). In that case, assuming that the energy density of the Universe at TD is dominated by X particles (73)
ρX MX nX ,
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Leptogenesis
and that it is converted entirely into radiation at the reheating temperature TRH (74)
π2 4 g TRH , 30
ρ=
we obtain nX
(75)
π2 T4 g RH . 30 MX
We can therefore write the baryon number as B
(76)
3 TRH ΔB. 4 MX
We can relate TRH with the decay rate ΓX using the decay condition (77)
Γ2X H 2 (TD )
8πρX , 3MP2
and so we can write (78)
B
−1/2
ΓX M P 2 MX
g
1/2 ΔB.
For the case of strongly decaying particles (through renormalizable interactions) we obtain (79)
B
−1/2
g
αMP MX
1/2 ΔB,
while for the case of weakly decaying particles (through gravitational interactions) we obtain (80)
B
−1/2
g
MX
MP
1/2 ΔB.
In the other extreme regime K 1, one expects the abundance of X and X bosons to track the equilibrium values as ΓX H for T ∼ MX . If the equilibrium is tracked precisely enough, there will be no departure from thermal equilibrium and no baryon number may evolve. The intermediate regime, K ∼ 1, is more interesting and to address it one has to invoke numerical analysis involving Boltzmann equations for the evolution of B. The numerical analysis essentially confirms the qualitative picture we have described so far and its discussion is beyond the scope of these lectures.
360
A. Riotto
f
f
1
f
3
Y
X
X f
f
f2
1
(d)
(c)
(b)
4
Y
f4
2
(a)
f
3
Fig. 1. – Couplings of X and Y to fermions fi .
f3
f1 f3 X
f4
3
f4
f1
f3
Y
X
Y
Y X f2
f2
f4
f4
f2 (a)
f
X f1
f1 (c)
(b)
f2 (d)
Fig. 2. – One-loop corrections to the Born amplitude of fig. 1.
. 4 2.1. An explicit example. Let us consider first two massive boson fields X and Y coupled to four fermions f1 , f2 , f3 and f4 through the vertices of fig. 1 and describing the decays X → f 1 f2 , f 3 f4 and Y → f 3 f1 , f 4 f2 . We will refer to these vertices as f2 |X|f1 , f4 |X|f3 , f1 |Y |f3 and f2 |Y |f4 , and their CP conjugate X → f 2 f1 , f 4 f3 and Y → f 1 f3 , f 2 f4 by their complex conjugate. In the Born approximation ΔB = 0 because from (70) one finds (81)
2
12 |f2 |X|f1 | = Γ(X → f 2 f1 )Born , Γ(X → f 1 f2 )Born = IX
12 accounts for the kinematic structures of the processes X → f 1 f2 and X → where IX f 2 f1 and the same may be found for the other processes contributing to ΔB. This shows that, to obtain a nonzero result for ΔB, one must include (at least) corrections arising from the interference of Born amplitudes of fig. 1 with the one-loop amplitude of fig. 2. For example, the interference of the diagrams in fig. 1(a) and fig. 2(a) (in the square amplitude) is shown in fig. 3(a), where the thick dashed line is the unitarity cut (equivalent to say that each cut line represents on-shell mass particles). The amplitude of 1234 1234 the diagram in fig. 3(a) is given by IXY Ω1234 , where the kinematic factor IXY accounts for the integration over the final state phase space of f2 and f 1 and over momenta of the internal states f4 and f 3 , and
(82)
Ω1234 = f1 |Y |f3 ∗ f4 |X|f3 f2 |Y |f4 f2 |X|f1 ∗ .
361
Leptogenesis (a) f3 X
f1 X
Y
f4
(c)
(b) f1 X
X
Y
f2
f2
f1
f3
(d) f3
X
f3 X
X
Y
f4
f2
f1
X Y
f4
f4
f2
Fig. 3. – Intereference between the diagrams of fig. 1 and fig. 2 for the square amplitudes of X decay.
The complex-conjugate diagram of fig. 3(b) has the complex-conjugate amplitude. Therefore, the contribution from the diagrams in figs. 3(a) and (b) to the decay X → f 1 f2 is (83)
1234 Ω1234 + h.c. Γ(X → f 1 f2 )interference = IXY
To obtain the CP conjugate amplitude X → f 2 f1 , all couplings must be complex conjugated, although the kinematic factors IXY are unaffected by CP conjugation. Therefore the interference contribution to the X → f 2 f1 decay rate is given by (84)
1234 ∗ Ω1234 + h.c. Γ(X → f 2 f1 )interference = IXY
and the relevant quantity for baryogenesis is given by (85)
1234 Im [Ω1234 ] . Γ(X → f 1 f2 ) − Γ(X → f 2 f1 ) = −4 Im IXY
The diagrams of the decays X → f 3 f4 and X → f 3 f4 differ from the one in figs. 3(a) and (b) only in that the unitarity cut is taken through f3 and f4 instead of f1 and f2 . One easily obtains (86)
3412 Im [Ω∗1234 ] . Γ(X → f 3 f4 ) − Γ(X → f 4 f3 ) = −4 Im IXY
The kinematic factors IXY for loop diagrams may have an imaginary part whenever any internal lines may propagate on their mass shells in the intermediate states, picking the pole of the propagator (87)
p2
1 PP = 2 + iπδ(p2 − m2 ), 2 − m + i p − m2
where PP stands for the principal part. This happens if MX > m1 + m2 and MX > m3 + m4 . This means that with light fermions, the imaginary part of IXY will be always 1234 3412 nonzero. The kinematic factors Im[IXY ] and Im[IXY ] are therefore obtained from diagrams involving two unitarity cuts: one through the lines f1 and f2 and the other through the lines f3 and f4 . The resulting quantities are invariant under the interchanges f1 ↔ f3 and f2 ↔ f4 and consequently (88)
3412 1234 = Im IXY = Im [IXY ] . Im IXY
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A. Riotto
Defining Bi the baryon number of the fermion fi , the net baryon number produced in the X decays is therefore (89)
(ΔB)X =
4 Im [IXY ] Im [Ω1234 ] [B4 − B3 − (B2 − B1 )] . ΓX
To compute the baryon asymmetry (ΔB)Y one may observe that the set of vertices in fig. 1 is invariant under the transformations X ↔ Y and f1 ↔ f4 . These rules yield (90)
(ΔB)Y =
4 Im [IY X ] Im [Ω∗1234 ] [B4 − B3 − (B2 − B1 )] ΓY
and the total baryon number is therefore given by (91)
(ΔB) = (ΔB)X + (ΔB)Y ! " Im [IXY ] Im [IY X ] =4 Im [Ω1234 ] [B4 − B3 − (B2 − B1 )] . − ΓX ΓY
We can notice a few things: – If the X and Y couplings were B conserving, the two possible final states in X and Y decays would have the same baryon number, i.e. B4 − B3 = B2 − B1 and therefore ΔB = 0. Therefore the baryon number must be violated not only in X decays but also in the decays of the particle exchanged in the loop. – Some coupling constants in the Lagrangian must be complex to have Im[Ω1234 ]. – Even if (ΔB)X and (ΔB)Y are both nonvanishing, the sum can vanish if the first bracket in (91) cancels out. This happens if the X and Y particles have the same mass and ΓX = ΓY . . 4 3. Baryon number violation within the SM and out-of-equilibrium baryogenesis. – At this point, we are ready to discuss the implications of the baryon number violation in the early Universe for the baryogenesis scenarios discussed so far. The basic lesson we have learned previously is that any asymmetry (B + L) is rapidly erased by sphaleron transitions as soon as the temperature drops down at ∼ 1012 GeV. Now, we can always write the baryon number B as (92)
B=
B+L B−L + . 2 2
This equation seems trivial, but is dense of physical significance! Sphaleron transitions only erase the combination (B + L), but leave the orthogonal combination (B − L) untouched. This means that the only chance for a GUT baryogenesis scenario to work is to produce at high scale an asymmetry in (B − L). However, we have learned that there is no possibility of generating such an asymmetry in the framework of SU (5). This is because the fermionic content of the theory is the one of the SM and there is no violation of (B −L). Sphaleron transitions are therefore the killers of any GUT baryogenesis model
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Leptogenesis
based on the supersymmetric version of SU (5) with R parity conserved (the nonsupersymmetric version is already ruled out by experiments on the proton decay lifetime). This is a striking result. However, theories with (B − L) violation incorporated in them may be successful. Indeed, any asymmetry in (B − L) will correspond to an asymmetry in B once sphalerons are allowed to act, see eq. (92). This is, in a nutshell, the idea of leptogenesis, which we now describe. 5. – Baryogenesis via leptogenesis: one-flavour approximation Baryogenesis through leptogenesis [9] is a simple mechanism to explain this baryon asymmetry of the Universe. As we said, a lepton asymmetry is dynamically generated and then converted into a baryon asymmetry due to (B + L)-violating sphaleron interactions which exist in the SM. The reader is forwarded to refs. [11] and [12] for more details. A simple model in which this mechanism can be implemented is the “see-saw”(type I) [13], consisting of the SM plus two or three right-handed (RH) Majorana neutrinos. In this simple extension of the SM, the usual scenario that is explored (referred to as “thermal leptogenesis”) consists of a hierarchical spectrum for the RH neutrinos, such that the lightest of the RH neutrinos is produced by thermal scattering after inflation, and subsequently decays out-of-equilibrium in a lepton number and CP -violating way, thus satisfying Sakharov’s constraints. This section introduces notations and reviews the calculation of the lepton asymmetry when the charged lepton Yukawa couplings are neglected. As we shall see, the commonly used formulae for the final lepton asymmetry, which we report here, may not be appropriate once flavours are considered. The reader should be patient, flavour effects will be discussed later. Our starting point is the Lagrangian of the SM with the addition of three righthanded neutrinos Ni (i = 1, 2, 3) with heavy Majorana masses M3 > M2 > M1 and Yukawa couplings λαi . Working in the basis in which the Yukawa couplings for the charged leptons are diagonal, the Lagrangian reads (93)
L = LSM +
Mi 2 N + λαi Lα HNi 2 i
+ h.c.
Here Lα is the lepton doublet with flavour (α = e, μ, τ ), and H is the Higgs doublet whose neutral component has a vacuum expectation value (VEV) equal to v = 175 GeV. After spontaneous symmetry breaking, a Dirac mass term mD = λv, is generated by the VEV of the Higgs boson. In the see-saw limit, M mD , the spectrum of neutrino mass eigenstates splits in two sets: three very heavy neutrinos, N1 , N2 and N3 , respectively with masses M1 ≤ M2 ≤ M3 , almost coinciding with the eigenvalues of M , and three light neutrinos with masses m1 ≤ m2 ≤ m3 , the eigenvalues of the light neutrino mass matrix given by the see-saw formula [13] (94)
mν = −mD
1 T m . M D
364
A. Riotto
Neutrino oscillation experiments measure two neutrino mass-squared differences. For normal schemes one has m23 − m22 = Δm2atm and m22 − m21 = Δm2sol , whereas for inverted schemes one has m23 − m22 = Δm2sol and m22 − m21 = Δm2atm . For m1 matm ≡ 2 = (0.050 ± 0.001) eV [14] the spectrum is quasi-degenerate, while for Δm2atm + Δm sol m1 msol ≡ Δm2sol = (0.00875 ± 0.00012) eV [14] it is fully hierarchical (normal or inverted). Here we will restrict ourselves to the case of normal schemes. The most stringent upper bound on the absolute neutrino mass scale comes from cosmological observations. Recently, a conservative upper bound on the sum of neutrino masses, i mi ≤ 0.61 eV (95% CL), has been obtained by the WMAP collaboration combining CMB, baryon acoustic oscillations and supernovae type-Ia observations [2]. Considering that it falls in the quasi-degenerate regime, it straightforwardly translates into (95)
m1 < 0.2 eV (95% CL).
It proves sometimes useful to adopt the bi-unitary parametrization (96)
mD = VL† DmD UR ,
where VL and UR are two unitary matrices that diagonalize mD and DmD is the diagonal matrix whose elements are the eigenvalues of mD : DmD ≡ diag(lD1 , lD2 , lD3 ). This shows that in the process of see-saw, 9 parameters are lost: at high energies there 18 real parameters (3 from the eigenvalues of mD , 3 from Mi , and 12 from VL and VR ). At low energy there are only 9 (3 from mi and 6 from the matrix U diagonalizing mν ). We now assume that right-handed neutrinos are hierarchical, M2,3 M1 so that studying the evolution of the number density of N1 suffices. The final amount of (B − L) asymmetry can be parametrized as YB−L = nB−L /s, where s = 2π 2 g T 3 /45 is again the entropy density and g counts the effective number of spin-degrees of freedom in thermal equilibrium (g = 217/2 in the SM with a single generation of right-handed neutrinos). After reprocessing by sphaleron transitions, the baryon asymmetry is related to the L asymmetry by [5] (97)
YB = −
8nG + 4nH 14nG + 9nH
YL ,
where nH is the number of Higgs doublets, and nG the number of fermion generations (in equilibrium). It is also useful to define an efficiency factor η which tells how efficient is the production of the baryon asymmetry, e.g. how much of the asymmetry per RH neutrino decay remais after wash-out processes are accounted for (98)
YB 1.38 · 10−3 1 η,
where we have assumed nH = 1. Now, the idea of thermal leptogenesis is that RH neutrinos decay in the early Universe out of equilibrium, thus producing a lepton asymmetry.
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Leptogenesis
One defines the CP asymmetry generated by N1 decays as (99)
2 Im (λλ† )2j1 Mj [Γ(N → H ) − Γ(N → H )] 1 1 α 1 α . g 1 ≡ α = †] 8π [λλ M12 11 α [Γ(N1 → H α ) + Γ(N1 → H α )] j =1
It is the sum of two contributions, the vertex and the wave function ones. A calculation similar to the one performed in the toy model previously discussed leads to (100)
g(x) =
√ x
1 + 1 − (1 + x) ln 1−x
1+x x
3 −→ − √ . 2 x
x 1
Besides the CP parameter 1 , the final baryon asymmetry depends on a single washout parameter, (101)
K≡
α
Γ(N1 → H α ) ≡ H(M1 )
m A1 m A∗
,
where H(M1 ) denotes the value of the Hubble rate evaluated at a temperature T = M1 (m A ∗ ∼ 10−3 eV) and (102)
m ˜1 ≡
(λλ† )11 v 2 M1
is proportional to the total decay rate of the right-handed neutrino N1 . One could now proceed as in the previous sections by estimating the baryon asymmetry. However, we want to do a better job and we resort to the Boltzmann equations. By defining the variable z = M1 /T , the Boltzmann equations for the lepton asymmetry YL , and the right-handed neutrino number density YN1 (both normalised to the entropy s), may be written in a compact form as (103)
(104)
d(YN1 − YNEQ ) dYNEQ z YN1 1 1 =− (γD + γΔL=1 ) , − 1 − dz sH(M1 ) dz YNEQ 1 z dYL YL YN1 = − 1 1 (γD + γΔL=1 ) − EQ (γD + γΔL=1 ) . dz sH(M1 ) YNEQ YL 1
The processes taken into account in these equations are decays and inverse decays with rate γD , ΔL = 1 scatterings such as (qtc → N ), and ΔL = 2 processes mediated by heavy neutrinos (see fig. 4). The first three modify the abundance of the lightest righthanded neutrinos. The ΔL = 2 scatterings mediated by N2,3 are neglected in our analysis for simplicity. The various γ are thermally averaged rates, including all contributions summed over flavour (s, t channel interference, etc.); explicit expressions can be found in the literature. Notice that in this “usual” analysis, ΔL = 1 scattering contributes to the
366
A. Riotto L
L
H
N1
N1
L
N2, 3 H
H L
H
L N2, 3
H
L
L
N1, 2, 3
L N1
H
L
N1, 2, 3
H N1, 2, 3
H
H
H
L
L
H
N1
U3
N1
L
N1
L
H
H
H
L
Q3
U3
Q3
Q3
U3
N1
H
N1
H
N1
H
H
L
L
L
A
L
A
A
L
N1
L
N1
L
N1
L
L H
H A
H
H
A
A
H
Fig. 4. – Feynman diagrams contributing to thermal leptogenesis.
creation of N1 ’s and not to the production of a lepton asymmetry, only to the wash-out. can be obtained from Approximate analytic solutions for YL and ΔN1 ≡ YN1 − YNEQ 1 simplified equations. Calculating in zero-temperature field theory for simplicity, one obtains (105)
γD sYNEQ 1
K1 (z) ΓD , K2 (z)
YNEQ
1
1 2 z K2 (z). 4g
The Boltzmann equations can be approximated (106)
ΔN1 = −zK
(107)
YL = 1 Kz
K1 (z) f1 (z)ΔN1 − YNEQ , 1 K2 (z) K1 (z) 1 ΔN1 − z 3 KK1 (z)f2 (z)YL , K2 (z) 4
where K1 and K2 are modified Bessel functions of the second kind. The function f1 (z) accounts for the presence of ΔL = 1 scatterings, and f2 (z) accounts for scatterings in the
367
Leptogenesis
wash-out term of the asymmetry. They can be approximated, in interesting limits, as ⎧ ⎨1, for z 1, (108) f1 (z) 2 2 ⎩ N2c m2 t 2 , for z 1, 4π v z and f2 (z)
(109)
⎧ ⎨1,
for z 1,
2 2 ⎩ aK 2Nc2m2t , 8π v z
for z 1,
N 2 m2
where 8πc2 v2t ≡ Ks /K ∼ 0.1 parametrizes the strength of the ΔL = 1 scatterings and aK = 4/3(2) for the weak (strong) wash out case. A good approximation to the rate Kz(K1 (z)/K2 (z))f1 (z) is given by the function (Ks + Kz) while the wash-out term −(1/4)z 3 KK1 (z)f2 (z)YL is well approximated at small z by −aK Ks YL . . 5 1. Strong wash-out regime. – In the strong wash-out regime, the parameter K 1 and the right-handed neutrinos N1 ’s are nearly in thermal equilibrium. Under these circumstances, one can set ΔN1 0 and ΔN1 (zK2 /4g K). Exploiting a saddle-point approximation in eq. (107), we find that the lepton asymmetry is given by ∞ K 2 − R ∞ dz ((z )3 /4)K1 (z )K (110) YL 1 dz z e z . 4g 0 Using the steepest-descent method to evaluate the integral, one finds that it gets the major contribution at z such that z = log K + (5 ln z/2) when inverse decays become inefficient. The lepton asymmetry in the flavour α becomes 1 YL 0.3 g
(111)
0.55 × 10−3 eV m ˜1
1.16 .
. 5 2. Weak wash-out regime. – In this case all the K 1. We assume that right-handed neutrinos are not initially present in the plasma, but they are generated by inverse decays and scatterings. The equation of motion for YN1 is well approximated by (112) YN 1 = −(Ks + Kz) YN1 − YNEQ , 1 We split the solution into two pieces. Let us define zEQ the value of z at which YN1 (zEQ ) = YNEQ (zEQ ). This value has to be found a posteriori. For z zEQ , we may suppose that 1 and eq. (112) is solved by YN1 YNEQ 1 (113)
YN−1 (z)
z
dz (Ks + Kz 0
)YNEQ 1
1 = 4g K = 4g
z
0
dz (Ks + Kz )(z )2 K2 (z )
Ks I1 (z) + I2 (z) . K
368
A. Riotto
With I1 and I2 integral involving the modified Bessel functions, (114)
z
I1 (z) =
x2 K2 (x)dx f (z) + z 3 K2 (z),
0
where (115)
f (z) =
3πz 3 ((9π)c + (2z 3 )c )
1/c
,
c = 0.7.
The integral I2 is well known, and equals I2 (z) =
(116)
z
x3 K2 (x)dx = 8 − z 3 K3 (z).
0
Therefore (117)
YN−1 (z)
K
4g
Ks 3 3 (f (z) + z K2 (z)) + 8 − z K3 (z) . K
As expected for weak wash-out, we find that the maximum number density of N1 is proportional to K (recall Ks ∝ K). Let us now compute the value of zEQ . We expect it to be 1 and we therefore π −1/2 −z approximate, up to O(z −3/2 ): K2 (z) K3 (z) e . Imposing YN−1 (zEQ ) = 2z EQ YN1 (zEQ ), we find (118)
zEQ
3
ln zEQ − ln 2
) π Ks . K +3 2 π/2 8
This solution is a good approximation to the real value for K 1. For z > zEQ , we have (Ks + Kz) Kz and (119)
2 2 (zEQ )eK/2(zEQ −z ) . YN1 (z) YNEQ 1
Notice that we have included CP violation in ΔL = 1 scattering, unlike the usual analysis, so we expect our solution for YL to have a different scaling with K than in the traditional literature: if CP / in scattering is neglected, then YN ∝ K, and the N1 ’s decay out of equilibrium, so one expects YL ∝ K1 . However, if CP / in N1 production ( scattering) is included, and washout is neglected, then the equations for YN1 and YL are identical, so YL (z → ∞) vanishes. That is, for every |1/1 | N1 ’s that are created, be it by inverse decay or scattering, an (anti)-lepton is produced. This (anti-)asymmetry will approximately cancel against the lepton asymmetry generated later on, when the N1 ’s decay. However the cancellation will be imperfect, because the anti-asymmetry has more time to be washed out, so the final asymmetry should scale as K 2 . After integrating by
369
Leptogenesis SM
102
SM
1
dominant N1 thermal N1 zero N1
10− 4 −6
10− 1
10− 2
atm
efficiency η
10− 2
sun
efficiency η
1
10
10− 8 − 10 10
10− 8
10− 6 10− 4 ∼ in eV m 1
10− 2
10− 3 10− 3
1
10− 2 ∼ in eV m 1
10− 1
Fig. 5. – The efficiency factor as a function of m ˜ 1 . Notice that for small values of m ˜ 1 the slope as the CP asymmetry from scattering is not accounted for. On the left-hand side three is m ˜ −1 1 and dominant N1 . On cases are analyzed: zero N1 abundance, thermal abundance (nN1 ∼ neq N1 the right-hand side, there is a zoom). From [11].
parts, this is what we find for the lepton asymmetry, which is given by ∞ R ∞ 1 (120) YL 1 dz YN1 (z )g1 (z )e− z dz g1 (z ) , g1 (z) = z 3 KK1 (z)f2 (z), 4 0 2 m ˜1 1 .
1.5 g 3.3 × 10−3 eV Our findings hold provided that the nonresonant ΔL = 2 scattering rates, in particular those mediated by the N2 and N3 heavy neutrinos, are slower than decays and ΔL = 1 scatterings when most of the asymmetry is generated. We estimate that this applies when M1 (121) 10−1 . 1014 GeV This means that K should be larger than 10−4 . Our results can be summarized with simple analytical fits for the efficiency factor (122)
1 ≈ η
3.3 × 10−3 eV m ˜1
2
+
m ˜1 0.55 × 10−3 eV
1.16 ,
valid for MN1 1014 GeV and when one starts with no RH neutrinos in the plasma. Numerical results are presented in fig. 5. This enables the reader to study leptogenesis in neutrino mass models without setting up and solving the complicated Boltzmann equations. . 5 3. Implications of one-flavour leptogenesis. – Experiments have not yet determined the mass m3 of the heaviest mainly left-handed neutrinos. We assume m3 = max(m ˜ 1 , matm ). A crucial assumption we have so far is that right-handed neutrinos are
370
A. Riotto
very hierarchical. Under this hypothesis the CP asymmetry is bounded by the expression [15] that in the hierarchical and quasi-degenerate light neutrino limit simplifies as follows:
(123)
⎧ ˜ 1, if m1 m3 , 3 M1 (m3 − m1 ) ⎨1 − m1 /m |1 | ≤ × 2 ⎩ 1 − m2 /m 16π v2 1 ˜ 1 , if m1 m3 ,
where all parameters are renormalized at the high-energy scale ∼ M1 . The 3σ ranges of matm and of YB imply the lower bound (124)
M1 >
4.5 × 108 GeV > 2.4 × 109 GeV, η
in the case in which no RH neutrinos are present in the plasma at the beginning of the dynamics and we have assumed that the efficiency is maximal (and therefore η minimal). This bound is relevant because it tells us that a working model of thermal leptogenesis (in the one-flavour approximation) needs enough heavy RH neutrinos. This has implications for SO(10) grand-unified theories, commonly regarded as the most attractive way to embed the see-saw mechanism. Indeed, in a traditional version of leptogenesis, where the spectrum of right-handed (RH) neutrinos is hierarchical and the asymmetry is produced from the decays of the lightest ones, there is a stringent lower bound on their mass [15], M1 > O(109 ) GeV, for a sufficiently large baryon asymmetry to be produced. On the other hand, SO(10) grand-unified theories typically yield, in their simplest version and for the measured values of the neutrino mixing parameters, a hierarchical spectrum with the RH neutrino masses proportional to the squared of the up-quark masses, leading to M1 = O(105 ) GeV and to a final asymmetry that falls a few orders of magnitude below the observed one. We can also work out a bound on the light neutrino masses, i mi 0.15 eV. It can be understood to arise from the lower bound on the total decay rate m1 ≤ m ˜ 1 , and the upper bound on the total CP asymmetry (123). We assume the light neutrinos are √ degenerate, so |m1 | |m2 | |m3 | ≡ m/ ¯ 3 —but the masses can have different Majorana phases. Leptogenesis takes place in the strong wash-out regime, due to the lower bound on the total decay rate. The final baryon asymmetry can be roughly approximated as YB ∼ 10−4 /K ∝ Δm2atm /m ¯ 2 . As the light neutrino mass scale is increased, M1 and the temperature of leptogenesis must increase to compensate the Δm2atm /m ¯ 2 suppression. However, this temperature is bounded from above, from the requirement of having the ΔL = 2 processes out of equilibrium when leptogenesis takes place: (125)
m ¯ 2T 3 10T 2 , 12πv 4 MP
¯ 2 GeV. There is therefore an upper bound on the baryon asymmetry so M1 1010 (eV/m)
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Leptogenesis
SM 3σ ranges
10− 9
10− 10 0.08
0.1 0.12 0.14 heaviest ν mass m3 in eV
MSSM
10− 8 maximal nB / nγ
maximal nB / nγ
10− 8
0.16
3σ ranges
10− 9
10− 10 0.08
0.1 0.12 0.14 heaviest ν mass m3 in eV
0.16
Fig. 6. – The bound on the heaviest light neutrino mass for the SM and for the Minimal Supersymmetric Standard Model (MSSM).
which scales as 1/m ¯ 4 , and with our rough estimates, one finds (126)
√ 3 0.1 eV. m/ ¯
This result is confirmed by a full numerical analysis [11], see fig. 6. 6. – Comments on baryogenesis via leptogenesis when flavours are accounted for In recent years, a lot of work has been devoted to a thorough analysis of thermal leptogenesis giving limited attention to the issue of lepton flavour [16]. The dynamics of leptogenesis is usually addressed within the “one-flavour” approximation, where Boltzmann equations are written for the abundance of the lightest RH neutrino, responsible for the out of equilibrium and CP asymmetric decays, and for the total lepton asymmetry. However, this “one-flavour” approximation is rigorously correct only when the interactions mediated by charged lepton Yukawa couplings are out of equilibrium. Flavour effects have not been included in leptogenesis calculations till very recently [17-19]. This is perhaps because perturbatively, they seem to be a small correction. For instance, if the asymmetry is a consequence of the very-out-of-equilibrium decay of an initial population of right-handed neutrinos, then the total lepton asymmetry is of order /g∗ , where is the total CP asymmetry in the decay, and g∗ counts for the entropy dilution factor. Clearly the small charged lepton Yukawa couplings have no effect on . However, realistic leptogenesis is a drawn-out dynamical process, involving the production and destruction of right-handed neutrinos, and of a lepton asymmetry that is distributed among distinguishable flavours. The processes which wash out lepton number are flavour dependent, e.g., the inverse decays from electrons can destroy the lepton asymmetry carried by, and only by, the electrons. The asymmetries in each flavour are therefore washed out differently, and will appear with different weights in the final formula for the baryon asymmetry. This is physically inequivalent to the treatment of wash-out in the one-flavour approximation, where indistinguishable leptons propagate between decays and inverse decays, so inverse decays from all flavours are taken to wash out asymmetries in any flavour.
372
A. Riotto
We define Yαα to be the lepton asymmetry in flavour α, where the α are the lepton mass eigenstates at the temperature of leptogenesis. The Yαα are the diagonal elements of a matrix [Y ] in flavour space, whose trace is the total lepton asymmetry. In this lecture the off-diagonal elements are neglected. The equations of motion for the matrix [Y ] are more complicated than the Boltzmann equations, but at most temperatures are equivalent to Boltzmann equations written in the mass eigenstate basis of the leptons in the plasma. The off-diagonal elements of [Y ] could have some effect on the lepton asymmetry, if leptogenesis takes place just as a charged lepton Yukawa coupling is coming into equilibrium (so the mass eigenstate basis is changing). The mass eigenstates for the particles in the Boltzmann equations (BE) are determined by the interactions which are fast compared to those processes included in the BE. The interaction rate for Yukawa coupling hα can be estimated as Γα 5 × 10−3 h2α T,
(127)
so interactions involving the τ (μ) Yukawa coupling are out of equilibrium in the primeval plasma if T 1012 GeV (T 109 GeV)(1 ). Thermal leptogenesis takes place at temperatures on the order of M1 , and the asymmetry is generated when the rates H, so we conclude that the τ (μ) lepton doublet is a distinguishable mass eigenstate, for the purposes of leptogenesis, at T < 1012 (109 ) GeV. Suppose therefore that M1 is below 109 GeV and we forget for the time being about the bound (124). All flavours are distinguishable. The Boltzmann equations for the flavour asymmetries Yαα , are as follows. The equation for the N1 number density remains unchanged, and the equation for the flavoured lepton asymmetry is αα z YN1 Y dY αα αα αα (128) = − 1 αα (γD + γΔL=1 ) − EQ (γD + γΔL=1 ) . dz sH(M1 ) YNEQ YL 1
To obtain analytic solutions, we could simplify this, with the approximations introduced in the previous section, to (129) (130)
Yαα = αα Kz
K1 (z) 1 f1 (z)ΔN1 − z 3 K1 (z)f2 (z)Kαα Yαα , K2 (z) 4
ΔN1 = −zK
K1 (z) f1 (z)ΔN1 − YNEQ , 1 K2 (z)
ΔN1 = −zK
K1 (z) f1 (z)ΔN1 − YNEQ , 1 K2 (z)
where (131)
(1 ) The electron Yukawa coupling mediates interactions relevant in the early Universe only for temperatures below ∼ 105 GeV and can be safely disregarded.
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Leptogenesis
where (132)
Kαα
λ1α λ∗1α = K = 2 γ |λ1γ |
m ˜ αα 10−3 eV
,
K=
Kαα .
α
Kαα parametrizes the decay rate of N1 to the α-th flavour, and the trace α Kαα coincides with the K parameter defined in the previous section, see eq. (101). Notice, in particular, that the dynamics of the right-handed neutrinos is always set by the total K. The CP asymmetry in the α-th flavour is αα and is normalised by the total decay rate @ ? Mj2 1 1 † ∗ αα = (133) Im (λ1α )(λλ )1j λjα g (8π) [λλ† ]11 j M12 " ! 3 [m∗ ]βα (134) Im λ λ → 1β 1α , (16π)[λλ† ]11 v2 where the second line is in the limit of hierarchical NJ , and m = U ∗ Dm U † = v 2 λT M −1 λ is the light neutrino mass matrix. If m3 is the heaviest light neutrino mass (= matm for the nondegenerate case) and we define max = 3Δm2atm M1 /(8πv 2 mmax ) [15], then the flavour-dependent CP asymmetries are bounded by ) ) 3M1 m3 Kαα Kαα m23 max = , (135) αα ≤ 2 2 16πv K Δmatm K so the maximum CP asymmetry in a given flavour is unsuppressed for degenerate light neutrinos [17], but decreases as the square root of the branching ratio to that flavour = Kαα /K. The first consequence of this result is that there is no upper bound on the light neutrino masses when flavours are accounted for. The CP asymmetry αα can be written in terms of the diagonal matrix of the light neutrino mass eigenvalues m = Diag(m1 , m2 , m3 ), the diagonal matrix of the the right handed neutrino masses M = Diag(M1 , M2 , M3 ) and an orthogonal complex matrix [20] (136)
R = vM −1/2 λU m−1/2 ,
where the matrix U diagonalizes the light neutrino mass matrix mν , so that (137)
U † mν U = −Dm
and it can be identified with the lepton mixing matrix in a basis where the charged lepton mass matrix is diagonal. The asymmetry reads 1/2 3/2 ∗ m m U U R R Im ρ αρ β1 ρ1 αβ βρ β 3M1 (138) αα = − . 2 16πv 2 m |R β 1β | β
374
A. Riotto
For a real R matrix, the individual CP asymmetries αα may not vanish because of the presence of CP violation in the U matrix. On the contrary, the total CP asymmetry 1 = α αα vanishes. This is the second implication of the flavours: leptogenesis works even if the matrix R is real. Let us say this in other words. The neutrino Yukawa coupling can be written in its singular value decomposition, λ = VL† Diag(λ1 , λ2 , λ3 )UR . Hence, the CP violation in the right-handed neutrino sector is encoded in the phases in VR , that can be extracted from diagonalizing the combination λλ† = UR† Diag(λ21 , λ22 , λ23 )UR . In the one-flavour regime, only the phases of the matrix VR count, in the flavour regime the phase of the matrix VL also counts and the final lepton asymmetry does not vanish even if the matrix VR is real. Implications of this result can be found in [21-23]. 7. – Conclusions In this lecture we have presented the scenario of thermal leptogenesis. Because of the lack of time, we have touched in more details the one-flavour scenario, while we have left the more realistic flavourful scenario to the curiousity of students. Leptogenesis is very much linked to neutrino physics and therefore any future new discovery in this field will open for us a new window into the primordial Universe. ∗ ∗ ∗ It is with great pleasure that the author thanks the Directors of the School F. Ferroni and F. Vissani for the wonderful atmosphere they have created around the School.
REFERENCES [1] Kolb E. W. and Turner M. S., Front. Phys., 69 (1990) 1. [2] Komatsu E. et al. (WMAP Collaboration), arXiv:0803.0547 [astro-ph]. [3] Dodelson S., Modern Cosmology (Academic Press, Amsterdam, The Netherlands) 2003, p. 440. [4] Iocco F., Mangano G., Miele G., Pisanti O. and Serpico P. D., arXiv:0809.0631 [astro-ph]. [5] Riotto A., arXiv:hep-ph/9807454. [6] Riotto A. and Trodden M., Annu. Rev. Nucl. Part. Sci., 49 (1999) 35 [arXiv:hepph/9901362]. [7] Davidson S., Nardi E. and Nir Y., arXiv:0802.2962 [hep-ph]. [8] Sakharov A. D., Zh. Eksp. Teor. Fiz. Pis’ma, 5 (1967) 32; JETP Lett. B, 91 (1967) 24. [9] Fukugita M. and Yanagida T., Phys. Lett. B, 174 (1986) 45. [10] Lyth D. H. and Riotto A., Phys. Rep., 314 (1999) 1 [arXiv:hep-ph/9807278]. [11] Giudice G. F., Notari A., Raidal M., Riotto A. and Strumia A., Nucl. Phys. B, 685 (2004) 89 [arXiv:hep-ph/0310123]. [12] Buchmuller W., Di Bari P. and Plumacher M., Ann. Phys. (NY), 315 (2005) 305. [13] Minkowski P., Phys. Lett. B, 67 (1977) 421; Gell-Mann M., Ramond P. and Slansky R., Proceedings of the Supergravity Stony Brook Workshop, edited by Van Nieuwenhuizen P. and Freedman D. (New York) 1979; Yanagida T., Proceedings of the Workshop on Unified Theories and Baryon Number in the Universe, Tsukuba, Japan 1979, edited by
Leptogenesis
[14] [15] [16]
[17] [18] [19] [20] [21] [22] [23]
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Sawada A. and Sugamoto A.; Mohapatra R. N. and Senjanovic G., Phys. Rev. Lett., 44 (1980) 912. Strumia A., this volume, p. 21. Davidson S. and Ibarra A., Phys. Lett. B, 535 (2002) 25 [arXiv:hep-ph/0202239]. See also, Barbieri R., Creminelli P., Strumia A. and Tetradis N., Nucl. Phys. B, 575 (2000) 61; Endoh T., Morozumi T. and Xiong Z. h., Prog. Theor. Phys., 111 (2004) 123. Abada A., Davidson S., Josse-Michaux F. X., Losada M. and Riotto A., J. Cosmol. Astroparticle Phys., 0604 (2006) 004. Nardi E., Nir Y., Roulet E. and Racker J., JHEP, 0601 (2006) 164. Abada A., Davidson S., Ibarra A., Josse-Michaux F. X., Losada M. and Riotto A., JHEP, 0609 (2006) 010. Casas J. A. and Ibarra A., Nucl. Phys. B, 618 (2001) 171 [arXiv:hep-ph/0103065]. Pascoli S., Petcov S. T. and Riotto A., Phys. Rev. D, 75 (2007) 083511 [arXiv:hepph/0609125]. Branco G. C., Gonzalez Felipe R. and Joaquim F. R., Phys. Lett. B, 645 (2007) 432 [arXiv:hep-ph/0609297]. Pascoli S., Petcov S. T. and Riotto A., Nucl. Phys. B, 774 (2007) 1 [arXiv:hepph/0611338].
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DOI 10.3254/978-1-60750-038-4-377
Measurements of neutrino mass. Concluding remarks A. Yu. Smirnov International Centre for Theoretical Physics - Strada Costiera 11, Trieste, Italy and Institute for Nuclear Research, RAN - Moscow, Russia
Summary. — Measurements of the neutrino mass are considered in a general context of neutrino studies and searches for new physics beyond the standard model. The topics include: phenomenology of mass, origins and nature of neutrino masses, explanation of their smallness, relation between masses and mixing, implications of mass determination for fundamental theory and a possibility to predict neutrino mass or type of mass spectrum.
1. – Introduction It took more than 70 years since Pauli’s original idea (1930) that the neutrino mass is of the order of the electron mass (or smaller) and the first Fermi’s estimation (1934) that mν < 0.1me to conclude that at least one of neutrino masses is in the range (1)
m = (0.04–0.20) eV.
Behind this conclusion, one finds marvelous works of several generations of experimentalists and theoreticians, particle physicists and cosmologists [1]. The scale (1) is about 10−7 me , 10−10 mp , 10−12 mt , and the latter provides strong evidence that with neutrino masses we are touching something really new. c Societ` a Italiana di Fisica
377
378
A. Yu. Smirnov
Q3
Qe
Q2 Q1
MASS
QP
QW Q2
Q3
Q1 NORMAL
INVERTED
Fig. 1. – Neutrino mass and flavor spectra for the normal (left) and inverted (right) mass hierarchies. The distribution of flavors (parts of boxes with different shadowing) in the mass eigenstates corresponds to the best-fit values of the mixing parameters and sin2 θ13 = 0.05.
The long history of the direct kinematic measurements (better —searches) of neutrino mass will culminate next year with the start of the KATRIN experiment [2]. One can measure “time in neutrino physics” using the upper bounds on the mass of the electron neutrino with ∼ 105 eV as the starting point down to 2 eV now. The breakthrough came from a completely different side. About 50 years ago in the paper “Mesonium and antimesonium” [3] Bruno Pontecorvo mentioned a possibility of neutrino oscillations which implies non-zero neutrino mass and mixing. Incidentally, in the same year 1957, the discovery of the parity violation was announced. This led to establishing the V -A theory of the weak interactions, and the theory of the two-component massless neutrino which was the dominating idea for almost 40 years. As we know, since 1998, the first line —Pontecorvo’s idea— won. The lesson we can extract from this story is that the “non-standard” and “exotic” process (oscillations) led eventually to this discovery. The fact that neutrinos have mass has been established not from the direct kinematic measurements but indirectly, observing new processes related to the phenomenon of mixing which requires the existence of non-zero mass. Detailed analysis then showed that non-zero mass is the only explanation of data (or at least, the origin of the dominant mechanism behind observations). Without additional assumptions, the oscillations give only the lower bound on the mass. The upper bound in (1) follows from Cosmology, the analysis of the Larger Scale Structure (LSS) of the Universe. The interval (1) is amazingly small in comparison with our staring point. At the same time, it is still large from the point of view of theory. There is a big difference of implications of two different values of mass m = 0.04 eV and m = 0.2 eV. Furthermore, the key questions is: what is the flavor of this heaviest state, that is, what is the mass hierarchy (fig. 1)? How are we going to further improve the determination of the absolute mass scale? Remember the lesson: the answer may come from something unexpected which is based on a new and non-standard phenomenon.
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Fig. 2. – Cosmological bounds on the sum of neutrino masses obtained using different sets of data: CMB (dotted), CMB + LSS (dashed), CMB + Hubble Space Telescope + SN-Ia (dot-dashed), the same as previous + Baryonic acoustic oscillations (long dashed), the same as previous + Lyα (solid); from [4].
2. – Phenomenology of neutrino mass Phenomenology of neutrino masses (their absolute values) includes: 1) Kinematic features in processes with neutrino emission (kinks and shift of the end points in β-decay energy spectra, peaks in the energy distributions of the accompanying charged leptons in 2-body decays (e.g., π-decay). 2) Neutrino decays. 3) Double-beta decays. 4) Helicity flip effects. 5) Dispersion of neutrino signals from supernova and γ-bursters. 6) Neutrino interactions with radioactive nuclei (zero-threshold effects). 7) Neutrino pair emission from metastable atoms. 6) Spectrum of the Z 0 -bursts due to annihilation of high-energy cosmic neutrinos on relic neutrinos. The phenomenology is related to the large-scale structure of the Universe and leptogenesis. What else? What are other possible manifestations of neutrino mass? Which effects are sensitive to neutrino mass? In what follows I will comment on several mentioned phenomenological consequences. . 2 1. Three observables. – There are three parameters that depend on the absolute scale of masses and not on mass differences, as oscillations. These parameters (combinations of masses and mixings) show up in cosmology, beta decays and double-beta decays. 1) Cosmology. The bounds on the sum of neutrino masses, (2)
Σ≡
mi ,
i
follow from the analysis of various sets of cosmological data (see fig. 2 from [4]). The
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1 disfavoured by 0ν2β
Δm 223 < 0 10-2 Δm 223 > 0 10-3
99% CL (1 dof) 10-4 10-4 10-3 10-2 10-1 lightest neutrino mass in eV
disfavoured by cosmology
| m ee | in eV
10-1
1
Fig. 3. – The 90% CL range for mee as a function of the lightest neutrino mass for the normal (Δm223 > 0) and inverted (Δm223 < 0) mass hierarchies. The darker regions show the allowed range for the present best-fit values of the parameters with negligible errors; from [5].
bounds can be relaxed, e.g., by changing the equation of state of dark matter (the parameter ω). The existence of cosmic stings also weakens the bounds, etc. 2) Neutrinoless double-beta decay: Figure 3 (from [5]) shows the typical dependence of the effective mass of the electron neutrino 2 (3) mee ≡ mi Uei i
on the mass of the lightest neutrino. 3) The effective mass measured in beta decay equals (4)
me =
:
m2i |Uei |2 ,
i
according to [6], or me = i mi |Uei |2 according to [7]. In the case of quasi-degenerate spectrum, m1 ≈ m2 ≈ m3 = m0 , which is in the range of present day sensitivity, both definitions give the same result me = m0 . The difference matters for a non-degenerate spectrum if the energy resolution of the experiment becomes comparable with the mass difference. Consider the interplay of the observables me , Σ and mee . me versus mee : their relationship can be influenced by the nature of neutrinos. Apparently, there is no connection in the case of Dirac neutrinos. Even if neutrinos are Majorana particles, contributions to mee unrelated to light neutrino masses can exist. Also in this case the relation is essentially absent.
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me versus Σ: factors which can influence their relationship are i) non-standard neutrino properties (e.g., fast decay); ii) degeneracy of cosmological parameters; iii) nonstandard cosmology, e.g., the existence of cosmic strings. mee versus Σ: all the factors listed in the two previous cases are relevant. . 2 2. Mass-dependent processes. – The radiative decay (5)
ν2 → ν1 + γ
has a decay rate (6)
Γ ∝ m52
or
Γ ∝ m32 ,
depending on the model. If the neutrino spectrum is not strongly hierarchical, the rate is proportional to the mass difference again, but the combination is different from what we have in oscillations: Γ ∝ [(m22 − m21 )/m1 ]3 (m22 + m21 ). There are strong astrophysical bounds on this rate. The rate of non-radiative decay (7)
ν2 → ν1 + ν1 + ν¯1
equals Γ ∝ m52 . For the majoron decay, (8)
ν2 → ν1 + φ,
the rate is given by Γ = h2 m2 /16π. The charged current interactions with radioactive nuclei (9)
ν + 3 He → e− + 3 He
have zero threshold and the cross-section is proportional to the neutrino mass, σ ∝ mν [8]. Neutrino (pair) emission from metastable atoms looks intriguing in view of the closeness of the mass scales of atomic transitions and neutrino mass [9]. Here one can expect a strong enhancement of the processes: superradiance due to coherence in large volume. The processes of photon (laser) irradiated neutrino pair emission from metastable atoms, γ + Ai → νi νj + Af , and radiative pair emission, Ai → νi νj + γ + Af , have been considered [9]. Their rates are proportional to neutrino masses: Γ ∝ mi mj . . 2 3. Astrophysics and neutrino mass. – Z 0 burst: the annihilation of cosmic neutrinos on the relic neutrinos (10)
ν + ν¯ → Z 0 → hadrons
has a resonance character [10]. Since the relic neutrinos (at least two of them) are nonrelativistic, the energy of the cosmic neutrino should be E ≈ m2Z /2mi , and this same
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energy is then released as the energy of the burst of produced particles. So, the energy spectrum of the Z 0 -bursts should have three peaks which correspond to three neutrino masses. Unfortunately, the required energies are above 1021 eV, and the existence of significant fluxes of neutrinos with such an energies is very doubtful, especially after establishing the GZK cutoff. A possibility to get information on neutrino mass from the detection of supernova neutrinos has been proposed long time ago by Zatsepin. The time delay of supernova neutrinos (with respect to massless particles) equals (11)
Δt = 5.1 ms
L 10 kpc
10 MeV E
2
m 2 . 1 eV
Possible effects of this delay are: – time delay with respect to some benchmarks, e.g., neutronization peak, the lightest neutrinos arrival, gravitational waves arrival; – increase of the length of the burst; – smoothing fine structures of the burst (neutronization peak, initial steep rise, abrupt interruption of signal in the case of collapse into black hole); – energy ordering; – spread of the wave packets. The problem is that for the Galactic supernova Δt is too small (smaller than all expected structures of the burst); for SN burst from other galaxies one expects too small a number of events. A rather elaborated method of analysis of the data has been proposed in [11]: a) use the high-energy part of the spectrum to reconstruct the time dependence of flux, then b) analyze the whole spectrum using the reconstructed time dependence. The conclusion is that even with high statistics and sophisticated methods, it is difficult to test masses below 1 eV. Detection of neutrino signals from γ bursters may, in principle, give stronger bounds [12]. It is not clear if any of these astrophysical effects can be used to measure the absolute scale of neutrino mass. 3. – Analysing results . 3 1. Three conclusions. – There are three important conclusions which can be drawn from the existing results. 1) Lower bound on the mass of the heaviest neutrino. Apparently mh ≥
(12)
Δm232 .
Using recent MINOS [13] data (13)
Δm232 = (2.43 ± 0.13) × 10−3 eV2 ,
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we obtain from (12) mh ≥ 0.049 eV.
(14)
Cosmology gives mh < 0.2 eV. 2) Bound on mass hierarchy. The lower bound on the ratio of neutrino masses equals : m2 ≥ m3
(15)
Δm221 . Δm232
Taking Δm221 = (7.66 ± 0.35) × 10−5 eV2 and Δm232 from (13), we obtain from this equation rν ≡
(16)
m2 ≥ 0.179. m3
Comparing with the corresponding ratios of masses of quarks and leptons, we conclude that neutrinos have the weakest mass hierarchy (if any) among fermions. This may be related to the large lepton mixing. Notice that r(up quarks) < rν3 . 3) Bound on the type of mass spectrum. In terms of m2 (the mass of tri-maximal mixed state, see fig. 1) other neutrino masses can be written as (17)
m1 =
m22 − Δm221 ,
m3 =
Δm232 + m22 .
Using these relations and the known values of Δm2 , and taking criteria of the mass degeneracy, Δm/m < 0.1, we conclude that in the case of normal mass hierarchy the spectrum can be – completely (quasi-) degenerate for m2 > 0.1 eV; – partially degenerate for 0.07 < m2 < 0.1 eV; – hierarchical m2 < 0.05 eV. In the case of inverted mass ordering, the spectrum can be either completely degenerate or partially degenerate. It seems that a possibility of non-degenerate and nonhierarchical spectrum with Δmij ∼ mk is not realized. The spectrum with “anarchy” is probably excluded. Each of these spectra has different theoretical implications. . 3 2. Heidelberg-Moscow result. – Special comments on the positive claim of the Heidelberg-Moscow experiment on 76 Ge → 76 Se + e− + e− . The latest (2006) analysis [14] has reinforced the evidence of the neutrinoless double-beta decay. The new rate (18)
25 years T1/2 = (2.23+0.44 −0.31 ) × 10
has more that 6σ deviation from zero. Being interpreted as due to exchange of light Majorana neutrino, one obtains, using new values of nuclear matrix elements, (19)
mee = 0.16–0.52 eV (2σ).
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Fig. 4. – The allowed region of parameters from cosmological and oscillations data confronted with the HM positive claim; from [4].
This, in turn, implies a strongly degenerate mass spectrum with Σ > 0.5 eV and a serious tension with the cosmological bound, see fig. 4 from [4]. A possible way out is to assume that some other mechanism (which differs from the light Majorana neutrino exchange) is responsible for neutrinoless double-beta decay. Alternatively one may “play” with cosmological uncertainties. . 3 3. Beyond the Standard Model . – Two facts are on the basis of the claim that the discovery of neutrino oscillations is the evidence of physics beyond the SM. 1) Smallness of neutrino mass within a given generation: For the third generation we have (20)
m3 = (0.3–1) · 10−10 , mτ
which should be compared with another mass ratio from the same family mτ /mt ∼ 10−2 . Another scale which is the closest one to the neutrino mass is the scale of Dark Energy in the Universe, 10−3 eV, and there is a number of speculations on how these scales can be connected. 2) Pattern of neutrino mixing with two large angles. 4. – Nature of neutrino mass Here the main issues are: i) Majorana versus Dirac, ii) hard versus soft, iii) effective versus fundamental. Are neutrino masses qualitatively the same as quark masses? In
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fact, the observed smallness of neutrino masses may indicate something different: new contributions to neutrino mass which cannot be seen in masses of other particles. The main line of thinking is that masses and mixing have a pure vacuum origin; they are generated at the electroweak, and probably, higher energy scales. These are “hard” masses. The VEV’s involved are large or, if small, induced by other large VEV’s, as in the case of type-II see-saw. In general (21)
mν = mhard + msoft (E, n),
where msoft (E, n) is the medium-dependent soft component which can be substantial for neutrinos and not for other particles. For instance, according to the MaVaN scenario [15], soft neutrino masses can be generated due to the exchange of very light scalar particles with mφ = 10−8 –10−6 eV between neutrinos and particles of the medium: νL + fR → νR + fL (f = e, u, d, ν). Introducing the corresponding Yukawa couplings λf and number density nf , we find the mass msoft = λν λf nf /mφ . Another soft contribution to neutrino mass may come, e.g., from unparticles. According to the unparticle physics scenario [16], the hidden sector (HS) of theory includes the gauge theory with fermions. The number of fermions in the HS is such that the effective gauge coupling g increases with the decrease of energy and at the energies below a certain scale ΛU approaches the infrared fixed point g → g ∗ . If g ∗ 1 fermions form composite (confined) states. This transition is similar, to some extent, to the transition from quarks to hadrons below ∼ ΛQCD . Particles of HS couple to the SM particles via exchange of messenger fields with mass M ΛU . At energies below M the interaction of SM particles with HS particles is described by the effective interactions 1 OSM OU V , Mk
(22)
where OSM and OU V are operators which depend on the SM and HS fields, respectively. In analogy with QCD, one can consider, e.g., that OSM is the leptonic operator, whereas OU V is the quark operator. Below ΛU the operator OU V transforms into the operator of composite (confined) states OU (e.g., “pion”): OU V → OU and the interaction (22) becomes (23)
C
ΛdUU V −dU OSM OU , Mk
where dU V and dU are dimensions of operators OU V and OU , respectively. The key difference from the hadron case is that here, due to scale invariance (no energy gap), the confined states have a continuous mass spectrum [17,18]. In the QCD case the spectrum is discrete. As a result, individual mass modes have an infinitesimal effect. Finite effects of production and exchange of unparticles appear as a consequence of the integration over the spectrum (integration over the mass) of composite states.
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As far as applications to neutrinos are concerned, several processes have been considered: the neutrino decays νi → νj + U [19, 20], scattering on electrons να e → νβ e via an unparticle exchange [20]. The exchange of unparticles influences the refraction: modifies matter potential (in the case of vector operators). In the case of scalar unparticles, the effective neutrino mass is generated. This, in turn, modifies the conversion probabilities in matter [21]. An important phenomenological and experimental problem is to put model-independent experimental limits on (or discover) msoft . The existence of extra spatial dimensions opens qualitatively new possibilities to generate small Dirac masses of neutrinos. The left and the right components of neutrinos can have different localizations in extra dimensions. The value of the Yukawa coupling in 4D is then proportional to the degree of overlap of the LH and RH component of the wave functions or the overlap factor κ. Then neutrino mass in 4D can be written as (24)
mD = λvEW ξ,
where λ ∼ 1 and vEW is the electroweak VEV. Due to the fact that the RH neutrinos have no SM interactions, their localization can be substantially different, which leads to strong suppression of masses. This mechanism can be called overlap suppression. Let us evaluate the overlap (suppression) factor in different scenarios with extra dimensions. 1) Large flat extra dimensions; ADD-scenario. The 3D spatial brane is embedded in (3 + δ)D bulk of extra dimensions. Extra dimensions have large radii Ri 1/MPl which allows one to reduce the fundamental scale of theory down to M ∗ ∼ 10–100 TeV. The left-handed neutrino is localized on the brane, whereas the right-handed component (being a singlet of the gauge group) propagates in the bulk. For one extra D with coordinate y the normalization condition gives a typical value of the wave function √ νR (y) ∼ 1/ R. The width of the brane is of the order d ∼ 1/M ∗ , therefore the overlap factor with the LH component which is localized on the brane equals (25)
1 ξ = d1/2 νR ∼ √ . M ∗R
For δ extra dimensions, we get for the overlap factor ξ = 1/ M ∗δ Vδ , where Vδ is the volume of extra dimensions. 2) Warped extra dimensions; Randall-Sundrum setup. Two branes, the visible and the “hidden”, are localized in different points of the extra dimension with non-factorizable metric. The wave function of the RH neutrino νR (φ) is centered on the hidden brane, whereas the LH one on the visible brane. Due to warped geometry, νR exponentially decreases from the hidden to the observable brane. The overlap factor is given by the value of νR on the visible brane (26)
ξ = ν R (vis) ∼ ν−1/2 ,
= e−krc π =
vEW . MPl
Measurements of neutrino mass. Concluding remarks
387
Here MPl is the Planck scale, rc is the radius of extra dimension, k ∼ MPl is the curvature parameter. In (26) ν ≡ m/k and m ∼ MPl is the Dirac mass in 5D. For ν = 1.1–1.6, we obtain the mass in the required range. 5. – Masses and mixing What is the connection between masses and mixing? Can we obtain certain predictions for values of masses using information about mixing? . 5 1. Test equalities. – On pure phenomenological ground, one can connect the effective mass mee with the oscillation parameters making certain assumptions on neutrino mass spectrum and the Majorana CP phases. The experimental confirmation of such equalities which we can call the test equalities with high enough precision would testify for a given scenario. Some examples follow. 2 1) Normal mass hierarchy, Ue3
0.04: (27)
mee = sin2 θ12
Δm221 .
2) Inverted mass hierarchy, opposite CP parities of ν1 and ν2 : (28)
mee = cos 2θ12
Δm231 ;
the same CP parities give in this case (29)
mee =
Δm231 .
3) Degenerate spectrum with the same CP parities would lead to (30)
mee = me
and opposite parities give (31)
mee = cos 2θ12 me .
If it turns out that one of these equalities is satisfied with high precision, it will be difficult the believe that this is accidental. . 5 2. Three lines in the bottom-up approach. – Are small neutrino masses related to a strange mixing pattern? What is behind this pattern? Symmetry? If so, what are the implications of this symmetry for mass spectrum? There are three lines of studies in the bottom-up approach with different implications for fundamental physics and with different connections between masses and mixing.
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1) Tri-bimaximal mixing (TBM). The tri-bimaximal mixing matrix is defined as [22] √ ⎞ 2 2 √0 √ 1 m ⎠ ≡ U23 U12 (θ12 ) = √ ⎝−1 √2 √3 , 6 3 1 − 2 ⎛
(32)
Utbm
m is the maximal (π/4) rotation in the 2-3 plane and sin2 θ12 = 1/3. It agrees where U23 within 1σ with the present experimental results. An immediate implication of this possibility is flavor symmetry. The majority of models proposed so far are based on the discrete symmetry group A4 [23]. Other possibilities include models based on the groups T , D4 , S3 , S4 , Δ(3n2 ). Extension of these symmetries to quarks is however, problematic, it requires a further complication of the models. TBM may indicate that quarks and leptons are fundamentally different. Mixing is not related (at least in a straightforward way) to masses. Relations between masses and mixing may appear if some bigger structure exists which includes the proposed discrete symmetries and also predicts masses. In this case one would expect some particular relations between masses. 2) Quark-Lepton Complementarity (QLC). QLC [24] is based on observations that q q l l θ12 + θ12 ≈ π/4 and θ23 + θ23 ≈ π/4. It is difficult to expect exact equalities but a certain correlation exists: the 1-2 leptonic mixing deviates from maximal substantially because the quark 1-2 mixing (Cabibbo angle) is not small, the 2-3 leptonic mixing is close to maximal because the 2-3 quark mixing (Vcb ) is small. A general scheme for QLC: “lepton mixing = bi-maximal mixing − CKM”, where the bi-maximal mixing matrix, Ubm , is defined as √ ⎛√ ⎞ 2 2 √0 1 m m (33) Ubm ≡ U23 U12 = ⎝−1 1 √2⎠ . 2 2 1 −1
Two extreme realizations of the complementarity, QLCν and QLCl , are determined by the order of the bi-maximal and CKM rotations: (34)
† UPMNS = Ubm UCKM (QLCl ),
† UPMNS = UCKM Ubm (QLCν ).
Implications: – Quark-lepton symmetry or unification (apparently leptons should know about quark mixing). Alternatively, the information about quark mixing can be communicated to the lepton sector via the horizontal (flavor) symmetry. – Existence of a structure which produces the bi-maximal mixing. It can be the see-saw itself with certain properties of the RH neutrino mass matrix. The latter may require a certain symmetry. Again there is no straightforward connection of mixing to masses. 3) Quark-lepton universality. This approach does not rely on any specific symmetry in the lepton sector. The mass (Yukawa coupling) matrices of quarks and leptons have
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no fundamental distinction. The whole difference is related probably to the see-saw mechanism itself which explains simultaneously the smallness of neutrino mass and large lepton mixing. Mass matrices of quarks and leptons are constructed on the basis of the same principles, and furthermore, masses and mixing are related with each other. “The same principles” could be the Froggatt-Nielsen mechanism based on U (1)-flavor symmetry. Large lepton mixing can be related to weak mass hierarchy of neutrinos. The mass spectrum is expected to be hierarchical unless a screening mechanism is realized (see later). TBM and the two versions of QLC differ by predictions of the 1-2 and 1-3 mixing angles (θ12 , θ13 ): (35)
QLCν : (35.4◦ , 9◦ ),
TBM : (35.2◦ , 0),
QLCl : (32.2◦ , 1.5◦ ).
Notice that θ12 (QLCl ) = π/4 − θC and θ12 (QLCν ) ≈ θ12 (TBM). All three possibilities agree with the present data within 1σ. Clearly, the combination of future precise measurements of these angles will disentangle the schemes. At the same time the predictions can be changed due to the appearance of CP violation phases and by RGE effects. Furthermore, in specific models, some additional corrections appear due to violation of the underlying symmetry. Small deviations from the predictions do not exclude the context. Exact confirmation would be very demanding and restrictive. . 5 3. Absolute scale without measurements. – In the following situations we can conclude about the absolute scale without measurements of masses, e.g., by kinematic method (recall, the answer may come from an unexpected side): 1) Establish that the spectrum is hierarchical. 2) Put the upper bound on the lightest neutrino mass (studying, e.g., the dispersion of the burst from SN). 3) Confirm the scenario which predicts the hierarchical mass spectrum such as – “ νMSM [25]; – some GUT models. 4) Measure the rates of processes such as neutrino decay, further study of the LSS of the Universe. 5) Using LHC, uncover the electroweak-scale mechanisms of neutrino mass generation e.g., with Higgs triplet bosons, measure its couplings. It would be interesting to check whether positive results from these indirect studies will coincides with results of the direct measurements. 6. – Predicting neutrino mass Do we have at least some models or context which lead to the predictions of neutrino mass? . 6 1. Koide relations. – The Koide relations are pure mass relations [26]. (In fact, there is no reason to consider the tri-bimaximal mixing but ignore the Koide relations.
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A. Yu. Smirnov
Furthermore, they may be related.) Recall that the equality (36)
2 m e + mμ + mτ = √ √ √ ( me + mμ + mτ )2 3
is satisfied with an accuracy of 10−5 for mass shell and with 10−3 at Mz . All three families are involved substantially: no perturbation approach is possible. The relation (36) has been obtained in attempts to explain (37)
tan θC =
√
√ √ mμ − me 3 √ . √ √ 2 mτ − me − mμ
Both relations can be reproduced if (38)
2
mi = m0 (zi + z0 ) ,
zi = 0,
z0 =
i
:
zi2 /3.
i
Brannen [27] has generalized the relation to neutrinos: (39)
2 m1 + m2 + m3 = , √ √ 2 (− m1 + m2 + m3 ) 3 √
where the minus sign in front of the first term in the denominator is crucial. According to (39), neutrinos have a hierarchical spectrum with m1 = 3.9 · 10−4 eV. Non-Abelian flavor symmetry and specific VEV alignment can be behind the relations. . 6 2. Main line. – The dominant line of thinking is that 1) The smallness of the neutrino mass is related to the Majorana nature of neutrinos and eventually to zero values of the conserved charges (electric and QCD). 2) The see-saw scenario is realized: “small = (normal)2 /Large”, where small = masses of usual neutrinos; normal = the electroweak scale 100 GeV; large = masses of “right neutrinos”. 3) The same mechanism explains the large lepton mixing. The large-scale Λ is the scale of new physics. Different realizations of the see-saw depend on which is the carrier of this new scale (physics) and on the size of the scale. In the simplest version Λ is just the bare mass of the RH neutrino. In general, there is some particle sector and dynamics behind. Various realizations have been proposed with Λ √ equal to MPl (which requires many RH neutrinos), or MGUT , or MPl MEW , or MEW . In the νMSM scenario [25] the mass Λss ∼ 0.1–0.5 GeV. Even an extreme possibility, Λ = few eV, is not excluded [28]. All this means that physics behind the neutrino mass is not yet identified. The scale of the RH neutrinos is in favor of GUT. In fact, the value of the mass of the heaviest RH neutrino can coincide with the GUT scale MR ≈ MGUT ∼ 1016 GeV, which can be achieved in the presence of mixing of three generations. Alternatively,
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391
the scale of RH neutrino masses can be related to MGUT via the Planck scale MPl : 2 MR ≈ MGUT /MPl ∼ 1014 GeV. (The latter is realized, e.g., in the double-see-saw scenario [29].) 7. – Implications of the neutrino mass measurements Why measurements of mν are important? There are three reasons: 1) Knowledge of mν opens new possibilities of the determination of other neutrino parameters, e.g. the type of the neutrino mass spectrum. 2) It is important for implications for fundamental physics: – mechanism of neutrino mass generation, explanation of its smallness; – identification of the underlying physics; – origin and nature of neutrino mass; – flavor symmetry; – difference of mixing patterns of the quarks and leptons. 3) It provides a new way to search for new physics: A comparison of the values of neutrino masses determined in different processes may reveal something new. . 7 1. Type of spectra and implications. – Different types of mass spectra may have different implications for fundamental theory. Some examples follow: A) Normal mass ordering: – degenerate spectrum (m2 > 0.1 eV), m1 ≈ m2 ≈ m3 , would imply a non-Abelian flavor symmetry; – partially degenerate spectrum, m2 = 0.02–0.06 eV, m1 ≈ m2 ∼ m3 , testifies for pseudo-Dirac neutrino and weakly broken U (1); – hierarchical spectrum m2 < 0.01 eV, m1 m2 m3 , may indicate on U (1) symmetry, in the Froggatt-Nielsen scenario. B) In the case of inverted ordering: – degenerate spectrum m2 > 0.1 eV, m1 ≈ m2 ≈ m3 , would imply a non-Abelian flavor symmetry; – partially degenerate spectrum m2 = 0.02–0.06 eV, m1 ≈ m2 > m3 , again, pseudoDirac neutrino and weakly broken U (1); – partially hierarchical (hierarchical-degenerate) spectrum m2 < 0.01 eV, m2 ≈ m1 m3 , would testify for U (1) (Le − Lμ − Lτ ), or non-Abelian symmetries. . 7 2. Quasi-degenerate spectrum. – This spectrum is of special interest: the HeidelbergMoscow result may testify for that. It is in the range of sensitivity of the forthcoming experiments. This spectrum is the most interesting and unusual from the point of view of what we know now. Being identified this possibility would imply different sources of neutrino masses, unrelated to the Dirac mass matrices of the charged leptons and quarks. There are two possible realizations: 1) Higgs triplet, type-II see-saw;
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A. Yu. Smirnov
2) Screening of Dirac structure in the double see-saw. Here three additional singlets, S, which belong to families, couple to the RH neutrinos. In the basis (ν, ν c , S) the mass matrix has the form ⎛ (40)
0 ⎝mTD 0
mD 0 MT
⎞ 0 M ⎠, MS
due to certain symmetries including the lepton number one. It leads to the light neutrino masses (41)
m = −mTD (M −1 )T MS M −1 mD .
As a consequence of some horizontal symmetry or Grand unification, the two Dirac mass matrices can be proportional to each other: (42)
MD = A−1 mD ,
A ≡ vEW /VGU .
They cancel each other in (41) and for the light neutrinos we obtain (43)
mν = A2 MS .
That is, the structure of the light neutrino mass matrix is determined by MS immediately and does not depend on the Dirac mass matrix. If MS = I, the light neutrinos will be degenerate. Corrections to this scheme can lead to the desirable mass splitting. . 7 3. Applications. – The role of mν measurements in a more general framework can be outlined in the following way. 1) In the Standard neutrino scenario mν is the key element of the picture, it determines the type of mass spectrum. 2) Searches for physics beyond standard scenario: New neutrino states can lead, e.g., to additional kinks in the energy spectra of decays. Non-standard interactions and new dynamics can generate additional contributions to neutrino mass. Violation of the fundamental symmetries may show up first via neutrino masses (e.g., CPT violation may imply different mass spectra of neutrino and antineutrinos). 3) Understanding the fermion masses. Properties of neutrino mass matrix. 4) Unification of particles and forces. Let us underline two aspects. Cosmology: Knowledge of neutrino mass is important for the determination of other cosmological parameters. The mass determines the neutrino structure of the Universe which could include neutrino stars and clouds, neutrino halos of galaxies and clusters of Galaxies [30]. The existence of new neutrino interactions can lead to instabilities in the neutrino sea, neutrino condensation, superfluidity [31], etc. All this determines
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a local (near the Earth) concentration of neutrinos and therefore important for possible detection of relic neutrinos. Kinematics: as we have already mentioned, the neutrino mass enters the probabilities of the processes. However, they are too small and irrelevant for existing and probably future measurements. The hope is that they still can show up in some exotic cases. Neutrino mass determines the rates of rare processes like neutrino decay, zero-threshold scattering, etc. 8. – In conclusion of concluding remarks In conclusion, instead of a summary, I would like to ask several questions. What is the mass of the heaviest neutrino? What are the processes sensitive to neutrino mass? Is something missed? What is the type of neutrino mass spectrum: degenerate, partially degenerate, hierarchical? What is the nature of neutrino mass? Majorana versus Dirac; hard versus soft. What are the bounds on soft contributions? How light is the lightest neutrino? Is mixing related to mass? Can we establish the absolute mass scale without direct measurements of neutrino masses? Can we find some more test equalities or even create a catalog of these equalities? This question is for experimentalists: What are the fundamental limitations of the sensitivity of different experimental techniques: spectrometers, bolometric measurements [32]? ∗ ∗ ∗ I would like to thank F. Vissani for hospitality during my stay in Varenna. REFERENCES [1] For a comprehensive list of references, see the reviews: Strumia A. and Vissani F., arXiv:hep-ph/0606054; Gonzalez-Garcia M. C. and Maltoni M., Phys. Rep., 460 (2008) 1; Mohapatra R. N. and Smirnov A. Y., Annu. Rev. Nucl. Part. Sci., 56 (2006) 569; Altarelli G. and Feruglio F., New J. Phys., 6 (2004) 106; King S. F., Rep. Prog. Phys., 67 (2004) 107. [2] Osipowicz A. et al. (KATRIN Collaboration), arXiv:hep-ex/0109033. [3] Pontecorvo B., Zh. Eksp. Theor. Fiz., 33 (1957) 549; Zh. Eksp. Theor. Fiz. 34 (1958) 247. [4] Fogli G. L. et al., Phys. Rev. D, 78 (2008) 033010. [5] Strumia A. and Vissani F., in [1]. [6] Vissani F., hep-ph/0102235. [7] Farzan Y. and Smirnov A. Y., Phys. Lett. B, 557 (2003) 224. [8] Weinberg S., Phys. Rev., 128 (1962) 1457; Cocco A. G., Mangano G. and Messina M., J. Cosmol. Astroparticle Phys., 0706 (2007) 015 (J. Phys. Conf. Ser., 110 (2008) 082014) [arXiv:hep-ph/0703075]; Lazauskas R., Vogel P. and Volpe C., J. Phys. G, 35 (2008) 025001 [arXiv:0710.5312 [astro-ph]]. [9] Yoshimura M., Phys. Rev. D, 75 (2007) 113007; Yoshimura M. et al., arXiv:0805.1970.
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[10] Weiler T. J., Astropart. Phys., 11 (1999) 303. [11] Nardi E. and Zuluaga J. I., Nucl. Phys. B, 731 (2005) 140; Phys. Rev. D, 69 (2004) 103002. [12] Choubey S. and King S. F., Phys. Rev. D, 67 (2003) 073005. [13] Adamson P. et al. (MINOS Collaboration), arXiv:0806.2237 [hep-ex]. [14] Klapdor-Kleingrothaus H. V. and Krivosheina I. V., Mod. Phys. Lett. A, 21 (2006) 1547. [15] Hung P. Q., hep-ph/0010126; Peihong Gu, Xiulian Wang and Xinmin Zhang, Phys. Rev. D, 68 (2003) 087301; Fardon R., Nelson A. E. and Weiner N., J. Cosmol. Astroparticle Phys., 0410 (2004) 005; hep-ph/0507235; Kaplan D. B., Nelson A. E. and Weiner N., Phys. Rev. Lett., 93 (2004) 091801. [16] Georgi H., Phys. Rev. Lett., 98 (2007) 221601; Phys. Lett. B, 650 (2007) 275. [17] Krasnikov N. V., Phys. Lett. B, 325 (1994) 430; arXiv:0707.1419. [18] Stepanov M. A., Phys. Rev. D, 76 (2007) 035008. [19] Anchordoqui L. and Goldberg H., ArXiv:0709.0678 [hep-ph]. [20] Shun Zhou, ArXiv:0706.0302 [hep-ph]. [21] Gonzalez-Garcia M. C., de Holanda P. C. and Zukanovich-Funchal R., ArXiv:0803.1180 [hep-ph]. [22] Wolfenstein L., Phys. Rev. D, 18 (1978) 958; Harrison P. F., Perkins D. H. and Scott W. G., Phys. Lett. B, 458 (1999) 79; Phys. Lett. B, 530 (2002) 167. [23] Ma E., Mod. Phys. Lett. A, 17 (2002) 2361; Rajasekaran G., Phys. Rev. D, 64 (2001) 113012; Babu K. S., Ma E. and Valle J. W. F., Phys. Lett. B, 552 (2003) 207. [24] Smirnov A. Yu., hep-ph/0402264; Raidal M., Phys. Rev. Lett., 93 (2004) 161801; Minakata H. and Smirnov A. Yu., Phys. Rev. D, 70 (2004) 073009. [25] Asaka T., Blanchet S. and Shaposhnikov M., Phys. Lett. B, 631 (2005) 151; Asaka T. and Shaposhnikov M., Phys. Lett. B, 620 (2005) 17. [26] Koide Y., Lett. Nuovo Cimento, 34 (1982) 201; Phys. Rev. D, 28 (1983) 252. [27] Brannen C. A., http://brannenwork.com (2006). [28] de Gouvea A., Jenkins J. and Vasudevan N., Phys. Rev. D, 75 (2007) 013003. [29] Mohapatra R. N., Phys. Rev. Lett., 56 (1986) 561; Mohapatra R. N. and Valle J. W. F., Phys. Rev. D, 34 (1986) 1642. [30] Ringwald A. and Wong Y. Y. Y., J. Cosmol. Astroparticle Phys., 0412 (2004) 005 [arXiv:hep-ph/0408241]. [31] Kapusta J. I., Phys. Rev. Lett., 93 (2004) 251801; Bhatt J. R. and Sarkar U., arXiv:0805.2482. [32] Andreotti E. et al., Nucl. Instrum. Methods A, 572 (2007) 208.
POSTERS
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DOI 10.3254/978-1-60750-038-4-395
νe and ν¯e disappearance in Gallium and reactor experiments M. A. Acero Dipartimento di Fisica Teorica, Universit` a di Torino e INFN, Sezione di Torino - Torino, Italy Laboratoire d’Annecy-le-Vieux de Physique Th´ eorique LAPTH, Universit´ e de Savoie Annecy-le-vieux Cedex, France
C. Giunti INFN, Sezione di Torino - Torino, Italy
M. Laveder Dipartimento di Fisica “G. Galilei”, Universit` a di Padova, e INFN, Sezione di Padova Padova, Italy
Summary. — The disappearance of electron neutrinos observed in the Gallium radioactive source experiments is analyzed in the effective framework of two-neutrino mixing. We found an indication of neutrino disappearance due to neutrino oscillations with a square-mass difference much larger than those observed in solar and atmospheric neutrino experiments. We studied the compatibility of this result with the data of the Bugey and Chooz reactor short-baseline antineutrino disappearance experiments. We found an indication in favor of neutrino oscillations with 1.8 eV2 Δm2 1.9 eV2 , from the Bugey data, which is compatible with the Gallium allowed region of the mixing parameters. This indication persists in the combined analyses of Gallium, Bugey, and Chooz data.
1. – Introduction Solar, atmospheric, reactor and accelerator neutrino experiments give very robust evidence of three-neutrino mixing [1, 2]. However, data from LSND, MiniBooNE (at low energy) and the Gallium radioactive source experiments show some anomalies which open a window to the possible existence of exotic neutrino physics beyond three-neutrino mixing. The Gallium radioactive source experiments (GALLEX [3] and SAGE [4, 5]) measured a lower electron neutrino flux than the expected one. This can be interpreted as an indication of the disappearance of electron neutrinos due to neutrino oscillations [6]. c Societ` a Italiana di Fisica
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Fig. 1. – Left: allowed region in the oscillation parameter space for the individual Gallium radioactive source experiments. Right: allowed regions in the oscillation parameter space and marginal Δχ2 ’s for the combined fit of the results of the four radioactive source experiments.
2. – Gallium experiments The Gallium radioactive source experiments consist in the detection of electron neutrinos produced by artificial 51 Cr and 37 Ar radioactive sources which decay through electron capture e− + 51 Cr → 51 V + νe and e− + 37 Ar → 37 Cl + νe . The neutrinos are detected through the reaction νe + 71 Ga → 71 Ge + e− . We present the results of the fit of the data of Gallium radioactive source experiments in terms of effective two-neutrino oscillations. The survival probability of electron (anti)neutrinos with energy E at a distance L from the source is given by Δm2 L 2 2 (1) P(−) (−)(L, E) = 1 − sin 2ϑ sin , 4E νe →νe where ϑ is the mixing angle and Δm2 is the squared-mass difference (see refs. [1, 2]). We use the theoretical value of the ratio R of the predicted 71 Ge production rates in each of the Gallium radioactive source experiments in the cases of presence and absence of neutrino oscillations given by B dV L−2 i (B.R.)i σi Pνe →νe (L, Ei ) B (2) R= , −2 i (B.R.)i σi dV L where σi is the cross-section and (B.R.)i the branching ratio of the i-th νe line emitted in Cr and 37 Ar decays. The result of the individual and combined least-squares analysis of the four Gallium source experiments is shown in fig. 1. One can see that there is an allowed region in the sin2 2ϑ–Δm2 plane at 1σ for Δm2 0.6 eV2 and 0.08 sin2 2ϑ 0.4. 51
3. – Bugey and Chooz reactor experiments Reactor neutrino experiments detect antineutrinos through the reaction ν¯e + p → n + e+ . In this process, the neutrino energy is related to the positron energy by Eν = Ee+ +
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Table I. – Values of χ2min , number of degrees of freedom (NDF) and best-fit values of sin2 2ϑ and Δm2 from the fit of different combinations of the results of the Gallium radioactive source experiments and the Bugey and Chooz reactor experiments. χ2min NDF GoF sin2 2ϑbf Δm2bf (eV2 )
Ga
Bu
Ga + Bu
Bu + Ch
Ga + Ch
Ga + Bu + Ch
2.69 2 0.26 0.23 2.00
46.55 53 0.72 0.043 1.85
52.59 57 0.64 0.057 1.85
47.12 54 0.73 0.036 1.85
6.57 3 0.087 0.079 1.73
53.40 58 0.65 0.05 1.85
1.8 MeV. The Bugey experiment used three source-detector distances (Lj = 15, 40, 95 m for j = 1, 2, 3), while in Chooz, the distance was about 1 km. For the Bugey experiment we use the ratio of observed and expected (in the case of no oscillation) positron spectra given in fig. 17 of ref. [7], in which there are Nj = 25, 25, 10 energy bins (data). We analyze the data with the following χ2 : ⎧ ⎫ Nj exp 2 3 ⎨ 2⎬ 2 the (Aa + b (E − E )) R − R (a b2 (A − 1) − 1) j ji 0 ji j ji + + , + (3) χ2 = 2 2 ⎩ σji σa2j ⎭ σA σb2 j=1 i=1 where Eji is the central energy of the i-th bin in the positron kinetic energy spectrum measured at the Lj source-detector distance; the coefficients (Aaj + b(Eji − E0 )) are introduced to account for the systematic uncertainty of the positron energy calibration [7]; exp Rji is the measures ratio and B +∞ B E +ΔE /2 B dL L−2 Ejiji−ΔEjj/2 dE −∞ dTe F (E, Te ) Pν¯e →¯νe (L, Eν ) the B , (4) Rji = ΔEj dL L−2 here F (E, Te ) is the energy resolution function of the detector. The Chooz experiment gives constrains on sin2 2ϑ for Δm2 10−3 eV2 . Therefore, for our purpose, the Chooz experiment is only sensitive to the average survival probability P(−) (−) = 1 − 12 sin2 2ϑ. The allowed regions obtained from our fits are shown in fig. 2, νe →νe
while the best fit values of the mixing parameters are reported in table I. 4. – Conclusions In the framework of two-neutrino mixing, we found that, from the analysis of the Gallium radioactive source experiments, there is an indication of electron neutrino disappearance due to neutrino oscillations with sin2 2ϑ 0.03 and Δm2 0.1 eV2 . This result is compatible with the data form Bugey and Chooz reactor experiments. Besides, the Bugey data present an indication in favor of neutrino oscillations with 0.01 sin2 2ϑ 0.07 and 1.8 eV2 Δm2 1.9 eV2 . Such a disappearance of electron neutrinos due to Δm2 0.1 eV2 is an indication of the possible existence of at least one light sterile neutrino with a mass of the order of about 1 eV.
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Fig. 2. – Top: allowed region in the oscillation parameter space and histograms with the best fit the the ; dashed: Rji , obtained from for the Bugey reactor experiment (solid: (Aaj + b(Eji − E0 ))Rji see eq. (3)). Lower left: allowed regions in the oscillation parameter space from the combined fit of the Bugey and Chooz reactor experiments. Lower right: allowed regions from the combined fit of the Gallium radioactive source experiments and the Bugey and Chooz reactor experiments.
∗ ∗ ∗ We would like to express our gratitude to Y. Declais for giving us detailed information on the Bugey experiment. M.A.A. would like to thank the International Doctorate on AstroParticle Physics (IDAPP) for financial support. C.G. would like to thank the Department of Theoretical Physics of the University of Torino for hospitality and support. REFERENCES [1] Bilenky S. M. and Petcov S. T., Rev. Mod. Phys., 59 (1987) 671. [2] Giunti C. and Kim C. W., Fundamentals of Neutrino Physics and Astrophysics (Oxford University Press) 2007. [3] GALLEX, Hampel W. et al., Phys. Lett. B, 420 (1998) 114. [4] SAGE, Abdurashitov J. N. et al., Phys. Rev. C, 59 (1999) 2246, arXiv:hep-ph/9803418. [5] Abdurashitov J. N. et al., Astropart. Phys., 25 (2006) 349, arXiv:nucl-ex/0509031. [6] Acero M. A., Giunti C. and Laveder M., arXiv:0711.4222. [7] Bugey, Achkar B. et al., Nucl. Phys. B, 434 (1995) 503.
DOI 10.3254/978-1-60750-038-4-399
Feasibility study for a measurement of the QE νμ CC cross-section with the ArgoNeuT liquid-argon TPC M. Antonello on behalf of the ArgoNeuT Collaboration INFN, Laboratori Nazionali del Gran Sasso - Assergi (AQ), Italy
Summary. — The present paper reports a feasibility study for the measurement of the QE CC (Quasi Elastic Charged Current) cross-section for the νμ -nucleus interaction in liquid argon in the few GeV region, using the ArgoNeuT detector on the NuMI Beam and the MINOS near detector as muon catcher. The number of QE νμ CC events expected in 180 days is about 4700 and the relative statistical error between 1 and 4 GeV is at the level of 4%, below the beam systematics.
1. – Introduction Following the recent discovery of neutrino oscillation in the atmospheric neutrino sector and the subsequent confirmation by the long-baseline K2K and MINOS experiments, the next step in studying neutrino mass and mixing are: 1) determine the rate of νμ → νe oscillation in the “atmospheric” oscillation length, characterized by the mixing angle θ13 ; 2) determine the ordering of the three neutrino mass states, known as the “mass hierarchy”; 3) determine whether there is CP violation in the neutrino sector. Assuming typical values of θ13 5◦ , θ23 45◦ , Δm213 2.4 × 10−3 eV2 and a baseline of about 1000 km (typical of the current long-baseline experiments) the oscillation probability becomes maximum in the few GeV region. 2. – The cross-section problem At the few GeV energy scale, that long-baseline experiments are supposed to probe, the main uncertainties come from the poor knowledge of the neutrino-nucleus interacc Societ` a Italiana di Fisica
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tion cross-section. Measurements for ν cross-section determination are available from old data [1,2] with errors around 20–30% in the GeV region. Moreover, only the simplest exclusive channels have been considered. A series of dedicated experiments for cross-section measurements is considered as a high-priority goal for the next few years. A detector based on the Liquid Argon TPC (LAr TPC) technology, developed by the ICARUS Collaboration [3], is one of the best candidate for such a measurement [4], to be exposed to an intense neutrino beam with a well-controlled beam profile. It provides unique features in energy and direction measurements for muons (electrons) and protons (the final-state particles emerging from Quasi Elastic (QE) νμ (νe ) interactions) as well as for pions (emerging from ν-induced nucleon resonances decay). Clean particle identification properties are accomplished when the full containment of the particle tracks in LAr is guaranteed in relatively modest volumes. 3. – The ArgoNeuT detector at the NuMI beam ArgoNeuT [5] is a joint NSF/DOE R&D project at Fermilab (USA) to expose a small liquid-argon TPC (350 kg of LAr) on-axis to the NuMI (Neutrinos at the Main Injector) low-energy neutrino beam [6]. By taking measurements in the 0.1 to 10 GeV range, ArgoNeuT will produce the first ever data for low-energy neutrino interactions within a LAr TPC. ArgoNeuT will also serve as a stepping stone to larger detectors (such as MicroBooNE [7] and LAr5 [8]). The ArgoNeuT detector is presently under construction, with data taking expected to start in July 2008 and last several months. ArgoNeuT will sit upstream the MINOS Near Detector (ND) in the NuMI Tunnel (100 m underground). In fig. 1 the energy spectrum of the NuMI on-axis beam is shown. The flux is known with a 5–10% precision and the mean muon neutrino energy is about 3.7 GeV. The neutrino intensity is Φν = 3 × 1010 ν/cm2 for 1018 POT. 4. – The QE νμ CC cross-section measurement In the few GeV energy range the ν-nucleus interactions are dominated by the QE channel. A feasibility study for the QE νμ CC cross-section measurement is reported hereafter. As a first issue, the total number of νμ CC neutrino events per day (assuming 8 · 1017 POT/day) and the relative energy distribution, as a function of incident neutrino energy, have been calculated taking into account the actual active mass of the ArgoNeuT detector. The cross-sections (as a function of the incident neutrino energy) have been calculated independently for the three channels (QE, RES and DIS) from two independent codes/models (GENEVE [9] and LIPARI [2]) and convoluted with the onaxis NuMI beam spectrum. The expected number of νμ CC events (referred to events with the vertex contained in the LAr active volume) obtained using the LIPARI MC code are: 26.6 events/day in the QE channel, 21.8 events/day in the RES channel and 118.5 events/day in the DIS channel, for a total of 166.9 events/day. Using GENEVE MC code the number of QE νμ CC events is expected to be 25.2 events/day (in agreement with LIPARI code result).
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Fig. 1. – NuMI on-axis beam energy spectrum. Fig. 2. – Top: QE νμ CC events energy distribution (GENEVE Monte Carlo). Bottom: relative statistical error (bin size 0.5 GeV).
Assuming a run period of 180 days (i.e. the time needed to collect 1.4 × 1020 POT), the QE νμ CC events distribution (GENEVE MC) as a function of incident neutrino energy has been calculated (fig. 2, top). About 4700 QE νμ CC interactions are expected in the ArgoNeuT sensitive volume. Notice (fig. 2, bottom) that in the energy region between 1 and 4 GeV the relative statistical error is reduced to the level of 4–5%, below the NuMI Beam systematics. The QE cross-section, as a function of the incident neutrino energy, can be directly measured combining the experimental information from the final-state reconstructed kinematics with the neutrino beam profile. We implemented a GEANT4 MC simulation (particle level) of QE events generated (GENEVE MC) with the on-axis NuMI beam, including the geometry of the ArgoNeuT detector (see fig. 3) as designed in the CAD project. In fig. 4 the experimental QE cross-section distribution is shown assuming that all the collected events are somehow fully reconstructed. Due to the limited dimensions of the detector, only in a small fraction of the events the muon and proton tracks are fully contained in the LAr sensitive volume. In 54% of the QE events with the vertex in ArgoNeuT the proton is completely cointained in LAr, while 86% of the muons escape the detector. 47% of the QE events have the proton completely contained and the muon entering the immediately downstream MINOS ND. ArgoNeuT will profit from the MINOS energy reconstruction capability to recover these events. 5. – Conclusions One of the main uncertainties for the next generation of long-baseline neutrino oscillation experiments is given by the neutrino-nucleus interaction cross-section in the few GeV region, where the dominant reaction is the QE CC. The QE νμ cross-section is
M. Antonello on behalf of the ArgoNeuT Collaboration
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Fig. 3. – ArgoNeuT geometry imported in the GEANT 4 simulation as designed in the CAD project. The first Iron Plate of the MINOS ND is also reported. Fig. 4. – Experimental QE cross-section distribution, expected from MC simulation (histogram, GENEVE MC). Complete reconstruction of all the events and full detector efficiency are assumed. Error bars refer to the statistical error and full dots indicate the theoretical cross-section value in the corresponding bin.
presently known with 20–30% errors. The ArgoNeuT experiment will measure the QE νμ cross-section in LAr with a precision of 4–5%. The detector is now under construction at Fermilab and will be exposed on-axis to the NuMI beam. The expected number of QE νμ CC events in a 180 days run is about 4700. REFERENCES Bleve C. et al., Astroparticle Phys., 16 (2001) 145. Lipari P., Lusignoli M. and Sartogo F., Phys. Rev. Lett., 74 (1995) 4384. Amerio S. et al., Nucl. Instrum. Methods Phys. Res. A, 527 (2004) 329. Cavanna F. and Palamara O., Nucl. Phys. B Proc. Suppl., 112 (2002) 265. ArgoNeuT: Mini LArTPC Exposure to Fermilab’s NuMI Beam; http://t962.fnal.gov. Kopp S., arXiv:physics/0508001v1, 30 Jul 2005; Kopp S., arXiv:0709.2737v1, 18 Sep 2007. Fleming B. T. et al., A Proposal for a New Experiment Using the Booster and NuMI Neutrino Beamlines: MicroBooNE, 15 Oct 2007; http://www-microboone.fnal.gov. [8] Rameika R. et al., LAr5 - A Liquid Argon Neutrino Detector for Long Baseline Neutrino Physics, DRAFT Letter of Intent, 13 Mar 2008, http://www.fnal.gov/directorate/ program planning/Mar2008PACPublic/LAr5 LOI.pdf. [9] Cavanna F. and Palamara O., Nucl. Phys. B Proc. Suppl., 112 (2002) 183. See also ICARUS Proposal LNGS-94/99, Vol. I.
[1] [2] [3] [4] [5] [6] [7]
DOI 10.3254/978-1-60750-038-4-403
Search for ββ decay modes at LNGS by DAMA experiment F. Cappella on behalf of DAMA Collaboration Dipartimento di Fisica, Universit` a “La Sapienza” Roma and INFN, Sezione di Roma I - Italy
Summary. — In this paper a short summary of results achieved by the DAMA experiment in the investigation of double-beta decay modes in various isotopes is presented.
1. – Introduction The DAMA experiment (working at LNGS of the INFN) has investigated several double-beta decay modes in various isotopes using and developing low-background scintillators. In particular, the double-beta decay modes of 40 Ca [1], 46 Ca [1], 48 Ca [2], 64 Zn [3], 106 Cd [4], 108 Cd [5], 114 Cd [5], 130 Ba [6], 136 Ce [7], 138 Ce [7], 142 Ce [7], 134 Xe [8] and 136 Xe [8] have been studied exploiting either the active or the passive source technique. Such searches allowed to improve (sometimes of several orders of magnitude) the limits set by previous experiments or to set new limits on unexplored isotopes or decay modes. It is worth noting that DAMA has obtained the best experimental limit on double-beta plus decay modes, investigating the 22ν decay mode of 40 Ca [1], and one of the highest experimental limits for double-beta minus decay modes studying the 2β − 0ν of 136 Xe (T1/2 > 1.2 × 1024 y, 90% C.L. [8]). Figure 1 shows a summary of the T1/2 limits achieved by DAMA on ββ decay (red bars) and by previous experiments (blue bars). 2. – Search for ββ decay with DAMA/LXe set-up The DAMA/LXe set-up (6.5 kg, i.e., 2 litres of liquid-xenon pure scintillator), its radiopurity and its performances have been described in detail in ref. [9]; here, we briefly mention only some main points. The inner vessel, which contains the liquid xenon, is made of OFHC low radioactive copper. The scintillation light is collected by three EMI photomultipliers (PMTs) with MgF2 windows working in coincidence. The PMTs are housed in the insulation vacuum and are fully surrounded by low radioactive Cu. They collect the scintillation light through three windows (3 in diameter) made of special cultured crystal quartz (total transmission of the LXe ultraviolet scintillation light is c Societ` a Italiana di Fisica
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Fig. 1. – (Colour online) Summary of the T1/2 limits achieved by DAMA on ββ decay (red bars in colour or light gray in black and white printing) and by previous experiments (blue bars in colour or dark bars in black and white printing) on various double-beta decay processes and candidate isotopes.
≈ 80%). The passive shield is composed of 5–10 cm of low radioactive copper inside the vacuum insulation vessel, 2 cm of steel (wall of the insulation vacuum vessel), 5–10 cm of low radioactive copper, 5 cm of Polish lead and 10 cm of Boliden lead, 1 mm cadmium and ≥ 10 cm polyethylene. The whole set-up is enclosed in a sealed plexiglas box maintained in a high purity nitrogen atmosphere. The cryogenic and vacuum system is described in detail in ref. [9]. For each event the shape of the sum pulse is recorded by a LeCroy transient digitizer. The data collected over 8823.54 hours with 6.5 kg Kr-free xenon containing 17.1% of 134 Xe and 68.8% of 136 Xe (statistics of 1.1 kg × y for the 134 Xe and of 4.5 kg × y for the 136 Xe) have been analysed to investigate the 134 Xe and the 136 Xe ββ decay modes [8]. In particular, a joint analysis of the ββ0ν decay mode in 134 Xe and in 136 Xe has been carried out as suggested in ref. [10]; in principle, this kind of analysis could improve the information obtained when separately studying the two isotopes. In the case of the 134 Xe ββ0ν(0+ − 0+ ) decay mode the limit T1/2 > 5.8 × 1022 y at 90% C.L. is obtained, while in the case of the 136 Xe ββ0ν(0+ − 0+ ) decay mode the limit is: T1/2 > 1.2 × 1024 y at 90% C.L. The last one represents one of the highest experimental limits for doublebeta minus decay modes and allowed to set an upper limit on effective light Majorana neutrino mass, mν , ranging from 1.1 to 2.9 eV depending on the adopted model [8]. Other decay modes have been also studied (see ref. [8] and fig. 1). 3. – Search for ββ decay with DAMA/R&D set-up The DAMA/R&D installation is a low background set-up used for measurement on low-background prototype scintillators and PMTs, realized in various R&D efforts
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with industries. It is regularly used to perform small-scale experiments also in collaboration with INR Kiev. The used detectors in each experimental measurement are surrounded by low-radioactive Cu bricks and sealed in a low-radioactive air-tight Cu box continuously flushed with high-purity nitrogen gas to avoid presence of residual environmental radon. The Cu box is surrounded by a passive shield made of high-purity Cu, 10 cm of thickness, 15 cm of low radioactive lead, 1.5 mm of cadmium and 4 to 10 cm of polyethylene/paraffin to exclude the external background. The shield is contained inside a plexiglas box, also continuously flushed by high-purity nitrogen gas. In the following we briefly summarize some of the results obtained with DAMA/R&D set-up in the search for ββ decay processes. An experiment to study the ββ decay processes in 106 Cd has been performed using a low-background set-up with two large low-background NaI(Tl) crystals and cadmium samples (total mass 154 g), enriched in 106 Cd at 68% (passive source technique) [4]. The β + /EC and β + β + processes would manifest themselves by the simultaneous emission of two or four 511 keV γ rays. Therefore, the two low-background NaI(Tl) scintillators placed around the 106 Cd source in DAMA/R&D are well suitable to measure these γ-rays in coincidence. New limits on the half-lives for the different decay channels in 106 Cd have been obtained. They are in the range (0.3–4)×1020 y at 90% C.L. [4], which is significantly higher (by a factor 6 to 60) than those previously published for this nuclide. The development of highly radiopure CaF2 (Eu) crystal scintillators has been performed aiming at a substantial sensitivity enhancement of the 2β decay investigation of Ca isotopes. Two CaF2 (Eu) crystals (each 370 g mass) developed with Bicron company were used in ref. [1] to study ββ processes in 40 Ca and 46 Ca. The locations and amounts of the radioactive contaminations have been estimated and a background model has been used to estimate half-life limits for the double EC capture of 40 Ca and the ββ0ν decay of 46 Ca [1]. This experiment allowed to obtain the best experimental limit on doublebeta plus decay modes, investigating the 22ν decay mode of 40 Ca: T1/2 > 5.9 × 1021 y at 90% C.L. [1]. The performances of a CeF3 crystal scintillator with a mass of 49.3 g have been investigated in ref. [7]. In particular, the α/β light ratio, the possibility of a pulse-shape discrimination between α particles and γ quanta and the radioactive contamination of the crystal have been studied. The application of the obtained results in the search for the two-neutrino double-electron capture in 136 Ce and 138 Ce allowed to obtain new T1/2 limits for these processes (see ref. [7] and fig. 1). The potentiality of the coincidence technique to search for ββ decay processes in 130 Ba has been studied [6]. A BaF2 crystal scintillator with a mass of 3615 g and two low-background NaI(Tl) detectors have been used for this purpose. The performances of the BaF2 crystal scintillator have preliminarily been investigated, in particular: i) the α/β light ratio; ii) the pulse shape discrimination capability between α and β particles; iii) the main radioactive contaminations of the crystal. As a result, new limits (90% C.L.) on the lifetimes of double-beta decay processes in 130 Ba have been determined (see ref. [6] and fig. 1). The new experimental limits obtained for the 0ν(0+ → 0+ ) decay modes represent a significant improvement with respect to those previously available for this
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isotope, while those involving excited levels were set there for the first time. The potentiality of the β-γ coincidence technique has also been exploited in the search for the ordinary highly forbidden β decay, for the lepton number violating ββ0ν(0+ −2+ ) decay and for the ββ2ν(0+ − 2+ ) decay of 48 Ca [2]. For this purpose, a 1.11 kg CaF2(Eu) detector partially surrounded by low-background NaI(Tl) detectors has been used [2]. A low-background experiment to search for double-electron capture and electron capture with positron emission in 64 Zn was carried out over 1902 h by using a ZnWO4 scintillation detector with mass of 117 g [3]. New improved limits were set for different modes of double-beta decay of 64 Zn by analyzing the energy distribution measured with the ZnWO4 detector (see ref. [3] and fig. 1). In particular, the 2K2ν and β + 2ν processes were restricted at the level of T1/2 > 6.2 × 1018 y and T1/2 > 2.1 × 1020 y at 90% C.L., respectively. The positive indication on the β + (2ν + 0ν) decay of 64 Zn suggested in [11] is discarded by this experiment [3]. Finally, a search for ββ processes in 108 Cd and 114 Cd has been realized by using data of a low-background experiment with CdWO4 crystal scintillator (434 g mass) [5]. New improved half-life limits on double-beta processes were established (see ref. [5] and fig. 1). The obtained values are higher than those reached in the previous experiments and demonstrate the possibility of a scintillation experimental technique to search for double-beta processes in 108 Cd and 114 Cd. At present an experiment to further investigate ββ2ν decay of 100 Mo to the first 100 excited 0+ Ru is in progress using a 1 kg sample enriched in 100 Mo at 95.5% 1 level of and 4 low-background HPGe detectors [12]. Moreover, purification of enriched 106 Cd is in progress towards the construction of 106 CdWO4 low background scintillator [13]. Other small-scale experiments are in preparation in collaboration with INR Kiev and IIT Kharagpur. REFERENCES [1] Belli P. et al., Nucl. Phys. B, 563 (1999) 97; Bernabei R. et al., Astropart. Phys., 7 (1997) 73. [2] Bernabei R. et al., Nucl. Phys. A, 705 (2002) 29. [3] Belli P. et al., Phys. Lett. B, 658 (2008) 193. [4] Belli P. et al., Astropart. Phys., 10 (1999) 115. [5] Belli P. et al., Eur. Phys. J. A, 36 (2008) 167. [6] Cerulli R. et al., Nucl. Instrum. Methods Phys. Res. A, 525 (2004) 535. [7] Belli P. et al., Nucl. Instrum. Methods Phys. Res. A, 498 (2003) 352; Bernabei R. et al., Nuovo Cimento A, 110 (1997) 189. [8] Bernabei R. et al., Phys. Lett. B, 546 (2002) 23; ibidem, 527 (2002) 182. [9] Bernabei R. et al., Nucl. Instrum. Methods Phys. Res. A, 482 (2002) 728. [10] Simkovic F. et al., hep-ph/0204278. [11] Bikit I. et al., Appl. Radiat. Isot., 46 (1995) 455. [12] Belli P. et al., in Current Problems in Nuclear Physics and Atomic Energy (INR-Kiev) 2006, p. 479. [13] Belli P. et al., preprint ROM2F/2008/17, to be published in Proceedings of the II International Conference NPAE, June 2008, Kiev, Ukraine.
DOI 10.3254/978-1-60750-038-4-407
Development of nuclear emulsions for the OPERA experiment E. Carrara INFN, Sezione di Padova and Universit` a di Padova - Padova, Italy
Summary. — The ambitious aim of the OPERA experiment is to detect for the first time the appeareance of tauonic neutrinos out of an artificial beam of pure muonic neutrinos. This is achieved with the mature nuclear emulsion technique, greatly improved by the R&D of the OPERA collaboration both in quality and analisys speed. The huge number of emulsions (over 9 million sheets) of the OPERA experiment requires a completely automated development facility, able to perform photographic development process on up to 3000 emulsion sheets per working day. I will present the project of the development facility and report on the current status.
1. – The OPERA experiment OPERA (Oscillation Project with Emulsion-tRacking Apparatus) [1] is a long-baseline experiment designed primarily to make a conclusive test of the νμ → ντ oscillation hypothesis by means of the direct observation of ντ in an initially pure νμ beam. The beam is produced at CERN, in Geneva, and fired towards the OPERA detector, which is located in the Hall C of the Gran Sasso Underground Laboratory, 730 km away [2]. The average energy of the neutrinos is well above the τ lepton production threshold, in the oscillation parameter region indicated by atmospheric neutrino experiments. OPERA is a hybrid detector made of electronic subdetectors and lead/nuclear emulsion target. The c Societ` a Italiana di Fisica
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Table I. – The CNGS beam main characteristics. νμ (m−2 /pot) νμ CC events/pot/kton Eνμ (GeV) νe /νμ ν¯μ /νμ ν¯e /νμ
7.36 × 10−9 5.05 × 10−17 17 0.8% 4% 0.07%
target is composed of Emulsion Cloud Chambers (ECC), whose unit is called brick. The total nominal mass of the experiment exceeds 1.5 kton, which is required to reach the desired sensitivity in the parameters to be measured. When a neutrino hits the detector, the electronics show in which brick the event occurred. The brick is then removed from the experiment target, the emulsions are chemically developed and sent to the scanning laboratories for interaction vertex analysis. The scanning stations use high-performance automated microscopes for the identification of the tracks inside the emulsions. . 1 1. The CNGS neutrino beam. – The CNGS [2] neutrino beam was designed and optimized for the study of νμ → ντ oscillations in appearance mode, by maximizing the number of charged current (CC) ντ interactions at the LNGS site (see table I). The average neutrino energy at the LNGS location is 17 GeV. The ν¯μ contamination is 4%, the νe and ν¯e contaminations are lower than 1%, while the number of prompt ντ from Ds decay is negligible. The average L/Eν ratio is 43 km GeV−1 . Assuming a CNGS beam intensity of 4.5 × 1019 pot per year and a five year run, about 31000 charged current (CC) plus neutral current (NC) neutrino events will be collected by OPERA from interactions in the lead-emulsion target. Out of them 148 (214) CC ντ interactions are expected for Δm2 = 2.5×10−3 eV2 (3×10−3 eV2 ) and sin2 2θ23 = 1. Taking into account the overall τ detection efficiency, the experiment should gather 12–15 signal events with a background of less than one event. 2. – Emulsion cloud chambers The ντ appearance search is based on the observation of τ − events produced by CC interactions, with the τ − decaying in all possible decay modes: (1a)
τ − → e− ντ ν e ,
(1b)
τ − → μ− ντ ν μ ,
(1c)
τ − → h− ντ (nπ 0 ),
(1d )
τ − → 3πντ .
Given the τ lifetime of ∼ 3 × 1013 s, hence a typical path of ∼ 1 mm, and the small expected event rate, it is crucial to efficiently separate the ντ CC events from all the
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other flavor neutrino events, and to keep the background at a very low level. The basic structure of the emulsion detector is called Emulsion Cloud Chamber (ECC) and is made of ECC bricks. Each brick consists of 57 emulsion sheets; in order to have large mass (1.3 kton) and high granularity, the emulsion sheets are interleaved with 1 mm thick lead plate. The emulsions used to build ECC are 44 μm-thick layers placed at the upper and bottom surfaces of a 205 μm-thick plastic base. The dimensions of a brick are 12.5 × 10.2 × 7.5 cm3 , the weight is 8.3 kg each. In terms of radiations length, a brick corresponds to a thickness of 10X0 which is long enough to allow the identification of electrons by electromagnetic showers and to measure momentum by multiple Coulomb scattering following the tracks in consecutive emulsion sheets [4]. . 2 1. Emulsion development. – Emulsions suspected to contain neutrino interactions are developed with photographic technique. The so-called latent image impressed in the emulsion is made visible by the development process. In chemical development, silver ions are provided from the silver halide crystals containing the latent image center. If silver halide solvents, such as sulphite, are present in the chemical developer, some physical development may occur, in which silver ions are provided from the developer solution. A deeper understanding of the development processes is given by the quantitative methods of electrochemistry [5]. The developers generally used for nuclear emulsions are combined chemical and physical developers, sulphite and bromide are solvents of the silver halides. The conventional photographic processing steps are: Presoak; Development; Stop; Fixation; Wash; Dry. In the OPERA experiment, standard procedures need to be modified due to the great thickness of the emulsions and by the requirement of the development to be as uniform as possible and the resulting emulsion must be distortion-free [6]. 3. – Development facility The core of the development facility of the OPERA experiment are the development lines, 6 robots which perform the necessary steps for emulsion development. Their goal is to provide high and uniform quality films to the scanning laboratories worldwide. Up to 5 bricks are handled simultaneously by each machine, with a possible overall rate of over 3000 emulsions per day, making it a hardware and software challenge. The mechanical structure of the lines is made of aluminum rods, which are approximately 4.5 meters long, 80 cm wide and 1.2 m high, each one is equipped with 8 inox steel tanks, 5 of which have a capacity of 30 liters, the remaining 3 have a capacity of 45 liters. A mechanical arm, mounted on an aluminum bridge, can slide horizontally forth and back for the whole length of the line. The arm can slide vertically along the bridge, and is equipped with a jaw which allows to carry the film holders containing the emulsions and deposit them into the tanks. The machines are connected by cables to racks that contain the higher level logics of the machines and the power supplies. The racks offer
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an interface to the remote control computer through a collection of Field Point modules. The Field Points are then connected with an ethernet cable to the control PC, which runs the scheduling program, that automatically executes the full development procedure, without the operator intervention. The control software relies on middle-ware component, provided by National Instruments, which is a OPC (OLE for Process Control, where OLE stands for Object-Linking and Embedding) server, to communicate seamlessly with the Field Points. The program is internally structured in different layers, which are independent of each other. The lowlevel library is a collection of classes(1 ) written in C# that provides a direct interface to the available hardware. The high-level layer of the software takes care of the scheduling of processes to be performed. It adapts at the current status of the development line, reschedules tasks in case of clashes and manages different priorities of the operations. Currently the control software is able to handle an arbitrary number of processes simultaneously on any number of development lines, and operate in the so-called set & forget mode, where the operator feeds the machine with a bunch of emulsions, and hours later retrieves the developed films. In conclusion the development facility of the OPERA experiment is able to provide high-quality emulsions at a sustained rate that can cope with the OPERA expected extraction rate. The emulsions are then sent to the scanning laboratories where the neutrino interactions are reconstructed. The discovery of the ντ appearance ultimately and doubtlessly will prove the neutrino oscillation hypothesis suggested by several experiments in the last decades. ∗ ∗ ∗ I deeply thank professors L. Stanco and R. Brugnera for their priceless support and guidance. This work is dedicated to the dear memory of my grandparents Regina and Antonio. REFERENCES [1] OPERA Collaboration, Guller M. et al., OPERA experiment proposal, CERN-SPSC2000-028. [2] Acquistapace G. et al., The CERN neutrino beam to Gran Sasso (CNGS), Conceptual Technical Design (CERN 98-02 and INFN/AE-98/05) 1998. [3] OPERA Collaboration, Guller M. et al., CNGS project CERN-SPSC-2001-025. [4] Arrabito L. et al., Nucl. Instrum. Methods Phys. Res. A, 568 (2006) 578. [5] Barkas W. H., Nuclear Research Emulsion (Academic Press, New York and London) 1963. [6] Sirignano C., R&D on OPERA ECC: studies on emulsion handling and event reconstruction techniques, PhD Thesis (Salerno University) 2005.
(1 ) In object-oriented programming, a class is a language construct used to group related fields and methods.
DOI 10.3254/978-1-60750-038-4-411
First events from the OPERA detector at Gran Sasso D. Di Ferdinando and G. Sirri INFN, Sezione di Bologna - Bologna Italy
Summary. — OPERA is a long-baseline experiment designed to be the conclusive proof of the νμ → ντ oscillation hypothesis by means of the direct observation of ντ in an initially pure νμ beam. The detector is located at the underground Gran Sasso laboratory, 730 km from CERN, on the CNGS neutrino beam. It consists of a lead/emulsion film target complemented by magnetic spectrometers and electronic detectors. We give a report on the OPERA detector and we show the first neutrino events detected in the emulsions.
1. – The OPERA experiment In the last decades, several disappearance experiments [1-5] using neutrinos coming from cosmic rays and then by accelerators have shown a deficit in the measured νμ neutrino flux with respect to the predicted one. This behavior was explained by the neutrino oscillation hypothesis which requires neutrino mixing and mass. The Oscillation Project with Emulsion tRacking Apparatus (OPERA) was built in hall C of the Gran Sasso National Laboratory (LNGS) to prove without any doubt the neutrino oscillation by the appearance of ντ in a pure νμ beam (CNGS) coming from CERN . The average neutrino energy is about 17 GeV. This optimizes the number of ντ CC interactions in the detector for values of Δm2 ∼ 2.5 × 10−3 eV2 indicated by the atmospheric neutrinos experiments. The prompt ντ contamination is negligible; the ν¯μ contamination is ∼ 4%; the total νe and ν¯e contamination is below 1%. The commissioning of the CNGS beam started in July 2006 and was completed at low intensity c Societ` a Italiana di Fisica
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Fig. 1. – Schematic view of the OPERA detector. The neutrino beam arrives from the left side.
on August 2006, when the first 319 neutrinos interactions were detected by OPERA at LNGS [6]. The first physics run with real bricks (October 2007) allowed to test the whole strategy of the experiment, from the electronic detector triggers to the neutrino vertex interactions fully reconstructed in the emulsions. 2. – The detector The OPERA detector is 10 m × 10 m × 20 m and with a target mass of 1.35 kton. To satisfy the requirements of high tracking capabilities (the short τ decay length is O(1 mm)) and large target mass (small ν interaction cross-section) the technique of the Emulsion Cloud Chamber (ECC) is used. The basic ECC element of the OPERA detector is the brick composed of a sequence of 57 nuclear emulsion films interleaved with 56 1 mm-thick lead plates packaged in a light tight and high mechanical precision way. Each emulsion, that acts as sub-micrometric resolution tracker, is composed of two 44 μm sensitive layers deposited onto a 205 μm plastic base and has an area of 12.7 × 10.2 cm2 . The thickness of the whole brick is about 7.5 cm, corresponding to ∼ 10 radiation lengths. The 150000 bricks of the OPERA detector are arranged in two identical supermodules, each one composed of a target section and a muon spectrometer fig. 1. The target section is composed of vertical structures with transverse size ∼ 6.6×6.7 m2 (walls). Each wall is a matrix 52 × 51 bricks, organized in horizontal trays; the bricks are moved by the Brick Manipulator System (BMS) for insertion and extraction. The 29 brick walls are interleaved with planes of plastic scintillator strips (690 cm × 2.6 cm × 1 cm) and readout by WaveLength-Shifting fibers (WLS) and multi-anode 64-pixel PMTs at both ends. The Target Tracker (TT) provides the trigger and identifies the brick where the event vertex should be found. The detection efficiency of each plane is ∼ 99%. More information about the TT is provided in [7]. The spectrometer is an instrumented dipolar magnet (∼ 8.75 × 8 m2 ) made of two magnetized iron walls producing a field of 1.52 T in the tracking region with vertical lines of opposite directions in the two walls. The 12 iron slabs of magnet walls are interleaved with planes of bakelite RPCs (2.9 × 1.1 m2 ) that provide the range of stopping charge particles and crude tracking information inside
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Fig. 2. – Left: detector display of the first event registered in the OPERA target. Right: reconstruction of the neutrino interaction inside the triggered ECC brick.
the magnet [8, 9]. Six planes of precision drift tubes with 38 mm diameter and 8 m length, placed in pairs in front, behind and between magnet walls measure the charge and momentum of the muons [10]. Planes of glass RPCs are placed in front of the apparatus to tag the interactions occurring in the upstream rock [11]. 3. – Strategy for ν event location The CNGS beam provides 2 fast extractions of 10.5 μs duration separated by 50 ms per cycle and two cycles per super-cycle of the SPS. Two GPS, one located at CERN and the other at LNGS, allow the time correlation between the beam and the events recorded in the detector. The TT data provide the trigger and the event reconstruction; this in turn fixes the bricks with the highest probability to host the event; these bricks are promptly removed from the target for further analysis. Before disassembling the extracted brick, the presence of the event is validated by matching the tracks reconstructed in the TT with those located in a doublet of emulsion films, called Changeable Sheets (CS) [12], attached on the downstream surface of each brick. The validated brick is exposed to cosmic rays, to have a precise reference frame for film-to-film alignment [13]. The brick is then dismounted, in an automated developing chain and sent to the scanning laboratories. By using custom-made microscopes [14-16], the tracks measured in the CS are connected to the brick and followed back film by film up to the interaction point. A volume scan around this point is used for full event reconstruction, decay topology validation and kinematic analysis (using momentum measurement by multiple Coulomb scattering [17], electron-pion identification and energy reconstruction [18, 19]). 4. – First physics run The first OPERA physics run was held in October 2007. At that time about 40% of the target was installed for a total mass of 550 tons. Due to failures of ventilation control units of the proton target, caused by high radiation level in the area were electronic was installed, the beam was available only for few days with a global intensity of 0.79 ×
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1018 pot. In a few hours the first neutrino interactions were successfully located and reconstructed. 38 events (29 CC and 9 NC) were registered in the OPERA target to be compared with (32 ± 6) expected. Figure 2 shows the first located event, as detected by the elctronic (left) and as reconstructed in emulsions (right). Despite of its short duration the run allowed a successful testing of the electronic detectors, data acquisition and brick finding algorithms, proved the ability of the matching between the target tracker and the bricks and validated the full scanning strategy. 5. – Conclusions The first neutrino interactions in the OPERA target were recorded in the 2007 short run and the strategy of the experiment was successfully tested. The target is now complete and the beam is expected to reach the detector during this summer. The expected number of interactions in the OPERA target in 2008 is about 2200. ∗ ∗ ∗ To the memory of Romolo Diotallevi who left us while we were attending this School. His technical competence in emulsion development and overall sincere humanity represent an unbridgeable loss for all who had the luck and joy to share moments with him. REFERENCES [1] Ambrosio M. et al., Phys. Lett. B, 434 (1998) 451; 566 (2003) 35; Eur. Phys J. C, 36 (2004) 323. [2] Allison W. W. M. et al., Phys. Lett. B, 449 (1999) 137. [3] Fukuda J. et al., Phys. Rev. Lett., 81 (1998) 1562; Hosaka J. et al., Phys. Rev. D, 74 (2006) 032002; Abe K. et al., Phys. Rev. Lett., 97 (2006) 171801. [4] Ahn M. H. et al., Phys. Rev. D, 74 (2006) 072003. [5] Michael D. G. et al., Phys. Rev. Lett., 97 (2006) 191801. [6] Acquafredda R. et al., New. J. Phys., 8 (2006) 303. [7] Adam T. et al., Nucl. J. Phys., 8 (2006) 303. [8] Bergnoli A. et al., Nucl. Phys. B (Proc. Suppl.), 158 (2006) 35. [9] Adinolfi Falcone R. et al., Nucl. Phys. Conf. Suppl., 172 (2007) 165. [10] Zimmermann R. et al., Nucl. Instrum. Methods Phys. Res. A, 555 (2005) 435; 557 (2006) 690. [11] Candela A. et al., Nucl. Instrum. Methods Phys. Res. A, 581 (2007) 206. [12] Anokhina A. et al., JINST, 3 (2008) P07002; P07005. [13] Barbuto E. et al., Nucl. Instrum. Methods Phys. Res. A, 525 (2004) 485. [14] De Serio M. et al., Nucl. Instrum. Methods Phys. Res. A, 554 (2005) 247. [15] Armenise N. et al., Nucl. Instrum. Methods Phys. Res. A, 551 (2005) 261. [16] Nakano T., Automated Emulsion Read-out System, in III International Workshop on Nuclear Emulsion Techniques, 24-25 January 2008, Nagoya, Japan. [17] De Serio M. et al., Nucl. Instrum. Methods Phys. Res. A, 512 (2003) 539. [18] Kodama K. et al., Rev. Sci. Instrum., 74 (2003) 53. [19] Arrabito L. et al., JINST, 2 (2007) P02001.
DOI 10.3254/978-1-60750-038-4-415
Automated scanning of OPERA emulsion films D. Di Ferdinando and G. Sirri INFN, Sezione di Bologna - Bologna, Italy
Summary. — Large neutrino experiments with an accuracy of better than 1 micron are possible thanks to the recent improvements in the nuclear emulsion detectors. The European Scanning System (ESS) is a fast automatic system developed for the mass scanning of the emulsions of the OPERA experiment. Improvements in the automatic scanning technique and performance of ESS are reported.
1. – Introduction Nuclear emulsion is the detection technique with the best known position resolution (< 1 μm). This technique is connected to many early particle physics discoveries despite the huge scanning effort and the consequent limited mass of the experiments. The development of fast automated scanning systems have made possible the use of nuclear emulsion in large scale experiments like OPERA at the INFN Gran Sasso Underground Laboratories (LNGS). OPERA is a long-baseline experiment [1-3] designed to search for νμ → ντ oscillations as suggested by atmospheric neutrino experiments [4-6]. The aim is to observe the appearance of ντ through the identification of the τ leptons created in their CC interactions in a pure νμ beam produced by the CNGS facility at CERN [7]. This requires track position and angular measurements with accuracies of ∼ 1 μm and a few milliradiants, respectively. c Societ` a Italiana di Fisica
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Fig. 1. – Exploded view of the OPERA lead/emulsion target.
2. – Nuclear emulsions. OPERA target Nuclear emulsions are composed of microcrystals of silver halides (AgBr) dispersed in a gelatin layer. The energy released by ionizing particles to the crystals produces the so-called latent image. After a chemical development process the particle trajectory is visible as a sequence of black silver grains about 0.5 μm in size [8]. Nuclear emulsions are ideal for the detection of short-lived particles (like τ decay); they provide i) threedimensional spatial information, ii) excellent resolution (< 1 μm, < 5 mrad), iii) high hit density (∼ 300 hits/mm for a minimun ionizing particle). OPERA is an hybrid apparatus with electronic detectors and a massive lead-emulsion target. The OPERA target is segmented into ∼ 150000 bricks (fig. 1). A brick is a sequence of 56 lead plates interleaved with 57 emulsion films with an area of 12.7×10.2 cm2 and satisfies the need of both a large-mass and a high-precision tracking capability. OPERA emulsions are used as thin films: pairs of emulsion layers (44 μm thick) are mounted on both sides of a plastic base (205 μm thick) [9]. In 5 years data taking ∼ 1000 emulsion films per day will be (totally or partially) scanned in order to find and analyze the neutrino interations. In total, ∼ 6000 cm2 per day have to be analyzed with sub-micrometric precision. 3. – Automatic scanning system New fast automatic scanning systems have been developed for OPERA: the European Scanning System (ESS) [10] and the S-UTS in Japan [11]. The Japanese S-UTS system uses a dedicated hardware while the ESS (fig. 2) is based on the use of commercial hardware components. The ESS, described further, is able to scan an emulsion volume of 44 μm thickness with a speed of 20 cm2 /h (10 times more than past scanning systems [12, 13]). By adjusting the focal plane of the objective, the whole 44 μm emulsion thickness is spanned and a sequence of 15 tomographic images of each field of view, taken at equally
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Fig. 2. – Schematic layout of the European Scanning System microscope (left). A photograph of one of the microscopes of the European Scanning System (right).
spaced depth levels (3 μm), is obtained. Emulsion images are digitized, converted into a grey scale of 256 levels, sent to a vision processor board and analyzed to recognize sequences of aligned grains. The three-dimensional structure of a track in an emulsion layer is reconstructed by combining clusters belonging to images at different levels and searching for geometrical alignments. Tracks are then reconstructed in the entire brick, after film-to-film alignment. (fig. 3). 4. – Scanning performances Several test exposures at pions beams were performed to estimate the scanning performances. The scanning systems are successfully running with high efficiency (> 90% in the [0, 400] mrad angular range), good purity (∼ 2 fake tracks/cm2 /[angle < 0.4 rad])
Fig. 3. – Track reconstruction: for each field of view, several emulsion images are taken by moving the optical axis and track segments are found by connecting aligned grains (left). Tracks are reconstructed by linking both sides of the emulsion film (center) and then all films of the entire brick (right).
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and the design speed of 20 cm2 /h. Position and angular resolutions at small incident angles are σposition = 1 μm and σangle = 2 mrad [14]. 5. – Conclusions The features and performances of the European Scanning System (ESS) have been described. The resulting tracking efficiencies have been evaluated to be above 90% in the [0, 400] mrad angular range with resolutions of ∼ 1 μm and ∼ 2 mrad for vertical tracks. The ESS has reached the speed of ∼ 20 cm2 /h in an emulsion volume 44 μm thick. This represents an improvement of more than an order of magnitude with respect to the systems developed in the past. The scanning performances satisfy the requirements of the OPERA experiment. About 25 ESSs have been installed in European laboratories collaborating in the OPERA experiment. Eight more have been installed at the Gran Sasso Laboratory (LNGS). ∗ ∗ ∗ This paper is dedicated to the memory of our friend Romolo Diotallevi. REFERENCES [1] Eskut E. et al., Nucl. Instrum. Methods Phys. Res. A, 401 (1997) 7. [2] Kodama K. et al., Phys. Lett. B, 504 (2002) 218. [3] Acquafredda R. et al., New J. Phys., 8 (2006) 303; Anokhina A. et al., JINST, 3 (2008) P07002; P07005. [4] Fukuda Y. et al., Phys. Rev. Lett., 81 (1998) 1562; Ashie Y. et al., Phys. Rev. Lett., 93 (2004) 101801. [5] Ambrosio M. et al., Phys. Lett. B, 434 (1998) 451; 566 (2003) 35; Eur. Phys. J. C, 36 (2004) 323. [6] Allison W. W. M. et al., Phys. Lett. B, 449 (1999) 137; Phys. Rev. D, 72 (2005) 052005. [7] Bailey R., CERN-SL/99-034 (DI) Geneva (1999) http://proj-cngs.web.cern.ch. [8] Powell C. F. et al., The Study of Elementary Particles by the Photographic Method (Pergamon, New York) 1959; Barkas W. H., Nuclear Research Emulsions (Academic Press, New York and London) 1963. [9] Nakamura T. et al., Nucl. Instrum. Methods Phys. Res. A, 556 (2006) 80. [10] D’Ambrosio N. et al., Nucl. Phys. B Proc. Suppl., 125 (2003) 22. [11] Nakano T., Automated Emulsion Read-out System, in III International Workshop on Nuclear Emulsion Techniques, 24-25 January 2008, Nagoya, Japan. [12] Rosa G. et al., Nucl. Instrum. Methods Phys. Res. A, 394 (1997) 357. [13] Aoki S., Nucl. Instrum. Methods Phys. Res. A, 473 (2001) 192. [14] Armenise N. et al., Nucl. Instrum. Methods Phys. Res. A, 551 (2005) 261; Arrabito L. et al., Nucl. Instrum. Methods Phys. Res. A, 568 (2006) 578; JINST, 2 (2007) P05004; Kreslo I. et al., JINST, 3 (2008) P04006.
DOI 10.3254/978-1-60750-038-4-419
The GERDA experiment and first results from phase-I detector operation in LAr/LN2 A. di Vacri for the GERDA Collaboration INFN, Laboratori Nazionali del Gran Sasso - Assergi (AQ), Italy
Summary. — GERDA (GERmanium Detector Array) is designed to search for the neutrinoless double-beta (0νββ) decay of 76 Ge. The experiment is being installed in Hall A of the INFN-Laboratori Nazionali del Gran Sasso. Aiming at a background index of 10−3 counts/(kg y keV) at the Qββ value of 2039 keV, GERDA will operate bare high-purity germanium detectors enriched in 76 Ge immersed directly in the liquid Argon (LAr). The discovery potential of the GERDA experiment will be presented together with the first results from the operation of natural HPGe detectors directly immersed in LAr.
1. – GERmanium Detector Array at LNGS The GERDA experiment [1] searches for 0νββ decay of 76 Ge. High-Purity Germanium (HPGe) detectors, isotopically enriched in 76 Ge (∼ 86%), are operated naked in LAr, which acts both as the cooling medium for the detectors and as a shield against γ radiation. Moreover, the cryostat is surrounded by a stainless-steel tank containing ultra-pure water providing an additional shield against external background. The experiment aims to reduce the background at the level of 10−3 counts/(kg y keV) and energy resolution ≤ 4 keV at the Qββ (2039 keV). The layout of the experiment follows the idea proposed several years ago [2] and is similar to the GENIUS [3] and GEM [4] proposals. c Societ` a Italiana di Fisica
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2. – 0νββ searches with HPGe detectors: experimental considerations Germanium spectrometry is an established technology adopted since the ’60 to search for 0νββ of 76 Ge. The parameters determining the sensitivity of such an experiment are the mass of the candidate isotope M , the measurement time t, the background index b, the energy resolution R, the efficiency and the fraction a of isotope enrichment. The kinetic energy spectrum of the two electrons is measured and the number of events in 0ν the region of the Qββ is counted. In case of non-zero background, the T1/2 corresponding to the minimal number of detectable events above background at a given confidence
t level varies as a M bR . Germanium is considered as a good choice since: it can be used both as source and detector assuring a high efficiency; the natural abundance of 76 Ge is 7.44%, but it can be successfully enriched up ∼ 85%; it is characterized by high intrinsic purity and excellent energy resolution can be achieved with HPGe detectors; Ge density is 5.3 g cm−3 allowing compact setup; Ge has low atomic mass (1 kg of 76 Ge equal 7.9 · 1024 nuclei).
3. –
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Ge 0νββ decay: present status
Evidence for 76 Ge neutrinoless double-beta decay has been claimed by the group of Klapdor-Kleingrothaus, at 4.2σ based of the data of the Heidelberg-Moscow (HdM) experiment [5] at LNGS. The total achieved exposure was 72 kg y, with b = 0ν 0.11 cts/(kg keV y) before pulse shape analysis and R = 3.27 keV, producing T1/2 = 25 1.2 · 10 y. Still no positive indication has been achieved from the IGEX experiment [6] at Canfranc laboratory with an exposure of 8.9 kg y and b = 0.2 cts/(kg keV y) before pulse shape analysis (0.1 cts/(kg keV y) after pulse shape analysis). The derived limit on 0ν the half-life is T1/2 > 1.57 · 1025 y corresponding to mee < 0.33–1.3 eV, depending on the assumed nuclear matrix element calculation. The first purpose of the GERDA experiment is to confirm or to reject with high statistical significance the claim for evidence of 0νββ decay. 4. – GERDA phases and discovery potential GERDA experiment is foreseen to proceed in two phases: I) In the GERDA phase I, eight reprocessed enriched (∼ 85%) HPGe detectors from the past HdM [7] and IGEX [8] experiments with a total mass of ∼ 18 kg and six reprocessed natural HPGe detectors (∼ 15 kg) from the Genius Test-Facility [9] will be deployed in strings. The goal is to achieve b ∼ 10−2 cts/(kg keV y) in the Qββ range. Assuming an exposure of ∼ 15 kg y and R ∼ 3.6 keV, if no event is observed, 0ν the limit on the half-life will be T1/2 > 3 · 1025 y (90% C.L.) resulting in an upper limit of mee < 270 meV [10]. II) In the GERDA phase II, new diodes, able to discriminate between single-site (typical of 0νββ decay) and multi-site events, will be added to reach a total mass of
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Ge of ∼ 40 kg. Reducing in such a way the background index of a further order of magnitude, the resulting limit on the half-life, with an exposure of about 120 kg y, 0ν is expected to be T1/2 > 1.5 · 1026 y (90% C.L.) with mee < 110 meV [10]. 76
III) A possible experiment on 1 ton scale on worldwide collaboration is considered. In case the HdM signal was 0νββ decay, this would produce in about 1 year of GERDA phase I data taking (∼ 15 kg y) 7 counts above background of 0.5 counts, but if no event is observed, the claim would be ruled out with 99.6% C.L. [1]. 5. – GERDA phase-I prototype detectors Natural Ge detectors are operated in a GERDA underground facility at LNGS (GDL) to investigate the effect of the phase-I detector assembly (low-mass holder (80 g) made of low-activity copper, PTFE and silicon), the protocol of detector handling and the refurbishment technology on long-term stability and spectroscopy performance. Three natural p-type HPGe detectors are available at LNGS. They differ by their passivation layer design: prototype 1 (1.6 kg) has a passivation layer that covers all the borehole side, prototype 2 (2.5 kg) has a passivation layer limited to the groove, prototype 3 (2.5 kg) has no passivation layer. The investigations started on prototype 1 and it has been observed that γ-irradiation from 60 Co of the detector in LAr resulted in a reversible increase of the leakage current (LC). In particular, after the expected spontaneous increase of the bulk current when the source is inserted (Ibulk = 2.95C E eV/e− , where C is the counting rate (Hz), E is the average energy deposited in the HPGe and 2.95 eV is the energy necessary to produce an e− -hole pair in Ge at 80 K) a continuous increase depending on the ionization rate in the LAr volume facing the detector borehole side, is observed. Typical value is 40 pA/d, for the source placed in such a way to produce a LAr ionization rate of ∼ 2 kHz in the volume facing the passivation layer (Monte Carlo estimation). The process is totally reversible: LC recovers at few pA value after irradiating without HV and after warming cycles. Prototype 2 and 3 have been irradiated in LAr to check geometrical influences of the passivation layer and measurements with prototype 1 have been repeated in LN2 : – prototype 2 (passivation layer only in the groove) in LAr shows a LC increase rate lower (1.4 pA/d) than the case of prototype 1; – prototype 3 (no passivation layer) does not show any increase of LC; – prototype 1 operated in LN2 does not show any increase. The most likely interpretation of such an increase is that ionization of the LAr volume facing the passivation layer, due to γ irradiation, produces pairs Ar+ /e− . The diode bias electric field dispersed in LAr (numerically calculated by the Maxwell 2D code [11]) drifts both Ar+ and e− toward the passivation layer (along their path e− are eventually
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trapped by electro-negative species like O2 and CO2 ). Charge collected on the passivation layer induces a decrease of resistivity (ΔR = 1014 Ω for ΔI = 40 pA and HV = 4 kV), causing an increase of the detector surface LC. The rate of the increase depends on the charge collection rate, on the density of trapped charge and on the starting value of the passivation layer resistivity. As concerns the cure of the detector by irradiation without HV, this can be explained either by γ-ray ionization in the passivation layer itself or by effect of the UV (128 nm) scintillation light of the LAr. 6. – Conclusions GERDA experiment is in construction at LNGS. It will operate HP 76 Ge detectors naked in LAr, deployed in strings. After two years of operation of bare HPGe detectors in LN2 /LAr (3 prototype detectors with different passivation layer designs) we conclude that the detector assembly has been successfully tested: energy resolution in LN2 with warm electronics and a 40 cm long cable is FWHM = 2.2 keV at 1330 keV (same resolution in LAr), while energy resolution measured in GDL in LAr with warm electronics and an 80 cm long cable is FWHM = 3 keV at 1330 keV; the detector handling protocol has been defined (more than 50 cooling/warming cycles) and the detector parameters are stable over long-term measurement and not deteriorated after one year of continuous operation in LAr. The results of the one year deep and extensive investigation on the effect of γ irradiation on the detector LC are that irradiation results in an increase of the leakage current (LC) in detectors having PL. The increase depends both on the ionization rate of LAr volume facing the borehole side of the detector and on the design of passivation layer. The radiation-induced LC is reversible by γ irradiation with HV off. From measurements, we have indications that UV light (LAr scintillation light) can de-trap the charge, restoring the LC at the initial value. Reducing the size of the passivation layer strongly suppresses γ radiation-induced LC. GERDA will be able to calibrate about once a week for several minutes with a negligible increase of LC during the experiment live-time (< 10 pA). REFERENCES [1] GERDA Collaboration, Proposal 2004 http://www.mpi-hd.mpg.de/GERDA/proposal.pdf. [2] Heusser G., Annu. Rev. Nucl. Part. Sci., 45 (1995) 543. [3] Klapdor-Kleingrothaus H. V. et al., hep-ph/9910205 and Baudis L. et al., Nucl. Instrum. Methods Phys. Res. A, 426 (1999) 425 [4] Zdesenko Yu. G. et al., J. Phys. G, 27 (2001) 2129. [5] Klapdor-Kleingrothaus H. V. et al., Phys. Lett. B, 586 (2004) 198. [6] Aalseth C. E. et al., Phys. Rev. D, 65 (2002) 092007. [7] Balysh C. et al., Phys. Rev. D, 66 (1997) 54. [8] Aalseth C. E. et al., Phys. At. Nuclei, 63 (2000) 1225. [9] Klapdor H. V. et al., Nucl. Instrum. Methods Phys. Res. A, 481 (2002) 149. [10] Rodin V. A. et al., Nucl. Phys. A, 766 (2006) 107. [11] See web page http://www.ansoft.com/maxwell.
DOI 10.3254/978-1-60750-038-4-423
MARE-1: A next-generation calorimetric neutrino mass experiment E. Ferri on behalf of the MARE Collaboration University of Milano-Bicocca and INFN Milano-Bicocca - Milano, Italy
Summary. — The only model-independent experiments dedicated to neutrino mass determination are the kinematic ones from single β-decay. In this context an international collaboration is growing around the project of Microcalorimeter Arrays for a Rhenium Experiment (MARE) for a direct calorimetric measurement of the neutrino mass with sub-electronvolt sensitivity. MARE is divided into two phases. The first phase consists of two independent experiments using the presently available detector technology to reach a sensitivity of the order of 1 eV, and to improve the understanding of the systematic uncertainties specific of the microcalorimetric technique. The two experiments are: MARE-1 in Milan, in collaboration with NASA/GSFG and the University of Wisconsin at Madison, and MARE-1 in Genoa. The goal of the second phase (MARE-2) is to achieve a sub-electronvolt sensitivity on the neutrino mass. The Milan MARE-1 arrays are based on semiconductor thermistors and dielectric silver perrhenate absorbers, AgReO 4 . To optimize the detector performance, crystals of silver perrhenate have been glued to the thermistors with different epoxy resins in order to determine the best thermal coupling. Now a 72 channel measurement is starting. To identify a shielding configuration that minimizes radioactivity background in the energy region of the 187 Re β spectrum, a preliminary study of the cryogenic laboratory environmental background has been performed. Using a planar germanium detector for the energy range below 10 keV, different shielding configurations have been realized to find the best thickness and material to shield the microcalorimeter arrays.
1. – Introduction Neutrino oscillation experiments have shown that neutrinos are massive particles, but they are not able to determine the absolute neutrino mass scale. Therefore, the neutrino mass is still an open question in elementary particle physics. Nowadays the only modelindependent experiments dedicated to neutrino mass determination are the kinematic c Societ` a Italiana di Fisica
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ones based on the measurement of momentum and energy of electrons from single βdecay. The most stringent results come from electrostatic spectrometers on tritium decay (E0 = 18.6 keV). The Mainz/Troitsk collaboration has reached mν ≤ 2 eV/c2 [1]. The next-generation experiment KATRIN is designed to reach a sensitivity of mν ≤ 0.2 eV/c2 . To scrutinize the current and future results of Mainz/Troitsk and KATRIN, an entirely different method to determine the neutrino mass from single β-decay has been investigated. Thermal microcalorimeters measure the temperature rise induced by the energy deposition of the β-electron in a low heat capacity absorber. This method uses the nuclide with the lowest known end-point energy: 187 Re (E0 = 2.47 keV). It has been demonstrated in the past that observing the β-decay spectrum of 187 Re provides a suitable method to determine the mass of the anti-neutrino from the Kurie-plot end-point. At the beginning, a sensitivity of mν ≤ 15 eV/c2 was achieved with the experiments MIBETA in Milan and MANU in Genoa [2]. The MANU and MIBETA results together with the constant advance in the performance of low-temperature detectors open the door to a new large-scale experiment able to explore the sub-eV neutrino mass range. In this context, an international collaboration is growing around the project of Microcalorimeter Arrays for a Rhenium Experiment (MARE) for a direct calorimetric measurement of the neutrino mass with sub-electronvolt sensitivity. 2. – MARE MARE is divided into two phases. The first phase consists of two independent experiments using the presently available detector technology to reach a sensitivity of the order of 1 eV and to improve the understanding of the systematic uncertainties specific of the microcalorimetric technique. The two experiments are: MARE-1 in Milan, in collaboration with NASA/GSFG and the University of Wisconsin at Madison, and MARE-1 in Genoa. The goal of the second phase is to achieve a sub-electronvolt sensitivity on the neutrino mass. The Milan MARE-1 arrays (288 detectors) are based on semiconductor thermistors coupled to dielectric silver perrhenate (AgReO4 ) absorbers. These arrays, provided by the NASA/Goddard group, consist of 6 × 6 implanted Si:P thermistors with a size of 300 × 300 × 1.5 μm3 . An energy resolution of 2 eV for the Mn Kα line has been obtained with these thermistors. On their top single crystals of AgReO4 are glued with epoxy resins. The crystals, the mass of which is around 500 μg, are cut in regular shape of 600 × 600 × 250 μm3 . To realize a defined thermal coupling between the thermistor and the rather large absorber, silicon pieces of 300 × 300 × 10 μm3 are glued between them. With all 288 detectors with an energy and time resolution of about 25 eV and of 250 μs, respectively, a sensitivity of 3.3 eV at 90% CL on the neutrino mass will be reached within 3 years. This corresponds to a statistics of about 7 × 109 decays [3]. The read-out electronics of MARE-1 in Milan is characterized by two stages: the preamplifier cold stage, based on JFETs which work at about 135 K, and the amplifier stage at room temperature. To electrically connect the cold electronics to the detectors with low thermal conductance wires, microbridges are fabricated by the ITC-irst in Trento, Italy. The microbridges are thin wires (thickness around 200 nm) made of Ti or
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Al deposited onto polyamid. The microbridges provide thermal decoupling between detectors and JFETs, but they do not guarantee mechanical stability. Therefore, materials with very low thermal conductivity are used as mechanical support, namely Kevlar and Vespel. Ti microbridges and Kevlar are used in order to thermally decouple the detectors from the JFET holder (4 K) and Al microbridges and Vespel to thermally decouple the JFET box from the JFETs. The cryogenic setup is mounted in the Kelvinox KX400 dilution refrigerator located in the Cryogenic laboratory in the Physics Departement of Milano-Bicocca University. The dilution unit is equipped with a copper rod which is used to hold the structure of the JFET electronics at 4.2 K, very close to the arrays at 35 mK. The energy calibration system is located between the detector holder and the JFETs boxes. The calibration system consists of four fluorescence sources with 10 mCi of 55 Fe as a primary source movable in and out of a lead shield [4]. The fluorescence targets are made of Al, Ti, CaF2 and NaCl to allow a precise energy calibration around the end-point of 187 Re with the Kα X-rays at about 1.49, 1.74, 2.31, 2.62 and 3.69 keV. 3. – Detector performance To optimize the detector performance we have equipped a test array to determine the best thermal coupling between Si thermistors and AgReO4 absorbers. Therefore different kinds of glues were tested to attach silicon spacers on the thermistors and AgReO4 crystals on the spacers, respectively. The masses and the resins combinations are listed in table I together with the baseline and the energy resolution achieved near the 187 Re end-point [5]. The results of table I indicate that the combination Araldit R/Araldit R is favourable over the other glue combinations, but this epoxy resin deteriorates during the years and probably also due to thermal cycling. Therefore, the ST2850 has to be used to glue AgReO4 absorbers on the silicon spacers. We would like to point out that MIBETA also used ST2850 to directly glue the AgReO4 crystals onto the thermistors; no spacers were needed in that experiment. Currently the last test is ongoing, in order to evaluate the crystal quality. In spring 2009, the 72 channel measurement is expected to start. 4. – Enviromental background To identify the best shielding approach that minimizes radioactivity background in the energy region of the 187 Re β spectrum, a preliminary study of the Cryogenic laboratory environmental background was performed. Using a planar germanium detector for the energy range below 10 keV, we have set up different shielding configurations to find the ideal thickness and the best material to shield the microcalorimeter arrays. The detector, characterized by an energy resolution of 195 eV at 5.9 keV, is covered by a very thin Be window, the thickness of which is around 0.127 mm. The detection threshold is around 1 keV. The background spectrum shows peaks at the position of Pb Kα1 , Kα2 and Kβ1 . These peaks rise above a continuous background. In order to reduce the environmental
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Table I. – Listed are different absorber masses and their respective resin combinations. The first resin is between the thermistor and the silicon spacer and the second one between silicon spacer and AgReO4 absorber. (Araldit R = Araldit Rapid and Araldit N = Araldit Normal). The asterisk marks the same detector as the first line but measured two years later. AgReO4 mass (μg)
resin
baseline (eV)
ΔEFWHM (eV)
402 388 456 470 406 442 506 430 390 386 273 300 427 ∗ 402
Araldit R/Araldit R Araldit R/Araldit R Araldit R/ST2850 Araldit R/ST2850 ST1266/ST2850 ST1266/ST2850 ST2850/ST2850 ST2850/ST2850 SU8/ST2850 SU8/ST2850 Araldit R/Araldit R Araldit R/Araldit R Araldit N/Araldit N Araldit R/Araldit R
14 28 21 33 22 30 113 132 131 190 18 12 36 35
28 36 37 38 38 41
22 17 44 36
background, we have built different shields around the germanium detector made of lead, copper and lead plus copper (like “onion” shielding). It turned out that the best configuration is a lead shield. But in the past, background spectra acquired in the MIBETA experiment have shown that a shield made of lead increases the background due to escape peaks resulting from the interaction of the Pb X-rays with Ag and Re atoms in the AgReO4 absorbers. Therefore, since copper is better than lead for shielding, we have studied the background reduction due to a copper shield [2]. We have shielded the Germanium detector putting a copper cup with variable thickness. With a thickness of around 40 mm, the maximum space in the cryostat, we achieve a reduction of 70%. To investigate the background suppression in the microcalorimeters inside the cryostat, a Monte Carlo simulation has been performed. Simulating the two detector types and the incoming X-rays it is possible to compare the respective detection efficiences for X-rays for energies from 0 to 19 keV. From these Monte Carlo simulations it can be said that the background reduction in microcalorimeters should be even more efficient than in the germanium detector. REFERENCES [1] [2] [3] [4] [5]
Angrik J., Katrin design report 2004. Sisti M. et al., Nucl. Instrum. Methods Phys. Res. A, 520 (2004) 125. Nucciotti A., J. Low Temp. Phys., 151 (2008) 597. Schaeffer D. et al., J. Low Temp. Phys., 151 (2008) 623. Kraft-Bermuth S. et al., J. Low Temp. Phys., 151 (2008) 619.
DOI 10.3254/978-1-60750-038-4-427
Different approaches to CUORE background analysis L. Gironi, M. A. Carrettoni, C. G. Maiano and L. M. Pattavina Dipartimento di Fisica “G. Occhialini”, Universit` a di Milano-Bicocca and INFN, Sezione di Milano-Bicocca - Milano, Italy
Summary. — Neutrino oscillation experiments have unequivocally demonstrated that neutrinos have mass and that neutrino mass eigenstates mix. One possible way to determine the scale of the neutrino mass and its nature is to investigate the neutrinoless double-beta decay(0νdbd). The CUORE experiment is designed with a sensitivity capable of probing all but a small portion of the Majorana electron neutrino effective mass range. CUORE is an array of 988 TeO 2 bolometers. The signal of neutrinoless double-beta decay for 130 Te would be a sharp peak at 2530 keV. The rarity of the process under consideration makes its identification very difficult. The main task in 0νdbd searches is to understand and to diminish the background as much as possible. In order to reach this goal, different techniques are employed: Monte Carlo simulations, alpha-spectroscopy and pulse shape discrimination.
1. – Introduction - 0νdbd Although the existence of neutrino oscillations (and consequently of massive neutrinos) seems now to be well proved, several properties of the neutrino family have still to be fixed. Measuring the masses, the mixing angles (and phases) as well as accessing the Dirac/Majorana nature of neutrinos will be the goal of the next-generation experiments. In this scenario an important role is played by neutrinoless double-beta decay searches c Societ` a Italiana di Fisica
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which will probe the Majorana nature of neutrinos (as foreseen by the majority of theories) allowing in the meantime to obtain information on the neutrino mass hierarchy and scale. The challenge of these experiments is the realization of a large mass (of the order of tons) counting facility with extremely low background. 2. – Double-beta decay with TeO2 bolometers Natural tellurium contains about 33.8% of the isotope 130 Te that, given its high Qββ value (2530 keV) and its favorable nuclear factor of merit, is one of the most interesting candidates for study. To scale sensitivity toward the next-generation experiment frontier (approximately tens of meV) a favorable candidate is not enough: a technique that guarantees large operating masses, high energy resolution and low achievable background is indeed mandatory. The so-called “source ⊆ detector” configuration, where the candidate nuclei are contained within the active mass of the detector, makes mass scaling toward high values easy and natural. In this configuration the decay signal would appear in the background spectrum of the detector as a peak at the Qββ value of the decay and the sensitivity is determined on one side by the detector mass and energy resolution and on the other by the background level. CUORE (Cryogenic Underground Observatory for Rare Events) is a proposed tightly packed array of 988 TeO2 bolometers [1], each being a 5 × 5 × 5 cm3 cube with a mass of 750 g operating at 8–10 mK. The CUORE know-how is based on the experience gained in almost 25 years by the Milano group in bolometers technology [2] and culminated in CUORICINO experiment [3]. CUORE detector array will consist of 19 vertical towers, each divided into 13 layers of 4 crystals. 3. – Coincidences and pulse shape analysis The array is designed in order to have the most compact structure reducing to a minimum the distance between the crystals and the amount of inert material interposed between them. This will maximize the efficiency of the anticoincidence cut in order to reduce the background produced by degraded alphas present on the crystal surfaces. An important role in the recognition of these coincidences patterns is played by the high sensitivity of bolometers to any energy depositions which offers, for example, the possibility to discard surface contamination by identifying the nucleus recoil on a crystal and the emitted alpha in another one. Such powerful detection efficiency is at the same time a disadvantage of the bolometric approach: any physical process, such as cryostat instabilities or vibrations, may cause a temperature rise that can trigger the acquisition. All these spurious signals can be difficult to recognize, expecially in the lower part of the spectrum, where the signal-to-noise ratio is low. Since the shape discrimination of these low-energy pulses is so crucial, each acquired event will be processed with an optimum filtering technique, in order to maximise the signal-to-noise ratio. This technique has been fully tested and used during Cuoricino analysis, showing a strong noise reduction.
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After the filtering, a set of shape-sensitive parameters are computed from the acquired signals. This procedure takes advantage from another feature of bolometers: the rise time of particle-induced signals is proportional to the sound velocity in the crystal and to the interface between the absorber and the thermistor used for temperature-to-voltage conversion, while the decay time depends mainly on the effective thermal impedance of the link with the thermal bath. This implies that the particle signals shape depends mainly on the structure of the crystal and on the detector assembly and it is independent, at least at the first order of magnitude, of the incident energy. This offers the possibility to identify regions in the parameters space where good signals show similar structures. 4. – CUORE Background The possibility of studying rare events such as 0νdbd is strongly influenced by the background in the region of interest of the energy spectrum. In a deep underground laboratory the background is ascribed to different sources such as environmental γ radioactivity, cosmic rays, neutrons, radon and contamination of materials with which detectors and their shielding are made. The reduction of these background sources is possible by the construction of shields and placing the experiment in a deep underground laboratory. For this reason CUORE will be located in the underground hall A of Laboratori Nazionali del Gran Sasso (LNGS) at a depth of ∼ 3400 m.w.e. In order to shield the detectors from environmental γ radioactivity (the most energetic γ is 208 Tl line at 2615 keV) about 25 cm of lead are planned to be placed outside the cryogenic apparatus to shield the detector from the bottom and from the sides radiations. An equivalent shielding from the up-coming radiation has to be placed inside the cryostat, a 30 cm thick lead disk. A lead ring-shaped shield closes the gap between the lead disk above the detector and the 6 cm internal lead layer positioned near the sides and near the bottom of the detector. In order to define and optimize the shielding for CUORE, different Monte Carlo simulations were carried out with the GEANT4 code, which is able to simulate transport and interaction of particles in matter. The results of these analyses have allowed also to evaluate the maximum levels of contamination acceptable for the cryostat and for the shielding itself, imposing limits on the choice of construction materials and geometry of shielding. Outside the lead shield an 18 cm polyethylene shield will be placed in order to thermalize environmental neutrons that will be then absorbed by the 2 cm of boric acid located between the lead and the polyethylene itself. Monte Carlo simulations indicate that such shield will be able to limit neutron-induced background to a rate far below the required level for CUORE. 5. – Surface contaminations and recontaminations Previous studies of the background, like Cuoricino’s one, allow to state that surface alpha contamination is the predominant source of avoidable background, even in the
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0νdbd region. An accurate identification and localization of α contamination seems to be mandatory. With SBD (Silicon Barrier Detectors) it is possible to perform a detailed screening of the materials which are candidates to be employed for CUORE. The SBD are semiconductor detectors suitable for charged-particle radiation: they are made of a thin dead layer (few tens of nm) and semiconductor surface area of 1200 mm2 and active layer of about hundreds of microns. They offer an energy resolution of about 30 keV at 5 MeV. The study of the surface contamination is made mainly out of three steps: – Calibration and identification of the alpha counts and peaks. This is the most difficult part of the analysis because we have to deal with low background measurements, about 10−7 Bq/cm2 . – Comparison between different sample spectra in order to estimate the detector background and identify the effective material contamination. SBD are always faced to some inert material (the chamber in which the measure is performed), so a thorough analysis is required to recognize the detector internal contaminations. – Evaluation of the contamination depth exploiting Monte Carlo simulations. Once chosen and applied the best cleaning technique, recontaminations caused by the exposure to the natural atmosphere (during shipment and storage) is still possible. In fact, as it is well known, the environment is rich of 238 U and of its daughters which are the major components of natural radioactivity. One of the elements in the 238 U decay chain is 222 Rn which is a gas; this can diffuse through the atmosphere and through different materials. The radon decays and it becomes 210 Pb with a half-life of 22.3 years, this is much longer than all the others 222 Rn daughters. This radioactive element, with such a long half-life, will affect the radiopurity of the experiment since it will be present in the experimental set-up for many τ (210 Pb). 210 Po is a 210 Pb daughter that decays emitting a 5.3 MeV alpha particle. Now, if the polonium is implanted in the materials then the alphas may lose part of their energy in the materials and this energy loss can produce some counts in the 0νdbd region. In order to keep under control any possible radon contamination, a thorough analysis of these possible recontaminations is carried out, taking into account radon/polonium diffusion profile related to the time exposure of the different CUORE materials. REFERENCES [1] Ardito R. et al. (The CUORE Collaboration), arXiv:hep-ex/0501010 (2005). [2] Fiorini E. and Niinikoski T., Nucl. Instrum. Methods, 224 (1984) 83. [3] Arnaboldi C. et al. (The Cuoricino Collaboration), arXiv:hep-ex/0802.3439v1 (2008) and references therein.
DOI 10.3254/978-1-60750-038-4-431
Intermediate-energies π-induced reactions studied with a streamer chamber I. Gnesi on behalf of PAINUC Collaboration Centro studi e Ricerche “E. Fermi” - Roma, Italy Dipartimento di Fisica Generale, Universit` a di Torino e INFN, Sezione di Torino - Torino, Italy Joint Institute for Nuclear Research - Dubna, Russia
Summary. — The interaction with two charged particles in the final states between negative pions at 106 MeV and helium nuclei has been studied with the Self Shunted Streamer Chamber at the Joint Institute for Nuclear Research in Dubna. In the neutron knockout channel we observed the first evidence of Δ− excitation. In the channel with the emission of a γ we observed a γ energy spectrum that is in good agreement with a Planck black-body radiation distribution at a temperature of 14 MeV. A Monte Carlo simulation of inflight pion decays, within our chamber, has been performed to study the possibility of upgrading the upper limit of the muon neutrino mass; at 90% c.l. we obtained a prevision of Mν < 7 MeV.
1. – Introduction Experimental studies on the various π ±4 He interaction channels are important for undestanding the pion-nucleus dynamics and for addressing unresolved issues about the nuclear structure, which is far from being understood. The optical model [1] was the first attempt for modeling πp scattering while subsequent studies were dedicated to the description of complexes of nucleons (see [2]). Later, the effects of the so-called “nuclear matter” were observed in elastic and inelastic pion-nucleus scattering. Excitation of the Δ resonance in elastic reactions was found to be mass-dependent and angle-dependent (see ref. [3]) in agreement with the hypothesis of a collective isobaric resonance activation. The inelastic channels revealed the existence of several nuclear medium effect; among them, clustering effects, multiple scattering effects, pion absorption on multiple nucleon systems (see ref. [4]) provided information about the quasi-deuteron and quasi-helium substructure of nuclei, highlighted c Societ` a Italiana di Fisica
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the activation of resonances, also in inelastic reactions, and allowed the study of Coulomb barrier effects [5]; but, nevertheless, there actually exists no comprehensive model. The investigation on π ±4 He interaction at energies below the QCD transition phase [6] (on which much effort is focused, at present) and in the region where the hadron-gas state transition is expected in heavy nuclei (see [7]) should reveal effects about this intermediate state of nuclear matter in light and deeply bound nuclei, such as 4 He. The most recent experimental studies (see ref. [8] and references therein) of pion interactions performed at energies in the region of and below the Δ(1232) resonance have provided high-statistics information on various multinucleon pion absorption and charge exchange reaction channels. Although a semiclassical pick-up model was applied in these recent works to explain the observed knockout and breakup reaction channels, no agreement with experimental data was achieved. Nowadays, the field of experimental low-energy nuclear physics is of interest for measuring the mass of neutrinos, which are fundamental quantities for the completion of SM. 2. – Brief description of the experimental apparatus The experimental apparatus we used is a magnetic spectrometer with a Self Shunted Streamer Chamber (SSSC) placed in a magnetic field of 0.65 T. The streamer chamber serves simultaneously as a low-density target and a triggerable track detector (as well as a vertex detector), thus combining the useful features of the visualizing techniques and of electronic detectors (such as MWPC). The chamber consists of a 470 × 600 × 160 mm3 fiducial volume filled with 4 He at atmospheric pressure. The triggering system based on 7 scintillators selects the events occurring within the fiducial volume. The electronics controls a high-voltage power generator (250 kV). The emitted light from the discharge channels along the particle tracks is captured by a high-resolution and sensitivity stereoscopic CCD system. Data acquisition system provides digitized CCD videoimages of nuclear events, that can be handled on-line during experimental runs. The triggering signal was driven by pions or protons with energies 6 MeV and 18 MeV, respectively. We detected the particles scattered within [20,160] degrees with respect to the incident pions direction. The experimental apparatus is described in detail elsewhere [9]. It must be stressed that in the present study advantage is taken of the following: with the DUBTO streamer chamber filled with a gas at atmospheric pressure a complete kinematical analysis of nuclear reactions involving very low-energy short-ranged charged secondaries is possible. Thus, the path range of a 5.0 MeV α-particle in 4 He at atmospheric pressure exceeds 20 cm, resulting in a readily measurable track. 3. – Experimental results Analysing π ±4 He interaction at T = 106 MeV we obtained the first evidence for the existence of π ± (4 He, 4 He)π ± γ channel in which a high-energy γ is produced. In fig. 1, upper-right plot, the differential cross-sections for π ± (4 He, 4 He)π ± and π ± (4 He, 4 He)π ± γ channels are shown: the emission of the γ is nearly isotropic. The upper-left plot shows the γ energy distributions, both for π + and π − fitted with a Planck black-body radiation
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Fig. 1. – (Colour on-line) Recent PAINUC collaboration results (see text).
curve. The isotropic γ emission, their energies (higher than the 4 He binding energy) and the high branching ratio of the channel are well explained with a black-body model,
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revealing a temperature T = 14–16 MeV for the 4 He nucleus when excited with a T = 106 MeV pion beam (see ref. [10]). The π − (4 He, 3 He)π − n knockout channel shows a bimodal θn3 He outgoing angle distribution, plotted in fig. 1, second row, left plot, in contradiction with the expected phase-space (green curve). The event at intermediate angles (red area) gives a resonant π − n invariant mass distribution, with Mπ− n ∼ 1160 MeV and Γ/2 ∼ 20 MeV. The mass spectra is shown in fig. 1, second row, right plot, red curve. This is the first esperimental evidence for Δ− in medium excitation and the left shift of the mass w.r.t. the free Δ− mass (1232 MeV) can be explained with a collective isobaric resonance excitation (see refs. [3] and [11]). The collaboration has also analysed π + (107 Ag, 105 Ag)π − pp and π + (107 Ag, 103 Ag) π − ppnn DCX reactions on nuclear photoemulsion (ref. [12]). The π − pp spectrum, shown in fig. 1, third row, left plot, is a confirmation of the existence of a dybarionic resonance, J P = 0− , B = 2 and mass M = 2050 MeV, called d . The π + nn spectrum obtained with the SSSC in π + (4 He, pp)π + nn reaction channel is shown in fig. 1, third row, right plot and confirms previous results on nuclear photoemulsion. The lower-left plot in fig. 1 shows the image of a p ¯20 Ne annihilation event with a + + + + complete π → μ νμ , μ → e ν¯μ νe decay chain. From such event an upper limit of mν (μ) < 2.2 MeV at 90% of c.l. has been obtained (see ref. [13]). A study has been performed to evaluate the possibility of increasing the limit measuring π ± decays in SSSC, with the possibility of observing both νμ and ν¯μ . All the error sources have been taken into account, and the influences of the hypothetical ν mass and measurement error have been studied. In fig. 1, lower-right plot, the νμ mass upper limit vs. the given confidence level is shown. At 90% of c.l., it is mν (μ) < 7 MeV, half of the recently published bound by MINOS collaboration with a 734 km baseline experiment ν beam TOF measurement. The collaboration is studying the possibility to increase the momentum resolution of the SSSC, in order to obtain a new upper limit for the muon neutrino mass. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Fernbach S., Serber R. and Taylor T. B., Phys. Rev., 75 (1949) 1352. Budagov Yu. A. et al., Sov. Phys. JETP, 15 (1962) 824. Balestra F. et al., Nuovo Cimento A, 55 (1980) 273. Balestra F. et al., Nucl. Phys. A, 340 (1980) 372. Balestra F. et al., Lett. Nuovo Cimento, 12 (1975) 351; 13 (1975) 673; Balestra F. et al., Nuovo Cimento A, 78 (1983) 331. Rapp R. and Wambach J., Adv. Nucl. Phys., 25 (2000) 1. Viola V. et al., Nucl. Phys. A, 734 (2004) 487. Alteholz T. et al., Nucl. Instrum. Methods Phys. Res. A, 373 (1998) 374; Mateos A. O. et al., Phys. Rev. C, 58 (1998) 942. Andreev E. M. et al., Nucl. Instrum. Methods Phys. Res. A, 489 (2002) 99. Angelov N. et al., Nuovo Cimento B, 121 (2006) 771. Angelov N. et al., Eur. Phys. J. A, 34 (2007) 255269. Batusov Yu. A. et al., Eur. Phys. J. A, 28 (2006) 11. Angelov N. et al., Nucl. Phys. A, 780 (2006) 78.
DOI 10.3254/978-1-60750-038-4-435
Features of SN signal for massive neutrinos using LVD simulated events A. A. Machado CNPq - Brasilia, Brazil INFN, Laboratori Nazionali del Gran Sasso - Assergi (AQ), Italy
G. Pagliaroli University of L’Aquila - L’Aquila e INFN, Laboratori Nazionali del Gran Sasso Assergi (AQ), Italy
F. Vissani INFN, Laboratori Nazionali del Gran Sasso - Assergi (AQ), Italy
W. Fulgione Istituto Nazionale di Astrofisica, IFSI - Torino e INFN, Sezione di Torino - Italy
Summary. — We consider the neutrino signal from a future core collapse supernova detected by the LVD experiment at the Gran Sasso National Laboratory. We generate a sample of Monte Carlo events accounting for the detector response. The effect of neutrino masses on the expected signal will be studied for a supernova exploding at different distances.
1. – The LVD experiment The Large Volume Detector (LVD), located in the INFN Gran Sasso National Laboratory, at the depth of 3600 m.w.e., is a 1 kt liquid scintillator detector whose major purpose is to study neutrino bursts from a Gravitational Stellar Collapse (GSC) in our Galaxy [1]. In spite of the lack of a standard model of GSC, the correlated neutrino emission appears to be well established. At the end of its burning phases a massive star (M > 8M ) explodes into a Supernova (SN), originating a neutron star which emits a large amount of neutrinos that carry away 99% of the binding energy EB ∼ 3 × 1053 ergs. The largest part of this energy, almost equipartitioned among neutrino and antineutrino c Societ` a Italiana di Fisica
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Fig. 1 Fig. 1. – LVD efficiency.
A. A. Machado, G. Pagliaroli, F. Vissani and W. Fulgione
Fig. 2
Fig. 2. – The integrated rate of background events on LVD.
species, is emitted in the cooling phase: Eν¯e ∼ Eνe,μ,τ ∼ EB /6. The energy spectra are approximate Fermi-Dirac distributions with different temperatures. The experiment has been taking data since 1992. LVD consists of an array of 840 scintillator counters, 1.0 × 1.0 × 1.5 m3 each one. The whole array is divided into three identical towers with independent high-voltage power supply, trigger and data acquisition. Since 2005, all the LVD counters are operating at the same energy thresholds (EH = 4 MeV and EL ∼ 1 MeV). The LVD counter energy resolution has been studied by comparing the energy spectra obtained by different sources (γ from (n, p) capture, μ-decay electrons, γ’s from (n, Ni) capture) with the Monte Carlo simulation. The following function represents the best approximation of the single counter energy resolution in terms of σE /E: (1)
σE /E = 0.07 + 0.23 × (E/MeV)−0.5 ,
that means FWHM/MAX = 0.33 for 10 MeV electrons(1 ). The observation of neutrinos is made mainly through the inverse beta decay reaction of electron anti-neutrinos on scintillator protons ν¯e + p → n + e+ which gives two detectable signals: the prompt one due to the e+ (visible energy Evis = Eν¯e − 0.8 MeV) followed by the neutron capture n + p → d + γ (Eγ = 2.2 MeV), mean Δt = 185 μs. The efficiency function η(E) used in the MC simulations, and displayed in fig. 1, reaches 90% at Evis = 7 MeV, and 95% at Evis = 10 MeV. (1 ) Because of its geometry and dimensions, the energy counter response in not uniform and spectra are not symmetric.
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Fig. 3 Fig. 4 Fig. 3. – Number of events as a function of the distance considering the two threshold energy at 7 and 10 MeV. The Galactic Center is located at 7.7 kpc and the Large Magellanic Cloud at 50 kpc of distance. Fig. 4. – Signal and background simulated events for a SN collapse at the Galactic Center.
The expected signal event rate from a SN as a function of the distance is given by (2)
dN (D) = Np × σIBD (Eν ) × Φν (Eν , t, D) × η(E), dEν dt
where Np is the number of target protons, σIBD is the inverse beta decay cross-section, η(E) is the LVD efficiency, and Φν (Eν , t, D) is the ν¯e flux constructed following the analysis reported in [2]. The number of expected events is obtained by integrating this function in energy and time. The integrate rate of background events is shown in fig. 2. For a time interval of 20 s, Evis ≤ 50 MeV and a distance D = 20 kpc, we obtain 4 events for Evis ≥ 7 MeV and 0.5 for Evis ≥ 10 MeV. The functions that give the number of expected events in LVD (scaled by 10 kpc) are 2 10 Nevents (D) = 226.7 × (3) + 4, for Eth = 7 MeV, D 2 10 (4) + 0.5, for Eth = 10 MeV. Nevents (D) = 213.5 × D In fig. 3 we can see these two functions, where the upper line corresponds to a 7 MeV threshold and the lower line at 10 MeV. The detector sensitivity decreases in the dark region of the figure, namely for D > 50 kpc. 2. – Events simulation We generated a sample of Monte Carlo events accounting for the experiment response for a SN collapse in the Galactic Center (GC). The distance of GC was calculated by different authors, see [3] for a review. We adopted the value 7.7 kpc for the simulation displayed in fig. 4. Background events are contained in the region between the two
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Fig. 5 Fig. 6 Fig. 5. – MC events at D = 20 kpc for mν = 0, 1, 3 and 5 eV. Fig. 6. – MC events at D = 7.7 kpc for mν = 0, 3, 5 and 7 eV.
energy threshold values (7 and 10 MeV) and the neutrino burst is concentred in a time window of 14 s. . 2 1. Time delay. – The shape of the neutrino signal can be affected by the delay time correlated to different mass values. The delay time written as m 2 E −2 D ν (5) Δt(mν , E) = 0.51 s eV MeV 10 kpc was taken into account to construct the flux of ν¯e in eq. (2). In order to analyze this effect, we generated two samples of events, one at 20 kpc (the distance for which LVD still have good sentitivity), and one at 7.7 kpc. For a distance of 20 kpc, the Monte Carlo produced 57 events on average. Using mass values of 0, 1, 3 and 5 eV, we observe a variation in the distribution of signals in the low-energy region see fig. 5. The same effect is seen, in a bigger sample (368 events), when the distance is 7.7 kpc. In the latter case, the neutrino masses considered were 0, 3, 5 and 7 eV, see fig. 6. A more accurate analysis in the low-energy range is underway. 3. – Conclusions We showed, through a Monte Carlo simulation, the possibility to detect by LVD experiment the correlation between energy, arrival times of the signals and ν mass values for a GSC at different distances. The presence of a finite neutrino mass produces a distortion of ν¯e spectra as a consequence of the delay time introduced in eq. (5) that is a function of the ν energy. REFERENCES [1] LVD Collaboration, Nuovo Cimento A, 105 (1992) 1793; Agafonova N. Yu. et al., Astroparticle Phys., 28 (2008) 516; Porta A., Ph.D. Thesis, Torino University (2005) and INAF IFSI-TO, Int. Rep. n. 15/2007. [2] Pagliaroli G. et al., Astroparticle Phys., 31 (2009) 163. [3] Groenewegen M. A. T., Udalski A. and Bono G., Astron. Astrophys., 481 (2008) 441.
DOI 10.3254/978-1-60750-038-4-439
Minimal renormalizable SO(10) splits supersymmetry M. Nemevˇ sek, B. Bajc and I. Dorˇ sner Joˇzef Stefan Institute - Jamova cesta 39, 1000 Ljubljana, Slovenia
Summary. — We have revisited the minimal renormalizable supersymmetric grand unified theory (MSGUT) based on the group SO(10) with two Higgs superfields, 10 and 126, responsible for the masses of the fermions. The model has previously been shown to be inconsistent with current data on fermion masses. After a careful analysis, taking into account the calculated spectrum of all the heavy particles which influence the running of the gauge and Yukawa couplings, we find out that the overall scale of neutrino mass is an order of magnitude too small, when unification takes place in the low SUSY regime. However, if one allows for the splitting between the masses of sfermions and the other SUSY partners, preliminary results show that one is able to obtain a fit of all the light fermion masses, including the large enough neutrino mass scale. This step also eliminates the dangerous contribution of d = 5 proton decay operator which in turn leaves us with a clear prediction of the proton decay rate mediated by the d = 6 operator and a nonzero Ue3 .
1. – Minimal renormalizable supersymmetric SO(10) We have addressed the viability of the minimal renormalizable supersymmetric grand unified theory (MSGUT) model in the context of SO(10) gauge group [1]. The theory is attractive because it contains the right-handed neutrino in the spinorial 16F representation and an SU (2)L triplet in the 126H , which gets a small vacuum expectation value, therefore neutrinos naturally receive a small mass via type I+II seesaw. c Societ` a Italiana di Fisica
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The model is dubbed minimal due to the small number of couplings in the superpotential and although large representations are employed, the model is predictive. The superpotential consists of all renormalizable terms with Higgs representations 10H , 126H , 126H and 210H : (1)
W = 16F (Y10 10H + Y126 126H )16F λ M η m 126H 126H + 126H 210H 126H + 2102H + 2103H + 4! 4! 5! 5! 1 +mH 10H 2 + 210H 10H (α126H + α126H ). 4!
This superpotential has been carefully studied and after a fine-tuning which is required to have two light Higgs doublets, we are left with only eight real parameters in the Higgs sector: (2)
m,
α,
α,
|λ|,
|η|,
φ = arg(λ) = − arg(η),
x = Re(x) + i Im(x).
The whole mass spectrum of the heavy Higgses has been given in [2] and we have determined the one loop beta coefficients and masses of the gauge bosons in terms of the above parameters. Thus we can calculate the running of gauge couplings, fermion masses and mixing angles from MZ to MGUT including all the threshold corrections. The matter fermions in 16F are coupled to only two Higgses, namely 10H and 126H , therefore the Yukawa matrices Y10 and Y126 in eq. (1) are not independent in contrast to the standard (or MSSM) model. They are complex symmetric matrices and their dependence can be conveniently expressed using a sum rule for charged fermions: (3)
Mu =
Nu tan β[Md + ξ(Md − Me )] Nd
and also for neutrino masses (4)
v sin2 β α Mn = m cos β
:
|λ| Nu2 [mI fI + mII fII ]. |η| Nd
Here Nu,d and fI,II are functions of the parameters (2) and mI,II are mass matrices from type I and II contributions (exact expressions are given in [3]). To count the number of free parameters in the Yukawa sector, we use a congruent transformation to go to the basis where one of the matrices is diagonal and real and therefore has 3 real parameters and the other is complex symmetric with 12 real parameters. The issue at hand is whether one can satisfy the above relations with these 15 parameters and accommodate all the masses and mixing angles of charged and neutral fermions in a perturbative unified theory with a slow enough proton decay rate. Since the above relations are valid at the GUT scale, such an analysis is further complicated by a system of nonlinear differential equations describing the running, therefore our approach is
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numerical. For each set of parameters we run the gauge couplings to the GUT scale and if they unify, we use the central values of fermion masses and mixing angles in our fit to determine the validity of the above relations. 2. – The overall neutrino scale and proton decay lifetime The main issue concerning this model is the connection between the proton decay rate and the overall scale of neutrino masses. One can think of neutrino masses coming from d = 5 Weinberg operator yeff (LH)(LH)/Λ and it turns out that the cutoff is around 1014 GeV for Yukawas of order one and neutrino masses of 0.1 eV. The role of Λ in this particular model is realized by the only mass parameter m in eq. (4). However, the heavy gauge bosons are proportional to the same parameter, which is therefore directly connected to the proton decay rate. The bound on d = 6 operator coming from p → π 0 e+ decay channel is around 1016 GeV and this creates a tension between the proton decay rate and the neutrino mass scale. As it was argued in [3], this tension is a serious issue, unless one is able to find a cancellation in type I case or one tries to find the solution for mass fits near singular points of fI,II , enhancing the overall scale. This has been investigated [4, 5] and the conclusion was that either neutrino mass is too low, or unification does not take place. We show how this tension is realized in the low SUSY regime, therefore eliminating this part of the parameter space and then we solve the problem by splitting the masses of gauginos and sfermions. 3. – MSSM vs. split supersymmetry The only “fast” parameter in the theory is the complex parameter x, meaning that the masses of gauge and Higgs bosons vary rapidly with it. By setting λ = η = α = α = 1, we find that the region in the complex x-plane where unification takes place does not coincide with any of the singular points of fI , therefore pure type I dominance is excluded. We further focus on type II and find regions where neutrino mass is enhanced, but the overall scale is roughly two orders of magnitude too small, when unification takes place. We further proceed by taking into account the variation of all the superpotential parameters to enhance the overall scale and we scan the complex x-plane for a good fit with different values of tan β. We are able to obtain good fits, however the mass of the heaviest neutrino is still a factor of 6 too small. We therefore conclude that, although our approach of neutrino scale maximization in terms of superpotential parameters (and not some special singular points) gives an improvement of more than one order of magnitude, the minimal theory still fails to accommodate neutrino masses and mixing angles with a low-SUSY spectrum. We can get out of this impasse by considering a split between gauginos and sfermions, setting the former at around TeV and the latter at 1012 GeV. By using the appropriate equations for running, we repeat the above procedure and preliminary results give good solutions with a low χ2 and large enough neutrino masses with a dominant type II
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Fig. 1. – Split-SUSY scenario results with a good fit and large enough neutrino mass scale is shown for three different values of tan β. Scattered points correspond to different values of x.
contribution, as shown in fig. 1. The benefit of heavy sfermions is that our analysis is not affected by finite threshold corrections to fermion masses and moreover the dangerous d = 5 proton decay operator is suppressed. Therefore only the d = 6 operator is present, which depends on the masses of heavy gauge bosons and mixing matrices resulting from the fit. We also present the nonzero value of sin θ13 = 0.1 ± 0.01, the only parameter in the PMNS mixing matrix yet to be measured. ∗ ∗ ∗ ´ for discussions and encouragements. This It is a pleasure to thank G. Senjanovic work has been supported by the Slovenian Research Agency (B.B. and M.N.) and by the Marie Curie International Incoming Fellowship within the 6th European Community Framework Program (I.D.). REFERENCES [1] Bajc B., Dorˇ sner I. and Nemevˇ sek M., JHEP, 0811 (2008) 007 (arXiv:hepph/0809.1069) [2] Bajc B., Melfo A., Senjanovic G. and Vissani F., Phys. Rev. D, 70 (2004) 035007 (arXiv:hep-ph/0402122). [3] Bajc B., Melfo A., Senjanovic G. and Vissani F., Phys. Lett. B, 634 (2006) 272 (arXiv:hep-ph/0511352). [4] Aulakh C. S. and Garg S. K., Nucl. Phys. B, 757 (2006) 47 (arXiv:hep-ph/0512224). [5] Bertolini S., Schwetz T. and Malinsky M., Phys. Rev. D, 73 (2006) 115012 (arXiv:hepph/0605006).
DOI 10.3254/978-1-60750-038-4-443
Absolute neutrino mass from helicity measurements C. C. Nishi Institute of Physics “Gleb Wataghin”, University of Campinas, UNICAMP 13083-970, Campinas, SP, Brasil
Summary. — The possibility to access the absolute neutrino mass scale through the measurement of the wrong helicity contribution of charged leptons is investigated in pion decay. Through this method, one may have access to the same effective mass m2β extractable from the tritium beta decay experiments for electron neutrinos as well as the analogous effective mass (m2νμ )eff for muon neutrinos. In the channel π − → e− ν¯, the relative probability of producing an antineutrino with left helicity is enhanced if compared with the naive expectation (mν /2Eν )2 .
1. – Introduction After the confirmation that neutrinos have non-null masses and non-trivial mixing among the various types, the knowledge of the absolute scale of neutrino masses is one of the most urgent questions in neutrino physics. In recent times, the greatest advances in the understanding of neutrino properties were boosted by neutrino oscillation experiments which are only capable of accessing the two mass-squared differences and, in principle, three mixing angles and one Dirac CP -violating phase of the Maki-NakagawaSakata (MNS) leptonic mixing matrix. Too large masses for the light active neutrinos may alter significantly the recent cosmological history of the Universe. Indeed, the most stringent bounds for the value of c Societ` a Italiana di Fisica
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the sum of neutrino masses come from Cosmology [1]:
(1)
mν < 0.17 eV.
ν
Despite of the stringent bound (1) coming from cosmological analyses, terrestrial direct search experiments establish much looser bounds [2]: (2)
mνe ≤ 2 eV,
mνμ ≤ 190 keV,
mντ ≤ 18.2 MeV.
These bounds are based on ingeniously planned experiments [3], but their intrinsic difficulties rely on the fact that they should probe, essentially, the kinematical effects of tiny neutrino masses. Nevertheless, it is always desirable to have a direct measurement of neutrino masses because cosmological bounds may be quite model dependent. For the electron neutrino, there are ongoing experiments planning to reduce the respective bound to 0.2 eV [4]. The main goal of this note is to investigate the possibility of accessing the absolute neutrino mass scale through one of the most natural consequences of massive fermions, i.e., the dissociation of chirality and helicity. Consider the pion decay π − → μ− ν¯μ . Since the pion is a spin-zero particle, in its rest frame, the decaying states should have the following form from angular momentum conservation: (3)
←
→
→
←
|π → |μ :←|¯ ν :→ + δ|μ :←|¯ ν :→,
where the arrows represent the momentum direction (longer arrow) and the spin direction (shorter arrow). The normalization of the state is arbitrary and the coefficient δ is of the order of mν /Eν , which will be calculated in sect. 2. Thus, by measuring the wrong helicity contribution of the charged lepton, it is possible to have access to the neutrino mass. Such possibility was already suggested in ref. [5] but we intend here a focused reanalysis of the possibility considering the present experimental bounds. Further analyses such as the precision in the polarization measurement necessary to extract the wrong helicity can be found in ref. [6]. The possibility to constrain scalar/pseudoscalar interactions in two-Higgs-doublet models is also investigated in the same reference. It is interesting to notice that 50 years have passed since the first measurement of the helicity of the electron neutrino [7]. At that time the primary concern was to confirm the V -A theory of weak interactions. Nowadays, we can try to invert their roles to obtain new information about the neutrinos from the well-established weak interaction part of the Standard Model (SM). 2. – Pion decay − Pion decay π − → l√ ¯j can be effectively described by the four-point Fermi interaction i ν Lagrangian, LCC = 2 2GF ¯li γ μ LUij νj Jμ + h.c., where L = 12 (1 − γ5 ), {Uij } denotes the MNS matrix while Jμ = Vud u ¯L γμ dL for pion decay.
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The amplitude for π(p) → li (q)νj (k), i, j = 1, 2, gives ¯li (q) pLUij vνj (k), −iM(π → li ν¯j ) = 2GF Fπ Vud u
(4)
√ by using 0|¯ uγ5 γμ d|π − = ipμ 2Fπ , where Fπ ≈ 92 MeV is the pion decay constant. Let ˜ ij = u us denote the spinor-dependent amplitude as M ¯li (q) pLUij vνj (k). The amplitude squared summed over all spins yields (5)
˜ ij |2 = 4(p · qi )(p · kj )−2p2 (qi · kj ) = Mi2 (Mπ2 − Mi2 )+ m2j (Mπ2 + 2Mi2 − m2j ). |M
spins
We can calculate the amplitude squared, summed over the neutrino spin, but depenˆ of the charged lepton in its rest frame dent on the polarization n (6)
Pij (ni ) ≡
˜ ij |2 = M 2 [qi · kj + Mi (kj · ni )]+2M 2 m2 + m2 [qi · kj − Mi (kj · ni )], |M i i j j
νj spin
ˆ )ˆ ˆ = hi q ˆ , we − 1 (ˆ n·q q . For the particular directions n single out the positive (hi = 1) and negative (hi = −1) helicity for the charged lepton. For the pion at rest we obtain
where nμi =
(7)
q·ˆ n ˆ Mi , n
+
Eli Mi
Pij (hi = 1) = Mi2 (Mπ2 − Mi2 ) + O(m2j ),
Pij (hi = −1) = m2j
Mπ4 + O(m4j ). Mπ2 − Mi2
Considering numerical values for Mi = Mμ and Mi = Me , respectively, the ratio between the squared amplitudes for left-handed (h = −1) and right-handed (h = 1) helicities is (8)
Rμj =
m2j × 4.92 × 10−6 , (100 keV)2
Rej =
m2j × 3.83 × 10−6 . (1 eV)2
Considering the actual direct bounds for the neutrino masses in eq. (2), we need a precision of 10−6 in the helicity measurement to reach those bounds either in the case of muons or electrons. Therefore, the coefficient δ in eq. (3) has exactly the modulus (9)
If we rewrite |δμj | =
|δμj |2 = Rμj = m j Mπ 2Eνj Mμ ,
where Eνj =
m2j Mπ4 . Mμ2 (Mπ2 − Mμ2 )2 2 Mπ2 −Mμ 2Mπ
+O(m2j ), we see that |δμj | is modified
π by the factor M Mμ when compared to the naive estimate mν /2Eν . We can also conclude that for the channel π → e¯ ν the real factor is enhanced considerably (∼ 274×).
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C. C. Nishi
Considering the leptonic mixing, the measurement of the wrong helicity for muons probes (10)
|M(π → μ¯ ν : hμ = −1)|2 = |C|2
Mπ4 (m2 )eff , Mπ2 − Mμ2 νμ
where C ≡ 2GF Fπ Vud and (m2νμ )eff ≡ j |Uμj |2 m2j , is an effective mass for the muon neutrino, analogous to m2β [3] inferred from the tritium beta decay experiments for the electron neutrino. In fact, m2β can be extracted from π − → e− ν¯ by measuring the electron with negative helicity. REFERENCES [1] Seljak U., Slosar A. and McDonald P., J. Cosmol. Astropart. Phys., 0610 (2006) 014 (arXiv:astro-ph/0604335). [2] Eidelman S. et al. (Particle Data Group), Phys. Lett. B, 592 (2004) 1. [3] Giunti C., Acta Phys. Polonica B, 36 (2005) 3215 (arXiv:hep-ph/0511131). [4] Bornschein L. (KATRIN Collaboration), KATRIN: Direct measurement of neutrino masses in the sub-eV region, in Proceedings of 23rd International Conference on Physics in Collision (PIC 2003), Zeuthen, Germany, 26-28 Jun 2003, pp FRAP14 (arXiv:hepex/0309007). [5] Shrock R. E., Phys. Lett. B, 96 (1980) 159; Phys. Rev. D, 24 (1981) 1232. [6] Nishi C. C., Mod. Phys. Lett. A, 24 (2009) 219. [7] Goldhaber M., Grodzins L. and Sunyar A. W., Phys. Rev., 109 (1958) 1015.
DOI 10.3254/978-1-60750-038-4-447
First year of Borexino data acquisition: Background analysis P. Risso(∗ ) on behalf of the Borexino Collaboration Dipartimento di Fisica, Universit` a di Genova and INFN, Sezione di Genova - Italy
Summary. — Borexino is a large-scale real-time detector for sub-MeV solarneutrino spectroscopy. It is located underground at the Laboratori Nazionali del Gran Sasso, LNGS. Its main application is the flux measurement of mono-energetic (862 keV) 7 Be neutrinos as well as pp and CNO solar-neutrino fluxes. The detection mechanism is neutrino-electron elastic scattering, ES, in ultra-pure organic liquid scintillator. Internal, external, and cosmogenic backgrounds are under detailed analysis by means of energy and position reconstruction with techniques like shape analysis of scintillation light and fast-coincidences tagging.
1. – Introduction Borexino has been built by a collaboration of 80 physicists of 14 different Institutions, coming from Italy, France, Germany, Poland, Russia and USA. The idea of studying realtime solar neutrinos, with energies below 2 MeV was born in 1990. First feasibility studies indicated a required radio-purity of materials of the order of 10−16 g/g to be sensitive at 2 MeV thresholds. After some years spent in R&D of purification procedures, the collaboration built at LNGS a prototype detector, named Counting Test Facility (CTF). This detector showed that in principle extreme radio-purity levels were achievable. In 1996 the Borexino project was approved. 2. – Physics of Borexino In spite of the extensive investigation carried on by experiments like SuperKamiokande or SNO, many issues concerning solar neutrinos still remain unsolved. Rel(∗ ) E-mail: [email protected] c Societ` a Italiana di Fisica
447
448
P. Risso on behalf of the Borexino Collaboration
ative rates at which various nuclear reactions in the Sun produce neutrinos are in many cases theoretically uncertain at a level of 10% or greater. Moreover, neutrinos with E < 2 MeV, that constitute ∼ 99% of the solar ν flux and are so critical for neutrino oscillations, have been observed only in radiochemical experiments, not in a real time data taking experiments. Low-energy neutrinos have an oscillation wavelength much smaller than the Sun core, so their observation probability is of the order of 57%. For the high-energy ones there are no oscillation phenomena, as MSW effect explains, and their probability is of the order of 31%. The neutrino fluxes dominating in the transition region around 2 MeV, between vacuum oscillations and MSW effects, are coming from 7 Be and pep. Until May 2007, when Borexino started taking data, no experiment had been able to inspect this region and reduce these uncertainties. 3. – Borexino detector The main challenge [1] is the suppression of the background derived not only from the surrounding environment but also from the detector materials and the scintillator itself. Its design is based on a gradual shielding, with inner shells of increasing radio-purity. In fig. 1 a schematic view of the detector is shown. The detection mechanism is based on the elastic scattering of ν on electrons in the scintillator. The induced photons carry energy information but lose the directional one. For instance, mono-energetic 7 Be ν’s of 0.862 MeV produce a Compton-like edge with recoil electrons of 0.665 MeV. The liquid scintillator adopted is PC/PPO because of elevate scintillation yield (104 photons/MeV), high transparency (mean free path is of the order of 10 m) and fast decay time (3 ns). The latter is critical since it allows the distinction between scintillation caused by α vs. β/γ events. The inner vessel (IV) is made of a 125 μm thick nylon layer, of 4.2 m radius, chosen for its good optical clearance, mechanical strength and high radio-purity. It separates the scintillating liquid from the buffer liquid, a mixture of PC and DMP, a light quencher. The presence of another nylon vessel in the buffer region, placed at a radius of 6 m, acts as former barrier, to prevent inner diffusion of radioactive contaminants coming from Stainless Steel Sphere (SSS), and the PMTs. The total mass of the liquid scintillator inside the IV is of the order of 280 ton. To increase rejection from natural radioactivity, we chose to analyse data spatially reconstructed in an inner sphere of 3 m of radius, called Fiducial Volume, FV. Its mass is equal to 100 ton. 2212 PMTs are uniformly distributed over the inner surface of the sphere, 384 of them are installed without a light concentrator. Gran Sasso rock leads to a 106 reduction of muon rate, but still high-energy μ can reach the detector, with a daily rate of 4 × 103 . The sphere is surrounded by a water tank that acts as Cherenkov detector, with 208 PMTs, same model as in the SSS, installed. Combining information coming from this outer muon detector and from the inner one, we reach a muon rejection efficiency bigger than 99.5%.
First year of Borexino data acquisition: Background analysis
Fig. 1
449
Fig. 2
Fig. 1. – Borexino view. Fig. 2. – Energy spectrum of data.
4. – Background components The 7 Be ν event rate in the 100 ton FV is expected to be of the order of 50 events/day. Since scintillation phenomena do not permit to distinguish between neutrino signal and background, we evaluated the most important radio contaminants with detailed studies a priori via CTF. The background levels measured in the detector are in general better than expected. 232 Th family background, investigated by the 212 Bi-212 Po (τ = 433 ns) sequence, assuming a secular equilibrium and excluding nylon IV surface-emanated 220 Rn events, indicates an intrinsic scintillator contamination of (6.8 ± 1.5)10−18 g/g. 238 U family contaminations have been investigated by detecting the coincidence 214 214 Bi- Po (τ = 236 μs). Restricting the analysis to the FV region, minimizing the pollution from IV emanation and assuming secular equilibrium, the contamination is (1.6 ± 0.3)10−17 g/g. 222 Rn belongs to this radioactive chain but it is also a common gas, present also in LNGS cavern. It is impossible, up to now, to distinguish its contribution to pollution from a complete 238 U chain, so results are kept together. Since now 210 Po is decaying matching the half-life of about 200 days; these results support the hypothesis that the mother nuclide 210 Bi contamination should be very small. It is difficult to estimate its concentration, so its value is kept like a free parameter of the fit of the 7 Be spectrum. 85 Kr, an obstacle for low-energy solar ν spectroscopy, is studied directly via 85 Kr85 mRb-85 Rb, (τ = 1.46 ns, BR = 0.43%). Our best estimation for the total activity of 85 Kr is (29 ± 14) counts/(day·100 ton). 14 C (β − Q = 156 keV) is present in the detector coming from the organic scintillator solvent. Its decay is useful to study detector energy response and to calibrate spatial reconstruction, however it limits the detection of neutrinos with energies less than 200 keV. 11 C (β + Q = 1.98 MeV) is created by residual cosmic rays under Gran Sasso Mountain
450
P. Risso on behalf of the Borexino Collaboration
Fig. 3
Fig. 4 210
Po.
Fig. 4. – Energy spectrum without
210
Fig. 3. – Energy spectrum with
Po.
and is identified with the three-fold coincidence with the muon parent, the associated neutron emission/capture and the β + decay. 5. – Data analysis and results Now we show the energy spectrum of data [2]. The X-axis unit is Borexino charge, the number of photo-electrons that constitute a scintillation event. To fit the spectrum in the energy window of 2 MeV, the following restrictions were applied: removal of all muons and all events within a time window of 2 ms after; inclusion of events belonging to a unique cluster, to avoid pile-up; removal of all events coming from 222 Rn daughters; events must be reconstructed within the FV to reject external gamma background and the z-coordinate satisfy |z| < 1.7 m, to remove background near the poles of the IV. Figure 2 shows the effect of the above criteria starting from the total raw spectrum, “no cuts” line. The curve “after μ cut” shows the subtraction of the muons and their correlated activities, while the last one shows the effects of FV cuts and of removal of 238 U/222 Rn daughters. The shape distribution is in remarkable agreement with what we can expect from the most important signal and unavoidable background: 7 Be ν, 14 C, 11 C. Below 100 p.e., or 200 keV, the spectrum is dominated by 14 C intrinsic to the scintillator, and the peak at 200 p.e. is due to the 210 Po. Prominent features of the spectrum are the Compton-like edge due to 7 Be solar neutrinos (300–350 p.e.) and the spectrum of 11 C. 85 Kr component was constrained in the region between the 14 C end-point and the 210 Po peak. The submitted analysis of 192 days of lifetime is focused on the 7 Be region. The shoulder in the spectrum between 560 and 800 keV was fitted considering the expected
First year of Borexino data acquisition: Background analysis
451
Compton-like shape of the electron-recoil spectrum. The pep neutrino components were fixed to the expected SSM/LMA value and the contributions from CNO solar neutrino and from 210 Bi were taken as a single free parameter in the fit. No other background component was included in the final fit because they were found negligible (fig. 3). A second fit was performed with the α statistical subtraction, via the α/β discrimination based upon the Gatti parameter (reference in parentheses) (fig. 4). Our results, for the solar neutrinos from the electron-capture decay of 7 Be neutrinos, is (49 ± 3 stat ± 4 sys) counts/(day·100 ton), outcome fully compatible with MSW-LMA scenario. The statistical error is determined by the χ2 profile, the systematic errors are mostly dominated by the uncertainty in the FV mass definition as by software cut and the uncertainty in the detector response function. 6. – Future perspectives To improve the measure, Borexino goal is to reduce the global systematic uncertainty to values below ±5% (1σ), performing a deployment of known radiation sources, inside the inner vessel, to calibrate position reconstruction algorithms and to study the in-depth detector energy response function. In addition to the 7 Be flux, we entrust our ability to measure also the pep, CNO and 8 B fluxes: the main problem for the study of these two fluxes is the production, of μ-induced 11 C nuclides within the scintillator. Also pp flux could be studied because the 14 C spectrum ends well below 200 keV. Another subject concerns the search for anti-ν from the Sun, Earth (geoneutrinos) and nuclear reactors. We expect 7–17 events per year for geoneutrinos, depending on the Earth models. The background due to nuclear reactors is particularly favourable at Gran Sasso, since they are absent in Italy. Borexino is also a good observatory for a Supernova explosion in our Galaxy. It is almost unique for measuring ν-p elastic scattering. Due to quenching, the p is pushed toward the low energies and, to detect it, a very low threshold is needed. REFERENCES [1] Borexino Collaboration, The Borexino detector at the Laboratori Nazionali del Gran Sasso, arXiv:0806.2400 [physics.ins-det]. [2] Borexino Collaboration, New results on solar neutrino fluxes from 192 days of Borexino data, arXiv:0805.3843 [astro-ph].
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International School of Physics “Enrico Fermi” Villa Monastero, Varenna Course CLXX 17–27 June 2008
“Measurements of Neutrino Mass” Directors Fernando FERRONI Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-49914613 Fax: ++39-06-4957697 [email protected] Francesco VISSANI Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-437-205 Fax: ++39-0862-437-570 [email protected]
Scientific Secretary Chiara BROFFERIO Dipartimento di Fisica Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-6448-2426 Fax: ++39-02-6448-2463 [email protected] c Societ` a Italiana di Fisica
Lecturers Flavio GATTI INFN, Sezione di Genova e Dipartimento di Fisica Universit` a di Genova Via Dodecaneso 33 16146 Genova Italy Tel.: ++39-010-3536280/461 Fax: ++39-010-3536499 [email protected] Andrea Ernesto Guido GIULIANI INFN, Sezione di Milano-Bicocca e Dipartimento di Fisica e Matematica Universit` a dell’Insubria Via Valleggio 11 22100 Como Italy Tel.: ++39-031-238-6217 Fax: ++39-031-238-6119 [email protected] Sergio PASTOR IFIC-Instituto de Fisica Corpuscular Edificio Institutos de Investigaci´ on Apartado de Correos 22085 E-46071 Valencia Spain Tel. ++34-963543510 [email protected] 453
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´ Goran SENJANOVIC ICTP-The International Centre for Theoretical Physics High Energy, Cosmology and Astroparticle Physics Section Strada Costiera 11, Miramare 34014 Trieste Italy Tel. ++39-040-2240303 Fax ++39-040-2240304 [email protected] Alessandro STRUMIA Dipartimento di Fisica “E. Fermi” Universit` a di Pisa Largo B. Pontecorvo 3 56127 Pisa Italy Tel.: ++39-050-2214905 Fax: ++39-050-2214887 [email protected] Petr VOGEL Mail Stop 106-38 Kellogg Radiation Laboratory and Physics Department California Institute of Technology Caltech 1200 E. California Blvd Pasadena, CA 91125 USA Tel.: ++1-626-395-4303 Fax: ++1-626-564-8708 [email protected] Christian WEINHEIMER Institut f¨ ur Kernphysik Westf¨alischen Wilhelms-Universit¨ at M¨ unster Wilhelm-Klemm-Strasse 9 D-48149 M¨ unster Germany Tel: ++49-251-833-4970 Fax ++49-251-8334962 [email protected]
Elenco dei partecipanti
Seminar Speakers Alessandro BETTINI INFN, Sezione di Padova e Dipartimento di Fisica “G. Galilei” Universit` a di Padova Via F. Marzolo 8 35131 Padova Italy Tel.: ++39-049-8277090 Fax: ++39-049-8277102 [email protected] Paolo DE BERNARDIS Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-49914271 Fax: ++39-06-4957697 [email protected] Ettore FIORINI Dipartimento di Fisica “G. Occhialini” Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-64482424/2340 Fax: ++39-02-64482463 [email protected] Dieter FREKERS Institut f¨ ur Kernphysik Westf¨alischen Wilhelms-Universit¨ at M¨ unster Wilhelm-Klemm Strasse 9 D-48149 M¨ unster Germany Tel.: ++49-251-833-4996 Fax: ++49-251-833-4962 [email protected]
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Elenco dei partecipanti Alfredo POVES Departamento de F´ısica Te´orica Universidad Aut´ onoma de Madrid Carretera Colmenar Viejo km. 15,500 E-28049 Madrid Spain Tel.: ++34-91-497-4883 Fax: ++34-91-497-3936 [email protected] Antonio RIOTTO INFN, Sezione di Padova e Dipartimento di Fisica “G. Galilei” Universit` a di Padova Via F. Marzolo 8 35131 Padova Italy Tel.: ++39-049-8277256 Fax: ++39-049-8277102 [email protected] Alexei SMIRNOV ICTP-The International Centre for Theoretical Physics High Energy, Cosmology and Astroparticle Physics Section Strada Costiera 11, Miramare 34014 Trieste Italy Tel.: ++39-040-2240413 Fax: ++39-040-2240304 [email protected] [email protected]
Students Mario Andres ACERO ORTEGA Dipartimento di Fisica Universit` a di Torino Via Giuria 1 10125 Torino Italy Tel.: ++39-011-6707066 [email protected]
Maddalena ANTONELLO Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-437517 Fax: ++39-0862-437-570 [email protected]
Daniela BAGLIANI INFN, Sezione di Genova e Dipartimento di Fisica Universit` a di Genova Via Dodecaneso 33 16146 Genova Italy Tel.: ++39-010-3536333 Fax: ++39-010-3536499 [email protected]
Fabio BELLINI Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-49914338 Fax: ++39-06-4463158 [email protected]
Adam BRYANT Lawrence Berkeley National Laboratory 1 Cyclotron Rd Mail Stop 50R5008 Berkeley, CA 94720-8185 USA Tel.: ++1-510-4865017 Fax: ++1 510 4866738 adam [email protected]
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Fabio CAPPELLA Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-72594825 [email protected]
Enrico CARRARA Dipartimento di Fisica “G. Galilei” Universit` a di Padova Via F. Marzolo 8 35131 Padova Italy [email protected]
Marco Andrea CARRETTONI Dipartimento di Fisica Universit` a di Milano-Bicocca e INFN, Sezione di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-6448-2463 [email protected]
Matteo DE GERONE INFN, Sezione di Genova e Dipartimento di Fisica Universit` a di Genova Via Dodecaneso 33 16146 Genova Italy Tel.: ++39-010-3536333 Fax: ++39-010-3536499 [email protected]
Elenco dei partecipanti Sergio DI DOMIZIO INFN, Sezione di Genova e Dipartimento di Fisica Universit` a di Genova Via Dodecaneso 33 16146 Genova Italy Tel.: ++39-010-3536468 Fax: ++39-010-314218 [email protected]
Donato DI FERDINANDO INFN, Sezione di Bologna Viale Berti Pichat 6/2 40127 Bologna Italy Tel.: ++39-051-2095230 Fax: ++39-051-22095269 [email protected]
Assunta DI VACRI Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-437532 Fax: ++39-0862-437-570 [email protected]
Michelle DOLINSKI Department of Physics University of California 94720 Berkeley, CA USA Tel.: ++1-510-2823523 Fax: ++1-510-4866738 [email protected]
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Elenco dei partecipanti Elena FERRI Dipartimento di Fisica Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-64482463 [email protected]
Eike FRANK Laboratorium fur Hochenergiephysik Universit¨ at Bern Sidlerstrasse 5 3012 Bern Switzerland Tel.: ++41-31-6314064 Fax: ++41-31-6314487 [email protected]
Ivan GNESI Dipartimento di Fisica Universit` a di Torino Via Giuria 1 10125 Torino Italy Tel.: ++39-011-6707049 Fax: ++39-011-6707269 [email protected]
Pawin ITTISAMAI ICTP-The International Centre for Theoretical Physics High Energy, Cosmology and Astroparticle Physics Section Strada Costiera 11, Miramare 34014 Trieste Italy [email protected]
Andrea GIACHERO Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-437318 Fax: ++39-0862-437570 [email protected]
Laura KOGLER Lawrence Berkeley National Laboratory 1 Cyclotron Rd Mail Stop 50B5239 94720 Berkeley, CA USA Tel.: ++1-510-4865017 Fax: ++1-510-4866738 [email protected]
Luca GIRONI Dipartimento di Fisica Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-64482463 [email protected]
Conradin LANGBRANDTNER MPI f¨ ur Kernphysik Saupfercheckweg 1 69117 Heidelberg Germany Tel.: ++49-6221-516472 Fax: ++49-6221-516802 [email protected]
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Jing LIU MPI M¨ unchen Room 114, Fohringer Ring 6 80805 Munchen Germany Tel.: ++49-89-32354415 [email protected]
Ana Amelia MACHADO Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-4203 Fax: ++39-0862-437570 [email protected]
Elenco dei partecipanti Celso Chikahiro NISHI Departamento de Raios C´osmicos e Cronologia Instituto de Fisica Gleb Watagin Universitdade Estadual de Campinas 13083-970 Campinas, SP Brazil Tel.: ++55-19-3521-5270 [email protected] Filippo ORIO Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-49914338 Fax: ++39-06-4463158 [email protected]
Cecilia Giovanna MAIANO Dipartimento di Fisica Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-64482435 Fax: ++39-02-64482463 [email protected]
Giulia PAGLIAROLI Istituto Nazionale di Fisica Nucleare Laboratori Nazionali del Gran Sasso Strada Statale 17/bis, km. 18+910 67010 Assergi (AQ) Italy Tel.: ++39-0862-437499 Fax: ++39-0862-437570 [email protected]
Miha NEMEVSEK Jozef Stefan Institute Jamova cesta 39 1000 Ljubljana Slovenia Tel.: ++386-1-4773334 Fax: ++386-1-4773900 [email protected]
Luca Maria PATTAVINA Dipartimento di Fisica Universit` a di Milano-Bicocca Piazza della Scienza 3 20126 Milano Italy Tel.: ++39-02-64482435 Fax: ++39-02-64482463 [email protected]
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Elenco dei partecipanti Marisa PEDRETTI INFN, Sezione di Milano-Bicocca e Dipartimento di Fisica e Matematica Universit` a dell’Insubria Via Valleggio 11 22100 Como Italy Tel.: ++39-031-2386233 Fax: ++39-031-2386209 [email protected]
Paolo RISSO INFN, Sezione di Genova e Dipartimento di Fisica Universit` a di Genova Via Dodecaneso 33 16146 Genova Italy [email protected]
Vladimir TELLO ICTP-The International Centre for Theoretical Physics High Energy, Cosmology and Astroparticle Physics Section Strada Costiera 11, Mirama¡re 34014 Trieste Italy [email protected] Marco VIGNATI Dipartimento di Fisica “G. Marconi” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 00185 Roma Italy Tel.: ++39-06-49914338 Fax: ++39-06-4463158 [email protected]
Observers Chiara SALVIONI INFN, Sezione di Milano-Bicocca e Dipartimento di Fisica e Matematica Universit` a dell’Insubria Via Valleggio 11 22100 Como Italy Tel.: ++39-031-2386245 Fax: ++39-031-2386209 [email protected]
Gabriele SIRRI INFN, Sezione di Bologna Viale Berti Pichat 6/2 40127 Bologna Italy Tel.: ++39-051-2095228 Fax: ++39-051-2095269 [email protected]
Philip M. GORE 629 Country Club Lane 37205 Nashville, TE USA Tel.: ++1-615-3566470 [email protected] Elisabeth F. JONES 629 Country Club Lane 37205 Nashville, TE USA Tel.: ++1-615-3568848 [email protected] Xavier-Francois NAVICK Centre d’Etudes de Saclay CEA/IRFU/SEDI bat534 91191 Gif sur Yvette France Tel.: ++33-16908-9442 Fax: ++33-16908-024 [email protected]
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PROCEEDINGS OF THE INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”
Course I (1953) Questioni relative alla rivelazione delle particelle elementari, con particolare riguardo alla radiazione cosmica edited by G. Puppi Course II (1954) Questioni relative alla rivelazione delle particelle elementari, e alle loro interazioni con particolare riguardo alle particelle artificialmente prodotte ed accelerate edited by G. Puppi Course III (1955) Questioni di struttura nucleare e dei processi nucleari alle basse energie edited by C. Salvetti Course IV (1956) Propriet` a magnetiche della materia edited by L. Giulotto Course V (1957) Fisica dello stato solido edited by F. Fumi Course VI (1958) Fisica del plasma e relative applicazioni astrofisiche edited by G. Righini Course VII (1958) Teoria della informazione edited by E. R. Caianiello
Course XIII (1959) Physics of Plasma: Experiments and Techniques ´n edited by H. Alfve Course XIV (1960) Ergodic Theories edited by P. Caldirola Course XV (1960) Nuclear Spectroscopy edited by G. Racah Course XVI (1960) Physicomathematical Aspects of Biology edited by N. Rashevsky Course XVII (1960) Topics of Radiofrequency Spectroscopy edited by A. Gozzini Course XVIII (1960) Physics of Solids (Radiation Damage in Solids) edited by D. S. Billington Course XIX (1961) Cosmic Rays, Solar Particles and Space Research edited by B. Peters Course XX (1961) Evidence for Gravitational Theories edited by C. Møller
Course VIII (1958) Problemi matematici della teoria quantistica delle particelle e dei campi edited by A. Borsellino
Course XXI (1961) Liquid Helium edited by G. Careri
Course IX (1958) Fisica dei pioni edited by B. Touschek
Course XXII (1961) Semiconductors edited by R. A. Smith
Course X (1959) Thermodynamics of Irreversible Processes edited by S. R. de Groot
Course XXIII (1961) Nuclear Physics edited by V. F. Weisskopf
Course XI (1959) Weak Interactions edited by L. A. Radicati
Course XXIV (1962) Space Exploration and the Solar System edited by B. Rossi
Course XII (1959) Solar Radioastronomy edited by G. Righini
Course XXV (1962) Advanced Plasma Theory edited by M. N. Rosenbluth
Course XXVI (1962) Selected Topics on Elementary Particle Physics edited by M. Conversi Course XXVII (1962) Dispersion and Absorption of Sound by Molecular Processes edited by D. Sette Course XXVIII (1962) Star Evolution edited by L. Gratton Course XXIX (1963) Dispersion Relations and their Connection with Casuality edited by E. P. Wigner Course XXX (1963) Radiation Dosimetry edited by F. W. Spiers and G. W. Reed Course XXXI (1963) Quantum Electronics and Coherent Light edited by C. H. Townes and P. A. Miles Course XXXII (1964) Weak Interactions and High-Energy Neutrino Physics edited by T. D. Lee Course XXXIII (1964) Strong Interactions edited by L. W. Alvarez Course XXXIV (1965) The Optical Properties of Solids edited by J. Tauc Course XXXV (1965) High-Energy Astrophysics edited by L. Gratton
Course XLI (1967) Selected Topics in Particle Physics edited by J. Steinberger Course XLII (1967) Quantum Optics edited by R. J. Glauber Course XLIII (1968) Processing of Optical Data by Organisms and by Machines edited by W. Reichardt Course XLIV (1968) Molecular Beams and Reaction Kinetics edited by Ch. Schlier Course XLV (1968) Local Quantum Theory edited by R. Jost Course XLVI (1969) Physics with Intersecting Storage Rings edited by B. Touschek Course XLVII (1969) General Relativity and Cosmology edited by R. K. Sachs Course XLVIII (1969) Physics of High Energy Density edited by P. Caldirola and H. Knoepfel Course IL (1970) Foundations of Quantum Mechanics edited by B. d’Espagnat Course L (1970) Mantle and Core in Planetary Physics edited by J. Coulomb and M. Caputo Course LI (1970) Critical Phenomena edited by M. S. Green
Course XXXVI (1965) Many-body Description of Nuclear Structure and Reactions edited by C. L. Bloch
Course LII (1971) Atomic Structure and Properties of Solids edited by E. Burstein
Course XXXVII (1966) Theory of Magnetism in Transition Metals edited by W. Marshall
Course LIII (1971) Developments and Borderlines of Nuclear Physics edited by H. Morinaga
Course XXXVIII (1966) Interaction of High-Energy Particles with Nuclei edited by T. E. O. Ericson
Course LIV (1971) Developments in High-Energy Physics edited by R. R. Gatto
Course XXXIX (1966) Plasma Astrophysics edited by P. A. Sturrock
Course LV (1972) Lattice Dynamics and Forces edited by S. Califano
Course XL (1967) Nuclear Structure and Nuclear Reactions edited by M. Jean and R. A. Ricci
Course LVI (1972) Experimental Gravitation edited by B. Bertotti
Intermolecular
Course LVII (1972) History of 20th Century Physics edited by C. Weiner
Course LXXII (1977) Problems in the Foundations of Physics edited by G. Toraldo di Francia
Course LVIII (1973) Dynamics Aspects of Surface Physics edited by F. O. Goodman
Course LXXIII (1978) Early Solar System Processes and the Present Solar System edited by D. Lal
Course LIX (1973) Local Properties at Phase Transitions ¨ller and A. Rigamonti edited by K. A. Mu Course LX (1973) C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory edited by D. Kastler
Course LXXIV (1978) Development of High-Power Lasers and their Applications edited by C. Pellegrini Course LXXV (1978) Intermolecular Spectroscopy and Dynamical Properties of Dense Systems edited by J. Van Kranendonk
Course LXI (1974) Atomic Structure and Mechanical Properties of Metals edited by G. Caglioti
Course LXXVI (1979) Medical Physics edited by J. R. Greening
Course LXII (1974) Nuclear Spectroscopy and Nuclear Reactions with Heavy Ions edited by H. Faraggi and R. A. Ricci
Course LXXVII (1979) Nuclear Structure and Heavy-Ion Collisions edited by R. A. Broglia, R. A. Ricci and C. H. Dasso
Course LXIII (1974) New Directions in Physical Acoustics edited by D. Sette
Course LXXVIII (1979) Physics of the Earth’s Interior edited by A. M. Dziewonski and E. Boschi
Course LXIV (1975) Nonlinear Spectroscopy edited by N. Bloembergen
Course LXXIX (1980) From Nuclei to Particles edited by A. Molinari
Course LXV (1975) Physics and Astrophysics of Neutron Stars and Black Hole edited by R. Giacconi and R. Ruffini
Course LXXX (1980) Topics in Ocean Physics edited by A. R. Osborne and P. Malanotte Rizzoli
Course LXVI (1975) Health and Medical Physics edited by J. Baarli
Course LXXXI (1980) Theory of Fundamental Interactions edited by G. Costa and R. R. Gatto
Course LXVII (1976) Isolated Gravitating Systems in General Relativity edited by J. Ehlers
Course LXXXII (1981) Mechanical and Thermal Behaviour of Metallic Materials edited by G. Caglioti and A. Ferro Milone
Course LXVIII (1976) Metrology and Fundamental Constants edited by A. Ferro Milone, P. Giacomo and S. Leschiutta
Course LXXXIII (1981) Positrons in Solids edited by W. Brandt and A. Dupasquier
Course LXIX (1976) Elementary Modes of Excitation in Nuclei edited by A. Bohr and R. A. Broglia
Course LXXXIV (1981) Data Acquisition in High-Energy Physics edited by G. Bologna and M. Vincelli
Course LXX (1977) Physics of Magnetic Garnets edited by A. Paoletti
Course LXXXV (1982) Earthquakes: Observation, Theory and Interpretation edited by H. Kanamori and E. Boschi
Course LXXI (1977) Weak Interactions edited by M. Baldo Ceolin
Course LXXXVI (1982) Gamow Cosmology edited by F. Melchiorri and R. Ruffini
Course LXXXVII (1982) Nuclear Structure and Heavy-Ion Dynamics edited by L. Moretto and R. A. Ricci
Course CII (1986) Accelerated Life Testing and Experts’ Opinions in Reliability edited by C. A. Clarotti and D. V. Lindley
Course LXXXVIII (1983) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics edited by M. Ghil, R. Benzi and G. Parisi
Course CIII (1987) Trends in Nuclear Physics edited by P. Kienle, R. A. Ricci and A. Rubbino
Course LXXXIX (1983) Highlights of Condensed Matter Theory edited by F. Bassani, F. Fumi and M. P. Tosi
Course CIV (1987) Frontiers and Borderlines in ManyParticle Physics edited by R. A. Broglia and J. R. Schrieffer
Course XC (1983) Physics of Amphiphiles: Micelles, Vesicles and Microemulsions edited by V. Degiorgio and M. Corti Course XCI (1984) From Nuclei to Stars edited by A. Molinari and R. A. Ricci Course XCII (1984) Elementary Particles edited by N. Cabibbo Course XCIII (1984) Frontiers in Physical Acoustics edited by D. Sette Course XCIV (1984) Theory of Reliability edited by A. Serra and R. E. Barlow Course XCV (1985) Solar-Terrestrial Relationship and the Earth Environment in the Last Millennia edited by G. Cini Castagnoli
Course CV (1987) Confrontation between Theories and Observations in Cosmology: Present Status and Future Programmes edited by J. Audouze and F. Melchiorri Course CVI (1988) Current Trends in the Physics of Materials edited by G. F. Chiarotti, F. Fumi and M. Tosi Course CVII (1988) The Chemical Physics of Atomic and Molecular Clusters edited by G. Scoles Course CVIII (1988) Photoemission and Absorption Spectroscopy of Solids and Interfaces with Synchrotron Radiation edited by M. Campagna and R. Rosei
Course XCVI (1985) Excited-State Spectroscopy in Solids edited by U. M. Grassano and N. Terzi
Course CIX (1988) Nonlinear Topics in Ocean Physics edited by A. R. Osborne
Course XCVII (1985) Molecular-Dynamics Simulations of Statistical-Mechanical Systems edited by G. Ciccotti and W. G. Hoover
Course CX (1989) Metrology at the Frontiers of Physics and Technology edited by L. Crovini and T. J. Quinn
Course XCVIII (1985) The Evolution of Small Bodies in the Solar System ˇ Kresa `k edited by M. Fulchignoni and L.
Course CXI (1989) Solid-State Astrophysics edited by E. Bussoletti and G. Strazzulla
Course XCIX (1986) Synergetics and Dynamic Instabilities edited by G. Caglioti and H. Haken
Course CXII (1989) Nuclear Collisions from the Mean-Field into the Fragmentation Regime edited by C. Detraz and P. Kienle
Course C (1986) The Physics of NMR Spectroscopy in Biology and Medicine edited by B. Maraviglia
Course CXIII (1989) High-Pressure Equation of State: Theory and Applications edited by S. Eliezer and R. A. Ricci
Course CI (1986) Evolution of Interstellar Dust and Related Topics edited by A. Bonetti and J. M. Greenberg
Course CXIV (1990) Industrial and Technological Applications of Neutrons edited by M. Fontana and F. Rustichelli
Course CXV (1990) The Use of EOS for Studies of Atmospheric Physics edited by J. C. Gille and G. Visconti
Course CXXIX1 (1994) Observation, Prediction and Simulation of Phase Transitions in Complex Fluids edited by M. Baus, L. F. Rull and J. P. Ryckaert
Course CXVI (1990) Status and Perspectives of Nuclear Energy: Fission and Fusion edited by R. A. Ricci, C. Salvetti and E. Sindoni
Course CXXX (1995) Selected Topics in Nonperturbative QCD edited by A. Di Giacomo and D. Diakonov
Course CXVII (1991) Semiconductor Superlattices and Interfaces edited by A. Stella Course CXVIII (1991) Laser Manipulation of Atoms and Ions edited by E. Arimondo, W. D. Phillips and F. Strumia
Course CXXXI (1995) Coherent and Collective Interactions of Particles and Radiation Beams edited by A. Aspect, W. Barletta and R. Bonifacio Course CXXXII (1995) Dark Matter in the Universe edited by S. Bonometto and J. Primack
Course CXIX (1991) Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky
Course CXXXIII (1996) Past and Present Variability of the SolarTerrestrial System: Measurement, Data Analysis and Theoretical Models edited by G. Cini Castagnoli and A. Provenzale
Course CXX (1992) Frontiers in Laser Spectroscopy ¨nsch and M. Inguscio edited by T. W. Ha
Course CXXXIV (1996) The Physics of Complex Systems edited by F. Mallamace and H. E. Stanley
Course CXXI (1992) Perspectives in Many-Particle Physics edited by R. A. Broglia, J. R. Schrieffer and P. F. Bortignon
Course CXXXV (1996) The Physics of Diamond edited by A. Paoletti and A. Tucciarone
Course CXXII (1992) Galaxy Formation edited by J. Silk and N. Vittorio
Course CXXXVI (1997) Models and Phenomenology for Conventional and High-Temperature Superconductivity edited by G. Iadonisi, J. R. Schrieffer and M. L. Chiofalo
Course CXXIII (1992) Nuclear Magnetic Double Resonsonance edited by B. Maraviglia Course CXXIV (1993) Diagnostic Tools in Atmospheric Physics edited by G. Fiocco and G. Visconti Course CXXV (1993) Positron Spectroscopy of Solids edited by A. Dupasquier and A. P. Mills jr. Course CXXVI (1993) Nonlinear Optical Materials: Principles and Applications edited by V. Degiorgio and C. Flytzanis Course CXXVII (1994) Quantum Groups and their Applications in Physics edited by L. Castellani and J. Wess Course CXXVIII (1994) Biomedical Applications of Synchrotron Radiation edited by E. Burattini and A. Balerna 1 This
Course CXXXVII (1997) Heavy Flavour Physics: a Probe of Nature’s Grand Design edited by I. Bigi and L. Moroni Course CXXXVIII (1997) Unfolding the Matter of Nuclei edited by A. Molinari and R. A. Ricci Course CXXXIX (1998) Magnetic Resonance and Brain Function: Approaches from Physics edited by B. Maraviglia Course CXL (1998) Bose-Einstein Condensation in Atomic Gases edited by M. Inguscio, S. Stringari and C. E. Wieman Course CXLI (1998) Silicon-Based Microphotonics: from Basics to Applications edited by O. Bisi, S. U. Campisano, L. Pavesi and F. Priolo
course belongs to the NATO ASI Series C, Vol. 460 (Kluwer Academic Publishers).
Course CXLII (1999) Plasmas in the Universe edited by B. Coppi, A. Ferrari and E. Sindoni
Course CLIV (2003) Physics Methods in Archaeometry edited by M. Martini, M. Milazzo and M. Piacentini
Course CXLIII (1999) New Directions in Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky
Course CLV (2003) The Physics of Complex Systems (New Advances and Perspectives) edited by F. Mallamace and H. E. Stanley
Course CXLIV (2000) Nanometer Scale Science and Technology edited by M. Allegrini, N. Garc´ıa and O. Marti
Course CLVI (2003) Research on Physics Education edited by E.F. Redish and M. Vicentini
Course CXLV (2000) Protein Folding, Evolution and Design edited by R. A. Broglia, E. I. Shakhnovich and G. Tiana
Course CLVII (2003) The Electron Liquid Model in Condensed Matter Physics edited by G. F. Giuliani and G. Vignale
Course CXLVI (2000) Recent Advances in Metrology and Fundamental Constants edited by T. J. Quinn, S. Leschiutta and P. Tavella
Course CLVIII (2004) Hadron Physics edited by T. Bressani, U. Wiedner and A. Filippi
Course CXLVII (2001) High Pressure Phenomena edited by R. J. Hemley, G. L. Chiarotti, M. Bernasconi and L. Ulivi
Course CLIX (2004) Background Microwave Radiation and Intracluster Cosmology edited by F. Melchiorri and Y. Rephaeli
Course CXLVIII (2001) Experimental Quantum Computation and Information edited by F. De Martini and C. Monroe
Course CLX (2004) From Nanostructures to Nanosensing Applications edited by A. D’Amico, G. Balestrino and A. Paoletti
Course CXLIX (2001) Organic Nanostructures: Science and Applications edited by V. M. Agranovich and G. C. La Rocca Course CL (2002) Electron and Photon Confinement in Semiconductor Nanostructures ´dran, A. Quatedited by B. Deveaud-Ple tropani and P. Schwendimann Course CLI (2002) Quantum Phenomena in Mesoscopic Systems edited by B. Altshuler, A. Tagliacozzo and V. Tognetti Course CLII (2002) Neutrino Physics edited by E. Bellotti, Y. Declais and P. Strolin Course CLIII (2002) From Nuclei and their Constituents to Stars edited by A. Molinari, L. Riccati, W. M. Alberico and M. Morando
Course CLXI (2005) Polarons in Bulk Materials and Systems with Reduced Dimensionality edited by G. Iadonisi, J. Ranninger and G. De Filippis Course CLXII (2005) Quantum Computers, Algorithms and Chaos edited by G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti Course CLXIII (2005) CP Violation: From Quarks to Leptons edited by M. Giorgi, I. Mannelli, A. I. Sanda, F. Costantini and M. S. Sozzi Course CLXIV (2006) Ultra-Cold Fermi Gases edited by M. Inguscio, W. Ketterle and C. Salomon Course CLXV (2006) Protein Folding and Drug Design edited by R. A. Broglia, L. Serrano and G. Tiana
Course CLXVI (2006) Metrology and Fundamental Constants ¨nsch, S. Leschiutta, A. edited by T. W. Ha J. Wallard and M. L. Rastello
Course CLXVIII (2007) Atom Optics and Space Physics edited by E. Arimondo, W. Ertmer, W. P. Schleich and E. M. Rasel
Course CLXVII (2007) Strangeness and Spin in Fundamental Physics edited by M. Anselmino, T. Bressani, A. Feliciello and Ph. G. Ratcliffe
Course CLXIX (2007) Nuclear Structure far from Stability: New Physics and New Technology edited by A. Covello, F. Iachello, R. A. Ricci and G. Maino
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