Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
Подождите немного. Документ загружается.
474 A Appendix
for E DjEjW
N u
e
i.prEt/
D N
p
jEjCm
e
i.prCjEjt/
(A.53b)
where N D
q
mCjEj
2jEj
is a normalization factor derived from the condition
D
1. It is easy to show that the four solutions (A.53) are mutually orthogonal. The
meaning of the four component spinors u
C
; u
is evident in the limit p ! 0:
u
C
D
0
I u
D
0
: (A.54)
Since ; are eigenstates of
3
, we interpret u
C
; u
as eigenstates of the spin
operator, which in a generalized form is
Q
3
D
3
0
0
3
: (A.55)
Relativistic limit. The relativistic limit is represented by the condition m ! 0I
jEj!p. It easy to show that the spinors u of (A.53) become
u
C
D
D
p
p
(A.56a)
u
D
D
p
p
: (A.56b)
The function in the lower part of the four-dimensional spinor u
C
is the eigenstate
of the helicity operator
p
p
. Indeed, taking the z axis coincident with the direction
of the momentum p D .0;0;p
3
/,wehave
p
p
D
3
D˙; (A.57)
depending on D
1
0
R
,or D
0
1
L
. Similarly, in the upper
spinor component u
represents the helicity eigenstate, though with reversed signs
of the eigenvalues, namely,
p
p
3
D˙: (A.58)
With the choices we made, the two upper and lower components of the four-
dimensional spinor u
C
(u
) are respectively associated with the eigenstates of the
spin and helicity operators (helicity and spin). Below, to highlight the different
orientation of the spins, we will use the notations u
Cr
; u
r
; u
Cl
and u
l
.
A.4 Dirac Equation and Formalism 475
A.4.4 Helicity Operator and States
The two pairs of Dirac equation solutions (A.48) are functions of the momentum p
and correspond to both positive and negative energy eigenvalues. Their eigenstates
are the two component spinors (A.48). Because the four solutions are independent
(as seen in Sect. A.4.2), another independent observable that commutes with the
Hamiltonian exists. This is the generalized helicity
2
operator:
D
p
p
0
0
p
p
!
: (A.59)
The operator commutes with p and the Hamiltonian: Œ; p D 0, Œ; H D 0,
and therefore it represents a constant of motion (Sect. 6.3). The helicity operator
has two eigenvalues, D˙1, which distinguish eigenstates with the same energy
eigenvalue.
Relativistic case (particle with null mass). Let us now verify the effect of the gen-
eralized helicity (A.59) on the helicity eigenstates (A.57) of the spinor u
C
(A.56a).
This corresponds to the relativistic limit where the mass of the particle is negligible.
In particular, if we consider the momentum oriented along the z-axis, one has
p D .0;0;p
3
/,and
u
C
D
0
B
B
@
1000
0 10 0
0010
1001
1
C
C
A
0
B
B
B
@
1
0
p
3
jEjCm
0
1
C
C
C
A
DC
0
B
B
B
@
1
0
p
3
jEjCm
0
1
C
C
C
A
D u
C
: (A.60)
In practice, the spinor u
C
(solution with positive energy) with the spin along the
direction of the momentum is the eigenstate of the operator with eigenvalue
DC1. We call this state .u
C
/
DC1
u
Cr
. Similarly, the other three states will
be u
Cl
; u
r
; u
l
.
With this momentum orientation, it is easy to verify that the operator (A.59) can be
expressed as D
R
C
L
,where
R
1 C
5
2
D
1
2
11
11
(A.61)
L
1
5
2
D
1
2
1 1
11
: (A.62)
2
Note that the polarization and the helicity operators
p
p
are defined on two-component spinors
(separately describing the fermions and antifermions). Sometimes, the generalized helicity opera-
tor is called chirality. The chirality is an operator acting on a four-component spinor.
476 A Appendix
The matrix array
5
is defined in (A.25c). The operators defined above are Hermitian
and have the properties of projection, that is,
R
C
L
D 1,
R
L
D
L
R
D 0,
R
R
D
R
,
L
L
D
L
. Applying the projectors to the eigenstates u
C
and u
(A.56), it is easy to verify that
L
u
Cr
D 0I
R
u
Cr
D
1
1
R
(A.63a)
L
u
Cl
D
1
1
L
I
R
u
Cl
D 0: (A.63b)
Depending on the fact that the four-components spinor u has spin parallel or
antiparallel to the momentum
3
, one has
u
Cr
D
R
u
Cr
(A.64a)
u
Cl
D
L
u
Cl
: (A.64b)
L
is the projection operator of the negative helicity state (or Left-handed, L)and
the operator
R
is the projection operator of the positive helicity state (or Right-
handed, R). Taking into account (A.57, A.58), where the particle mass is m D 0,
the spinor u is intrinsically the eigenstate of helicity C1 or of helicity 1. This is
not true in the case of massive particles.
The projection operators applied to the spinor of negative energy produce similar
effects to (A.63). For jEj <0, the operator
L
extracts from u
4 the state of
positive helicity (right-handed) and
R
the state of negative helicity (left-handed).
Remember that the matrix
0
is the parity operator for a Dirac spinor: u changes
its helicity state under the application of this operator:
0
u D
10
0 1
D
: (A.65)
Particle with mass. Let us assume once again that the momentum p of the particle
is along the z axis and initially consider positive energy solutions (A.48). In this
case, one can show that
p
R
D p
R
I p
L
Dp
L
; (A.66)
3
Sometimes, the states with up and down spins are indicated as spin " or spin #.
A.4 Dirac Equation and Formalism 477
and it can be verified that
L
u
Cl
D
1
2
1 C
p
jEjCm
1
1
L
(A.67a)
R
u
Cl
D
1
2
1
p
jEjCm
1
1
L
: (A.67b)
Being both projections that are not null, u
Cl
D
L
u
Cl
C
R
u
Cl
appears to be
composed of a superposition of positive and negative helicity states, with different
weights. If we define the probability of finding the lepton described by the spinor
u
Cl
in the left and right-handed helicity state with (respectively) the relative weights
P
l
and P
r
,thatis,
P
l
j
L
u
Cl
j
2
D
1
2
jEjCm C p
jEjCm
2
(A.68a)
P
r
j
R
u
Cl
j
2
D
1
2
jEjCm p
jEjCm
2
; (A.68b)
the average helicity of the state can be defined as
hhi
P
r
P
l
P
r
C P
l
: (A.69)
Introducing the (A.68)into(A.69), we obtain that the average helicity for a fermion
(D spinor with positive energy) with left-handed helicity is
hhiD
p
jEj
D
v
c
Dˇ: (A.70)
Using similar calculations, it can be verified that the other spinors are a superpo-
sition of positive and negative helicity states. The corresponding average helicity
values are reported here below.
Particle Antiparticle
u
Cl
u
Cr
u
r
u
Cl
h.mD 0) 1 C1 C1 1
hhi .m ¤ 0) ˇ Cˇ Cˇ ˇ
As discussed in Sect. 8.16, the experiments agree with the fact that neutrinos
(antineutrinos) (case m D 0) have helicity 1(C1), while the electron (positron)
has helicity ˇ (Cˇ). The Hamiltonian describing the interaction must take into
account the suppression of the unobserved states, as it was done in the V A theory.
478 A Appendix
A.5 Physical and Astrophysical Constants [P08]
A.5 Physical and Astrophysical Constants [P08] 479
References
[Note] Writing a book is a complex task, simply because it requires the knowledge of the
existing literature, from which we are inevitably in great debt to. Below is a list of
texts, reviews and some specialized works. The works used in different parts of this
book are referred to within square brackets in the general bibliography section and
ordered by year (the last two characters). Those more specific references are divided
between chapters, whose number is denoted by the first character. We frequently refer
to web pages: in this case, they are indicated by the lower case character [w], followed
by a progressive number. We avoided mentioning old historical papers because they are
generally difficult to find. In the book, the Nobel laureates are quoted with the indication
of the year, and the reference is not duplicated. We strongly recommend reading the
Nobel lectures, which are all available at: http://nobelprize.org/nobel
prizes/physics/
laureates/
General Bibliography
[H84] Halzen, F., Martin, A.D.: Quarks and Leptons. Wiley, New York (1984). ISBN: 978-
0471887416
[S86] Segr
`
e, E.: Nuclei and Particles. Benjamin-Cummings Pub Co. (1977). ISBN: 978-
0805386011. Also in Italian: Nuclei e particelle. Zanichelli, Bologna (1986). ISBN:
978-8808056283
[L87] Leo, W.R.: Techniques for Nuclear and Particle Physics Experiments. Springer, Berlin
(1994). ISBN: 978-3540572800
[P87] Perkins, D.H.: Introduction to High Energy Physics, 4th edn. The Edimburgh building,
Cambridge CB2 8RU, UK (2000). ISBN: 978-0521621960
[K88] Krane, K.S.: Introductory Nuclear Physics. Wiley, New York (1988). ISBN: 978-
0471805533
[A89] Aitchison, I.J.R., Hey, A.J.G.: Gauge Theories in Particle Physics, 3rd edn. Taylor &
Francis, New York/Philadelphia (2002). ISBN: 978-0750308649
[B89] Bachall, J.N.: Neutrino Astrophysics. Cambridge University Press, Cambridge (1989).
ISBN: 978-0521379755. The Solar Standard Model is also described on the website:
http://www.sns.ias.edu/
jnb/
[C89] Cahn, R.N., Goldhaber, G.: The Experimental Foundations of Particle Physics. The
Edimburgh building, Cambridge CB2 8RU, UK (1989). ISBN: 978-0521521475
[G90] Gaisser, T.K.: Cosmic Rays and Particle Physics. Cambridge University Press, Cam-
bridge (1991). ISBN: 978-0521339315
[G91] Guidry, M.: Gauge Field Theories. Wiley, New York (1991). ISBN-13: 978-0471353850
S. Braibant et al., Particles and Fundamental Interactions: An Introduction to Particle
Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-94-007-2464-8,
© Springer Science+Business Media B.V. 2012
481
482 References
[H91] Hughes, I.S.: Elementary Particles, 3rd edn. The Edimburgh building, Cambridge CB2
8RU, UK (1991). ISBN: 978-0521407397
[W91] Williams, W.S.C.: Nuclear and Particle Physics (1991). ISBN:
978-0198520467
[L92] Longair, M.S.: High Energy Astrophysics, 2nd edn. Cambridge University Press,
Cambridge (1992). ISBN: 978-0521387736 (Vol. I); ISBN: 978-0521435840 (Vol. II)
[D94] Das, A., Ferbel, T.: Introduction to Nuclear and Particle Physics. Wiley, New York
(1994). ISBN: 978-0471571322
[M94] Martin, B.R., Shaw, G.: Particle Physics, 2nd edn. Wiley, New York (1997). ISBN: 978-
0471972853
[P95] Povh, B., Rith, K., Scholz, C., Zetsche, F.: Particles and Nuclei. An Introduction to the
Physical Concepts, 6th edn. Springer, Berlin (2008). ISBN: 978-3540793670
[J99] Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1999). ISBN: 978-
0471309321
[G03] Gasiorowicz, S.: Quantum Physics, 3rd edn. Wiley, New York (2003). ISBN: 978-
0471057000
[B04] Basdevant, J.L., Rich J., Spiro, M: Fundamentals in Nuclear Physics. Springer, 2004,
ISBN 0-387-01672-4
[B08] Bettini, A.: Introduction to Elementary Particle Physics. Cambridge University Press,
Cambridge (2008). ISBN: 978-0521880213
[Be08] Bendiscioli, G.: Fenomeni Radioattivi. Springer, Milan (2008) (in Italian). ISBN:978-
8847008038
[P08] Amsler, C., et al., (Particle Data Group).: Review of particle physics. Phys. Lett. B667,
1 (2008)
[W08] Weinberg, S.: Cosmology. Oxford University Press, Oxford (2008)
[P10] Nakamura, K., et al., (Particle Data Group).: Review of particle physics J. Phys. G. 37,
075021 (2010). Phys. Lett. B667, 1 (2008). Available at: http://pdg.lbl.gov/
Chapter 2
[2C70] Charpak, G., Rahm, D. Steiner, M.: Some developments in the operation of multiwire
proportional chambers. Nucl. Instrum. Methods. 80, 13 (1970)
[2S77] Sauli, F.: Principles of Operation of Multiwire Proportional and Drift Chambers.
CERN/EP 77–09. European Organization for Nuclear Research, Geneva (1977). http://
cdsweb.cern.ch/record/117989
[2L07] Lagniel, J.-M.: Status of the hadron therapy projects in Europe. DOI 10.2
109/PAC.2007.4440335. Also in http://hal.in2p3.fr/in2p3-00169261/en/
[2w1] http://www.tera.it/
[2w2] “The Particle Detector BriefBook” http://physics.web.cern.ch/Physics/ParticleDetector/
BriefBook/
Chapter 3
[3w1] Didactic website on bubble chambers (Introduction to the BC): http://teachers.web.cern.
ch/teachers/archiv/HST2005/bubble
chambers/BCwebsite/
[3w2] http://cdsweb.cern.ch/record/39469
Chapter 4
[4T73] ’t Hooft, G., Veltman, M.: Diagrammar. CERN Report 73-9, Geneva (1973). http://
cdsweb.cern.ch/record/186259/files/
[4A08] Avignone, F.T., Elliott, S.R., Engel, J.: Double beta decay, Majorana neutrinos, and neu-
trino mass. Rev. Mod. Phys. 80, 481–516 (2008) arXiv:0708.1033 [nucl-ex]
References 483
Chapter 7
[7S63] Shafer, J.B., et al.: Spin and parity of the 1385-MeV resonance. Phys. Rev. Lett. 10, 179
(1963)
[7G76] Giacomelli, G.: Total cross sections and elastic scattering at high energies. Phy. Rep.
23C, 124 (1976)
[7B79] Barber, D.P., et al.: Test of universality of charged leptons. Phys. Rev. Lett. 43, 1915–
1918 (1979)
[7P04] Perl, M.L., Lee, E.R., Loomba, D.: A brief review of the search for isolatable fractional
charge elementary particles. Mod. Phys. Lett. A. 19, 2595-2610 (2004)
Chapter 8
[8B78] The Pauli letter is extracted from: Brown, L.M.: The idea of the neutrino. Phys. Today
31, 23–28 (1978)
[8B83] Bizzeti, P.G.: Weak interactions in nuclei. La Rivista del Nuovo Cimento 6(12), 1–64
(1983)
[8A84] Ademollo, M.: Cabibbo theory and current algebra. La Rivista del Nuovo Cimento 7(1),
(1984). doi: 10.1007/BF02724333
[8B84] Bertin, A., Vitale, A.: Pure leptonic weak processes. La Rivista del Nuovo Cimento 7(7),
(1984). doi: 10.1007/BF02724331
[8F84] Focardi, S., Ricci, R.A.: Historical introduction. The beta-decay and the fundamental
properties of weak interactions. La Rivista del Nuovo Cimento 6(11), (1983). doi:
10.1007/BF02740918
[8G84] Goggi, G.: Gauge vector bosons and unified electroweak interaction. La Rivista del
Nuovo Cimento 7(4), (1984). doi: 10.1007/BF02724354
[8P84] Palmonari, F.: Nonleptonic weak interactions. La Rivista del Nuovo Cimento 7(9),
(1984). doi: 10.1007/BF02724346
[8Pu84] Pullia, A.: Structure of charged and neutral weak interactions at high energy. La Rivista
del Nuovo Cimento 7(2), (1984). doi: 10.1007/BF02724329
[8V84] Bertin, A., Vitale, A.: Strangeness-conserving semi-leptonic weak processes. La Rivista
del Nuovo Cimento 7(8), (1984). doi: 10.1007/BF02724332
[8G10] Grodzins, L.: The tabletop measurement of the helicity of the neutrino, Il Nuovo
Saggatore 26(5–6), 23 (2010) . On line on: http://prometeo.sif.it/papers/online/sag/026/
05-06/pdf/05-percorsi.pdf
Chapter 9
[9BH82] Bloom, E., Feldman, G.: Quarkonium. Sci Am. 246, 66–77 (1982)
[9M98] Montagna, G., Nicrosini, O., Piccinini, F.: Precision physics at LEP. La Rivista del
Nuovo Cimento 21(9), 1–162 (1998). Also: hep-ph/9802302
[9B00] Brandt, D., Burkhardt, H., Lamont, M., Myers, S., Wenninger, J.: Accelerator physics at
LEP. Rep. Prog. Phys. 63, 939–1000 (2000). Also: CERN-SL-2000-037-DI
[9A01] Abbiendi, G., et al., (OPAL Coll.): Measurement of the mass and width of the
W boson in e+e- collisions at 189 GeV. Phys. Lett. B 507, 29 (2001). Also:
arXiv:hep-ex/0009018v1
[9L03] LEP and SLD Coll. A Combination of preliminary electroweak measurements and
constraints on the Standard Model. LEPEWWG/2003-02, hep-ex/0312023.
[9H04] Heister, A., et al., (ALEPH Coll.): Studies of QCD at e
C
e
centre-of-mass energies
between 91 GeV and 209 GeV. Eur. Phys. J. C 35, 457 (2004)
[9w1] The LEP Electroweak working group. http://lepewwg.web.cern.ch/LEPEWWG/
Welcome.html
[9w2] The LEP QCD annihilations working group. http://lepqcd.web.cern.ch/LEPQCD/
annihilations/
484 References
Chapter 10
[10G93] Giacomelli, G., Giacomelli, P.: Particle searches at LEP. La Rivista del Nuovo Cimento
16(3), 1 (1993). doi: 10.1007/BF02832334
[10P02] Hagiwara, K., et al., (Particle Data Group): Review of particle physics. Phys. Rev. D 66,
010001 (2002)
[10A09] ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD Collaborations, the LEP Working Group,
the Tevatron Electroweak Working Group, and the SLD Electroweak and Heavy flavor
Group. Precision electroweak measurements and constraints on the standard model
arXiv:0911.2604 [hep-ex].
[10C11] The CMS Collaboration. Measurement of W
C
W
production and search for the
Higgs Boson in pp Collisions at
p
s D 7 TeV. Phys. Lett. B699 (2011) 25–47.
arXiv:1102.5429 [hep-ex]
[10L11] LHC Higgs Cross Section Working Group, Dittmaier, S., Mariotti, C., Passarino, G.,
Tanaka, R. (eds.): Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables,
CERN-2011-002 CERN, Geneva (2011), arXiv:1101.0593 [hep-ph]
[10w1] The Physics of RHIC. http://www.bnl.gov/rhic/physics.asp
Chapter 11
[11A84] Altarelli, G.: Future perspectives in particle physics. La Rivista del Nuovo Cimento 7(3),
(1984). doi: 10.1007/BF02724328
[11W04] Wilczek, F.A.: Asymptotic freedom: from paradox to paradigm. http://nobelprize.org/
nobel
prizes/physics/laureates/2004/wilczek-lecture.pdf
Chapter 12
[12M79] Mikheyev, S.P., Smirnov, A.Y.: Nuovo Cimento 19, 17 (1986) ; Wolfenstein, L.: Phys.
Rev. D 20, 2634 (1979)
[12B95] Paver, N.: Aspects of CP violation in neutral kaon decays. In: Proceedings of the 4th
School on Non–accelerator Particle Astrophysics, p. 34. World Scientific, Singapore
(1995)
[12A98] Ambrosio, M., et al., (MACRO Coll.): Measurement of the atmospheric neutrino–
induced upgoing muon flux using MACRO. Phys. Lett. 434, 451 (1998)
[12C98] Carrigan, R.A., Giacomelli, G., Paver, N. (eds.): Proceedings of the 5th School on Non-
Accelerator Particle Astrophysics. Ed. Universit
`
a di Trieste, Trieste (1999):
– Giacomelli, G., Spurio, M.: Atmospheric neutrinos and neutrino oscillations, p. 146
(hep-ph/9901355);
– Cecchini, S., Sioli, M.: Cosmic ray muon physics, p. 201;
[12F98] Fukuda, Y., et al., (SuperKamiokande coll.): Evidence for oscillation of atmospheric
neutrinos. Phys. Rev. Lett. 81, 1562 (1998)
[12S98] Sanchez, M., et al., (Soudan 2 Coll.): Measurement of the L/E distributions of
atmospheric in Soudan 2 and their interpretation as neutrino oscillations. Phys. Rev.
D 68, 113004 (2003)
[12A01] Abe, K., et al.: Observation of large CP violation in the neutral B meson system. Phys.
Rev. Lett. 87, 091802 (2001)
[12D01] Dorfan, J.: BaBar results on CP violation. Int. J. Mod. Phys. A 17, 2925 (2002)
[12L01] Lipari, P.: Introduction to neutrino physics. 1st CERN - CLAF School of High-energy
Physics, Itacuruca, Brazil, (2001). http://cdsweb.cern.ch/record/677618
[12H03] Harrison, P.: Nature’s flawed mirror. Phys. World 16(7), 27 (2003)
[12G07] Gershon, T.: A triangle that matters. Phys. World 20(4), 24 (2007)
[12B08] Bellini, G., et al., (BOREXino Coll.): Measurement of the solar
8
B neutrino flux with
246 live days of Borexino and observation of the MSW vacuum-matter transition. Phys.
Rev. D 82, 033006 (2010) . Also: arXiv:0808.2868 [astro-ph]