Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics

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392 13 Microcosm and Macrocosm

energy of baryons (940 MeV per baryon) dominates the total visible energy.

GUT theories are currently the only ones that easily explain the small n



value.

13.2 Supersymmetry (SUSY)

The transformations seen thus far link particles of the same type; in general,

they “rotate” bosonic (or fermionic) states in other bosonic (or fermionic) states.

Supersymmetries transform (rotate) a bosonic state into a fermionic one and vice

versa. If these transformations exist, they would imply that bosons and fermions

are different manifestations of a uniﬁed state: fermions and bosons would be in

the same multiplet. These transformations are foreseen by supersymmetric theories

which represent a new form of uniﬁcation. A supersymmetric operation changes by

1=2 the spin of particles, leaving the electric and color charges unchanged.

Supersymmetry has a cultural interest in itself; it also addresses some of

the difﬁculties of Grand Uniﬁed theories. Without supersymmetry, it is indeed

difﬁcult to understand why the known fundamental particles are so light with

respect to the Grand Uniﬁcation scale at 10

GeV. Supersymmetry solves this

hierarchy problem as well as the problem of divergences (for example, the radiative

corrections to the Higgs boson mass) [13L94]. In supergravity theories (SUGRA),

the supersymmetric concepts lead to the uniﬁcation with gravity.

According to supersymmetry, every known boson has a fundamental fermionic

partner with spin differing by 1=2 unit, and any fundamental fermion has a bosonic

partner with spin differing by 1=2. The supersymmetric partners of the particles are

called sparticles. However, it seems impossible to connect the known fundamental

bosons and fermions. The sparticles must be new objects: they are expected to

be heavier than any known particle. This is a consequence of a supersymmetry

breaking: corresponding particles and sparticles should otherwise have the same

mass.

Supersymmetries postulate the existence of an operator U which transforms a

fermion into a boson by varying the spin by 1=2 unit, and vice versa:

U jfermioniDjbosoni

U jbosoniDjfermioni:

(13.6)

The boson-fermion symmetry implies the existence of doublets containing a

quark q and its supersymmetric partner, denoted as squark Qq (scalar quark), and

similarly for a lepton and the corresponding boson, the slepton

l (scalar lepton):





;





with spin



1=2



: (13.7)

13.2 Supersymmetry (SUSY) 393

Table 13.1 Fundamental particles and their supersymmetric partners. Only

the Higgs bosons of the MSSM are considered

Particle, R DC1 Sparticle, R D1

Particle Spin Charge Sparticle Spin S-name

e 1/2 1 Qe 0 selectron

 1/2 1 Q 0smu

 1/2 1 Q 0stau

 1/2 0 Q 0 sneutrino

q 1/2 2=3; 1=3 Qq 0 squark

g 10 Qg 1/2 gluino

 10 Q 1/2 photino

Z 1/2 zino

;

1/2 neutral higgsino



;



1/2 charged higgsino

W 1/2 wino

G 20

G 3/2 gravitino

Sparticles couple with ﬁelds with the same coupling constant as the particles. For

example, the couplings qqg, Qq Qq Qg, ggg, Qg Qg Qg are always ˛

. Similarly, there exist

doublets for every vector boson and its supersymmetric partner, that is,





Q



;





;





;





with spin



1=2



: (13.8)

The Higgs boson has spin S D 0; its supersymmetric partner has spin S D 1=2.

The notation for boson partners is as follows: starting from the ordinary name,

the sufﬁx ino is added, that is, photon ! photino, Z

! zino, W

! wino, Higgs

! higgsino.

13.2.1 Minimal Standard Supersymmetric Model (MSSM)

Table 13.1 gives an overview of fundamental particles and their supersymmetric

partners. It refers to the Minimal Supersymmetric Standard Model (MSSM), the

simplest supersymmetric model (the only one we consider in the following). Also,

Table 13.2 refers to the MSSM. In this model, at least two complex doublets of

Higgs bosons



;



and



;





should be introduced to generate the masses of

“up” and “down” type quarks and the masses of the charged leptons. Supersym-

metric neutral states should mix as we saw in Chap. 12 for other systems. The

four neutral fermions Q;

;

are not mass eigenstates; the latter are the

neutralinos Q

; Q

, expressed as a mixing of the type:

Q

1;2;3;4

D a Q Cb

Z C c

C d

(13.9)

394 13 Microcosm and Macrocosm

Table 13.2 “Normal” fundamental particles and supersymmetric partners; right-handed particles

are included. The mass eigenstates resulting from the mixing between sparticles are also indicated

Quark Squark

(spin 1/2)





(spin 0)





















!

1;2

Leptons Sleptons

(spin 1/2)







(spin 0)



Q















Q



Q



Q













Q



Q



Q

! Q

1;2

Gauge bosons Gauginos

(spin 1) g (spin 1/2) Qg

 Q Neutralinos

Z ! 

1;2;3;4

fQ;

;

Higgs bosons Higgsinos Charginos

(spin 0) h

(spin 1/2)

;

! 

1;2



;



;

with different coefﬁcients for Q

, Q

. Similarly, the two charged Higgsinos



and the two winos



are not mass eigenstates; the mass eigenstates

are the charginos Q

and Q



expressed as mixings of the type:

Q

D a

C b

; Q



D a



C b



: (13.10)

Since no sparticle has been experimentally observed, a new quantum number,

the R-parity, is introduced in order to provide the supersymmetric particles with

some properties that make them (currently) inaccessible. R-parity is equal to

R D .1/

3BCLC2S

,whereB is the baryon number, L the lepton number, and S

the spin. Most SUSY models require the conservation of R andthenpredictthe

existence of a stable supersymmetric particle with a minimum mass (the “Lightest

Supersymmetric Particle,” LSP). There are other models where R can be violated.

In the MSSM, R-parity is conserved; R DC1 is assigned to known particles and

R D1 to the supersymmetric partners, that is,

Rjsupersymmetric particleiDjsupersymmetric particlei

RjparticleiDCjparticlei:

The general situation can be more complex, as the right-handed states, such as, e

, could give sparticles different from the left-handed ones, as shown in Table 13.2.

13.2 Supersymmetry (SUSY) 395

WW W

Fig. 13.4 (a) Fundamental Feynman diagrams in the SM and (b) with the introduction of

supersymmetric partners. Antiparticles are not indicated. Here L (Left) represents the particle

helicity

13.2.1.1 The Higgs Sector of the MSSM

In the MSSM, two complex doublets of Higgs bosons provide eight degrees of

freedom. Three degrees of freedom are used to give mass to the W



The Higgs sector in the MSSM contains ﬁve bosons (and their supersymmetric

partners) that correspond to the remaining ﬁve degrees of freedom. The Higgs

bosons have been called h



with even CP and A

with odd CP .

With m

0 <m

,theh

boson can be identiﬁed with the scalar Higgs

boson H

of the Standard Model, introduced in Chap. 11.

In summary, one can observe from Table 13.2 that in the MSSM:

1. The number of particles doubles.

2. The supersymmetric partner of known gauge bosons corresponds to the

gauginos.

3. Three sfermion families correspond to the three fermion families.

4. In the Higgs sector, there are two doublets leading to ﬁve Higgs bosons and ﬁve

Higgsinos.

13.2.1.2 Fenomenology

Calculations in the MSSM use the same Feynman rules of the Standard Model,

adding the contribution of diagrams in which particles are replaced by their super-

partners. The substitution of particles with sparticles must be made in pairs for an-

gular momentum conservation. Boson-fermion couplings are illustrated in Fig. 13.4.

Since in the MSSM model the R-parity is a conserved quantum number, all

vertices include superpartner pairs. Three conclusions can be drawn:

1. The supersymmetric partners are produced in pairs from normal particles.

2. There is always a supersymmetric particle in the decay products of a supersym-

metric particle.

3. The lightest supersymmetric particle (LSP) is stable; it is usually assumed that

the LSP is the neutralino Q

which is neutral both for electric and color charges;

396 13 Microcosm and Macrocosm

–

γ, Z

∼

−

∼

–

Fig. 13.5 (a) Feynman diagram for the slepton–antislepton pair production followed by the

slepton decay:



! `



Q

and

! `

Q

.` D e; ;/.For` D e, the diagram where

a Q

is exchanged in the t-channel also contributes. (b)Sketchofaneventwhichproducesan

acoplanar slepton–antislepton pair. The acoplanarity angle  is the angle between the projections

of the `



and `

momenta in the plane perpendicular to the e

and e



beams

the neutralino could therefore be an important component of dark matter in

the Universe (see Sect. 13.6). For special mixing conditions, the LSP could

correspond to the sneutrino.

Supersymmetry is not visible in nature and therefore must be “broken,” at

least at the currently accessible energies. As the Standard Model, supersymmetric

theories cannot predict the values of the particle masses, but once they have

been experimentally determined, one can calculate all other quantities. Recent

phenomenological supersymmetric models predict supersymmetric particles with

masses between 100 and 1,000GeV; these are searched for at the LHC.

Many searches for supersymmetric particles at

pp, ep, e



colliders have been

carried out, up to now with negative results; only lower limits on their mass were

determined. These limits (at the 95% conﬁdence level) are presented in Table 13.3.

With the introduction of supersymmetry, the extrapolation of the interaction

coupling constants to 10

GeV have better behavior: the three constants appear to

converge towards a common point, as shown in Fig. 13.1b.

13.2.1.3 Searches for Charged Sleptons

Let us brieﬂy discuss an example of a typical SUSY particle search. Figure 13.5a

shows the Feynman diagram for the production of charged scalar leptons e



; Z

!Qe



; !Q

Q



The sleptons decay in the corresponding charged leptons plus a neutralino,

Q

;the Q

(subject only to weak interaction) leaves the detector unobserved. The

topology of the signal events (Fig. 13.5b) consists of two acollinear and acoplanar

charged leptons, plus missing energy and momentum. The sleptons

and

are

different particles with different masses. The most stringent experimental limits

13.3 Composite Models 397

generally apply to

, whose mass is expected to be smaller than that of

There may also be special conditions in the MSSM parameters space for which

D m

13.2.2 Supergravity (SUGRA). Superstrings

GUT theories, including normal supersymmetric theories, do not incorporate the

gravitational interaction. Gravitation becomes important for energies of the order

of the Planck mass, 10

GeV, corresponding to a size of 10

33

cm. Such energies

existed in the early Universe until the Planck time, 10

43

A complete theory must contain quantum gravity. Its inclusion presents important

difﬁculties, connected both to the quantization and to divergences. The latter cannot

be easily removedby renormalization procedures, as in the supersymmetric theories.

The supersymmetric theory of gravity, called quantum supergravity or SUGRA,

has local gauge invariance. In some models, the lightest supersymmetric particle is

assumed to be the gravitino.

Finally, theories in which the particles are very small size objects, for example,

strings, in particular closed strings, have been formulated. These theories succeeded

in incorporating quantum gravity. The string theories concern both bosonic states

and supersymmetric theories (superstrings theories). They should be formulated in

a space with many dimensions (at least ten); it is not obvious how to switch to the

four dimensions of the ordinary space (the other six dimensions must be “hidden”)

[13G00].

13.3 Composite Models

The existence of three families of quarks and leptons suggests the existence of

possible substructures in quarks and leptons. The situation could be similar to

that already encountered with atoms, nuclei and hadrons. There are other reasons

to consider substructures. In some models quarks, leptons and gauge bosons are

composite objects. For example, in the model of Salam and Pati, quarks and leptons

are composed of three objects called prequark or preons; in the model of Harari,

they are composed of three objects called rishons, which means “fundamental”

in Hebrew. There are several difﬁculties in these theories associated with the very

small dimensions of prequarks (much less 10

16

cm, also their orbits must be less

than 10

16

cm) and with the need for an additional force that binds the prequarks.

In another model, prequarks are dions, namely, particles with both electric and

magnetic charges; in other models, the prequarks are so close together that the

gravitational force is also involved.

398 13 Microcosm and Macrocosm

Table 13.3 Searches for new particles. Lower mass limits in GeV [P10, 13w1,

13w2] at the 95% conﬁdence level obtained in searches at LEP and other high-

energy colliders (Tevatron and HERA). The LEP limits are obtained by combining

the results of the four LEP experiments. If the combination does not exist, we refer

to the best limit from a single experiment

Particle Symbol LEP limit (GeV) Other limits (GeV)

Charged sleptons

Qe, Q, Q 99:9; 94:9; 86:6

Sneutrino Q 94 (DELPHI)

Neutralino (LSP) Q

50.3

Charginos Q

103.5 117 (D0)

Scalar quarks

t 99, 98 222, 176 (D0)

Gluinos Qg 26.9

308 (D0)

SM Higgs H

114:4

MSSM Higgs h

92:8 100 (CDF)

93:4 100 (CDF)

78:6

Higgs (charge D 2) H

˙˙

99 (OPAL) 136 (CDF)

Exited leptons e



;



;



208; 190; 185 (OPAL) 255 (H1), 221 (CDF)

Exited neutrinos 



190 (L3) 158 (ZEUS)

Exited quarks q



200 775 (D0)

Scalar leptoquarks

LQ 98 (OPAL) 229 (D0)

Vector leptoquarks

LQ 98 (OPAL) 240 (D0)

For supersymmetric particles, the limits are usually obtained within the MSSM;

in some cases, the limits are within a speciﬁc model

Limit for a light gluino (stable)

Associated pair production

Several searches for particle substructures or excited states have been carried

out, all with negative results so far. The lower limits on their masses are presented

in Table 13.3.

Excited leptons. They would correspond to excited states of known leptons that

could be produced in pairs or individually: e



! e

C



, e



! e

C



Exited leptons could decay electromagnetically,



! e; 



! ; 



! ; (13.11)

without violating the three lepton numbers. These decays were not observed at a

sensitivity level of about 10

4

Decays with violation of lepton numbers. Quarks and leptons of the second and

third family can be thought of as excited states of quarks and leptons of the ﬁrst

family. In this case, decays of the type

 ! e;  ! e;  ! eee (13.12)

13.3 Composite Models 399

could be possible and the electron and muon, or electron and tau leptonic numbers

are violated. These decays were not observed, resulting in the following stringent

upper limit [13M10]:

.! e/

.! e/

< 310

11

: (13.13)

The  could also decay in ,  ! , violating the muon and tau leptonic

numbers (also this decay channel has never been observed).

Excited neutrinos could be the weak isospin partners of excited charged leptons.

They could be produced in pairs or individually and could then decay with the

emission of a photon: e



! 







! , e



! 



 ! .

Excited quarks. An excited quark could decay into a quark and a photon, q



! q,

or a quark and a gluon, q



! qg; the relative frequency should be approximately

1–10. In the ﬁrst case, q



! q, an isolated high energy photon would be present

in the ﬁnal state. In e



collisions, events with one or more high energy photons

could come from excited quarks, but also from the bremsstrahlung of a beam e



(or

) or from the bremsstrahlung from a q.q/ in the ﬁnal state.

Leptoquarks. Leptons and quarks are classiﬁed in a similar way with respect to

the family structures and weak isospin multiplets. The Standard Model does not

explain this parallelism, although it is necessary for the cancellation of divergent

terms. This suggests that there could be a deeper correlation between quarks and

leptons. In models that go beyond the SM, new particles that mediate the interaction

between quarks and leptons may exist. This occurs naturally in GUT models, at very

high energies. In some SM extensions, the existence of leptoquark bosons .LQ/

are foreseen. LQ are color triplets with both lepton and baryon number, electric

charge 1=3, 2=3 and relatively low masses. These LQs decay and couple with

a lepton and a quark. According to various models, LQs are scalar (spin D0) or

vector (spin D1) particles, and their coupling with leptons and quarks depends on

the model. LQs were introduced for the ﬁrst time [13P74] in an attempt to consider

the lepton number as a fourth color. In the simplest model, the interactions involving

LQs conserve baryon and lepton numbers and respect the symmetry of the Standard

Model. In this case, the only free parameters are the LQ masses and their couplings

with the fermions.

LQs produced in pairs have been searched for in e



collisions, as shown in

Fig. 13.6a:

C e



! LQ C LQ; LQ ! ` C q; LQ ! ` C q: (13.14)

In this case, the LQ decay modes into eq, q, q were used. These combinations

should conserve the “family number,” i.e., the pairs must involve ` and q only of the

same family.

400 13 Microcosm and Macrocosm

–

γ, Z

–

γ,Z

LQ, q

abcd

Fig. 13.6 Feynman diagrams for leptoquark production and decay in e



collisions. The

leptoquark can be produced (a) in pairs (any LQ type) (b)and(c) individually (in this case, the

leptoquarks belong to the ﬁrst family and only couple with charged leptons). (d) Indirect effects

due to the (virtual) exchange of a leptoquark of the ﬁrst family in the t-channel

–

Fig. 13.7 Feynman diagrams for leptoquarks of the ﬁrst family produced in ep collisions: (a) LQ

formation, (b) LQ exchange in the t-channel

Individually produced LQs were also searched for in (see Fig. 13.6b,c):

C e



! e

C q CLQ ! e

C e



C q Cq (13.15)

as well as indirect effects due to the exchange of a leptoquark of the ﬁrst family

in the t-channel (Fig. 13.6d). Electron–positron (ep) collisions, as in the HERA

collider (Chap. 10), offer an easier way to search for leptoquarks of the ﬁrst family

(Fig. 13.7), that is,

e Cp ! LQ C q Cq ! e

C d Cq C q: (13.16)

In this case, a resonant system (eq) with an enhanced cross-section could be formed.

The events should have a high energy lepton, a jet of hadrons at large angles and a

“spectator jet” in the ﬁnal state. Events are therefore similar to those of a typical

deep inelastic collision.

13.4 Particles, Astrophysics and Cosmology

Cosmology has recently acquired the characteristics of an experimental discipline

[13W07]. Particles with their fundamental interactions, astrophysics and cosmology

13.4 Particles, Astrophysics and Cosmology 401

have become closely related ﬁelds. The submicroscopic phenomena allow us to

better understand the evolution of the Universe and astrophysics, and vice versa.

For example:

• The understanding of atomic and molecular phenomena clariﬁed the light

spectrum emitted by the Sun (and more generally by the stars), the chemical

composition of the solar atmosphere and led to the discovery of helium.

• The knowledge of nuclear phenomena explain the energy source of the Sun and

of the stars. At the center of a star there is a “furnace” where nuclear fusion

reactions take place (Sect. 14.10).

• Nuclear and subnuclear physics allow us to understand the structure of particular

celestial bodies in extreme conditions, for example, white dwarfs, neutron stars

and stellar gravitational collapses (Problem 13.6).

A deep connection exists between cosmic rays and subnuclear physics [G90].

Primary cosmic rays mainly consist of high energy protons and atomic nuclei which,

coming from outside of the solar system, interact with the nuclei of the Earth’s

atmosphere. In an interaction, many particles (some of them unstable) are produced

(secondary cosmic rays). The understanding of the origin of cosmic rays, their

acceleration mechanisms, the interaction processes with interstellar material and the

Earth’s atmosphere is based on the knowledge of nuclear and subnuclear physics.

However, the reverse is also true: many particles were discovered in secondary

cosmic rays.

Particle and astrophysics: the example of neutrino telescopes [13C10].

Astrophysics has experienced an extraordinary development in recent

decades thanks to new techniques that allowed the transition from limited

observations of the visible band of the electromagnetic spectrum to other

wavelengths: from the radio and infrared waves on one side, to X -rays

and  -rays on the high energy side. The information that these observations

can provide to the mechanisms that take place in astrophysical objects such

as supernova remnants (SNR), pulsars (PLS), active galactic nuclei (AGN)

and others is unfortunately incomplete, as it is limited to electromagnetic

processes. Cosmic particle accelerators (in the galaxy and outside our

galaxy) capable of accelerating protons and nuclei as observed in the cosmic

rays (CR) must therefore exist .

The CR spectrum extends with amazing regularity up to 10

eV, with a

power law of the form d˚=dE ' E



, with  D 2:7 for energies up to 

eV. This behavior can be explained by the so-called “Fermi mechanism”

acceleration process. Accelerated protons and nuclei (hadronic processes)

may interact with ambient nuclei or with the gas of photons giving rise to

 and K mesons. The 

(with very short lifetime) produces a photon pair,

the 

decay in charged leptons, neutrinos and antineutrinos (Problem13.1).

It is not an easy task to distinguish the photons produced in hadronic

processes from those produced in purely electromagnetic processes (by high