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

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340 11 The Standard Model of the Microcosm

Fig. 11.12 (a)Vertexgggg and (b) illustration in terms of lines of color

With three colors and three anticolors, one can form a color octet plus a color

singlet. The color singlet

r C gg C bb

(11.89)

does not carry color and cannot mediate an interaction between color charges. This

interaction can only be mediated by the remaining 8 gluons of the octet.

As previously said, the color charge for the strong interaction is the analogue

of the electric charge for the electromagnetic interaction. Both forces are mediated

by massless vector bosons (photon or gluons); however, for the electromagnetic

interaction, there are 2  1 charge types (positive and negative) and one neutral

mediator boson (the photon), and in QCD, there are 2 3 charge types (three colors

and three anticolors) and 8 colored mediator bosons (with color and anticolor).

These considerations lead to signiﬁcant differences between QCD and QED.

A fundamental difference between QED and QCD is due to the fact that gluons

carry color and anticolor charges: they can thus interact with each other. In addition

to the vertex qqg, there is a vertex ggg as shown in Fig. 5.3. The vertex gggg

(see Fig. 11.12a, b) is also foreseen. These vertices make QCD richer than QED and

allow for the possibility of hadronic states formed only by gluons (the glueballs)and

of hybrid states such as q

qg. However, they also make QCD mathematically more

complex. Note that in QED, “photon-photon” interactions are not directly possible,

but only through pairs of electric charges, as shown, for example, in the diagram of

Fig. 11.13.

In addition to cases similar to QED (e.g., the repulsive force between two quarks

of the same color and the attractive force between quarks with color and anticolor),

in QCD, different colors may give rise to an attractive force if the quantum state

is antisymmetric, and a repulsive force if it is symmetric under the interchange of

quarks. This means that the favorite state of three quarks is the state with three

quarks of different colors, q

, that is, the colorless state of baryons.

At small distances (corresponding to large momentum transfers, i.e., at high Q

is small enough to allow perturbative methods of calculation in analogy with

QED. However, at large distances (low Q

), one has ˛

 1 and perturbative

methods (and ﬁrst order Feynman diagrams) can no longer be used to make

calculations. The hadron masses and most of the hadronic processes at low Q

cannot be simply derived from QCD.

11.9 The Strong Interaction 341

Fig. 11.13  interaction

through a charged fermion

box

qq qq qq

qq qqqq

abc de

Fig. 11.14 Modiﬁcations to the simplest gluon propagator (a). In (b), one has a fermion loop (qq)

as in QED. In (c), (d), and (e), the boson contributions not present in QED are shown

11.9.2 Color Charge Screening in QCD

In QCD, the color charge screening is similar to that of an electric charge in

QED. However, besides fermionic loops, there are also bosonic loops producing

an “antiscreening” which dominates the situation (see Fig. 11.14).

It follows that ˛

is a “running coupling constant” [11W04]. At the lowest

order of perturbative QCD (i.e., ﬁrst order in ˛

), the structure functions receive

contributions from the following elementary processes (V



is a virtual boson,

; Z

V*q q

V*q qg

V*g qq

–

Elastic scattering

QCD Compton

Boson-gluon fusion

Several parameterizations of ˛

as a function of Q

are possible, but all depend

on a free parameter to be determined experimentally since it cannot be predicted by

342 11 The Standard Model of the Microcosm

the theory (as the mass and electric charge of the electron in QED). In the version

in which the free parameter (dimensionally, an energy in the natural unit system) is

called 

QCD

, one has

/ D

12

.11N

 2N

/ ln





QCD



(11.90)

with N

D number of colors D 3andN

D number of kinematically accessible

ﬂavors (for example, N

D 5 at Q

D m

since the top quark mass is larger than

the Z

boson mass). From measurements (see Sect. 11.9.4), it results that 

QCD

200 MeV and ˛

decreases with increasing Q

. While it has a value  10 (1) for

energies of the order of 100 (1,000) MeV, one has ˛



/ ' 0:36, ˛

/ ' 0:12

(Fig. 11.15). In the limit Q

!1, one has ˛

! 0, that is, the so-called asymptotic

freedom which consists of a fundamental proof regarding the non-abelian nature of

QCD (due to the interaction terms between gluons).

The increase in effective color charge with the separation distance between

quarks can be considered as an effect due to the vacuum. It could be the cause of the

conﬁnement of quarks and gluons in hadrons. For r  10

13

cm, the attractive force

between two quarks is large and roughly constant with r (a situation similar to that

in electromagnetism between the two plates of a capacitor). This is called infrared

slavery.

Due to the antiscreening effect, the vacuum around an isolated color charge is

unstable. This may result in the hadronization phenomenon, i.e, the transformation

of quarks in hadrons. However, the conﬁnement is still not understood at the

fundamental level and the existence of free quarks is not fully excluded. Note that

in conditions of high energy and high density, the deconﬁnement phase with the

formation of a quark-gluon plasma should take place.

11.9.3 Color Factors

In QED, the electromagnetic coupling “force” between two quarks of electric

charge e

, e

is equal to e

˛ (e

DC2=3 or 1=3). Similarly, in QCD, the

strong coupling “force” between two color charges via the exchange of a gluon is

j˛

D C

,whereC

is the so-called color factor.

The color factor for the interaction between two blue quarks can be calculated

as follows, see (11.88): only gluons that contain the color term b

b, i.e., only .rr C

g  2bb/

6, have to be taken into consideration. In this case, one has c











. Therefore, the color factor is C



By analyzing all other possibilities, it is found that for two quarks of the same color

(or of different color), c

D P 

, with P D˙1 depending on whether the two

quarks are in a symmetric or antisymmetric color state.

11.10 The Standard Model: A Summary 343

In a baryon, the interaction between each quark pair concerns antisymmetric

states because it is necessary to have an antisymmetric ﬁnal state. Therefore, for

each pair, one has c

D

and thus C

For a b

b meson changing into rr or gg, the color factor is C

. For a meson

which exchanges a gluon in a singlet state, the color factor is instead C

.In

conclusion, the general rule is



for q

qW ˛

is replaced by

for qqqW ˛

is replaced by

11.9.4 The Strong Coupling Constant ˛

The strong coupling constant, ˛

, as the ﬁne structure constant and the universal

Fermi constant, is a free parameter of the theoretical model and must be experi-

mentally determined. QCD predicts a well-deﬁned dependence of ˛

on the on the

energy scale Q of the interaction, and makes predictions about various phenomena

that allow one to determine its value. Experimental methods which measure ˛

and

verify the QCD predictions are (the value of Q corresponding to the process is also

indicated):

• Hadronic decays of the  lepton:  ! 



C hadrons (Q D1.77 GeV)

• Evolution of the nucleon structure functions measured in inelastic scattering of

e; ; on nucleons (Q D2  50 GeV)

• Jet production in the inelastic scattering ep ! eX (Q D2  50 GeV)

• Analyses of the energy levels of bound states q

q (quarkonium) (Q D1:55 GeV)

• Decays of the vector mesons  (Q D5GeV)

• Hadronic cross-section of the annihilation e



! hadrons (Q D10200 GeV)

• Fragmentation function of jets produced in e



! hadrons (Q D10 

200 GeV)

• Hadronic decays of the Z

boson (Q D91 GeV)

• Jet production in pp; p

p interactions (Q D50  300 GeV)

• Photon production in pp; p

p interactions (Q D30  150 GeV)

The results for some of these processes are illustrated in Fig. 11.15,whichshows

that the “constant” ˛

decreases with increasing of the energy scale Q.

11.10 The Standard Model: A Summary

Let us summarize the main points of the Standard Model (SM) of the micro-

cosm. It includes the electroweak interaction described by the symmetry group

[SU(2)

SU(1)

] and the strong interaction described by the symmetry group

SU(3)

344 11 The Standard Model of the Microcosm

Fig. 11.15 Strong coupling

“constant” ˛

as a function of

the centre–of–mass energy.

Thelineshowsaﬁttothe

experimental data [P08]

QCD α

(Μ

) = 0.1184±0.0007

0.1

0.2

0.3

0.4

0.5

(Q)

1 100

Q [GeV]

Heavy Quarkonia

–

Annihilation

Deep Inelastic Scattering

July 2009

Table 11.4 Couplings in the Standard

Model

Vertex Coupling proportional to:

f

fe



2 cos 

sin 





a



The couplings of the vector boson Z

to fermions and the couplings of the

scalar Higgs boson H

to the W

and Z

vector bosons are listed in Table 11.4.

Electroweak and strong interactions have a similar structure and are mediated via

the vector bosons shown in Table 11.5. The strong interaction mediators, the gluons,

carry a color charge and an anticolor charge and can interact between them, giving

rise to the ggg and gggg vertices. The mediator of the weak interaction, the vector

bosons Z

and W

, carry a weak charge and can also interact between themselves

(vertex Z



). The gravitational interaction is not included in the SM.

The electromagnetic and weak interactions are interpreted as two different as-

pects of a uniﬁed theory describing the electroweak interaction. The electromagnetic

and weak charges are related via the Weinberg angle.

The range of the electromagnetic interaction is inﬁnite because the photon is

massless. The range of the weak interaction at low energies is about 10

3

D 10

16

cm because of the large mass of the mediator bosons W

.The

mediators of the strong interaction, the gluons, are also massless. However, the range

of the strong force is limited to about 1 fm because of the strong interaction between

11.10 The Standard Model: A Summary 345

Table 11.5 The three fundamental interactions of the SM of the

microcosm have couplings through different charge types and are

mediated by vector bosons .J D 1/ which are massless or have

a mass around 100 GeV; the parity P is conserved in two of the

three interactions

Exchanged Mass

Interaction Coupling with bosons (GeV/c

) J

EM Electric charge Photon . / 0 1



Weak Weak charge W

10

Strong Color charge 8 gluons .g/ 0 1



Table 11.6 In the SM, the fundamental fermions are leptons and quarks with spin 1/2; they are

grouped into three “families” (three “generations”). Quarks have a fractional electric charge and

red, blue or yellow color charge. There are no right-handed neutrinos. The right-handed quarks

and charged leptons have I

D 0

Family Weak isospin

Fermions 1 2 3 Electric charge Color Left-handed Right-handed Spin

Leptons 









0  1/2  1/2

e1  1/2 0 1/2

Quarks u ctC2=3 r;b;g 1/2 0 1/2

ds b1=3 r; b; g 1/2 0 1/2

gluons. This interaction is responsible for the conﬁnement of quarks and gluons in

hadrons, and for the nonexistence of free quarks and gluons.

The fundamental fermions are the quarks and leptons, as shown in Table 11.1.

They have spin 1/2 and are grouped in three families (generations), as illustrated

in Table 11.6. Remember that each fermion has a corresponding antifermion

as mentioned in the introduction of this chapter. On the basis of the precision

measurements of the Z

width, one can conclude that there are only three families

of massless (or light) neutrinos. The strong isospin is only important for the strong

interaction; for the strong isospin, the six different quark ﬂavors form a doublet

(I D 1=2, quarks u;d) and four singlets (I D 0, quarks s; c; b; t).

The Standard Model has received strong experimental conﬁrmation. However, an

essential element of the SM, the Higgs boson .H

/, has not yet been observed. The

search is now started at the LHC, see Sect. 10.10. Precision measurements in e



collisions at

s  m

constrain the H

mass to a relatively low value. There are

many unanswered questions surrounding the Standard Model: (1) There is a large

number of free parameters (about 18, depending on how they are counted; they are

the fundamental fermion and boson masses, the coupling constants g; g

;˛

,the

coefﬁcients of the CKM matrix; these parameters are not given in the SM and must

be measured experimentally). (2) Why are there exactly three families of fermions?

(3) How can the neutrino masses be introduced? (4) Does the Higgs boson really

exist? (5) In vacuum, the Higgs ﬁeld acquires a nonzero value (11.60) that permeates

the whole space. Does it affect the energy density of the universe?

Chapter 12

CP-Violation and Particle Oscillations

12.1 The Matter-Antimatter Asymmetry Problem

The CP transformation combines the charge conjugation C and parity P operators

(Sect. 6.8). Under CP, a left-handed electron (e



) becomes a right-handed positron

). If CP is an exact symmetry, the laws of nature would be completely identical

for matter and antimatter. Most of the observed phenomena are symmetric with

respect to C and P , and consequently symmetric to CP. Exceptions are the weak

interactions (WI), which violate C and P maximally.ThismeansthataW boson

couples with a left-handed electron e



, but not with the particle P -conjugated (e



)

or C -conjugated (e

). However, the same boson couples with the CP-conjugated

particle, the e

. This seems to suggest that the weak interaction preserves CP.On

the contrary, it has been known for many years that the CP symmetry is violated in

some rare processes, as found for neutral K in 1964. In particular, the K

meson

decays more often in the 





channel than in the 





, with a very small

asymmetry of about 0.3%. Recently, a similar effect was found for beauty mesons

with a larger asymmetry.

Closely linked to CP is the time reversal transformation T :theCPT transfor-

mation is a fundamental symmetry of physical laws. The violation of symmetry T

is also observed in the decay of neutral K. Within the Standard Model (Chap. 11),

the CP symmetry violation is introduced through complex phases. In particular, a

simple phase factor appears in the 3  3 CKM (Sect. 8.14.3) unitary matrix that

describes the transformation of an up-type quark into a down-type quark by the

exchange of a W vector boson.

In 1967, Sakharov realized that the CP-violation is a necessary condition for

baryogenesis (Sect. 13.6), i.e., the dynamic process generating matter-antimatter

asymmetry in the Universe. The Universe seems to be made of matter, with

almost no antimatter. This implies a particle-antiparticle asymmetry and sug-

gests that CP could not be a symmetry of all fundamental interactions. Despite

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

12,

347

348 12 CP-Violation and Particle Oscillations

the phenomenological success of the mechanism that describes the CP-violation

in the Standard Model, there are probably additional, still unknown, sources of

CP-violation that explain the matter-antimatter asymmetry in the Universe.

The recent discovery that neutrinos have small mass seems to imply that there

is a source of CP-violation in the leptonic sector in addition to the hadronic one.

The studies of neutrino oscillations could provide information on leptogenesis.The

search for new processes that would violate the CP symmetry is currently one of

the most important experimental efforts in particle physics. These studies involve

the decay of mesons, the measurement of the electric dipole moment of neutrons,

electrons and nuclei, and neutrino oscillations.

In Sects. 12.2 and 12.3, we will ﬁrst describe the K

 K

system followed by

the similar B

B

system (Sect. 12.5). Neutrino oscillations, namely, the possible

transformation of a neutrino of one type into another type (for example, 



! 



simultaneously violating two leptonic numbers, will be described in Sect. 12.6.

While K

 K

and B

 B

mixings are included in the Standard Model, the

neutrino mixings require theories with massive neutrinos that enlarge the Standard

Model in some way.

12.2 The K

 K

System

The K

and K

mesons are part of the meson nonet J

D 0



and are

strangeness eigenstates. An important property of the two neutral mesons is that

they have different production energy thresholds. The K

meson can be produced

in association with the 

hyperon in the strong interaction reaction



p ! 

: (12.1)

The threshold energy of this reaction is 0.91GeV in the c.m. The

meson can

only be produced at higher energies by (for example) the strong interaction reaction



p ! K

p (12.2)

which has an energy threshold of 1.5 GeV in the c.m. Even higher energies are

needed to produce a K

in combination with an antihyperon. K

and K

are

therefore distinguishable particles. It is possible to determine if a K

or a K

has

been produced by observing the associated particles. Furthermore, once produced,

it is possible to distinguish one from the other because when interacting with target

nuclei, they produce particles with opposite strangeness. Also the cross-sections are

different with .K

N/ < .K

N/ because there are much more available ﬁnal

states for

interactions. For the K

p ! K

p; K

n ! K

12.2 The K

 K

System 349

For the K

p ! K

p; 



;

n ! K

n; K



p; 



;



The

can be considered as the antiparticle of K

,theK



as the antiparticle of

: this is evident when the K are expressed in terms of their valence quarks and

antiquarks, that is, K

D us, K



D us; K

D d s; K

D ds. K

and K

are

strangeness eigenstates.

The two neutral K mesons are observed to decay by WI into  or in 

states, following the same rules observed for other meson and baryon decays. The

decay into two pions in the ﬁnal state has a lifetime much shorter than that with

three pions in the ﬁnal state.

We know that the same particle cannot decay into either two or three pions:the

ﬁnal states with two and three pions are CP eigenstates with different eigenvalues.

The pions are produced in a state of total angular momentum JD0. In Sect. 6.5,

we have seen that the parity of a state with n pions is P.n/ D .1/

while the pions are C eigenstates with eigenvalue C1. The  state is then

CP eigenstate with eigenvalue C1, while the state  is a CP eigenstate with

eigenvalue 1.

Virtual transitions via WI are allowed between K

and K

. Virtual transitions

between a charged particle and the corresponding antiparticle are forbidden by

conservation of the electric charge; they are also forbidden between a baryon and

the corresponding antibaryon by conservation of the baryonic number. Transitions

between K

and K

violate strangeness .S D 2), which is conserved in the

electromagnetic and strong interactions, but not in the WI. Virtual transitions

$ K

may be described by special Feynman diagrams, as discussed in the

next section.

A pure K

beam can be obtained at c.m. energies of 1 GeV, that is, just above

the threshold of reaction (12.1), though below the threshold for reaction (12.2). Due

to the allowed WI transitions between K

and K

, this pure K

beam at tD0evolves

after some time into a superposition of K

and K

expressed as

jK.t/iD˛.t/jK

iCˇ.t/jK

i : (12.3)

Physical states decaying through WI are, in principle, not eigenstates of the charge

conjugation C , parity P , or strangeness S operators. They may eventually be

eigenstates of the CP operator. An appropriate phase can be chosen so that the

application of the CP operator on the K

and K

states at rest gives

CPjK

iDCjK

i (12.4)

CPj

iDCjK

i : (12.5)

350 12 CP-Violation and Particle Oscillations

50% K

decay

{{

re-generator

decay

Fig. 12.1 Schematic diagram of an experiment highlighting the ﬂuctuations in strangeness

From Eqs. 12.4 and 12.5, it is evident that K

and K

are not CP eigenstates. Let us

now build two linear combinations that are CP eigenstates, for example,

iD.jK

iCjK

i/=

2; jK

iD.jK

ijK

i/=

2: (12.6a)

From (12.6a), one has

iCjK

;

ijK

: (12.6b)

We can now verify that jK

iand jK

iare CP eigenstates with eigenvalues C1 e 1,

that is,

(

CPjK

iD.CPjK

iCCPjK

i/=

2 D .CjK

iCK

i/=

2 DjK

CPjK

iD.CPjK

iCPjK

i/=

2 D .jK

ijK

i/=

2 DjK

(12.7)

The K

and K

states do not have deﬁnite strangeness. The  ./ state has

S D 0 and is a CP eigenstate with eigenvalue C1(1). For this reason, if CP is

conserved in the weak interaction, the K

decays into 2,andtheK

decays in 3.

Because of the larger phase space available for K

! 2 than for K

! 3,the

lifetime is much smaller than that of K

12.2.1 Time Development of a K

Beam. K

Regeneration.

Strangeness Oscillations

Suppose one has an initially pure K

beam produced in the reaction (12.1)just

above threshold (see Fig. 12.1). For the strong interaction, the K

beam is a pure

strangeness S DC1 state. For the weak interaction, the beam consists of 50% of

mesons and 50% of K

. After about 10

9

s, measured in the K

rest system,

almost all K

have disappeared (if there are no relativistic effects, it corresponds to

about 30 cm in the laboratory frame). Now, the intensity of the K

beam is halved

and the beam is predominantly composed of K

. For the strong interaction, the