bilenky, hosek. glashow-weinberg-salam theory of electroweak interactions and their neutral currents

HOSEK* Joint Institute for Nuclear Research, Head Post Office PO Box 79, Moscow, USSR 23 The Higgs mechanism 24 The standard electroweak theory the case of leptons 25 The standard th

GLASHOW-WEINBERG-SALAM THEORY

OF ELECTROWEAK INTERACTIONS AND THE NEUTRAL CURRENTS

S.M BILENKY and J HOSEK Joint Institute for Nuclear Research, Head Post Office P.O Box 79, Moscow, USSR

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 90, No 2 (1982) 73-157 North-Holland Publishing Company

GLASHOW-WEINBERG-SALAM THEORY OF ELECTROWEAK

INTERACTIONS AND THE NEUTRAL CURRENTS

S.M BILENKY and J HOSEK*

Joint Institute for Nuclear Research, Head Post Office PO Box 79, Moscow, USSR

23 The Higgs mechanism

24 The standard electroweak theory (the case of leptons)

25 The standard theory of electroweak interactions (the

33 The leptonic processes v(v)+e > v(v)+e

34 P-odd asymmetry in the deep inelastic scattering of

longitudinally polarized electrons by nucleons

35 Neutral currents and the processes e* + e > 17 +1

Appendix A General rule for the evaluation of deep inelastic

cross sections in the quark-parton approximation

Appendix B On the generation of fermion and gauge particle

In the first part of the review we expound in detail the unified theory of weak and electromagnetic interactions of Glashow, Weinberg and

Salam In the second part, on the basis of this theory a number of the neutral current induced processes are discussed We consider in detail the deep

inelastic scattering of neutrinos on nucleons, the P-odd asymmetry 1n the deep inelastic scattering of longitudinally polarized electrons by nucleons,

the scattering of neutrinos on electrons, the elastic scattering of neutrinos on nucleons, and the electron-positron annihilation into leptons

Copies of this issue may be obtained at the pnce given below All orders should be sent directly to the Publisher Orders must be

0 370-1573/82/0000-0000/$21 25 @ 1982 North-Holland Publishing Company

SM Builenky and J Hosek, GWS theory of electroweak interactions and the neutral currents 75

1 Introduction

The invention of unified renormalizable theories of electroweak interactions 1s indisputably one of

the outstanding successes of elementary particle physics The first of these theories was the theory of

Glashow, Weinberg and Salam [1-3] (abbreviated as GWS, or referred to as the standard electroweak

theory, in the following) A beautiful exposition of the historical development of the ideas underlying

unified electroweak theories can be found in the Nobel lectures of Weinberg [4], Salam [5] and

Glashow [6]

The GWS theory predicts the existence of neutral currents Their structure follows from the fact that

the weak and the electromagnetic interactions are unified into a unique electroweak interaction in the

framework of a gauge theory based upon the SU(2) x U(1) group Namely, the neutral current of the

GWS theory 1s a linear combination of the third component of the V—A isovector current (the “‘plus-

component’’ of which is the charged weak current) and the electromagnetic current The only free

parameter, which enters in the definitions of the neutral current in the standard theory, 1s sin? @w

(where Ow 1s the Weinberg angle)

Neutral currents were discovered at CERN in 1973 in an experiment using the large bubble chamber

‘“‘Gargamelle”’ [7, 8] In these experiments the muonless neutrino processes v,,(¥,,)+ N> v,(¥,.)+ X and

the process v, +e pv, +e were observed

After the pioneering work of the ““Gargamelle” collaboration a large number of experiments was

done in a relatively short time investigating various neutral current induced processes A bibliography

of the experimental works on neutral currents can be found in the review paper [9]

An exceptional role in the investigation of the neutral currents has been played by the discovery of

the weak interactions between electrons and nucleons (Novosibirsk [10] and Stanford [11], 1978) After

these works it became possible to perform a complete phenomenological analysis of all the neutral

current data Such an analysis was the object of many papers [9, 12-16] As a result one could uniquely

determine all the coefficients appearing in the most general phenomenological V, A expressions written

for hadron and lepton neutral currents It was shown that this umique solution 1s in agreement with the

GWS theory

In 1980-81, in experiments on the e* —e” colliding beams [17], information has been obtained on the

contribution of neutral currents to the cross sections of the processes e* +e > 1*+T1° (Ï = e w 7) These

new data also agree with the standard electroweak theory

The GWS theory predicts the values of the charged and neutral intermediate boson masses

Experiments searching for W and Z bosons using the p—p colliding beams at CERN are underway at

present [18] These experiments will provide the crucial test of the standard theory of electroweak

TCL AUTIoris

In the GWS theory sin* 6w is a free parameter Its value is predicted by Grand Unified Theories [19]

“h Ci J alraire cá Mir s 4 wv, YT) £3 k2 ne ha 6i Œ ATT) wiv a L) FWA = WG aT L}LĐ(C] = kỉ r1 Gi = a 3 4 G a c kí Sa a CC ir) = a _

on neutral currents Besides increasing the experimental accuracy it necessitates also the correct

This review is devoted to the exposition of the standard model of the electroweak theory and of the

basic concepts underlying it, both at the elementary level (section 2), and to the detailed discussion of

a number of neutral current induced processes (section 3)

At present, the best investigated processes with neutral currents are deep inelastic neutrino

(antineutrino) scattering on nucleons and deep inelastic scattering of longitudinally polarized electrons

76 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

on deuterons The first process is considered In section 31 The detailed discussion of the second

process 1s the object of section 3.4

The theoretically simplest neutral current induced process 1s the scattering of muon neutrinos

(antineutrinos) on electrons Neutral currents contribute also to other purely leptonic processes,

⁄¿(P„) + e—> w¿(P.) + e and e° +e' >/”+/Ƒ (l=e, 4,7) All these processes are considered in detail in

sections 3.3 and 3 5

Next to that, the simplest exclusive neutrino-hadron process is elastic scattering of neutrinos

(antineutrinos) on protons In the framework of the GWS theory the matnx element relevant to this

process is determined by the weak and electromagnetic formfactors of the nucleons The processes

1„(P„) + p—> „(P„)+ p are discussed in detail in section 3 2

Finally, in appendix A, a derivation of the basic formula 1n the quark-parton approxtmation for

the calculation of the cross section of neutrino (antineutrino}-quark (antiquark) scattering is given It

enables one to easily get the cross sections for deep inelastic neutrino (antineutrino) scattering on

nucleons in this approximation In appendix B, the basic ideas of dynamical symmetry breaking

are described Many theores, which are practically indistinguishable from the GWS theory in

the energy region accessible to present experiments, have been considered 1n the last few years These

alternative theories will not be discussed here and we refer the reader to an existing brilliant review

[23]

Our review was actually written so as to be complementary to other reviews [9, 12, 24, 25]

It 1s pleasant to acknowledge very useful discussions with B Pontecorvo and F Niedermayer

concerning the problems considered 1n this review

2 The Glashow—Weinberg—Salam theory

21 Introduction

In this section we describe in detail the unified theory of weak and electromagnetic interactions of

Glashow, Weinberg and Salam [1-3] In 1961 Glashow [1] constructed a model for the weak and

electromagnetic interactions of leptons which was based on the gauge SU(2) x U(1) invariance This

theory 1s based on the assumption that, together with the photon, there exist also charged W and

neutral Z intermediate bosons The masses of the W and Z bosons were introduced ‘‘by hand" As a

result, the model was unrenormalizable In 1967-68 Weinberg [2] and Salam [3] constructed the

3E A, moge a 212C a = 1 2 ais O aifais ry arall no 5 DOntaneou D a gown œO

the gauge e symmetry In 1971-72 1t was proved by t’Hooft [26] and others [27] that models of this type

cre renormaltZap ne modet was generalized (oO quark 8 sing ne mechanism proposed D

Glashow, Ilhopoulos and Maianm [29]

The GWS theory 1s based on the assumption of the existence of charged and neutral intermediate

vector bosons and it 1s constructed so that, for massless fundamental fermions (leptons and quarks), a

local SU(2) x U(1) gauge invariance takes place Then the interaction (again locally gauge invariant) of

Higgs scalar fields [30], with both gauge vector bosons and fermions, 1s introduced As a consequence of

the spontaneous breakdown of the underlying symmetry, leptons, quarks and intermediate bosons all

SM Bilenky and J HoSek, GWS theory of electroweak interachons and the neutral currents 77

2.2 Gauge invariance

Quantum electrodynamics is the gauge invanant theory which describes all relevant experimental

data We start this subsection with a short review of this theory As an example, consider the electron

field (x) The free Lagrangian of this field has the standard form*

where mm is the mass of the electron, 0, = 6/dxX_ The Lagrangian (2.1) is invariant with respect to the global

gauge transformation

Ú(x)> ứ(x)= €` W(x), (2.2)

where A is an arbitrary real constant It is obvious that the Lagrangian (2.1) 1s not invariant with respect

to the local gauge transformation

where U(x) = exp{iA(x)} and A(x) is an arbitrary real function of x The derivative đ„ý(x) is mndeed not

transformed under (2.3) as the field /(x) itself Really, we have

9„ (x)= U(x) (ð„ + 19„À(x)) ý(x)

As 1s well known, the local gauge invariance (2.3) can be maintained provided that the interaction of the

field y with the electromagnetic field A, is introduced Consider the quantity (0, —ieA,)w (e is the

electron charge) We have

is now invariant with respect to the gauge transformations (2 3) and (2 5) To construct the complete

aprangian of the em under consideration, we have to add also the gauge invariant Lagrangian o

the electromagnetic field The tensor of the electromagnetic field is given as

78 SM Bilenky and J Hosek GWS theory of electroweak interactions and the neutral currents

Clearly

Fig = Fup Consequently, the gauge invariant Lagrangian of the fields of electrons and photons takes the form

The substitution of the derivative 0,“ by the covariant derivative (0, — 1eA,)wW in the free Lagrangian of

the field y leads to the following interaction Lagrangian for electrons and photons

where J„ = y„ý 1s the electromagnetic current Thus the substitution (2 6) fixes uniquely the form of

the interaction Lagrangian Such an interaction 1s called a minimal electromagnetic interaction Let us

note however that the principle of gauge invariance alone does not fix the interaction Lagrangian uniquely

For example, the addition of the Pauli term pivougiF.g to the Lagrangian (2.8) does not spoil the gauge

invariance of the theory (Gag = (1/21)(Ya¥g — Ye¥a), & 18 the anomalous magnetic moment )

All available experimental data confirm that the Lagrangian (29) 1s the true Lagrangian which

governs the interactions of electrons and photons It 1s also well known that electrodynamics, with the

minimal interaction (2 9), 1s a renormalizable theory

The modern theory of weak interactions 1s constructed in analogy with quantum electrodynamics

This can be done provided that intermediate vector bosons exist* We know from the experiment that

the Hamiltonian of weak interactions contains charged currents Therefore, to construct a theory of

weak interactions we have to start with a gauge theory containing fields of charged vector particles

Such a theory does exist It 1s the Yang-Mills theory [31] which we will now briefly present

Consider the doublet

* Experiments with the p-p colliding beams at CERN should enable one to check this fundamental assumption of the present theory of weak

interactions in the near future [13]

SM Bilenky and J Hoiek, GWS theory of electroweak interactons and the neutral currents 79 respect to the local SU(2) transformations

where

U(x) = expfizr A(x)}

and where the A,(x) are arbitrary real functions of x It 1s sufficient to consider only infinitesimal

transformations (2.12) The parameters A, will be taken as infinitesimal and in all expansions in powers of A,

we shall keep only the linear terms Thus, we have

Next, we get

It 1s clear from (2.14) that the Lagrangian (2.10) 1s not Invariant with respect to the transformation

(2.12) To construct a gauge invariant theory in analogy with electrodynamics, we thus introduce, besides

the field yf, the vector fields A, Consider the quantity

(2„ — 1g37 A„) Ú, (2.15)

where g 1s a dimensionless constant Using (2.13) and the commutation relations (57,, 37] = l€yy 27k we find

(da — gat Aa(x)) W(x) = U'(x) U(x) (da — 1837 Aa) U~'(x) (x)

with

A‘ (x) = A, (x) + ; aud (x) — A(x) X Ag (x) (2.17)

The field A,(x)1s called a Yang-Mills field It is seen from (2 17), that under global SU(2) transformations

the field_A, transforms_as—a_triplet—

Thus, as it 1s seen from the expression (2 16), the covariant derivative 0, —ig37A, applied to the field ứ,

ansforms under the gauge transformation and as the field elf (a primed quantity 1

obtained from an unprimed one through its multiplication by the matrix U) This means that the

substitution of the derivative 0, in the free Lagrangian by the covariant derivative (2.15) leads to a

Lagrangian, which is invariant with respect to the gauge transformations (2 12) and (2.17)

To construct the gauge invariant Lagrangian of the field A, consider the quantity

80 SM Bilenky and J Hogek, GWS theory of electroweak interactions and the neutral currents

With the help of (2 17) it is easy to check that

It is immediately seen that the quantity FigF.g 1s a group scalar In analogy with electrodynamics we

take the Lagrangian of the field A, in the form

L = —F pF op

(2 20)

Thus, 1f the interaction of the fields ~ and A, 1s introduced through the ‘‘minimal”’ substitution

dat > (J, —1957A,), the total Lagrangian of the system under consideration has the form

Consequently the interaction Lagrangian of the fields w and A, 1s as follows:

The constant g introduced before becomes the interaction constant Therefore, the ‘“‘mimimal’’ sub-

stitution dot > (da — 183TA,)& fixes uniquely the interaction Lagrangian of the fields y and A, We have

arrived at the “minimal” interaction Lagrangian for the fields y and A,, which ts compatible with gauge

Invariance Note also that owing to the non-linear term gA, X Ag appearing in the expression (2 18)

written for the field tensor Fg, the Lagrangian (2 21) contains terms, which are responsible for the

self-interaction of the field A,

Notice that a mass term —3m7A,A, for the gauge field cannot be added to the Lagrangian of the

fields of electrons and photons because its presence would destroy the gauge invariance of the theory

This means that the mass of the photon 1s equal to zero In the case of the Yang-Mills theory, the

imposed gauge invariance also does not allow a mass term of the type —3m%A,Aq Consequently, the

particles of the field A, are all massless (see appendix C)

We conclude this section with the following remark Consider several fields y% (1=1, ,n)

interacting with the gauge field A, We can write

e, are the constants of interaction between the fields y, and the gauge field A, It 1s clear from (2 25) that the

local gauge invariance 1s guaranteed provided that

SM Biuenky and J HoSek, GWS theory of electroweak interactions and the neutral currents 81

(A(x) 1s an arbitrary real function of x) Gauge invariance does not impose any restriction on the

coupling constants @,

In a non-Abelian Yang-Mills theory the situation 1s completely different If there are several field

multiplets interacting with one Yang-Mills gauge field, the coupling constant of all the fields with the

gauge field is unique It follows immediately from the fact that the coupling constant enters into the

expression for the field tensor Fig (eq (2.18)) because of the non-Abelian character of the Yang-Mills

group

2.3 The Higgs mechanism

The Lagrangian mass terms are introduced into the GWS theory via the so-called Higgs mechanism

for the spontaneous breakdown of the gauge symmetry To illustrate how this mechanism works, we

consider in this subsection classical examples of spontaneous symmetry breakdown in relativistic field

We now look for the minimum of the energy of the system Obviously, the Hamiltonian (2.29) is

minimal at ¢ = const., a value obtained from the condition

where a is an arbitrary real parameter

Thus, the minimum of the Hamiltonian (2 29) is infinitely degenerate The degeneracy 1s obviously

82 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

connected with the fact that the Lagrangian (227) is invariant with respect to the global U/(1)

transformations

$(x) > $'(x) =e" 6(x) (2 32)

The energy minimum of the system under consideration corresponds to an arbitrary value of ø in

(2 31) Due to the gauge invariance (2 32) it 1s always possible to taket

This is the typical example of spontaneously broken symmetry{—the Lagrangian of the field @ 1s

invariant with respect to the global U(1) transformations, while the value of the field ¢, corresponding

to the minimal energy, 1s just one of the many possible choices

We further introduce two real fields y; and y2 as

b= Toten ta) (2 34)

It follows from (2 33) that the energy of the system reaches 1ts minimum value when the fields y; and x2

have vanishing values Substituting (2 34) into (2 27), and omitting the unimportant constant Av*/4, we

get the Lagrangian of the system in the following form

L = ~30aX1 IaX1~ 20aX2 IaX2— 4À(40 7y 1+ 4oy1† x1† 4oyix2+ 2y 1X) + X)) (235)

It now describes the interaction of two neutral scalar fields The mass term of the fñeld x; 1s

Consequently, in the case of quantized fields, the mass of the field quantum y, equals m, = V2Au?°=

V2 There 1s no term quadratic in the field x2 This means that the particle corresponding to the

quantum of the field x2 1s massless

We have assumed that the constants A and yz” in the Lagrangian (2.27) are postrve Consequently, the

term quadratic in the field @ appears in (2 27) with positive, 1e ‘‘wrong”™ sign§ This leads to the

spontaneous breaking of the symmetry The degeneracy of the ground state is a characteristic of this

phenomenon We have howev ntroduced new real fields_y;and-y> for which the ground state O

degenerate This leads to the spontaneous breakdown of the original U(1) global symmetry of the

+ We note that in the quantum case the conditions (2 30) and (2 33) become |(0|ø|0)|ˆ= 0°/2 and (0|¢|0) = v/V2, where |0) 1s the vacuum state

+ Spontaneous symmetry breakdown 1s a phenomenon well known in many-body physics [33] A typical example 1s ferromagnetism The

corresponding Hamiltonian of the spins of the electrons 1s invariant with respect to the global SU(2) spin rotations The ground state, that 1s the state

with all spins parallel in an arbitrary but particular direction 1s clearly invariant only with respect to U(1) rotations around the direction of the

macroscopic magnetization

§ We can rewrite V also in the form V(d*¢) = A(b*¢ — 7/2A) — w7/4A_ From this expression it 1s obvious that the condition A > Ú ts necessary

tor } to be bounded from below Note also, that for «?<0 the potential has its mintmum at ¢ = 0 In this case the Lagrangian (2 27) describes the

massive complex scalar field with the interaction - A(o* 6)

SM Buenky and J HoSek, GWS theory of electroweak interachons and the neutral currents 83

Lagrangian As a result, the quanta of one field are massive, while the mass of the second field is equal to

zero

With spontaneous breakdown of a continuous symmetry massless spin zero particles always appear

This statement 1s quite general, and 1t comprises the content of the Goldstone theorem [32, 34, 35] The

corresponding massless spin zero bosons are therefore called Goldstone bosons

Scalar massless particles are not observed This might imply that the ideas of spontaneous symmetry

breakdown are useless in constructing realistic physical theories in elementary particle physics

However, it will be shown in the following, how the spontaneous breakdown of a local gauge symmetry

results in massive gauge quanta due to the disappearance of Goldstone bosons

Let us assume that the complex field ¢ with the Lagrangian (2.27) interacts minimally with the gauge

field A, This interaction is introduced by the substitution 6.6 —> (ó„ — 1gA„)ó 1n (2.27)

The complete Lagrangian of the system 1s [30, 36]:

where

The Lagrangian (2 37) is invariant with respect to the local gauge transformations

$(x)> (x) =e? P(x),

where A(x) 1s an arbitrary real function of x

As in the previous example, the minimum of the energy corresponds to a value of the field @ equal

to (v/V2) e'* (where a is an arbitrary parameter, v/V2= V u?/2A) Due to the gauge invariance of the

Lagrangian (2.37) the “vacuum” value of the field ¢ can always be taken as

It is clear that due to the local gauge invariance of the theory the function @(x) appearing 1n (2.41)

has no physical meaning It can always be eliminated by an appropriate gauge transformation Thus, we

have

+ Restricting the expansion of the exponential exp{16(x)/v} by terms linear in 1/v we get }(x) = (v + x(x) + 1(x))/V2 In this approximation we

have x(x) = xi(x), 0(x) = x2(x), where yi(x) and y2(x) are the functions introduced in eq (2 34)

84 SM Biuenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

Substituting (2.42) into (2 37) and omitting the unimportant constant, we get the Lagrangian of the

system under consideration in the following form:

The Lagrangian (2.43) contains the mass term of the vector field A,(—3g*v7A,A,,) and the mass term of

the scalar field y(—32Av7y*) Consequently, the masses of the quanta of the fields A, and y are equal to

ma = gv and m, = V2Av’, respectively

Before spontaneous symmetry breakdown the Lagrangian of the system contained a complex scalar

field (two real functions) and a massless gauge field (two independent real functions) After spontaneous

breakdown of the local symmetry we arrived at the Lagrangian of an interacting real massive scalar field

(one real function) and a massive vector field (three real functions) The degree of freedom, which

would correspond to the massless Goldstone boson (in the absence of the gauge field A.), had

been transformed through the spontaneous breakdown of the local gauge symmetry of the Lagrangian

(2.37), into the additional degree of freedom of the vector field

The mechanism thus discussed for mass generation 1s called the Higgs mechanism The scalar

particle, corresponding to the quantum of the field y, 1s called the Higgs particle For a more general

discussion of particle mass generation we refer the reader to appendix B

We have explained the basic principles which are used in constructing models of electroweak

interactions Now we turn to the detailed discussion of the standard SU(2) x U(1) theory of Glashow,

Weinberg and Salam

24 The standard electroweak theory (the case of leptons)

The phenomenological V-A current X current theory [37] was capable of describing the vast amount of

existing experimental data Consequently, any new theory of weak interactions has to be built up so as to

reproduce the results of this theory

The GWS theory is based on the assumption that there exist intermediate vector bosons To

reproduce the results of the phenomenological, current X current theory at low energies it 1s therefore

necessary to assume that at least part of the ‘‘true’’ weak interaction Lagrangian 1s of the form:

2V27

where W, ¡s the field of the charged vector bosons and ,;{? is the charged weak current The

dimensionless coupling constant g is related to the Fermi constant by

go _ Gr

where my 1s the mass of the charged intermediate boson The charged current 1s the sum of lepton and

hadron (quark) current In this subsection we shall consider the GWS theory of leptons Consequently,

SM Bllenky and J HoSek, GWS theory of electroweak interacnons and the neutral currents 85

we will be interested only in the lepton current It follows from all available data that the charged

lepton current is

Œ2= p3 „(1+ ys)e + P„y„(1 + ys)¿ + P,y„(1 + 3)T, (2.46)

where e, ¿ and 7 are the fteld operators of the electron, muon and 7-lepton, respectively; »., v,, and v,

are the field operators of the electron-, muon- and 7-neutrinos, respectively*

At the beginning we consider the case of massless fields In order to get the term (2 44) in the

interaction Lagrangian of leptons and vector bosons we assume that

ina (ir) een) (2.47)

forms a doublet of the SU(2) group and

are singlets** of this group Here

u =3(1+ x)( 7) and I£=ÄXI—ys)!", zie=XI—+ai (2.49)

are the left-handed (L) and the right-handed (R) components of the corresponding fields

The free field lepton Lagrangian

l=e,y,T

is clearly invariant with respect to the global SU(2) group We demand now for massless fields the local

Yang-Mills invariance with respect to

W(x)? win (x) = exp{i27À (x)#iŒ),

where the A,(x) are arbitrary real functions of x(r = 1, 2, 3), and where Aj is a tri of vector

We assume the interaction of leptons and vector bosons to be minimal Such an interaction is

—————mtroduced via the substitutton (see sectHon 2.2)

* A direct proof of the existence of the v, 1s still lacking All available data are, however, consistent with its existence [38]

** We assume that the masses of the neutrinos are different from zero Such a possibility is widely discussed in the literature at present [39]

86 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

(g is the dimensionless constant) From (2 50) and (2 52) we get the interaction Lagrangian of leptons

and vector bosons as

1s the field of charged vector bosons and”

I=? =2 a Wit YaT+ Win = a ViYa(1 + ys)l’ (2 57)

is the charged current Therefore, the interaction Lagrangian (2.55) which follows from the local gauge

invariance does contain the term (2 44) describing the interaction of leptons with charged intermediate

boson

The second term in the Lagrangian (2.55) describes the interactions of neutrinos and charged leptons

with the neutral vector boson:

Ly = 1g A > (zry.( + Ys)U1— ly + +:)!)Aa (2.58)

|

“he Ẳ heo a umified theory of weak and electromànetic ¡nteracttons ObvIous he

interaction (2 58) is not the electromagnetic interaction For a unification of weak and electromagnetic

interactions it is necessary, therefore, to require the invariance of the Lagrangian of the system with

respect to a larger group than the local SU(2) The simplest possibility 1s the group SU(2) x U(1) which

makes the basis of the GWS theory

To construct the locally SU(2)X U(1) invariant Lagrangian we perform in (2.50) the minimal

substitutions (see section 2 2)t

* The primes put on lepton fields indicate that these fields do not necessarily correspond to lepton fields with well defined masses which will be

generated later through spontaneous breakdown of the underlying symmetry

¥ It 1s appropriate to express the coupling constants of the group U(1) as g’ VL g '$y§ Đ and g 2y

SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents 87

Ôxửn > (dq ~~ 1827A„ ~ ig’syL Bo) Win ’

Ô„U1 —> (ô„ —1Ø'2VR`B„)U1n „

where A, is a triplet of gauge fields with respect to the group SU(2), B, is the gauge field associated

with the symmetry group U(1) The complete gauge invariant Lagrangian of leptons and vector bosons

The current j, 1s given by (2.54) and

Ja = > Yi Vath + > yk Pläy„lä + >» YRPinYaV ir - (2.63)

The U(1) invariance does not impose any constraints on the coupling constants between the leptons

and the field B, (see the discussion at the end of section 2.2) This freedom 1n the choice of the coupling

constants for the U(1) gauge group can then be used to unify weak and electromagnetic interactions

We will choose y,, y&” and y® so as to satisfy the Gell-Mann-Nishijima relation*

88 SM Buenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

is the electromagnetic current of leptons and where /2 1s the third component of the isovector j,

Using eq (2.67) the interaction Lagrangian (2.62) can be rewritten as

ov! giới (2 69)

where

LY = 19/2 A, +12'9S" — 72) Ba (2 70)

is the interaction Lagrangian of the leptons and the neutral vector bosons To single out the

electromagnetic interaction from (2 70), we rewrite this expression as

orthogonal to Z, Elementary algebra Jmphes that the ñeld Ag IS coupled only to 7Š", while the field Z2,

Lagrangian of the electromagnetic interactions and that A, is the electromagnetic field Indeed, we

SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents 89

Thus, there are four vector boson fields associated with the gauge SU(2) x U(1) group Two fields

correspond to charged vector bosons (W* and W ), two fields correspond to neutral ones One neutral

field is identified with the electromagnetic field, the other 1s the field of the neutral intermediate boson

Consequently, the unification of weak and electromagnetic interactions based on the group SU(2) x

U() is possible provided that not only charged vector bosons and charged currents but also neutral

vector bosons and neutral currents, do exist

As is well known, the neutral currents were discovered at CERN in 1973 [7, 8] The second part of

this review is devoted to the detailed study of selected processes which are due to neutral currents Now

we will continue our constructing of the unified electroweak theory of GWS The Weinberg angle Øw 1s

electromagnetic interactions The first term in eq (2 78) is the third component of the isovector, whose

“plus-component” is identified with the charged weak current The second term in eq (2.78) 1s

proportional to the electromagnetic current The parameter sin* @w 1s thus the only parameter which

enters the expression for the neutral current Its value can be determined from the data on neutral

cu induced processes Note values of the parameter sin’ 6y, determined from the data o

most diverse experiments, all coincide within the experimental errors

90 SM Bulenky and J Hosek GWS theory of electroweak interactions and the neutral currents

The theory we have considered up to now satisfies the requirements of a local SU(2) x U(1) gauge

invariance Mass terms of the vector boson fields cannot be introduced into the Lagrangian of such a

theory It 1s also obvious that the SU(2) x U(1) invariance with left-handed fields in doublets ứ„ and

right-handed fields /p in singlets also forbids the introduction of lepton mass terms into the Lagrangian

In the standard electroweak theory the Lagrangian mass terms of both the vector boson and fermion

fields are introduced by the Higgs mechanism of spontaneous breakdown of the gauge symmetry (see

section 23) The theory 1s built up so that, at the beginning, the complete Lagrangian, including the

Higgs sector, 1s locally SU(2)x U(1) invariant It 1s then necessary to assume that the Higgs fields

transform according to a definite (non-trivial) representation of the gauge group Further, due to the

spontaneous breakdown of gauge invariance charged (W* and W_) as well as neutral Z intermediate

bosons have to acquire masses That 1s, three Goldstone degrees of freedom of Higgs fields can

transform at the spontaneous breakdown of the gauge invariance into the additional degrees of freedom

of vector fields Thus, we are forced to assume that the Higgs fields form at least a doublet It 1s

this “‘minimal” assumption which 1s at the basis of the GWS theory

Hence we assume that the Higgs fields form a doublet of the SU(2) group [2|

“TL: 089

where the complex functions „ and óo are the fields of charged and neutral bosons respectively Weak

hypercharge of the doublet (2 8l) 1s defined so as to fulfil the Gell-Mann-Nishijima relation (2.64) We

where #«* and A are positive constants CC CC C7 C7S7õ7õ7õ7ẽ7õ7ẽõẽõẽSẽSS———=ee=

Taking into account (2 82), we get from (2 83) by the standard substitution

SM Bilenky and J Hosek, GWS theory of electroweak interacnons and the neutral currents 91

where Ø,(x) and y(x) are real functions Finally, the functions Ø,(x), which correspond to the “would

be” Goldstone bosons, can always be eliminated owing to the gauge invariance of the Lagrangian (2 85)

by appropriately fixing the gauge (the so-called unitary gauge) Thus we have

Let us substitute (2 89) into (2.85) Taking into account that

(zA„)(zZA„)= 2W„W„ + A$A}, ó*(zA„) B„¿ = — A$ B„340 + xŸ

we get

2= —3ô„y ô„x — 3(0 + x)°[äg?2W„W„ + 2(g? + Ø2)Z.Z.Ì- 3Ax {v + 20 (2.90)

Here W, = Al-?/V2 and W, = A1*?/V2 are the fields of the charged vector bosons and Z, 1s the field

of the neutral vector bosons

As a result of the spontaneous breakdown of the symmetry, mass terms for the intermediate bosons

have emerged in the Lagrangian’

Symmetry was broken 1n such a way that the photon remained a massless particle

The function y(x) 1s a field of neutral scalar particles (the so-called Higgs particles) It follows from

(2.90) that their mass 1s equal to

92 SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

Note that the Lagrangian (2.90) contains also a term describing the interaction of the Higgs particles

with the intermediate bosons

We find from (2 77) and (2 92) that the mass squared of the Z boson 1s related to that of the W boson

and to the parameter cos” Ow by

It should be stressed that this relation 1s satisfied only if the Higgs fields form doublets In the case of

higher Higgs multiplets no relation between masses of neutral and charged intermediate bosons does

The value of this parameter can consequently be determined from the neutral current data Available

data are consistent with p = 1 (see section 3.1), 1.e they agree with the ‘minimal’ assumption with a

doublet representation of the Higgs fields

Thus far, we have considered the Higgs mechanism of W and Z mass generation To get also lepton

mass terms in the Lagrangian via the spontaneous breakdown of symmetry, it is necessary to introduce the

interaction of the Higgs fields with the lepton fields The corresponding Lagrangian has to be SU(2) x U(1)

invariant The most general renormalizable SU(2) x U(1) invariant Lagrangian for the interaction of lepton

and Higgs fields has the Yukawa form

V2~— = ;

h;b

Here ¢ 1s the doublet of Higgs fields and Mj, are constants (complex, in general) Substituting

further (2.89) into (2 96) (1.e breaking spontaneously the symmetry), we obtain

Here V is a untary matrx and (m2) = m;ô¿ We defne the diagonal matnx m as follows m m= m2, mụ„= +Vim2 We have, dentically

M = V`mU where U = m™'VM It 1s easy to see that U 1s a unitary matnx Indeed, using (2 98') we have ƯUˆ = m1VMM}Vˆm"!= ]

SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents 93

Here V, and Vz are unitary matrices and m’ is the diagonal matrix with positive entries

Substituting (2.98) into (2 97), we obtain

The first term in the Lagrangian (2.99) is the standard lepton mass term (m, is the lepton mass), lepton

fields with a given mass being linear combinations of primed lepton fields (i.e fields entering SU(2)

multiplets)

The second term in (2.99) is the Lagrangian of the Yukawa interaction of lepton fields with the scalar

Higgs field As seen from (2.101), the couplings are proportional to lepton masses (This is one of

characteristic features of the standard electroweak theory.)

We have considered the Higgs mechanism of generation of charged lepton masses If the neutrino

masses are different from zero, it is necessary to assume that an additional interaction of the Higgs fields

with the lepton fields exists besides (2.96)

Having the doublet ¢@ we have also the doublet

94 SM Bilenky and J Hošek, GWS theory of electroweak interactions and the neutral currents

Therefore, if /” 1s present in the complete Lagrangian of the system, neutrinos acquire masses* due

to the spontaneous symmetry breakdown (in (2 106) », 1s the field of the neutrino with the mass m,)

Let us go back to the expressions for charged and neutral currents For the charged lepton current

we get with the help of (2 57), (2 100) and (2 107)

where U 1s the unitary mixing matrix (U= V_U{) Therefore, if neutrino masses are different from

zero, the standard Higgs mechanism of mass generation results in neutrino mixing Neutrino fields

entering the charged current (current neutrino fields) are linear combinations of neutrino fields with

definite masses Let us note that neutrino mixing means nonconservation of lepton charges In

particular, the neutrino oscillations first considered by Pontecorvo [41] become possible We will not

consider these problems here and refer the reader to existing reviews [39]

For neutral lepton current we get with the help of (2 47), (2 57), (2 78), (2 100) and (2 107) (and using

the unitarity property of the matrices Vip and U,)

Jo=5 2, P.y„(l + Ys) — 5 2, lyz(++‹:)l—=2sn 6y 3, (Dl yal (2.111)

Hence, the neutral current of the standard electroweak theory 1s diagonal in the lepton fields (it

conserves lepton flavour)

We will complete this consideration of the GWS theory in the case of leptons by the following

remark If the fundamental fermions would be only leptons, the GWS theory would not be renormaliz-

SM Bulenky and J Hošek, GWS theory of electroweak interactons and the neutral currents 95

electric charges of the fundamental fermions 1s zero [44] Leptons taken separately do not fulfil this

requirement

We now turn to the consideration of the GWS theory in the case of quarks We will assume that

besides three neutrinos and three charged leptons there exist three colored quarks with charge § (u, c t)

and three colored quarks with charge — š (d, s, b) In this case the condition for the absence of triangular

anomalies 1s clearly satisfied:

-3+ 336) + 3-3) =0

25 The standard theory of electroweak interactions (the case of quarks)

We start this subsection by several remarks of historical character As 1s well known, the huge

amount of weak interaction data obtained before the discoveries of neutral currents and of charmed

particles (1973-1975) was described within the framework of the V-A theory with the Cabibbo [45] current

When written in terms of quark fields, the Cabibbo current has the form

where

and đc ¡s the Cabibbo angle

If we attempt to build up a SU(2) x U(1) theory with the charged current identical with the Cabibbo

current, we have to assume that the relevant doublet is

In such a case the neutral current would clearly contain (in its 72 part) the term

di Yad} = cos” Ốc dy Vad + sin? 6c SLYaSL + sin 6c Cos 6c (d YoSt + SLYady)

which does not conserve strangeness We would thus arrive at a contradiction with the experimental

ˆ Bí /^ ¬ "^^ ry ry ^^ ^ of se rie + eet : yp with hœ/m maìiankm Py te cy

LIata L} ũ _}) ° LÌ 9 k2 ya L} KỈ ũ k} Fa ow L2

of K*-decay It however follows from the data, that [46]

I(K* Tai <0-5% 10 > a" vb) —6

The conflict became particularly sharp in 1973, after the discovery of neutral currents at CERN [7, 8]

A possible solution of this puzzle was provided in fact already in 1970 by Glashow, Iliopoulos and

Maiani in their celebrated paper [29] They assumed that the Cabibbo current is not the full hadronic

current The authors of ref [29] accepted the hypothesis [47] of the existence of a fourth (charmed)

quark c with electric charge 3 The additional term in the charged hadronic current was assumed to be

96 SM Bilenky and J Hosek GWS theory of electroweak interactions and the neutral currents

forms a doublet Since

dt Yad} + SLY„ŠL — di Yad + SLYaSL

the neutral current of such a theory 1s flavor diagonal (the GIM mechanism does work)

The discovery of the r-lepton in 1975 [48] was an indication that the number of quarks might be

bigger than four The discovery of the Y family in 1977 [49] was the first justification of the existence of

a new b quark with charge —3 Recently, particles containing the b quark have been discovered 1n e*-e~

beams experiments [50]

The standard theory of electroweak interactions 1s based upon the assumption of the existence of three

neutrinos, three charged leptons, and six quarks (d, s, b, u, c, t)*

The GWS theory for quarks 1s constructed in full analogy with that developed for leptons We

assume that the left-handed components of the quark fields form SU(2) doublets

"1" su

and that the right-handed components of the quark fields

are SU(2) singlets

The free Lagrangian of massless quarks reads 2 CC 77777

SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents 97

Oot > (0 — igstA — 1ø'2yLB,) Wis

Jak > (a —18'2VR Baga; In = Ak, Sk, Dp are made in 5%

(A, and B, are the gauge fields of the symmetry groups SU(2) and U(1), respectively, g and g’ are the

corresponding coupling constants, y, is the hypercharge of the left-handed doublets, y&’ (y&") is the

hypercharge of the right-handed quark fields with charge 3 (—3).)

As a result of the substitution (2.121) we get the following minimal interaction Lagrangian for quark

and vector fields:

Note that due to the non-Abelian character of the SU(2) group the constant g, characterizing the

interaction of leptons with A,, has to be the same as that for the interaction of quarks with the field A,

Therefore, an electroweak theory can be based upon the SU(2) x U(1) group provided that weak

interactions of leptons and quarks are universal As 1s well known, experimental data agree with the

universality of weak interactions*

The group U(1) does not impose any restriction on the weak hypercharges of lepton and quark

multiplets To unify the weak and electromagnetic interactions into an electroweak one, as already done

in the lepton case, weak quark hypercharges are chosen so as to obey the relation

is the electromagnetic quark current (e, is the charge of the quark in units of the proton charge) Thus,

hypercharges of quarks have to be chosen such that the Gell-Mann—Nishijima relation be satisfied The

* Universality of weak interactions of leptons and quarks 1s exhibited, in particular, by the equality of the coupling constants of ¿- and Ø-decays

98 SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

hypercharge of doublets must be equal to the sum of the charges of upper and lower components of the

Let us single out from Y; the term which describes the interactions of quarks with charged

intermediate bosons We obtain

L,= củ sul Wo +he )* 10/1+A2 +i1øg'(2"—/2)B., (2 129)

where

/Ðg)=2 > Wi YaX(T1 + 172) ir

is the weak charged quark current and W, = (Az - 1A2yV2 is the field of the charged vector bosons

Instead of the fields A and B, let us introduce the field of the neutral intermediate bosons Z, and the

electromagnetic field A, (see eqs (2.72) and (2.73)), into the Lagrangian (2 129) Taking into account

the condition (2 76) for the full interaction Lagrangian for quarks and vector boson fields, we have

The first term in the expression C 131) is the interaction Lagrangian of quarks and charged

intermediate bosons, the second term he interaction Lagrangian of quarks and neutral intermediate

bosons, and the third term is the Lagrangian of the electromagnetic interactions of quarks

Now we have to introduce the SU(2) x U(1) invariant interactions of quarks and vector bosons with

the scalar Higgs particles and then to break spontaneously the symmetry The interaction of vector

bosons with a doublet of Higgs particles was already considered 1n section 2.4 We have seen that, due

to the spontaneous symmetry breakdown, the intermediate bosons W~ and Z acquired masses while the

photon remained massless Here we consider the mechanism for the generation of quark masses under

the “minimal” assumption (already made in section 2 4) of one Higgs doublet

The most general SU(2) x U(1) invariant Yukawa interaction Lagrangian of quarks with the Higgs

SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents 99

doublet can be written as

>) SX baMGPqre- 2S DY da MEP gad +he (2 133)

Here

are SU(2) doublets with hypercharges equal 1 and —1, respectively, M“' and M°” are complex 3 x 3

matrices Setting (1 e breaking spontaneously the symmetry)

Therefore, as a result of spontaneous symmetry breaking, a quark mass term appears in the

Lagrangian To put it in its usual form, it is necessary to diagonalize the complex matrices MW and

M°” This can be done with the help of bi-unitary transformations (see (2 98)) We have

(~1/3)

Here Vx and U_p are unitary matrices and m and m@” are diagonal matrices with positive

entries Substituting (2.137) into (2.136) we obtain

100 SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

Thus it follows from (2 138) that g = đu + gris the quark field with mass m, The constants f, appearing

in the Lagrangian (2 138) are the coupling constants for the interaction of quarks and neutral Higgs

particles We have

]

The coupling constant ƒ„ 1s proporttonal to the mass mụ Thìs proportionality emerges as a result of

the diagonalization of the matrices M~" and M°

Both expressions for charged and neutral currents (2.130) and (2 132) are written in terms of primed

quark fields Using the relations (2.139) we now rewrite these currents in terms of quark fields with

definite masses For the charged current we have

đá sb

where

1s the unitary mixing matrix

The unitary 3 x 3 matrix U in the charged current (2 141) can be parametrized as follows {22]

U =| —5¡Œ C1€2C3 — $253€ C1C253+ 52¢3e"" J (2.143)

S82 — Ci82€aT— €a§a e'° — Ci§2§4 + C;ạCa ©'Š

where c, = cos Ø,„ s, = sn Ø„ ¿= 1,2,3 Thus in the case of three quark doublets the mixing matrix 1s

characterized by three angles and one phase (responsible for CP violation) The matrix (2 143) 1s the

known mixing matrix of Kobayashi and Maskawa [22] and it 1s the generalization of the Cabibbo-GIM

mixing matrix to the case of six quarks

Owing to the unitarity property of the matrices V, zp and U_ pr, the neutral quark current is equal to

R= 5 > ấy„(l+yz)q— 5 > Feallt ys)q —2 sin? Oy D suy.4 (2 144)

Therefore, the neutral current of the standard electroweak theory ¡s diaponal in the quark fñields († 1s

flavour conserving) We note that the available experimental data are all in agreement with this

consequence of the standard theory As an example we give the results of searches for the process

„+ ÌN>ư„+C+X

It follows from the data obtained in ref [52] that the ratio of the cross section for this process to the

cross section for the process v, + N> vy, + X 1s less than 0.026

SM Bilenky and J Hošck GWS theory dƒ electroweak interactions and the neutral currents 101

2.6 The effective weak interaction Hamiltonian

The complete interaction Lagrangian of leptons and quarks with charged W and neutral Z

intermediate bosons within the SU(2) x U(1) gauge theory, discussed in sections 2 4 and 2.5, is

The full charged current of leptons and quarks in the standard electroweak theory is given by

IP= DS Grit ysUean+ Š Hya(1+ ys)l (2.146)

The effective Hamiltonian for processes with virtual W and Z bosons to second order in perturbation

theory in g has, in the region q* << mw, m3, the following form:

If the Higgs fields form SU(2) doublets, the relation (2.94) holds and, consequently,

102 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

where v 1s the vacuum expectation value of the Higgs field @o It follows from (2.150) and (2 152) that

Therefore, the theory enables us to calculate the parameter v Substituting the numerical value of Gr,

we find

p = 2462 GeV (2.154)

The coupling constants of Higgs particles to leptons and quarks are given by the expressions (2.101)

and (2 140) Using (2 153), we obtain

fi=38x107 m/M, — f,=38x 10° m,/M (2 155)

(where M 1s the proton mass)

Weak and electromagnetic interactions are unified provided that (see section 2 4)

The value of the parameter sin 6w 1s determined from experimental data on neutral current induced

processes (see the second part of the review) Therefore, theory enables us to predict the value of the W

boson mass If the Higgs fields form doublets, the masses of the charged and neutral intermediate

bosons are related by eq (2.94) In this case we have

From the analysis of the world data on the deep inelastic processes „ +ÌN v, +X and „„+N—

„ + X one could deduce [53] the value*

* 2 _

‘A nine apove d C_O n? Ay, Ne MaSSes O Naryed and Ne dl 1(CFfm€CGIatC ĐOSODS ñ GIN

radiative corrections) turn out to be:

SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents 103

Experiments searching for the charged and neutral intermediate bosons in p-f colliding beams are

being performed at CERN at present [18] If these intermediate bosons are found, the test of the

predictions (2 160) and (2 161) will represent the most serious test of the GWS theory

The neutral current is a sum of neutrino, lepton and quark pieces

The neutrino neutral current is written as

l=, 7

In the general case of V and A interactions the lepton neutral current can be parametrized as

Ja= 3; lyalgv + ays)! (2.164)

Ì=€,u,T

In the standard electroweak theory the constants gy and ga are equal to

Finally, the neutral quark current in the general V, A case can be written in the form

The constants ¢,(q) and er(q) are given in the GWS theory as

er(q)=3—4sin’ Ow, £en(g)= —-iSn Øwy; q=ứGt

£¡(g)= —53+3 sn” Øwy, Enq(4)= šssinˆ0y; q=đs,b

These relations can be written in a more compact form as follows:

is t f the g-quark and J4 1s the third component of weak isospin for the field

It follows from (2.148) and (2.163) that the part of the effective weak interaction Hamiltonian, which

contains the neutral currents, can be expressed as a sum of the following terms:

104 SM Bilenky and J HoSek, GWS theory of electroweak interactions and the neutral currents

The processes (2.177) are governed by the neutrino—quark Hamiltonian #”*? while the processes (2.178) are

governed by the neutrino—-lepton Hamiltonian #””

Measurement performed at Novosib DỊ of the optical rotation of the plane of polarization

which arises when a polarized photon beam passes through a vapour of ””Bi, as well as the

measurement performed at SLA of the P-odd asymmetry in the deep 1nelasti attering 0

longitudinally polarized electrons on nucleons represent important results in the investigation of neutral

currents These effects are due to the interference of the electromagnetic and lepton—quark interactions

SM Bdenky and J Hošek, GWS theory of electroweak interactions and the neutral currents 105

Recently, information has been obtained [17] on the contribution of the neutral current j, 1n purely

leptonic processes e* +e > 1° +T° (1=e, ws, 7) using the e* —e° colliding beams at PETRA

The interaction Hamiltonian #77 contributes to the P-odd effects in nuclear transitions, to which

charged currents also contribute P-odd effects in nuclei have been observed already for a long time

[55] Their quantitative interpretation remains, however, a difficult and still unsolved problem [56]

Finally, information on »y-v interactions (the Hamiltonian #”:”) 1s practically missing Searching for

the decay K* > 4 *3v an experimental bound has been obtained [57] on the effective coupling constant

of the v—v interaction

F„ <2Xx 10° Gg

A slightly more stringent restriction on F,,

F,, <3x 10* G, can be obtained [58] from the analysis of neutrino experiments at high energies The standard

electroweak theory described in this section will now be applied to a detailed study of those neutral

current-induced processes, which have been best investigated experimentally

Processes (3.1) and (3.2) were first observed in 1973 with the bubble chamber “‘Gargamelle”’ [7] The

observation of these processes has marked the discovery of neutral currents At present, (3.1) and (3.2)

are still the best investigated processes with neutral currents

First we will obtain the general expressions for the cross sections for processes (3.1) and (3.2) The

evant na O ne effec weak 1D n on Hamutonian nas th orm

= SE Gaya — —?= v2“ a * 5 : 19M) œ

where 7$ is the hadronic neutral current Let us start with the process (3.1) The relevant diagram for

this process is shown in fig 1

The matrix element for this process is written as

(f|(S — Dl) =—i a NuNx G(K') YoA1 + ys) ukXp]J¿|ÌpX2a}` ô — p- 4) (3.4)

106 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

Fig 1 Dhragram of the process y», +N>», + X

Here k and k' are the momenta of initial and final neutrino, respectively, p 1s the momentum of the initial

nucleon, p’ 1s the momentum of the final hadron system, q = k — k', N, = 1/42)?2V2k¿} s the standard

normalization factor and J§ 1s the neutral hadron current tn the Heisenberg representation From (3 4) we

have for the cross section

Waa(p 4) = -0z)'ft 5 [ (y Ja|pXp|J2|p) - ô(p'~ p~ q)dÏ 37)

(dÏ” is the phase space element for the final particles, M 1s the proton mass)

The matrix element of the process v, +N- p, + X 1s written as

(HS — Dị) =7 MAM ñCk) (1+ 9) HC) (pJ2|p) -2n)' ô(g'= p + 4) G8

where k and k’ are the momenta of the initial and the final antineutrino Obviously,

ñ{(~k) ya( + +) (Ck)= tte(k') Yad — Ys) Me) (3.9)

The expressions (3.5) and (3 10) differ only by a sign of the pseudotensor L2,(k, k’) The difference 1s

due to the opposite helicities of neutrino and antineutrino

SM _ Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents 107

The quantity W,(p, q) is the sum of a tensor and a pseudotensor and has the following general

structure:

Has (0A) + (mB) (me Bh a)

+ 2ME 6apsơ P› Ve W3+ M? q„qa W¿ + Mã (Pada + JaPa) Ws + Mã (Dx4s — đapPa) W< (3.11)

where W, (1=1, ,6) are real functions of the scalar vartiables pg and ” The functions W, have the

dimension M~' The cross sections for the inclusive processes considered depend upon three in-

dependent variables, which are conveniently taken as:

“ã _Pq_” _ _ pk

(E 1s the energy of the initial neutrino, in the lab system and v = —pq/M) It 1s easy to show that the

variables x and y scan the intervals

1

0<x<t, 0<y= 11 Mv2E

Further, since Logga = LagGg = 0, it is clear from (3.5), (3.10) and (3 11) that the functions W,, Ws; and

W, do not enter the expressions for the cross sections In terms of the variables x, y and EF, the cross

sections for the processes (3.1) and (3.2) have the form

are the dimensionless structure functions

Note that the expressions (3.12) for the cross sections for deep inelastic scattering of neutrino on

nucleons and antineutrino on nucleons contain the same structure functions This 1s a consequence of

our assumption that the initial neutrino (antineutrino) and the final neutrino (antineutrino) (which are

not detected experimentally) are the same particles The Hamiltonian is given in such a case by the

expression (3.3) and since it is hermutian, it follows that

0 0

a (3.15)

108 SM Bilenky and J HoSek, GWS theory of electroweak interacnons and the neutral currents

where

(na = 1; 4, =—-1, 1 = 1,2, 3)

If we now assume that the initial and final neutrino (antineutrino) in the processes (3.1) and (3.2) are

different particles, the corresponding V, A Hamiltonian becomes

(v' # v,) The condition (3 15) then need not be fulfilled* The matrix element for the process of deep

inelastic scattering of antineutrino on nucleons in the case of the Hamiltonian (3.17) 1s given by the

expression (3.8), in which the substitution (p'|J°|p) > (p'|J|p) is performed Consequently, the structure

functions for the processes (3.1) and (3.2), in the case of nonidentical initial and final neutrino

(antineutrino), can be different [59]

If follows from (3 12) that

da;

y>0

If this relation 1s not valid, 1t 1s clear from the previous discussion that the initial and final neutrino

(antineutrino) in (3 1) and (3.2) cannot be identical particles The relation (3 18) was experimentally

verified Available data [60] agree with (3 18)

Up to now we have only assumed that the neutral current has a V, A structure We shall now assume

the validity of the GWS theory, 1.e we take the neutral hadronic current to be of the form (2 144) First,

we shall derive a relation [61] which connects the parameter sin* 6w with the total cross sections of deep

inelastic processes

U„(P„) + N- U„(P„) + xX,

Use will be made of transformation properties of the neutral hadronic current of the GWS theory

Let us write the neutral hadronic (quark) current of the GWS theory as

SM Bdenky and J HoSek, GWS theory of electroweak interactions and the neutral currents 109

and where the dots stand for the contribution of the s quark and other heavier quarks to the neutral

current Under a usual isospin transformations, N transforms as a doublet Therefore, v2 and a2 are

the third components of isotopic vectors

It follows from experiments with high energy neutrinos that the contribution of s and other heavier

quarks to the cross sections for deep inelastic processes comprises a few percent of the contribution of u

and d quarks [62] We will neglect this contribution in the following With this approximation

Further let us consider deep inelastic scattering of neutrino on nuclei, with an approximately equal

number of neutrons and protons Such targets are indeed used in most neutrino experiments From such

experiments available information corresponds to cross sections averaged over p and n

dơ_ _ 1[/_dơ dơ

It is clear that the interference between isoscalar and isovector currents cannot contribute to such

averaged cross sections For the total cross sections for the processes v, +N>v, +X and p, +N>

v, + X, we obtain the following expressions:

And 47 FOS 2 Q to 1 =; c%¿ Dn Trip MA on =, KZ a 1 7 ` k2 Œ as r1 AZ i 3 a7 D (

‘he qc antitie S To(\ + Fo (Tai A ( Fo

vector-axial-vector interference and finally isoscalar v= to the cross sections For example,

110 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

In the u, d approximation, the charged quark current reads

where 6c 1s the Cabibbo angle Since sin* 6c = () 04, we shall approximate cos 6¢ in (3.31) by unity The

charged current becomes

[P= =v eg ree

(3 32)

irre =Ja+1%) Therefore, within the approximations we have made, the charged current 1s the

"plus-component” of an isovector

The cross section for the process (3 29) ((3.30)) 1s given by the expression (3 5) ((3 10)) in which the

substitution Wig > WS? (Wg > WS) 1s made, where

2=-Cry > | @|Jz"°(x) J2°®(0)|p)e "“* dx (3 33)

The quantities WS? obey the charge symmetry relations *

(W Cp = =(W%

(3 34)

(tensor (WS), enters the cross section for the process v, +p— +X etc.) Averaged over p and n

cross sections of processes (3.29) and (3.30) contain the quantities

It 1s clear from (3.34) and (3.36) thatT

It follows from (332) and (3.37) that the cross sections for the processes v, +N +X and

v, +N-> pw” +X, averaged over p and n, have the following form:

* Indeed, we have UJ ?U7! =~ J", where U =exp{iaT>} 1s the charge conjugation operator (7; 1s the isospin operator) For the matnx

HƠ,p /H}ÿ|[y¬0 ~ OO, UY] ys0

We note that the available data [63] agree with this relation

SM Buenky and J Hošek, GWS theory of electroweak interactions and the neutral currents 111

The quantities o(V), o(A) and o(I) are the contributions of the vector v}"", axial-vector a)”, and of

the vector and axial-vector interference term to the cross sections, respectively We have

ơ(V)=~2mGïm | L.ạ(k k) ® (plVs*(2) VEO) ey dx dy dz (3.39)

etc

Using the isospin invariance of strong interactions and the transformation properties of the charged

and neutral currents, it 1s easy to see that for quantities entering the expressions (3 27) and (3.38) the

following relations hold

where T, is the isospin operator Using (3.41) we get

For the matrix elements we find therefore

p.n(D| Và ”(2) V2”°(0)Ìp);.a = 2p,n(p| V2(z) VS(0)|p);.a + ;(p|Và(z) V2”°(0)|p)a (3.43)

With the help of (3.28), (3.39) and (3.43) we find for the cross sections averaged over p and n the first

relation (3.40) The other relations can be derived analogously

We find from (3.27) and (3.38) that:

oN©— o © = 2(1 -2 sin? 8y) ơo(I),

(3.44)

ơÿ -—ơÿ- =2ơ()

Then, using also the third equality (3.40), the Paschos—Wolfenstein relation [61]

gon get = Al —2 sin” Ow) (3.45)

easily follows.*

* We have assumed that the parameter p which enters in general the effective weak interaction Hamiltoman of the GWS theory (see (2 148))

equals one If p# 1, then instead of (3 45) we have

(ơẶC — g©)/(œ€C— ơ§©) = 3ø2(1 ~ 2 sin? 8y)

112 SM Bilenky and J Hosek, GWS theory of electroweak interactions and the neutral currents

This relation connects the parameter sin? @w of the GWS theory with experimentally measurable

quantities, namely the total cross sections for deep inelastic processes v,(¥,)+N— v,(v,)+X and

Before starting a more detailed discussion of the experimental data, we get expressions for the cross

sections under consideration in the parton approximation For cross sections averaged over p and n we

find easily [66] with the help of the formula (A.23)

NC

da>s

a dy = PMGi nl(u(x) + d(x) + (A(x) + dex) Œ — y}]

where u(x), d(x), 1s the probability density of finding u,d, | quarks in the proton, respectively,

Oy = (Gz/7)ME Note that in (3 48) we have taken into account only the light u, d and s quarks and

antiquarks and we have assumed that s(x) = 5(x)

The processes v,(v,)+N-v,(%,)+X are studied with neutrino detectors in parallel with the

charged current induced processes v,(¥,.)+N—> mw (u*)+ X With the help of (A 23) we get in the

parton approximation the following cross sections, averaged over p and n for the latter processes

fey = aox{[u(x) + d(x) + 25(x)] + [ñ(x) + d(x)\(1 - y¥} (3 50)

Tay = ơox{[w(x)+ đ(x)](1 — yŸ + [d(x)+ ø(x) + 25(x)]) (351)

From (3.48), (3 50) and (3 51) it follows that the cross sections z„(7„)+ N— z„(ð„)+ X and v,(¥,)+N>

$

dơ3;_ ;¿ dơ¿c ; dơ$c dơ(s)

dxdy 5 ®qrdy ZRLdydy' dxdy' (3 32)