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Chapter 5 Colour and gluons
5.1 Colour and its appearence in particle physics
Hypothesis of colour has been the beginning of creation of the modern theory of strong interaction that is quantum chromodynamics. We discuss in the beginning several experimental facts which haveforced physicists to accept an idea of existence of gluons - quanta of colour eld. 3 1) Problem of statistics for states uuu ddd sss with J P = 2 + As it is known fermion behaviour follows Fermi-Dirac statistics and because of that a total wave function of a system describing half-integer spin should be antisymmetric. But in quark model quarks forming resonances ++ = (uuu), ; = (ddd) and the particle ; = (sss) should be in symmetric S states either in spin or isospin spaces which is prohibited byPauli principle. One can obviously renounce from Fermi-Dirac statistics for quarks, introduce some kind of 'parastatistics' and so on. (All this is very similar to some kind of 'parapsychology' but physicists are mostly very rational people.) So it is reasonable to try to maintain fundamental views and principles and save situation by just inventing new 'colour' space external to space-time and to unitary space (which include isotopic one). As one should antisymmetrize qqq and wehave the simplest absolutely antisymmetric tensor of the 3rd rank abc which (as we already know) transforms as singlet representation of the group SU (3) the reduction abcqaqbqc would be a scalar of SU (3) in new quantum number called 'colour' (here a b c =1 2 3 are colour indices 83


annihilation

and have no relation to previous unitary indices in mass formulae, currents and so on!) Thus the fermion statistics is saved and there is no new quantum number (like strangeness or isospin) for ordinary baryons in accord with the experimental data. But quarks become coloured and number of them is tripled. Let it be as we do not observe them on experiment. 2) Problem of the mean life of 0 meson Wehave already mentioned that a simple model of 0 meson decaybased on Feynman diagram with nucleon loop gives very good agreementwith experiment though it looks strange. Nucleon mass squared enters the denominator in the integral over the loop. Because of that taking now quark model (transfer from mN =0:940 GeV to mu =0:300GeV of the constituent quark) wewould have an extra factor 10! In other words quark model result would give strong divergence with the experimental data. How is it possible to save situation? Triplicate number of quark diagrams by introducing 'colour' !!! Really as 32 = 9 the situation is saved. 3) Problem with the hadronic production cross-section in e+e; Let us consider the ratio of annihilation to the well-known annihilation: R= the hadronic production cross-section of e+e cross section of the muon production in e+e (e+e; ! hadrons) : (e+e; ! + ; ) It is seen from the Feynman diagrams in the lowest order in
; ;

e+ e;

q q

e e

+

+

;

;

that the corresponding processes upon neglecting 'hadronization' of quarks are described by similar diagrams. The di erence lies in di erentcharges of electrons(positrons) and quarks. In a simple quark model with the pointlike quarks the ratio R is given just by the sum of quark charges squared that is for the energy interval of the electron-positron rings up to 2-3 GeV it should 84


1 be R =( 2 )2 +(; 1 )2 +(; 3 )2)= 2 : However experimentgives in this interval 3 3 3 the value around 2.0. As one can see, many things could be hidden into the not so understandable 'hadronization' process (we observe nally not quarks but hadrons!). But the simplest way to do has been again triplication of the 2 number of quarks, then one obtains the needed value: 3 3 =2: With the 'discouvery' of the charmed quark we should recalculate the value of R for energies higher then thresholds of pair productions of 2charm particles 2 that is for energies 3 GeV, R =3 ( 3 )2 +(; 1 )2 +(; 1 )+ 2 )2)] = 10 . 3 3 3 3 Production of the pair of (bb) quarks should increase the value R by 1/3 which proves to hard ly note experimental ly. Experiment gives above 4 GeV and p to e+e; energies around 35-40 GeV the value '4. At higher energies in uence on R of the Z boson contribution (see diagram ) is already seen

e e

+

Z
;

0

ql ql

Thus introduction of three colours can help to escape several importantand even fundamental contradictions in particle physics. But dynamic theory appears only there arrives quant of the eld (gluon) transferring colour from one quark to another and this quant in some way acts on experimental detectors. Otherwise everything could be nished at the level of more or less good classi cation as it succeeded with isospin and hyper charge with no corresponding quanta) There is assumption that dynamical theories are closely related to local gauge invariance of Lagrangian describing elds and its interactions with respect to well de ned gauge groups.

5.1.1 Gluon as a gauge eld

Similar to cases considered above with photon and meson let us write a Lagrangian for free elds of 3-couloured quarks qa where quark qa a =1 2 3 85


is a 3-spinor of the group SU (3)C in colour space L0 = qa(x)@ qa(x) ; mq qa(x)qa(x): This Lagrangian is invariant under global gauge transformation q0a(x)= ei( ) qb(x) , where k k =1 :::8 are known to us Gell-Mann matrices but now in colour space. Let us require invariance of the Lagrangian under similar but local gauge transformation when k are functions of x: q0a(x)= ei( (x) ) qb(x) or q0(x)= U (x)q(x) U (x)= ei (x) : But exactly as in previous case L0 is not invariant under this local gauge transformation L00 = qa(x)@ qa(x)+ qa(x)(U (x) @ U (x))a(x)qb(x)+ b +mqqa(x)qa(x): In order to cancel terms breaking gauge invariance let us introduce massless vector elds Gk k =1 :::8 with the gauge transformation k G0k = U k Gk U y ; 1 @U U y : gs @x Let us de ne interaction of these elds with quarks by Lagrangian L = gsqa(x)(Gk k )a qb(x) b to which corresponds Feynman graphs
k ka b k ka b k k

G q
(a)

ba

q
86

(b)


0 G +1=p3G 3 1 p2 B G1 ; iG2 G= @ G4 ; iG5

8

1 G1 + iG2 p3G8 G46 + iG57 C = ;G3 +1= G + iG A G6 ; iG7 ;2=p3G8 1 0 D1 G21 G31 B G12 D2 G23 C =@ A 13 23 G G D3
22

(5.1)

where

)+ 1 (G11 + G22 ; 2G33) 6 1 11 22 )+ 6 (G11 + G22 ; 2G33) 2 11 + G22 ; 2G33): Final expression for the Lagrangian invariant under local gauge transformations of the non-abelian group SU (3)C in colour space is:
1 11

D = 1 (G ; G 2 D = ; 1 (G ; G 2 ; 2 (G 6
L
SU
(3)C

qa(x)+ mqqa(x)qa(x)+ ~ gs qa(x)(Gk k )a qb(x) ; 1 F k F k b 4 where F k k = 1 2 :::8 is the tensor of the free gluon eld transforming under local gauge transformation as
= qa(x)@
k

F k0 = U y(x) l(x)F l U (x):

It is covariant under some detail tensor of 2i ijk k i j k =1 ~ F or

~ ~ gauge transformations U y F 0 U = F . Let us write in ~ ~ the free gluon eld ~ F = k F k F , ( i j ]= 2 :::8):
k

=(@ G

k

;

@ Gk ) ; 2gs if kij Gi G
~ ~~ @ G ) ; gs G G ] 87

j

~ ~ F =(@ G

;


and prove that this expression in a covariant way transforms under gauge transformation of the eld G: ~ ~ U y(@ G0 ; @ G0 )U = ~ ~ ~ ~ =(@ G ; @ G )+ U y@ U G ] ; U y@ U G ] ~~ ~~ ~ ~ U y G0 G0 ]U = G G ]+ g1 U y@ U G ] ; g1 U y@ U G ]: s s Finally ~ ~ ~ ~~ U yF 0 U = U y(@ G0 ; @ G0 ; gs G0 G0 ])U = ~ ~ ~~ ~ = @ G ; @ G + gs G G ]= F : The particular property of non-Abelian vector eld as wehave already seen on the example of the eld is the fact that this eld is autointeracting that ~ is in the Lagrangian in the free term (;1=4)jF j2 there are not only terms bilinear in the eld G as it is in the case of the (Abelian) electromagnetic eld but also 3-and 4- linear terms in gluon eld G of the form G G @ G and G2 G2 to which the following Feynman diagrams correspond: Gi Gj Gi Gk

G

k

Gi

Gk

This circumstance turns to be decisive for construction of the non-Abelian theory of strong interaction - quantum chromodynamics. The base of it is the asymptotic freedom which can be understood from the behaviour of the e ective strong coupling constant of quarks and gluons 2 s = gs =4 for which
s

(Q2)

1+(11NC ; 2nf )ln(Q2= 2)

s

2

where Q2 is momentum transfer squared, 2 is a renormalization point, is a QCD scale parameter, NC being number of colors and nf number of avours. With Q2 going to in nity coupling constant s tends to zero! Just 88


this property is called asymptotic freedom.(Instead in QED (quantum electrodynamics) with no colors it grows and even have a pole.) But one should also have in mind that in the di erence from QED where we have two observable quantities electron mass and its charge (or those of -and -leptons ) in QCD wehave none. Indeed we could not measure directly either quark mass or its coupling to gluon. Here we shall not discuss problems of the QCD and shall giveonly some examples of application of the notion of colour to observable processes. By introducing colour wehave obtained possibility to predict ratios of many modes of decays and to prove once more validity of the hypothesis on existence of colour. Let us consider decay modes of lepton discouvered practically after J= which has the mass 1800 MeV (exp.(1777,1 +0,4-0,5) MeV). Taking quark model and assuming pointlike quarks (that is fundamental at the level of leptons) we obtain that ; lepton decays emitting neutrino either to lepton (twochannels, e; e or ; ) or to quarks (charm quark is too heavy, strange quark contribution is suppressed by Cabibbo angle and we are left with u and d quarks).
; ; e; ;

5.1.2 Simple examples with coloured quarks

d W

C

W

e

u

From our reasoning it follows that in absence of color wehavetwolepton modes and only one quark mode and partial hadron width Bh should be equal to 1/3 of the total width while with colour quarks wehavetwolepton modes and three quark modes leading to Bh 3/5. Or in other words de nite lepton mode would be 33 % in absence of colour and 20 % with the colour. Experiment gives ; ! ; + + = (17 37 0 09)% and ; ! e; + e + = (17 81 0 07)% supporting hypothesis of 3 colours. 89


The W decays already in three lepton pairs and two quark ones

W

;

e

W

;

dC s uc

C

e

;;;

This means that in absence of colour hadron branching ratio Bh would be 40% while with three colours around 66%. Experiment gives Bh (67 8 1 5)% once more supporting hypothesis of 3-coloured quarks. Z boson decays already along 6 lepton and 5 quark modes,

Z

0

e

+

+

+

Z

0

e

e;

;;

e

90


Z

0

udscb udscb

Thus it is possible to predict at once that in absence of colour hadron channel should be 5/11'45% of the total width of Z while with three colours number of partial lepton and colour quark channels increase up to 6+3 5=21, and hadron channel would be 15/21'71% . Experimentally it is (69:89 0:07%).

91


Chapter 6 Conclusion
In these chosen chapters on group theory and its application to the particle physics there have been considered problems of classi cation of the particles along irreducible representations of the unitary groups, have been studied in some detail quark model. In detail mass formulae for elementary particles have been analyzed. Examples of calculations of the magnetic moments and axial-vector weak constants have been exposed in unitary symmetry and quark model. Formulae for electromagnetic and weak currents are given for both models and problem of neutral currents is given in some detail. Electroweak current of the Glashow-Salam-Weinberg model has been constructed. The notion of colour has been introduced and simple examples with it are given. Introduction of vector bosons as gauge elds are explained. Author has tried to write lectures in such a way as to give possibility to eventual reader to evaluate by him- or herself many properties of the elementary particles.

92