QUARKS, LEPTONS AND WEAK INTERACTIONS

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QUARKS, LEPTONS AND WEAK INTERACTIONS

Y. Achiman

To cite this version:

Y. Achiman. QUARKS, LEPTONS AND WEAK INTERACTIONS. Journal de Physique Colloques, 1971, 32 (C3), pp.C3-113-C3-116. �10.1051/jphyscol:1971317�. �jpa-00214598�

JOURNAL DE PHYSIQUE ColZoque C3, supplkment au no 10, Tome 32, Octobre 1971, page C3-113

QUARKS, LEPTONS AND WEAK INTERACTIONS

Y . ACHIMAN

Laboratoire de Physique ThCorique et Hautes Energies, Orsay - France

Rksumk. - En utilisant un quatrieme quark neutre, on classifie les particules 'a interactions faibles, sans changer le contenu en interactions fortes du modele des quarks. Les leptons sont dans l'octet 3 x 3 x I, qui comprend aussi bien les <(leptons lourds ^)). Les interactions faibles sont representees graphiquement par des diagrammes de quarks d'une maniere analogue aux interactions fortes. Elles sont pratiquement de la forme courant-courant. L'octet des courants hadroniques faibles i( nus ⁾⁾ a exactement la meme structure en quarks que celui des courants leptoniques (<( Uni- versalit6 ¹⁾ g6neralisee). Les regles de selection usuelles des interactions faibles en dkoulent sans hypothese supplementaire. La representation traite de nombres quantiques internes seulement et peut Stre utilisee comme un cadre nature1 pour diffkrentes theories des interactions faibles, en utili- sant diverses hypotheses sur la nature dynamique ⁽⁽ externe ⁾⁾ de la theorie.

Abstract. - Using a fourth neutral quark, we classify weakly interacting particles, without changing the strong interactions content of the quark model. Leptons are in the octet 3 x 3 x 1, which includes ~c heavy leptons ⁾⁾ as well. Weak interactions are represented graphically by quark diagrams in an analogous way to the strong ones. They are practically of current x current nature.

The octet of ^<( bare ⁾⁾ weak hadronic currents has exactly the same quark structure as the leptonic ones (generalized ⁽⁽ Universality ^D). The usual selection rules of weak interactions follow without any additional assumption. ^{AQ = - A S} is possible, but only for meson-lepton interactions, in accord with the only experimental possibility open, also K; -t+ il. The representation deals with internal quantum numbe~s only and may be used as a natural frame-work for different theories of weak interactions, using different assumptions about the ⁽⁽ external ⁾⁾ dynamical nature of the theory.

I n spite of the successful application of algebric methods t o classification of hadrons and strong interactions, leptons remain an isolated family which d o not participate in the ⁽⁽ symmetry game )). We would like to suggest here a classification of weakly interact- ing particles, assuming maximal symmetry between hadrons and leptons, as well as between weak hadronic and leptonic currents. To do this we use cr Universa- lity ⁾⁾ [I] as a guide. This requires the charges of the leptonic currents t o generate the same SU, algebra (SU:) as the cabibbo rotated hadronic currents.

Cabibbo current may be represented using quarks, as

~r~~~~~

+

1 (nA) > ⁼ cos 8 ( n >

+

One can formally relate (nA) t o a rotation in the n, A plane by adding

I (;A) > ⁼ - sin 8 I n >

+

p, (nA) and

(2)

I t is assumed that a particle having q0 as one of its compounds do not interact strongly ^(I). Hence, the strong interactions content of the quark model is not changed. In the following, we are going t o use the internal quantum numbers of the quarks only, hence one may look at quarks as a formal representation of SU, tensor indices. While looking for analogy between hadrons and leptons, one should not worry about the contrast between the big family of hadrons and the small number of observed leptons. It is well known that leptons with a mass heavier than that of kaons cannot participate in usual weak decays. Only c( high energy weak interactions ⁾⁾ experiments may reveal the full famiIy of leptons and << weak resonances >), if they exist.

Due to the existence of one leptonic number L = N(qO) - N ( a , leptons will be classified as was suggested first by Konopinski and Mahmond [2], i. e. using the fact that conjugate neutrinos have diffe- rent chirality. Also, weak interactions mix n, A, and

(1) This may be related to the fact that the probability of creation or destruction of such a state is considerably lower than that of a << nonexotic >> particle, built of p, n, A. In other words, four point functions in which ^{q o} is involved, have by definition at least two exotic channels. Those carrying the quantum num- bers of q o .

8

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971317

C3-114 Y. ACHIMAN

this may be generally written as a rotation (2a, b). we shall see that the charges of the weak currents Weakly interacting particles will be built therefore of generate the S U ~ algebra, so that the above classifica-

H

(n, A) and (n, A). Hence, one may classify them in tion has a certain meaning.

irreductible representations of S U ~ . This, obviously, is Leptons and antileptons are in the octets 3 x 3 ^{x 1}

not a symmetry because it is violantly broken, and one and 3 x 3 x ⁱ⁷respectively- expects big mass differences within multiplets. However, Explicitly in terms of ⁽⁽ quarks ⁾⁾

Leptons in the center of the octet, v and xO, are probably mixed. We use the simplest mixing which allows a graphical representation of weak interac- tions (2).

It is possible to choose a representation for leptons in which electrons and antielectrons are in one octet, while muons and antimuons are in the another. This is done by using coupling of the rotated 3 to

One may show, using the graphical representation of weak interactions, that this representation is practically equivalent to the use of different lepton numbers for electrons and muons. However the second representa- tion is also practically equivalent to the first one.

Higher irreducible representations of S U ~ containing 1 and having nonfractional charges, like 3 x 3 x 3 x 1 or 3 x 3 x 1 x 1, may also be realized as hadron-lepton or lepton-lepton weak resonances. As heavy leptons contain always

(a),

their high mass may indicate that (;A) is relatively heavy. It may happen that q0 is coupled only to the cabibbo combination (nA) and not to the orthogonal one (;A). In this case x0 will be the only existing heavy lepton.

The explicit representation of leptons in terms of quarks, allows us to draw quark diagrams for weak interactions in a way analogous to the quark ⁽⁽ duality diagrams ⁾⁾ of strong interactions [3]. We shall require that a weak process exists if it can be described as a quark diagram. To draw such a diagram one represents external baryons, mesons and leptons (3) by

their quark content. Then, one writes all planer

(( connected ⁾⁾ diagrams using the following rules : a) a quark line cannot change its nature or direction, b) quarks and antiquarks run in opposite directions,

c) (( hadronic ^)> exotic channels are not allowed, d) a double quark line (i. e. (nA) or (2)) must start or end its way in a lepton. If the other end runs into a hadron, it represents a combination of hadrons.

Let us limit ourselves in the following to weak processes containing leptons (4). As a first step we discuss four point functions related to first order in weak interactions. In this case weak quark diagrams cannot be confused with strong duality diagrams because of the q0 line.

( 2 ) Note, however, that using different angles, one may - L e ~ t o n - z e ~ t o n interactions. - is described

face different experimental predictions. as follows

(3) For L. H. (R. H.) neutrinos, the (d) or

(z)

positive (negative) direction in time should be near q O . This is (4) The description of non leptonic weak interactions is more related to the opposite helicities, and will be discussed elsewhere. complicate and will be discussed elsewhere.

QUARKS, LEPTONS A N D WEAK INTERACTIONS C3-115

This diagram describes also processes like :

x0 ^{+ p+ e - v,+.} Other possible processes of usual leptons are described in a similar way, for example

These diagrams represent ^(< neutral currents D.

The existence of a

(z)

Note, that charged heavy leptons cannot decay in a way analogous to p decay and a heavy neutral lepton is always present. Another result is that due to the fact that q0 must run through two leptons they can couple only in currents.

- Baryon-lepton interactions. - To have a baryon- lepton interaction, according to our rules, one must replace one of the leptonic ^<( currents ⁾⁾ by a suitable baryonic current. /3 decay, for example, is described as follows :

Similar diagrams can be drawn for all baryon-lepton weak interactions.

Heavy leptons may couple to a hadronic current

orthogonal to the cabibbo one, and this is possible for heavy leptons only, for example :

In general, one obtains for semileptonic interactions all the usual AS, A I selection rules, because only the following transformation in the hadron currents are possible : p e, n, p ^o A, n ^o A. Also the AQ = AS rule is automatically obtained for baryon-lepton inter- actions. The leptonic currents appearing in the semi- leptonic interactions are exactly those taking part in the pure leptonic processes. Cabibbo theory [l] is obtained as a special case, when we limit ourselves to : usual leptons, V-A currents and C . V. C. ((Bare)) hadronic currents have exactly the same quark repre- sentation as the leptonic ones. For example, the usual charged current has the form

p0,(n1) where Oi suitable dirac tensors .

It has matrix elements between leptons as well as between hadrons. The interaction between two such currents may be described as a skeleton ^)> diagram, which can be cr dressed ⁾⁾ in different ways.

In general, one has octets of currents J , =

Goi

where

The charges of those currents generate the SU: aIgebra (for suitable 0,). In a situation where heavy leptons are allowed, one is led in a trivial way to a ^(< generalized Universality D, since leptonic and hadronic currents have exactly the same structure.

- Meson-lepton interactions. - One class of meson lepton interactions behave exactly as in the baryon- lepton case, for example :

C3-116 Y . ACHIMAN However, in this case, it is possible to have a new

class of diagrams, which allow A Q = - A S (as well as A1 =

s).

\ '."-.+".

(AQ = AS)

These diagrams are not of a ((current x current ^>) nature and we do not expect them to have the same coupling constant as that of the other class. It is interest- ing that experimentally the only place where

is possible is also in meson decays. The AQ = - A S amplitude (if it exists) will give the coupling constant related to the new class of diagrams. This class may be also used to introduce CP violation.

Neutral leptonic currents can only couple to hadronic currents transforming like :

pp, (a)

(z)

Hence, the 1 A S I ⁼ 1 ones must transform like : - nA + ^nx -- ⁽⁽ K: D. In our representation, therefore,

KO, cannot decay into 11 independent of any assumption about the dynamical or kinematical nature of the theory. If we assume that the leptonic currents are V-A ones, K: also cannot decay into 11 [4]. In the A S = 0 case, experimental limits on neutral leptonic currents [5], can be explained by a Clebsch-Gordan coefficient. However, a suitable mixing angle for the physical v and X o , may suppress the contribution of A S = 0 neutral currents by two orders of magnitude, without changing the previous results.

Finally, let us note that the limitation to four point functions can be easily generalized. One may <( sand- wich ^>) hadronic currents between many particle states

or vacuum. Leptonic currents, on the other hand, can be ⁽⁽ sandwiched ⁾⁾ between one particle states only (if we limit ourselves to ⁽⁽ first order ^)> in weak inter- actions). This is due to the role played by qO. Let us give two examples :

To summerize, we have suggested a representation for weakly interacting particles which leads to a great analogy between hadrons and leptons, as well as between weak hadronic and leptonic currents. Weak interactions may be represented graphically in a way which incorporates their observed ^<( internal ⁾⁾ pro- perties. The representation may be used as a frame- work for different theories of weak interactions, while using different assumptions on the external dynamical properties of these interactions ( 5 ) . cr Heavy leptons ))

and weak resonances ^)> are needed for internal consistency ; their internal quantum numbers and possible interactions follow in a natural way. However, one should not expect to explain all the observed experimental facts using only internal quantum numbers .

Acknowledgements. - The author would like to thank Drs. L. Michel, 0. Nachtmann, Y. Ne'Eman, E. de Rafael and B. Sakita for discussions and impor- tant comments.

( 5 ) This will be discussed in detail elsewhere.

References

[I] CABIBBO (N.), Phys. Rev. Letters, 1963, 10, 531.

GELL-MANN (M.). Physics. 1964. 1. 63.

121 KONOPINSKI (J.) and MAHMOND (H.), Phys. Rev., 1953,

92, 1045.

[3] HARARI.(H.), Phys. Rev. Letters, 1969, 22, 563.

ROSNER (J. L.), Phys. Rev. Letters, 1969, 22, 689.

[4] DE RAFAEL (E.), Phys. Rev., 1967, 157, 1486.

[5] PERKINS (D. H.), Proceedings of the Topical Conference on Weak interactions, CERN, Geneve, January,

1969, CERN, 69-7.