NEW RESULTS IN THE QUARK PARTON MODEL

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NEW RESULTS IN THE QUARK PARTON MODEL

O. Nachtmann

To cite this version:

O. Nachtmann. NEW RESULTS IN THE QUARK PARTON MODEL. Journal de Physique Collo- ques, 1971, 32 (C3), pp.C3-99-C3-103. �10.1051/jphyscol:1971314�. �jpa-00214595�

NEW RESULTS IN THE QUARK PARTON MODEL

0. NACHTMANN

Laboratoire de Physique ThCorique et Hautes Energies, Orsay, France (*)

Rhum6. - Le modkle des partons pour la diffusion profondement inelastique leptons-hadrons est analyse en employant les moyens puissants de l'invariance sous SU(3) et de la positivitk. On trouve que le modele prkdit certains domaines de positivitk pour les fonctions de structure d'dlectro- production et neutrinoproduction. Ces domaines sont donnks explicitement. On fait aussi la compa- raison avec les donnees expkrimentales existantes et des predictions pour des experiences neutrino.

Abstract. - The quark parton model for deep inelastic lepton-hadron scattering is analysed using the powerful tools of SU(3)-invariance and positivity. It is found that the model predicts the structure functions for electro- and neutrino production to lie in certain positivity domains, which are given explicitly. A comparison with available data and predictions for neutrino experiments are made.

1 . Introduction. - In this contribution to the

<< Rencontres de Moriond )>I would like to outline some results which hold in all quark-parton models with or without SU(3)-neutral gluons as binding stuff (I).

All cinematics and the definition of electropro- duction structure functions will be taken from ref. [I].

My convention for neutrino structure functions fol- lows from the cross-section formula below (for A S = 0 transitions)

u , E + E'

+ W ~ ( G ) = ~ C O S ~ ( ~ / ~ ) T W3AS=0 -

M sin 2(9/2) ] ^(1.1)

where G is the Fermi constant and 9, the Cabbibo angle. For AS = 1 transitions :

Instead of the W's we shall use the following struc- ture functions :

FY, = 2 M W 1

for electroproduction, and

for neutrino production.

(*) Laboratoire associk au Centre National de la Recherche Scientifique.

Postal address : Laboratoire de Physique Theorique et Hautes Energies, Bdtiment 21 1, Universitk Paris Sud, 91, Orsay, France.

(1) A detailed account of this work will be published elsewhere.

The quark-parton model as formulated by Bjorken and Paschos ( ~ e f . [2]) predicts the structure functions Fi, Si to be functions of the one variable x only.

Moreover the Callan-Gross relation is satisfied if only quarks have electromagnetic and weak interac- tions.

F2(x) = xF,(x) S,(x) = xS,(x) . (1.5) In all that follows the validity of these relations will be assumed. The expression one gets for F: f. i. is : (eq. (2.151, ref. [I 21)

N

where P, is the probability for the N-parton configu- ration, Q i and f,'(x) the charge and density distribu- tion of the i-th parton. Similar expressions hold for the neutrino structure functions [3,4] assuming a V - A current for quarks.

The main steps of our investigation will be : a) Write expression (1.6) and the analogous ones for the other structure functions as expectation values of certain operators.

b) Identify the SU(3) transformation properties of these operators and apply the Wigner-Eckart theorem. This leads in a systematic way to all the relations and sumrules which have been found earlier 151.

c) Define two SU(3)-invariant matrices, and exploite their positive semi-definiteness. This defines a posi- tivity domain for the structure functions.

d ) Apply these results to the available expetimen- tal information.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971314

C3-100 0. NACHTMANN It should be stressed that all our results are inde- ^N

V% v,

pendent of the density functions f i ( x ) appearing in 6(x -xi) (2.9)

N i = l

eq. (1.6). The physical property which is exploited

is the incoherence of the scattering on partons. ^N

- ) a(x-xi) (2.10)

N i = l VF ^{v i i}

2. Operator formalism, application of Wigner-Eckart

theorem. Sumrules. - consider the nucleon made where T , = $(A1 + in,), V , = MI, + ^{iI,) and ;1}

out of a one-dimensional parton gas. The nucleon- the usual Gell-Mann matrices.

wave function will be of the type The structure functions are always the expectation values of the corresponding operators in the one-

I ^N > ⁼ C U N ( X I , ..-, X N ) xir, ..., iN (2.1) nucleon state.

A complete basis for all operators of the type where uN(x,, . . ., xN) is the longitudinal momentum

(2.2), (2.7) to (2.10) is given by the following set of space wave function in the N-particle subspace. The

operators.

probability distribution P, f,'(x) isjust ] u,(x,, ..., xN) 1'

with all coordinates, except the i-th, integrated over

xi,, ^{..., i N ,} the SU(3) part of the wave function is a = ( ) S ( x - x i ) (2.11)

N i = l 1 ⁱ

combination of quarks and antiquarks. It can be thought of as an element in the N-th tensor product

of a six-dimensional space containing as basis vectors E(x) = z _{N i = l} 2 (+ ^- _{3 i} ^{5(x - xi) (2 .12)}

quarks and antiquarks. Let us define the following

operator (') : -

m N

~ a t x ) = c N i = l 2

p2

a:(,) ⁼ N = O i = l z ^(Q2 ') i ^{a(. - x.) (2.2)}

a = c ^{- ,I2) - xi)}

where the 6 x 6 matrix ^{N i = l}

a ⁼ 1, ..., 8 . (2.13) (2.3)

The operators for Baryon number and the SU(3) acts in the i-th component of the tensor product. generators arejust

Q and are the usual charge matrices for quarks

and antiquarks B =

I

-

Q=[

+ J ;

(which of course depend on x).

F ; ~ ( X ) = < N 1 @:(x) 1 N > . (2.6) Let 1 a > ^, I b > be states of the nucleon octet, then : Similarly one introduces for the neutrino and anti- < ^a 1 fi(x) 1 b > ⁼ n(x)

neutrino structure functions the following operators :

< a I E(x) I b > = dab b(x)

N

N i = 1 b(x-xi> (2.7) < a I GC(x) I b > ⁼ ifacbql(x) + dacbq2(~)

< a i R , ( x ) l b > = i f , c , r l ( x ) + d , , , r 2 ( x ) . - (2.16) b(x-xi) (2.8)

N i = l

All structure functions F : P ( ~ ) etc. are linear combi-

(2) The whole thing could also he formulated in a manner nations of the six independent functions n(x), b(x), analogous to a second quantized nonrelativistic Schriidinger q ~ ( ~ ) , r ~ ( x ) , r7.(x), with coefficients given by

theory. simple Clebsch-Gordan algebra :

2 1 1 else. We claim that is a positive semidefinite FiP(x) = - n(x) + 3 r l ( x ) + g r,(x)

9 24 x 24 matrix, i. e.

2 M!: C; Cgb 2 0

9 r2(x) (3 -2)

for arbitrary C,,. To see this, consider one term of

1 2 1 eq. (3.1) :

5 ( ~ ? ' ( x ) + FT(x)) = -- 3 n(x) + ^{r1(x) -} r2(x)

O I,-, 1 b > ⁼

FP(x) - F';(x) = - 2 ql(x) - 2 q2(x)

The six functions on the left hand side of eq. (2.17) constitute an independent set of structure functions.

All the others are therefore expressible in terms of them. As a particular case, one finds f. i.

and all other relations given earlier [5].

Relations (2.14) and (2.15) impose conditions on the integrals of three reduced functions, namely :

This leads to the well-known sumrules :

where t runs over a complete set of states in the (N - 1)-th tensor product space, and insert eq. (3.3) in eq. (3.2).

Our matrix M is invariant under SU(3) in a space carrying the representation 3 x 8 = + 6 + 3.

Therefore M has three different eigenvalues with multiplicities 15, 6, 3, and the condition of eq. (3.2) requires these eigenvalues to be bigger or equal to zero. To extract these eigenvalues which are linear combinations of our six reduced quantities introduced in sec. 2, requires some gymnastics with SU(3) indices and is conveniently done by constructing projection operators on the different subspaces. We get three more conditions by inter-changing the role of quarks and antiquarks. So we end up with six conditions for the six quantities n, b, q,, q,, r , , r,, of the general form

where li i = 1 , ..., 6 are six linearly independent linear functionals. Eq. (3.4) defines a convex cone K in six dimensions. It is convenient to introduce the reciprocal basis ti ^{i =} 1, ..., 6

Eq. (2.20) is Adler's sumrule [6] which coincides with l i ( t j ) = c j d i j c j > 0 . (3.6) Bjorken's backward sumrule 171 in our model. Eq. he ti span the cone K

(2.21) has been derived by Gross and Llewellyn-

Smith [3]. K = ( t , t = x c j C j c j 2 0 ) . (3.7)

Here I shall only give the vectors which span K i n the 3. Positivity restrictions. - Let us define a matrix

(( physical basis ^)> given by eq. (3.8) below M by

w h e r e a , p = 1, 2, 3, a, b = 1 ,..., 8 a n d I e , > < e P I is the 3 x 3 matrix which has 1 at the intersection of

the a-th line and the P-th column and zeros everywhere

C3-102 0. NACHTMANN

- 3 1 1 0 - 2

0 0 1 6 0 2 1 2 4/

K = { q , q = x c j q j cj > O ) . (3.10) To get the positivity domain for any subset of struc- ture functions we just have to project the cone K on this subspace.

4. Applications. - A) RATIO FYlF:'. - Looking at the first two lines of the matrix eq. (3.9) we get immediately bounds for the ratio F:"/F:' : (valid for all x )

1 4 < --. FY/FiP < 3 . (4.1) This is not very stringent, and experiments are well within these limits 181.

I

1, =

5

0

~h~ positivity domain for I,, l2 is shown in figure 1 Also shown are the line corresponding to Bjorken's inequality 191, and present lower limits for 11 > 0,58, I2 > ^0,45 ( ~ e f . 181).

where use has been made of the relations (2.19).

Also shown is the line corresponding to Bjorken's

inequality 19). Ref. 181 gives the following experimen- In the high energy limit (neglecting AS = 1 tran-

tal values : sitions)

Figure 1 shows that the parton model requires a Ref. [8] gives the following information sustantial contribution from the small x region. Appli- 1

cation of the parton model in the small x region, dx xF;'(x) = 0.14 where coherent phenomena should come into play ^{1 / 1 2}

is however dubious. Also, experimentally there is

little evidence for scaling in this region [lo].

1'

1 / 1 2 (4.7)

C) TOTAL NEUTRINO AND ANTINEUTRINO CROSS

SECTIONS. - Set Assuming XF;' = xF? = 0.35 for 0 < x ,< 1/12, we find

Yl = 0.17 Y2 = 0.13. (4.8) From ref. [ l 11 we get

Z = 0.51 f 0.12. (4.9)

1

Z = z 1 dx xFP + FY - $

1:

3 o 2 2 is shown in figure 2 and figure 3. It is seen that pre-

XF? + FY sently available information gives already quite

+

5 1:

2 2 . stringent limits on Z ^:

(4.4) 0.13 < 2 < 0.31 (4.10)

FIG. 2. - Positivity domain for Y2/Yl,Z/Y1, Z/YI as defined in the text. The figure is to be understood in the sense of descriptive geometry. P I , P2, Pq, P5, P,j define a polyhedron as positivity domain. The intersection of this polyhedron with the plane a) Y2/Y1 = 0,77 (experimental value) gives the polygon drawn in full. Its projection on the Z/Yr, Z / Y ~ plane defines the positi- vity domain for these variables, given Y ~ / Y I . The experimental value, 6) for Z/Y I, with error, gives the limits, c) on Z / Y ~ resp. z

as given in the text.

where however no error for the electroproduction data has been taken into account. It is concluded that better values for the neutrino, together with a determination of antineutrino cross section would constitute a rather stringent test for the parton model.

5. Conclusion. - We have found that the parton

FIG. 3. - The same as figure 2, but in perspective view.

model has quite a predictive power, coming essen- tially from the assumption that contributions of partons add up incoherently. We found rather strin- gent limits on the total antineutrino cross section

$(uP + 2"). Our method also predicts positivity domains for the structure functions F;'(x) + ^Fy(x)

and F;'(x) + F;"(x) using the electroproduction data at the same x as input. So Gargamelle experiments will be very valuable in testing the parton model.

AcknowIedgements. - The author would like to thank Prof. L. Michel and Dr. E. de Rafael for useful discussions on positivity, and Prof. Ph. Meyer and Dr. Cl. Bouchiat for discussions and reading of the manuscript.

References

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at High Energies. _[6] ADLER (S. L.), Phys. Rev., 1966,143,1144.

[2] BJORKEN (J. D.) and PASCHOS (E. A.), Phys. Rev., 1969, [7] BJORKEN (J. D.), Phys. Rev., 1967, 163, 1767.

185, 1975. [8] BLOOM (E. D.) et al., SLAC PUB 796,1970.

13] ^{(*' J') and} LLEwELLYN-SMITH (''

[9] BJORKEN (J. D.), Phys. Rev. Letters, 1966, 16, 408.

Physics, 1969, B 14, 337.

[4] BJORKEN (J. D.) and PGSCHOS (E. A.), Phys. Rev., 1970, [lo] NAcHTMANN (O.), lg719 B283 283.

D 1 , 3151. [I 11 BUDAGOV (I.) et a]., Phys. Letters, 1969, 308, 364.