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Light cone analysis of parton model relations
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
LIGHT CONE ANALYSIS OF PARTON MODEL RELATIONS L. GOMBEROFF and L.P. HORWITZ Dept. of Physics and Astronomy, Tel-A viv University, Ramat A viv, Israel Received 24 December 1972 The validity of relations among deep inelastic scattering structure functions originally derived by parton methods is examined from the point of view of light cone analysis. The presence of Schwinger terms in the equal time limit can be associated with violations of these relations. Fritzsch and Gell-Mann [1] have proposed an asymptotic form for the commutator of current densities in the neighborhood of the light cone. This expansion was abstracted from a free quark model. It constitutes a generalization for any SU(3) × SU(3) current, of the light cone expansion for electromagnetic currents proposed by several authors [2], based on earlier work by Wilson on short distance expansion [3]. It accounts for scaling in deep inelastic electron scattering and establishes a link with parton approaches. It has been pointed out recently [4] that the Fritzsch-Gell-Mann expansion does not, however, lead to the Schwinger term (ST) required for consistency [5] between the Lee-Doshen-Gell-Mann [7] sum rule and that of Cabibbo and Radicati [8]. An additional singularity in the light cone expansion has been proposed [4] which provides the required ST and preserves the equal time current algebra, as well as Bjorken scaling for the structure functions of deep inelastic electron scattering. It does, however, lead to a scaling behavior for W3 (one of the structure functions of deep inelastic neutrino scattering) which is different from that implied by the Ffitzsch-Gell-Mann expansion, namely, W3 rather than vW 3 scales. It is the purpose of this note to study the consequences of this term, and in particular to show that the validity of the Llewllyn Smith relation [9] depends on the existence of the ST under consideration. However, the related sum rule due to Gross and Llewllyn Smith [10], may remain unchanged. Actually, these relations provide an experimental way of checking the presence of higher order singularities both at equal times and on the light cone. The question of the validity of these relations, originally derived by parton meth. ods, is therefore connected, through the light cone expansion, with the presence of Schwinger terms. It has been shown [4] that the following light cone expansion of current commutators leads to the ST discussed in ref. [5]:
[J~(x), jb(0)] -~i(a~ a~ -g~ []) (e(Xo) 8(x2)) ~dabc(Co (x,, 0) + Cc(0,x)) + ifabc (Cc(X, 0) -- Cc(0,x))) + Suvoo a ° (e (Xo) 5 (x2)) (dabe(Fac(x,O) - Fae(O,x)) + ifab c (F°e(x, 0)) + F°c(O,x)) + ieuvo o ap (e (Xo) (5(x2)) {dab e (F5°(x, O) + F5°(O,x)) + ifab c (F5°(x, O) - F5°(O,x))}
+ i~...o a~ (c2
(~)
where C2(x) = (x 2 - iexo)-2 - (x 2 + iexo)-2
(2)
The relevant ST arises from the antisymmetric part of the last term in eq. (1). As shown in ref. [4], the required Schwinger term, which must be antisymmetric in both space and unitary spin indices [5, 11], cannot be derived from the third term ofeq. (1) (the only possible source of such a term in the Gell-Mann Fritzsch expansion). This term leads, in the equal time limit, only to unitary spin symmetric 5-functionk 408
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
contributions to the space-space commutator. The antisymmetric part of the higher order singularity in the last term of eq. (1) leads to contributions of the required form. proportional to B H 50
ei]ko ~
B OH5k a [a,b] --axk 63(x) and eo.ok --Bxm [a,b] ~xm 8 3(x),
where the index [a, b] means antisymmetry in unitary spin indices. Addition of singularities of still higher order to the light cone expansion lead to higher order Schwinger terms and to divergences in the equal time limit. If the equal time limit of the symmetric part of the last term does not vanish [4], it leads to a singularity of order xo2. The zeroth order term in the Taylor expansion of the symmetric part leads to a second order ST in the equal time commutator between two space current components [4]. Considering eq. (1) between proton states of equal momentum, spin averaged, we obtain in the Bjorken region (v = -2Mp .q ~ a, q2 ~ a with the ratio ~ = -q2/2Mp" q fixed) the structure functions for deep inelastic neutrinoproton and electron-proton scattering relevant to our discussion, 2 F~0 (~) 2 0 2 8 F~ p (~,u) = 2F3 (~) - 2 X/~--] - $2 x / ~ F 8S (~) + v {2HA3(~) - 2 V/-~ HS (~) - ~- x / ~ H s (~)),
(3)
and F~p(~)_2 , F/~0 (~) + ~' X/3-F8(~)+ ' F3(~) - ~ x/~--~
(4)
In order to obtain these results, we have used (see, for example ref. [11] )
fd4x exp (iqx) Cd(x) F(p 2,p "x) ~j v d- 2 dp(~)
(5)
Thus, for example,
fd4x exp (iqx) euvoo bo(C2(x)) ~ s
Cp,s[II~(x,O) +H~(O,x)lp, s) ~ euuoo qO pO H~ (~)
(6)
The first three terms of eq. (3) arise from the third term of eq. (1), which is less singular then the last. Since, according to eqs. (3) and (4), F~P - F e2 n -_ z~ FA3(~),
(7)
and F~ p - F ~ = 4 [ F 3 (~) + v H 3 (~)]
(8)
we conclude that the Llewellyn Smith relation, namely (F~p - Fin ) - g' ~(F~p - F~ n)
(9)
follows from eqs. (7) and (8) if and only i f H 3 (~) = 0. However, this is the term which leads to the ST discussed above. Moreover, it follows from eq. (3) that 1
1
f d~(F~ p + F ~ ) = - 2 f d~ {-2 V~(F°(~)+uHO(~))--}x/'3(F~(~)+uHS(~))} (10) -1 -1 I f H ° andH 8 are not present, eq. (10) reduces to the Gross-Llewllyn Smith sum rule.Hs(0, 0) is associated with second order ST's at the tip of the light cone. Therefore, if the light cone expansion provides a good description for deep inelastic processes, then relation (10) constitutes a check on the existence of higher than first order ST's at equal times. 409
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
References [l] H. Fritzsche and M. Gell-Mann, Proc. Intern. Conf. on Duality and symmetry in hadron physics, ed. E. Gotsman (The Weizmann Science Press of Israel). [2] Y. Frishman, Phys. Rev. Lett. 25 (1970) 966. G. Altarelli, R. Brandt and G. Preparata, Phys. Rev. Lett. 26 (1971) 42. [3] K. Wilson, Phys. Rev. 179 (1969) 1499. [4] L. Gomberoff and L.P. Horwitz, Phys. Rev., to be published. [S] F. Buccella, R. Gatto, S. Okubo and G. Veneziano, Phys. Rev. 149 (1966) 1268. [6] B.W. Lee, Phys. Rev. Lett. 14 (1965) 613. [7] R.F. Dashen and M. Gell-Mann, Phys. Lett. 17 (1965) 142. [8) N. Gabibbo and A.L. Radiati, Phys. Lett. 19 (1966) 697. [9] C.H. Llewllyn Smith, Nucl. Phys. B17 (1970) 277. [lo] D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B14 (1969) 337. [ll] L. Gomberoff, Phys. Rev. D4 (1971) 3228.
410
PHYSICS LETTERS
5 March 1973
LIGHT CONE ANALYSIS OF PARTON MODEL RELATIONS L. GOMBEROFF and L.P. HORWITZ Dept. of Physics and Astronomy, Tel-A viv University, Ramat A viv, Israel Received 24 December 1972 The validity of relations among deep inelastic scattering structure functions originally derived by parton methods is examined from the point of view of light cone analysis. The presence of Schwinger terms in the equal time limit can be associated with violations of these relations. Fritzsch and Gell-Mann [1] have proposed an asymptotic form for the commutator of current densities in the neighborhood of the light cone. This expansion was abstracted from a free quark model. It constitutes a generalization for any SU(3) × SU(3) current, of the light cone expansion for electromagnetic currents proposed by several authors [2], based on earlier work by Wilson on short distance expansion [3]. It accounts for scaling in deep inelastic electron scattering and establishes a link with parton approaches. It has been pointed out recently [4] that the Fritzsch-Gell-Mann expansion does not, however, lead to the Schwinger term (ST) required for consistency [5] between the Lee-Doshen-Gell-Mann [7] sum rule and that of Cabibbo and Radicati [8]. An additional singularity in the light cone expansion has been proposed [4] which provides the required ST and preserves the equal time current algebra, as well as Bjorken scaling for the structure functions of deep inelastic electron scattering. It does, however, lead to a scaling behavior for W3 (one of the structure functions of deep inelastic neutrino scattering) which is different from that implied by the Ffitzsch-Gell-Mann expansion, namely, W3 rather than vW 3 scales. It is the purpose of this note to study the consequences of this term, and in particular to show that the validity of the Llewllyn Smith relation [9] depends on the existence of the ST under consideration. However, the related sum rule due to Gross and Llewllyn Smith [10], may remain unchanged. Actually, these relations provide an experimental way of checking the presence of higher order singularities both at equal times and on the light cone. The question of the validity of these relations, originally derived by parton meth. ods, is therefore connected, through the light cone expansion, with the presence of Schwinger terms. It has been shown [4] that the following light cone expansion of current commutators leads to the ST discussed in ref. [5]:
[J~(x), jb(0)] -~i(a~ a~ -g~ []) (e(Xo) 8(x2)) ~dabc(Co (x,, 0) + Cc(0,x)) + ifabc (Cc(X, 0) -- Cc(0,x))) + Suvoo a ° (e (Xo) 5 (x2)) (dabe(Fac(x,O) - Fae(O,x)) + ifab c (F°e(x, 0)) + F°c(O,x)) + ieuvo o ap (e (Xo) (5(x2)) {dab e (F5°(x, O) + F5°(O,x)) + ifab c (F5°(x, O) - F5°(O,x))}
+ i~...o a~ (c2
(~)
where C2(x) = (x 2 - iexo)-2 - (x 2 + iexo)-2
(2)
The relevant ST arises from the antisymmetric part of the last term in eq. (1). As shown in ref. [4], the required Schwinger term, which must be antisymmetric in both space and unitary spin indices [5, 11], cannot be derived from the third term ofeq. (1) (the only possible source of such a term in the Gell-Mann Fritzsch expansion). This term leads, in the equal time limit, only to unitary spin symmetric 5-functionk 408
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
contributions to the space-space commutator. The antisymmetric part of the higher order singularity in the last term of eq. (1) leads to contributions of the required form. proportional to B H 50
ei]ko ~
B OH5k a [a,b] --axk 63(x) and eo.ok --Bxm [a,b] ~xm 8 3(x),
where the index [a, b] means antisymmetry in unitary spin indices. Addition of singularities of still higher order to the light cone expansion lead to higher order Schwinger terms and to divergences in the equal time limit. If the equal time limit of the symmetric part of the last term does not vanish [4], it leads to a singularity of order xo2. The zeroth order term in the Taylor expansion of the symmetric part leads to a second order ST in the equal time commutator between two space current components [4]. Considering eq. (1) between proton states of equal momentum, spin averaged, we obtain in the Bjorken region (v = -2Mp .q ~ a, q2 ~ a with the ratio ~ = -q2/2Mp" q fixed) the structure functions for deep inelastic neutrinoproton and electron-proton scattering relevant to our discussion, 2 F~0 (~) 2 0 2 8 F~ p (~,u) = 2F3 (~) - 2 X/~--] - $2 x / ~ F 8S (~) + v {2HA3(~) - 2 V/-~ HS (~) - ~- x / ~ H s (~)),
(3)
and F~p(~)_2 , F/~0 (~) + ~' X/3-F8(~)+ ' F3(~) - ~ x/~--~
(4)
In order to obtain these results, we have used (see, for example ref. [11] )
fd4x exp (iqx) Cd(x) F(p 2,p "x) ~j v d- 2 dp(~)
(5)
Thus, for example,
fd4x exp (iqx) euvoo bo(C2(x)) ~ s
Cp,s[II~(x,O) +H~(O,x)lp, s) ~ euuoo qO pO H~ (~)
(6)
The first three terms of eq. (3) arise from the third term of eq. (1), which is less singular then the last. Since, according to eqs. (3) and (4), F~P - F e2 n -_ z~ FA3(~),
(7)
and F~ p - F ~ = 4 [ F 3 (~) + v H 3 (~)]
(8)
we conclude that the Llewellyn Smith relation, namely (F~p - Fin ) - g' ~(F~p - F~ n)
(9)
follows from eqs. (7) and (8) if and only i f H 3 (~) = 0. However, this is the term which leads to the ST discussed above. Moreover, it follows from eq. (3) that 1
1
f d~(F~ p + F ~ ) = - 2 f d~ {-2 V~(F°(~)+uHO(~))--}x/'3(F~(~)+uHS(~))} (10) -1 -1 I f H ° andH 8 are not present, eq. (10) reduces to the Gross-Llewllyn Smith sum rule.Hs(0, 0) is associated with second order ST's at the tip of the light cone. Therefore, if the light cone expansion provides a good description for deep inelastic processes, then relation (10) constitutes a check on the existence of higher than first order ST's at equal times. 409
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
References [l] H. Fritzsche and M. Gell-Mann, Proc. Intern. Conf. on Duality and symmetry in hadron physics, ed. E. Gotsman (The Weizmann Science Press of Israel). [2] Y. Frishman, Phys. Rev. Lett. 25 (1970) 966. G. Altarelli, R. Brandt and G. Preparata, Phys. Rev. Lett. 26 (1971) 42. [3] K. Wilson, Phys. Rev. 179 (1969) 1499. [4] L. Gomberoff and L.P. Horwitz, Phys. Rev., to be published. [S] F. Buccella, R. Gatto, S. Okubo and G. Veneziano, Phys. Rev. 149 (1966) 1268. [6] B.W. Lee, Phys. Rev. Lett. 14 (1965) 613. [7] R.F. Dashen and M. Gell-Mann, Phys. Lett. 17 (1965) 142. [8) N. Gabibbo and A.L. Radiati, Phys. Lett. 19 (1966) 697. [9] C.H. Llewllyn Smith, Nucl. Phys. B17 (1970) 277. [lo] D.J. Gross and C.H. Llewellyn Smith, Nucl. Phys. B14 (1969) 337. [ll] L. Gomberoff, Phys. Rev. D4 (1971) 3228.
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