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Structure functions of nucleons inside nuclei
Volume 128B, n u m b e r 3,4
PHYSICS LETTERS
25 August 1983
STRUCTURE FUNCTIONS OF NUCLEONS INSIDE NUCLEI J. SZWED Institute of Computer Science, Jagellonian University, Cracow, Poland Received 17 February 1983 Revised manuscript received 19 May 1983
The structure functions of nucleons inside large nuclei is discussed. The presence o f A isobars in nuclei, which simulate nuclear interactions, is able to explain the recently measured difference between the structure o f b o u n d and free nucleons.
The recently measured [1] ratio of the nucleon structure function in iron and deuterium has brought an unexpected result. It turns out that the nucleon bound in a large nucleus consists of slower constituents than a nearly free nucleon in deuterium (see figs. 1 and 2). This effect cannot be explained, neither by the Fermi motion, nor by the neutron excess in iron [1 ]. The deviation of the ratio of the nucleon structure functions
r(x, 0 2) = FFe(x, Q2)/l~2(x, 0 2)
(I)
010 - F~"°" -F~
0.05
0
0;5 ~- . , ( , _ ~
I£
Fig. 2. The difference between the nucleon structure functions in iron and deuterium. The curves correspond to those o f fig. 1.
from unity and their difference * A(x, 0 2) =
F~e(x, 02) - F~(x, 0 2)
(2)
from zero seems to avoid a convincing explanation. • t The data extracted from refs. [ 1 - 3 ] .
Fl~O~ r - ~-~--~
1.2 1.1 1.C O.g 0.8
0.~
Fig. 1. The ratio o f the nucleon structure functions in iron and deuterium at Q2 = 50 GeV 2 as a function o f x [11. The solid line represents the model prediction with 15% o f the A isobars in iron. The dashed line assumes 9% o f the A isobars and 3.3% decrease o f the gluon m o m e n t u m .
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
In this letter I propose a simple mechanism which can be responsible for the observed effects. It consists of taking into account the nuclear interactions which at nuclear distances are effectively described by the pion exchange. This pionic component of the nucleus has necessarily to be taken into account in a deep inelastic process. In the following we include it by assuming the pions to resonate with the nucleons in the A isobars. The contamination of nuclei by the A (1230 MeV), isopin - 3/2 resonances was proposed a long time ago ,2. It is not large in deuterium [5] where the nucleons are loosely bound and can be treated as approximately free, It may be however considerable in large nuclei. Its calculation or measurement is at present a difficult task [5]. One constructs the A structure functions in analogy , 2 For references see ref. [4].
245
Volume 128B, number 3,4
PHYSICS LETTERS
to the nucleon ones. This can be done by writing the proton and neutron structure functions.
FP(x,
0 2) = {ho(x, Q2) + {A 1(x, 0 2) + S(x, Q2),
FN(x,
Q2) = {Ao(x, 0 2) +_39A1(x, 0 2)
+S(x, 02),
(3)
where AI, I = 0, 1, denotes the contribution in which the valence quark is struck by the photon and the remaining (spectator) valence quarks are in spin and isospin I configuration. The Al(x , Q2) are normalized by
[6] 1
A1
3
f ~x ~---~,
25 August 1983
A 1 falls down faster at large x. This fact is crucial to produce the required effect. At large x, where the valence contribution dominates, any admixture of A isobars in the nucleus shifts the structure function towards lower x values. Thus a "bound nucleon" in a large nucleus has a softer quark distribution at large x than a (nearly) free one in deuterium. One can also expect an increase of the structure function at low x when compared with that of a free nucleon. I illustrate my qualitative argument with a quantitative example defined as follows (1 omit the variable Q2 which is fixed at 50 GeV 2, at which value the data are available):
0
xuV(x) + xdV(x) = [3/B(0.35, 4)] x0.35(1 - x) 3 ,
and can be expressed by the valence quark distributions in the proton
xdV(x) [1/B(0.65, 6)1x0.65(1 - x) 5 .
Ao(x ' Q2) = xuV(x, Q2)
The above relation can be translated for the functions
_ ~xdV(x, Q2),
A 1(x, Q2) = _~x dr(x, 0 2 ) .
(4)
By S(x, Q2) I denote the SU(3) symmetric sea quark contribution. Following this distinction one can write the ~x structure function, remembering that any two valence quarks are there in spin and isospin 1 state
F~(x,
0 2 ) = ~Al(X, 0 2 )
+S'(x,
02),
F~*(x, Q2) = ]A 1(x, Q2) + S'(x,
Q2),
F z~O 2 (x, Q2)=~AI(x, Q2)+S F~-(x,
(x, QZ),
Q2) = ~A 1(x, Q2) + S'(x, Q2),
(5)
The assumption made in eq. (5) which states that the functions Al(X ) depend only on spin and isospin of the spectator quarks, seems to be a natural extension of the Carlitz-Kaur model [6], it also holds in the perturbative approach to structure funcions at x ~ 1 [7]. The functions A 0 and A 1 are separately tested in experiment by measuring e.g. the ratio of the proton to neutron structure functions, the ratio of down to up quarks in the proton, etc. [8]. From these experiments it is known that the SU(6) symmetric relation A0(x, Q2) =Al(X ' Q2) is broken as x -+ 1. The models which contain SU(6) symmetry breaking [6,9] and account for most of the experimental results assume usually Al(X ) ~ (l - x)~ao(x) at large x, with a = 1 or 2 - The isospin I function 246
(7)
(6)
A 1 by using eqs. (4). The sea contribution to the structure function S(x) and S'(x) are assumed to take the same form
S(x) = c(l -
x) 8 ,
S'(x) = c'(1 -
x) 8 .
(8)
In both cases the normalisation is given by the requirement that the momentum carried by all quarks (valence and sea) is 48%. The A component can appear in many possible ways. One can imagine pairs of protons producing a A++_ A0 system, protons converting to A+ and neutrons to A0 etc. -- the effect under consideration depends little on which possibility I choose. The construction o f r ( x , Q2 = 50 GeV 2) and A(x, Q2 = 50 GeV 2) from expressions (3)-(5) and t7), (8) is straightforward. The only free parameter is the percentage of A isobars in the iron nucleus. The curves (solid lines in fig. 1 and 2) are shown for 15% of nucleons in iron being changed to A isobars (A+÷ and A0). One sees that our single assumption describes the data for x >/0.2 well. It should be stressed that the decrease in r(x) at large x does not depend strongly on the detailed choice of powers in eq. (7), provided the relation (6) is preserved. One can also try to use the structure functions resulting from the QCD fit to the data [10]. The curves are of equal quality for x > 0.3 and the same percentage of A isobars. The disagreement of our model structure functions with the data at x < 0.2 proves the need for other effects which are responsible for the low x region. This
Volume 128B, number 3,4
PHYSICS LETTERS
may be the decrease o f the m o m e n t u m carried b y the gluons, mentioned by Jaffe [2]. In the above considerations I have kept the fraction o f gluon momentum f'txed - this quantity, measured and calculated from QCD*3 seems to be an universal property o f q u a r k gluon matter. If however one gives back this assumption and accepts a decrease o f a few percent o f the gluon momentum, one obtains very good agreement with the data (dashed line in figs. 1 and 2) with a lowel percentage o f isobars (9% A and 1.1% decrease o f gluon momentum) , 4 . It should be stressed however that the dashed curves go beyond the proposed mechanism. It requires an additional study why the A isobar sea is so much larger than the nucleon one. The presented estimates for the A contamination in iron should be considered as upper limits. I am aware o f other effects thay may also contribute in the same direction leading consequently to the decrease o f the A percentage. One of them, multiquarks bags in nuclei, was proposed by Jaffe [2]. At first sight the effect is obvious - the structure funcion o f a multiquark state falls down very fast with quark momentum because o f phase space. The counting rules [12], for instance, which predict a (1 - x ) 3 behaviour for the nucleon structure function, would give (1 - x ' ) 9 for the six-quark structure functions. These two functions differ however very little because x' = x/2. Therefore only a detailed study o f multiquark states can shed more light on their role. In the present analysis I neglected the influence o f the Fermi motion inside the nucleus. Its estimates [1 ] show that it is not able to explain the measured effect. On the contrary, the ratio r(x, Q2) is expected to increase because o f the Fermi motion, making the effect even stronger. However quantitative evaluation of the Fermi motion involves off shell kinematics which requires assumptions which are by no means obvious. In particular the definition o f the recoil system in deep inelastic scattering off nuclei influences strongly the results [13]. For these reasons it is hard to esti-
,3 For a review and references see ref. [1 I]. $4 The gluon momentum can be transferred to the quarks in many ways. In our example 1.1% of the momentum was added to the A sea which changes the coefficient c' in eq. (8) by 66%. The effect of such a large sea increase was also noticed by Jaffe [2].
25 August 1983
mate the importance o f the Fermi motion in the considered problem. To summarize, I suggest that the admixture o f A isobars inside large nuclei, which is a way o f taking into account nuclear interactions, can be responsible for the observed differences between bound and free nucleon structure functions. If this is the case, the pro. posed explanation builds up a bridge between the physics at high energies and short distances - and low energy nuclear structure. It also means that the leptoproduction experiments off nuclei, e.g. n e u t r i n o production, do not measure the pure nucleon structure function. I would like to thank T. Chmaj for drawing my attention to this problem and B. Kamys and A. Kotaftski for discussions.
References [ I ] EMC Collab., A, Edwards, Communication XXth Intern. Conf. on High energy physics (Paris, 1982), [2] R.L. Jaffe, MIT preprint CTP 1029 (1982). [3] J.J. Aubert et ',d., Phys. Lett. 105B (1982) 322. [4] J. Hiifner. Phys. Rep. C21 (1975) 1; A,M. Green and L.S. Kislinger, in: Mesons in Nuclei, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979). [5] H.J. Wever and H. Arenhovel, Phys. Rep. 36C (1978) 278. [6] R. Catlitz, Phys. Lett. 58B (1975) 345; J. Kaur, Nucl. Phys. B128 (1977) 219. [7] G. Farrar and D.R. Jackson, Phys. Rev. Lett. 35 (1975) 1416. [8] A. Bodek et al., Phys. Rev. Lett. 30 (1973) 1087; J.S. Poucher et al., Phys. Rev. Lett. 32 (1974) 118; P. Allen et al., Phys. Lett. 103B (1981) 71; BEBC CoUab., P. Schmid, talk XXlth Intern. Conf. on High energy Ed. physics (Paris, 1982); CDHS Collab., .1.P. Merlo, talk XXth Intern. Conf. on High energy Ed physics (Paris, 1982). EMC CoUab., A. Edwards, talk XXth Intern. Conf. on High energy physics (Paris, 1982). [9] F.E. Close, Phys. Lett 43B (1973) 422; J. Szwed, Acta Phys. Polon. BI4 (1983) 55. [10] J.F. Owens and E. Reya, Phys. Rev. DI7 (1978) 3003. [11] A. Burns, Rev. Mod. Phys. 52 (1980) 199. [12] S.J. Brodsky and G. Farrar, Phys. Rev. Lett. 31 (1973) 1153; A.V. Matveev, R.M. Muradyan and A.V. Tavkhelidze, Lett. Nuovo Cimento 7 (1973) 19. [13] A. Bodek and J.L. Ritchie, Phys. Rev. D23 (1981) 1070.
247
PHYSICS LETTERS
25 August 1983
STRUCTURE FUNCTIONS OF NUCLEONS INSIDE NUCLEI J. SZWED Institute of Computer Science, Jagellonian University, Cracow, Poland Received 17 February 1983 Revised manuscript received 19 May 1983
The structure functions of nucleons inside large nuclei is discussed. The presence o f A isobars in nuclei, which simulate nuclear interactions, is able to explain the recently measured difference between the structure o f b o u n d and free nucleons.
The recently measured [1] ratio of the nucleon structure function in iron and deuterium has brought an unexpected result. It turns out that the nucleon bound in a large nucleus consists of slower constituents than a nearly free nucleon in deuterium (see figs. 1 and 2). This effect cannot be explained, neither by the Fermi motion, nor by the neutron excess in iron [1 ]. The deviation of the ratio of the nucleon structure functions
r(x, 0 2) = FFe(x, Q2)/l~2(x, 0 2)
(I)
010 - F~"°" -F~
0.05
0
0;5 ~- . , ( , _ ~
I£
Fig. 2. The difference between the nucleon structure functions in iron and deuterium. The curves correspond to those o f fig. 1.
from unity and their difference * A(x, 0 2) =
F~e(x, 02) - F~(x, 0 2)
(2)
from zero seems to avoid a convincing explanation. • t The data extracted from refs. [ 1 - 3 ] .
Fl~O~ r - ~-~--~
1.2 1.1 1.C O.g 0.8
0.~
Fig. 1. The ratio o f the nucleon structure functions in iron and deuterium at Q2 = 50 GeV 2 as a function o f x [11. The solid line represents the model prediction with 15% o f the A isobars in iron. The dashed line assumes 9% o f the A isobars and 3.3% decrease o f the gluon m o m e n t u m .
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
In this letter I propose a simple mechanism which can be responsible for the observed effects. It consists of taking into account the nuclear interactions which at nuclear distances are effectively described by the pion exchange. This pionic component of the nucleus has necessarily to be taken into account in a deep inelastic process. In the following we include it by assuming the pions to resonate with the nucleons in the A isobars. The contamination of nuclei by the A (1230 MeV), isopin - 3/2 resonances was proposed a long time ago ,2. It is not large in deuterium [5] where the nucleons are loosely bound and can be treated as approximately free, It may be however considerable in large nuclei. Its calculation or measurement is at present a difficult task [5]. One constructs the A structure functions in analogy , 2 For references see ref. [4].
245
Volume 128B, number 3,4
PHYSICS LETTERS
to the nucleon ones. This can be done by writing the proton and neutron structure functions.
FP(x,
0 2) = {ho(x, Q2) + {A 1(x, 0 2) + S(x, Q2),
FN(x,
Q2) = {Ao(x, 0 2) +_39A1(x, 0 2)
+S(x, 02),
(3)
where AI, I = 0, 1, denotes the contribution in which the valence quark is struck by the photon and the remaining (spectator) valence quarks are in spin and isospin I configuration. The Al(x , Q2) are normalized by
[6] 1
A1
3
f ~x ~---~,
25 August 1983
A 1 falls down faster at large x. This fact is crucial to produce the required effect. At large x, where the valence contribution dominates, any admixture of A isobars in the nucleus shifts the structure function towards lower x values. Thus a "bound nucleon" in a large nucleus has a softer quark distribution at large x than a (nearly) free one in deuterium. One can also expect an increase of the structure function at low x when compared with that of a free nucleon. I illustrate my qualitative argument with a quantitative example defined as follows (1 omit the variable Q2 which is fixed at 50 GeV 2, at which value the data are available):
0
xuV(x) + xdV(x) = [3/B(0.35, 4)] x0.35(1 - x) 3 ,
and can be expressed by the valence quark distributions in the proton
xdV(x) [1/B(0.65, 6)1x0.65(1 - x) 5 .
Ao(x ' Q2) = xuV(x, Q2)
The above relation can be translated for the functions
_ ~xdV(x, Q2),
A 1(x, Q2) = _~x dr(x, 0 2 ) .
(4)
By S(x, Q2) I denote the SU(3) symmetric sea quark contribution. Following this distinction one can write the ~x structure function, remembering that any two valence quarks are there in spin and isospin 1 state
F~(x,
0 2 ) = ~Al(X, 0 2 )
+S'(x,
02),
F~*(x, Q2) = ]A 1(x, Q2) + S'(x,
Q2),
F z~O 2 (x, Q2)=~AI(x, Q2)+S F~-(x,
(x, QZ),
Q2) = ~A 1(x, Q2) + S'(x, Q2),
(5)
The assumption made in eq. (5) which states that the functions Al(X ) depend only on spin and isospin of the spectator quarks, seems to be a natural extension of the Carlitz-Kaur model [6], it also holds in the perturbative approach to structure funcions at x ~ 1 [7]. The functions A 0 and A 1 are separately tested in experiment by measuring e.g. the ratio of the proton to neutron structure functions, the ratio of down to up quarks in the proton, etc. [8]. From these experiments it is known that the SU(6) symmetric relation A0(x, Q2) =Al(X ' Q2) is broken as x -+ 1. The models which contain SU(6) symmetry breaking [6,9] and account for most of the experimental results assume usually Al(X ) ~ (l - x)~ao(x) at large x, with a = 1 or 2 - The isospin I function 246
(7)
(6)
A 1 by using eqs. (4). The sea contribution to the structure function S(x) and S'(x) are assumed to take the same form
S(x) = c(l -
x) 8 ,
S'(x) = c'(1 -
x) 8 .
(8)
In both cases the normalisation is given by the requirement that the momentum carried by all quarks (valence and sea) is 48%. The A component can appear in many possible ways. One can imagine pairs of protons producing a A++_ A0 system, protons converting to A+ and neutrons to A0 etc. -- the effect under consideration depends little on which possibility I choose. The construction o f r ( x , Q2 = 50 GeV 2) and A(x, Q2 = 50 GeV 2) from expressions (3)-(5) and t7), (8) is straightforward. The only free parameter is the percentage of A isobars in the iron nucleus. The curves (solid lines in fig. 1 and 2) are shown for 15% of nucleons in iron being changed to A isobars (A+÷ and A0). One sees that our single assumption describes the data for x >/0.2 well. It should be stressed that the decrease in r(x) at large x does not depend strongly on the detailed choice of powers in eq. (7), provided the relation (6) is preserved. One can also try to use the structure functions resulting from the QCD fit to the data [10]. The curves are of equal quality for x > 0.3 and the same percentage of A isobars. The disagreement of our model structure functions with the data at x < 0.2 proves the need for other effects which are responsible for the low x region. This
Volume 128B, number 3,4
PHYSICS LETTERS
may be the decrease o f the m o m e n t u m carried b y the gluons, mentioned by Jaffe [2]. In the above considerations I have kept the fraction o f gluon momentum f'txed - this quantity, measured and calculated from QCD*3 seems to be an universal property o f q u a r k gluon matter. If however one gives back this assumption and accepts a decrease o f a few percent o f the gluon momentum, one obtains very good agreement with the data (dashed line in figs. 1 and 2) with a lowel percentage o f isobars (9% A and 1.1% decrease o f gluon momentum) , 4 . It should be stressed however that the dashed curves go beyond the proposed mechanism. It requires an additional study why the A isobar sea is so much larger than the nucleon one. The presented estimates for the A contamination in iron should be considered as upper limits. I am aware o f other effects thay may also contribute in the same direction leading consequently to the decrease o f the A percentage. One of them, multiquarks bags in nuclei, was proposed by Jaffe [2]. At first sight the effect is obvious - the structure funcion o f a multiquark state falls down very fast with quark momentum because o f phase space. The counting rules [12], for instance, which predict a (1 - x ) 3 behaviour for the nucleon structure function, would give (1 - x ' ) 9 for the six-quark structure functions. These two functions differ however very little because x' = x/2. Therefore only a detailed study o f multiquark states can shed more light on their role. In the present analysis I neglected the influence o f the Fermi motion inside the nucleus. Its estimates [1 ] show that it is not able to explain the measured effect. On the contrary, the ratio r(x, Q2) is expected to increase because o f the Fermi motion, making the effect even stronger. However quantitative evaluation of the Fermi motion involves off shell kinematics which requires assumptions which are by no means obvious. In particular the definition o f the recoil system in deep inelastic scattering off nuclei influences strongly the results [13]. For these reasons it is hard to esti-
,3 For a review and references see ref. [1 I]. $4 The gluon momentum can be transferred to the quarks in many ways. In our example 1.1% of the momentum was added to the A sea which changes the coefficient c' in eq. (8) by 66%. The effect of such a large sea increase was also noticed by Jaffe [2].
25 August 1983
mate the importance o f the Fermi motion in the considered problem. To summarize, I suggest that the admixture o f A isobars inside large nuclei, which is a way o f taking into account nuclear interactions, can be responsible for the observed differences between bound and free nucleon structure functions. If this is the case, the pro. posed explanation builds up a bridge between the physics at high energies and short distances - and low energy nuclear structure. It also means that the leptoproduction experiments off nuclei, e.g. n e u t r i n o production, do not measure the pure nucleon structure function. I would like to thank T. Chmaj for drawing my attention to this problem and B. Kamys and A. Kotaftski for discussions.
References [ I ] EMC Collab., A, Edwards, Communication XXth Intern. Conf. on High energy physics (Paris, 1982), [2] R.L. Jaffe, MIT preprint CTP 1029 (1982). [3] J.J. Aubert et ',d., Phys. Lett. 105B (1982) 322. [4] J. Hiifner. Phys. Rep. C21 (1975) 1; A,M. Green and L.S. Kislinger, in: Mesons in Nuclei, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979). [5] H.J. Wever and H. Arenhovel, Phys. Rep. 36C (1978) 278. [6] R. Catlitz, Phys. Lett. 58B (1975) 345; J. Kaur, Nucl. Phys. B128 (1977) 219. [7] G. Farrar and D.R. Jackson, Phys. Rev. Lett. 35 (1975) 1416. [8] A. Bodek et al., Phys. Rev. Lett. 30 (1973) 1087; J.S. Poucher et al., Phys. Rev. Lett. 32 (1974) 118; P. Allen et al., Phys. Lett. 103B (1981) 71; BEBC CoUab., P. Schmid, talk XXlth Intern. Conf. on High energy Ed. physics (Paris, 1982); CDHS Collab., .1.P. Merlo, talk XXth Intern. Conf. on High energy Ed physics (Paris, 1982). EMC CoUab., A. Edwards, talk XXth Intern. Conf. on High energy physics (Paris, 1982). [9] F.E. Close, Phys. Lett 43B (1973) 422; J. Szwed, Acta Phys. Polon. BI4 (1983) 55. [10] J.F. Owens and E. Reya, Phys. Rev. DI7 (1978) 3003. [11] A. Burns, Rev. Mod. Phys. 52 (1980) 199. [12] S.J. Brodsky and G. Farrar, Phys. Rev. Lett. 31 (1973) 1153; A.V. Matveev, R.M. Muradyan and A.V. Tavkhelidze, Lett. Nuovo Cimento 7 (1973) 19. [13] A. Bodek and J.L. Ritchie, Phys. Rev. D23 (1981) 1070.
247