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The structure function of the nucleon
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
THE STRUCTURE FUNCTION OF THE NUCLEON A.I. S I G N A L and A.W. T H O M A S
DepartmentofPhysicsandMathematicalPhysics,The UniversityofAdelaide,Adelaide,SA 5001,Australia Received 20 June 1988
We present a novel method of calculation of the structure function of the "nucleon" for the two-dimensional MIT bag model which overcomes the defects inherent in previous calculations. In particular we naturally obtain the correct support and normalization. We also find that there is an intrinsic quark-antiquark sea associated with the MIT bag.
Ever since the discovery of scaling and identification of partons with quarks there have been attempts to calculate the structure function o f the nucleon for a given model. This endeavour has recently received impetus for two reasons. First the discovery o f the EMC effect [ 1 ] stimulated a great deal o f theoretical work aimed at understanding how, if at all, the quark structure o f the nucleon is altered inside the nucleus. Clearly no unambiguous conclusions can be drawn from any analysis o f the EMC data until the problem of relating the structure functions to quark models is solved. The second stimulus is the variety of models of hadron structure which have been created in the last decade - notably with enormous variation in the typical confinement region for the quarks. It has been argued [ 2 ] that deep inelastic scattering is probably the only possible tool for unambiguously distinguishing between the models. The M I T bag model [ 3 ] was constructed with the SLAC data on deep inelastic scattering (DIS) in mind, and one o f the earliest attempts to calculate structure functions was made for this model by Jaffe [4]. This calculation was of fundamental importance and served to stimulate much further work. Nevertheless, it suffered from two major problems. While physical structure functions must vanish for x outside [0, 1 ] (i.e. they have support only on [0, 1 ] ), the bag model calculation had support on the interval [0, ~ ) . Secondly the theory seemed to imply the presence o f a negative sea, which is again unphysical. Subsequent work [ 5 ] based on the Lo approximation of Krapchev [ 6 ] appeared to solve the problem
o f support, but led to unrealistically oscillatory quark distributions. Even worse, it had no physical basis that is, it was simply the first term in a mathematical expansion which was poorly understood (and still is). No solution has hitherto been offered to the negative sea, and indeed the work of Bell and others [ 7 ] suggested even worse problems. In this paper we suggest a physically sensible solution to these problems. While full details will be provided elsewhere it is possible to explain the essence o f our approach very quickly. In view o f the current interest, it seems worthwhile to us to do this in a letter format, and to present some of the main numerical results for the bag in one space dimension. Although we used a much more circuitous route, it is most convenient to begin with some formal results of Ellis, Furmanski, and Petronzio [ 8 ], and Jaffe [ 9 ]. They write the key structure functions F2 and F 3 measured in v DIS as F2(x)
=x[H(x) - H ( - x ) ] ,
F3(x) =
[H(x)+H(-x)] .
(1)
As explained by Jaffe, for positive x it is sensible to use the expression for H(x):
H(x)=~
i d,-exp(-ixp+,-)
X ( p I ~'*+( ~ - ) q/+ (0) Ip)c,
x>0.
(2)
In eq. (2) C denotes a connected matrix element, p + 481
Volume 211, number 4
PHYSICS LETTERS B
is the plus-component of m o m e n t u m of the nucleon, in the rest frame, ~ - is the light-cone coordinate ~ - = (~o_ ~, )/x/~, and q/+ is given by
p +=M/x/2
~+ = ½ ( l + a ) ~ ,
(3)
8 September 1988
H(x) = -~ ~ Jlap; 4~pO Xi
d~- exp[i(p+-xp+-p+)~ -]
where o~ is the 2 X 2 matrix (~ _o). In order to evaluate H ( - x ) , with x positive, we should use the form
X I ( n , p ~ l q / + ( 0 ) l p ) I2
H(-x)=~n
--
4hE f 4~p dp~On~J(p + -- ( 1-- x )p + ) 1
; d~-exp(ixp+~-)
X[-(Plq/+(O)q/t+(~-)lP)c],
x>0.
(4)
Following the arguments of Jaffe we may define an antiquark distribution/-/(x), for x positive as
tl(x)=-H(-x), so that, for x >
(5)
0,
Fz(X ) =x[H(x) F3(x) = [ H ( x )
x>0,
+ Ii (x) ] , -H(x) ] .
(6)
Changing the variable of integration ~- to - ~ - in eq. (4) and using translational invariance we find
I~(x)=~n i d~-exp(-ixp+~-) --oo
X (Pl q/+ ( ~ - ) q/*+(0) Ip)c,
x>0.
(8)
where/~ is the full m o m e n t u m and energy operator for the strongly interacting system. To exploit this we introduce a complete set of states n, with m o m e n t u m p" and rest mass M~ into eqs. (2) and (4). Because both IP) and In; p ' ) are eigenstates of/~ we find, again for x > 0, 482
(9)
wherep + is (,/.~.[ 2. .,4_. p , ,t 2 ~ 1 / 2 _ 1 - -p , .t A similar expression is obtained f o r / ~ ( x ) with q/t+(0) replacing q/+ (0). As p+ is positive definite both H and H are guaranteed to be zero beyond x = 1. It is only that this stage that one can think of using a model ofhadron structure to evaluate H a n d / ~ . The quark field operators are taken to be those of the MIT bag model. For IP) we use the Peierls-Yoccoz (PY) procedure [ 10 ] to construct a translationally invariant state with zero momentum corresponding to three quarks in the 1si/2 level. The most natural choice for the states n are MIT bags of the same radius as the proton, but with a different number of quarks. Again the PY procedure is used to construct an approximate m o m e n t u m eigenstate of m o m e n t u m p ' :
(7)
We stress that eqs. (3), (6) and (7) can be evaluated in any frame, including the rest frame of the target. The problem of bad support arises if one approximates the quark field operator evaluated at ~- (e.g. qP+ ( ~ - ) ) by a bag field operator at this stage - e.g. with the time dependence exp ( + ie~°), where is the quark eigenenergy. For correct support it is essential that m o m e n t u m and energy conservation be guaranteed exactly. That is, translation invariance in space and time must be retained. For this reason we write q/t+ (~) =exp(+i/~.~)q/*+ (0) exp(-i/~.~) ,
X I ( n , p " I q/+ (0)IP)12 ,
oo
In; p" ) =
1
¢(p,~ f dzexp(ip'~z)ln;z)
(10)
--oo
Here In; z) is a bag with contents labelled n which is centered at z, and I¢(P')12 is proportional to the Fourier transform of the Hill-Wheeler overlap function [ 11 ] for bag states. The key question is which state In ) must be considered. In previous work (using either bags or other models) the only state considered was that most obviously involved when q/(0) acts on a three-quark ground state, namely two quarks in the lowest energy state (we shall call this n = 2 ) . The resulting form H(x) is shown in fig. 1 for a cavity "radius" of 1 fro. The area under the curve [i.e f~ H(x) dx] is 91%. Increasing the cavity radius by 10% gives a sharper distribution function, with slightly more area under the curve (92%). This agrees quantitatively with what we would expect from the Heisenberg uncertainty principle, that as the cavity expands the distribution function becomes sharper
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
2'5. . ' " ' ' " ... 2"0
:"
1-5
O'
p
'.
/
1'0
0"5
)
"
~
".
0.2
)p*
p
Fig. 2. The parton model diagram for H(x). A quark-like parton of flavor a and momentum fraction x is removed from the target, then later inserted.
."
0
) p~ : ( 1 - x
0.4
0-6
0"8
x
Fig. 1. The two contributions to H(x) for a cavity "radius" of 1 fm. The dotted line is the contribution when the intermediate state contains two quarks, and the dashed line is when the intermediate state contains three quarks and an antiquark. about the most probable m o m e n t u m fraction x. Decreasing the mass o f the state In> by 10% pushes H(x) to higher x and increases the area under the curve to 95%. Since one expects one valence quark ( o f each colour), the usual procedure has been to simply renormalize H(x) to unit probability. We believe this is incorrect, and in suggesting an alternative solution we also resolve the second puzzle mentioned earlier, namely the negative sea. To see the resolution we return to eq. (2). The operator V*+(~-)~v+(0) involves four types of terms, btb, ddt, bid t and d*bt (where b and d are quark and antiquark annihilation operators in the bag ). The n = 2 contribution to H ( x ) , just calculated, arises from the b*b term, and the last three are usually assumed to give vanishing contributions. This is true of bid* and dtb *, but not ofddt. There is indeed a valid contribution to the structure function o f the target, from the term where a b o u n d antiquark is inserted into the target (dr), interacts with it, and is later removed (d). In a free field theory it could not contribute at positive x, but in the interacting case it can. The usual parton diagram for H(x) show in fig. 2,
is correctly interpreted as first removing a quark-like parton from the target and later inserting it. In the interacting case the same physical interpretation applies to either removing and reinserting a bound quark or inserting and removing a bound antiquark. The sum of the two is the total quark-like parton content o f the target. Since the mass o f a 3q-el bag (M4) is greater than that of the nucleon, the peak of the dd* contribution occurs outside the physical region (at x ~ - ( m - M 4 ) / m < O ) . Thus, in the physical region ( x > 0) it contributes only at small x, as shown in fig. 1. Note that whereas the bib term has three components, dd* gives rise to six (two flavours and three colours). The antiquark t e r m / ~ ( x ) has the same shape but only three contributions (form bbt), because three of the 1s states are occupied (by the original quarks). N o w it should be clear that half o f the six dd* terms should be identified as our missing valence quark. Numerically we find 5.2% and 4.4% for R = 1.0 fm and 1.1 fm, respectively, to be added to the 91% and 92% calculated earlier. Increasing M by 10% pushes H(x) to lower values o f x and lowers its contribution to 1.5%. The remaining half o f the dd ~ term and the bb* term constitutes a positive, unavoidable sea o f q Ctpairs intrinsic to the bag. Physically we believe it is associated with the fact that the bag vacuum differs from that o f free space. We have calculated a valence quark distribution for the cavity approximation using the distributions H(x) a n d / / ( x ) . In all our calculations we find that the total area under the valence distribution is approximately 96%. Presumably the missing 4% is lost because the Peierls-Yoccoz projection for the intermediate state breaks down at large x, however this effect is not large. This gives us some confidence that our method of calculating structure functions can also be applied to the M I T bag in three dimensions. 483
Volume 211, number 4
PHYSICS LETTERS B
This work was supported by the ARGS and the Commonwealth Department of Education, Employment and Training. References [ 1 ] EM Collab., J.J. Aubert et al., Phys. Lett. B 123 ( 1983 ) 275; R.G. Arnold et al., Phys. Rev. Lett. 54 (1984) 1431. [2] A.W. Thomas, Prog. Nucl. Part. Phys. 20 (1988) 21. [ 3 ] A. Chodos et al., Phys. Rev. D 9 ( 1974 ) 3471; D 10 (1974) 2599.
484
8 September 1988
[4] R.L. Jaffe, Phys. Rev. D 11 (1975) 1953. [5] R.U Jaffe, Ann. Phys. 132 ( 1981 ) 32. [6] V. Krapchev, Phys. Rev. D 13 (1976) 329. [ 7 ] J.S. Bell and A.J.G. Hey, Phys. Lett. B 74 (1978) 77; J.S. Bell, A.C. Davis and J. Rafelski, Phys. Lett. B 78 ( 1978 ) 67. [8] R.K. Ellis, V. Furmanski and R. Petronzio, Nucl. Phys. B 207 (1982) 1;B212 (1983) 29. [9] R.L. Jaffe, Nucl. Phys. B 229 (1983) 205. [10] R.E. Peierls and J. Yoecoz, Proc. Phys. Soc. A 70 (1957) 381. [ 11 ] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1102.
PHYSICS LETTERS B
8 September 1988
THE STRUCTURE FUNCTION OF THE NUCLEON A.I. S I G N A L and A.W. T H O M A S
DepartmentofPhysicsandMathematicalPhysics,The UniversityofAdelaide,Adelaide,SA 5001,Australia Received 20 June 1988
We present a novel method of calculation of the structure function of the "nucleon" for the two-dimensional MIT bag model which overcomes the defects inherent in previous calculations. In particular we naturally obtain the correct support and normalization. We also find that there is an intrinsic quark-antiquark sea associated with the MIT bag.
Ever since the discovery of scaling and identification of partons with quarks there have been attempts to calculate the structure function o f the nucleon for a given model. This endeavour has recently received impetus for two reasons. First the discovery o f the EMC effect [ 1 ] stimulated a great deal o f theoretical work aimed at understanding how, if at all, the quark structure o f the nucleon is altered inside the nucleus. Clearly no unambiguous conclusions can be drawn from any analysis o f the EMC data until the problem of relating the structure functions to quark models is solved. The second stimulus is the variety of models of hadron structure which have been created in the last decade - notably with enormous variation in the typical confinement region for the quarks. It has been argued [ 2 ] that deep inelastic scattering is probably the only possible tool for unambiguously distinguishing between the models. The M I T bag model [ 3 ] was constructed with the SLAC data on deep inelastic scattering (DIS) in mind, and one o f the earliest attempts to calculate structure functions was made for this model by Jaffe [4]. This calculation was of fundamental importance and served to stimulate much further work. Nevertheless, it suffered from two major problems. While physical structure functions must vanish for x outside [0, 1 ] (i.e. they have support only on [0, 1 ] ), the bag model calculation had support on the interval [0, ~ ) . Secondly the theory seemed to imply the presence o f a negative sea, which is again unphysical. Subsequent work [ 5 ] based on the Lo approximation of Krapchev [ 6 ] appeared to solve the problem
o f support, but led to unrealistically oscillatory quark distributions. Even worse, it had no physical basis that is, it was simply the first term in a mathematical expansion which was poorly understood (and still is). No solution has hitherto been offered to the negative sea, and indeed the work of Bell and others [ 7 ] suggested even worse problems. In this paper we suggest a physically sensible solution to these problems. While full details will be provided elsewhere it is possible to explain the essence o f our approach very quickly. In view o f the current interest, it seems worthwhile to us to do this in a letter format, and to present some of the main numerical results for the bag in one space dimension. Although we used a much more circuitous route, it is most convenient to begin with some formal results of Ellis, Furmanski, and Petronzio [ 8 ], and Jaffe [ 9 ]. They write the key structure functions F2 and F 3 measured in v DIS as F2(x)
=x[H(x) - H ( - x ) ] ,
F3(x) =
[H(x)+H(-x)] .
(1)
As explained by Jaffe, for positive x it is sensible to use the expression for H(x):
H(x)=~
i d,-exp(-ixp+,-)
X ( p I ~'*+( ~ - ) q/+ (0) Ip)c,
x>0.
(2)
In eq. (2) C denotes a connected matrix element, p + 481
Volume 211, number 4
PHYSICS LETTERS B
is the plus-component of m o m e n t u m of the nucleon, in the rest frame, ~ - is the light-cone coordinate ~ - = (~o_ ~, )/x/~, and q/+ is given by
p +=M/x/2
~+ = ½ ( l + a ) ~ ,
(3)
8 September 1988
H(x) = -~ ~ Jlap; 4~pO Xi
d~- exp[i(p+-xp+-p+)~ -]
where o~ is the 2 X 2 matrix (~ _o). In order to evaluate H ( - x ) , with x positive, we should use the form
X I ( n , p ~ l q / + ( 0 ) l p ) I2
H(-x)=~n
--
4hE f 4~p dp~On~J(p + -- ( 1-- x )p + ) 1
; d~-exp(ixp+~-)
X[-(Plq/+(O)q/t+(~-)lP)c],
x>0.
(4)
Following the arguments of Jaffe we may define an antiquark distribution/-/(x), for x positive as
tl(x)=-H(-x), so that, for x >
(5)
0,
Fz(X ) =x[H(x) F3(x) = [ H ( x )
x>0,
+ Ii (x) ] , -H(x) ] .
(6)
Changing the variable of integration ~- to - ~ - in eq. (4) and using translational invariance we find
I~(x)=~n i d~-exp(-ixp+~-) --oo
X (Pl q/+ ( ~ - ) q/*+(0) Ip)c,
x>0.
(8)
where/~ is the full m o m e n t u m and energy operator for the strongly interacting system. To exploit this we introduce a complete set of states n, with m o m e n t u m p" and rest mass M~ into eqs. (2) and (4). Because both IP) and In; p ' ) are eigenstates of/~ we find, again for x > 0, 482
(9)
wherep + is (,/.~.[ 2. .,4_. p , ,t 2 ~ 1 / 2 _ 1 - -p , .t A similar expression is obtained f o r / ~ ( x ) with q/t+(0) replacing q/+ (0). As p+ is positive definite both H and H are guaranteed to be zero beyond x = 1. It is only that this stage that one can think of using a model ofhadron structure to evaluate H a n d / ~ . The quark field operators are taken to be those of the MIT bag model. For IP) we use the Peierls-Yoccoz (PY) procedure [ 10 ] to construct a translationally invariant state with zero momentum corresponding to three quarks in the 1si/2 level. The most natural choice for the states n are MIT bags of the same radius as the proton, but with a different number of quarks. Again the PY procedure is used to construct an approximate m o m e n t u m eigenstate of m o m e n t u m p ' :
(7)
We stress that eqs. (3), (6) and (7) can be evaluated in any frame, including the rest frame of the target. The problem of bad support arises if one approximates the quark field operator evaluated at ~- (e.g. qP+ ( ~ - ) ) by a bag field operator at this stage - e.g. with the time dependence exp ( + ie~°), where is the quark eigenenergy. For correct support it is essential that m o m e n t u m and energy conservation be guaranteed exactly. That is, translation invariance in space and time must be retained. For this reason we write q/t+ (~) =exp(+i/~.~)q/*+ (0) exp(-i/~.~) ,
X I ( n , p " I q/+ (0)IP)12 ,
oo
In; p" ) =
1
¢(p,~ f dzexp(ip'~z)ln;z)
(10)
--oo
Here In; z) is a bag with contents labelled n which is centered at z, and I¢(P')12 is proportional to the Fourier transform of the Hill-Wheeler overlap function [ 11 ] for bag states. The key question is which state In ) must be considered. In previous work (using either bags or other models) the only state considered was that most obviously involved when q/(0) acts on a three-quark ground state, namely two quarks in the lowest energy state (we shall call this n = 2 ) . The resulting form H(x) is shown in fig. 1 for a cavity "radius" of 1 fro. The area under the curve [i.e f~ H(x) dx] is 91%. Increasing the cavity radius by 10% gives a sharper distribution function, with slightly more area under the curve (92%). This agrees quantitatively with what we would expect from the Heisenberg uncertainty principle, that as the cavity expands the distribution function becomes sharper
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
2'5. . ' " ' ' " ... 2"0
:"
1-5
O'
p
'.
/
1'0
0"5
)
"
~
".
0.2
)p*
p
Fig. 2. The parton model diagram for H(x). A quark-like parton of flavor a and momentum fraction x is removed from the target, then later inserted.
."
0
) p~ : ( 1 - x
0.4
0-6
0"8
x
Fig. 1. The two contributions to H(x) for a cavity "radius" of 1 fm. The dotted line is the contribution when the intermediate state contains two quarks, and the dashed line is when the intermediate state contains three quarks and an antiquark. about the most probable m o m e n t u m fraction x. Decreasing the mass o f the state In> by 10% pushes H(x) to higher x and increases the area under the curve to 95%. Since one expects one valence quark ( o f each colour), the usual procedure has been to simply renormalize H(x) to unit probability. We believe this is incorrect, and in suggesting an alternative solution we also resolve the second puzzle mentioned earlier, namely the negative sea. To see the resolution we return to eq. (2). The operator V*+(~-)~v+(0) involves four types of terms, btb, ddt, bid t and d*bt (where b and d are quark and antiquark annihilation operators in the bag ). The n = 2 contribution to H ( x ) , just calculated, arises from the b*b term, and the last three are usually assumed to give vanishing contributions. This is true of bid* and dtb *, but not ofddt. There is indeed a valid contribution to the structure function o f the target, from the term where a b o u n d antiquark is inserted into the target (dr), interacts with it, and is later removed (d). In a free field theory it could not contribute at positive x, but in the interacting case it can. The usual parton diagram for H(x) show in fig. 2,
is correctly interpreted as first removing a quark-like parton from the target and later inserting it. In the interacting case the same physical interpretation applies to either removing and reinserting a bound quark or inserting and removing a bound antiquark. The sum of the two is the total quark-like parton content o f the target. Since the mass o f a 3q-el bag (M4) is greater than that of the nucleon, the peak of the dd* contribution occurs outside the physical region (at x ~ - ( m - M 4 ) / m < O ) . Thus, in the physical region ( x > 0) it contributes only at small x, as shown in fig. 1. Note that whereas the bib term has three components, dd* gives rise to six (two flavours and three colours). The antiquark t e r m / ~ ( x ) has the same shape but only three contributions (form bbt), because three of the 1s states are occupied (by the original quarks). N o w it should be clear that half o f the six dd* terms should be identified as our missing valence quark. Numerically we find 5.2% and 4.4% for R = 1.0 fm and 1.1 fm, respectively, to be added to the 91% and 92% calculated earlier. Increasing M by 10% pushes H(x) to lower values o f x and lowers its contribution to 1.5%. The remaining half o f the dd ~ term and the bb* term constitutes a positive, unavoidable sea o f q Ctpairs intrinsic to the bag. Physically we believe it is associated with the fact that the bag vacuum differs from that o f free space. We have calculated a valence quark distribution for the cavity approximation using the distributions H(x) a n d / / ( x ) . In all our calculations we find that the total area under the valence distribution is approximately 96%. Presumably the missing 4% is lost because the Peierls-Yoccoz projection for the intermediate state breaks down at large x, however this effect is not large. This gives us some confidence that our method of calculating structure functions can also be applied to the M I T bag in three dimensions. 483
Volume 211, number 4
PHYSICS LETTERS B
This work was supported by the ARGS and the Commonwealth Department of Education, Employment and Training. References [ 1 ] EM Collab., J.J. Aubert et al., Phys. Lett. B 123 ( 1983 ) 275; R.G. Arnold et al., Phys. Rev. Lett. 54 (1984) 1431. [2] A.W. Thomas, Prog. Nucl. Part. Phys. 20 (1988) 21. [ 3 ] A. Chodos et al., Phys. Rev. D 9 ( 1974 ) 3471; D 10 (1974) 2599.
484
8 September 1988
[4] R.L. Jaffe, Phys. Rev. D 11 (1975) 1953. [5] R.U Jaffe, Ann. Phys. 132 ( 1981 ) 32. [6] V. Krapchev, Phys. Rev. D 13 (1976) 329. [ 7 ] J.S. Bell and A.J.G. Hey, Phys. Lett. B 74 (1978) 77; J.S. Bell, A.C. Davis and J. Rafelski, Phys. Lett. B 78 ( 1978 ) 67. [8] R.K. Ellis, V. Furmanski and R. Petronzio, Nucl. Phys. B 207 (1982) 1;B212 (1983) 29. [9] R.L. Jaffe, Nucl. Phys. B 229 (1983) 205. [10] R.E. Peierls and J. Yoecoz, Proc. Phys. Soc. A 70 (1957) 381. [ 11 ] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1102.