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Deep inelastic scattering from a momentum conserving two-dimensional cavity
Deep Inelastic Scattering from a Momentum Conserving Two-dimensional Cavity A. S I G N A L and A. W. T H O M A S Department of Physics, University of Adelaide, P.O. Box 498, G.P.O. Adelaide, S.A. 5001, Australia
I.
INTRODUCTION
Historically the bag model in two spacetlme dimensions has been important in discussion of the deep inelastic nucleon structure functions. (Jaffe 1975, Bell and Hey 1978, Jaffe 1981). In two dimensions the light core nature of deep inelastic processes is apparent, and the existence of simple wave functions in the bag enables one to explicitly calculate structure functions. This program is not without its problems. In particular the cavity approximation to the bag model, which is frequently used, badly violates translation invarlance. This leads to support for the structure functions in the non-physical region x > i, where x is the BJorken scaling variable. To overcome this problem one might attempt to restore translation invarlanee using the Pelerls-Yoccoz projection (Peierls and Yoccoz 1957). Alternatively one might explicitly ensure momentum conservation throughout the scattering process. In this paper we apply both of these methods and compare the resulting structure functions and their support.
2. BASIC ~ E O R Y To leading order in Q2, the structure f ~ c t l o n is given h7 (Jaffe 1985): fa(X)where we have i ~ o r e d
M
d~- eiq ~
of a q ~ r k ,
flavour a, in a target at rest
< P I ~a+ ( ~ - ) ~ a + (0) I P >c
~+ - 0
the quark's charge and where we have used light cone co-ordinates a-+ - (a,o ± al)/2 #+ = % (I + ~ ) @ ,
x-
and a -
Q2 /2p. q
IlOl 0-i
(2.2)
(2.3)
with q+ - - xp + in the BJorken Limit (Qa ~ ~, q- ~ ~), and M the ~ s s of the target. We use the n o ~ a l l z a t l o n < p' [ P > " 2~6(p-p'). The subscript c denotes that only connected diagrams are included. Note that this structure function does not include any a n t i - q ~ r k contributions as we will be considering only target states consisting of valence q ~ r k s . We now insert a c o ~ l e t e set of states b e ~ e e n the quark field operators in eq. (2.1), and translate ~- dependence out of ~ + . Integrating over (-now gives
283
284
A. Signal and A. W. Thomas fa
(x) - p+ Z
6 (p+-xp +-
n +) [ < n [ #a+ [ p > [2
(2.4.)
n
which is familiar from the parton model. Now we turn to the description of the target state. The quark field operator in a cavity of length 2~ centered in light-core co-co-ordlnates at the origin is
#a ( ~+, ~-)
1
m ~> ~
2 ~
[ e- i~m~ -
vm'a
+]
(2.5)
(-i)m e-i~m~ ]
with , - ~(mm + ~)/2
(2.6)
and colour is suppressed.
The mass of the cavity plus quarks is given by the relation Mi " ~
z
Z
Nm(m + ~)
(2.7)
m where Nm is the number of quarks in the mth mode. Taking the double Fourier transform of #(x +, x-) gives the momentum space representation of the operator.
4a(P-' P+) " 2'It "]~- Z bm'a 6(P- + P+ - EI~ m -> 0
- 2~ 7
b
[[sin ((P+ } L
6(p + + p
sin
- ('!!m)' )/(p+,p~e )/((P+ + 2 - 'm)2 ) )
- E ) 4+ (p)
m,a
m
(2.8)
(2.9)
m
mE0 In the following sections we will consider scattering from a cavity at rest populated by three valence quarks in the ground (m - 0) state with separable wave functions.
3. PEIERLS-YOCCOZ PROJECTION The Peierls-Yoccoz projection approximates a translation invariant momentum eigenstate by a weighted integral over cavity states in all positions (Wong 1981) r (3.0) ] p > - [4(0)] -1 Id~ e-ip~ [ B(~)>
J in one space dimension where [ B(~) > is a cavity centered at ~ and 4(P) is the Fourier transform of the Hill-Wheeler overlap function (Hill and Wheeler 1953) denoted by Im(~) for m independent quarks. We now put this state into equ. (2.1) for the quark structure function and for a quark in the ground state we obtain
fa(X)-
M
-i(*p÷-
(3.1
where E(~-) is a llght-cone correlation function given in terms of Hill-Wheeler overlap functions E({) - [ 43 (p - 0) [-2 ] d~- 12 (~-) I, (~- - {)
(3.2)
The two factors in the integral of equ. (3.3) come from the requirement that the two spectator quarks and the struck quark are all found in both the initial and final state of the virtual Compton process.
Deep Inelastic Scattering
285
After doing the integrations in equs. (3.3) and (3.2) we arrive at the result fa(X)= P~+
sin2(x l p+~ P+~ - ./2) - "/2 2( )
{ 2(2 x x p+~ - "/2)-2
[I + c°s 2 x (2 p+~x P+~) - ~/2
which is shown in fig. i. The term outside the braces is the same as obtained by Jaffe (Jaffe 1975) and by Bell and Hey (Bell and Hey 1978). The term in the braces damps the structure function for X > 1 and arises from the Pelerls-Yoccoz projection and oru requirements that the two spectators be found in both the initial and final states. We find that the ratio
I dx fa(X) / I dx fa(X) o o
(3.4)
> 99.5%
which is an improvement on the value of = 92% obtained by Jaffe (Jaffe 1975). The fact that f, (x) ~ 0 for x > 1 arises because we do not ensure that the spectators form an onmass-shell state after the struck quark is removed. We can examine the number and momentum sum rules. We find ~o I
(3.5)
dx fa(X) - 0.92 o
and 3 I
(3.6)
dx x fa(X) - 0.92 o
2.5 - - -
2.0
`%
Cavity approximation Peierls-Yoccoz correction to cavity approximation
1.5 f(x) 1.o
/"
'k'%
•
'%
0.5 I
I
0.25
0.50
I 0.75
1.00
.----I--1.25
Figure i The quark structure function f(x) of a bag containing three valence quarks for the cavity approximation and for the cavity with the Feierls-Yoccoz correction eq. (3.4).
286
A. Signal and A. W. Thomas
which shows that all the momentum is carried by the quark fields, as expected from the cavity approximation. The first sum rule should give unity, but is less than one because we have ignored any contributions from negative x. The contributions from the region x < 0 arise because the struck quark is in a bound system and has a small probability of having p÷ < O.
4.
STRUCTURE FUNCTION WITH MOMENTUM CONSERVATION ENFORCED
We noted above t h a t unphysical contributions to t h e structure function f,(x) (notably beyond x - i) arise if the spectator quarks do not form an on-mass-shell state. We can address this problem by examining the intermediate states I n > in eq. (2.5) and applying appropriate constraints. As I n > is formed by the removal of a quark with plus component of momentum xp + from the state I P >, conservation of two-momentum implies that the plus component of momentum of the state I n >, n + is equal to (l-x)p +. Also for the state I n > to be onmass-shell n+ must be greater than or equal to zero. These two constraints on I n > imply that the only possible values that x can take are the physical values 0 ~ x ~ i. To a first approximation we may treat [ n > as a state made of two quarks both on the mass shell with wavefunctions which are the projections of eq. (2.9) onto free spinors:
~a(p , p+) - 2~
i
~
sin ((p+- Em)~)
bm, a 6(p- + p+ - E m)
- 2x I
m~O
(p+ _ ~ )~
Um
bm, a~ (p- + p+- Em) Sm(P+)
(4.1)
(4.2)
Where the U m are two-component spinors. This approximation neglects on-shell intermediate states where one spectator quark has p÷ < 0 i.e. we have neglected binding in the intermediate state and treated the spectator quarks as "quasl-free" The structure function eq. (2.5) may now be written
fa (x) " ~ P
I ~O,a+ (xp+) 12
dq +1 o
x
dq +2 o
+ ) 12 I -~ o ( q +l ) 12 I ~o(q2
6 ((l-x)p+-q~ - q;)]
(4.3)
where ql and q2 are the momenta of the two spectator quarks and the delta function ensures that the two spectators conserve momentum. Substituting in the wave functions (2.10) and (4.2) and using the delta function to evaluate one integral gives: +
f a ( x ) _ / _l ~
[~13.1~,.2
sin
(x ~ ~ - ~/2)
(xP +~ _ "/2) 2 (l-x)p + I
2+ + sin (ql ~ - ~/2)
dql
( q ; ~ - ~/2) 2
x
+ sln2((l-x)P+~ - ql ~ w/2)
((1-x)p+~ - q ~ - ~/2) 2
(4.4)
o which is shown in fig. 2. As with the modified by function to of m o m e n t u m
Pelerls-Yoccoz projection, we obtain the same structure function as 3affe, a term which damps the contributions at large x and sifts shifts the structure lower x. In this case the contributions for x > 1 disappear as a consequence conservation. If we examine the sum rules again we find: 1 dx f a (x) - 0.87
I
o
(4.5)
Deep Inelastic Scattering 2.5
--
-
-
-
287
Lo o p p r o x i m o t i o n
Cavity approximation with 2.0 -
ntum
conservotion
1.5
f(x) 1.0
05
I 0.25
0.50
0.75
1.00
X Figure 2 The quark structure function f(x) of a bag containing three valence quarks for the L o approximation and for the cavity aproximation with the correction for momentum conservation eq. (4.4).
f 1 3 I dx x fa(X) - 0.87 (4.6) J o which again indicates that all the momentum of the cavity appears to be carried by the quark fields. The first integral should be unity if our normalization is correct. The difference gives an indication that, by dropping intermediate states with one spectator quark having p+ < 0, we have substracted out about 10% of the possible intermediate states.
5.
DISCUSSION
We have presented two procedures for restoring translation invariance to the cavity approximation in the context of deep inelastic scattering. Both procedures yield reasonable structure functions which improve on earlier cavity approximation calculations (Jaffe 1975). We may also compare our results with those of the translation invarlant "L o approximation" (Jaffe and Ross 1980, Jaffe 1981), see fig. 2. The L o approximation gives structure functions with the correct support, however at large x the structure function has zeroes which are not seen, nor expected, physically and the structure function diverges as x ~ 1 rather than going to zero. In both of the procedures we have used, the spectator quarks have an important role, in limiting the momentum of the struck quark. In contrast the spectators are of no importance in the naive cavity or L o approximations. The importance of the spectators comes from the physical constraints or the forward virtual Compton process, that the intermediate state be on shell and that the final state is formed, and from translation invarlance (or momentum conservation) imposed on the cavity.
288
A. Signal and A. W. Thomas
It has been pointed out earlier (Thomas 1987, Jaffe and Ross 1980) that quark models such as the bag do not really apply in the Bjorken limit (Q2 ~ ~, v ~ ~ ). However it is postulated that quark model matrix elements are those of a full field theory renormalized at a low momentum scale. That is, we have calculated the leading twist two part of the structure function at a value of Q2 somewhat less than I GeV 2, which is well out of the range of experimental data. However we could connect our calculations with experiments, which also measure the leading twist two part of the structure at large Q 2 by taking moments of our distributions and evolving to higher Q2 using the Alteralli-Parisl equation~ (see Bickerstaff and Thomas 1987, and Jaffe and Ross 1980 for details). In conclusions, restoring translation invariance to the cavity or enforcing momentum conservation in deep inelastic scattering leads to structure functions with good qualitative properties including the damping, and even complete suppression, of the unphysical contributions for x > i. This work was supported by the Commonwealth Department of Education and Youth Affairs and the ARGS.
REFERENCES J.S. Bell and A.J. Hey Phys. (1978), Phys. Lett. 74B, 77. R.P. Bickerstaff and A.W. Thomas (1987), University of Adelaide preprint "The structure function of a swollen nucleon", ADP-87-1/T29. D.L. Hill and J.A. Wheeler (1953), Phys. Rev. 89. 1102 R.L. Jaffe and G.G. Ross (1980), Phys. Lett. 93B, 313. R.L. Jaffe (1981), Ann. Phys 132, 32. R.L. Jaffe (1985).
Lectures presented at 1985 Los Alamos School on Quark nuclear Physics.
R.E. Peierls and J. Yoccoz (1957), Proc. Phys. Soc. London, A70, 381. A.W. Thomas (1987), Lectures presented at this school. C.W. Wong (1981), Phys. Rev. D24, 1416.
I.
INTRODUCTION
Historically the bag model in two spacetlme dimensions has been important in discussion of the deep inelastic nucleon structure functions. (Jaffe 1975, Bell and Hey 1978, Jaffe 1981). In two dimensions the light core nature of deep inelastic processes is apparent, and the existence of simple wave functions in the bag enables one to explicitly calculate structure functions. This program is not without its problems. In particular the cavity approximation to the bag model, which is frequently used, badly violates translation invarlance. This leads to support for the structure functions in the non-physical region x > i, where x is the BJorken scaling variable. To overcome this problem one might attempt to restore translation invarlanee using the Pelerls-Yoccoz projection (Peierls and Yoccoz 1957). Alternatively one might explicitly ensure momentum conservation throughout the scattering process. In this paper we apply both of these methods and compare the resulting structure functions and their support.
2. BASIC ~ E O R Y To leading order in Q2, the structure f ~ c t l o n is given h7 (Jaffe 1985): fa(X)where we have i ~ o r e d
M
d~- eiq ~
of a q ~ r k ,
flavour a, in a target at rest
< P I ~a+ ( ~ - ) ~ a + (0) I P >c
~+ - 0
the quark's charge and where we have used light cone co-ordinates a-+ - (a,o ± al)/2 #+ = % (I + ~ ) @ ,
x-
and a -
Q2 /2p. q
IlOl 0-i
(2.2)
(2.3)
with q+ - - xp + in the BJorken Limit (Qa ~ ~, q- ~ ~), and M the ~ s s of the target. We use the n o ~ a l l z a t l o n < p' [ P > " 2~6(p-p'). The subscript c denotes that only connected diagrams are included. Note that this structure function does not include any a n t i - q ~ r k contributions as we will be considering only target states consisting of valence q ~ r k s . We now insert a c o ~ l e t e set of states b e ~ e e n the quark field operators in eq. (2.1), and translate ~- dependence out of ~ + . Integrating over (-now gives
283
284
A. Signal and A. W. Thomas fa
(x) - p+ Z
6 (p+-xp +-
n +) [ < n [ #a+ [ p > [2
(2.4.)
n
which is familiar from the parton model. Now we turn to the description of the target state. The quark field operator in a cavity of length 2~ centered in light-core co-co-ordlnates at the origin is
#a ( ~+, ~-)
1
m ~> ~
2 ~
[ e- i~m~ -
vm'a
+]
(2.5)
(-i)m e-i~m~ ]
with , - ~(mm + ~)/2
(2.6)
and colour is suppressed.
The mass of the cavity plus quarks is given by the relation Mi " ~
z
Z
Nm(m + ~)
(2.7)
m where Nm is the number of quarks in the mth mode. Taking the double Fourier transform of #(x +, x-) gives the momentum space representation of the operator.
4a(P-' P+) " 2'It "]~- Z bm'a 6(P- + P+ - EI~ m -> 0
- 2~ 7
b
[[sin ((P+ } L
6(p + + p
sin
- ('!!m)' )/(p+,p~e )/((P+ + 2 - 'm)2 ) )
- E ) 4+ (p)
m,a
m
(2.8)
(2.9)
m
mE0 In the following sections we will consider scattering from a cavity at rest populated by three valence quarks in the ground (m - 0) state with separable wave functions.
3. PEIERLS-YOCCOZ PROJECTION The Peierls-Yoccoz projection approximates a translation invariant momentum eigenstate by a weighted integral over cavity states in all positions (Wong 1981) r (3.0) ] p > - [4(0)] -1 Id~ e-ip~ [ B(~)>
J in one space dimension where [ B(~) > is a cavity centered at ~ and 4(P) is the Fourier transform of the Hill-Wheeler overlap function (Hill and Wheeler 1953) denoted by Im(~) for m independent quarks. We now put this state into equ. (2.1) for the quark structure function and for a quark in the ground state we obtain
fa(X)-
M
-i(*p÷-
(3.1
where E(~-) is a llght-cone correlation function given in terms of Hill-Wheeler overlap functions E({) - [ 43 (p - 0) [-2 ] d~- 12 (~-) I, (~- - {)
(3.2)
The two factors in the integral of equ. (3.3) come from the requirement that the two spectator quarks and the struck quark are all found in both the initial and final state of the virtual Compton process.
Deep Inelastic Scattering
285
After doing the integrations in equs. (3.3) and (3.2) we arrive at the result fa(X)= P~+
sin2(x l p+~ P+~ - ./2) - "/2 2( )
{ 2(2 x x p+~ - "/2)-2
[I + c°s 2 x (2 p+~x P+~) - ~/2
which is shown in fig. i. The term outside the braces is the same as obtained by Jaffe (Jaffe 1975) and by Bell and Hey (Bell and Hey 1978). The term in the braces damps the structure function for X > 1 and arises from the Pelerls-Yoccoz projection and oru requirements that the two spectators be found in both the initial and final states. We find that the ratio
I dx fa(X) / I dx fa(X) o o
(3.4)
> 99.5%
which is an improvement on the value of = 92% obtained by Jaffe (Jaffe 1975). The fact that f, (x) ~ 0 for x > 1 arises because we do not ensure that the spectators form an onmass-shell state after the struck quark is removed. We can examine the number and momentum sum rules. We find ~o I
(3.5)
dx fa(X) - 0.92 o
and 3 I
(3.6)
dx x fa(X) - 0.92 o
2.5 - - -
2.0
`%
Cavity approximation Peierls-Yoccoz correction to cavity approximation
1.5 f(x) 1.o
/"
'k'%
•
'%
0.5 I
I
0.25
0.50
I 0.75
1.00
.----I--1.25
Figure i The quark structure function f(x) of a bag containing three valence quarks for the cavity approximation and for the cavity with the Feierls-Yoccoz correction eq. (3.4).
286
A. Signal and A. W. Thomas
which shows that all the momentum is carried by the quark fields, as expected from the cavity approximation. The first sum rule should give unity, but is less than one because we have ignored any contributions from negative x. The contributions from the region x < 0 arise because the struck quark is in a bound system and has a small probability of having p÷ < O.
4.
STRUCTURE FUNCTION WITH MOMENTUM CONSERVATION ENFORCED
We noted above t h a t unphysical contributions to t h e structure function f,(x) (notably beyond x - i) arise if the spectator quarks do not form an on-mass-shell state. We can address this problem by examining the intermediate states I n > in eq. (2.5) and applying appropriate constraints. As I n > is formed by the removal of a quark with plus component of momentum xp + from the state I P >, conservation of two-momentum implies that the plus component of momentum of the state I n >, n + is equal to (l-x)p +. Also for the state I n > to be onmass-shell n+ must be greater than or equal to zero. These two constraints on I n > imply that the only possible values that x can take are the physical values 0 ~ x ~ i. To a first approximation we may treat [ n > as a state made of two quarks both on the mass shell with wavefunctions which are the projections of eq. (2.9) onto free spinors:
~a(p , p+) - 2~
i
~
sin ((p+- Em)~)
bm, a 6(p- + p+ - E m)
- 2x I
m~O
(p+ _ ~ )~
Um
bm, a~ (p- + p+- Em) Sm(P+)
(4.1)
(4.2)
Where the U m are two-component spinors. This approximation neglects on-shell intermediate states where one spectator quark has p÷ < 0 i.e. we have neglected binding in the intermediate state and treated the spectator quarks as "quasl-free" The structure function eq. (2.5) may now be written
fa (x) " ~ P
I ~O,a+ (xp+) 12
dq +1 o
x
dq +2 o
+ ) 12 I -~ o ( q +l ) 12 I ~o(q2
6 ((l-x)p+-q~ - q;)]
(4.3)
where ql and q2 are the momenta of the two spectator quarks and the delta function ensures that the two spectators conserve momentum. Substituting in the wave functions (2.10) and (4.2) and using the delta function to evaluate one integral gives: +
f a ( x ) _ / _l ~
[~13.1~,.2
sin
(x ~ ~ - ~/2)
(xP +~ _ "/2) 2 (l-x)p + I
2+ + sin (ql ~ - ~/2)
dql
( q ; ~ - ~/2) 2
x
+ sln2((l-x)P+~ - ql ~ w/2)
((1-x)p+~ - q ~ - ~/2) 2
(4.4)
o which is shown in fig. 2. As with the modified by function to of m o m e n t u m
Pelerls-Yoccoz projection, we obtain the same structure function as 3affe, a term which damps the contributions at large x and sifts shifts the structure lower x. In this case the contributions for x > 1 disappear as a consequence conservation. If we examine the sum rules again we find: 1 dx f a (x) - 0.87
I
o
(4.5)
Deep Inelastic Scattering 2.5
--
-
-
-
287
Lo o p p r o x i m o t i o n
Cavity approximation with 2.0 -
ntum
conservotion
1.5
f(x) 1.0
05
I 0.25
0.50
0.75
1.00
X Figure 2 The quark structure function f(x) of a bag containing three valence quarks for the L o approximation and for the cavity aproximation with the correction for momentum conservation eq. (4.4).
f 1 3 I dx x fa(X) - 0.87 (4.6) J o which again indicates that all the momentum of the cavity appears to be carried by the quark fields. The first integral should be unity if our normalization is correct. The difference gives an indication that, by dropping intermediate states with one spectator quark having p+ < 0, we have substracted out about 10% of the possible intermediate states.
5.
DISCUSSION
We have presented two procedures for restoring translation invariance to the cavity approximation in the context of deep inelastic scattering. Both procedures yield reasonable structure functions which improve on earlier cavity approximation calculations (Jaffe 1975). We may also compare our results with those of the translation invarlant "L o approximation" (Jaffe and Ross 1980, Jaffe 1981), see fig. 2. The L o approximation gives structure functions with the correct support, however at large x the structure function has zeroes which are not seen, nor expected, physically and the structure function diverges as x ~ 1 rather than going to zero. In both of the procedures we have used, the spectator quarks have an important role, in limiting the momentum of the struck quark. In contrast the spectators are of no importance in the naive cavity or L o approximations. The importance of the spectators comes from the physical constraints or the forward virtual Compton process, that the intermediate state be on shell and that the final state is formed, and from translation invarlance (or momentum conservation) imposed on the cavity.
288
A. Signal and A. W. Thomas
It has been pointed out earlier (Thomas 1987, Jaffe and Ross 1980) that quark models such as the bag do not really apply in the Bjorken limit (Q2 ~ ~, v ~ ~ ). However it is postulated that quark model matrix elements are those of a full field theory renormalized at a low momentum scale. That is, we have calculated the leading twist two part of the structure function at a value of Q2 somewhat less than I GeV 2, which is well out of the range of experimental data. However we could connect our calculations with experiments, which also measure the leading twist two part of the structure at large Q 2 by taking moments of our distributions and evolving to higher Q2 using the Alteralli-Parisl equation~ (see Bickerstaff and Thomas 1987, and Jaffe and Ross 1980 for details). In conclusions, restoring translation invariance to the cavity or enforcing momentum conservation in deep inelastic scattering leads to structure functions with good qualitative properties including the damping, and even complete suppression, of the unphysical contributions for x > i. This work was supported by the Commonwealth Department of Education and Youth Affairs and the ARGS.
REFERENCES J.S. Bell and A.J. Hey Phys. (1978), Phys. Lett. 74B, 77. R.P. Bickerstaff and A.W. Thomas (1987), University of Adelaide preprint "The structure function of a swollen nucleon", ADP-87-1/T29. D.L. Hill and J.A. Wheeler (1953), Phys. Rev. 89. 1102 R.L. Jaffe and G.G. Ross (1980), Phys. Lett. 93B, 313. R.L. Jaffe (1981), Ann. Phys 132, 32. R.L. Jaffe (1985).
Lectures presented at 1985 Los Alamos School on Quark nuclear Physics.
R.E. Peierls and J. Yoccoz (1957), Proc. Phys. Soc. London, A70, 381. A.W. Thomas (1987), Lectures presented at this school. C.W. Wong (1981), Phys. Rev. D24, 1416.