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A simple model for the variation with x of quark transverse momentum
Volume 69B, number 2
PHYSICS LETTERS
1 August 1977
A SIMPLE MODEL FOR THE VARIATION WITH x OF QUARK TRANSVERSE
MOMENTUM
Anne C. DAVIS and Euan J. SQUIRES.
Department of Mathematics, University of Durham, Durham, UK Received 28 May 1977 We show that elementary kinematics yield a k T distribution which varies with x. The nature of the relationship depends on the form of the quark wavefunction. We calculate explicitly in the MIT bag model.
There has recently been considerable interest in the relationship between the transverse momentum of quarks and their longitudinal momentum. Theoretical papers [1, 2] have predicted contradictory behaviour and both have claimed experimental support. In this paper we discuss a trivial kinematic model for deep inelastic scattering which gives the required distributions directly in terms of the wavefunction of the quarks in the hadron. We take for the hadron a bag of confined, but otherwise free, massless quarks and work in the hadron rest frame. For initial orientation we ignore the confinement and consider the collision of a photon (q0, q) with a free massless quark (k 0, k; k 02 = Ikl2). After the collision we have
associated with intermediate values ofx. (Note that by k T we mean the momentum perpendicular to q; this gives an upper limit to the momentum perpendicular to a particular plane.) We now take account of conf'mement by assuming instead of a single value for k a distribution P(k) = [(k [~b)]2, where [~) is the quark state in the bag. This means that eq. (2) is replaced by
x = kO/M - (k/M) cos O,
and the probability of a given value o f x is proportional to
f(x) = f - k2dkP(k) +l fd(cos0)~Ix 0
(k 0 +q0)2 = ik+ql 2 '
(2)
(3)
If we take a spherically symmetric distribution and integrate over all angles this yields a uniform distribution in x from zero to 2kO/M. We refer to this region o f x as the mass-shell-allowed (MSA) region. In this model the transverse momentum is uniquely determined by x, i.e.
k T = k 0 sin 0 = (2Mxk 0 - M2x2) 1/2.
kO k 1 --~+~cos0 (6)
oo
where x is defined in the usual way
2MqOx = iql 2 _ q02.
-1
(1)
which, for large q0, gives k0(1 cos 0) = Mx
(5)
(4)
This is plotted in fig. 1 and exhibits the behaviour suggested in ref. [2], namely that large k T values are
=M ( kdkP(k). LMx-.k°l
(7)
These equations are based on the assumption that appropriate hadronic final states are available for all values o f x.
kT t
0
2 k° M
X
Fig. 1. The maximum transverse momentum as a function of x for the free quark model. 249
Volume 69B, number 2
PHYSICS LETTERS
In the cavity approximation [3] to the MIT bag model of hadrons [4] the wavefunction for massless quarks in the ground state is
t~(r, t) = e-ik°tt~(r) (8)
(i/o(k°r)Uu) =N(kO)e-ik°t \_Jl(kOr) a"
duction diagrams where one of the pair is in an occupied state in the bag. These diagrams will suppress the small x region (see the discussion of Jaffe [6]). If we ignore these defects of our simple model we find an average value o f x given by
(x) ~. kO/M -- 0.43.
where U is a two-component Pauli spinor and N(k O) is taken such that fbag~ffd3r = 1. The boundary condition requires kOR = 2.04, where R is the radius. This wavefunction gives
e(k) =
1 August 1977
I(kl~b)l2
Since there are three quarks in the hadron this does not seem to be compatible with the data which requires 3 (x) ~ ~. Better agreement would be obtained by reducing k 0 to about half the above value. The resulting f(x), with k 0 = 0.204, is shown as the broken line in fig. 2. The probability of a given x and k T is now given by
(9)
[-sin(k-ko)R _ _ kko(k _ ko)
sinkoR s i n k R ] Rk2k 2 .2
The resultingf(x) is shown in fig. 2 for k 0 = 0.408, the value used in the MIT bag model fit to hadronic states [5]. Note that when x is near unity the final hadron has a mass close on M so the approximation of assuming a uniform density of final states is clearly inappropriate in this region. Thus the curve should fall much faster as x approaches unity. Similarly in the small x region we expect contributions from pair pro-
kTF(X, k T ) = / k 2 d k P ( k ) X [ x - - ~kO +
-1
--k*= ~
....
0.4 k° ~ 0 . 2
U
t
I i ....
\
!
x \.,.
i
MSA
kcosO]5 ~sinO-kT]
=M'kTP((k2 + (k 0 -
Mx)2)l/2).
(12)
This shows how the k T distribution depends upon x. The resulting values for (kT), defined by
(kT) = / k 2 dkTF(X, kT)//kTdkTF(X, kT),
(13)
0
P(k) = [R4(k 2 -
k2) 2 + 1]-1
(14)
I i
~:o region
Fig. 2. The structure function f ( x ) in the MIT bag model for k ° = 0.408 and k ° = 0.204.
250
(11)
are plotted as a function o f x in fig. 3. We see that the effect in (4) is considerably reduced by the off-massshell corrections. There is also an interesting effect outside the MSA region. Here the "scale" of k T is no longer governed by k 0 but by Mx; hence, particularly for the smaller value o f k 0, we obtain much larger (kT). We have also studied the effect of a different P(k) chosen to permit the integration to be performed analytically,
t(x)~
\
~ d(cos 0)5
0
0
°o
(10)
This gives
f(x) =rr+ 2 tan_l (RC)2 and
(15)
Volume 69B, number 2
~
PHYSICS LETTERS
- -
k*= 0 . 4
....
k* = O. 2
R = 10 GeV - I . The resultingf(x) and (k T) are shown in fig. 4. Their features are similar to those of the MIT bag model. It is amusing to note that the two "scales" remain in evidence even when R -~ oo. In fact (16) yields:
/i/ /
0.6~-
/ I
/
i
1 August 1977
, (k T)
/ 0.4. /
/
/
' [k 02 - (Mx - k0) 2 ] 1/2 < k 0 R--~
in the MSA range
// /
R.-,~ 2 [(Mx - kO) 2 - k02l 1/2 < - ~/T M x
0.2.i
(18)
outside the MSA range 0 L_ 0
.....
MSA
] -~1.0 x
region
Fig. 3. The mean value ofk T as a function ofx in the MIT bag model for k ° = 0.408 and k 0 = 0.204. =____~ {[(RC~ 4 + 1] 1/2 _ (RC)2}-I/2 (k-r )
Rx/~
f(x)
(16)
where C 2 = k 02 -
(Mx - kO) 2 .
(! 7)
We take k 0 = 2 G e V and have used R = 5 G e V - 1 and
t(x)
/ /
//
- -
R=
....
R = 10 G e V -1
5
GeV
-~
\ \
f~f (x)
\;,
'
//
,,\, / o;
MSA
--
region
Fig. 4. The structure functionf(x) and the mean value ofk T
in the model specified by eq. (14).
We can summarise the results of this work in the following conclusions. 1. The simple "on-mass-shell" model yields a rapidly varying (kT) , which peaks in the centre of the x distribution (ko/M). 2. Off-mass-shell effects caused by confinement considerably suppress this variation within the MSA region, but predict a large increase o f ( k T) with x outside this region. 3. With the MIT bag model of confinement, using the parameter of the best fit to hadron masses, the quarks appear to carry too much momentum for the model to be compatible with deep inelastic data. This point has also been noted by Jaffe [6]. 4. The increase o f ( k T) with x which is required [2] by experiment, at least up to x ~ 0.4, is not apparent in our model when we use realistic forms for P(k). The off-mass-shell effects which effectively kill this variation might be suppressed by a mechanism analogous to that discussed by Bell [7]. We propose to investigate this. 5. If the appropriate k 0 is such that f ( x ) peaks at a value o f x significantly less than ½, then the dramatic increase of (k T) for x tending to 1 predicted in the model would be an interesting effect to observe. Although our derivation is somewhat different it turns out that our final result could also be obtained directly from the work of Jaffe [6]. For example, our x, k T distribution would follow from Jaffe's eq. (B6) by omitting the integral over Ok. Comparison with other work is less easy. Landshoff [ 1] imposes the condition that the system "hadron minus struck quark" has a positive mass. Close et at. [2] makes the assumption that the x, k T distributions factorise. In both these papers, however, the intimate connection
251
Volume 69B, number 2
PttYSICS L['~TTERS
between x and k T which follows from spherical s y m m e t r y o f the initial hadron is lost. We are grateful to J.S. Bell, F.E. Close, F. Halzen, P.V. Landshoff and D.M. Scott for helpful c o m m e n t s on an earlier version of this note and wish to acknowledge receipt o f a research grant from the S.R.C.
References [1] P.V. Landshoff, Phys. Lett. 66B (1977) 452.
252
1 August 1977
12} I:.E. Close, F. l-Ialzen and D.M. Scott, Phys. Lett. 68B (1977) 447. [31 A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. DI0 (1974) 2599. [41 A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. 1)9 (1974) 3471. [5] T. l)eGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. 1)12 (1975) 2060. 161 R.L. Jaffc, Phys. Rcv. D l l (1975) 1953. [7] J.S. Bell, The Schr6dinger handbag model, CIIRN preprint 2291 (1977).
PHYSICS LETTERS
1 August 1977
A SIMPLE MODEL FOR THE VARIATION WITH x OF QUARK TRANSVERSE
MOMENTUM
Anne C. DAVIS and Euan J. SQUIRES.
Department of Mathematics, University of Durham, Durham, UK Received 28 May 1977 We show that elementary kinematics yield a k T distribution which varies with x. The nature of the relationship depends on the form of the quark wavefunction. We calculate explicitly in the MIT bag model.
There has recently been considerable interest in the relationship between the transverse momentum of quarks and their longitudinal momentum. Theoretical papers [1, 2] have predicted contradictory behaviour and both have claimed experimental support. In this paper we discuss a trivial kinematic model for deep inelastic scattering which gives the required distributions directly in terms of the wavefunction of the quarks in the hadron. We take for the hadron a bag of confined, but otherwise free, massless quarks and work in the hadron rest frame. For initial orientation we ignore the confinement and consider the collision of a photon (q0, q) with a free massless quark (k 0, k; k 02 = Ikl2). After the collision we have
associated with intermediate values ofx. (Note that by k T we mean the momentum perpendicular to q; this gives an upper limit to the momentum perpendicular to a particular plane.) We now take account of conf'mement by assuming instead of a single value for k a distribution P(k) = [(k [~b)]2, where [~) is the quark state in the bag. This means that eq. (2) is replaced by
x = kO/M - (k/M) cos O,
and the probability of a given value o f x is proportional to
f(x) = f - k2dkP(k) +l fd(cos0)~Ix 0
(k 0 +q0)2 = ik+ql 2 '
(2)
(3)
If we take a spherically symmetric distribution and integrate over all angles this yields a uniform distribution in x from zero to 2kO/M. We refer to this region o f x as the mass-shell-allowed (MSA) region. In this model the transverse momentum is uniquely determined by x, i.e.
k T = k 0 sin 0 = (2Mxk 0 - M2x2) 1/2.
kO k 1 --~+~cos0 (6)
oo
where x is defined in the usual way
2MqOx = iql 2 _ q02.
-1
(1)
which, for large q0, gives k0(1 cos 0) = Mx
(5)
(4)
This is plotted in fig. 1 and exhibits the behaviour suggested in ref. [2], namely that large k T values are
=M ( kdkP(k). LMx-.k°l
(7)
These equations are based on the assumption that appropriate hadronic final states are available for all values o f x.
kT t
0
2 k° M
X
Fig. 1. The maximum transverse momentum as a function of x for the free quark model. 249
Volume 69B, number 2
PHYSICS LETTERS
In the cavity approximation [3] to the MIT bag model of hadrons [4] the wavefunction for massless quarks in the ground state is
t~(r, t) = e-ik°tt~(r) (8)
(i/o(k°r)Uu) =N(kO)e-ik°t \_Jl(kOr) a"
duction diagrams where one of the pair is in an occupied state in the bag. These diagrams will suppress the small x region (see the discussion of Jaffe [6]). If we ignore these defects of our simple model we find an average value o f x given by
(x) ~. kO/M -- 0.43.
where U is a two-component Pauli spinor and N(k O) is taken such that fbag~ffd3r = 1. The boundary condition requires kOR = 2.04, where R is the radius. This wavefunction gives
e(k) =
1 August 1977
I(kl~b)l2
Since there are three quarks in the hadron this does not seem to be compatible with the data which requires 3 (x) ~ ~. Better agreement would be obtained by reducing k 0 to about half the above value. The resulting f(x), with k 0 = 0.204, is shown as the broken line in fig. 2. The probability of a given x and k T is now given by
(9)
[-sin(k-ko)R _ _ kko(k _ ko)
sinkoR s i n k R ] Rk2k 2 .2
The resultingf(x) is shown in fig. 2 for k 0 = 0.408, the value used in the MIT bag model fit to hadronic states [5]. Note that when x is near unity the final hadron has a mass close on M so the approximation of assuming a uniform density of final states is clearly inappropriate in this region. Thus the curve should fall much faster as x approaches unity. Similarly in the small x region we expect contributions from pair pro-
kTF(X, k T ) = / k 2 d k P ( k ) X [ x - - ~kO +
-1
--k*= ~
....
0.4 k° ~ 0 . 2
U
t
I i ....
\
!
x \.,.
i
MSA
kcosO]5 ~sinO-kT]
=M'kTP((k2 + (k 0 -
Mx)2)l/2).
(12)
This shows how the k T distribution depends upon x. The resulting values for (kT), defined by
(kT) = / k 2 dkTF(X, kT)//kTdkTF(X, kT),
(13)
0
P(k) = [R4(k 2 -
k2) 2 + 1]-1
(14)
I i
~:o region
Fig. 2. The structure function f ( x ) in the MIT bag model for k ° = 0.408 and k ° = 0.204.
250
(11)
are plotted as a function o f x in fig. 3. We see that the effect in (4) is considerably reduced by the off-massshell corrections. There is also an interesting effect outside the MSA region. Here the "scale" of k T is no longer governed by k 0 but by Mx; hence, particularly for the smaller value o f k 0, we obtain much larger (kT). We have also studied the effect of a different P(k) chosen to permit the integration to be performed analytically,
t(x)~
\
~ d(cos 0)5
0
0
°o
(10)
This gives
f(x) =rr+ 2 tan_l (RC)2 and
(15)
Volume 69B, number 2
PHYSICS LETTERS
- -
k*= 0 . 4
....
k* = O. 2
R = 10 GeV - I . The resultingf(x) and (k T) are shown in fig. 4. Their features are similar to those of the MIT bag model. It is amusing to note that the two "scales" remain in evidence even when R -~ oo. In fact (16) yields:
/i/ /
0.6~-
/ I
/
i
1 August 1977
, (k T)
/ 0.4. /
/
/
' [k 02 - (Mx - k0) 2 ] 1/2 < k 0 R--~
in the MSA range
// /
R.-,~ 2 [(Mx - kO) 2 - k02l 1/2 < - ~/T M x
0.2.i
(18)
outside the MSA range 0 L_ 0
.....
MSA
] -~1.0 x
region
Fig. 3. The mean value ofk T as a function ofx in the MIT bag model for k ° = 0.408 and k 0 = 0.204. =____~ {[(RC~ 4 + 1] 1/2 _ (RC)2}-I/2 (k-r )
Rx/~
f(x)
(16)
where C 2 = k 02 -
(Mx - kO) 2 .
(! 7)
We take k 0 = 2 G e V and have used R = 5 G e V - 1 and
t(x)
/ /
//
- -
R=
....
R = 10 G e V -1
5
GeV
-~
\ \
f~f (x)
\;,
'
//
,,\, / o;
MSA
--
region
Fig. 4. The structure functionf(x) and the mean value ofk T
in the model specified by eq. (14).
We can summarise the results of this work in the following conclusions. 1. The simple "on-mass-shell" model yields a rapidly varying (kT) , which peaks in the centre of the x distribution (ko/M). 2. Off-mass-shell effects caused by confinement considerably suppress this variation within the MSA region, but predict a large increase o f ( k T) with x outside this region. 3. With the MIT bag model of confinement, using the parameter of the best fit to hadron masses, the quarks appear to carry too much momentum for the model to be compatible with deep inelastic data. This point has also been noted by Jaffe [6]. 4. The increase o f ( k T) with x which is required [2] by experiment, at least up to x ~ 0.4, is not apparent in our model when we use realistic forms for P(k). The off-mass-shell effects which effectively kill this variation might be suppressed by a mechanism analogous to that discussed by Bell [7]. We propose to investigate this. 5. If the appropriate k 0 is such that f ( x ) peaks at a value o f x significantly less than ½, then the dramatic increase of (k T) for x tending to 1 predicted in the model would be an interesting effect to observe. Although our derivation is somewhat different it turns out that our final result could also be obtained directly from the work of Jaffe [6]. For example, our x, k T distribution would follow from Jaffe's eq. (B6) by omitting the integral over Ok. Comparison with other work is less easy. Landshoff [ 1] imposes the condition that the system "hadron minus struck quark" has a positive mass. Close et at. [2] makes the assumption that the x, k T distributions factorise. In both these papers, however, the intimate connection
251
Volume 69B, number 2
PttYSICS L['~TTERS
between x and k T which follows from spherical s y m m e t r y o f the initial hadron is lost. We are grateful to J.S. Bell, F.E. Close, F. Halzen, P.V. Landshoff and D.M. Scott for helpful c o m m e n t s on an earlier version of this note and wish to acknowledge receipt o f a research grant from the S.R.C.
References [1] P.V. Landshoff, Phys. Lett. 66B (1977) 452.
252
1 August 1977
12} I:.E. Close, F. l-Ialzen and D.M. Scott, Phys. Lett. 68B (1977) 447. [31 A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, Phys. Rev. DI0 (1974) 2599. [41 A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. 1)9 (1974) 3471. [5] T. l)eGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. 1)12 (1975) 2060. 161 R.L. Jaffc, Phys. Rcv. D l l (1975) 1953. [7] J.S. Bell, The Schr6dinger handbag model, CIIRN preprint 2291 (1977).