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Scaling law for fast pion production by 20–70 GeV protons
Volume
29B. number
SCALING
PHYSICS
10
LAW
FOR
FAST
PION
LETTERS
18 August 1969
PRODUCTION
BY
20-70
GeV
PROTONS
A. LILAND and H. PILKUHN Inslilut
fiiiir theovetische
Kernphysik. Received
Uniz~ersiliit
Karlsmhe.
Germany
7 July 1969
Negative pion production with inelasticity > 0.5 is shown to follow a simple scaling law. The law is used to predict the Al+ and K+ production by 70 GeV protons. For 300 GeV protons. it predicts fast pion intensities considerably smaller than those of the thermodynamic model.
Recently, negative particle production bv 70 GeV p-Al interactions has been measured [ 11, and it has oeen pointed out that the K-/nratio at that energy is close to that of 19.2 GeV pp-
Suppose that d2a(Pp. P, 0) depends only on the square of the 4-momentum transfer, t. from the incident proton to the outgoing pion, and on the pion’s transverse momentum, PT q P. .Q. At high
collisions P/‘Pmax,
energies,
[2] when plotted as a function of where P is the meson momentum and P max its kinematical limit. Actually, there appears to exist a separate scaling law for d2c/dS2dP for n- production (Pp = incident proton momentum) d2r(Pp,
P, B)/dS2dP = d2a(Po, P’, O’)/dfi’dP’
rP
(1)
one has
t -mi(l-P/Pp)
i.e.,
+ m,2(1-Pp/P)
t is only a function of P/Pp
viation of t from 0,
which vanishes
- P$P/Pp, and P+
(3) is -m2m 2,/2P Pr
quadratically
P
(3)
(the de-
P for small
for Pp m-m and
rO
d20/d%lP depends only on the meson P/Pp and the transverse momentum = P. 8. This law is evident from fig. 1. which
In words:
elasticity PT
the curves for d20(pA +n-. . . ) dS2dP 19.2 GeV protons [3] for 0 = 12.5, 20, 30, 40, 50, 60 and 70 mrad, together with the points from the 70 GeV p-Al collisions. For the 70 GeV points, 8 has been replaced by 0.70/19.2. Obviously, the agreements between the 70 GeV points (error f 30% [4]) and the 19.2 GeV points is complete below P/P = 0.8. For larger elasticities, the 70 GeV po?nts come higher. This effect appears to be due to the kinematical limit and should disappear if one uses P/P,, instead of P/Pp. However, the kinematical limit is smeared out by the nucleons’ Fermi motion inside the aluminium nucleus. For a free pn collision, it would occur at a missing mass of 2mp (Pmax = Pp - 1.4 GeV), while for a free pp collision, it would occur at 2mp + mn (or, more realistically, at mp + mA, Pmax = Pp - 2.0 GeV). Below P/Pp = 0.8, however, it appears that the scaling law is better fulfilled for P/P, than for Theoretically, this is rather plausible. P/pmax. contains
d3
OL
0’5
a8
P/Pp
O9
Fig. 1. Differential cross sections for r- and K- production in 19.2 GeV/c p-Al collisions 131. plotted verAlso given are the measuresus the elasticity P/P with their producments of 70 GeV/c p- R’1 collisions, tion angles multiplied by 70/19.2. The dashed curve refers to 0 mrad at 70 GeV/c [l], the dash-dotted curves to the calculations of Hagedorn and Ranft [5] at 30 and 300 GeV. 663
Volume 29B, number 10
PHYSICS
fixed, whereas the difference between P/Pp max and P/Pp vanishes only linearly). One can also test the assumption that d20(Pp, P, B)/dS2dP depends only on t, i.e., that all PT- dependence comes through (3). This is not the case. At 19.2 GeV, P = 16 GeV and 0 = = 12.5 mrad give the same t-value as P = 10 GeV and .Q= 40 mrad, whereas the cross sections of these two points differ by a factor 47. Before applying the scaling law to other reactions and energies, one should note that it cannot possibly be correct for small values of P/P . Integrating (1) over dW (- 2nPT dP’/P2), we 07Jtain
LETTERS
P/P
du(Pp, P)/dP
=($
i.e., da(Pp)/dP also over P,
da(Po, P’)/dP’
decreases
like l/Pi.
atot
,
Integrating
!nn - u(p) ) =
K’ilI’
i
0157
01
005
1
Fig. 3. The K+/n.’
(5)
we see that the whole integral goes to zero as l/P . For constant atot( this would mean thatpthe 7r- multiplicity tends to zero as l/Pp,
Fig. 2. Differential cross sections for X+ and Kt production in 19.2 p-Al collisions [3]. as function of the elasticity P/Pp. To obtain the cross sections at primary momentum Pp. one multiplies the angles by 19. 2/Pp.
664
02
18 August 1969
ratio at production Pp mrad.
angle 12.5 19.2/
whereas experimentally it certainly increases with Pp. Therefore there must be a region below P/Pp = 0.5 where du(Pp, P)/dP decf;egases much more slowly, roughly like (PO/P,) * . Remember that the average pion elasticity goes to zero as the multiplicity increases. For outgoing protons of course, a constant elasticity would imply an extra factor Pp/Po on the right hand side of eq. (1). Fig. 2 gives the 7~~and K+ production spectra of Allaby et al. [3] for 19.2 GeV p-Al collisions. In the region 0.5 EdP/Pp e: 0.8 the 7r+ spectra should be identical with the r+ spectra of 70 GeV p-Al collisions, provided the angles are reduced by a factor 19.2/70 = 0.274. For K’, the scaling law is less reliable. However, since the kinematic limit Pmax is closer to Pp for K+ than for K-, one can hope that the scaling law works better for Kf than for K-. The K+/n+ ratio for 8 = 12.5 mrad x 19.2/Pp is shown in fig. 3. Contrary to the K-/n- ratio, it is a rising function of P/Pp up to P/Pp = 0.8. Finally, we compare the thermodynamic model [5,6] with the scaling law. In fig. 1, we have included the predicted 0 mrad 7~~production spectra at 30 and 300 GeV, multiplied with the experimental ratio of r- production at 12.5 mrad in pA1 and pp collisions (this ratio varies between 8 and 10). We see that the ther modynamic model does not exhibit the scaling law. In particular, the predicted flux of fast rat 300 GeV is one order of magnitude too large.
Volume 29B, number 10
PHYSICS
At larger even worse,
disagreement becomes since the experimental pAl/pp ratio increases with angle (au effect of multiple collisions inside the nucleus). For n+ production, the situation is similar.
18 August 1969
References 1. Bushnin et al., Phys. Letters 29B (1969) 48. 2. J. V. Allaby et al., paper submitted to the 14th Intern. Conf. on High-energy physics, Vienna, 1968 (unpublished). 3. J. V. Allaby et al., to be published. 4. Yu. D. Prokoshkin, private communication by J.-P. Stroot. 3, R. Hagedorn and J. Ranft, Suppl. Nuovo Cim. 6 (1968) 169. 6. J. Ranft. Rutherford Lab. Report RHEL/R 165 (1968).
We wish to thank the authors of ref. 2 for giving us their unpublished results. Drs. R. Ha-
gedorn and K. Schliipmann have kindly with explanations and suggestions.
LETTERS
helped us *****
665
29B. number
SCALING
PHYSICS
10
LAW
FOR
FAST
PION
LETTERS
18 August 1969
PRODUCTION
BY
20-70
GeV
PROTONS
A. LILAND and H. PILKUHN Inslilut
fiiiir theovetische
Kernphysik. Received
Uniz~ersiliit
Karlsmhe.
Germany
7 July 1969
Negative pion production with inelasticity > 0.5 is shown to follow a simple scaling law. The law is used to predict the Al+ and K+ production by 70 GeV protons. For 300 GeV protons. it predicts fast pion intensities considerably smaller than those of the thermodynamic model.
Recently, negative particle production bv 70 GeV p-Al interactions has been measured [ 11, and it has oeen pointed out that the K-/nratio at that energy is close to that of 19.2 GeV pp-
Suppose that d2a(Pp. P, 0) depends only on the square of the 4-momentum transfer, t. from the incident proton to the outgoing pion, and on the pion’s transverse momentum, PT q P. .Q. At high
collisions P/‘Pmax,
energies,
[2] when plotted as a function of where P is the meson momentum and P max its kinematical limit. Actually, there appears to exist a separate scaling law for d2c/dS2dP for n- production (Pp = incident proton momentum) d2r(Pp,
P, B)/dS2dP = d2a(Po, P’, O’)/dfi’dP’
rP
(1)
one has
t -mi(l-P/Pp)
i.e.,
+ m,2(1-Pp/P)
t is only a function of P/Pp
viation of t from 0,
which vanishes
- P$P/Pp, and P+
(3) is -m2m 2,/2P Pr
quadratically
P
(3)
(the de-
P for small
for Pp m-m and
rO
d20/d%lP depends only on the meson P/Pp and the transverse momentum = P. 8. This law is evident from fig. 1. which
In words:
elasticity PT
the curves for d20(pA +n-. . . ) dS2dP 19.2 GeV protons [3] for 0 = 12.5, 20, 30, 40, 50, 60 and 70 mrad, together with the points from the 70 GeV p-Al collisions. For the 70 GeV points, 8 has been replaced by 0.70/19.2. Obviously, the agreements between the 70 GeV points (error f 30% [4]) and the 19.2 GeV points is complete below P/P = 0.8. For larger elasticities, the 70 GeV po?nts come higher. This effect appears to be due to the kinematical limit and should disappear if one uses P/P,, instead of P/Pp. However, the kinematical limit is smeared out by the nucleons’ Fermi motion inside the aluminium nucleus. For a free pn collision, it would occur at a missing mass of 2mp (Pmax = Pp - 1.4 GeV), while for a free pp collision, it would occur at 2mp + mn (or, more realistically, at mp + mA, Pmax = Pp - 2.0 GeV). Below P/Pp = 0.8, however, it appears that the scaling law is better fulfilled for P/P, than for Theoretically, this is rather plausible. P/pmax. contains
d3
OL
0’5
a8
P/Pp
O9
Fig. 1. Differential cross sections for r- and K- production in 19.2 GeV/c p-Al collisions 131. plotted verAlso given are the measuresus the elasticity P/P with their producments of 70 GeV/c p- R’1 collisions, tion angles multiplied by 70/19.2. The dashed curve refers to 0 mrad at 70 GeV/c [l], the dash-dotted curves to the calculations of Hagedorn and Ranft [5] at 30 and 300 GeV. 663
Volume 29B, number 10
PHYSICS
fixed, whereas the difference between P/Pp max and P/Pp vanishes only linearly). One can also test the assumption that d20(Pp, P, B)/dS2dP depends only on t, i.e., that all PT- dependence comes through (3). This is not the case. At 19.2 GeV, P = 16 GeV and 0 = = 12.5 mrad give the same t-value as P = 10 GeV and .Q= 40 mrad, whereas the cross sections of these two points differ by a factor 47. Before applying the scaling law to other reactions and energies, one should note that it cannot possibly be correct for small values of P/P . Integrating (1) over dW (- 2nPT dP’/P2), we 07Jtain
LETTERS
P/P
du(Pp, P)/dP
=($
i.e., da(Pp)/dP also over P,
da(Po, P’)/dP’
decreases
like l/Pi.
atot
,
Integrating
!nn - u(p) ) =
K’ilI’
i
0157
01
005
1
Fig. 3. The K+/n.’
(5)
we see that the whole integral goes to zero as l/P . For constant atot( this would mean thatpthe 7r- multiplicity tends to zero as l/Pp,
Fig. 2. Differential cross sections for X+ and Kt production in 19.2 p-Al collisions [3]. as function of the elasticity P/Pp. To obtain the cross sections at primary momentum Pp. one multiplies the angles by 19. 2/Pp.
664
02
18 August 1969
ratio at production Pp mrad.
angle 12.5 19.2/
whereas experimentally it certainly increases with Pp. Therefore there must be a region below P/Pp = 0.5 where du(Pp, P)/dP decf;egases much more slowly, roughly like (PO/P,) * . Remember that the average pion elasticity goes to zero as the multiplicity increases. For outgoing protons of course, a constant elasticity would imply an extra factor Pp/Po on the right hand side of eq. (1). Fig. 2 gives the 7~~and K+ production spectra of Allaby et al. [3] for 19.2 GeV p-Al collisions. In the region 0.5 EdP/Pp e: 0.8 the 7r+ spectra should be identical with the r+ spectra of 70 GeV p-Al collisions, provided the angles are reduced by a factor 19.2/70 = 0.274. For K’, the scaling law is less reliable. However, since the kinematic limit Pmax is closer to Pp for K+ than for K-, one can hope that the scaling law works better for Kf than for K-. The K+/n+ ratio for 8 = 12.5 mrad x 19.2/Pp is shown in fig. 3. Contrary to the K-/n- ratio, it is a rising function of P/Pp up to P/Pp = 0.8. Finally, we compare the thermodynamic model [5,6] with the scaling law. In fig. 1, we have included the predicted 0 mrad 7~~production spectra at 30 and 300 GeV, multiplied with the experimental ratio of r- production at 12.5 mrad in pA1 and pp collisions (this ratio varies between 8 and 10). We see that the ther modynamic model does not exhibit the scaling law. In particular, the predicted flux of fast rat 300 GeV is one order of magnitude too large.
Volume 29B, number 10
PHYSICS
At larger even worse,
disagreement becomes since the experimental pAl/pp ratio increases with angle (au effect of multiple collisions inside the nucleus). For n+ production, the situation is similar.
18 August 1969
References 1. Bushnin et al., Phys. Letters 29B (1969) 48. 2. J. V. Allaby et al., paper submitted to the 14th Intern. Conf. on High-energy physics, Vienna, 1968 (unpublished). 3. J. V. Allaby et al., to be published. 4. Yu. D. Prokoshkin, private communication by J.-P. Stroot. 3, R. Hagedorn and J. Ranft, Suppl. Nuovo Cim. 6 (1968) 169. 6. J. Ranft. Rutherford Lab. Report RHEL/R 165 (1968).
We wish to thank the authors of ref. 2 for giving us their unpublished results. Drs. R. Ha-
gedorn and K. Schliipmann have kindly with explanations and suggestions.
LETTERS
helped us *****
665