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Absorptive effects in the diffractive dissociation of nucleons
Volume 59B, number 4
PHYSICS LETTERS
ABSORPTIVE EFFECTS IN THE DIFFRACTIVE
24 November 1975
D I S S O C I A T I O N O F N U C L E O N S ¢r
E.L. BERGER and P. PIRIL,~
High Energy Physics Division, Argonne National Laboratory, Argonne, IIL 60439, USA Received 26 August 1975 Absorptive effects are found to be important in diffraction dissociation. For pp ~ (mr+)p and np ~ (pit-)p, at small excitation mass, a significant dip structure is created in the production momentum transfer distribution do/dtdM near t = -0.3 (GeV/c) 2, in agreement with data from Fermilab and the CERN-ISR. The mass-slope correlation, seen in the data, is also reproduced. Most other distributions are affected much less by absorption.
Recent studies of reactions pp ~ pnTr+ at the CERN-ISR [1] and np-~ p p n - at Fermilab [2] have shown that the momentum transfer dependence in diffractive production of a low-mass (Nlr) state is clearly different from that observed in elastic scattering. In elastic scattering at ISR energies [3], do/dr falls in featureless, roughly exponential fashion by over 6 orders of magnitude from its maximum at t = 0, before encountering a sharp minimum at It[ ~ 1.4 (GeV/c) 2. When recast in impact parameter language, these data imply that diffractive elastic scattering is a central process [3], concentrated about zero impact parameter. In contrast, in inelastic diffraction at small values of excitation mass, M, the Fermilab [2] and ISR [1] data show a dip or at least a break in do/dtdMat Itl ~ 0.2 (GeV/c) 2, (after a precipitous fall from the maximum at t = 0). A dip at such small Itl implies peripheral structure in impact parameter [4], with the cross-section peaking at about 1 fermi. Although a break near It I ~ 0.2 (GeV/c) 2 had been observed in some lower energy data [5], below 30 GeV/c, the possibility existed that the effect was a non-asymptotic, non-diffractive phenomenon. The ISR and Fermilab results demonstrate the diffractive nature of the inelastic structure. These new results are not reproduced by the unabsorbed pion-exchange Deck model [6] which had provided a successful picture of many features of earlier data. In particular, according to the model [6], the distribution in momentum transfer to the diffractively scattered proton should resemble that for elastic scattering, thus showing no structure at Itl ~ 0.2 Work performed under the auspices of the United States Energy Research and Development Administration.
(GeV/c) 2. A related problem concerns the variation with mass M of the small t slope of do/dtdM. A marked decrease of slope is observed in the data as M is increased above the threshold value (m~r + mN). While explained qualitatively by the model, this "mass slope" correlation has not been satisfactorily reproduced quantitatively in calculations to date [7]. In addition to these difficulties in practice, there are questions of principle which lead on to reexamine the model. In inelastic two-body and quasi-two-body reactions, absorptive corrections [8] are found to be often important. Such effects should a priori be included also in a proper calculation of two to threebody reactions. The role, indeed necessity of such absorption terms is further apparent when one adopts the optical interpretation of diffraction dissociation [9]. Since absorptive effect in two-body reactions often lead to structure near [t I ~ 0.2 (GeV/c) 2 in s-channel spin nonflip amplitudes [8], the motivation for seeking an absorptive explanation of the structure in inelastic diffraction dissociation is increased. In this letter we report results of an explicit absorbed pion exchange Deck model. A more detailed description of the results will be published elsewhere [10]. We achieve good agreement with the recent FNAL and ISR data [1,2]. Our results support the Deck interpretation of diffractive threshold enhancements, as well as the necessity for absorptive corrections to the model. The unabsorbed pion-exchange Deck amplitude which we employ in our calculations is the strippeddown expression Ao(S 1 , t 2, t 1) = ign(t2)cInPs 1 exp(B I t 1).
(1) 361
Volume 59B, number 4
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P2 s2
iP3
24 November 1975
two mass-shell delta functions, consistently dropping correction terms of order (MN~r/x/s), we obtain
Pl
sI
--
i
Aab s - 87t2Sl 2 t'
I'
2
f dZq3TA~P(s12,t3)Ao(S,t'2,S'l,S'2) (5)
l
PN
with
P~
t3 -- --qIT
(6)
Fig. 1. (a) The pion exchange Deck-graph for Np ~ (Nlr) p. The wavy line labelled P denotes the full elastic scattering
t'l = tl --(q~T + 2q3T " P l T )
(7)
amplitude. (b) Diagram illustrating the kinematic variables for the absorptive graph. All wavy lines denote elastic scattering.
t2 = t2 -- ~22 (q3T -- 2q3T" P2T )
(8)
x 2 = P2L/Pcm = 2P2L/N/~
(9)
(G)
(b)
Variables are indicated in fig. l(a). The factor is 1 oTrP exp(B 1 t l ) represents the essential subenergy and exponential momentum transfer dependence of the (off-shell) rrp diffractive elastic scattering amplitude imbedded in the unabsorbed amplitude, fig. l(a); o~P is the total ~rp cross-section. The function g~r(t2) is the pion propagator and coupling function. Explicitly,
g.(t2)=~¢C2g~
exp(t2 -M2)/(M2 - t2 ).
(2)
We ignore Reggeization of the pion, which would introduce dependence on s 2 into eq. (1), as well as a t 2 dependent phase variation. These omissions are inessential to the major conclusions of our investigation [10]. Arguments based on both the optical approach to diffraction dissociation and the double-peripheral model lead to the conclusion that the important absorptive amplitude Aab s is the "final state" rescattering term, represented in fig. l(b). This is correct to the extent that the ratio (MN~r/~4~-) is a small parameter, which is true for low mass diffractive excitation. We use on-shell propagators for the intermediate state particles N and p. Detailed arguments are presented in ref. [10]. We adopt a conventional asymptotic exponential parametrization of the elastic state rescattering Np Ael (S12, t 3) = iSl2oNP exp(B3t3),
(3)
s12 = (Pl + P2 )2"
(4)
A loop integral must be done to evaluate Aab s. After some uninspiring algebra and integrating to remove 362
,
1
2
The two-dimensional transverse vectors Pl T and P2T are the transverse components of the momenta of final state particles p and N, respectively; P2L is the center of mass longitudinal component of N. The eq. (5) is rather general, independent of the specific choices for A o and A el made above. Substituting the parametrizations (1) and (3) into eq. (5), we can perform one angular integration analytically; the remaining one-dimensional integral must be done numerically [10]. As can be expected, the absorption term (5) is smaller than the unabsorbed amplitude (1) for t 1 t 2 ~ 0. On the other hand the t 1 and t 2 dependence ofAab s is weaker than that o f A o. Therefore the full amplitude A = A o + Aab s
(12)
vanished for some specific values of t 1 and t 2 in our model (the real parts have been set to zero). We now select the specific dissociation process pp -+ p(nlr +) at 100 GeV/c in order to specify parameters. For small values of Mnrr+, the typical value of ~ 1 is V~-]2 and s12 ~ s. Thus, the elastic scattering parameters B 1 = 4.5 (GeV/e) -2 and B 3 = 5.5 (GeV/e) -2 are reasonable, with onP ~ 38 mb = 98 GeV - 2 . Because these parameters are taken directly from measured elastic and total cross-section data, the dip positions we calculate for do[dtdM, and the slopes are absolute predictions. In figs. 2 and 3, and table 1, we
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PHYSICS LETTERS
compare results obtained from our unabsorbed models for pp ~ p(nlr +) at 100 GeV/c. The same apply to np ~ (prr-)p. The most dramatic effect of absorption is observed in the m o m e n t u m transfer distribution do/dMdt for production of the low-mass diffractive enhancement. We select the mass value MnTr+ = 1.3 GeV because this corresponds roughly to the center of the peak in mass generated by the model. In the unabsorbed model, fig. 2 (a), do/dt falls roughly exponentially, whereas in the absorbed model, fig. 2(b), a pronounced break structure is observed near Itl ~ 0.4 (GeV/c) 2. This structure is caused by the cancellation between A o and Aab s in eq. (12). Near the threshold value M = (m N + m~), d o / d t d M shows a pronounced dip near Itl = 0.3 (GeV/c) 2 in the model. This dip gradually transforms into the break structure seen in fig. 2(b). Perceptible structure disappears altogether at higher M. These results agree at least qualitatively with experiment [ 1 , 2 ] . We decompose our full amplitude for the process pp ~ Mp into partial amplitudes for each of the angular m o m e n t u m and helicity states which contribute to
'o
,
,
,
I
,
24 November 1975
Table 1 Locations in Itl of the zeros in the amplitudes for producing various states of s-channel spin and helicity [L, hs] are given as a function of massM of the (nrr÷) system, from our absorbed Deck model of pp ~ p(mr +) at 100 GeV/c. The first column lists values of [L, hs]. Zeros for three different mass values are listed in columns 2-4. Dip positions in (GeV/c)2 as a function of mass
Lh s
Mmr+ = 1.1 GeV
1.3 GeV
1.5 GeV
00 10 11 20 21 22
0.29 0.33 0.48 0.17 0.63 0.66
0.37 0.47 0.67 0.33 > 1 0.93
0.43 0.63 0.80 0.45 > 1 > 1
the system M. We investigate how absorption affects each of these partial amplitudes individually. In fig. 2, for both the absorbed and unabsorbed differential cross-sections, we show the total do/dtdM, as well as a decomposition of this quantity into the portions for each of the spin and helicity states which contribute to the system of massM. The recoil system M in
uo4 ~ m
,
]
pp ~p(nTr*)
E
,oo Gev/c
\ I0 ~
_
XX UNABSORBED
~
XX,XS-CHANNEL HELICITIEE
~
O0~.,X~j TOTAL
,o
\\ \
I P I p p ~ p (n~r +)
I
:o?::';°ev ABSORBEO
,o OTAL
I0 -- 22
b
-~ I0
20
/ 0.1
0.2
0.3
0.4
Itl (GeV/c) 2
0.5 0.6
0,7
, 0.1
0.2
0.3 0.4 0.5 Itl (GeV/c) z
0.6
0.7
Fig. 2. Doubly differential cross-section da/dtdM for pp ~ p(mr÷) at 100 GeV/c and its decomposition into partial cross-sections for individual spin-helicity states of the (n~r+) system. The states are labelled by the orbital angular momentum L and s-channel helicity hs of the (n~r+) system, with the intrinsic spins of the nucleons ignored. For hs 4: O, curves denote the sum of cross-sections for (L, hs) and (L, -hs). Results for the unabsorbed model are given in part (a); these for the absorbed model in (b). The overall normalization is arbitrary, but the relative normalization of curves within part (a) and with part (b) is fixed by the model. The relative normalization between parts (a) and (b) is also determined by the model. 363
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pp ~ pM is not a state of unique spin, although the s-wave component is certainly dominant. Our Deck amplitudes specify the relative strengths of the different spin states. In fig. 2, the partial wave decomposition is presented in terms of s-channel helicities of system M (the quantization z axis is the direction of the final proton). We have ignored intrinsic spins of particles. Therefore the spin, helicity labels refer to orbital angular momentum only in the rest system of M. The quantity labeled o 1,1 is the sum o f o 1,1 and o 1,-1 . A comparison of figs. 2 (a) and 2 (b) shows that the amplitudes with s-channel helicity I~1 ~> 1 are not much absorbed. The states with Xs = 0 suffer t h e greatest absorption. At M,r÷n = 1.3 GeV, in the state [L = 0, Xs = 0], a zero of the full amplitude is generated at [tl = 0.37 (GeV/c) 2. The zero locations of some other amplitudes are listed in tal)le 1, for three values of mass. These explicit zero locations correspond crudely but not exactly to positions conjectured in ad-hoc geometric models [4, 11 ]. Most notable is the fact that all zero locations move to larger [tl asM increases. In impact parameter language, this means that for fixed (L, ;k) states of higher mass are produced less peripherally in our model. As a result of this motion of the zero location withM, structure in do/dtdM is washed out if a relatively large interval (e.g., AM 0.5 GeV) in M is averaged or integrated over. Thus, even doL,Xs/dtdM for a specific [1,, Xs] state may show little or no structure in t if too large an interval in M is selected in the data. High statistics data with good mass resolution are crucial. It is useful to recast the results of the above paragraphs in impact parameter language. The elastic scattering amplitude and the unabsorbed 7r exchange Deck amplitude have roughly exponential dependence on momentum transfers t 3 and t ] , respectively (c.f. eqs. (1) and (3)). Therefore, both are approximately Gaussian functions of impact parameter, and represent "central" collisions. Absorption generates a zero at t 1 ~ - 0 . 3 (GeV/c) 2, as discussed. When translated to impact space, this means that the central partial waves are depleted. At small values of the mass of the excited system, the resulting diffractive inelastic absorbed Deck amplitude has a peripheral impact parameter structure. In fig. 3 we present the variation withMn~r÷ of the small t slopes of do/dtdMn~r.. The slope b is defined 364
24 November 1975
{(]I) nj~__p~l.mp. ' \ \
24
1
pp ~pnTr* I00 GeVc
20
-+- FNAL- Rochesler- -
'~
16
Northwestern
\\",,~'~,-F-TOTAL
12 B b
4
-
÷ ÷
(b)
o,
241 \ 20 16 12
I
-
- - ABSORBED - - - UNABSORBED
~ __\ \ N \ ~
8 4
I
I
1.2 1.4
I
1.6
I
I
1.8 2.0
Mnlr+ (GeV)
Fig. 3. Logarithmic slope of the momentum transfer tpp dependence for nucleon dissociation Np ~ (Nlr)p at 100 GeV/c as a function of the mass of the (Nn) system. Results are shown for (a) the total cross-section do/dtdM and (b) the (NTr) s-wave component. Data from ref. [21on neutron dissociation are listed in (a). through the parametrization do dtdM = exp(bt). Fits were made over the range 0.05 ~< Itl ~< 0.2 (GeV/c) 2. The comparison of absorbed and unabsorbed results in fig. 3(a) indicates that absorption increases the threshold value of the slope by roughly 9 units, but causes only a modest increase at Mn~r+ ~ 2 GeV. Absorption accentuates the pronounced massslope correlation already present in the unabsorbed model. In fig. 3(a) we have placed slope values obtained in a recent Fermilab experiment [2 ] on np -* (prr-)p. The excellent agreement with our absolute predictions in the mass range up to 1.4 GeV seems to support strongly the Deck interpretation of kinematic nature of low-mass threshold enhancement, as well as the need for absorptive corrections in the model. At
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PHYSICS LETTERS
largerMp7r - , from 1.5 to 2.0 GeV, the model disagrees with data [2]. However, we call attention to the fact that in this region, obvious resonance effects are observed in the data. They are not included in the model. The resonances appear to be produced with a t slope which is substantially smaller than that o f the Deck background. In fig. 3(b), we present the slope o f do°O/dtdM. This is the differential cross-section for producing the s-wave part only o f the Deck enhancement. We observe that there is a pronounced mass-slope correlation in b o t h the unabsorbed and absorbed results. The resuits o f fig. 3(b) demonstrate that in our model the mass-slope correlation is present already in the dominant L = 0 partial wave, all by itself. An alternative interpretation [4, 11 ] o f the mass-slope correlation has been suggested repeatedly. In these approaches, the mass-slope correlation owes its existence to the presumed growth w i t h M of higher L and ~'s states, produced with systematically smaller slopes b. F o r a given (L, Xs), the slope b is assumed not to vary with M. While perhaps intuitively appealing, the approach suffers from a surplus of undetermined parameters and has not been tested quantitatively. It seems 1o us unlikely that the data (particularly the distribution in the s-channel azimuthal angle ¢s) would tolerate an increase o f ~s with M sufficiently rapid to achieve the result desired. In any case, the issue can be resolved experimentally. Some data on the mass-slope correlation in the L = 0 partial wave in ~rp ~ (p~r) p at 40 GeV/c have been published [12]. A decrease of slope from 12 + 1 to 7 -+ 1 (GeV/c) - 2 is observed [c.f. table 1, p. 157 o f r e f . 12] fromMp~ r = 1.1 to 1.3 GeV. Therefore, available results, while sketchy, surely are consistent with our viewpoint that the mass-slope correlation is an intrinsic property of each partial wave. As just summarized, the most dramatic effects of absorption are seen in the production t-distribution do/dMdt. The integrated mass distribution do/dM shows little modification in shape in the small M region o f interest here. Absorption does not yield the sharpening effect which might be desirable in improving fits to data on do/dM. The overall integrated crosssection is reduced to roughly one-half its unabsorbed value. At small t, the decay angular distributions in the rest frame o f M are essentially unchanged b y absorption. A t larger Itl, [ ~ 0.3 (GeV/c)2 ], where the absorption term and the unabsorbed amplitude are of
24 November 1975
comparable magnitude, the predicted partial wave structure is modified appreciably. The fig. 2 may be consulted for numerical estimates. Insofar as statistics allow, it is advisable to perform experimental partial wave analyses of the system M in several different regions of Itl. If all t values are included, little difference is seen between the absorbed and unabsorbed models. Because meson-nucleon total cross-sections are smaller than onP, we expect absorptive effect to be less important in diffractive reactions induced by a meson beam, e.g., 7rp ~ A 1 p ~ (prO p; Kp ~ Qp (K*~r)p; and lrp ~ 7r(NTr). The position of the dip or break in do/dt moves out to larger Itl, as shown explicitly in ref. [10]. Certain predictions based on the unabsorbed Deck model must be reevaluated in the light of our present conclusion that absorptive corrections are necessary. Several of these issues have already been treated above and some more are discussed in ref. [10]. We only remark here that no major changes appear in the crossover predictions [13], whereas the zero and change of sign in the imaginary part of the diffractive amplitude around Itl = 0.3 (GeV/c) 2 should manifest itself as a similar change of sign in the polarization o f the recoiling proton.
References
[ 1] CERN-Hamburg-Orsay-Vienna ISR Collaboration. E.
[2]
[3] [4]
[5]
[6 ]
Nagy et al., CERN report submitted to the 17th Intern. Conf. on High energy physics, London, July 1974. (pp p(nzr÷) at x/~-= 23 to 62 (GeV). Northwestern-Rochester-Fermilab Collaboration, J. Biel et al., reports submitted to the Intern. Conf. on High energy physics, Palermo, June 1975. (np ~ pplr- from 50 to 300 GeV/c). U. Amaldi, rapporteurs report, Aix-en-Provence Conf. 1973, J. de Physique C1 (1973) 241. N. Sakai and J.N.J. White, Nucl. Phys. B59 (1973) 511; Lett. Nuovo Cim. 8 (1973) 618; M. Jacob and R. Stroynowski, Nucl. Phys. B82 (1974) 189; G. Kane, Acta Phys. Pol. B3 (1972) 845. R.M. Edelstein et al., Phys. Rev. D5 (1972) 1073; J.V. Beaupr6 et al., Nucl. Phys. B66 (1973) 93; J. Hanlon et al., Vanderbilt report VAND-HEP 74(2); Scandinavian Bubble Chamber Collaboration, unpublished data on pp ~ pmr÷ at 19 GeV/c, as quoted by G. Kane, ibid. For a review and list of references to earlier work, consult E.L. Berger, A critique of the Reggeized Deck model, 365
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Argonne report ANL/HEP 7506, Daresbury Study Weekend Series no. 8, eds. J.B. Dainton and A.J.G. Hey; E.L. Berger, Phys. Rev. 166 (1968) 1525; 179 (1969) 1567. I7l H. Mtettinen and P. Piril~',Phys. Lett. 40B (1972) 127. 181 J.D. Jackson, Rev. Mod. Phys. 42 (1970) 12; F.S. Henyey, G.L. Kane, J. Pumplin and M. Ross, Phys. Rev. 182 (1969) 1579. I9l M.L. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857; E.L. Feinberg and I. Pomeranchuk, Suppl. Nuovo Cim. 3 (1956) 652; A. Bialas, W. Czyz and A. Kotanski, Ann. of Phys. (NY) 73 (1972) 439; J. Pumplin, Phys. Rev. D4 (1971) 3482; D7 (1973) 795.
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24 November 1975
[10] E.L. Berger and P. Piril/i, Absorptive effects in exclusive diffraction dissociation, Argonne report ANL-HEP-PR75-27, June 1975, submitted to Phys. Rev. Different approaches are discussed by V.A. Tsarev, Phys. Rev. D11 (1975) 1864, and G. Cohen-Tannoudji and U Maor, Phys. Lett. 57B (1975) 253. [11] S. Humble, Nucl. Phys. B76 (1974) 137; B86 (1975) 285. [12] Yu.M. Antipov et al., Nucl. Phys. B63 (1973) 153. We are grateful to R. Cutler for calling our attention to table 1 of this reference. [131 E.L. Berger, Systematics of cross-over effects in inelastic diffraction dissociation, Argonne report ANL/HEP 7464, to be published in Phys. Rev. D l l (1975) 3214.
PHYSICS LETTERS
ABSORPTIVE EFFECTS IN THE DIFFRACTIVE
24 November 1975
D I S S O C I A T I O N O F N U C L E O N S ¢r
E.L. BERGER and P. PIRIL,~
High Energy Physics Division, Argonne National Laboratory, Argonne, IIL 60439, USA Received 26 August 1975 Absorptive effects are found to be important in diffraction dissociation. For pp ~ (mr+)p and np ~ (pit-)p, at small excitation mass, a significant dip structure is created in the production momentum transfer distribution do/dtdM near t = -0.3 (GeV/c) 2, in agreement with data from Fermilab and the CERN-ISR. The mass-slope correlation, seen in the data, is also reproduced. Most other distributions are affected much less by absorption.
Recent studies of reactions pp ~ pnTr+ at the CERN-ISR [1] and np-~ p p n - at Fermilab [2] have shown that the momentum transfer dependence in diffractive production of a low-mass (Nlr) state is clearly different from that observed in elastic scattering. In elastic scattering at ISR energies [3], do/dr falls in featureless, roughly exponential fashion by over 6 orders of magnitude from its maximum at t = 0, before encountering a sharp minimum at It[ ~ 1.4 (GeV/c) 2. When recast in impact parameter language, these data imply that diffractive elastic scattering is a central process [3], concentrated about zero impact parameter. In contrast, in inelastic diffraction at small values of excitation mass, M, the Fermilab [2] and ISR [1] data show a dip or at least a break in do/dtdMat Itl ~ 0.2 (GeV/c) 2, (after a precipitous fall from the maximum at t = 0). A dip at such small Itl implies peripheral structure in impact parameter [4], with the cross-section peaking at about 1 fermi. Although a break near It I ~ 0.2 (GeV/c) 2 had been observed in some lower energy data [5], below 30 GeV/c, the possibility existed that the effect was a non-asymptotic, non-diffractive phenomenon. The ISR and Fermilab results demonstrate the diffractive nature of the inelastic structure. These new results are not reproduced by the unabsorbed pion-exchange Deck model [6] which had provided a successful picture of many features of earlier data. In particular, according to the model [6], the distribution in momentum transfer to the diffractively scattered proton should resemble that for elastic scattering, thus showing no structure at Itl ~ 0.2 Work performed under the auspices of the United States Energy Research and Development Administration.
(GeV/c) 2. A related problem concerns the variation with mass M of the small t slope of do/dtdM. A marked decrease of slope is observed in the data as M is increased above the threshold value (m~r + mN). While explained qualitatively by the model, this "mass slope" correlation has not been satisfactorily reproduced quantitatively in calculations to date [7]. In addition to these difficulties in practice, there are questions of principle which lead on to reexamine the model. In inelastic two-body and quasi-two-body reactions, absorptive corrections [8] are found to be often important. Such effects should a priori be included also in a proper calculation of two to threebody reactions. The role, indeed necessity of such absorption terms is further apparent when one adopts the optical interpretation of diffraction dissociation [9]. Since absorptive effect in two-body reactions often lead to structure near [t I ~ 0.2 (GeV/c) 2 in s-channel spin nonflip amplitudes [8], the motivation for seeking an absorptive explanation of the structure in inelastic diffraction dissociation is increased. In this letter we report results of an explicit absorbed pion exchange Deck model. A more detailed description of the results will be published elsewhere [10]. We achieve good agreement with the recent FNAL and ISR data [1,2]. Our results support the Deck interpretation of diffractive threshold enhancements, as well as the necessity for absorptive corrections to the model. The unabsorbed pion-exchange Deck amplitude which we employ in our calculations is the strippeddown expression Ao(S 1 , t 2, t 1) = ign(t2)cInPs 1 exp(B I t 1).
(1) 361
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PHYSICS LETTERS
P2 s2
iP3
24 November 1975
two mass-shell delta functions, consistently dropping correction terms of order (MN~r/x/s), we obtain
Pl
sI
--
i
Aab s - 87t2Sl 2 t'
I'
2
f dZq3TA~P(s12,t3)Ao(S,t'2,S'l,S'2) (5)
l
PN
with
P~
t3 -- --qIT
(6)
Fig. 1. (a) The pion exchange Deck-graph for Np ~ (Nlr) p. The wavy line labelled P denotes the full elastic scattering
t'l = tl --(q~T + 2q3T " P l T )
(7)
amplitude. (b) Diagram illustrating the kinematic variables for the absorptive graph. All wavy lines denote elastic scattering.
t2 = t2 -- ~22 (q3T -- 2q3T" P2T )
(8)
x 2 = P2L/Pcm = 2P2L/N/~
(9)
(G)
(b)
Variables are indicated in fig. l(a). The factor is 1 oTrP exp(B 1 t l ) represents the essential subenergy and exponential momentum transfer dependence of the (off-shell) rrp diffractive elastic scattering amplitude imbedded in the unabsorbed amplitude, fig. l(a); o~P is the total ~rp cross-section. The function g~r(t2) is the pion propagator and coupling function. Explicitly,
g.(t2)=~¢C2g~
exp(t2 -M2)/(M2 - t2 ).
(2)
We ignore Reggeization of the pion, which would introduce dependence on s 2 into eq. (1), as well as a t 2 dependent phase variation. These omissions are inessential to the major conclusions of our investigation [10]. Arguments based on both the optical approach to diffraction dissociation and the double-peripheral model lead to the conclusion that the important absorptive amplitude Aab s is the "final state" rescattering term, represented in fig. l(b). This is correct to the extent that the ratio (MN~r/~4~-) is a small parameter, which is true for low mass diffractive excitation. We use on-shell propagators for the intermediate state particles N and p. Detailed arguments are presented in ref. [10]. We adopt a conventional asymptotic exponential parametrization of the elastic state rescattering Np Ael (S12, t 3) = iSl2oNP exp(B3t3),
(3)
s12 = (Pl + P2 )2"
(4)
A loop integral must be done to evaluate Aab s. After some uninspiring algebra and integrating to remove 362
,
1
2
The two-dimensional transverse vectors Pl T and P2T are the transverse components of the momenta of final state particles p and N, respectively; P2L is the center of mass longitudinal component of N. The eq. (5) is rather general, independent of the specific choices for A o and A el made above. Substituting the parametrizations (1) and (3) into eq. (5), we can perform one angular integration analytically; the remaining one-dimensional integral must be done numerically [10]. As can be expected, the absorption term (5) is smaller than the unabsorbed amplitude (1) for t 1 t 2 ~ 0. On the other hand the t 1 and t 2 dependence ofAab s is weaker than that o f A o. Therefore the full amplitude A = A o + Aab s
(12)
vanished for some specific values of t 1 and t 2 in our model (the real parts have been set to zero). We now select the specific dissociation process pp -+ p(nlr +) at 100 GeV/c in order to specify parameters. For small values of Mnrr+, the typical value of ~ 1 is V~-]2 and s12 ~ s. Thus, the elastic scattering parameters B 1 = 4.5 (GeV/e) -2 and B 3 = 5.5 (GeV/e) -2 are reasonable, with onP ~ 38 mb = 98 GeV - 2 . Because these parameters are taken directly from measured elastic and total cross-section data, the dip positions we calculate for do[dtdM, and the slopes are absolute predictions. In figs. 2 and 3, and table 1, we
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compare results obtained from our unabsorbed models for pp ~ p(nlr +) at 100 GeV/c. The same apply to np ~ (prr-)p. The most dramatic effect of absorption is observed in the m o m e n t u m transfer distribution do/dMdt for production of the low-mass diffractive enhancement. We select the mass value MnTr+ = 1.3 GeV because this corresponds roughly to the center of the peak in mass generated by the model. In the unabsorbed model, fig. 2 (a), do/dt falls roughly exponentially, whereas in the absorbed model, fig. 2(b), a pronounced break structure is observed near Itl ~ 0.4 (GeV/c) 2. This structure is caused by the cancellation between A o and Aab s in eq. (12). Near the threshold value M = (m N + m~), d o / d t d M shows a pronounced dip near Itl = 0.3 (GeV/c) 2 in the model. This dip gradually transforms into the break structure seen in fig. 2(b). Perceptible structure disappears altogether at higher M. These results agree at least qualitatively with experiment [ 1 , 2 ] . We decompose our full amplitude for the process pp ~ Mp into partial amplitudes for each of the angular m o m e n t u m and helicity states which contribute to
'o
,
,
,
I
,
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Table 1 Locations in Itl of the zeros in the amplitudes for producing various states of s-channel spin and helicity [L, hs] are given as a function of massM of the (nrr÷) system, from our absorbed Deck model of pp ~ p(mr +) at 100 GeV/c. The first column lists values of [L, hs]. Zeros for three different mass values are listed in columns 2-4. Dip positions in (GeV/c)2 as a function of mass
Lh s
Mmr+ = 1.1 GeV
1.3 GeV
1.5 GeV
00 10 11 20 21 22
0.29 0.33 0.48 0.17 0.63 0.66
0.37 0.47 0.67 0.33 > 1 0.93
0.43 0.63 0.80 0.45 > 1 > 1
the system M. We investigate how absorption affects each of these partial amplitudes individually. In fig. 2, for both the absorbed and unabsorbed differential cross-sections, we show the total do/dtdM, as well as a decomposition of this quantity into the portions for each of the spin and helicity states which contribute to the system of massM. The recoil system M in
uo4 ~ m
,
]
pp ~p(nTr*)
E
,oo Gev/c
\ I0 ~
_
XX UNABSORBED
~
XX,XS-CHANNEL HELICITIEE
~
O0~.,X~j TOTAL
,o
\\ \
I P I p p ~ p (n~r +)
I
:o?::';°ev ABSORBEO
,o OTAL
I0 -- 22
b
-~ I0
20
/ 0.1
0.2
0.3
0.4
Itl (GeV/c) 2
0.5 0.6
0,7
, 0.1
0.2
0.3 0.4 0.5 Itl (GeV/c) z
0.6
0.7
Fig. 2. Doubly differential cross-section da/dtdM for pp ~ p(mr÷) at 100 GeV/c and its decomposition into partial cross-sections for individual spin-helicity states of the (n~r+) system. The states are labelled by the orbital angular momentum L and s-channel helicity hs of the (n~r+) system, with the intrinsic spins of the nucleons ignored. For hs 4: O, curves denote the sum of cross-sections for (L, hs) and (L, -hs). Results for the unabsorbed model are given in part (a); these for the absorbed model in (b). The overall normalization is arbitrary, but the relative normalization of curves within part (a) and with part (b) is fixed by the model. The relative normalization between parts (a) and (b) is also determined by the model. 363
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pp ~ pM is not a state of unique spin, although the s-wave component is certainly dominant. Our Deck amplitudes specify the relative strengths of the different spin states. In fig. 2, the partial wave decomposition is presented in terms of s-channel helicities of system M (the quantization z axis is the direction of the final proton). We have ignored intrinsic spins of particles. Therefore the spin, helicity labels refer to orbital angular momentum only in the rest system of M. The quantity labeled o 1,1 is the sum o f o 1,1 and o 1,-1 . A comparison of figs. 2 (a) and 2 (b) shows that the amplitudes with s-channel helicity I~1 ~> 1 are not much absorbed. The states with Xs = 0 suffer t h e greatest absorption. At M,r÷n = 1.3 GeV, in the state [L = 0, Xs = 0], a zero of the full amplitude is generated at [tl = 0.37 (GeV/c) 2. The zero locations of some other amplitudes are listed in tal)le 1, for three values of mass. These explicit zero locations correspond crudely but not exactly to positions conjectured in ad-hoc geometric models [4, 11 ]. Most notable is the fact that all zero locations move to larger [tl asM increases. In impact parameter language, this means that for fixed (L, ;k) states of higher mass are produced less peripherally in our model. As a result of this motion of the zero location withM, structure in do/dtdM is washed out if a relatively large interval (e.g., AM 0.5 GeV) in M is averaged or integrated over. Thus, even doL,Xs/dtdM for a specific [1,, Xs] state may show little or no structure in t if too large an interval in M is selected in the data. High statistics data with good mass resolution are crucial. It is useful to recast the results of the above paragraphs in impact parameter language. The elastic scattering amplitude and the unabsorbed 7r exchange Deck amplitude have roughly exponential dependence on momentum transfers t 3 and t ] , respectively (c.f. eqs. (1) and (3)). Therefore, both are approximately Gaussian functions of impact parameter, and represent "central" collisions. Absorption generates a zero at t 1 ~ - 0 . 3 (GeV/c) 2, as discussed. When translated to impact space, this means that the central partial waves are depleted. At small values of the mass of the excited system, the resulting diffractive inelastic absorbed Deck amplitude has a peripheral impact parameter structure. In fig. 3 we present the variation withMn~r÷ of the small t slopes of do/dtdMn~r.. The slope b is defined 364
24 November 1975
{(]I) nj~__p~l.mp. ' \ \
24
1
pp ~pnTr* I00 GeVc
20
-+- FNAL- Rochesler- -
'~
16
Northwestern
\\",,~'~,-F-TOTAL
12 B b
4
-
÷ ÷
(b)
o,
241 \ 20 16 12
I
-
- - ABSORBED - - - UNABSORBED
~ __\ \ N \ ~
8 4
I
I
1.2 1.4
I
1.6
I
I
1.8 2.0
Mnlr+ (GeV)
Fig. 3. Logarithmic slope of the momentum transfer tpp dependence for nucleon dissociation Np ~ (Nlr)p at 100 GeV/c as a function of the mass of the (Nn) system. Results are shown for (a) the total cross-section do/dtdM and (b) the (NTr) s-wave component. Data from ref. [21on neutron dissociation are listed in (a). through the parametrization do dtdM = exp(bt). Fits were made over the range 0.05 ~< Itl ~< 0.2 (GeV/c) 2. The comparison of absorbed and unabsorbed results in fig. 3(a) indicates that absorption increases the threshold value of the slope by roughly 9 units, but causes only a modest increase at Mn~r+ ~ 2 GeV. Absorption accentuates the pronounced massslope correlation already present in the unabsorbed model. In fig. 3(a) we have placed slope values obtained in a recent Fermilab experiment [2 ] on np -* (prr-)p. The excellent agreement with our absolute predictions in the mass range up to 1.4 GeV seems to support strongly the Deck interpretation of kinematic nature of low-mass threshold enhancement, as well as the need for absorptive corrections in the model. At
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largerMp7r - , from 1.5 to 2.0 GeV, the model disagrees with data [2]. However, we call attention to the fact that in this region, obvious resonance effects are observed in the data. They are not included in the model. The resonances appear to be produced with a t slope which is substantially smaller than that o f the Deck background. In fig. 3(b), we present the slope o f do°O/dtdM. This is the differential cross-section for producing the s-wave part only o f the Deck enhancement. We observe that there is a pronounced mass-slope correlation in b o t h the unabsorbed and absorbed results. The resuits o f fig. 3(b) demonstrate that in our model the mass-slope correlation is present already in the dominant L = 0 partial wave, all by itself. An alternative interpretation [4, 11 ] o f the mass-slope correlation has been suggested repeatedly. In these approaches, the mass-slope correlation owes its existence to the presumed growth w i t h M of higher L and ~'s states, produced with systematically smaller slopes b. F o r a given (L, Xs), the slope b is assumed not to vary with M. While perhaps intuitively appealing, the approach suffers from a surplus of undetermined parameters and has not been tested quantitatively. It seems 1o us unlikely that the data (particularly the distribution in the s-channel azimuthal angle ¢s) would tolerate an increase o f ~s with M sufficiently rapid to achieve the result desired. In any case, the issue can be resolved experimentally. Some data on the mass-slope correlation in the L = 0 partial wave in ~rp ~ (p~r) p at 40 GeV/c have been published [12]. A decrease of slope from 12 + 1 to 7 -+ 1 (GeV/c) - 2 is observed [c.f. table 1, p. 157 o f r e f . 12] fromMp~ r = 1.1 to 1.3 GeV. Therefore, available results, while sketchy, surely are consistent with our viewpoint that the mass-slope correlation is an intrinsic property of each partial wave. As just summarized, the most dramatic effects of absorption are seen in the production t-distribution do/dMdt. The integrated mass distribution do/dM shows little modification in shape in the small M region o f interest here. Absorption does not yield the sharpening effect which might be desirable in improving fits to data on do/dM. The overall integrated crosssection is reduced to roughly one-half its unabsorbed value. At small t, the decay angular distributions in the rest frame o f M are essentially unchanged b y absorption. A t larger Itl, [ ~ 0.3 (GeV/c)2 ], where the absorption term and the unabsorbed amplitude are of
24 November 1975
comparable magnitude, the predicted partial wave structure is modified appreciably. The fig. 2 may be consulted for numerical estimates. Insofar as statistics allow, it is advisable to perform experimental partial wave analyses of the system M in several different regions of Itl. If all t values are included, little difference is seen between the absorbed and unabsorbed models. Because meson-nucleon total cross-sections are smaller than onP, we expect absorptive effect to be less important in diffractive reactions induced by a meson beam, e.g., 7rp ~ A 1 p ~ (prO p; Kp ~ Qp (K*~r)p; and lrp ~ 7r(NTr). The position of the dip or break in do/dt moves out to larger Itl, as shown explicitly in ref. [10]. Certain predictions based on the unabsorbed Deck model must be reevaluated in the light of our present conclusion that absorptive corrections are necessary. Several of these issues have already been treated above and some more are discussed in ref. [10]. We only remark here that no major changes appear in the crossover predictions [13], whereas the zero and change of sign in the imaginary part of the diffractive amplitude around Itl = 0.3 (GeV/c) 2 should manifest itself as a similar change of sign in the polarization o f the recoiling proton.
References
[ 1] CERN-Hamburg-Orsay-Vienna ISR Collaboration. E.
[2]
[3] [4]
[5]
[6 ]
Nagy et al., CERN report submitted to the 17th Intern. Conf. on High energy physics, London, July 1974. (pp p(nzr÷) at x/~-= 23 to 62 (GeV). Northwestern-Rochester-Fermilab Collaboration, J. Biel et al., reports submitted to the Intern. Conf. on High energy physics, Palermo, June 1975. (np ~ pplr- from 50 to 300 GeV/c). U. Amaldi, rapporteurs report, Aix-en-Provence Conf. 1973, J. de Physique C1 (1973) 241. N. Sakai and J.N.J. White, Nucl. Phys. B59 (1973) 511; Lett. Nuovo Cim. 8 (1973) 618; M. Jacob and R. Stroynowski, Nucl. Phys. B82 (1974) 189; G. Kane, Acta Phys. Pol. B3 (1972) 845. R.M. Edelstein et al., Phys. Rev. D5 (1972) 1073; J.V. Beaupr6 et al., Nucl. Phys. B66 (1973) 93; J. Hanlon et al., Vanderbilt report VAND-HEP 74(2); Scandinavian Bubble Chamber Collaboration, unpublished data on pp ~ pmr÷ at 19 GeV/c, as quoted by G. Kane, ibid. For a review and list of references to earlier work, consult E.L. Berger, A critique of the Reggeized Deck model, 365
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Argonne report ANL/HEP 7506, Daresbury Study Weekend Series no. 8, eds. J.B. Dainton and A.J.G. Hey; E.L. Berger, Phys. Rev. 166 (1968) 1525; 179 (1969) 1567. I7l H. Mtettinen and P. Piril~',Phys. Lett. 40B (1972) 127. 181 J.D. Jackson, Rev. Mod. Phys. 42 (1970) 12; F.S. Henyey, G.L. Kane, J. Pumplin and M. Ross, Phys. Rev. 182 (1969) 1579. I9l M.L. Good and W.D. Walker, Phys. Rev. 120 (1960) 1857; E.L. Feinberg and I. Pomeranchuk, Suppl. Nuovo Cim. 3 (1956) 652; A. Bialas, W. Czyz and A. Kotanski, Ann. of Phys. (NY) 73 (1972) 439; J. Pumplin, Phys. Rev. D4 (1971) 3482; D7 (1973) 795.
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[10] E.L. Berger and P. Piril/i, Absorptive effects in exclusive diffraction dissociation, Argonne report ANL-HEP-PR75-27, June 1975, submitted to Phys. Rev. Different approaches are discussed by V.A. Tsarev, Phys. Rev. D11 (1975) 1864, and G. Cohen-Tannoudji and U Maor, Phys. Lett. 57B (1975) 253. [11] S. Humble, Nucl. Phys. B76 (1974) 137; B86 (1975) 285. [12] Yu.M. Antipov et al., Nucl. Phys. B63 (1973) 153. We are grateful to R. Cutler for calling our attention to table 1 of this reference. [131 E.L. Berger, Systematics of cross-over effects in inelastic diffraction dissociation, Argonne report ANL/HEP 7464, to be published in Phys. Rev. D l l (1975) 3214.