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Low mass diffractive dissociation in a simple t-dependent dual bootstrap model
Volume
81 B, number
2
PHYSICS
LOW MASS DIFFRACTIVE
LETTERS
12 February
1979
DISSOCIATION IN A SIMPLE r-DEPENDENT DUAL BOOTSTRAP MODEL*
Mordechai BISHARI Rutherford Laboratory I, Chilton, Didcot, Oxon, UK and Weizmann Institute of Science, Rehovot, Israel Received
13 November
1978
The smallness of inelastic diffractive dissociation is explicitly demonstrated, in the framework of the “l/N dual unitarization” scheme, by incorporating a Deck type mechanism with the crucial planar bootstrap equation. Althoueh both inelastic and elastic pomeron couplings are of the same order in /N, the origin for their smallness, however, is not identical.
One of the prominent aspects of the “1 /N Dual Unitarization” (“l/N D.U.“) approach is that it satisfactorily explains the physics of hadronic processes at small momentum transfers [ I,3 ] . Especially, a considerable understanding of the bare pomeron, which controls diffractive processes at high energies, has been achieved in the framework of the “1 /N D.U.” scheme. Here [2] we would like to investigate in more detail the dynamics of low muss diffractive processes. In a previous study [4] of elastic scattering it has been clarified how the underlying dynamics lead to a small oeQ/oT, that is, o,Q/oT = ae,*1/N2,
aeP = 0.7-1,
(1)
with N2 = 7 being the square of the effective quark flavor number. It is natural then to extend the previous study of elastic scattering to low mass diffractive production, both having, of course, the same 1/N dependence: o&T
=ag’l/N2.
(2)
The central issue in this work is the understanding and determination of @Din eq. (2) from the dynamics of low multiplicity diffractive dissociation. * Supported in part by the Israel Commission for Basic Research and the United States-Israel Binational Science Foundation. ’ Summer Visitor.
rr+
IT+
’
(a) Fig. 1. The diagram representing the generalized mechanism” for diffractive dissociation.
(b) “dual Deck
Why a theoretical study of low mass diffraction is interesting? The most important reason concerns the self-consistency of the “l/N D.U.” scheme, for which it is crucial to (a posteriori) justify the neglect of the intermediate diffractive states in the evaluation of the elastic overlap function. The neglect of the elastic diffractive state has been justified in ref. [4] and here we provide also the justification for the neglect of the inelastic diffractive states. We employ the “generalized dual Deck model”, recently constructed by Cohen-Tannoudji et al. [S] , where its implication for the present theoretical study is fully described in ref. [2]. We are then led to consider the two diagrams depicted in fig. 1, with reggeized exchanges, which constitute the adopted Deck type model. The quark duality diagram representation of the dominant mechanism (fig. la) is depicted in fig. 2. The internal quark line in this figure can have i = u, d, s, c. 241
Volume
SlB, number
PHYSICS
2
12 February
LETTERS
ll’
n+
I$. 2. The quark representation of the other flavors.
Fig. 3. The overlap of fig. 2, corresponding plicity diffractive cross section.
of fig. la with the addition
Thus the contribution from fig. 1a must be multiplied by the effective quark flavor number N, in which the symmetry breaking effects are taken into account. The amplitude corresponding to fig. la is written as
1979
to the low-multi-
T=A pn+.pn(~, t2)eb”1’2P,,‘~p~~(U1) (9) x (,‘~2)crPft2)(c,‘)l/2(Q’sl)ap(U1)-aP(r2). The low mass inelastic diffractive cross section, oD, is related to the transition amplitude in eq. (9) through (at high energies): 1
1
uD(s)=2,s
where in the elastic diffractive amplitude A.Lp,.n-tpn, one of the legs is a Reggeon having “mass” ul. Also, in eq. (3) a possible Toller angle dependence has been averaged out. The Reggeon-meson-meson residue function is determined from the triple Reggeon couplingg(0, t, t) via a linear planar relation [4] which reads &‘p
,,(ur) e (g/cw’)2ebur,
(4)
with g(0; f, t) = g(cr,(O) - 2a,(t)
+ 1) ebf.
(5)
Concerning the off-shell elastic amplitude one is tempted from eq. (4) to write: A ~~p~3?T-tpn(s3, fz3 ul) =Apn+pn(~33
in eq. (3),
t2) e bu1/2.
(10)
The factor of 2 in the numerator is due to the dissociation of also the lower pion in fig. 2 and N is the above mentioned effective quark flavor number, originating from the free loop in fig. 3. It is most useful to express [6] the phase-space volume in terms of the three invariants t2, ~1, u1 and one azimuthal angle which can be immediately integrated upon to give 271.One then readily obtains, in the limit of a zero mass pion, using eqs. (9) and (10) dt2
(6)
Fortunately, the results are not sensitive to the offshell extrapolation in eq. (6) as we shall see below. As the elastic amplitude is dominated by the bare pomeron P, one may write:
&Wpn
(S> t2)12)
(11)
X’j
dsldul
2(olp(tQ-“p(t2))-l
(QI’s~)
167~~ A pn+p,&j> f2) = Ap,,&>
~~)(s~/s)~~(~~)~
(7) X ebulf12,,-p4q)N,
If riz is the average mass (expected to be -1 GeV) of the produced vector-tensor mesons in fig. 2, we have
with the following limits [6] of integration:
s3 = h2SjSl.
-(sl
Finally,
eqs. (6) (7), (8) and (3) lead to
(8)
-?+)
i$
-m
It is remarked that eq. (11) indeed indicates the in242
Volume
81B, number
PHYSICS
2
sensitivity of oD to the off-shell extrapolation of the “elastic” amplitude AGA,,P>~_,~~ in eq. (6) since a substantial damping in u1 already arises from fl~,C~~(ul). To simplify the evaluation of oD we neglect the slopes of the Reggeon and bare pomeron trajectories in eq. (11). Also note that the term in bracket in eq. (11) is just oea(s) which has been studied before. Hence we have from eq. (4) (12) and (11) I- 2 201 g2Wb
0D(s)= 20ey(s)(a m ) ‘.
16~~’
71s 1
mdxx2(~~-‘+l(1
__,-2b@i2(X-1))
= 2 [~,&)b&>1
DK,
(14)
where D.= ((g2N/2b)/16rr(u’)(&z2)2”p,
(15) K = n-12biii2e2biii2E1(2bFi2),
with El(x) being [7] the exponential integral function. The factors D and K may be viewed as reflecting, respectively, dynamical and phase-space effects. Now, to demonstrate the smallness of oD/oT we must confirm that both D and K are so constrained as to not undermine the smallness arising from uep/uT. Indeed the “dynamical factor” D is directly constrained by the non-linear planar bootstrap equation [I] , relating the slope parameter b to the coupling strength g, namely
141 (g2N/2b)/16&
= 1,
(16)
and since ~1‘-m 2 <_ 1 (fi is prevented from being large primarily by. the produced p) and crP = 0.5 this leads to (see eq. (15)) D<
1.
(18)
Actually, knowing the value of b/a’ one can now estimate how the “kinematical factor” K is modified if the excited mass is not allowed to be very large, as should be the case, ipso facto, for low mass inelastic diffractive production. If for the highest allowed excited mass % we take the reasonable value M2 2 5 GeV2 one obtains [2] , 08’)
rather near to the value in eq. (18). From eqs. (14), (17) and (18’) we finally obtain 1.3 ‘se!?(s) -_ uT(s) - 7 oT(s) ’
‘D(‘)
where for our purposes the upper limit ofx = sl/fi2 can be taken 00 without committing a big error (see below). Since op - alp R 0.5, eq. (13) becomes simple, and will be written as u&)/o&)
1979
K = 0.84~~.
K = 0.65~l,
(13) Xl _
12 February
LETTERS
(17)
From a study of the cylinder topology, which is associated with the physical properties of the bare pomeron, we have obtained [4] b/a’ =S 2.3. With (U’iG = 1 eq. (15) then gives
(19)
thus deriving the central result in this work, namely that in addition to ueQ/uT also @D/UT is prevented from becoming large. The cross section for inelastic diffractive production of a state with a given mass, i.e. a resonance, is only part of @D. Consequently one may infer from eq. (19) that an inelastic coupling of the bare pomeron is significantly smaller than its elastic coupling, e.g. /&,, % /&Al, although both have the same l/N dependence. This is of course in accordance with experimental findings and illustrates the indispensibility of an explicit dynamics to complement the topological considerations. As discussed in detail in ref. [2] , the mechanism in fig. 1b and the diagrams where the upper produced pion in fig. la is replaced by vector-tensor states, are not expected to alter the basic significance of the derived restriction on UD/UT in eq. (19). Explicitly, with these additional mechanisms, eq. (19) may be modified to [2] ODcS) < 2.3 u.T(s)
ueP(s)
71 OT(S)
.
(20)
To summarize, we have verified the applicability of the iterative, or perturbative, approach to diffractive scattering in the “1 /N D.U.” scheme, where the input is provided by the non-diffractive component which is generated only from exchanged reggeons with no bare pomerons involved. The potentiality of the “1 /N D.U.” approach is further manifested in this work, and may be more appreciated if one notes that alternative schemes have con243
Volume 81B, number
2
PHYSICS
ceptual and practical difficulties [3] in coping with small momentum transfer phenomena. The author is grateful to R.J.N. Phillips for the extended hospitality at the Theory Division, Rutherford Laboratory. References
[I] Chan H.-M. and Tsou Sheung Tsun, Rutherford
Lab. preprint, RL-76-080/T.165 (1976); G.F. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) no. 5.
244
LETTERS
[21 M. Bishari, Weizmann
12 February
1979
Inst. preprint, WIS-77/54 PH (1977) and references therein. [31 G. Veneziano, TH.2311~CERN (1977). [41 M. Bishari, Phys. Lett. 66B (1977) 54. et al., Nucl. Phys. B125 (1977) 445. [Sl G. Cohen-Tannoudji [61 E. Byckling and K. Kajantie, Particle kinematics (Wiley, 1973). and LA. Stegun, Handbook of [71 E.g., M. Abramowitz mathematical functions (1964).
81 B, number
2
PHYSICS
LOW MASS DIFFRACTIVE
LETTERS
12 February
1979
DISSOCIATION IN A SIMPLE r-DEPENDENT DUAL BOOTSTRAP MODEL*
Mordechai BISHARI Rutherford Laboratory I, Chilton, Didcot, Oxon, UK and Weizmann Institute of Science, Rehovot, Israel Received
13 November
1978
The smallness of inelastic diffractive dissociation is explicitly demonstrated, in the framework of the “l/N dual unitarization” scheme, by incorporating a Deck type mechanism with the crucial planar bootstrap equation. Althoueh both inelastic and elastic pomeron couplings are of the same order in /N, the origin for their smallness, however, is not identical.
One of the prominent aspects of the “1 /N Dual Unitarization” (“l/N D.U.“) approach is that it satisfactorily explains the physics of hadronic processes at small momentum transfers [ I,3 ] . Especially, a considerable understanding of the bare pomeron, which controls diffractive processes at high energies, has been achieved in the framework of the “1 /N D.U.” scheme. Here [2] we would like to investigate in more detail the dynamics of low muss diffractive processes. In a previous study [4] of elastic scattering it has been clarified how the underlying dynamics lead to a small oeQ/oT, that is, o,Q/oT = ae,*1/N2,
aeP = 0.7-1,
(1)
with N2 = 7 being the square of the effective quark flavor number. It is natural then to extend the previous study of elastic scattering to low mass diffractive production, both having, of course, the same 1/N dependence: o&T
=ag’l/N2.
(2)
The central issue in this work is the understanding and determination of @Din eq. (2) from the dynamics of low multiplicity diffractive dissociation. * Supported in part by the Israel Commission for Basic Research and the United States-Israel Binational Science Foundation. ’ Summer Visitor.
rr+
IT+
’
(a) Fig. 1. The diagram representing the generalized mechanism” for diffractive dissociation.
(b) “dual Deck
Why a theoretical study of low mass diffraction is interesting? The most important reason concerns the self-consistency of the “l/N D.U.” scheme, for which it is crucial to (a posteriori) justify the neglect of the intermediate diffractive states in the evaluation of the elastic overlap function. The neglect of the elastic diffractive state has been justified in ref. [4] and here we provide also the justification for the neglect of the inelastic diffractive states. We employ the “generalized dual Deck model”, recently constructed by Cohen-Tannoudji et al. [S] , where its implication for the present theoretical study is fully described in ref. [2]. We are then led to consider the two diagrams depicted in fig. 1, with reggeized exchanges, which constitute the adopted Deck type model. The quark duality diagram representation of the dominant mechanism (fig. la) is depicted in fig. 2. The internal quark line in this figure can have i = u, d, s, c. 241
Volume
SlB, number
PHYSICS
2
12 February
LETTERS
ll’
n+
I$. 2. The quark representation of the other flavors.
Fig. 3. The overlap of fig. 2, corresponding plicity diffractive cross section.
of fig. la with the addition
Thus the contribution from fig. 1a must be multiplied by the effective quark flavor number N, in which the symmetry breaking effects are taken into account. The amplitude corresponding to fig. la is written as
1979
to the low-multi-
T=A pn+.pn(~, t2)eb”1’2P,,‘~p~~(U1) (9) x (,‘~2)crPft2)(c,‘)l/2(Q’sl)ap(U1)-aP(r2). The low mass inelastic diffractive cross section, oD, is related to the transition amplitude in eq. (9) through (at high energies): 1
1
uD(s)=2,s
where in the elastic diffractive amplitude A.Lp,.n-tpn, one of the legs is a Reggeon having “mass” ul. Also, in eq. (3) a possible Toller angle dependence has been averaged out. The Reggeon-meson-meson residue function is determined from the triple Reggeon couplingg(0, t, t) via a linear planar relation [4] which reads &‘p
,,(ur) e (g/cw’)2ebur,
(4)
with g(0; f, t) = g(cr,(O) - 2a,(t)
+ 1) ebf.
(5)
Concerning the off-shell elastic amplitude one is tempted from eq. (4) to write: A ~~p~3?T-tpn(s3, fz3 ul) =Apn+pn(~33
in eq. (3),
t2) e bu1/2.
(10)
The factor of 2 in the numerator is due to the dissociation of also the lower pion in fig. 2 and N is the above mentioned effective quark flavor number, originating from the free loop in fig. 3. It is most useful to express [6] the phase-space volume in terms of the three invariants t2, ~1, u1 and one azimuthal angle which can be immediately integrated upon to give 271.One then readily obtains, in the limit of a zero mass pion, using eqs. (9) and (10) dt2
(6)
Fortunately, the results are not sensitive to the offshell extrapolation in eq. (6) as we shall see below. As the elastic amplitude is dominated by the bare pomeron P, one may write:
&Wpn
(S> t2)12)
(11)
X’j
dsldul
2(olp(tQ-“p(t2))-l
(QI’s~)
167~~ A pn+p,&j> f2) = Ap,,&>
~~)(s~/s)~~(~~)~
(7) X ebulf12,,-p4q)N,
If riz is the average mass (expected to be -1 GeV) of the produced vector-tensor mesons in fig. 2, we have
with the following limits [6] of integration:
s3 = h2SjSl.
-(sl
Finally,
eqs. (6) (7), (8) and (3) lead to
(8)
-?+)
i$
-m
It is remarked that eq. (11) indeed indicates the in242
Volume
81B, number
PHYSICS
2
sensitivity of oD to the off-shell extrapolation of the “elastic” amplitude AGA,,P>~_,~~ in eq. (6) since a substantial damping in u1 already arises from fl~,C~~(ul). To simplify the evaluation of oD we neglect the slopes of the Reggeon and bare pomeron trajectories in eq. (11). Also note that the term in bracket in eq. (11) is just oea(s) which has been studied before. Hence we have from eq. (4) (12) and (11) I- 2 201 g2Wb
0D(s)= 20ey(s)(a m ) ‘.
16~~’
71s 1
mdxx2(~~-‘+l(1
__,-2b@i2(X-1))
= 2 [~,&)b&>1
DK,
(14)
where D.= ((g2N/2b)/16rr(u’)(&z2)2”p,
(15) K = n-12biii2e2biii2E1(2bFi2),
with El(x) being [7] the exponential integral function. The factors D and K may be viewed as reflecting, respectively, dynamical and phase-space effects. Now, to demonstrate the smallness of oD/oT we must confirm that both D and K are so constrained as to not undermine the smallness arising from uep/uT. Indeed the “dynamical factor” D is directly constrained by the non-linear planar bootstrap equation [I] , relating the slope parameter b to the coupling strength g, namely
141 (g2N/2b)/16&
= 1,
(16)
and since ~1‘-m 2 <_ 1 (fi is prevented from being large primarily by. the produced p) and crP = 0.5 this leads to (see eq. (15)) D<
1.
(18)
Actually, knowing the value of b/a’ one can now estimate how the “kinematical factor” K is modified if the excited mass is not allowed to be very large, as should be the case, ipso facto, for low mass inelastic diffractive production. If for the highest allowed excited mass % we take the reasonable value M2 2 5 GeV2 one obtains [2] , 08’)
rather near to the value in eq. (18). From eqs. (14), (17) and (18’) we finally obtain 1.3 ‘se!?(s) -_ uT(s) - 7 oT(s) ’
‘D(‘)
where for our purposes the upper limit ofx = sl/fi2 can be taken 00 without committing a big error (see below). Since op - alp R 0.5, eq. (13) becomes simple, and will be written as u&)/o&)
1979
K = 0.84~~.
K = 0.65~l,
(13) Xl _
12 February
LETTERS
(17)
From a study of the cylinder topology, which is associated with the physical properties of the bare pomeron, we have obtained [4] b/a’ =S 2.3. With (U’iG = 1 eq. (15) then gives
(19)
thus deriving the central result in this work, namely that in addition to ueQ/uT also @D/UT is prevented from becoming large. The cross section for inelastic diffractive production of a state with a given mass, i.e. a resonance, is only part of @D. Consequently one may infer from eq. (19) that an inelastic coupling of the bare pomeron is significantly smaller than its elastic coupling, e.g. /&,, % /&Al, although both have the same l/N dependence. This is of course in accordance with experimental findings and illustrates the indispensibility of an explicit dynamics to complement the topological considerations. As discussed in detail in ref. [2] , the mechanism in fig. 1b and the diagrams where the upper produced pion in fig. la is replaced by vector-tensor states, are not expected to alter the basic significance of the derived restriction on UD/UT in eq. (19). Explicitly, with these additional mechanisms, eq. (19) may be modified to [2] ODcS) < 2.3 u.T(s)
ueP(s)
71 OT(S)
.
(20)
To summarize, we have verified the applicability of the iterative, or perturbative, approach to diffractive scattering in the “1 /N D.U.” scheme, where the input is provided by the non-diffractive component which is generated only from exchanged reggeons with no bare pomerons involved. The potentiality of the “1 /N D.U.” approach is further manifested in this work, and may be more appreciated if one notes that alternative schemes have con243
Volume 81B, number
2
PHYSICS
ceptual and practical difficulties [3] in coping with small momentum transfer phenomena. The author is grateful to R.J.N. Phillips for the extended hospitality at the Theory Division, Rutherford Laboratory. References
[I] Chan H.-M. and Tsou Sheung Tsun, Rutherford
Lab. preprint, RL-76-080/T.165 (1976); G.F. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) no. 5.
244
LETTERS
[21 M. Bishari, Weizmann
12 February
1979
Inst. preprint, WIS-77/54 PH (1977) and references therein. [31 G. Veneziano, TH.2311~CERN (1977). [41 M. Bishari, Phys. Lett. 66B (1977) 54. et al., Nucl. Phys. B125 (1977) 445. [Sl G. Cohen-Tannoudji [61 E. Byckling and K. Kajantie, Particle kinematics (Wiley, 1973). and LA. Stegun, Handbook of [71 E.g., M. Abramowitz mathematical functions (1964).