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Meson spectrum in a simple planar bootstrap model
Volume 71 B, number 1
PHYSICS LETTI-RS
7 November 1977
MESON SPECTRUM IN A SIMPLE PLANAR BOOTSTRAP MODEL Louis A.P. BAL.~ZS Department of Physics, Purdue University, West Lafayette Indiana 4 790 7, USA Received 3 August 1977 A simple planar cluster-multiperipheral model with a finite-energy sum rule constraint is set up, and self-consistency is imposed. This dynamically generates a zero-parameter infinitely-rising v e c t o r - t e n s o r Regge traj¢ctory which is in good agreement with experiment.
Experimentally it has been known for a long time that hadron spectra can be described by ahnost linear Regge trajectories. Such trajectories are required for dual-resonance models. Dynamically they are often explained as qrq bound states, assuming quark confinement -- a consequence of the "infrared slavery" which is believed to be a feature of certain non-abelian gauge theories. Recently an alternative dynamical approach has been proposed, in which physical quarks do not have to be introduced [ 1 , 2 ] . It is based on a combination of unitarity and duality and has led to a number of successful predictions, mainly for slnall values of I tl. In first approximation the hadron spectrum is exchange degenerate and is supposed to be determined by the "planar bootstrap" [3]. In general this is a rather difficult problem which up to now has not led to a fully self-consistent spectrum, although preliminary calculations have been encouraging [ 4 . 5 ] . In the present paper we set up a simple planarbootstrap model which can bc solved analytically. A multiperiphcral model is combined with duality and self-consistency is imposed to generate the Regge trajectory a(t). We find that an infinitely-rising v e c t o r tensor trajectory arises quite naturally in this kind o f model. No arbitrary input parameters are needed to calculate it. (A) A tnultiperipheral cluster model. We will consider 7rTrscattering for simplicity, with m 2 ~- 0. It will be convenient to work with the generalized Mellin transform of the absorptive part A(s, t)
At(t)= ;
dsu I l A ( s , t ) ,
A I = Wl + B l + ....
(2)
where W, B .... are given by fig. l(a), (b) . . . . . To avoid double-counting, we will assume that the mass of the cluster a is bounded, since higher-mass effects are already taken into account by fig. l(b), (c) . . . . . If we represent it by a single effective mass X/~a, we then have
W(s, t) = l'(t)6(s
Sa),
(3)
which, from eq. ( I ), gives
Wl(t ) = F(t)(s a + t/2) - l - 1
(4)
It has been argued that the series (2) can be well represented by its [1, 1] Pad~ approximant [5, 6]
At(t) = Wt(t)/D(l, t),
(5)
where
D(l, t) = 1
Bt(t)/Wt(t).
(6)
In particular, this satisfies t-channel unitarity and is exact if we have a factorizable model [6]. It also takes into account threshold effects which are known to be important [7], but are often ignored in approximate treatments.
(1)
0
where ~, = ~(s-u) ~-s + ~ t is the usual crossing-symmetric variable used in finite-energy sum rules. Eq. (1) 216
can be obtained, for example, if we start from a Froissart-Gribov formula [6] and use the asymptotic form Qt(z) c~ z -1-1" In our model A is given by
(o)
fb)
Ic)
t:ig. 1. The absorptive part for a multiperipheral cluster model.
Volume 71 B, number 1
PHYSICS LETTERS
The function B(s, t) is in general quite complicated. From specific models, however, we find that it builds up quite rapidly above its threshold [5]. When we take its Mellin transform (1), we therefore expect the dominant contribution to come from the threshold region, since the factor v - l - 1 cuts down its large-s contribution, at least for I > a c. We should thus be able to make the approximation B(s, t) = ~ ( t ) 6 ( s
(7)
- So),
where s o "" 4s a is the threshold o f B. This approximation fails, of course, for l < a c. If we take the Mellin transform ( 1) of eq. (7), eqs. (4) and (6) now give
D(I, t) = 1 - k ( t ) [ x ( t ) ] - t - l ,
(8)
If we require this b and a to be consistent with the ones in eq. (I 1), and if we combine eqs. (4), (8), (9), (11), (12) and (13), we obtain a(t) = - 1 + 2 l n - 1 ( 2 } ~ t t ] . \ sa ]
x(t) = (s o + t/2)(s a + t/2) -1
(9)
Note that our planar amplitude does not contain the usual Regge cut [8]. (B) Duality. If we assume the Regge behavior bv a for large s, we can relate F(t) to the Regge residue b(t) through a finite-energy sum rule N f dv[W(s, t) - b( t ) v c~(t)] O. (10) o This sort o f constraint on sums of ladder graphs was first applied a number o f years ago within a pionexchange model [9], and has recently become an integral part of the dual-multiperipheral approach [2]. To avoid any ambiguity involving the upper limit we shall simply set N = s a + t/2. This corresponds to s = sa and so we must only include half the integral over the delta function when we integrate over W(s, t). We then obtain
(14)
Since s o = 4Sa, eq. (14) only depends on the constant s a. This is not, of course, a true arbitrary parameter in our model since one mass is always needed to fix the energy scale. We will require it to be such that a(m 2) = 1 with the experimental p-mass value m o = 0.77"0 GeV. This gives X/~a = 0.630 GeV, a value not too different from mp itself. Eq. (14) then predicts resonances with masses (in GeV units) mf = 1 . 3 1 2 ,
where k = k/l" and
7 November 1977
mg
=
1.697,
m h=2.013.
The corresponding experimental values are mf = 1.270,
mg
1.690,
m h = 2.040.
Fig. 2 gives a plot of Re a(t). For small It[ we have a(t) = 0.443 + 0.982 t,
(15)
so that we have a reasonable value for the intercept a(0). Asymptotically,
=
{r(t)
_
b(t) ~(t) + i Sa +
.
(11)
ot(t) -+ 0.838 t,
as
Itl ~oo,
so that we have an infinitely rising trajectory. From eq. (14) we see that there are singularities in ~ t ) at t = - 2 s a and t = -2s0, which can be joined by a cut. These occur in the region a = 1 < 0, however, where, as we have seen already, the approximation (7) is not warranted.
4 3 Z
We have repeated our calculation with other finiteenergy sum rule prescriptions, however, and find that it does not alter our results in any basic way. (C) Self-consistency. Eq. (6) gives rise to an output Regge pole at l = a(t) if D ( ~ t ) , t) = 0.
(16)
1
/ 1
~
3
4
i
, (aeV ~)
(12)
The corresponding residue is
b(t) = w ( t ) l a D ( , ~ , t)/a,~l -1
(13)
Fig. 2. Plot of Re a(t) as given by eq. (14) with so = 4sa and "~a = 0.630 GeV. 217
Volume 71 B, number 1
PHYSICS LETTERS
As it stands, our model does not give any information on the Regge residue b(t), which simply cancels out when we combine eqs. (4), (11), and (13). This kind of cancellation occurs in any planar bootstrap of the type we have been considering. If we were to evaluate fig. l(b) more explicitly, however [5], we could extract triple-Reggeon coupling parameters from k(t), which is determined by eqs. (8) and (12). The author would like to thank Professor G.F. Chew for his hospitality at the Lawrence Berkeley Laboratory as well as for several very helpful comments. He would also like to thank Dr. Chan Hong Mo, Dr. Tsou Sheung Tsun, Dr. D. Tow, Dr. M. Bishari, Dr. C. Rosenzweig and Dr. B. Nicolescu tbr useful conversations. The work was supported in part by ERDA.
218
7 November 1977
References [1 ] G. Veneziano, Nucl. Phys. B74 (1974) 365; Phys. Lett. 52B (1974) 220. [2] Chan ltong-Mo, J.E. Paton and Tsou Sheung Tsun, Nucl. Phys. B86 (1975) 479. [31 C. Rosenzweig and G. Vencziano, Phys. I.ett. 52B (1974) 335.' [4] M. Schaap and G. Vencziano, Letl. Nuovo Cimento 12 (1975) 204; K. Konishi, Nucl. Phys. BII6 (1976) 356; J. Millan, unpublished. [5] L.A.P. Bal:~zs,Phys. Rev. D15 (1977) 319. [6] L.A.P. Bal:~zs,Planar bootstrap without tile dual tree approximation, to be published in Phys. Rev. [71 G.F. Chew and D. Snider, Phys. Lett. 31B (1970) 75; G.F. Chew and J. Koplik, Phys. Lett. 48B (1974) 221. [8] M. Bishari and G. Veneziano, Phys. I.ett. 58B (1975) 445. 191 L.A.P. Bahlzs, Phys. Rev. D2 (1970) 999; Phys. Lett. 29B (1969) 228.
PHYSICS LETTI-RS
7 November 1977
MESON SPECTRUM IN A SIMPLE PLANAR BOOTSTRAP MODEL Louis A.P. BAL.~ZS Department of Physics, Purdue University, West Lafayette Indiana 4 790 7, USA Received 3 August 1977 A simple planar cluster-multiperipheral model with a finite-energy sum rule constraint is set up, and self-consistency is imposed. This dynamically generates a zero-parameter infinitely-rising v e c t o r - t e n s o r Regge traj¢ctory which is in good agreement with experiment.
Experimentally it has been known for a long time that hadron spectra can be described by ahnost linear Regge trajectories. Such trajectories are required for dual-resonance models. Dynamically they are often explained as qrq bound states, assuming quark confinement -- a consequence of the "infrared slavery" which is believed to be a feature of certain non-abelian gauge theories. Recently an alternative dynamical approach has been proposed, in which physical quarks do not have to be introduced [ 1 , 2 ] . It is based on a combination of unitarity and duality and has led to a number of successful predictions, mainly for slnall values of I tl. In first approximation the hadron spectrum is exchange degenerate and is supposed to be determined by the "planar bootstrap" [3]. In general this is a rather difficult problem which up to now has not led to a fully self-consistent spectrum, although preliminary calculations have been encouraging [ 4 . 5 ] . In the present paper we set up a simple planarbootstrap model which can bc solved analytically. A multiperiphcral model is combined with duality and self-consistency is imposed to generate the Regge trajectory a(t). We find that an infinitely-rising v e c t o r tensor trajectory arises quite naturally in this kind o f model. No arbitrary input parameters are needed to calculate it. (A) A tnultiperipheral cluster model. We will consider 7rTrscattering for simplicity, with m 2 ~- 0. It will be convenient to work with the generalized Mellin transform of the absorptive part A(s, t)
At(t)= ;
dsu I l A ( s , t ) ,
A I = Wl + B l + ....
(2)
where W, B .... are given by fig. l(a), (b) . . . . . To avoid double-counting, we will assume that the mass of the cluster a is bounded, since higher-mass effects are already taken into account by fig. l(b), (c) . . . . . If we represent it by a single effective mass X/~a, we then have
W(s, t) = l'(t)6(s
Sa),
(3)
which, from eq. ( I ), gives
Wl(t ) = F(t)(s a + t/2) - l - 1
(4)
It has been argued that the series (2) can be well represented by its [1, 1] Pad~ approximant [5, 6]
At(t) = Wt(t)/D(l, t),
(5)
where
D(l, t) = 1
Bt(t)/Wt(t).
(6)
In particular, this satisfies t-channel unitarity and is exact if we have a factorizable model [6]. It also takes into account threshold effects which are known to be important [7], but are often ignored in approximate treatments.
(1)
0
where ~, = ~(s-u) ~-s + ~ t is the usual crossing-symmetric variable used in finite-energy sum rules. Eq. (1) 216
can be obtained, for example, if we start from a Froissart-Gribov formula [6] and use the asymptotic form Qt(z) c~ z -1-1" In our model A is given by
(o)
fb)
Ic)
t:ig. 1. The absorptive part for a multiperipheral cluster model.
Volume 71 B, number 1
PHYSICS LETTERS
The function B(s, t) is in general quite complicated. From specific models, however, we find that it builds up quite rapidly above its threshold [5]. When we take its Mellin transform (1), we therefore expect the dominant contribution to come from the threshold region, since the factor v - l - 1 cuts down its large-s contribution, at least for I > a c. We should thus be able to make the approximation B(s, t) = ~ ( t ) 6 ( s
(7)
- So),
where s o "" 4s a is the threshold o f B. This approximation fails, of course, for l < a c. If we take the Mellin transform ( 1) of eq. (7), eqs. (4) and (6) now give
D(I, t) = 1 - k ( t ) [ x ( t ) ] - t - l ,
(8)
If we require this b and a to be consistent with the ones in eq. (I 1), and if we combine eqs. (4), (8), (9), (11), (12) and (13), we obtain a(t) = - 1 + 2 l n - 1 ( 2 } ~ t t ] . \ sa ]
x(t) = (s o + t/2)(s a + t/2) -1
(9)
Note that our planar amplitude does not contain the usual Regge cut [8]. (B) Duality. If we assume the Regge behavior bv a for large s, we can relate F(t) to the Regge residue b(t) through a finite-energy sum rule N f dv[W(s, t) - b( t ) v c~(t)] O. (10) o This sort o f constraint on sums of ladder graphs was first applied a number o f years ago within a pionexchange model [9], and has recently become an integral part of the dual-multiperipheral approach [2]. To avoid any ambiguity involving the upper limit we shall simply set N = s a + t/2. This corresponds to s = sa and so we must only include half the integral over the delta function when we integrate over W(s, t). We then obtain
(14)
Since s o = 4Sa, eq. (14) only depends on the constant s a. This is not, of course, a true arbitrary parameter in our model since one mass is always needed to fix the energy scale. We will require it to be such that a(m 2) = 1 with the experimental p-mass value m o = 0.77"0 GeV. This gives X/~a = 0.630 GeV, a value not too different from mp itself. Eq. (14) then predicts resonances with masses (in GeV units) mf = 1 . 3 1 2 ,
where k = k/l" and
7 November 1977
mg
=
1.697,
m h=2.013.
The corresponding experimental values are mf = 1.270,
mg
1.690,
m h = 2.040.
Fig. 2 gives a plot of Re a(t). For small It[ we have a(t) = 0.443 + 0.982 t,
(15)
so that we have a reasonable value for the intercept a(0). Asymptotically,
=
{r(t)
_
b(t) ~(t) + i Sa +
.
(11)
ot(t) -+ 0.838 t,
as
Itl ~oo,
so that we have an infinitely rising trajectory. From eq. (14) we see that there are singularities in ~ t ) at t = - 2 s a and t = -2s0, which can be joined by a cut. These occur in the region a = 1 < 0, however, where, as we have seen already, the approximation (7) is not warranted.
4 3 Z
We have repeated our calculation with other finiteenergy sum rule prescriptions, however, and find that it does not alter our results in any basic way. (C) Self-consistency. Eq. (6) gives rise to an output Regge pole at l = a(t) if D ( ~ t ) , t) = 0.
(16)
1
/ 1
~
3
4
i
, (aeV ~)
(12)
The corresponding residue is
b(t) = w ( t ) l a D ( , ~ , t)/a,~l -1
(13)
Fig. 2. Plot of Re a(t) as given by eq. (14) with so = 4sa and "~a = 0.630 GeV. 217
Volume 71 B, number 1
PHYSICS LETTERS
As it stands, our model does not give any information on the Regge residue b(t), which simply cancels out when we combine eqs. (4), (11), and (13). This kind of cancellation occurs in any planar bootstrap of the type we have been considering. If we were to evaluate fig. l(b) more explicitly, however [5], we could extract triple-Reggeon coupling parameters from k(t), which is determined by eqs. (8) and (12). The author would like to thank Professor G.F. Chew for his hospitality at the Lawrence Berkeley Laboratory as well as for several very helpful comments. He would also like to thank Dr. Chan Hong Mo, Dr. Tsou Sheung Tsun, Dr. D. Tow, Dr. M. Bishari, Dr. C. Rosenzweig and Dr. B. Nicolescu tbr useful conversations. The work was supported in part by ERDA.
218
7 November 1977
References [1 ] G. Veneziano, Nucl. Phys. B74 (1974) 365; Phys. Lett. 52B (1974) 220. [2] Chan ltong-Mo, J.E. Paton and Tsou Sheung Tsun, Nucl. Phys. B86 (1975) 479. [31 C. Rosenzweig and G. Vencziano, Phys. I.ett. 52B (1974) 335.' [4] M. Schaap and G. Vencziano, Letl. Nuovo Cimento 12 (1975) 204; K. Konishi, Nucl. Phys. BII6 (1976) 356; J. Millan, unpublished. [5] L.A.P. Bal:~zs,Phys. Rev. D15 (1977) 319. [6] L.A.P. Bal:~zs,Planar bootstrap without tile dual tree approximation, to be published in Phys. Rev. [71 G.F. Chew and D. Snider, Phys. Lett. 31B (1970) 75; G.F. Chew and J. Koplik, Phys. Lett. 48B (1974) 221. [8] M. Bishari and G. Veneziano, Phys. I.ett. 58B (1975) 445. 191 L.A.P. Bahlzs, Phys. Rev. D2 (1970) 999; Phys. Lett. 29B (1969) 228.