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Generalized vector dominance and the pion form factor
Volume 50B, number 4
PHYSICS LETTERS
24 June 1974
GENERALIZED VECTOR DOMINANCE AND THE PION FORM FACTOR M. BOHM I
CERN, Geneva, Switzerland and M. KRAMMER
DESY, Hamburg, Germany Received 28 February 1974 We have calculated the pion form factor using the bound state Bethe-Salpeter amplitudes and the quark form factor of a relativistic quark model. We obtain a ~eneralized vector dominance structure and an asymptotic behaviour in the space-like region Fn(Q2 ) ~ - 0.33 GeV2/Q2. At present there is sufficient experimental information on the pion form factor in the time-like [1 ] and spacelike [2] region to test dynamical models [3]. We consider the pion form factor to be dominated by the vector mesons [generalized VDM [4-6] ] and to depend on the wave function describing a composite structure of the pion. It is the purpose of this paper to discuss the interplay of generalized vector meson dominance and bound state structure in a relativistic quark model [7]. In this model the meson mass spectrum and wave functions were obtained by solving the bound state Bethe-Salpeter equation [8]. We now use the BS amplitude of the pion ×n(k, P) and the quark form factor Pu(k, Q) to determine the pion form factor with help of the triangle diagram
Qu =(PI +P2)u Xtr .'~P~
(1)
Xn k
"¢"P2
"
'' 1
1
1
Fu (Q2) = (2 Tr)-9/2 i f d 4 k Tr Fz(k, Q) x n (k + -~ P2, PI ) (~ - ~ trl + -f ~r2 - m) Xn( k - -i P1, P2 )" The quark form factor is given by the integral equation
rz(k, Q)=ZTu +ifdk'K(k-k') [~-~'+g'-m]-1 ru(k r' Q) [ ~1. . _
.~t
+ m]-l.
(2)
The BS kernel K describes the quark-antiquark interaction and Z 7u results from the ansatz ]~l = Z [ ~(x), 7 u ~(x)] for the electromagnetic current of the quarks. This inhomogeneous BS equation can be solved in good approximation for Q2 ~ 4 m 2 (where m is the mass of the heavy quarks) by using the completeness of the BS amplitudes. We obtain in the rest frame of the photon (Q = (Qo, 0))* t
r
1
Fz(k, Q) = Z(21r)-4 i ~ Tr 7~ f d k ×N (k, MN) (70 ~ MN +~ - m) ×N(k, MN) (70 ~ MN - t~ + m) N e2 = ~ rl ss
(3)
e~3gvr z(7° ~Mr +'l~-m)x?(k, Mr)(7o ~1 M r - f ~ +m ) Q2 __M 2
i On leave of absence from the University of Wiirzburg, Germ any. $ N is a shorthand for the quantum numbers of the mesons, r denotes the radial quantum number. 457
Volume 50B, number 4
PHYSICS LETTERS
24 June 1974
This expression shows a generalized vector dominance structure for the quark form factor. Because of the local current ]~! (x) only the radially excited [6] vector mesons contribute; the residues factorize into BS amplitude and vector meson-photon coupling constant gv ~" From the mathematical point of view a~unction is determined by its poles and residues up to an additive entire function only. We assume, following duality arguments, that the unsubtracted Mittag-Leffler expansion (3) may be used i.e., that the poles without a background term build up the quark form factor correctly. Now we use the quark form factor, eq. (3) to evaluate the pion form factor with help of eq. (1) F~(Q2) = (2n) -9/2 i ~
gVrv e~3 r
02 - M 2 Jdk Try(Sr3(k,MN)
r, s3
X (~-½~, oMN -m)x~r(k+lp = 2,P1)(~ ~- - -1~+~2P9 -' - -m)x'r(k-}P1, P2)(~+'~ 7oMN - m )
r,sa
(4)
Q2 _ M~r
The poles of the quark form factor also show up in the pion form factor. The residues are determined by the BS amplitudes of the pion and the heavy vector mesons. The explicit result depends on the form and spin structure ofthe BS kernel. In ref. [9] we found that a smooth kernel, approximated by a harmonic interaction and with a 3,5X3,5 Dirac part yields a singlet-triplet structure for the spectrum with the mass formula
M2=M2o + 2Kx/~N;
N=n +2r;
n,r = 0,1,2 ....
(5)
and describes the strong decays of the meson resonances reasonably well for r --- 3 and x/rff= 0.2 GeV2 . We use the Wick-rotated BS amplitudes of this model + 2rrX/~ X"(k,P) = -~b'5(~M,,X1-'/~2m) +51.l'Ys[~c/m,P/Mr]]
x°rO(k,P) = ~
exp ( - k2/2x/~)lpfi}
[ ~ ( 1 - ~ / 2 m ) -iek/m] exp (-k2/2,~/~)
Lrl(k2/V~-)
(6)
(Ip~) -Infi)).
for the evaluation of the integrals in eqs. (3) and (4) and obtain
gor7 = 2,yor = ( - l l r Z
F"(Q2)=g°°~g°°~
x/rr-+T~
,
k 3-r(-1)rer/2L l(r+c) = M2o + 1 2 r ~ '
(7)
M2° - 4 M 2 c=
12X/~
=0.21.
(8)
For comparison with experiment in the time like region one has to take into account the finite widths of the vector mesons. We have used Breit Wigner forms with Fp = 150 MeV and Pp, = 500 MeV and choose the only free parameter, the renormalization constant Z such that F,r(0 ) = 1. The results are given in fig. i. The sum of the residues in eq. (7) converges, we therefore obtain for large space-like values of Q2. F,r(Q 2) ~ - 0.33 GeV2/Q2 .
(9)
We find asymptotically a simple pole behaviour. Thisresult differs substantially from calculations which take into account only the pion wave function and not the quark form factor, where one would obtain F(Q 2) = exp (Q2[ const.), reflecting the Gaussians of the wave functions. Our result, eqs. (8) and (9) is a consequence of the duality 458
Volume 50B, number 4
PHYSICS LETTERS
S -I
Q° z [OeV z] i
24 June 1974
4
9
Fig. 1. Comparison of our result for the pion form factor with experiment. The data points are from ref. [1, 2]. assumption that the p i o n f o r m factor can be built up by resonances w i t h o u t a background t e r m and the s p e c t r u m and the coupling constants o f the relativistic quark model. It is a great pleasure t o t h a n k Hans J o o s for clarifying discussions.
References [1] V.L. Auslander et al., Soy. J. Nucl. Phys. 9 (1,969) 69; D. Benaksas et al., Phys. Lett. 39B (1972) 289; V.E. Balakin et al., Phys. Lett. 41B (1972) 205; M. Bernardini et al., Phys. Lett. 46B (1973) 261 ; S.F. Berezhnev et al., Yadern. Fiz. 18 (1973) 102; [2] C.W. Akerlof et al., Phys. Rev. 163 (1967) 1482; C. Mistretta et al., Phys. Rev. 184 (1969) 1487; E. Amaldi et al., Nuovo Cim. 65A (1970) 377; C. Driver et al., Phys. Lett. 35B (1971) 77, 81; P.S. Kummer et al., Lett. Nuovo Cim. 1 (1971) 1026; C.N. Brown et al., Phys. Rev. D8 (1973) 92. [3] The pion form factor has been discussed f.i. by taking into account the finite width of the po : G.J. Goundaris and J.J. Sakurai, Phys. Rev. Lett. 21 (1968) 244; by modifying the rho propagator through inelastic ehan~; nels: F.M. Renard, Phys. Lett. 47B (1973) 361; in Bethe-Salpeter type models: J. Fleischer and F. Gutbrod, Nuovo Cim. 10A (1972) 235, in the dual resonance model: Y. Oyanagi, Progr. Theor. Phys. 42 (1969) 898; H. Suura, Phys. Rev. Lett. 23 (1969) 551;
R. Jengo and E. Remiddi, Nucl. Phys. BI5 (1970) 1; by generalizing rho dominance: M. Greeo and Y.N. Srivastava, LNF-73/7; in the parton model: P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B53 (1973) 473; F. Ravndal, CALT-68-381 ; in the massive quark model: 1L Gatto and G. Preparata, DESY-73/15; by dimensional counting: S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153. [4] A. Bramon, E. Etim and M. Greco, Phys. Lett. 41B (1972) 609. [5 ] J.J. Sakurai and D. Schildknecht, Phys. Lett. 40B (1972) 121. [6] M. B/~hm, H. Joos and M. Krammer, Acta Phys. Austriaca 38 (1973) 123 and DESY 72/62. [71 M. Bfhm, H. Joos and M. Krammer, Acta Phys. Austriaea Suppl. XI (1973) 3 (Schladming Lectures 1973). [ 8 ] M. B6hm, H. Joos and M. Krammer, Nucl. Phys. B51 (1973) 397; M. Krammer, Internal report DESYT-73/1. [91 M. B6hm, H. Joos and M. Krammer, Ref. TH. 1715-CERN (to appear in Nucl. Phys.).
459
PHYSICS LETTERS
24 June 1974
GENERALIZED VECTOR DOMINANCE AND THE PION FORM FACTOR M. BOHM I
CERN, Geneva, Switzerland and M. KRAMMER
DESY, Hamburg, Germany Received 28 February 1974 We have calculated the pion form factor using the bound state Bethe-Salpeter amplitudes and the quark form factor of a relativistic quark model. We obtain a ~eneralized vector dominance structure and an asymptotic behaviour in the space-like region Fn(Q2 ) ~ - 0.33 GeV2/Q2. At present there is sufficient experimental information on the pion form factor in the time-like [1 ] and spacelike [2] region to test dynamical models [3]. We consider the pion form factor to be dominated by the vector mesons [generalized VDM [4-6] ] and to depend on the wave function describing a composite structure of the pion. It is the purpose of this paper to discuss the interplay of generalized vector meson dominance and bound state structure in a relativistic quark model [7]. In this model the meson mass spectrum and wave functions were obtained by solving the bound state Bethe-Salpeter equation [8]. We now use the BS amplitude of the pion ×n(k, P) and the quark form factor Pu(k, Q) to determine the pion form factor with help of the triangle diagram
Qu =(PI +P2)u Xtr .'~P~
(1)
Xn k
"¢"P2
"
'' 1
1
1
Fu (Q2) = (2 Tr)-9/2 i f d 4 k Tr Fz(k, Q) x n (k + -~ P2, PI ) (~ - ~ trl + -f ~r2 - m) Xn( k - -i P1, P2 )" The quark form factor is given by the integral equation
rz(k, Q)=ZTu +ifdk'K(k-k') [~-~'+g'-m]-1 ru(k r' Q) [ ~1. . _
.~t
+ m]-l.
(2)
The BS kernel K describes the quark-antiquark interaction and Z 7u results from the ansatz ]~l = Z [ ~(x), 7 u ~(x)] for the electromagnetic current of the quarks. This inhomogeneous BS equation can be solved in good approximation for Q2 ~ 4 m 2 (where m is the mass of the heavy quarks) by using the completeness of the BS amplitudes. We obtain in the rest frame of the photon (Q = (Qo, 0))* t
r
1
Fz(k, Q) = Z(21r)-4 i ~ Tr 7~ f d k ×N (k, MN) (70 ~ MN +~ - m) ×N(k, MN) (70 ~ MN - t~ + m) N e2 = ~ rl ss
(3)
e~3gvr z(7° ~Mr +'l~-m)x?(k, Mr)(7o ~1 M r - f ~ +m ) Q2 __M 2
i On leave of absence from the University of Wiirzburg, Germ any. $ N is a shorthand for the quantum numbers of the mesons, r denotes the radial quantum number. 457
Volume 50B, number 4
PHYSICS LETTERS
24 June 1974
This expression shows a generalized vector dominance structure for the quark form factor. Because of the local current ]~! (x) only the radially excited [6] vector mesons contribute; the residues factorize into BS amplitude and vector meson-photon coupling constant gv ~" From the mathematical point of view a~unction is determined by its poles and residues up to an additive entire function only. We assume, following duality arguments, that the unsubtracted Mittag-Leffler expansion (3) may be used i.e., that the poles without a background term build up the quark form factor correctly. Now we use the quark form factor, eq. (3) to evaluate the pion form factor with help of eq. (1) F~(Q2) = (2n) -9/2 i ~
gVrv e~3 r
02 - M 2 Jdk Try(Sr3(k,MN)
r, s3
X (~-½~, oMN -m)x~r(k+lp = 2,P1)(~ ~- - -1~+~2P9 -' - -m)x'r(k-}P1, P2)(~+'~ 7oMN - m )
r,sa
(4)
Q2 _ M~r
The poles of the quark form factor also show up in the pion form factor. The residues are determined by the BS amplitudes of the pion and the heavy vector mesons. The explicit result depends on the form and spin structure ofthe BS kernel. In ref. [9] we found that a smooth kernel, approximated by a harmonic interaction and with a 3,5X3,5 Dirac part yields a singlet-triplet structure for the spectrum with the mass formula
M2=M2o + 2Kx/~N;
N=n +2r;
n,r = 0,1,2 ....
(5)
and describes the strong decays of the meson resonances reasonably well for r --- 3 and x/rff= 0.2 GeV2 . We use the Wick-rotated BS amplitudes of this model + 2rrX/~ X"(k,P) = -~b'5(~M,,X1-'/~2m) +51.l'Ys[~c/m,P/Mr]]
x°rO(k,P) = ~
exp ( - k2/2x/~)lpfi}
[ ~ ( 1 - ~ / 2 m ) -iek/m] exp (-k2/2,~/~)
Lrl(k2/V~-)
(6)
(Ip~) -Infi)).
for the evaluation of the integrals in eqs. (3) and (4) and obtain
gor7 = 2,yor = ( - l l r Z
F"(Q2)=g°°~g°°~
x/rr-+T~
,
k 3-r(-1)rer/2L l(r+c) = M2o + 1 2 r ~ '
(7)
M2° - 4 M 2 c=
12X/~
=0.21.
(8)
For comparison with experiment in the time like region one has to take into account the finite widths of the vector mesons. We have used Breit Wigner forms with Fp = 150 MeV and Pp, = 500 MeV and choose the only free parameter, the renormalization constant Z such that F,r(0 ) = 1. The results are given in fig. i. The sum of the residues in eq. (7) converges, we therefore obtain for large space-like values of Q2. F,r(Q 2) ~ - 0.33 GeV2/Q2 .
(9)
We find asymptotically a simple pole behaviour. Thisresult differs substantially from calculations which take into account only the pion wave function and not the quark form factor, where one would obtain F(Q 2) = exp (Q2[ const.), reflecting the Gaussians of the wave functions. Our result, eqs. (8) and (9) is a consequence of the duality 458
Volume 50B, number 4
PHYSICS LETTERS
S -I
Q° z [OeV z] i
24 June 1974
4
9
Fig. 1. Comparison of our result for the pion form factor with experiment. The data points are from ref. [1, 2]. assumption that the p i o n f o r m factor can be built up by resonances w i t h o u t a background t e r m and the s p e c t r u m and the coupling constants o f the relativistic quark model. It is a great pleasure t o t h a n k Hans J o o s for clarifying discussions.
References [1] V.L. Auslander et al., Soy. J. Nucl. Phys. 9 (1,969) 69; D. Benaksas et al., Phys. Lett. 39B (1972) 289; V.E. Balakin et al., Phys. Lett. 41B (1972) 205; M. Bernardini et al., Phys. Lett. 46B (1973) 261 ; S.F. Berezhnev et al., Yadern. Fiz. 18 (1973) 102; [2] C.W. Akerlof et al., Phys. Rev. 163 (1967) 1482; C. Mistretta et al., Phys. Rev. 184 (1969) 1487; E. Amaldi et al., Nuovo Cim. 65A (1970) 377; C. Driver et al., Phys. Lett. 35B (1971) 77, 81; P.S. Kummer et al., Lett. Nuovo Cim. 1 (1971) 1026; C.N. Brown et al., Phys. Rev. D8 (1973) 92. [3] The pion form factor has been discussed f.i. by taking into account the finite width of the po : G.J. Goundaris and J.J. Sakurai, Phys. Rev. Lett. 21 (1968) 244; by modifying the rho propagator through inelastic ehan~; nels: F.M. Renard, Phys. Lett. 47B (1973) 361; in Bethe-Salpeter type models: J. Fleischer and F. Gutbrod, Nuovo Cim. 10A (1972) 235, in the dual resonance model: Y. Oyanagi, Progr. Theor. Phys. 42 (1969) 898; H. Suura, Phys. Rev. Lett. 23 (1969) 551;
R. Jengo and E. Remiddi, Nucl. Phys. BI5 (1970) 1; by generalizing rho dominance: M. Greeo and Y.N. Srivastava, LNF-73/7; in the parton model: P.V. Landshoff and J.C. Polkinghorne, Nucl. Phys. B53 (1973) 473; F. Ravndal, CALT-68-381 ; in the massive quark model: 1L Gatto and G. Preparata, DESY-73/15; by dimensional counting: S.J. Brodsky and G.R. Farrar, Phys. Rev. Lett. 31 (1973) 1153. [4] A. Bramon, E. Etim and M. Greco, Phys. Lett. 41B (1972) 609. [5 ] J.J. Sakurai and D. Schildknecht, Phys. Lett. 40B (1972) 121. [6] M. B/~hm, H. Joos and M. Krammer, Acta Phys. Austriaca 38 (1973) 123 and DESY 72/62. [71 M. Bfhm, H. Joos and M. Krammer, Acta Phys. Austriaea Suppl. XI (1973) 3 (Schladming Lectures 1973). [ 8 ] M. B6hm, H. Joos and M. Krammer, Nucl. Phys. B51 (1973) 397; M. Krammer, Internal report DESYT-73/1. [91 M. B6hm, H. Joos and M. Krammer, Ref. TH. 1715-CERN (to appear in Nucl. Phys.).
459