- Email: [email protected]
Electromagnetic form-factors of pions and kaons in the time-like region
' ~ .
E~E~ER
NUCLEAR PHYSICS
A
Nuclear Physics A623 (1997) 357c-360c
Electromagnetic
form-factors
o f p i o n s a n d k a o n s in t h e t i m e - l i k e r e g i o n
A. Bramon Grup de Fisica Tebrica, Universitat A u t b n o m a de Barcelona, 08193 Bellaterra (Barcelona), Spain The a"+, K + and ~-0 charge radii and form factors in the time-like region are analysed in an effective lagrangian context. This lagrangian, written in terms of the pseudoscalar, vector and photon fields, incorporates Vector Meson Dominance ideas and the main features of flavour symmetry. A reasonable description of the d a t a is achieved. 1. I N T R O D U C T I O N An e+e - machine such as D A O N E is the ideal tool to measure the electromagnetic form-factors ( F F ) of the light pseudoscalar mesons, P = 7r+, K + and K °, in the timelike region. Each F F is defined by the function Fp(s) depending exclusively on the square of the e+e - invaxiant-mass, s = q2, which controls the dynamics of the total e+e - --+ P P cross section
~$2 ~,+e-_~pp(S) =
2
4 m e )a/2
IFp(s)f(1 - - - 7 -
(1)
"
General physical arguments require Fp(s) = - F p ( s ) - - and hence the vanishing of the FF's of the truely neutral mesons P = P = ~r°,,7... - and that Fp(s) has an imaginary part only above the PP production threshold, as in D A ~ N E . We thus have to consider just three different FF's, F,~+ K+ K0(s), in three regions along the s > 0 axis: the first one near s = qZ = 0, where FF's are real; a second one from PP-thresholds up to ~_ 1.1 GeV, where the p,~o and ¢ resonances dominate the FF dynamics; and a third region at higher energies which is poorly known both experimentally and theoretically and will not be discussed in the present note. In the low-energy region the data are usually described in terms of the e.m. charge radii, < r~, >. These are defined as the slope at q2 = 0 of the corresponding FF, < r~ > 6dFp(q2)/dq21e=o,and are related to the spatial extension of the pseudoscalar P. From [1,2] we take the following experimental values for the various charge radii < r ~.+ > < r~¢+ > < r 2~+ > - < r ~ + >
= = =
0.44 4- 0.03 f m 0.31 -4- 0.05 fm 0.13+0.04fm
2, 2, 2.
- < r~o > < r~,~ >
= =
0.054 4- 0.026 0.36 4- 0.02
fm 2, fm 2, (2)
We will achieve a fully consistent theoretical understanding of these values in the context of Chirai Perturbation Theory ( C h P T ) [3]. 0375-9474/97/$17.00 © 1997 - Elsevier Science B.V. All rights reserved. PII: S0375-9474(97)00455-7
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
358c
There are also accurate data for Fp(s) in the region around the p,w and ¢ resonances (see [4] for the pion FF, and the reviews [5,6]). But the understanding of these data is possible only at a phenomenological level in terms of the old and successful notion of Vector Meson Dominance (VMD) [7]. This amounts to introduce in eq.(1)
Fp(s)
= ~ m~
gVPP "gv7
(3)
where gvPP and gv.y are the coupling constants of the relevant vector mesons V : p , w , ¢ to the final P P state and to the initial virtual 7. Asuming SU(3) honer symmetry for all these couplings and VMD universality [7], and introducing the physical (SU(3)-broken) masses and widths of V's, one obtains a surprisingly reasonable and economical description of all salient features of the above data. Other details of these data require further improvements on the conventional VMD model as we now proceed to discuss. 2. T H E L A G R A N G I A N The key ingredient is an effective lagrangian written in terms of the pseudoscalar, vector and photon fields, f ( P , Vv, A~,), incorporating the main successful features of VMD and flavour symmetry. DetMled discussions on this type of lagrangians can be found in the reviews [8,9], in the recent analysis by Harada and Schechter [10] or in ref.[ll], whose notation we axe going to follow. The effective lagrangiazl can be written as
£
:
£A+A£a+£v+A£v
-
~Tr{(mu~.~t+mu~.~t)2[l+(~ev~+~'ev~t)l},
(4)
where, as in ChPT, the pseudoscalar octet appears in ~ - (~ - e z p ( 2 i P / f ) , with ] ~_ 132 MeV for the pion decay constant. V, is the SU(3) matrix containing the vector honer (normalized as P) which appears together with the photon field in the covaziant derivatives D~,~ = (0, - igV,)~ + i e ~ A , . Q leading to .
_
.
(D~,~ . ~t + D~,~ . ~t)
=
¥(O.P
+ l e A . I F , QI +
PO.PP
+ ...)
:
- 2 i g V , + 2ieA, Q + -fi[O,P,P] + ...
(5)
The pieces f a y preserve SU(2)- and SU(3)-symmetry, which is broken only by the other two terms, AfA.v, containing the hermitean combinations (~eA,v~ + ~teA.V~t) of the matrices eA.v ~- k.4.v diag(1 - y, 1 + y, ~). These flavour-symmetry breaking matrices are proportional to the quark mass matrix f14 = diag(m,,,ma, m,) [11,10] and, from the recent determination of the quark-mass ratios [14], one can fix z = 24.4 + 1.5 and y : 0.29 + 0.03. We also fix g = 4.15 to obtain from (4) the correct value for the tightest V-mass, rn~ = 292 f 2 = 0.60GeV 2. This lagrangian is only of indirect interest when discussing the charge radii. Around q2 = 0 the ChPT series expansion is expected to converge rapidly thus making the ChPT
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
359c
approach the most convenient one. However the C h P T lagrangian [3] contains several terms closely related to the ones in (4). Thanks to this one can improve the well-known one-loop C h P T calculation [3] with corrections from the SU(3)-breaking counter-terms. One obtains (see refs.[3,11] for details)= ~+ 16-~-f2
3 + 2 1 n m~'+In #2
+--=0.46fm M~
2
-1 in_~2K+ 2 [1 ( 1 + 2cv)l = _0.043fro 2 < r~o > = 16~-f2 m~ ~ M~ ~/~ J 1 + cv 6 < r ~ > = chiral loops + ~ ME" - 0.33fro 2 < r~+ > - < r~+ > = 161r'f'
m,~
~
i + C----A
(6)
~-M-~-~ J = O'14fm2"
The numerical values follow from taking cv = k},z = 0.28, cA -- kAz = 0.36, as fixed in ref.[ll] when accounting for the SU(3)-breaking effects (but with no SU(2)-breaking, y = 0) in eq.(6), in the ratio fK/f,~ and in the V-meson mass spectrum. The agreement with the data is clearly a success of ChPT but it is also an indication of the possibilities of the lagrangian (4) to account for symmetry breaking corrections. The FF's for q2 _ mp.o,,¢2 cannot be understood in terms of C h P T (see ref.[12]) and thus the above lagrangian (4) becomes essential. It contains the vertices gypp and gv7 appearing in eq. (3). However, mixing among the ideal p, w and ¢ fields - - defined by the diagonal terms in V~,, namely, (p + w)/v~, (p - ¢v)/,¢~, ¢ - - requires a further refinement. One simply has to add to the lagrangian (4) the term +
+
and fix the new parameter v to 2v = 0.035 via the m~2 _ mp2 experimental mass difference [10]. With the corresponding values for m ~ , the V - 7 - V' conversions and the isospin violating m ~2 terms in (4), one then obtains the real part of the non-diagonal terms of the V mass matrix. The corresponding imaginary parts proceed exclusively through unitarity and can safely be computed. The mixing formalism of ref.[13] can then be applied to work with the physical (mixed) vector meson states V in our main eq.(3). 3. R E S U L T S Having fixed all the free parameters in our lagrangian we now proceed to confront additional predictions with the corresponding data for on-shell V-meson properties [2].
3.1. V-decays to PP-pairs Since the PDG compilation quotes the V ~ P P decay widths only for the sum of the various SU(2)-rotated charge states, we correspondingly ignore SU(2)-breaking effects. Using r ( V --. P P ) = g2pplffpl2/6~rm~ one obtains F(p --* 7r~r) = 142 MeV, F(K* ~ K~r) = 49 MeV, F(¢ --+ K / ( ) = 3.7 MeV, to be compared with [2] 150 4- 1, 50 4- 1 and 3.6 -4- 0.1 MeV, respectively.
(8)
360c
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
3.2. V-decays to e+e From r ( Y -~ e+e -) = 4 ~ 2 m v / 3 / ~ one easily gets
r(p ~ e+e -) = 4.9 MeV,
r(w - . e+e - ) = 0.63 MeV,
r ( ¢ ~ e+e -) = 1.36 MeV,
(9)
to compare with [2] 6.77 i 0.32, 0.60 ± 0.32 and 1.37 ± 0.05 keV, respectively. Notice that finite-width effects should increase our prediction for r(p --~ e+e-). 3.3. w,C-decays to ~-+~rOne has measured the modulations over the p-dominated pion form-factor, F,+(s) = F(P)ts~ + ~ j, due to the w and ¢ resonances. For the w case the usual parameterization of the data is in terms of a appearing in the expression ]F~(~)(s)l'll + ,F,(7)(s)l ~ which is proportional to a~+~-_~+,-(s ~- m~). The phase for a is measured to be ¢ = +4.1 + 5 degrees and its modulus [al leads to [4] r(w --. l r + r - ) / r ( p ~ r + r -) = (1.23±0.15). 10-s. Our approach predicts ¢ = +7.9 degrees and r(w ~ r + v - ) / r ( p ~ 7r%r-) = 0.99.10 -s. The experimental parameterization for ¢ -~ r+~r - leads to [15] Z~p ~- +0.08 - 0.05i, where now one has a~+~-__.,+,-(s ~- m~) proportional to [F~(P+)(s)[2]I-Z(r¢/m¢)F(~C+)(s)[2. We obtain Z - +0.16 - 0.07i. REFERENCES
1. A. Quenzer et al., Phys. Lett. B76 (1977) 512. E.B. Dally et al., Phys. Rev. Lett. 39 (1977) 1176. E.B. Dally et al., Phys. Rev. Lett. 48 (1982) 375. S.R. Amendolia et al., Phys. Lett. B178 (1986) 435. W.R. Molzon et al., Phys. Rev. Lett. 41 (1978) 1231. C. Erkal and M.G. Olsson, J. Phys. G: Nucl. Phys. 13 (1987) 1355. 2. Particle Data Group, Phys. Rev. D45 (1992) 1. 3. J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465, 517, 539. 4. L.M. Barkov et al., Nucl. Phys. B256 (1985) 365. 5. T . H . Bauer et al., Rev. Mod. Phys. 50 (1978) 261; S. I. Dollnsky et. al., Phys. Rep. C202 (1991) 99. 6. "The Second DA~NE Physics Handbook", ed. Maiani et al., 1995. 7. J . J . Sakurai, Phys. Rev. Lett. 17 (1966) 1012; 8. U.-G. Meissner, Phys. Rep. 16__!1(1988) 213. 9. M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B259 (1985) 493, and Phys. Rep. 164 (1988) 217. 10. M. Harada and J. Schechter, Phys. Rev. D64 (1996) 3394. 11. A. Bramon, A. Grau and G. Pancheri, Phys. Lett. B345 (1995) 263. 12. K. Maltman, Phys. Rev. D53 (1996) 2573. 13. M. Gourdin, L. Stodolsky and F. M. Renard, Phys. Left. 3OB (1969) 347. 14. H. Leutwyler, Phys. Lett. 378B (1996) 313. 15. I. B. Vasserman et al., Phys. Left. 99B (1981) 62; V. B. Golubev et al., Yad. Phys. 44 (1986) 633.
E~E~ER
NUCLEAR PHYSICS
A
Nuclear Physics A623 (1997) 357c-360c
Electromagnetic
form-factors
o f p i o n s a n d k a o n s in t h e t i m e - l i k e r e g i o n
A. Bramon Grup de Fisica Tebrica, Universitat A u t b n o m a de Barcelona, 08193 Bellaterra (Barcelona), Spain The a"+, K + and ~-0 charge radii and form factors in the time-like region are analysed in an effective lagrangian context. This lagrangian, written in terms of the pseudoscalar, vector and photon fields, incorporates Vector Meson Dominance ideas and the main features of flavour symmetry. A reasonable description of the d a t a is achieved. 1. I N T R O D U C T I O N An e+e - machine such as D A O N E is the ideal tool to measure the electromagnetic form-factors ( F F ) of the light pseudoscalar mesons, P = 7r+, K + and K °, in the timelike region. Each F F is defined by the function Fp(s) depending exclusively on the square of the e+e - invaxiant-mass, s = q2, which controls the dynamics of the total e+e - --+ P P cross section
~$2 ~,+e-_~pp(S) =
2
4 m e )a/2
IFp(s)f(1 - - - 7 -
(1)
"
General physical arguments require Fp(s) = - F p ( s ) - - and hence the vanishing of the FF's of the truely neutral mesons P = P = ~r°,,7... - and that Fp(s) has an imaginary part only above the PP production threshold, as in D A ~ N E . We thus have to consider just three different FF's, F,~+ K+ K0(s), in three regions along the s > 0 axis: the first one near s = qZ = 0, where FF's are real; a second one from PP-thresholds up to ~_ 1.1 GeV, where the p,~o and ¢ resonances dominate the FF dynamics; and a third region at higher energies which is poorly known both experimentally and theoretically and will not be discussed in the present note. In the low-energy region the data are usually described in terms of the e.m. charge radii, < r~, >. These are defined as the slope at q2 = 0 of the corresponding FF, < r~ > 6dFp(q2)/dq21e=o,and are related to the spatial extension of the pseudoscalar P. From [1,2] we take the following experimental values for the various charge radii < r ~.+ > < r~¢+ > < r 2~+ > - < r ~ + >
= = =
0.44 4- 0.03 f m 0.31 -4- 0.05 fm 0.13+0.04fm
2, 2, 2.
- < r~o > < r~,~ >
= =
0.054 4- 0.026 0.36 4- 0.02
fm 2, fm 2, (2)
We will achieve a fully consistent theoretical understanding of these values in the context of Chirai Perturbation Theory ( C h P T ) [3]. 0375-9474/97/$17.00 © 1997 - Elsevier Science B.V. All rights reserved. PII: S0375-9474(97)00455-7
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
358c
There are also accurate data for Fp(s) in the region around the p,w and ¢ resonances (see [4] for the pion FF, and the reviews [5,6]). But the understanding of these data is possible only at a phenomenological level in terms of the old and successful notion of Vector Meson Dominance (VMD) [7]. This amounts to introduce in eq.(1)
Fp(s)
= ~ m~
gVPP "gv7
(3)
where gvPP and gv.y are the coupling constants of the relevant vector mesons V : p , w , ¢ to the final P P state and to the initial virtual 7. Asuming SU(3) honer symmetry for all these couplings and VMD universality [7], and introducing the physical (SU(3)-broken) masses and widths of V's, one obtains a surprisingly reasonable and economical description of all salient features of the above data. Other details of these data require further improvements on the conventional VMD model as we now proceed to discuss. 2. T H E L A G R A N G I A N The key ingredient is an effective lagrangian written in terms of the pseudoscalar, vector and photon fields, f ( P , Vv, A~,), incorporating the main successful features of VMD and flavour symmetry. DetMled discussions on this type of lagrangians can be found in the reviews [8,9], in the recent analysis by Harada and Schechter [10] or in ref.[ll], whose notation we axe going to follow. The effective lagrangiazl can be written as
£
:
£A+A£a+£v+A£v
-
~Tr{(mu~.~t+mu~.~t)2[l+(~ev~+~'ev~t)l},
(4)
where, as in ChPT, the pseudoscalar octet appears in ~ - (~ - e z p ( 2 i P / f ) , with ] ~_ 132 MeV for the pion decay constant. V, is the SU(3) matrix containing the vector honer (normalized as P) which appears together with the photon field in the covaziant derivatives D~,~ = (0, - igV,)~ + i e ~ A , . Q leading to .
_
.
(D~,~ . ~t + D~,~ . ~t)
=
¥(O.P
+ l e A . I F , QI +
PO.PP
+ ...)
:
- 2 i g V , + 2ieA, Q + -fi[O,P,P] + ...
(5)
The pieces f a y preserve SU(2)- and SU(3)-symmetry, which is broken only by the other two terms, AfA.v, containing the hermitean combinations (~eA,v~ + ~teA.V~t) of the matrices eA.v ~- k.4.v diag(1 - y, 1 + y, ~). These flavour-symmetry breaking matrices are proportional to the quark mass matrix f14 = diag(m,,,ma, m,) [11,10] and, from the recent determination of the quark-mass ratios [14], one can fix z = 24.4 + 1.5 and y : 0.29 + 0.03. We also fix g = 4.15 to obtain from (4) the correct value for the tightest V-mass, rn~ = 292 f 2 = 0.60GeV 2. This lagrangian is only of indirect interest when discussing the charge radii. Around q2 = 0 the ChPT series expansion is expected to converge rapidly thus making the ChPT
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
359c
approach the most convenient one. However the C h P T lagrangian [3] contains several terms closely related to the ones in (4). Thanks to this one can improve the well-known one-loop C h P T calculation [3] with corrections from the SU(3)-breaking counter-terms. One obtains (see refs.[3,11] for details)
3 + 2 1 n m~'+In #2
+--=0.46fm M~
2
-1 in_~2K+ 2 [1 ( 1 + 2cv)l = _0.043fro 2 < r~o > = 16~-f2 m~ ~ M~ ~/~ J 1 + cv 6 < r ~ > = chiral loops + ~ ME" - 0.33fro 2 < r~+ > - < r~+ > = 161r'f'
m,~
~
i + C----A
(6)
~-M-~-~ J = O'14fm2"
The numerical values follow from taking cv = k},z = 0.28, cA -- kAz = 0.36, as fixed in ref.[ll] when accounting for the SU(3)-breaking effects (but with no SU(2)-breaking, y = 0) in eq.(6), in the ratio fK/f,~ and in the V-meson mass spectrum. The agreement with the data is clearly a success of ChPT but it is also an indication of the possibilities of the lagrangian (4) to account for symmetry breaking corrections. The FF's for q2 _ mp.o,,¢2 cannot be understood in terms of C h P T (see ref.[12]) and thus the above lagrangian (4) becomes essential. It contains the vertices gypp and gv7 appearing in eq. (3). However, mixing among the ideal p, w and ¢ fields - - defined by the diagonal terms in V~,, namely, (p + w)/v~, (p - ¢v)/,¢~, ¢ - - requires a further refinement. One simply has to add to the lagrangian (4) the term +
+
and fix the new parameter v to 2v = 0.035 via the m~2 _ mp2 experimental mass difference [10]. With the corresponding values for m ~ , the V - 7 - V' conversions and the isospin violating m ~2 terms in (4), one then obtains the real part of the non-diagonal terms of the V mass matrix. The corresponding imaginary parts proceed exclusively through unitarity and can safely be computed. The mixing formalism of ref.[13] can then be applied to work with the physical (mixed) vector meson states V in our main eq.(3). 3. R E S U L T S Having fixed all the free parameters in our lagrangian we now proceed to confront additional predictions with the corresponding data for on-shell V-meson properties [2].
3.1. V-decays to PP-pairs Since the PDG compilation quotes the V ~ P P decay widths only for the sum of the various SU(2)-rotated charge states, we correspondingly ignore SU(2)-breaking effects. Using r ( V --. P P ) = g2pplffpl2/6~rm~ one obtains F(p --* 7r~r) = 142 MeV, F(K* ~ K~r) = 49 MeV, F(¢ --+ K / ( ) = 3.7 MeV, to be compared with [2] 150 4- 1, 50 4- 1 and 3.6 -4- 0.1 MeV, respectively.
(8)
360c
A. Bramon/Nuclear Physics A623 (1997) 357c-360c
3.2. V-decays to e+e From r ( Y -~ e+e -) = 4 ~ 2 m v / 3 / ~ one easily gets
r(p ~ e+e -) = 4.9 MeV,
r(w - . e+e - ) = 0.63 MeV,
r ( ¢ ~ e+e -) = 1.36 MeV,
(9)
to compare with [2] 6.77 i 0.32, 0.60 ± 0.32 and 1.37 ± 0.05 keV, respectively. Notice that finite-width effects should increase our prediction for r(p --~ e+e-). 3.3. w,C-decays to ~-+~rOne has measured the modulations over the p-dominated pion form-factor, F,+(s) = F(P)ts~ + ~ j, due to the w and ¢ resonances. For the w case the usual parameterization of the data is in terms of a appearing in the expression ]F~(~)(s)l'll + ,F,(7)(s)l ~ which is proportional to a~+~-_~+,-(s ~- m~). The phase for a is measured to be ¢ = +4.1 + 5 degrees and its modulus [al leads to [4] r(w --. l r + r - ) / r ( p ~ r + r -) = (1.23±0.15). 10-s. Our approach predicts ¢ = +7.9 degrees and r(w ~ r + v - ) / r ( p ~ 7r%r-) = 0.99.10 -s. The experimental parameterization for ¢ -~ r+~r - leads to [15] Z~p ~- +0.08 - 0.05i, where now one has a~+~-__.,+,-(s ~- m~) proportional to [F~(P+)(s)[2]I-Z(r¢/m¢)F(~C+)(s)[2. We obtain Z - +0.16 - 0.07i. REFERENCES
1. A. Quenzer et al., Phys. Lett. B76 (1977) 512. E.B. Dally et al., Phys. Rev. Lett. 39 (1977) 1176. E.B. Dally et al., Phys. Rev. Lett. 48 (1982) 375. S.R. Amendolia et al., Phys. Lett. B178 (1986) 435. W.R. Molzon et al., Phys. Rev. Lett. 41 (1978) 1231. C. Erkal and M.G. Olsson, J. Phys. G: Nucl. Phys. 13 (1987) 1355. 2. Particle Data Group, Phys. Rev. D45 (1992) 1. 3. J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465, 517, 539. 4. L.M. Barkov et al., Nucl. Phys. B256 (1985) 365. 5. T . H . Bauer et al., Rev. Mod. Phys. 50 (1978) 261; S. I. Dollnsky et. al., Phys. Rep. C202 (1991) 99. 6. "The Second DA~NE Physics Handbook", ed. Maiani et al., 1995. 7. J . J . Sakurai, Phys. Rev. Lett. 17 (1966) 1012; 8. U.-G. Meissner, Phys. Rep. 16__!1(1988) 213. 9. M. Bando, T. Kugo and K. Yamawaki, Nucl. Phys. B259 (1985) 493, and Phys. Rep. 164 (1988) 217. 10. M. Harada and J. Schechter, Phys. Rev. D64 (1996) 3394. 11. A. Bramon, A. Grau and G. Pancheri, Phys. Lett. B345 (1995) 263. 12. K. Maltman, Phys. Rev. D53 (1996) 2573. 13. M. Gourdin, L. Stodolsky and F. M. Renard, Phys. Left. 3OB (1969) 347. 14. H. Leutwyler, Phys. Lett. 378B (1996) 313. 15. I. B. Vasserman et al., Phys. Left. 99B (1981) 62; V. B. Golubev et al., Yad. Phys. 44 (1986) 633.