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Electromagnetic from factors of charged and neutral kaons
28 March 1996
PHYSICS LETTERS 6
ELSEVIER
Physics Letters B 37 1 ( 1996) 163- I68
Electromagnetic form factors of charged and neutral kaons C.J. Burden a, C.D. Roberts b, M.J. Thomson c a Department ofTheoretical Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia b Physics Division. Bldg. 203, Argonne National Laboratory, Argonne, IL 60439-4843, ’ School of Physics, University of Melbourne, Parkville, WC. 3052, Australia
USA
Received 13 November 1995; revised manuscript received 27 December 1995 Editor: C. Mahaux
Abstract The charged and neutral kaon form factors are calculated as a phenomenological application of model QCD DysonSchwinger equations. The results are compared with the pion form factor calculated in the same framework and yield F,+ ( Q2) > F,+ ( Q2) on Q2 E [0,3] GeV2; and a negative charge radius for the neutral kaon. These results are sensitive to the difference between the kaon and pion Bethe-Salpeter amplitude and the u- and s-quark propagation characteristics. Keywords:
Hadron physics FKf ( Q2 ) , Fe ( Q2), F,+ ( Q2) ; Dyson-Schwinger equations; Confinement; Nonperturbative QCD
phenomenology
1. Introduction The kaon is the simplest strangeness-carrying bound state. Hence studies of kaon observables and their
comparison with analogous pion properties provide information about SU, (3) -breaking in QCD. Particularly interesting are nonperturbative effects, defined as those whose source is the current-quark mass difference, m, -mu, but which cannot simply be described as a linear response to this difference. Examples are the difference between the kaon and pion Bethe-Salpeter amplitude and that between the U- and s-quark propagation characteristics at small and intermediate k2; i.e., k2 E [0, 1 N 21 GeV2. The electromagnetic form factors of the charged and neutral kaon are sensitive to these effects, are accessible to experiments at CEBAP [ I] and have been the subject of a recent study [ 21. Herein we report a
calculation of a range of pion and kaon observables via a phenomenological application of the DysonSchwinger equations (DSEs) [3] in QCD; our primary focus being the elastic charged and neutral kaon form factors. In calculating these form factors we employ a generalised-impulse approximation, in which the quark propagators (2-point Schwinger functions), meson Bethe-Salpeter amplitudes (meson-quark vertices) and quark-photon vertices are dressed quantities whose form follows from nonperturbative, model DSE studies in QCD [3]. In this way our calculation provides for an extrapolation of the known large spacelike-k2 behaviour of these Schwinger functions to the small spacelike-k2 region, where they are unknown and confinement effects are manifest. This facilitates an exploration of the relationship between physical observables and the nonperturbative, infrared behaviour of these Schwinger functions.
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved (96)00006-8
PII SO370-2693
164
CJ. Burden er ul./ Physics Lenrrs B 371 (1996) 163- 168
This calculation is a reanalysis and extension of a study of the pion [4]. The quark propagator has no Lehmann representation and hence may be interpreted as describing a confined particle since this feature is sufficient to ensure the absence of quark production thresholds in S-matrix elements describing colour-singlet to singlet transitions. The quark-photon vertices, which describe the coupling of a photon to the dressed U- and s-quarks, follow from extensive QED studies and satisfy the Ward-Takahashi identity. This necessarily entails that the y-K-K and y-r+ amplitudes are current conserving. In the chiral limit the quark-pseudoscalar vertex is completely determined by the scalar part of the quark self-energy. The extension to finite current-quark masses requires a minimal modification and preserves Dashen’s relation [ 51.
2. Genera&d
impulse approximation to FK(Q”)
In Euclidean space, with metric S,, = diag( 1, 1, 1, l), Yfi = yf* and {yr, yy } = 2 S,,, the generalised impulse approximation to the y-K+-K- vertex is given by A$* (p, -4) = $A;(& -4) + 31is p (p. -(I> t
(1)
where Ak(P, -4) = 2N,
s
d4k -Wh(k;p) (2r)4
qJP*
- 4);
s
= - f$(P,
-4)
+ jqp,
-9).
(4)
Herein, except for the electric charge, the u and d quarks are identical and hence AE(p, -4) = At(p, -q), defined in Eq. (2). For elastic scattering [ p2 = $1 and AE*IKO(p, -4) = (p + q)r FK*iKo(Q2), where F K*IKo(Q2) is the electromagnetic form factor.
2.1. Dressed quark propagator
-9)
Ss(k
- /%)I,
(2)
-4)
= 2N,
A; (P, -4)
&(k+ap)
xir”,(k+rrp,k--pp+q)S(k-Pp+q) x rK(k - P(P
mass momentum P; and Ii is the flavour dependent photon-quark vertex. The variable cx [/I = 1 a] allows for uneven partitioning of the total momentum between the u- and s-quark legs connected to the Bethe-Salpeter amplitude. The choice of the relative-momentum p as the argument of the BetheSalpeter amplitude entails (Y= 0.5. The requirement that Fc (Q2 = 0) = 0 is a measure of the efficacy of our approximation for the kaon Bethe-Salpeter amplitude. The first of the two contributions in Eq. ( 1) corresponds to the u-quark interacting with the photon and the s-quark acting as a spectator; in the second the roles are reversed. If the current-quark masses are set equal in Eq. ( 1) the generalised impulse approximation to the y-7r-r vertex [ 4,6] is recovered, provided that IK -t I,,, fK --f fr and cy = l/2, which is required by charge conjugation symmetry in this case. The generalised impulse approximation to the neutral-kaon-photon vertex is given by
* tro[IK(k;p) (27r)4
S,(k+ Pp)
xi$(k+fip,k-apfq)&(k--ap+qJ
xr~(k-‘~(p-q);-q)SU(k-ap)l.
in Eqs. (2) and Sf = -iy -p af(p2) + ai(p2) (3) can be obtained by solving the quark DysonSchwinger equation [ 33. Model studies using a gluon propagator and quark-gluon vertex that manifest the qualitative features suggested by Refs. [ 7,8] provide a basis for the following approximating algebraic, model forms for CT{and a{ 141:
(3)
Here q is the initial momentum of the kaon, p - q z Q is the photon momentum and only the Dirac trace remains to be evaluated. In Eqs. (2) and (3): Sf is the propagator for a quark of flavour f = u, s; r~ (p; P) is the kaon Bethe-Salpeter amplitude, with quark-antiquark relative momentum p and centre-of-
G(X) = CL,e-2x f
&
f
(1 _ e-2(x+4))
CJ. Burden et al./Physics
i?{(x) =
2(x +
i”;> -
1+
e--2(x+6)
2( n + r$,’
-fi,c~I (?-2X ’ (6)
and: &J(X) = 2Dav(p2); where x = p2/(2D) *s(n) = v%as(p2); and fif = mf/m, with D a mass scale. The parameters CL,, fif, bf,,,s are determined either by: 1) fitting a quark-DSE solution obtained with a realistic gluon propagator; or 2) performing a X2-fit to a range of hadronic observables. (A = 10e4 is included to decouple the small and large spacelike-p2 behaviour of the l/p4 term, characterised by bof and bl.) We write the inverse of the quark propagator as 57’(p)
= iy*pAf(p2)
+ Bf(p2).
(7)
The quark propagator described by Eqs. (5), (6) is an entire function in the finite complex-p2. It therefore admits the interpretation that it describes a confined particle [ 31. The N e-’ form that ensures this is suggested by the algebraic solution of the model DSE studied in Ref. [ 91, which employed a confining model gluon propagator and dressed quark-gluon vertex. Furthermore, the behaviour of this model form on the spacelike-p2 axis is such that, neglecting ln[p2] corrections associated with the anomalous dimension of the quark propagator in QCD, it manifests asymptotic freedom. It has a term associated with dynamical chiral symmetry breaking (N 1/x2) and a term associated with explicit chiral symmetry breaking (- m/x). 2.2. Pseudoscalar meson Bethe-Salpeter amplitude r~ in Eq. (1) is the solution of an homogeneous Bethe-Salpeter equation (BSE) . Many studies of this BSE suggest strongly that the amplitude is c( 3/s [ lo], Furthermore, in the chiral limit the pseudoscalar BSE and quark DSE are identical [ 111 and one has a massless excitation in the pseudoscalar channel with IO- (p; P2 = 0) = iYsB,a(p2)/f?r, where Bd(p*) is given in Eq. (7) with mf = 0. This is the realisation of Goldstone’s theorem in the DSE framework; i.e., in the chiral limit Eqs. (5) and (6) completely determine I’,-. Herein, based on these observations, we employ the approximations
Letters B 371 (1996) 163-168
165
T,(p;
P2 = -m:)
x iys $
r~ (p;
P2 = -m2,) x iy5 J- %,,0(p2). .fK
?T
B;JP2),
(8)
(9)
For the pion this is a good approximation, both pointwise and in terms of the values obtained for physical observables [ 121. For the kaon it is an exploratory Ansatz, one which need only be accurate as an approximation to the integrated strength.
2.3. Quark-photon vertex I7i (~1, p2 ) in Eq. ( 1) satisfies a DSE that describes both strong and electromagnetic dressing of the quarkphoton vertex. Solving this equation is a difficult problem, which has recently begun to be addressed [ 131. Much progress has been made in developing a realistic Ansatz for rcL(pl ,p2) [ 81. The bare vertex, Iif;(Pr ,p2) = ycc, is inadequate when the quark propagator has momentum dependent dressing because it violates the Ward-Takahashi identity. In Ref. [ 141 the following form was proposed
rfQ (p ’ 4) =Zf A (p2 +K,
’
42) Y P
[;r&(p2,q2)
with K =p+q,Ci(p2,q2) Af,(p2
q2)
=
[Af(p2)
(10)
-i&p2,q2)],
= [Af(p2)+Af(q2)]/2, - Af(q2)]/[p2
- q2] and
*L(p2:q2)
= [Bf(p2) - Bf(q2)]/[p2 - q2]. This Ansatz is completely determined by the dressed quark propagator; satisfies the Ward-Takahashi identity; has a well defined limit as p2 + q2; transforms correctly under C, P, T and Lorentz transformations; and reduces to the bare vertex in the manner prescribed by perturbation theory. Furthermore, it is relatively simple and hence an ideal form to be employed in our phenomenological studies. Using charge conjugation it is straightforward to show that for elastic scattering one has = 0 in generalised impulse (P - 9)pA;*D(P+-4)
approximation. The result FK* (-rni, Q2 = 0) = 1 follows because the quark-photon vertex satisfies the Ward identity.
166
CJ. Burden et al./Physic.s Letters B 37/ (1996) 163-168
3. Calculated
spacelike kaon form factors
In the Breit frame: p = (O,O, i Q, iEK), q = the calculaEK = dm; tion of each meson form factor involves the numerical evaluation of a three-dimensional integral via straightforward numerical quadrature. To fix the parameters in the quark propagators we have revised the study of Ref. [4] and re-fitted the pion observables using the pion mass formula derived in Ref. [ 121, which yields an accurate estimate of the mass obtained by solving the generalised-ladder approximation to the pion Bethe-Salpeter equation. This reanalysis, carried out with the constraint C& +a = 0, leads to
(WA-;Q,iEd,
Ci”=s = 0.121,
fiu = 0.00897,
b; = 0.131,
b; = 2.90,
b; = 0.603,
b’; = 0.185,
(11)
with the mass scale D = 0.160 GeV2 chosen so as to give fc = 92.4 MeV. 3.1. Difeerences between u- and s-quark propagators Based on the studies of Ref. [ 161 we set fi, = 12.5 (ti, + tid) 3 25 ti,,,
b; =0.8b&
b; = brf,
The current theoretical prejudice [ 181 is that (ss)“=c N (0.5 - 0.8) (au) “a. We have explored the response of our calculated kaon obsetvables to variations in (5s). The sensitivity is to0 weak ( < 1%
(14)
CA, = C& = 0,
(15)
and allowed variations only in the parameter b$ To provide for a difference between I,, and I’K we also allowed c;,$) # c;,*. Having fixed the u-quark parameters in the pion sector we then have a two-parameter extension of the model to the s-quark sector. These parameters are fixed by requiring that the model reproduce, as well as possible, the experimental values for the dimensionless quantities: f~/f$, = 1.22 f 0.02, rKf jr& = 0.88 f 0.07 and mK/ fK = 4.37 f 0.05. The kaon mass is obtained by solving IIK( P’) = 0 where =8N,
J
(12)
= -(2D)i
b; = b;.
To allow for a minimal residual difference between the u and s quark propagators we did not allow C& to vary, simply setting
&(P2)
where iir, = fid herein, which is consistent with the theoretical estimates summarised in Ref. [ 171. The results we report herein are qualitatively and quantitatively insensitive to halving or doubling this ratio: the most sensitive, rK*, changing by < 5%. Dyson-Schwinger equation studies show that there are differences between the U- and s-quark propagators that cannot be accounted for simply by changing the mass in Eqs. (5) and (6). An example is the vacuum quark condensate, which, using the model quark propagator defined in Eqs. (5)) (6) and the definition in Ref. [ 41 (with cy= 1 in this model), is given by
($qf)rf
on this range) to provide a robust, independent estimate of (is), however, our calculations favour larger values. Hence in the calculations reported herein we used (Ss)‘” = 0.8 (iiu)‘“, which was implemented by setting
- [B&,(k2)]2 +
d4k &,,,(k2)&,Ak2) (27r)2 ( Ik+.k-d”v,“~O(k+)~,~+o(k-)
~,~o(k+)a~,,,~o(k-)l
>
9
(16)
with k+ = k + aP, k- = k - PP. For P2 + rni N 0, &(P2) x f:: (P2 + m$) with
fi = $
WP2) (p2__ml ’ - I
(17)
which also ensures the correct (unit) normalisation of the charged kaon form factor. These expressions are straightforward generalisations of their analogues in Ref. [ 121. Similar expressions are obtained in Ref. [24]. Following this procedure we obtain C;“, = 1.69,
b; = 0.74.
(18)
With fi, # fi,, Fe(Q2=O) =O for a = 0.49 (” l/2), as anticipated in our approximation of the kaon
CJ. Burden et d/Physics
Table 1 n and K observables calculated using the parameters of Eqs. ( 11) and (18). with the constraints of Eqs. (12). (14) and (15); g,,+ is discussed in Ref. [4] and F3”(4mi) in Ref. [ 151. The quoted “‘experimental” values of n$‘&, mf ,_vz, (I@, oev~and (Ss) , cev~are representative of other theoret& estimates. Most experimental values are extracted from Ref. [ 171; r,,. rK* and rh are taken from Refs. [ 19-2 11, respectively. The r-n scattering lengths, a:, are discussed in Ref. [22] Calculated
a7
'nK ma= I c&v2
T?rf ‘Kf
fK/fn rK*
fr,,f
0.6
163-168
167
-
Experiment
0.0924 GeV 0.113 0.1385 0.4936 0.0051 0.128
0.0924 f 0.001 0.113 f 0.001 0.1385 0.4937 0.0075 0.1 N 0.3
0.221
0.220
0.205 0.56 fm 0.49 -0.020 fm2 0.505 (dimensionless) 1.04 0.17 -0.048 0.030 0.0015 -0.0002 1 1.22 0.87
Letters B 371 (1996)
-
/ I
0.6 0.0
0.1 d(Geti)
1. Calculated form of 1FKf ( Q2) I2 compared with the availdata [ 20,231.
0.175 - 0.205 0.663 f 0.006 0.583 f 0.041 -0.054 zt 0.026 0.504 f 0.019 1 (anomaly) 0.21 f 0.01 -0.040 f 0.003 0.038 f 0.003 0.0017 f 0.0003 1.22 f 0.01 0.88 f 0.06
_------_--___
Bethe-Salpeter amplitude as a function of the relative quark-antiquark momentum. A comparison between calculation and experiment is presented in Table 1. The calculated form factors are presented in Figs. 1,2. The difference between the calculated and measured values of the charge radii and scattering lengths in Table 1 is a measure of the importance of final-state, pseudoscalar rescattering interactions and photon-vector-meson mixing, which are not included in the generalised-impulse approximation [ 61. Calculations of such effects for the pion [6] and p- and w-mesons [ 251 suggest that they contribute less than N 15% and become unimportant for Q2 > 1 GeV*. Herein we find that the calculated values of f~/f,, and r~/r,, agree with the experimental values of these ratios, independent of the details of our parametrisation. This makes it plausible that such effects are no more important for the kaon than for the pion.
0.0
1.0
2.0
3.0
Q*(Geti) Fig. 2. Calculated form factors: Q2 FKf : solid line; Q2 F,+ : shofi-dashed line; Q2 Fe: long dashed line. The difference between FK* and F,* for Q* > 3 GeV* is small but is amplified in this figure because of the multiplication by Q2.
4. Summary and discussion We find that on the range of Q2 currently accessible to experiment FK* (Q*) > I?,( Q2). This is qualitatively consistent with Ref. [2], however, our calculated results, for both form factors, are uniformly smaller in magnitude; as is the difference between them. The peak in Q2F( Q*), which is most pronounced for the K*-meson, is a signal of quarkantiquark recombination into the final state meson in the exclusive elastic scattering process. For Q* > 3 GeV2 we have F,+ (Q*) < Fw(Q2)
168
CJ. Burden et d/Physics
but the difference is small and sensitive to the form of the kaon Bethe-Salpeter amplitude. The behaviour of FKI (Q*) at Q* > 2 N 3 GeV* is influenced by details of the Ansatz for the kaon Bethe-Salpeter amplitude, Eq. (9)) that are not presently constrained by data. The results obtained for Q* < 2 GeV* are not sensitive to details of our parametrisation. We therefore view the results for Q* > 2 GeV* with caution. These observations emphasise that measurement of the electromagnetic form factors is a probe of the bound state structure of the meson; i.e., its BetheSalpeter amplitude. For the neutral kaon r$ < 0. The fact that Fe (Q*) f 0 is a manifestation of the u-s currentquark mass difference. We note that charge conjugation symmetry ensures F+I (Q*) = 0. Our calculated results are not sensitive to changes in the M, in the range ms/maveE [ 15,301 nor to changes in (Ss) in the range (Ss)/(iiu) E [0.5,1.0]. Nevertheless, the requirement that the calculation reproduce known values of kaon observables does lead to differences between the u- and s-quark propagators. This emphasises that measurement of the form factors is also a probe of nonperturbatively generated differences between the u- and s-quark propagation characteristics.
Acknowledgments
We acknowledge useful conversations with A.J. Davies. This work was supported by the National Science Foundation under grant no. INT92-15223; the Australian Research Council under grant no. S02947481; and the US Department of Energy, Nuclear Physics Division, under contract number W-3 I109-ENG-38. The calculations described herein were carried out using the resources of the National Energy Research Supercomputer Center.
Letters B 371 (1996) 163-168
References
[II CEBAF experimentE-91-016, Spokesperson B. Zeidman; CEBAF experimentE-93-018, Spokesperson O.K. Baker. I21 W.W. Buck, R.A. Williams and H. Ito, Phys. Lea. B 351 (1995)
24.
[31 C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477. [41 C.D. Roberts, in: Chiral Dynamics: Theory and Experiment, A.M. Bernstein and B.R. Holstein @Ids.), Lecture Notes in Physics, Vol. 452, p. 68 (Springer, Berlin 1995). [51 R. Dashen, Phys. Rev. 183 (1969) 1245. [61 R. Alkofer, A. Bender and C.D. Roberts, Intern. J. Mod. Phys. A. 10 (1995) 3319. [71 N. Brown and M.R. Pennington, Phys. Rev. D 39 ( 1989) 2266. [81 A. Bashir and M.R. Pennington, Phys. Rev. D 50 ( 1994) 7679. [91 C.J. Burden, C.D. Roberts and A.G. Williams, Phys. Len. B 285 ( 1992) 347. II01 P Jain and H. Munczek, Phys. Rev. D 48 ( 1993) 5403, and references therein. [Ill R. Delbourgo and M.D. Scadron, J. Phys. G 5 (1979) 1621. [I21 M.R. Frank and C.D. Robetts, Model gluon propagator and pion and rho-meson observables, hep-phI9508225; to appear in Phys. Rev. C. M.R. Frank, Phys. Rev. C 51 (1995) 987. J.S. Ball and T.-W. Chiu, Phys. Rev. D22 (1980) 2542. R. Alkofer and CD. Roberts, hep-ph/9510284, Phys. Lett. B 369 (1996) 101. [I61 C.J. Burden, Lu Qian, C.D. Roberts, PC. Tandy and M.J. Thomson, work in progress. [I71 Particle Data Group, Phys. Rev. D 50 (1994) 1173. 1181 S. Narison, QCD Spectral Sum Rules (World Scientific, Singapore, 1989), and references therein. [191 S.R. Amendolia et al., Nucl. Phys. B 277 (1986) 168. [201 S.R. Amendolia et al., Phys. I_&. B 178 ( 1986) 435. [211 W.R. Molzon et al., Phys. Rev. L&t. 41 (1978) 1213. [221 C.D. Roberts, R.T. Cahill, M.E. Sevior and N. Iannella, Phys. Rev. D 49 (1994) 125. ~231 E.B. Dally et al., Phys. Rev. L&t. 45 (1980) 232. [241 PC. Tandy, private communication. 1251 L.C.L. Hollenberg, C.D. Roberts and B.H.J. McKellar, Phys. Rev. C 46 (1992) 2057.
PHYSICS LETTERS 6
ELSEVIER
Physics Letters B 37 1 ( 1996) 163- I68
Electromagnetic form factors of charged and neutral kaons C.J. Burden a, C.D. Roberts b, M.J. Thomson c a Department ofTheoretical Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia b Physics Division. Bldg. 203, Argonne National Laboratory, Argonne, IL 60439-4843, ’ School of Physics, University of Melbourne, Parkville, WC. 3052, Australia
USA
Received 13 November 1995; revised manuscript received 27 December 1995 Editor: C. Mahaux
Abstract The charged and neutral kaon form factors are calculated as a phenomenological application of model QCD DysonSchwinger equations. The results are compared with the pion form factor calculated in the same framework and yield F,+ ( Q2) > F,+ ( Q2) on Q2 E [0,3] GeV2; and a negative charge radius for the neutral kaon. These results are sensitive to the difference between the kaon and pion Bethe-Salpeter amplitude and the u- and s-quark propagation characteristics. Keywords:
Hadron physics FKf ( Q2 ) , Fe ( Q2), F,+ ( Q2) ; Dyson-Schwinger equations; Confinement; Nonperturbative QCD
phenomenology
1. Introduction The kaon is the simplest strangeness-carrying bound state. Hence studies of kaon observables and their
comparison with analogous pion properties provide information about SU, (3) -breaking in QCD. Particularly interesting are nonperturbative effects, defined as those whose source is the current-quark mass difference, m, -mu, but which cannot simply be described as a linear response to this difference. Examples are the difference between the kaon and pion Bethe-Salpeter amplitude and that between the U- and s-quark propagation characteristics at small and intermediate k2; i.e., k2 E [0, 1 N 21 GeV2. The electromagnetic form factors of the charged and neutral kaon are sensitive to these effects, are accessible to experiments at CEBAP [ I] and have been the subject of a recent study [ 21. Herein we report a
calculation of a range of pion and kaon observables via a phenomenological application of the DysonSchwinger equations (DSEs) [3] in QCD; our primary focus being the elastic charged and neutral kaon form factors. In calculating these form factors we employ a generalised-impulse approximation, in which the quark propagators (2-point Schwinger functions), meson Bethe-Salpeter amplitudes (meson-quark vertices) and quark-photon vertices are dressed quantities whose form follows from nonperturbative, model DSE studies in QCD [3]. In this way our calculation provides for an extrapolation of the known large spacelike-k2 behaviour of these Schwinger functions to the small spacelike-k2 region, where they are unknown and confinement effects are manifest. This facilitates an exploration of the relationship between physical observables and the nonperturbative, infrared behaviour of these Schwinger functions.
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved (96)00006-8
PII SO370-2693
164
CJ. Burden er ul./ Physics Lenrrs B 371 (1996) 163- 168
This calculation is a reanalysis and extension of a study of the pion [4]. The quark propagator has no Lehmann representation and hence may be interpreted as describing a confined particle since this feature is sufficient to ensure the absence of quark production thresholds in S-matrix elements describing colour-singlet to singlet transitions. The quark-photon vertices, which describe the coupling of a photon to the dressed U- and s-quarks, follow from extensive QED studies and satisfy the Ward-Takahashi identity. This necessarily entails that the y-K-K and y-r+ amplitudes are current conserving. In the chiral limit the quark-pseudoscalar vertex is completely determined by the scalar part of the quark self-energy. The extension to finite current-quark masses requires a minimal modification and preserves Dashen’s relation [ 51.
2. Genera&d
impulse approximation to FK(Q”)
In Euclidean space, with metric S,, = diag( 1, 1, 1, l), Yfi = yf* and {yr, yy } = 2 S,,, the generalised impulse approximation to the y-K+-K- vertex is given by A$* (p, -4) = $A;(& -4) + 31is p (p. -(I> t
(1)
where Ak(P, -4) = 2N,
s
d4k -Wh(k;p) (2r)4
qJP*
- 4);
s
= - f$(P,
-4)
+ jqp,
-9).
(4)
Herein, except for the electric charge, the u and d quarks are identical and hence AE(p, -4) = At(p, -q), defined in Eq. (2). For elastic scattering [ p2 = $1 and AE*IKO(p, -4) = (p + q)r FK*iKo(Q2), where F K*IKo(Q2) is the electromagnetic form factor.
2.1. Dressed quark propagator
-9)
Ss(k
- /%)I,
(2)
-4)
= 2N,
A; (P, -4)
&(k+ap)
xir”,(k+rrp,k--pp+q)S(k-Pp+q) x rK(k - P(P
mass momentum P; and Ii is the flavour dependent photon-quark vertex. The variable cx [/I = 1 a] allows for uneven partitioning of the total momentum between the u- and s-quark legs connected to the Bethe-Salpeter amplitude. The choice of the relative-momentum p as the argument of the BetheSalpeter amplitude entails (Y= 0.5. The requirement that Fc (Q2 = 0) = 0 is a measure of the efficacy of our approximation for the kaon Bethe-Salpeter amplitude. The first of the two contributions in Eq. ( 1) corresponds to the u-quark interacting with the photon and the s-quark acting as a spectator; in the second the roles are reversed. If the current-quark masses are set equal in Eq. ( 1) the generalised impulse approximation to the y-7r-r vertex [ 4,6] is recovered, provided that IK -t I,,, fK --f fr and cy = l/2, which is required by charge conjugation symmetry in this case. The generalised impulse approximation to the neutral-kaon-photon vertex is given by
* tro[IK(k;p) (27r)4
S,(k+ Pp)
xi$(k+fip,k-apfq)&(k--ap+qJ
xr~(k-‘~(p-q);-q)SU(k-ap)l.
in Eqs. (2) and Sf = -iy -p af(p2) + ai(p2) (3) can be obtained by solving the quark DysonSchwinger equation [ 33. Model studies using a gluon propagator and quark-gluon vertex that manifest the qualitative features suggested by Refs. [ 7,8] provide a basis for the following approximating algebraic, model forms for CT{and a{ 141:
(3)
Here q is the initial momentum of the kaon, p - q z Q is the photon momentum and only the Dirac trace remains to be evaluated. In Eqs. (2) and (3): Sf is the propagator for a quark of flavour f = u, s; r~ (p; P) is the kaon Bethe-Salpeter amplitude, with quark-antiquark relative momentum p and centre-of-
G(X) = CL,e-2x f
&
f
(1 _ e-2(x+4))
CJ. Burden et al./Physics
i?{(x) =
2(x +
i”;> -
1+
e--2(x+6)
2( n + r$,’
-fi,c~I (?-2X ’ (6)
and: &J(X) = 2Dav(p2); where x = p2/(2D) *s(n) = v%as(p2); and fif = mf/m, with D a mass scale. The parameters CL,, fif, bf,,,s are determined either by: 1) fitting a quark-DSE solution obtained with a realistic gluon propagator; or 2) performing a X2-fit to a range of hadronic observables. (A = 10e4 is included to decouple the small and large spacelike-p2 behaviour of the l/p4 term, characterised by bof and bl.) We write the inverse of the quark propagator as 57’(p)
= iy*pAf(p2)
+ Bf(p2).
(7)
The quark propagator described by Eqs. (5), (6) is an entire function in the finite complex-p2. It therefore admits the interpretation that it describes a confined particle [ 31. The N e-’ form that ensures this is suggested by the algebraic solution of the model DSE studied in Ref. [ 91, which employed a confining model gluon propagator and dressed quark-gluon vertex. Furthermore, the behaviour of this model form on the spacelike-p2 axis is such that, neglecting ln[p2] corrections associated with the anomalous dimension of the quark propagator in QCD, it manifests asymptotic freedom. It has a term associated with dynamical chiral symmetry breaking (N 1/x2) and a term associated with explicit chiral symmetry breaking (- m/x). 2.2. Pseudoscalar meson Bethe-Salpeter amplitude r~ in Eq. (1) is the solution of an homogeneous Bethe-Salpeter equation (BSE) . Many studies of this BSE suggest strongly that the amplitude is c( 3/s [ lo], Furthermore, in the chiral limit the pseudoscalar BSE and quark DSE are identical [ 111 and one has a massless excitation in the pseudoscalar channel with IO- (p; P2 = 0) = iYsB,a(p2)/f?r, where Bd(p*) is given in Eq. (7) with mf = 0. This is the realisation of Goldstone’s theorem in the DSE framework; i.e., in the chiral limit Eqs. (5) and (6) completely determine I’,-. Herein, based on these observations, we employ the approximations
Letters B 371 (1996) 163-168
165
T,(p;
P2 = -m:)
x iys $
r~ (p;
P2 = -m2,) x iy5 J- %,,0(p2). .fK
?T
B;JP2),
(8)
(9)
For the pion this is a good approximation, both pointwise and in terms of the values obtained for physical observables [ 121. For the kaon it is an exploratory Ansatz, one which need only be accurate as an approximation to the integrated strength.
2.3. Quark-photon vertex I7i (~1, p2 ) in Eq. ( 1) satisfies a DSE that describes both strong and electromagnetic dressing of the quarkphoton vertex. Solving this equation is a difficult problem, which has recently begun to be addressed [ 131. Much progress has been made in developing a realistic Ansatz for rcL(pl ,p2) [ 81. The bare vertex, Iif;(Pr ,p2) = ycc, is inadequate when the quark propagator has momentum dependent dressing because it violates the Ward-Takahashi identity. In Ref. [ 141 the following form was proposed
rfQ (p ’ 4) =Zf A (p2 +K,
’
42) Y P
[;r&(p2,q2)
with K =p+q,Ci(p2,q2) Af,(p2
q2)
=
[Af(p2)
(10)
-i&p2,q2)],
= [Af(p2)+Af(q2)]/2, - Af(q2)]/[p2
- q2] and
*L(p2:q2)
= [Bf(p2) - Bf(q2)]/[p2 - q2]. This Ansatz is completely determined by the dressed quark propagator; satisfies the Ward-Takahashi identity; has a well defined limit as p2 + q2; transforms correctly under C, P, T and Lorentz transformations; and reduces to the bare vertex in the manner prescribed by perturbation theory. Furthermore, it is relatively simple and hence an ideal form to be employed in our phenomenological studies. Using charge conjugation it is straightforward to show that for elastic scattering one has = 0 in generalised impulse (P - 9)pA;*D(P+-4)
approximation. The result FK* (-rni, Q2 = 0) = 1 follows because the quark-photon vertex satisfies the Ward identity.
166
CJ. Burden et al./Physic.s Letters B 37/ (1996) 163-168
3. Calculated
spacelike kaon form factors
In the Breit frame: p = (O,O, i Q, iEK), q = the calculaEK = dm; tion of each meson form factor involves the numerical evaluation of a three-dimensional integral via straightforward numerical quadrature. To fix the parameters in the quark propagators we have revised the study of Ref. [4] and re-fitted the pion observables using the pion mass formula derived in Ref. [ 121, which yields an accurate estimate of the mass obtained by solving the generalised-ladder approximation to the pion Bethe-Salpeter equation. This reanalysis, carried out with the constraint C& +a = 0, leads to
(WA-;Q,iEd,
Ci”=s = 0.121,
fiu = 0.00897,
b; = 0.131,
b; = 2.90,
b; = 0.603,
b’; = 0.185,
(11)
with the mass scale D = 0.160 GeV2 chosen so as to give fc = 92.4 MeV. 3.1. Difeerences between u- and s-quark propagators Based on the studies of Ref. [ 161 we set fi, = 12.5 (ti, + tid) 3 25 ti,,,
b; =0.8b&
b; = brf,
The current theoretical prejudice [ 181 is that (ss)“=c N (0.5 - 0.8) (au) “a. We have explored the response of our calculated kaon obsetvables to variations in (5s). The sensitivity is to0 weak ( < 1%
(14)
CA, = C& = 0,
(15)
and allowed variations only in the parameter b$ To provide for a difference between I,, and I’K we also allowed c;,$) # c;,*. Having fixed the u-quark parameters in the pion sector we then have a two-parameter extension of the model to the s-quark sector. These parameters are fixed by requiring that the model reproduce, as well as possible, the experimental values for the dimensionless quantities: f~/f$, = 1.22 f 0.02, rKf jr& = 0.88 f 0.07 and mK/ fK = 4.37 f 0.05. The kaon mass is obtained by solving IIK( P’) = 0 where =8N,
J
(12)
= -(2D)i
b; = b;.
To allow for a minimal residual difference between the u and s quark propagators we did not allow C& to vary, simply setting
&(P2)
where iir, = fid herein, which is consistent with the theoretical estimates summarised in Ref. [ 171. The results we report herein are qualitatively and quantitatively insensitive to halving or doubling this ratio: the most sensitive, rK*, changing by < 5%. Dyson-Schwinger equation studies show that there are differences between the U- and s-quark propagators that cannot be accounted for simply by changing the mass in Eqs. (5) and (6). An example is the vacuum quark condensate, which, using the model quark propagator defined in Eqs. (5)) (6) and the definition in Ref. [ 41 (with cy= 1 in this model), is given by
($qf)rf
on this range) to provide a robust, independent estimate of (is), however, our calculations favour larger values. Hence in the calculations reported herein we used (Ss)‘” = 0.8 (iiu)‘“, which was implemented by setting
- [B&,(k2)]2 +
d4k &,,,(k2)&,Ak2) (27r)2 ( Ik+.k-d”v,“~O(k+)~,~+o(k-)
~,~o(k+)a~,,,~o(k-)l
>
9
(16)
with k+ = k + aP, k- = k - PP. For P2 + rni N 0, &(P2) x f:: (P2 + m$) with
fi = $
WP2) (p2__ml ’ - I
(17)
which also ensures the correct (unit) normalisation of the charged kaon form factor. These expressions are straightforward generalisations of their analogues in Ref. [ 121. Similar expressions are obtained in Ref. [24]. Following this procedure we obtain C;“, = 1.69,
b; = 0.74.
(18)
With fi, # fi,, Fe(Q2=O) =O for a = 0.49 (” l/2), as anticipated in our approximation of the kaon
CJ. Burden et d/Physics
Table 1 n and K observables calculated using the parameters of Eqs. ( 11) and (18). with the constraints of Eqs. (12). (14) and (15); g,,+ is discussed in Ref. [4] and F3”(4mi) in Ref. [ 151. The quoted “‘experimental” values of n$‘&, mf ,_vz, (I@, oev~and (Ss) , cev~are representative of other theoret& estimates. Most experimental values are extracted from Ref. [ 171; r,,. rK* and rh are taken from Refs. [ 19-2 11, respectively. The r-n scattering lengths, a:, are discussed in Ref. [22] Calculated
a7
'nK ma= I c&v2
T?rf ‘Kf
fK/fn rK*
fr,,f
0.6
163-168
167
-
Experiment
0.0924 GeV 0.113 0.1385 0.4936 0.0051 0.128
0.0924 f 0.001 0.113 f 0.001 0.1385 0.4937 0.0075 0.1 N 0.3
0.221
0.220
0.205 0.56 fm 0.49 -0.020 fm2 0.505 (dimensionless) 1.04 0.17 -0.048 0.030 0.0015 -0.0002 1 1.22 0.87
Letters B 371 (1996)
-
/ I
0.6 0.0
0.1 d(Geti)
1. Calculated form of 1FKf ( Q2) I2 compared with the availdata [ 20,231.
0.175 - 0.205 0.663 f 0.006 0.583 f 0.041 -0.054 zt 0.026 0.504 f 0.019 1 (anomaly) 0.21 f 0.01 -0.040 f 0.003 0.038 f 0.003 0.0017 f 0.0003 1.22 f 0.01 0.88 f 0.06
_------_--___
Bethe-Salpeter amplitude as a function of the relative quark-antiquark momentum. A comparison between calculation and experiment is presented in Table 1. The calculated form factors are presented in Figs. 1,2. The difference between the calculated and measured values of the charge radii and scattering lengths in Table 1 is a measure of the importance of final-state, pseudoscalar rescattering interactions and photon-vector-meson mixing, which are not included in the generalised-impulse approximation [ 61. Calculations of such effects for the pion [6] and p- and w-mesons [ 251 suggest that they contribute less than N 15% and become unimportant for Q2 > 1 GeV*. Herein we find that the calculated values of f~/f,, and r~/r,, agree with the experimental values of these ratios, independent of the details of our parametrisation. This makes it plausible that such effects are no more important for the kaon than for the pion.
0.0
1.0
2.0
3.0
Q*(Geti) Fig. 2. Calculated form factors: Q2 FKf : solid line; Q2 F,+ : shofi-dashed line; Q2 Fe: long dashed line. The difference between FK* and F,* for Q* > 3 GeV* is small but is amplified in this figure because of the multiplication by Q2.
4. Summary and discussion We find that on the range of Q2 currently accessible to experiment FK* (Q*) > I?,( Q2). This is qualitatively consistent with Ref. [2], however, our calculated results, for both form factors, are uniformly smaller in magnitude; as is the difference between them. The peak in Q2F( Q*), which is most pronounced for the K*-meson, is a signal of quarkantiquark recombination into the final state meson in the exclusive elastic scattering process. For Q* > 3 GeV2 we have F,+ (Q*) < Fw(Q2)
168
CJ. Burden et d/Physics
but the difference is small and sensitive to the form of the kaon Bethe-Salpeter amplitude. The behaviour of FKI (Q*) at Q* > 2 N 3 GeV* is influenced by details of the Ansatz for the kaon Bethe-Salpeter amplitude, Eq. (9)) that are not presently constrained by data. The results obtained for Q* < 2 GeV* are not sensitive to details of our parametrisation. We therefore view the results for Q* > 2 GeV* with caution. These observations emphasise that measurement of the electromagnetic form factors is a probe of the bound state structure of the meson; i.e., its BetheSalpeter amplitude. For the neutral kaon r$ < 0. The fact that Fe (Q*) f 0 is a manifestation of the u-s currentquark mass difference. We note that charge conjugation symmetry ensures F+I (Q*) = 0. Our calculated results are not sensitive to changes in the M, in the range ms/maveE [ 15,301 nor to changes in (Ss) in the range (Ss)/(iiu) E [0.5,1.0]. Nevertheless, the requirement that the calculation reproduce known values of kaon observables does lead to differences between the u- and s-quark propagators. This emphasises that measurement of the form factors is also a probe of nonperturbatively generated differences between the u- and s-quark propagation characteristics.
Acknowledgments
We acknowledge useful conversations with A.J. Davies. This work was supported by the National Science Foundation under grant no. INT92-15223; the Australian Research Council under grant no. S02947481; and the US Department of Energy, Nuclear Physics Division, under contract number W-3 I109-ENG-38. The calculations described herein were carried out using the resources of the National Energy Research Supercomputer Center.
Letters B 371 (1996) 163-168
References
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