- Email: [email protected]
Calculation of the anomalous γπ∗ → ππ form factor
22 February 1996
PHYSICS LETTERS 6 Physics Letters B 369 (1996) 101-107
ELSEVIER
Calculation of the anomalous yr* -+ OTTform factor Reinhard Alkofer a, Craig D. Roberts b a Institut fiir Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-72076 Tiibingen, Germany b Physics Division, Argonne National Laboratory, Argonne, IL 60439-4843,
USA
Received 18 October 1995; revised manuscript received 1 December 1995 Editor: C. Mahaux
Abstract The form factor for the anomalous process ye* --+ VT, p”( s, t, u), is calculated as a phenomenological application of the QCD Dyson-Schwinger equations. The chiral-limit value dictated by the electromagnetic, anomalous chiral Ward identity, is reproduced, independent of the details of the modelling of the gluon and quark 2-point Schwinger functions. Using a parametrization of the dressed u-d quark 2-point Schwinger function that provides a good description of pion observables F3” (s, I, u) is calculated on a kinematic range that proposed experiments plan to explore. Our result confirms the general trend of other calculations, i.e., a monotonic increase with s at fixed t and u, but is uniformly larger and exhibits a more rapid rise with s. PACS: 13.40.G~; 14.40.Aq; 12.38.Lg; 24.85.+p Keywords: Hadron physics FyaTr (s); Dyson-Schwinger equations; Effects of quark and gluon confinement; QCD phenomenology;
Nonperturbative QCD
1. Introduction Hadronic processes involving an odd number of pseudoscalar mesons are of particular interest because they are intimately connected to the anomaly structure of QCD. The decay ?r” --+ yy is the primary example of such an anomalous process. That such pro-
cesses occur in the chiral limit (m$ = 0) is a fundamental consequence of the quantisation of QCD; i.e., of the non-invariance of the QCD measure under chiral transformations even in the absence of current quark masses [ 11. The transition form factor for the related process r*?ro -+ y can be measured experimentally [ 21 and has attracted keen theoretical interest [ 3,4] because it involves only one hadronic bound state and provides a good test of QCD-based models and their interpolation between the soft and hard do-
mains. Another anomalous form factor, accessible to experiment, is that which describes the transition ye* -+ TT, denoted by F3r( s, t, u). This provides additional constraints on QCD based-models because three hadronic bound states are involved. The form factor F3=( s, t, u) has been measured at Serpukhov in the Primakov reaction T-A --) r-‘n-‘A [5]. In this experiment the considerable uncertainty in both the kinematic range and result make it difficult to draw a conclusion regarding the accuracy of the theoretical prediction for the chiral limit value of F3”(0,0,0) [6]. New experiments are planned at CEBAF: [7] yp + r+#n, s/m; E [4,15]; and at FermiLab [ 81 via the Primakov reaction using a 600 GeV pion beam, s/m: E [4,6]. Herein, as a phenomenological application of the
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)01517-S
102
R. Alkofer. CD. Roberts/Physics Letters 3 369 (1996) 101-107
QCD Dyson-Schwinger equations (DSEs), we report a calculation of F3V( s, t, u) on the kinematic range to be explored in the CEBAF experiment [ 71. In calculating F3”(s, t, u) we employ a generalised-impulse approximation, in which the quark 2-point Schwinger function, quark-photon vertex and quark-pion vertex (pion Bethe-Salpeter amplitude) are dressed quantities whose form follows from the extensive body of nonperturbative, Dyson-Schwinger equations studies in QCD [9], In this way our calculation provides for an extrapolation of the known large spacelike-q2 behaviour of these QCD Schwinger functions to the small spacelike-q2 and timelike-q2 region, where they are unknown and confinement effects are manifest. This facilitates an exploration of the relationship of physical observables to the nonperturbative, infrared behaviour of these Schwinger functions. This calculation employs the model forms introduced in Ref. [ lo]. The quark 2-point Schwinger function 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 vertex, which describes the coupling of a photon to a dressed quark, follows from extensive QED studies and satisfies the Ward-Takahashi identity. This necessarily entails that the amplitude is current conserving. In the chiral limit the quarkpion vertex is completely determined by the quark 2point Schwinger function, which is the manifestation of Goldstone’s theorem in this approach. The extension to finite quark mass requires a minimal modification and preserves Dashen’s relation [ 11 I.
In this expression the colour and flavour traces have been evaluated, leaving only the Dirac trace, kUpy = k+ $YPI f iPp2 + iyp3 and q=pl +p2 +p3 is the photon momentum. We adopt the convention that the pions labelled 1 and 2 are on-shell whereas pion 3 is off-shell. The dressed quark-photon vertex is denoted by I+*(kl , kz 1, the pion Bethe-Salpeter amplitude by ysTT(k) and the dressed quark 2-point Schwinger function by S(k). Evaluated with a dressed quarkphoton vertex that satisfies the Ward-Takahashi identity, which is a minimal requirement, this expression is current conserving. The quark 2-point Schwinger function can be written S(P)
=
for yc~~*a~n
d4k
J
--trD[r,(k---,k+++)S(k+++) &-I4
x Ysra(ko++)S(k-++)rsr,(k_o+)S(k--+) x yJ~(k__o)S(k___)].
(1)
(2)
+ dP2)
1 iy-pA(p2)
&(x)
+m+B(p2)
(3)
’
= Cfi,e-2X
+ x fti2 + lLebix
[ 1 _ e-2 (r+rih ] hx 1 _ e-b,x -.b3x
(
bo + b:! ’ ie;*‘)
.
(4) 2(X + r7t2) _ 1 + &(x+til*) =
- iir C,, e-2.r .
In Euclidean space, with metric S,, = diag ( 1,l ,1,l ) and y@ = yi, the generalised impulse approximation to the yawn. vertex is
= 2eN,
P(+“(P2)
where m = m, = md is the current quark mass, and can be obtained by solving the quark Dyson-Schwinger equation (DSE) [ 91. The many studies of this equation ensure that the qualitative features of the functions vs and TV are well known for real, spacelike-p2. In Ref. [ IO] the following upproximaring algebraic forms are used:
&J(x) 2. Amplitude
= -iY
2(x + iit2)2 (5)
with k2 = 2Dx, $.s(x) = mcrs(k2), (TV(X) = 2 D cv( k2> and in = m/m. The quantity A = m is a mass-scale related to the infrared behavior of the gluon 2-point Schwinger function [ 91. The quark propagator described by Eqs. (4)-(5) is an entire function in the finite complex-p2 plane and hence does not have a Lehmann representation. It therefore admits the interpretation that it describes a confined particle [9]. The behaviour of this model form on the spacelike-p2 axis is such that, neglecting
R. Alkofer, C.D. Roberts/Physics Letters B 369 (1996) 101-107
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 (- 1/x2) and a term associated with explicit chiral symmetry breaking (N m/x). Both of these terms are present in solutions of the quark DSE using a realistic model gluon propagator [ 91. The expressions in Eqs. (4) and (5) provide a six-parameter model of the dressed quark 2-point Schwinger function in QCD: C,, ti, bo, . . . , b3. (A = 10M4is introduced simply to decouple b2 from the quark condensate.) These parameters can easily be fitted to experimental observables, as we discuss below. The Bethe-Salpeter amplitude, I?r in Eq. ( 1), is the solution of the homogeneous Bethe-Salpeter equation (BSE). Many studies of this BSE suggest strongly that the amplitude is dominantly pseudoscalar. Furthermore, in the chiral limit the pseudoscalar BSE and quark DSE are identical [ 121 and one has a massless excitation in the pseudoscalar channel with T,(p; P2 = 0) = $
*
Bmdl(p2)
,
(6)
where B-4 (p2) is given in Eq. (3) with m = 0. This is the realisation of Goldstone’s theorem in the DSE framework; i.e., in the chiral limit Eqs. (4) and (5) completely determine Ia. Herein we employ the approximation r,(p;
P2 = -rr$)
= f
r
Bm4(p2)
1
(7)
which, for small current quark masses, is a good approximation both pointwise and in terms of the values obtained for physical observables [ 131. The quark-photon vertex, rP (pr ,p2), satisfies a DSE that describes both strong and electromagnetic dressing of the vertex. Solving this equation is a difficult problem that has only recently begun to be addressed [ 141. However, much progress has been made in developing a realistic Ansatz [ 151. The bare vertex, IP(pr, ~2) = yr, is inadequate when the fermion 2-point Schwinger function has momentum dependent dressing because it violates the WardTakahashi identity. In Ref. [ 161 the following form was proposed:
103
+(p+k),[4y.(p+k)A~(p~,
k2) -&(p2,
k2)
13
(8) with Wp2,k2)
+4k2>l/'L
= [A(p2)
Ldp2,k2)= Mp2) - A(k2>l/[p2 -k21. b(p2,k2)
= W(p2> - W2)l/[p2
- k*l .
This Ansatz is completely determined by the dressed quark 2-point Schwinger function: satisfies the WardTakahashi identity; has a well defined limit as p2 -+ k2; transforms correctly under C, P, T and Lorentz transformations; and reduces to the bare vertex in the manner prescribed by perturbation theory. Furthermore, it is relatively simple and hence an ideal form to be employed in our phenomenological studies. Its efficacy is illustrated in Ref. [ lo].
3. Chiral limit: yn* * rrn At the soft point in the chiral limit (s = t = u = 0) the transition form factor of Eq. ( 1) is co
F3”(0,0,0) = f$
s
dSST,(S)%”
0
x (AaVgs+
;sA&q
+ +A’uvuS
- $sB’&.
- ;sAcr&
(9)
Defining C(s) = B(d2/bAW21
= ~sW2/ba4~,*1
,
Eq. (9) becomes
s00
P(O,O,O) =
-2
r,w3
ds
By
W)cw [I
0
+Cts)14’
(10) Recalling Eq. (6), it follows that co
eN,
F3”(0,0,0)= -2df3 eNc
=iGq’
rO
s dC
C (1 +C)4
(11)
104
R. Alkofer, C.D. Roberts/Physics
since C(s) is a monotonic function for s > 0 with C(s = 0) = 00 and C( s = 00) = 0. Hence, the chiral limit value [ 61 is reproduced independent of the details of the quark 2-point Schwinger function, S(p). In order to obtain the result in Eq, ( 1I) it is essential that the photon-quark vertex satisfy the Ward identity and the pion Bethe-Salpeter amplitude be proportional to the scalar part of the quark self energy in the chiral limit, Eq. (6). The fact that one must dress all of the elements in the calculation consistently is often overlooked. The subtle cancellations that are required to obtain this result also make it clear that it cannot be obtained in model calculations where an arbitrary cutoff function (or “form-factor”) is introduced into each integral. These features are also to be seen in the calculation [ 171 of the Wess-Zumino five pseudoscalar term and the ?rorr vertex [ lo]. The chiral limit result is a property of the QCD measure. Therefore any analysis beyond generalised impulse approximation, such as to include final state 7r-rr interactions for example, must preserve the chiral limit result. This places constraints on such analyses.
Letters B 369 (19%) 101-107
ii=(-1.54.28x+0.012x2) with x = (S/4-
1).
(12)
Eqs. (4), (5), (6) and (8) providea six-parameter model [C,,, m, be . . . b3] of the nonperturbative, dressed-quark substructure of the pion. These parameters are fixed by requiring that the model reproduce, as well as possible, the following pion observables fp/(~~);~~v2 = 0.42 f 0.02, j-g r, = 0.31 f 0.01,
= 0.40 & 0.03; the dimensionless n--r m2/@#$2 scattering lengths (discussed in Ref. [ 181 with current experimental values presented in Table 1) and the pion electromagnetic form factor on spacelikeq’ E [0,4] GeV2 [lo]. The values of observables are given by simple integral expressions involving the quark 2-point Schwinger function and pion Bethe-Salpeter amplitude; for example [ 131,
x {a$ - 2[a@$ + sU”a;]
- s[rrscrY - (f&)2]
- ?[(+“UI: - (fl+:>%]} )
J
4. Numerical results
dp2
We employ the following definition of the Mandelstam variables: s= -(pt
+p*)2 3 iniS,
t=-p&n&
which ensures that even though we work in Euclidean metric these variables have their conventional interpretation. The quantity t - rni provides a measure of the amount by which the (third) pion is off-shell. In the experiment proposed at CEBAF the photon energy in the proton rest frame is between 1 and 2 GeV, which suggests the following range of Mandelstam variables: 4 5 S 5 16, -9 5 I< -1, -16 < c F 5. We fix 7 = -1; i.e., we choose pion 3 to be as close as (experimentally) possible to its mass shell, f = 1. Within the range of s considered, the requirement of a fixed photon energy of 1 GeV in the proton rest frame entails
x
tB(p2)
(13)
Bo(P*)
p2 B(p2)
&> -
Bo(p2>
m(p2>1
7
(14)
which follows from the pion Bethe-Salpeter equation and is consistent with Dashen’s relation [ 111; (&4); j&d = 3bd/ and G&d,2 = In [P’/&,] [4dbl b2] and AQCD = 0.2 GeV. In these equations
the subscript or superscript “0” indicates that the labelled function has been evaluated with zero quark current mass. Following this procedure one obtains co = 0.121 ,
~=0.00897,
bo=0.131,
bt =2.90,
b2 =0.603,
b3 =0.185.
C,=O,
(15)
The mass scale is set by requiring that fn = 92.4 MeV, which yields D = 0.160 GeV2. The calculated values of observables are presented in Table 1. The form factor is discussed in Refs. [ IO,211. Our numerical results are shown in Fig. 1. The solid line is our calculated result for
R. Alkofer, C.D. Roberts/Physics
105
Letters B 369 (19%) 101-107
Table 1 Pion observables calculated using the parameter values in Eq. ( 15) Observable
Calculated
fir (CeV)
0.0924
0.0924 f 0.001
rnr
0.1385
0.1385
nF
0.0051
0.0075
- (44) $“?
0.221
0.220
r?i (fm)
0.55
0.663 i 0.006
g,+,
0.505
0.504 f 0.019
IGeVZ
(dimensionless)
Experiment
F” ( 4m2 n)
1.04
I (anomaly )
(1;
0.17
0.21
-0.048
-0.040
a; flf 0’: (1;
f 0.01 f 0.003
0.030
0.038 f 0.003
0.0015
0.0017 l 0.0003
-0.00021
The “experimental” values listed for mave and (qq) are an indication of other contemporary theoretical estimates. Experimental values not discussed in the text are taken from Ref. [ 191. The difference between the calculated and experimental values of r,, is a measure of the importance of final-state rr-rr interactions and photon-p-meson mixing[ 201; that between the calculated and experimental values of the pion scattering lengths is a measure of the importance of V-V final-state interactions in this case[ 181.
I
1.0: 10.0
12.0
s (mx2)
Fig. 1. The normalized yn* rrr amplitude P3a (s, 1, u) as function of the Mandelstam variable S for different values of ti as given by Eq. (12) at i = -1. Our result: solid line; our result with m, -+ ma/2: short-dash line; vector meson dominance: long-dash and dash-dot lines. see Eqs. ( 17) and ( 18) ; chiral expansion plus vector meson saturation Ansatz: dotted line, see Eq. ( 19). The data point is taken from Ref. [ 51.
sistent. For example, using vector meson dominance one obtains, for real photons [ 221, = 1
P(s,t,u) P(s,t,U>
8.0
6.0
4.0
= (47ry/e)F3”(s,t,u)
+CPeiS function of the Mandelstam variable I for different values of ii as given by E!q. (12) at f = -1. An excellent fit to this curve is given by as
t’=s-ttu-2mi,
f13’r(s, t, u) = 1.044 + 0.096 n + 0.006 x2 with x =(9/4-l).
with
(16)
Fixing the Mandelstam variables such that the energy of the incident photon is 2 GeV in the proton restframe changes our curve by an amount that is not visible on the scale of this plot. Importantly, the result is insensitive to the details of the parametrization of the quark 2-point Schwinger function. A quantitatively similar curve is obtained using earlier sets of the parameters in Eq. ( 15); i.e., our result is not sensitive to the details of the model. The form factor does depend on the pion mass. The short-dashed line is the result we obtain with m, = mFp/2. A comparison of our result with that obtained in other models reveals that all results are broadly con-
C, = 2gp,,gp,,/[m~F3T(0,
O,O) 1 = 0.434,
and 6 an unknown phase. This expression, with S = 0, appears as the long-dashed line in Fig. 1. Another vector meson dominance model estimate of this process [ 231 appears as the dash-dot line in Fig. 1. This result is obtained from
F3*(s, t, u) 2
=- mP
1
-++ 3 ( mz--s
1 m2p- t’
+-
I rns-u
) ’
(18)
A model based on chiral expansion techniques and employing a vector meson saturation Ansatz [24] yields the dotted line in Fig. 1, which is obtained from the expression
106
R. Akofeer, CD. Roberts/Physics
F3Ys,t, u)
where the non-divergent
Cpion loops
s+t’+u
= 1 + Cpionloops +
2rns
9
(19)
part of the coefficient
is
f
+ 4rf(s)
f f(t’)
t” +u)
+ f(ii )I
3
(20)
i
with
Analytic continuation is used to define f (b > 0), which has an imaginary part for d > 4; i.e., for s in the domain explored by the existing and proposed experiments. The imaginary part is due to the pion loop in the s-channel. We observe that this calculation provides the weakest s dependence. All of this s-dependence arises from r-r final state interactions. This suggests that such final state interaction corrections to our result will be small, as discussed in Ref. [20], We can compare our result with the one existing data point [ 51. Its statistical and systematic error as well as the uncertainty in s are also displayed in Fig. 1. The fact that this data point is well above the chiral limit prediction has caused some concern [7]. However, given the experimental errors and the prediction of our model this data point does not appear untenable.
Letters B 369 (1996) 101-107
calculation to be compared, without adjustment, with the results of that experiment. In this calculation the chiral limit value of F3rr( 0,0,O) is reproduced in&pendent of the details of the model. Compared with other model calculations, we obtain a form factor that is uniformly larger and has a more rapid increase with s. The models are otherwise broadly consistent. The variation should, however, be sufficient for the proposed experiments to be able to distinguish between them. Along with Ref. [ 3 1, this application of the QCD DSEs is one of the first to explore the model quark 2-point Schwinger function and pion Bethe-Salpeter amplitude well outside the domain of the complex plane on which they have been fitted; i.e., well into the timelike region, which is not accessible in perturbation theory. This region is important in the study of, for example, vector meson Bethe-Salpeter and baryon Fadde’ev amplitudes. Experimental data on F3”( s, t, u) can therefore be used to place important new constraints on the analytic structure of the QCD Schwinger functions.
Acknowledgements
This work was supported by the US Department of Energy, Nuclear Physics Division, under contract number W-31-109-ENG-38 and by the Deutsche Forschungsgemeinschaft (DFG) under contract number AL 279/2-l. The calculations described herein were carried out using a grant of computer time and the resources of the National Energy Research Supercomputer Center. R.A. thanks the Physics Division of ANL for their warm hospitality during two visits in which most of the work described herein was performed. References
5. Summary
and conclusions
Using a phenomenological approach based on the QCD Dyson-Schwinger equations, F3n( s, t, u) was calculated for a range of (s, t, u> that cover the kinematic region to be explored in a proposed experiment [ 7 1. The small photon virtuality in another proposed experiment [ 81 should also make it possible for our
[ 11 K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; see, also, for example, W. D&rich and M. Reuter, Selected Topics in Gauge Theories, Lecture Notes in Physics, Vol. 244 (Springer, Berlin, 1986). [2j A. Afanasev, in: Proc. Workshop on CEBAF at Higher Energies, eds. N. Isgur and F! Stoler (CEBAE Newport News, 1994), p. 185. [3] M.R. Frank, K.L. Mitchell, C.D. Roberts and PC. Tandy, Phys. Lett. B 359 (1995) 17.
R. Alkojkr,
C.D. R&errs/Physics
[ 41 W. Jaus, Phys. Rev. D 44 (199 I ) 2851; L. Ametller, J. Bijnens, A. Bramon and F. Comet, Phys. Rev. D 45 ( 1992) 986. [S] YuM. Antipov et al., Phys. Rev. D 36 (1987) 21. 161 S.L. Adler, B.W. Lee. S.B. Treiman and A. Zee, Phys. Rev. D 4 (1971) 3497. [ 7 1 R.A. Miskimen, K. Wang and A. Yegneswaran (spokesmen), Study of the Axial Anomaly using the yrr+ nfno Reaction Near Threshold. Letter of intent, CEBAFExperiment 94-015. [ 81 M.A. Moinester, in: Proc. Conf. on Physics with GeVParticle Beams (Jtllich, Germany, August 1994), eds. H. Machner and K. Sistemich (World Scientific, Singapore, 1995). [9] C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 ( 1994) 477. 101 C.D. Roberts, in: Chiral Dynamics: Theory and Experiment, eds. A.M. Bernstein and B.R. Holstein, Lecture Notes in Physics, Vol. 452 (Springer, Berlin, 1995)~. 68. 1 I 1 R. Dashen, Phys. Rev. 183 (1969) 1245. 12 1 R. Delbourgo and M.D. Scadron, J. Phys. G 5 (1979) 1621.
Leners B 369 (19961 101-107
[ IS] M.R. Frank and CD. Roberts, Model gluon propagator
[ 141 [ 151 1161
[ 171 [ 181 [ 191 [20] [21]
[22] 1231 1241
107
and pion and rho-meson observables, Preprint ANL-PHY-8072TH-95 and INT 95-00-97 ( 1995), to appear in Phys. Rev. C. M.R. Frank, Phys. Rev. C 51 (1995) 987. A. Bashir and M.R. Pennington, Phys. Rev. D SO ( 1994) 7679. J.S. Ball and T.-W. Chiu, Phys. Rev. D 22 ( 1980) 2542. J. Praschitka, CD. Roberts and R.T. Cahill, Phys. Rev. D 36 (1987) 209. C.D. Roberts, R.T. Cahill, M.E. Sevior and N. Iannella, Phys. Rev. D 49 (1994) 125. Particle Data Group, Phys. Rev. D 50 (1994) 1175. R. Alkofer, A. Bender and C.D. Roberts, Intern. J. Mod. Phys. A IO (1995) 3319. C.J. Burden, CD. Roberts and M.J. Thomson, Electromagnetic Form Factors of Charged and Neutral Kaons, Archive: nucl-th/9511012. M.V. Terent’ev, Phys. Lett. B 38 (1972) 419. S. Rudaz, Phys. Rev. D 10 (1974) 3857. J. Bijnens. A. Bramon and E Comet, Phys. Lett. B 237 (1990) 488.
PHYSICS LETTERS 6 Physics Letters B 369 (1996) 101-107
ELSEVIER
Calculation of the anomalous yr* -+ OTTform factor Reinhard Alkofer a, Craig D. Roberts b a Institut fiir Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-72076 Tiibingen, Germany b Physics Division, Argonne National Laboratory, Argonne, IL 60439-4843,
USA
Received 18 October 1995; revised manuscript received 1 December 1995 Editor: C. Mahaux
Abstract The form factor for the anomalous process ye* --+ VT, p”( s, t, u), is calculated as a phenomenological application of the QCD Dyson-Schwinger equations. The chiral-limit value dictated by the electromagnetic, anomalous chiral Ward identity, is reproduced, independent of the details of the modelling of the gluon and quark 2-point Schwinger functions. Using a parametrization of the dressed u-d quark 2-point Schwinger function that provides a good description of pion observables F3” (s, I, u) is calculated on a kinematic range that proposed experiments plan to explore. Our result confirms the general trend of other calculations, i.e., a monotonic increase with s at fixed t and u, but is uniformly larger and exhibits a more rapid rise with s. PACS: 13.40.G~; 14.40.Aq; 12.38.Lg; 24.85.+p Keywords: Hadron physics FyaTr (s); Dyson-Schwinger equations; Effects of quark and gluon confinement; QCD phenomenology;
Nonperturbative QCD
1. Introduction Hadronic processes involving an odd number of pseudoscalar mesons are of particular interest because they are intimately connected to the anomaly structure of QCD. The decay ?r” --+ yy is the primary example of such an anomalous process. That such pro-
cesses occur in the chiral limit (m$ = 0) is a fundamental consequence of the quantisation of QCD; i.e., of the non-invariance of the QCD measure under chiral transformations even in the absence of current quark masses [ 11. The transition form factor for the related process r*?ro -+ y can be measured experimentally [ 21 and has attracted keen theoretical interest [ 3,4] because it involves only one hadronic bound state and provides a good test of QCD-based models and their interpolation between the soft and hard do-
mains. Another anomalous form factor, accessible to experiment, is that which describes the transition ye* -+ TT, denoted by F3r( s, t, u). This provides additional constraints on QCD based-models because three hadronic bound states are involved. The form factor F3=( s, t, u) has been measured at Serpukhov in the Primakov reaction T-A --) r-‘n-‘A [5]. In this experiment the considerable uncertainty in both the kinematic range and result make it difficult to draw a conclusion regarding the accuracy of the theoretical prediction for the chiral limit value of F3”(0,0,0) [6]. New experiments are planned at CEBAF: [7] yp + r+#n, s/m; E [4,15]; and at FermiLab [ 81 via the Primakov reaction using a 600 GeV pion beam, s/m: E [4,6]. Herein, as a phenomenological application of the
0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)01517-S
102
R. Alkofer. CD. Roberts/Physics Letters 3 369 (1996) 101-107
QCD Dyson-Schwinger equations (DSEs), we report a calculation of F3V( s, t, u) on the kinematic range to be explored in the CEBAF experiment [ 71. In calculating F3”(s, t, u) we employ a generalised-impulse approximation, in which the quark 2-point Schwinger function, quark-photon vertex and quark-pion vertex (pion Bethe-Salpeter amplitude) are dressed quantities whose form follows from the extensive body of nonperturbative, Dyson-Schwinger equations studies in QCD [9], In this way our calculation provides for an extrapolation of the known large spacelike-q2 behaviour of these QCD Schwinger functions to the small spacelike-q2 and timelike-q2 region, where they are unknown and confinement effects are manifest. This facilitates an exploration of the relationship of physical observables to the nonperturbative, infrared behaviour of these Schwinger functions. This calculation employs the model forms introduced in Ref. [ lo]. The quark 2-point Schwinger function 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 vertex, which describes the coupling of a photon to a dressed quark, follows from extensive QED studies and satisfies the Ward-Takahashi identity. This necessarily entails that the amplitude is current conserving. In the chiral limit the quarkpion vertex is completely determined by the quark 2point Schwinger function, which is the manifestation of Goldstone’s theorem in this approach. The extension to finite quark mass requires a minimal modification and preserves Dashen’s relation [ 11 I.
In this expression the colour and flavour traces have been evaluated, leaving only the Dirac trace, kUpy = k+ $YPI f iPp2 + iyp3 and q=pl +p2 +p3 is the photon momentum. We adopt the convention that the pions labelled 1 and 2 are on-shell whereas pion 3 is off-shell. The dressed quark-photon vertex is denoted by I+*(kl , kz 1, the pion Bethe-Salpeter amplitude by ysTT(k) and the dressed quark 2-point Schwinger function by S(k). Evaluated with a dressed quarkphoton vertex that satisfies the Ward-Takahashi identity, which is a minimal requirement, this expression is current conserving. The quark 2-point Schwinger function can be written S(P)
=
for yc~~*a~n
d4k
J
--trD[r,(k---,k+++)S(k+++) &-I4
x Ysra(ko++)S(k-++)rsr,(k_o+)S(k--+) x yJ~(k__o)S(k___)].
(1)
(2)
+ dP2)
1 iy-pA(p2)
&(x)
+m+B(p2)
(3)
’
= Cfi,e-2X
+ x fti2 + lLebix
[ 1 _ e-2 (r+rih ] hx 1 _ e-b,x -.b3x
(
bo + b:! ’ ie;*‘)
.
(4) 2(X + r7t2) _ 1 + &(x+til*) =
- iir C,, e-2.r .
In Euclidean space, with metric S,, = diag ( 1,l ,1,l ) and y@ = yi, the generalised impulse approximation to the yawn. vertex is
= 2eN,
P(+“(P2)
where m = m, = md is the current quark mass, and can be obtained by solving the quark Dyson-Schwinger equation (DSE) [ 91. The many studies of this equation ensure that the qualitative features of the functions vs and TV are well known for real, spacelike-p2. In Ref. [ IO] the following upproximaring algebraic forms are used:
&J(x) 2. Amplitude
= -iY
2(x + iit2)2 (5)
with k2 = 2Dx, $.s(x) = mcrs(k2), (TV(X) = 2 D cv( k2> and in = m/m. The quantity A = m is a mass-scale related to the infrared behavior of the gluon 2-point Schwinger function [ 91. The quark propagator described by Eqs. (4)-(5) is an entire function in the finite complex-p2 plane and hence does not have a Lehmann representation. It therefore admits the interpretation that it describes a confined particle [9]. The behaviour of this model form on the spacelike-p2 axis is such that, neglecting
R. Alkofer, C.D. Roberts/Physics Letters B 369 (1996) 101-107
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 (- 1/x2) and a term associated with explicit chiral symmetry breaking (N m/x). Both of these terms are present in solutions of the quark DSE using a realistic model gluon propagator [ 91. The expressions in Eqs. (4) and (5) provide a six-parameter model of the dressed quark 2-point Schwinger function in QCD: C,, ti, bo, . . . , b3. (A = 10M4is introduced simply to decouple b2 from the quark condensate.) These parameters can easily be fitted to experimental observables, as we discuss below. The Bethe-Salpeter amplitude, I?r in Eq. ( 1), is the solution of the homogeneous Bethe-Salpeter equation (BSE). Many studies of this BSE suggest strongly that the amplitude is dominantly pseudoscalar. Furthermore, in the chiral limit the pseudoscalar BSE and quark DSE are identical [ 121 and one has a massless excitation in the pseudoscalar channel with T,(p; P2 = 0) = $
*
Bmdl(p2)
,
(6)
where B-4 (p2) is given in Eq. (3) with m = 0. This is the realisation of Goldstone’s theorem in the DSE framework; i.e., in the chiral limit Eqs. (4) and (5) completely determine Ia. Herein we employ the approximation r,(p;
P2 = -rr$)
= f
r
Bm4(p2)
1
(7)
which, for small current quark masses, is a good approximation both pointwise and in terms of the values obtained for physical observables [ 131. The quark-photon vertex, rP (pr ,p2), satisfies a DSE that describes both strong and electromagnetic dressing of the vertex. Solving this equation is a difficult problem that has only recently begun to be addressed [ 141. However, much progress has been made in developing a realistic Ansatz [ 151. The bare vertex, IP(pr, ~2) = yr, is inadequate when the fermion 2-point Schwinger function has momentum dependent dressing because it violates the WardTakahashi identity. In Ref. [ 161 the following form was proposed:
103
+(p+k),[4y.(p+k)A~(p~,
k2) -&(p2,
k2)
13
(8) with Wp2,k2)
+4k2>l/'L
= [A(p2)
Ldp2,k2)= Mp2) - A(k2>l/[p2 -k21. b(p2,k2)
= W(p2> - W2)l/[p2
- k*l .
This Ansatz is completely determined by the dressed quark 2-point Schwinger function: satisfies the WardTakahashi identity; has a well defined limit as p2 -+ k2; transforms correctly under C, P, T and Lorentz transformations; and reduces to the bare vertex in the manner prescribed by perturbation theory. Furthermore, it is relatively simple and hence an ideal form to be employed in our phenomenological studies. Its efficacy is illustrated in Ref. [ lo].
3. Chiral limit: yn* * rrn At the soft point in the chiral limit (s = t = u = 0) the transition form factor of Eq. ( 1) is co
F3”(0,0,0) = f$
s
dSST,(S)%”
0
x (AaVgs+
;sA&q
+ +A’uvuS
- $sB’&.
- ;sAcr&
(9)
Defining C(s) = B(d2/bAW21
= ~sW2/ba4~,*1
,
Eq. (9) becomes
s00
P(O,O,O) =
-2
r,w3
ds
By
W)cw [I
0
+Cts)14’
(10) Recalling Eq. (6), it follows that co
eN,
F3”(0,0,0)= -2df3 eNc
=iGq’
rO
s dC
C (1 +C)4
(11)
104
R. Alkofer, C.D. Roberts/Physics
since C(s) is a monotonic function for s > 0 with C(s = 0) = 00 and C( s = 00) = 0. Hence, the chiral limit value [ 61 is reproduced independent of the details of the quark 2-point Schwinger function, S(p). In order to obtain the result in Eq, ( 1I) it is essential that the photon-quark vertex satisfy the Ward identity and the pion Bethe-Salpeter amplitude be proportional to the scalar part of the quark self energy in the chiral limit, Eq. (6). The fact that one must dress all of the elements in the calculation consistently is often overlooked. The subtle cancellations that are required to obtain this result also make it clear that it cannot be obtained in model calculations where an arbitrary cutoff function (or “form-factor”) is introduced into each integral. These features are also to be seen in the calculation [ 171 of the Wess-Zumino five pseudoscalar term and the ?rorr vertex [ lo]. The chiral limit result is a property of the QCD measure. Therefore any analysis beyond generalised impulse approximation, such as to include final state 7r-rr interactions for example, must preserve the chiral limit result. This places constraints on such analyses.
Letters B 369 (19%) 101-107
ii=(-1.54.28x+0.012x2) with x = (S/4-
1).
(12)
Eqs. (4), (5), (6) and (8) providea six-parameter model [C,,, m, be . . . b3] of the nonperturbative, dressed-quark substructure of the pion. These parameters are fixed by requiring that the model reproduce, as well as possible, the following pion observables fp/(~~);~~v2 = 0.42 f 0.02, j-g r, = 0.31 f 0.01,
= 0.40 & 0.03; the dimensionless n--r m2/@#$2 scattering lengths (discussed in Ref. [ 181 with current experimental values presented in Table 1) and the pion electromagnetic form factor on spacelikeq’ E [0,4] GeV2 [lo]. The values of observables are given by simple integral expressions involving the quark 2-point Schwinger function and pion Bethe-Salpeter amplitude; for example [ 131,
x {a$ - 2[a@$ + sU”a;]
- s[rrscrY - (f&)2]
- ?[(+“UI: - (fl+:>%]} )
J
4. Numerical results
dp2
We employ the following definition of the Mandelstam variables: s= -(pt
+p*)2 3 iniS,
t=-p&n&
which ensures that even though we work in Euclidean metric these variables have their conventional interpretation. The quantity t - rni provides a measure of the amount by which the (third) pion is off-shell. In the experiment proposed at CEBAF the photon energy in the proton rest frame is between 1 and 2 GeV, which suggests the following range of Mandelstam variables: 4 5 S 5 16, -9 5 I< -1, -16 < c F 5. We fix 7 = -1; i.e., we choose pion 3 to be as close as (experimentally) possible to its mass shell, f = 1. Within the range of s considered, the requirement of a fixed photon energy of 1 GeV in the proton rest frame entails
x
tB(p2)
(13)
Bo(P*)
p2 B(p2)
&> -
Bo(p2>
m(p2>1
7
(14)
which follows from the pion Bethe-Salpeter equation and is consistent with Dashen’s relation [ 111; (&4); j&d = 3bd/ and G&d,2 = In [P’/&,] [4dbl b2] and AQCD = 0.2 GeV. In these equations
the subscript or superscript “0” indicates that the labelled function has been evaluated with zero quark current mass. Following this procedure one obtains co = 0.121 ,
~=0.00897,
bo=0.131,
bt =2.90,
b2 =0.603,
b3 =0.185.
C,=O,
(15)
The mass scale is set by requiring that fn = 92.4 MeV, which yields D = 0.160 GeV2. The calculated values of observables are presented in Table 1. The form factor is discussed in Refs. [ IO,211. Our numerical results are shown in Fig. 1. The solid line is our calculated result for
R. Alkofer, C.D. Roberts/Physics
105
Letters B 369 (19%) 101-107
Table 1 Pion observables calculated using the parameter values in Eq. ( 15) Observable
Calculated
fir (CeV)
0.0924
0.0924 f 0.001
rnr
0.1385
0.1385
nF
0.0051
0.0075
- (44) $“?
0.221
0.220
r?i (fm)
0.55
0.663 i 0.006
g,+,
0.505
0.504 f 0.019
IGeVZ
(dimensionless)
Experiment
F” ( 4m2 n)
1.04
I (anomaly )
(1;
0.17
0.21
-0.048
-0.040
a; flf 0’: (1;
f 0.01 f 0.003
0.030
0.038 f 0.003
0.0015
0.0017 l 0.0003
-0.00021
The “experimental” values listed for mave and (qq) are an indication of other contemporary theoretical estimates. Experimental values not discussed in the text are taken from Ref. [ 191. The difference between the calculated and experimental values of r,, is a measure of the importance of final-state rr-rr interactions and photon-p-meson mixing[ 201; that between the calculated and experimental values of the pion scattering lengths is a measure of the importance of V-V final-state interactions in this case[ 181.
I
1.0: 10.0
12.0
s (mx2)
Fig. 1. The normalized yn* rrr amplitude P3a (s, 1, u) as function of the Mandelstam variable S for different values of ti as given by Eq. (12) at i = -1. Our result: solid line; our result with m, -+ ma/2: short-dash line; vector meson dominance: long-dash and dash-dot lines. see Eqs. ( 17) and ( 18) ; chiral expansion plus vector meson saturation Ansatz: dotted line, see Eq. ( 19). The data point is taken from Ref. [ 51.
sistent. For example, using vector meson dominance one obtains, for real photons [ 221, = 1
P(s,t,u) P(s,t,U>
8.0
6.0
4.0
= (47ry/e)F3”(s,t,u)
+CPeiS function of the Mandelstam variable I for different values of ii as given by E!q. (12) at f = -1. An excellent fit to this curve is given by as
t’=s-ttu-2mi,
f13’r(s, t, u) = 1.044 + 0.096 n + 0.006 x2 with x =(9/4-l).
with
(16)
Fixing the Mandelstam variables such that the energy of the incident photon is 2 GeV in the proton restframe changes our curve by an amount that is not visible on the scale of this plot. Importantly, the result is insensitive to the details of the parametrization of the quark 2-point Schwinger function. A quantitatively similar curve is obtained using earlier sets of the parameters in Eq. ( 15); i.e., our result is not sensitive to the details of the model. The form factor does depend on the pion mass. The short-dashed line is the result we obtain with m, = mFp/2. A comparison of our result with that obtained in other models reveals that all results are broadly con-
C, = 2gp,,gp,,/[m~F3T(0,
O,O) 1 = 0.434,
and 6 an unknown phase. This expression, with S = 0, appears as the long-dashed line in Fig. 1. Another vector meson dominance model estimate of this process [ 231 appears as the dash-dot line in Fig. 1. This result is obtained from
F3*(s, t, u) 2
=- mP
1
-++ 3 ( mz--s
1 m2p- t’
+-
I rns-u
) ’
(18)
A model based on chiral expansion techniques and employing a vector meson saturation Ansatz [24] yields the dotted line in Fig. 1, which is obtained from the expression
106
R. Akofeer, CD. Roberts/Physics
F3Ys,t, u)
where the non-divergent
Cpion loops
s+t’+u
= 1 + Cpionloops +
2rns
9
(19)
part of the coefficient
is
f
+ 4rf(s)
f f(t’)
t” +u)
+ f(ii )I
3
(20)
i
with
Analytic continuation is used to define f (b > 0), which has an imaginary part for d > 4; i.e., for s in the domain explored by the existing and proposed experiments. The imaginary part is due to the pion loop in the s-channel. We observe that this calculation provides the weakest s dependence. All of this s-dependence arises from r-r final state interactions. This suggests that such final state interaction corrections to our result will be small, as discussed in Ref. [20], We can compare our result with the one existing data point [ 51. Its statistical and systematic error as well as the uncertainty in s are also displayed in Fig. 1. The fact that this data point is well above the chiral limit prediction has caused some concern [7]. However, given the experimental errors and the prediction of our model this data point does not appear untenable.
Letters B 369 (1996) 101-107
calculation to be compared, without adjustment, with the results of that experiment. In this calculation the chiral limit value of F3rr( 0,0,O) is reproduced in&pendent of the details of the model. Compared with other model calculations, we obtain a form factor that is uniformly larger and has a more rapid increase with s. The models are otherwise broadly consistent. The variation should, however, be sufficient for the proposed experiments to be able to distinguish between them. Along with Ref. [ 3 1, this application of the QCD DSEs is one of the first to explore the model quark 2-point Schwinger function and pion Bethe-Salpeter amplitude well outside the domain of the complex plane on which they have been fitted; i.e., well into the timelike region, which is not accessible in perturbation theory. This region is important in the study of, for example, vector meson Bethe-Salpeter and baryon Fadde’ev amplitudes. Experimental data on F3”( s, t, u) can therefore be used to place important new constraints on the analytic structure of the QCD Schwinger functions.
Acknowledgements
This work was supported by the US Department of Energy, Nuclear Physics Division, under contract number W-31-109-ENG-38 and by the Deutsche Forschungsgemeinschaft (DFG) under contract number AL 279/2-l. The calculations described herein were carried out using a grant of computer time and the resources of the National Energy Research Supercomputer Center. R.A. thanks the Physics Division of ANL for their warm hospitality during two visits in which most of the work described herein was performed. References
5. Summary
and conclusions
Using a phenomenological approach based on the QCD Dyson-Schwinger equations, F3n( s, t, u) was calculated for a range of (s, t, u> that cover the kinematic region to be explored in a proposed experiment [ 7 1. The small photon virtuality in another proposed experiment [ 81 should also make it possible for our
[ 11 K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195; see, also, for example, W. D&rich and M. Reuter, Selected Topics in Gauge Theories, Lecture Notes in Physics, Vol. 244 (Springer, Berlin, 1986). [2j A. Afanasev, in: Proc. Workshop on CEBAF at Higher Energies, eds. N. Isgur and F! Stoler (CEBAE Newport News, 1994), p. 185. [3] M.R. Frank, K.L. Mitchell, C.D. Roberts and PC. Tandy, Phys. Lett. B 359 (1995) 17.
R. Alkojkr,
C.D. R&errs/Physics
[ 41 W. Jaus, Phys. Rev. D 44 (199 I ) 2851; L. Ametller, J. Bijnens, A. Bramon and F. Comet, Phys. Rev. D 45 ( 1992) 986. [S] YuM. Antipov et al., Phys. Rev. D 36 (1987) 21. 161 S.L. Adler, B.W. Lee. S.B. Treiman and A. Zee, Phys. Rev. D 4 (1971) 3497. [ 7 1 R.A. Miskimen, K. Wang and A. Yegneswaran (spokesmen), Study of the Axial Anomaly using the yrr+ nfno Reaction Near Threshold. Letter of intent, CEBAFExperiment 94-015. [ 81 M.A. Moinester, in: Proc. Conf. on Physics with GeVParticle Beams (Jtllich, Germany, August 1994), eds. H. Machner and K. Sistemich (World Scientific, Singapore, 1995). [9] C.D. Roberts and A.G. Williams, Prog. Part. Nucl. Phys. 33 ( 1994) 477. 101 C.D. Roberts, in: Chiral Dynamics: Theory and Experiment, eds. A.M. Bernstein and B.R. Holstein, Lecture Notes in Physics, Vol. 452 (Springer, Berlin, 1995)~. 68. 1 I 1 R. Dashen, Phys. Rev. 183 (1969) 1245. 12 1 R. Delbourgo and M.D. Scadron, J. Phys. G 5 (1979) 1621.
Leners B 369 (19961 101-107
[ IS] M.R. Frank and CD. Roberts, Model gluon propagator
[ 141 [ 151 1161
[ 171 [ 181 [ 191 [20] [21]
[22] 1231 1241
107
and pion and rho-meson observables, Preprint ANL-PHY-8072TH-95 and INT 95-00-97 ( 1995), to appear in Phys. Rev. C. M.R. Frank, Phys. Rev. C 51 (1995) 987. A. Bashir and M.R. Pennington, Phys. Rev. D SO ( 1994) 7679. J.S. Ball and T.-W. Chiu, Phys. Rev. D 22 ( 1980) 2542. J. Praschitka, CD. Roberts and R.T. Cahill, Phys. Rev. D 36 (1987) 209. C.D. Roberts, R.T. Cahill, M.E. Sevior and N. Iannella, Phys. Rev. D 49 (1994) 125. Particle Data Group, Phys. Rev. D 50 (1994) 1175. R. Alkofer, A. Bender and C.D. Roberts, Intern. J. Mod. Phys. A IO (1995) 3319. C.J. Burden, CD. Roberts and M.J. Thomson, Electromagnetic Form Factors of Charged and Neutral Kaons, Archive: nucl-th/9511012. M.V. Terent’ev, Phys. Lett. B 38 (1972) 419. S. Rudaz, Phys. Rev. D 10 (1974) 3857. J. Bijnens. A. Bramon and E Comet, Phys. Lett. B 237 (1990) 488.