Physics Letters B 287 (1992) 23-28 North-Holland PHYSICS LETTERS B

The axial anomaly and the dynamical breaking of chiral symmetry in the °--, 7 reaction

Hiroshi Ito a, W . W . Buck b,c and Franz Gross o,c

a Center for NuclearStudies, Department of Physics, George Washington University, Washington, DC 20052, USA Department of Physics, Hampton University, Hampton, VA 23668, USA

c Continuous Electron Beam Accelerator Facility, 12000 Jefferson Avenue, Newport News, VA 23606, USA o Physics Department, College of William and Mary, Williamsburg, VA 23185, USA

Received 19 November 1991; revised manuscript received 26 May 1992

Using the quark triangle diagram for the Adler-Bell-Jackiw axial anomaly, we calculate the form factor for the ~n°-~y transi- tion. This form factor depends on the quark mass, and we predict the right behavior with mq ~ 250 MeV, the same quark mass generated by the dynamical breaking of chiral symmetry through a Nambu-Jona-Lasinio mechanism.

The size of the axial anomaly [ 1 ] which occurs in the two-photon decay amplitude of a neutral pion (~o~yy) can be fairly well explained in model independent approaches [ 1-6 ]. I f we introduce an additional variable, such as the square of the four-momentum of one of the (virtual) photons, Q2, or the temperature and density of hadronic matter, we may study the dynamics of the axial anomaly. In this letter we report on a study of the 0 2 dependence of the y*~°-~7 transition form factor, Fv~. We find that this dependence is very sensitive to the value of the dynamical quark mass used to calculate it, and that a value determined both from a fit to a number o f pion observables, and from a consideration of the dynamical breaking of chiral symmetry, gives an excellent account of the recent data.

Recently, the form factor for the y*~°-~7 transition was measured for space-like momentum (Q2> 0) of the virtual photon (y*) [7]. Motivated by the vector meson dominance (VMD) hypothesis, Behrend et al. para- meterized the observed form factor as a monopole. At Q2= 0, this form factor coincides with the two-photon decay amplitude of the neutral pion, which was well explained a long time ago by an explicit calculation o f the fermion triangle diagrams [2] shown in fig. 1. The dependence of these diagrams on the fermion (quark) mass at Q 2 = 0 can be eliminated from the expression [ 8 ], and the remaining dependence on the quark degrees o f freedom is trivial; the amplitude is multiplied by the number of colors (no) and a factor depending on the quark charges e~ - e 2 = ] e2).

The two-photon decay of a neutral pion is also very successfully described by the chiral lagrangians developed by Wess, Zumino [ 5 ] and Witten [ 6 ]. Beyond the tree level, however, the loop corrections are found to diverge if one of the two photons is virtual ( a2 :~ 0 ), while these corrections cancel if the two photons are real [ 9 ]. Thus,

k I k I

k 2 k 2

Fig. 1. Triangle diagrams for the ~r/amplitude, where the solid line, wavy line and double line are the fermion, photon and pion respectively.

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Volume 287, number 1,2,3 PHYSICS LETTERS B 6 August 1992

the use of chiral perturbation theory to calculate the form factor requires a renormalization procedure [10]. Two other independent approaches, one [ 3 ] based on the noninvariance of the path-integral measure under chiral transformations, and one [4] based on the operator product expansion [ 11 ], also do not determine the mass scale of the "off-shell" axial anomaly.

In this letter we assume that the constituent quark mass is generated through the spontaneous breaking of chiral symmetry in QCD, and that the Nambu-Goldstone boson generated by the same mechanism is the pion. Using these relations, we are able to determine the size of the dynamical quark mass from a study of pion properties, and then use the resulting information to predict the form factor F~.

To illustrate the sensitive dependence of the ~,*n°~'{ form factor on the quark mass, we will first calculate it from the simple linear o.-model [ 8 ], in which the quarks are coupled to n and c~ mesons through an interaction lagrangian of the simple form ~ i n t =glP(O"~-i~)57~)~.t with effective coupling constant g. In this model, both the pion decay amplitude (M~w) and the amplitude for the virtual process "[*n°---,'f(M~.,~o_~, ) can be calculated from the triangle diagrams shown in fig. 1. For the virtual photon process, these diagrams give

1 1--~

. _ e gmf f M~,~.~o~,----E puf.lpqa~-~-2 d x dy m2 + (q2 p2)xy+q2(y2_y) =e'"P~p.e~pq~Fr~(Q 2) , (1) 0 0

where p is the four-momentum of the pion (p2=(kl+ 2 2 k2) = m ~ = ( 1 3 8 M e V ) 2 ) , k 2 = - q (with k2= ( _ q ) 2= _ Q z) is the four-momentum of an incoming virtual photon in the initial state, k~ = p + q (with k~ = (p+ q)2= 0) is the four-momentum of a real photon in the final state, el is the polarization four-vector of the outgoing photon, and Fv~(Q 2) is the form factor for the virtual process. We assume that the fermion has an effective charge e 2 = nc (e 2 - e 2 ) and mass m.

The amplitude for two-photon decay can be obtained from eq. ( 1 ) using the real photon conditions k 2 = k 2 = 0, and adding the polarization vector of the second photon, e2,

v p o e2 g l i/ p2 "~-1 l' v p ,~ O~ Mrto__-t~=,ltvpo.,~,2klk2-~2- ~ 1-]-O~ l~m2)J~-ElZUpaf_l f. 2 k l k2 -~ , (2)

where ot=e2/4rt. The relation m=gf~ (the Goldberger-Treiman relation to lowest order [8] in the linear o.- model), and the soft pion limit, 0 (p 2/12m 2)__, 0, are used to obtain the second equality. This gives the decay width F~o~ w = 0¢2m 3 / 6 4 n 3f ~ = 7.8 eV, which is independent of the fermion mass m. [The next order correction is ~ O (p2/12m 2). This would be larger than 1 if the current quark masses were used for the fermion, but with constituent masses it is small. ] The numerical results for the form factor ( 1 ) are shown in fig. 2a for m = 100, 200 and 300 MeV, where g = rn/f~ has been used again. Note that the form factor in the simple linear a model is quite sensitive to the size of the fermion mass, and that the result with m = 200 MeV agrees with the experimen- tal data. This mass scale is also consistent with the soft pion approximation, O(p2/12m 2) << 1, used in the calculation of two-photon decay width. A more fundamental calculation requires that we find some other way to fix this mass.

To accomplish this, we introduce a mechanism for generating the quark mass, and relating it to the momen- tum distribution of the pion. The model, a generalization of the Nambu-Jona-Lasinio (NJL) model [12], employs a covariant, separable form for the quark-antiquark (q(t) interaction, with a single mass scale (A) [ 13 ]. Such a separable form is frequently used to treat the pairing problem in the BCS theory [ 14 ]. The dynam- ical quark mass which emerges from this model, and the qdl momentum distribution in the pion wave function are both related to the same parameters.

The Nambu-Jona-Lasinio (NJL) interaction is a contact interaction of the form V NJL-~. G[II-- (ySr) (ySr) ] with explicit chiral symmetry for the u- and d-quarks. A momentum cutoff [ 15 ] is introduced with A N J L = 0 . 7 -

1.0 GeV to avoid the divergence of loop integrals caused by the contact interaction. We modify this NJL inter- action by using a covariant separable interaction, still keeping the essential aspects of the original model. Our model interaction is

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Volume 287, number 1,2,3 PHYSICS LETTERS B 6 August 1992

10.0

S.O (a)

l.O

~ ' 0.5

~ o.t , I . . . . I . . . . I . . . . ×

,~ 10.0

v

5.0

1.0

0.5

0.1

(b)

. . . . I . . . . I 0 1 2 3 4 5

M o m e n t u m t r a n s f e r : qz [(GeV/c)Z]

Fig. 2. The theoretical predictions and the experimental data [ 7 ] for the 7"n°--,7 form factor, F,,JQ2), where Q2= _q2>0" (a) The calculation with the linear a model, eq. (1), with m=300MeV (crosses), m=200MeV (circles), m=100MeV (diamonds). (b) The result of this letter with the generalized Nambu-Jona-Lasinio model, eq. (9), with A=450MeV and rnq = 248 MeV. The dashed line is the dipole fit [ 7 ] to the exper- imental data, FvJQ2)~ 1/(Q2+2 z) with 2=748+30MeV, where the charge radius is 0.65 + 0.03 fro.

Vaa:a~(k', k) =gf( k'2)f( k 2) [ I,~fl~-- (75z),~a(75z)or] , (3)

where the function f ( k 2) ( f ( k '2) ) depends on the relative momen tum, k (k ' ) , of the q(t pairs in the initial (final) state. We chose a monopole function f ( k 2 ) = 1 / ( k 2 - A 2 ) with a scale parameter A. The relativistic pion vertex function will be proport ional to f and if the energy of the qCt pair of quarks in the pion is (on the average ) shared equally, the four-vector k 2 reduces to - k z and the function f b e c o m e s the familiar m o m e n t u m space Yukawa function for an S-state. Another reason for choosing this form is that its comparat ively slow fall-off at large m o m e n t u m insures that the pion form factor calculated f rom it will also fall off comparat ively slowly at large m o m e n t u m transfer, as required by the data.

Using this interaction, the Bethe-Salpeter (BS) equation for the bound state vertex function is

f d4 k G~(k';p)=i ~ Gp:~(k',k)S.,(k+½p)G,~,(k;p)S~,~(k-½p) , (4)

where the quark propagator is S(p)= [I~-mo -,v'(p2) q_ iE ]-1, with mo the current quark mass, and T,(p 2) the quark self-energy. The quark self-energy is given by the Schwinger-Dyson (SD) equation

f d4k /" d4k 2 t o o + Z ( k 2 ) S ( k '2) = i (--~n)4 V,~p:,~(k', k) S,ffk) =4in fg f (k '2) J (-~n)nf(k) k2 - [mo + S ( k 2) ]2, (5)

where nf--. 2 is the number of quark flavors. Now, note that eq. (4) implies that the solution for the pion (at the pion mass p 2= m 2 ) has the form F(k; p) = Y75f ( k 2 )Zf~c, where Y is the wave function normalizat ion deter- mined by charge normalization, and the normalized flavor and color wave functions are given by gf and Xc, respectively. The eigenvalue equation for the pion mass which follows from eq. (4) is

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l = - i n r g f d4k 2 2 4[mo+X(k2)] 2+p2-4k2 (-~n)4f ( k ) {(k_½p)2 - [mo+27(k2)]2}{(k+½p)2_ [mo+X(k2)]2}. (6)

In the chiral limit, when mo = 0, and the pion mass is zero so that p--, 0 (in the pion rest frame), eq. (6) reduces to the following constraint:

I d4 k f2 (k2 ) 1 =4infg (27r)4 kZ_S2(k2 ) . (7)

However, when mo= 0 this equation is identical to eq. ( 5 ) for the quark self-energy, provided X(k 2 ) = Cf(k 2 ), where C is a constant. In this limit, if a nontrivial solution of eq. (5) exists, so that -r(k 2) ~ 0 and a non-zero quark mass is generated, then a zero mass pion must also exist with the qq momentum distribution proportional to f ( k 2). For such solutions to exist, the coupling strength g >/&, where the critical value is gc depends on A and C.

Instead of using the full momentum dependent self-energy X(k 2 ), the size of which depends on the parameter C, we replace _r( k 2 ) by its mean value, (27) _~ M = mq - too, where this mean value is defined by the requirement that the constraint (6) be preserved

1 =- - infg ~ dak 2 2 4mZ+P 2-4k2 (~n)4 f (k ) [(k_½p)2_m2ql[(k+ lp)2_m2q ] . (8)

The parameter mq is our definition of the dynamical quark mass, and it replaces C. If all of the integrals to be evaluated had the form of (6), the introduction of the mean value would be completely equivalent to the use of 27(k 2 ) itself, but since some of the integrals have additional factors, the replacement of mo+ 27(k 2) by mq in all integrals represents an additional approximation, which should be fairly accurate for the observables discussed in this letter.

To determine the parameters A and rnq we calculated the weak decay constant (f~) and the two-photon decay width (F~o~w) for several choices of these parameters. Some results are (A[MeV], mq[MeV]: f~[MeV], F~o~w [ e V ] ) = (700, 200: 90.0, 11.0), (700, 250: 105.0, 7.2), (700, 300: 123.2, 5.8), (400, 200: 75.1, 12.0), (400, 250: 89.3, 8.0), and (400, 300: 100.0, 5.6). Note that these observables are quite stable under small variations of the parameters. The best fit is given by A=450MeV, mq=248MeV, f~=93.0MeV, and F~o w = 7.74 eV. (See ref. [ 13 ] for more details. )

With this best choice (A = 450 MeV and mq= 248 MeV) and using m~= 138.0 MeV, the coupling constant obtained from ( 8 ) is q = 2.795 n 2./12. Using this value in eq. ( 7 ) gives the mean value ( - r ) = M = 234 MeV. This gives a reasonable estimate for the current quark mass, mo = m q - M ~ 14 MeV.

As a further test of our method, including our choice of parameters, the pion charge form factor obtained using our best set of parameters is shown in fig. 3. The agreement with experimental data is very good, and our results support the view, popularized by Isgur and Llewellyn-Smith [ 17 ], that the pion form factor for Q2 ~< 10 (GeV) z can be largely explained by non-perturbative effects. The charge radius, r~, is found to be 0.74 fm. Additional effects from two-body interaction currents associated with the rank-1 separable interaction are not included in fig. 3, although it is found [ 13 ] that the interaction current reduces the form factor about 10 (40)% at Q2=2(10) GeV2/c 2.

The calculation of the xyy amplitude predicted by the generalized NJL model presented above differs from the simple sigma model result, eq. (2), primarily by the appearance of the BS vertex functions F(k; p) at the rcq~l vertices in the Feynman triangle diagrams shown in fig. 1. These vertex functions contain the qcl momentum distribution associated with the pion structure. The amplitude can be written

M~v~=T(k~, ~j :k2, ~2) + T(l'--.2) ,

where

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Volume 287, number 1,2,3 PHYSICS LETTERS B 6 August 1992

0 . 6

e~

C-~ 0 . 4 v

e~

0 . 2 r .

0 , 0 . . . . i, . . . . I . . . . I . . . . I . . . .

2 4 6 B

Q2 [(GeV/e)Z]

10

Fig. 3. The charge form factor of the pion, F~ (Q2). The solid line is the theoretical result obtained with the present model (A = 450 MeV and mq = 248 MeV), and the experimental data are from ref. [ 16]. The dashed line is given by the vector meson dominance model with the pole mass of) .= 748 MeV.

T(k~, ~ :k2, ~2)= - i y T r ( Q q Q q z f ) x / ~ c f d4k

(_.~n_~n~4f( k2 )Tr [~2S(k- ½ (kl -k2) )~,S( k + ½p )y'S( k - ½p ) l •

(9)

This prediction for the 7"n°--,7 form factor is shown in fig. 2b. Excellent agreement with the experimental data is obtained. For comparison, fig. 2b also shows the monopole fit used in the experimental analysis.

Our predictions are in remarkable agreement with the experimental data for both the charge and Fr~ form factors. The VMD model also works well for these two form factors, even though the vector mesons are far away from their mass shell at these space-like momenta. VMD effects are not explicitly included in our calculations, but duality [ 18,19 ] suggests that the qdl contribution to the quark loop in fig. 1 should give the same result as the propagation of vector mesons when we are far from the meson pole.

In conclusion, we emphasize that while the n°--,77 amplitude can be calculated in a model independent way in which any dependence on the quark mass can be removed, the same is not true for the form factor which determines the 7"n°-,7 reaction. Here it is essential to use a quark mass mq=200-250 MeV in order to predict the right behavior of these form factors, and this quark mass is determined by the low energy properties of the pion, consistent with the Schwinger-Dyson equation, and in agreement with the value used by Sakurai, Schilcher and Tran in their discussion of quark-meson duality [ 18 ].

The first author (H.I.) thanks Professor G.E. Brown for very useful and valuable advice. The work of H.I. is supported in part by Department of Energy Grant Number DE-FG05-86-ER40270. The work of W.B. was par- tially supported by the National Science Foundation, Grant RII-8704038. The work of F.G. was partially sup- ported by the Department of Energy Grant Number DE-FG05-88ER40435

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