- Email: [email protected]
Effects of the UA(1) anomaly on η → 2γ decay
_l!!i!J
5
cw3
0ctobe.r1995 PHYSICS
em
ELSEVIER
LETTERS
B
Physics Letters B 359 (1995) 210-216
Effects of the UA( 1) anomaly on
33 -+
2y decay
M. Takizawa a,1, M. Oka b~2 a Institute for Nuclear Study? University of Tokyo, Tanashi, Tokyo 188, Japan b Department of Physics, Tokyo Institute of Technology, Meguro. Tokyo 152. Japan Received 12 June 1995 Editor: M. Dine
Abstract We study the V,I-+ 27 decay using an extended three-flavor Nambu-Jona-Lasinio model that includes the ‘t Hooft instanton induced interaction. The 77 meson mass and the r] + 2y decay width are reproduced simultaneously with a rather strong instanton induced interaction. The calculated 7 decay constant is fq = 2.23 fir and it suggests that the 77meson is no longer a Goldstone boson.
1. Introduction The structure of the q meson gives us information about the mechanisms of spontaneous breaking of chiral symmetry, the pattern of the explicit breaking of chiral symmetry and the dynamics of the VA ( 1)anomaly. The r] meson is the eighth member of the low-lying nonet pseudoscalar mesons and is considered as a Goldstone boson associated with the spontaneous breaking of chiral symmetry in the QCD vacuum. The physics of the ~7and 7’ mesons has been extensively studied in the l/No expansion approach [ 11. In the NC + co limit, the U, ( 1) anomaly is turned off and then the q meson becomes degenerate with the pion and the q’ meson becomes pure Ss state with n$,,(Nc -+ 00) = 2rng- rni~11 (687 MeV)* [2]. So the U, ( 1) anomaly pushes up m,, by about 400 MeV and m,,l by about 300 MeV. Usually the r]’ meson is not considered as a Goldstone boson since it is ’ E-mail address: 2 E-mail address:
[email protected]. [email protected].
0370-2693/95/.$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0370-2693( 95)01068-8
considered as mostly flavor singlet meson and in this channel there exists the VA( 1) anomaly which explicitly breaks the chiral symmetry. On the other hand, the ~7 meson is considered as a Goldstone boson because its mass is close to the naive estimation of the pseudoscalar meson mass in A8 channel, i.e., m& = 4 2 ?mK -
!rni N (567 MeV) *. From the 1/NC expansion point of view, r]-meson mass is largely affected by the finite NC correction, i.e., the UA (1) anomaly, then it is natural to ask how much 7 meson loses the Goldstone boson nature. In order to answer to the above question, it may be important to study the ~7 + 2y decay because of relation to the Adler-Bell-Jackiw ( ABJ) triangle anomaly [3] through the partial conservation of axial-vector current (PCAC) hypothesis. One of the useful and widely used frameworks for studying the phenomena related to the axial-vector anomaly and the spontaneous chiral symmetry breaking is the chiral effective meson lagrangian given by Wess and Zumino [4] and developed by Witten [5]. The v, 7’ + 2y decays have been studied using
M. Takizawa. U. Oka /Physics
the Wess-Zumino-Witten (WZW) lagrangian with the corrections at one-loop order in the chiral perturbation and it has been shown that the two-photon decay widths can be explained with an 7-v’ mixing angle 0 N -20” [ 61. From the chiral perturbation [7] point of view, the WZW term is derived in the chiral limit and is of order p4. As discussed in [8], to reliably calculate SU(3) breaking effects of the 77,v’ + 2y decays, the low-energy expansion to order p6 has to be carried out. However in [ 61 full analysis of order p6 has not been performed. Furthermore, because of the UA( 1) anomaly, the singlet channel decay amplitude ~0 -+ 2y derived using PCAC + ABJ anomaly should be modified so as to become the renormalisation group invariant [ 91. The purpose of this paper is to study the r,i + 2y decay in the framework of the generalized NambuJona-Lasinio (NJL) model [ lo] as a chiral effective quark lagrangian of the low-energy QCD. The generalized three-flavor NJL model which involves the UL (3) x UR (3) symmetric four-quark interaction and the six-quark flavor-determinant interaction [ 111 incorporating effects of the UA( 1) anomaly is used widely in recent years to study such topics as the quark condensates in vacuum, the spectrum of lowlying mesons, the flavor-mixing properties of the lowenergy hadrons, etc. [ 12-161. In this approach the effects of the explicit breaking of the chiral symmetry by the current quark mass term and the U,t( 1) anomaly on the r) 4 2y decay amplitude can be calculated consistently with those on the v-meson mass, 7 decay constant and mixing angle within the model applicability.
2. Extended Nambu-Jona-Lasinio
model
We work with the NJL model lagrangian extended to three-flavor case:
L==,,
-
tk)
*
,
(2)
211
+det [c&Cl +YS)$~]
(4)
} .
Here the quark field (/I is a column vector in color, flavor and Dirac spaces and Aa is the U( 3) generator in flavor space. The free Dirac lagrangian .& incorporates the current quark mass matrix which breaks the chiral ??l = diag(m,, md, m,) U,( 3) x UR( 3) invariance explicitly. & is a QCD motivated four-fermion interaction, which is chiral UL(3) x UR(3) invariant. The ‘t Hooft determinant & represents the UA (1) anomaly. It is a 3 x 3 determinant with respect to flavor with i,j = u, d, Y. Quark condensates and constituent quark masses are self-consistently determined by the gap equations M, = m, - 2Gs(iiu)
- 2GD(cid)(Ss) ,
hfd = md - 2Gs(dd) M,
=
m,
-
~GD
(ss)
(ik)
,
2Gs(Ss) - 2G~(Eu)(dd),
(5)
with (aa) = -Tr(c.D)
[i&$(x = 0)]
A
d4pT~(C.D)
=-
J
i
pcLyp
(2rj4
-
M, f
ic I
*
(61
Here the covariant cutoff A is introduced to regularize the divergent integral and Tr(c,D) means trace in color and Dirac spaces. The pseudoscalar channel quark-antiquark scattering amplitudes (ps, ~4; outjpt , a; in) = (27r) 4# (p3 + p4 - pl - p2)'2& are then calculated in the ladder approximation. We assume the isospin symmetry too. In the 77 and 7’ channel, the explicit expression is ;r_=_ 44
T
U(p3)A8&5u(p4) ( dp3)A"iw(p4)
>(
>
(1)
+l4+L6,
LO = iJ (iJ,Yp
density
Letters B 359 (1995) 210-216
Ab2)
Wq2)
m2>
C(q2)
>
(7)
’
with 2
Nq’) = detD(q2) x (2WoGs - G,G,)Z"(q2) - G}
9
(8)
212
M. Takkawa, hf. Oka/Physics Letters B 359(19951210-216
2
B(s2)= detD(q2) x {-2(GoG*
- G,G,)Z”(&
9
- G,}
(9)
2 G(q2)
=
detD(q2)
x {2(GoG*
X
- G,G,)Z*(q*)
- Go} 9
( ~(P2M0~Y5~(Pl)
(10) = _
and Gr, = ;Gs - 3(2(iiu) ~((SS) - 4(fiu))Gn, The quark-antiquark
+ (js))Gn,
G,
G8 = &Gs -
= -&((ss)
~(P2)A8t?5dP1)
T
ti(P3)AViY5U(P4) ( fi(P3)A’l’iYsdP4)
- (ilu))Go. X
bubble integrals are
A
X
Ws2) 0
>
0
Dq’(q2> >
~(P2)~%W(PI)
(
~(P2)~“‘~~5dPl)
(11) Z8(q2)
d4pTr(dD)
J
(12)
cos0 ( sin0
-
= i J
d4pTr(Gf,D)
tan2e =
x [SF(P)AOiygSF(P+q)A*iyS]
(13)
,
with q = p1 + ~2. The 2 x 2 matrix D is
D(q2)= Ddq2) D21(q2)
D*2(q2) D22(q2)
-sin8 case
> ’
(14)
>’
with
Di,(q2)
= 2G*Z*(q2)
+ 2G,,,Z”(q*)
-
h(q2)
= 2G*Zm(q2)
+
2G,Z”(q?
,
(16)
&&x2)
= 2GoZYq’)
+ 2G,Z8(q2)
,
(17)
Ddq2)
= 2GoZ0(q2>
+ 2G,Zm(q2)
-
1
1
(15)
by
Wq*)
(21)
*
G(q2) - Ns2>
WI4
(20)
’
The rotation angle 8 is determined
A
Z”(q2)
)
with A’J E cos Bh* - sin 8A”, Aq’ E sin $A* + cos OA” and Ts =
(2r)4
= i
(19)
> ’
So 8 depends on q2. At 8 = m$ 0 represents the mixing angle of the A* and A0 components in the vmeson state. In the usual effective pseudoscalar meson lagrangian approaches, the v and 7’ mesons are analyzed using the q2-independent 17-q’ mixing angle. Because of the $-dependence, B cannot be interpreted as the v-v’ mixing angle. The origin of the $-dependence is that the v and ?I’ meson have the internal structures. The effective g-quark coupling constant gV is determined by the residue of the scattering amplitude at the v pole, i.e., gi = lim,z,,,,t, ( q2 - mf,) D, ( q2), and the ~7decay constant fV is determined by calculating the quark-antiquark one-loop graph,
(18)
From the pole position of the scattering amplitude Eq. (7), the T-meson mass m,, is determined. The scattering amplitude (7) can be diagonalized by rotation in the flavor space
X SF(P)iY5A9&(P
-
q)]
(y’=fn;
.
(22)
One can easily show that in the uA( 1) limit, i.e., GD = 0 and mu,d $ m,, the q meson becomes the ideal
M. Takizawa. M. Oka/Physics
mixing *,
= *,,
state composed of u and d-quarks, namely, go = g,, f,, = fr and tan0 = ---A.
3. r~ +
2y decay amplitudes
Let us now turn to the calculations of the ?r”, 77 -+ 2y decay widths. The TO, 7 ---f 2y decay amplitudes are given by (y(ki)y(kz)lM(q))
= i(2~)~s’(k,
x eCL~pa~ltLE~k~k~l~-2,(q2)
+ k:! - s) 3
(23)
where ~1 and E:! are the polarization vectors of the photon. By calculating the pseudoscalar-vector-vector type quark triangle diagrams, we get the following results: ?L2y
= +Yud?
Ii_2v
= cg ~
-
(24)
9
v$$c0Se{5wd
sin8&{5F(u,q)
-
+
~FWI))
F(s,v)}] .
(25)
Here cy is the fine structure constant of QED and F(a, M) (a = 14,s and M = 9,~) is defined as F(a, M) I
I
=
JJ dx
0
0
dy
2(1 -x)M, Mz-m&x(1
-x)(1-y)
’
(26) Then the M + 2y d_ecay width li(M -+ 2y) is given by I( M + 2y) = (‘L+2y12m~/(64r). In the chiral limit, the pion mass vanishes and F (u, TO> becomes 1/M,. In this limit, the GoldbergerTreiman (GT) relation at the quark level, M%= g,f,, holds in the NJL model and this leads to 7g_2y = a/ (rr f ,> which is same as the tree-level results in the WZW lagrangian approach. It should be mentioned that we have to integrate out the triangle diagrams without introducing a cutoff A in order to get the above result though the cutoff is introduced in Eqs. (6), (1 I)-( 13), (22) in the NJL model. If we introduce a cutoff A to the loop-integral of the triangle diagrams, the decay amplitude becomes too small and we lose the success of the model independent prediction for the I($ + 27). In the us X u(3)R
LettersB 3.59 (1995) 210-216
213
version of the NJL model, the WZW derived using the bosonization method kernel expansion [ 17,181. In their 0( l/A) term has been neglected and take the A + cc limit.
4. Numerical
term has been with the heatapproach, the it is same as to
results
The recent experimental results of the z-O, 7 ---t 2y decay widths are r(7r” -+ 2-y) = 7.7 & 0.6 eV and T(v + 2y) = 0.510 * 0.026keV [ 191 and the reduced amplitudes are ‘Z~I_.+~~= (2.5f0.1) II Iii27
I
From
I
=
x lo-”
(2.5 f 0.06) x IO-”
Eq.
(24)
and Eq.
(25),
[eV]-‘,
(27)
[eV] -’ .
(28)
we get 7&+2y
(5/3)1+_2,
in the UA( 1) limit. Therefore
to reproduce
the experimental
=
in order
value of ?q_2y,
the
effect of the UA( 1) anomaly should reduce ‘?&+, by a factor 315. In our theoretical calculations, the parameters of the NJL model are the current quark masses m,, = md, m,, the four-quark coupling constant Gs, the coupling constant of the ‘t Hooft instanton induced interaction Gn and the covariant cutoff A. First, we take Go as a free parameter and study the q-meson properties as functions of Go. For the light quark masses, m, = md = 8.0 MeV is taken to reproduce M, = Md N 330 MeV which is the value usually used in the nonrelativistic quark model. Other parameters ms, Go and A are determined so as to reproduce the isospin averaged observed masses, m, = 138.04 MeV, mK = 495.7 MeV and the pion decay constant f,,. The calculated constituent U, d-quark mass is M u,d = 324.9 MeV which is independent of CD. On the other hand the calculated constituent s-quark mass weakly decreases from M, = 556.3 MeV to M, = 503.3 MeV when Gn is changed from Go = 0 to GD = Gz where the observed m,, is reproduced. The fitted result of the current s-quark mass m, is almost independent of Gn and m,T = 192.95 MeV at Go = GL. The ratio of the current s-quark mass to the current u,d-quark mass is m,/m, = 24.1, which agrees wellwithm,/rit=25f2.5 (?%= $(m,+md)) derived from ChBT [20]. The kaon decay constant
M. Tukizawa, M. Oka / Physics Letters B 359 (199s) 210-216
214
600
250
, 20
I
200-
1505 E.
loo-
50-
0
1 0.0
I
I
I
I
I
I
,
0.2
0.4
0.6
0.8
1.0
12
1.4
G
D
’
-60
0 0.0
, 0.2
fK is the prediction and is almost independent of GD. We have obtained f~ = 96.6 MeV at Go = Gz which is about 15% smaller than the observed value. We consider this is the typical predictive power of the N.JL model in the strangeness sector. We next discuss the rr” -+ 2y decay. The calculated result is 7$1_,2~ = 2.50 x 10-i’ [ l/eV] which agrees well with the observed value given in Eq. (27). The current algebra result is I$x._,~~ = a/(rf,,) = 2.514 x lo-” [ l/eV], so the soft pion limit is a good approximation for 7r-O-+ 2y decay. There are two effects of the symmetry breaking on ?$0_+2~. One is the deviation from the GT relation and another is the matrix element of the triangle diagram F(u, 7~~). Our numerical results are g, = 3.44, MU/f,, = 3.52 and F(U,?rO)Mu = 1.015, therefore the deviations from the soft pion limit are very small both in the GT relation and the matrix element of the triangle diagram. Let us now turn to the discussion of the v-meson properties, The calculated results of the r]-meson mass m7] and the mixing angle 0 are shown in Fig. 1, the 17 decay constant f,,is given in Fig. 2 and the 7 -+ 2y decay amplitude &lY is given in Fig. 3 as functions of the non-dimensionalized coupling constant of ‘t Hooft instanton induced interaction c”df = -G~(h/2?r)~f@. As can be seen from Fig. 1, in order to reproduce the observed r]-meson mass, a rather strong instanton
I 0.6
I 0.8 G
Sfl
Fig. 1. Dependence of the calculated q-meson mass tn,, (solid line) and the mixing angle 13(dashed line) on the non-dimensionalized coupling constant of the ‘t Hooft instanton induced interaction G”,“. The observed v-meson mass ~1~= 547.45 MeV is reproduced at G”,” = 1.41.
I 0.4
D
I 1.0
I 1.2
I 1.4
elf
Fig. 2. Dependence of the calculated 7 decay constant fq on the non-dimensionalized coupling constant of the ‘t Hooft instanton induced interaction G’$.
induced interaction is necessary. For example, at Go = Gz, GD(&)/G, = 1.58, it means that the contribution of .Ccgto the dynamical mass of the U, d-quarks is about 60% bigger than that of 1c4. In the previous study of the r] and 7’ mesons in the extended NJL model [ 12161, the strength of the instanton induced interaction has been determined so as to reproduce the observed 7’ mass though the 7’ state has the unphysical decay mode of the q’ + UU, &. The strength determined from m,!, G$ is much smaller than G;),, about l/10 to l/5 of G;),. One of the shortcomings of the NJL model is the lack of the confinement mechanism. It is expected that theconfinement gives rise to an attractive force between quark and antiquark in the 71’meson to prevent the 7’ meson from decaying to the quark and antiquark pair. One of the important results is that the calculated 77 decay constant is very different from the pion decay constant, fv = 206 MeV = 2.23f, at CD = Gz. This suggests that the 7 meson loses the Goldstone boson nature very much. To see whether the 77meson is the Goldstone boson, the GT relation is another important information. For the r] meson, the naive GT relation at the quark level is 2g, f,, = (-&cos 0 -
2 5 sin8]M,
i sinB]M, and at + 1 - ‘jj cost? J$ Go = CL, our numerical results are 2g, fv = 3.202
GeVand/&cosB-2
$sinBIM,+]--&cosB$ 0.892 GeV. Therefore the GT relation
M. Takizawa, M. Oka /Physics Letters B 359 (1995) 210-216
215
able from our analysis. Our calculated result of the mixing angle is 8 = 15.1” which should be compared with 8 N -20” [22]. There are two major differences between our model calculations and the usual analysis. One is that, in the usual analysis, the energy independent mixing is assumed. Another point is that the effects of the U, ( 1) anomaly on the v-meson properties are not taken into account in the usual analysis.
5. Concluding I
00
0.2
I
04
I
06
1
I
I
0.6
1.0
1.2
G De” Fig. 3. Dependence non-dimensionalized induced interaction
remarks
14
of the ~7 -+ 2y decay amplitude ‘&zr on the coupling constant of the ‘t Hooft instanton c”,“.
dose not hold at all. Since the NJL model is known as the model that describes the Goldstone boson properties reasonably well, it is natural to ask whether the present model is applicable to the q meson. For the v’ meson, we expect that the confinement plays an important role since the $-meson pole of the scattering amplitude Eq. (7) appears above the iiu and (ldthreshold. On the other hand, the 17 meson is a tight bound state, so we expect that the NJL model can describe the essential feature of the g meson. In order to confirm it, we have studied the constituent u,dquark mass dependence of the r]-meson properties. By changing m,,d from 7.5 MeV to 8.5 MeV, we have changed Mu,d from about 300 MeV to 360 MeV and other parameters of the model have been chosen so as to reproduce the experimental values of mB, rnK, rn7 and ,fr. The changes of the calculated q decay constant and the mixing angle have been within 2%. We are now in the position to discuss the 77 ---f 2y decay amplitude. Our result is ‘2&zY = 2.73 x lo-” [ 1/eV] at GO = Gz, which is about 10% larger than the experimental value. Therefore the present model reproduces the v-meson mass and the 7 ---f 2-y decay width simultaneously. As for the effects of the symmetry breaking on ?v_~r, our results are F( u, 7) M, = 1.41 andF(u,rl)/F(s,v) = 1.96.Bernardetal. [21] calculated the 71 --t 2y decay width using a similar model. They used a rather weak instanton induced interaction and their result of T(q -+ 27) is about 50% bigger than the experimental value. It is understand-
Using an extended three-flavor Nambu-JonaLasinio model that includes the ‘t Hooft instanton induced interaction, we have studied the Z-O,T,J-+ 2y decays as well as the properties of the pion, the kaon and the v meson. The q-meson mass and the 7 --t 2y decay width have been reproduced simultaneously with a rather strong instanton induced interaction. The calculated 77decay constant is about twice of the pion decay constant. So it is rather hard to consider the r] meson as the Goldstone boson. Because of the above novel picture of the 17 meson, the situation of the mixing angle is also different from the usual analysis. In order to confirm the novel picture of the v meson we have obtained here, certainly further studies of the q-meson properties are necessary. One thing is to study other v-meson decay processes such as 7 ---f 7~O2y, r~ + 3~ in the present framework and such calculations are now in progress. Since the properties of the ~7meson and those of the 7’ meson are closely related, one has to construct the low-energy effective model of QCD which can apply to the 7’ meson. Such an attempt is left to future study. References [ 1] G. ‘t Hooft, Nucl. Phys. B 72 (1974) 461; for a review, see G.A. Christos, Phys. Rep. 116 ( 1984) 251. (21 G. Veneziano, Nucl. Phys. B 159 (1979) 213. [ 3 ] S. Adler, Phys. Rev. 177 ( 1969) 2426; J. Bell and R. Jackiw, Nuovo Cimento A 60 ( 1969) 47. [ 4) J. Wess and B. Zumino, Phys. L&t. B 37 (1971) 95. [51 E. Witten, Nucl. Phys. B 223 (1983) 422. [6] J.E Donoghue, B.R. Holstein and Y.-C.R. Lin, Phys. Rev. Lett. 55 (1985) 2766; 61 (1988) 1527 (E); J. Bijnens, A. Bramon and F. Comet, Phys. Rev. Lett. 61 (1988) 1453; J.F. Donoghue and D. Wyler, Nucl. Phys. B 316 ( 1989) 289.
216
hf. Takizawa, hf. Oka/Physics Letters B 359 (1995) 210-216
[ 71 J. Gasser and H. Leutwyler, Ann. of Phys. 158 (1984) 142.
[8] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [9] GM. Shore and G. Veneziano, Nucl. Phys. B 381 (1992) 3. [lo] Y. Nambu and G. Jona-I-asinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [ 111 M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. 45 (1971) 1955; G. ‘t Hooft, Phys. Rev. D 14 (1976) 3432. [ 121 T. Kunihiro and T. Hatsuda, Phys. Lett. B 206 (1988) 385; T. Hatsuda and T. Kunihiro, Z. Phys. C 51 ( 1991) 49; Phys. Rep. 247 (1994) 221. [ 131 V. Bernard, R.L. Jaffe and U.-G. Meissner, Nucl. Phys. B 308 (1988) 753. [ 14) Y. Kohyama, K. Kubodera and M. Takizawa, Phys. Lett. B 208 (1988) 165; M. Takizawa, K. Tsushima, Y. Kohyama and K. Kubodera,
Prog. Theor. Phys. 82 ( 1989) 48 1; Nucl. Phys. A 507 ( 1990) 611. [ 151 H. Reinhardt and R. Alkofer, Phys. Len. B 207 (1988) 482; R. Alkofer and H. Reinhardt, Z. Phys. C 45 (1989) 275. 1161 8.. Klimt, M. Lutz, U. Vogl and W. Weise, Nucl. Phys. A 516 (1990) 429; U. Vogl, M. Lutz, S. Klimt and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. [ 171 D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. ]18] M. Wakamatsu, Ann. of Phys. 193 (1989) 287. [ 191 Panicle Data Group, Phys. Rev. D 50 (1994) 1173. [20] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [21] V. Bernard, A.H. Blin, B. Hiller, U.-G. Meissner and MC. Ruivo, Phys. Lett. B 305 ( 1993) 163. [22] EJ. Gilman and R. Kauffman, Phys. Rev. D 36 ( 1987) 2761.
5
cw3
0ctobe.r1995 PHYSICS
em
ELSEVIER
LETTERS
B
Physics Letters B 359 (1995) 210-216
Effects of the UA( 1) anomaly on
33 -+
2y decay
M. Takizawa a,1, M. Oka b~2 a Institute for Nuclear Study? University of Tokyo, Tanashi, Tokyo 188, Japan b Department of Physics, Tokyo Institute of Technology, Meguro. Tokyo 152. Japan Received 12 June 1995 Editor: M. Dine
Abstract We study the V,I-+ 27 decay using an extended three-flavor Nambu-Jona-Lasinio model that includes the ‘t Hooft instanton induced interaction. The 77 meson mass and the r] + 2y decay width are reproduced simultaneously with a rather strong instanton induced interaction. The calculated 7 decay constant is fq = 2.23 fir and it suggests that the 77meson is no longer a Goldstone boson.
1. Introduction The structure of the q meson gives us information about the mechanisms of spontaneous breaking of chiral symmetry, the pattern of the explicit breaking of chiral symmetry and the dynamics of the VA ( 1)anomaly. The r] meson is the eighth member of the low-lying nonet pseudoscalar mesons and is considered as a Goldstone boson associated with the spontaneous breaking of chiral symmetry in the QCD vacuum. The physics of the ~7and 7’ mesons has been extensively studied in the l/No expansion approach [ 11. In the NC + co limit, the U, ( 1) anomaly is turned off and then the q meson becomes degenerate with the pion and the q’ meson becomes pure Ss state with n$,,(Nc -+ 00) = 2rng- rni~11 (687 MeV)* [2]. So the U, ( 1) anomaly pushes up m,, by about 400 MeV and m,,l by about 300 MeV. Usually the r]’ meson is not considered as a Goldstone boson since it is ’ E-mail address: 2 E-mail address:
[email protected]. [email protected].
0370-2693/95/.$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0370-2693( 95)01068-8
considered as mostly flavor singlet meson and in this channel there exists the VA( 1) anomaly which explicitly breaks the chiral symmetry. On the other hand, the ~7 meson is considered as a Goldstone boson because its mass is close to the naive estimation of the pseudoscalar meson mass in A8 channel, i.e., m& = 4 2 ?mK -
!rni N (567 MeV) *. From the 1/NC expansion point of view, r]-meson mass is largely affected by the finite NC correction, i.e., the UA (1) anomaly, then it is natural to ask how much 7 meson loses the Goldstone boson nature. In order to answer to the above question, it may be important to study the ~7 + 2y decay because of relation to the Adler-Bell-Jackiw ( ABJ) triangle anomaly [3] through the partial conservation of axial-vector current (PCAC) hypothesis. One of the useful and widely used frameworks for studying the phenomena related to the axial-vector anomaly and the spontaneous chiral symmetry breaking is the chiral effective meson lagrangian given by Wess and Zumino [4] and developed by Witten [5]. The v, 7’ + 2y decays have been studied using
M. Takizawa. U. Oka /Physics
the Wess-Zumino-Witten (WZW) lagrangian with the corrections at one-loop order in the chiral perturbation and it has been shown that the two-photon decay widths can be explained with an 7-v’ mixing angle 0 N -20” [ 61. From the chiral perturbation [7] point of view, the WZW term is derived in the chiral limit and is of order p4. As discussed in [8], to reliably calculate SU(3) breaking effects of the 77,v’ + 2y decays, the low-energy expansion to order p6 has to be carried out. However in [ 61 full analysis of order p6 has not been performed. Furthermore, because of the UA( 1) anomaly, the singlet channel decay amplitude ~0 -+ 2y derived using PCAC + ABJ anomaly should be modified so as to become the renormalisation group invariant [ 91. The purpose of this paper is to study the r,i + 2y decay in the framework of the generalized NambuJona-Lasinio (NJL) model [ lo] as a chiral effective quark lagrangian of the low-energy QCD. The generalized three-flavor NJL model which involves the UL (3) x UR (3) symmetric four-quark interaction and the six-quark flavor-determinant interaction [ 111 incorporating effects of the UA( 1) anomaly is used widely in recent years to study such topics as the quark condensates in vacuum, the spectrum of lowlying mesons, the flavor-mixing properties of the lowenergy hadrons, etc. [ 12-161. In this approach the effects of the explicit breaking of the chiral symmetry by the current quark mass term and the U,t( 1) anomaly on the r) 4 2y decay amplitude can be calculated consistently with those on the v-meson mass, 7 decay constant and mixing angle within the model applicability.
2. Extended Nambu-Jona-Lasinio
model
We work with the NJL model lagrangian extended to three-flavor case:
L==,,
-
tk)
*
,
(2)
211
+det [c&Cl +YS)$~]
(4)
} .
Here the quark field (/I is a column vector in color, flavor and Dirac spaces and Aa is the U( 3) generator in flavor space. The free Dirac lagrangian .& incorporates the current quark mass matrix which breaks the chiral ??l = diag(m,, md, m,) U,( 3) x UR( 3) invariance explicitly. & is a QCD motivated four-fermion interaction, which is chiral UL(3) x UR(3) invariant. The ‘t Hooft determinant & represents the UA (1) anomaly. It is a 3 x 3 determinant with respect to flavor with i,j = u, d, Y. Quark condensates and constituent quark masses are self-consistently determined by the gap equations M, = m, - 2Gs(iiu)
- 2GD(cid)(Ss) ,
hfd = md - 2Gs(dd) M,
=
m,
-
~GD
(ss)
(ik)
,
2Gs(Ss) - 2G~(Eu)(dd),
(5)
with (aa) = -Tr(c.D)
[i&$(x = 0)]
A
d4pT~(C.D)
=-
J
i
pcLyp
(2rj4
-
M, f
ic I
*
(61
Here the covariant cutoff A is introduced to regularize the divergent integral and Tr(c,D) means trace in color and Dirac spaces. The pseudoscalar channel quark-antiquark scattering amplitudes (ps, ~4; outjpt , a; in) = (27r) 4# (p3 + p4 - pl - p2)'2& are then calculated in the ladder approximation. We assume the isospin symmetry too. In the 77 and 7’ channel, the explicit expression is ;r_=_ 44
T
U(p3)A8&5u(p4) ( dp3)A"iw(p4)
>(
>
(1)
+l4+L6,
LO = iJ (iJ,Yp
density
Letters B 359 (1995) 210-216
Ab2)
Wq2)
m2>
C(q2)
>
(7)
’
with 2
Nq’) = detD(q2) x (2WoGs - G,G,)Z"(q2) - G}
9
(8)
212
M. Takkawa, hf. Oka/Physics Letters B 359(19951210-216
2
B(s2)= detD(q2) x {-2(GoG*
- G,G,)Z”(&
9
- G,}
(9)
2 G(q2)
=
detD(q2)
x {2(GoG*
X
- G,G,)Z*(q*)
- Go} 9
( ~(P2M0~Y5~(Pl)
(10) = _
and Gr, = ;Gs - 3(2(iiu) ~((SS) - 4(fiu))Gn, The quark-antiquark
+ (js))Gn,
G,
G8 = &Gs -
= -&((ss)
~(P2)A8t?5dP1)
T
ti(P3)AViY5U(P4) ( fi(P3)A’l’iYsdP4)
- (ilu))Go. X
bubble integrals are
A
X
Ws2) 0
>
0
Dq’(q2> >
~(P2)~%W(PI)
(
~(P2)~“‘~~5dPl)
(11) Z8(q2)
d4pTr(dD)
J
(12)
cos0 ( sin0
-
= i J
d4pTr(Gf,D)
tan2e =
x [SF(P)AOiygSF(P+q)A*iyS]
(13)
,
with q = p1 + ~2. The 2 x 2 matrix D is
D(q2)= Ddq2) D21(q2)
D*2(q2) D22(q2)
-sin8 case
> ’
(14)
>’
with
Di,(q2)
= 2G*Z*(q2)
+ 2G,,,Z”(q*)
-
h(q2)
= 2G*Zm(q2)
+
2G,Z”(q?
,
(16)
&&x2)
= 2GoZYq’)
+ 2G,Z8(q2)
,
(17)
Ddq2)
= 2GoZ0(q2>
+ 2G,Zm(q2)
-
1
1
(15)
by
Wq*)
(21)
*
G(q2) - Ns2>
WI4
(20)
’
The rotation angle 8 is determined
A
Z”(q2)
)
with A’J E cos Bh* - sin 8A”, Aq’ E sin $A* + cos OA” and Ts =
(2r)4
= i
(19)
> ’
So 8 depends on q2. At 8 = m$ 0 represents the mixing angle of the A* and A0 components in the vmeson state. In the usual effective pseudoscalar meson lagrangian approaches, the v and 7’ mesons are analyzed using the q2-independent 17-q’ mixing angle. Because of the $-dependence, B cannot be interpreted as the v-v’ mixing angle. The origin of the $-dependence is that the v and ?I’ meson have the internal structures. The effective g-quark coupling constant gV is determined by the residue of the scattering amplitude at the v pole, i.e., gi = lim,z,,,,t, ( q2 - mf,) D, ( q2), and the ~7decay constant fV is determined by calculating the quark-antiquark one-loop graph,
(18)
From the pole position of the scattering amplitude Eq. (7), the T-meson mass m,, is determined. The scattering amplitude (7) can be diagonalized by rotation in the flavor space
X SF(P)iY5A9&(P
-
q)]
(y’=fn;
.
(22)
One can easily show that in the uA( 1) limit, i.e., GD = 0 and mu,d $ m,, the q meson becomes the ideal
M. Takizawa. M. Oka/Physics
mixing *,
= *,,
state composed of u and d-quarks, namely, go = g,, f,, = fr and tan0 = ---A.
3. r~ +
2y decay amplitudes
Let us now turn to the calculations of the ?r”, 77 -+ 2y decay widths. The TO, 7 ---f 2y decay amplitudes are given by (y(ki)y(kz)lM(q))
= i(2~)~s’(k,
x eCL~pa~ltLE~k~k~l~-2,(q2)
+ k:! - s) 3
(23)
where ~1 and E:! are the polarization vectors of the photon. By calculating the pseudoscalar-vector-vector type quark triangle diagrams, we get the following results: ?L2y
= +Yud?
Ii_2v
= cg ~
-
(24)
9
v$$c0Se{5wd
sin8&{5F(u,q)
-
+
~FWI))
F(s,v)}] .
(25)
Here cy is the fine structure constant of QED and F(a, M) (a = 14,s and M = 9,~) is defined as F(a, M) I
I
=
JJ dx
0
0
dy
2(1 -x)M, Mz-m&x(1
-x)(1-y)
’
(26) Then the M + 2y d_ecay width li(M -+ 2y) is given by I( M + 2y) = (‘L+2y12m~/(64r). In the chiral limit, the pion mass vanishes and F (u, TO> becomes 1/M,. In this limit, the GoldbergerTreiman (GT) relation at the quark level, M%= g,f,, holds in the NJL model and this leads to 7g_2y = a/ (rr f ,> which is same as the tree-level results in the WZW lagrangian approach. It should be mentioned that we have to integrate out the triangle diagrams without introducing a cutoff A in order to get the above result though the cutoff is introduced in Eqs. (6), (1 I)-( 13), (22) in the NJL model. If we introduce a cutoff A to the loop-integral of the triangle diagrams, the decay amplitude becomes too small and we lose the success of the model independent prediction for the I($ + 27). In the us X u(3)R
LettersB 3.59 (1995) 210-216
213
version of the NJL model, the WZW derived using the bosonization method kernel expansion [ 17,181. In their 0( l/A) term has been neglected and take the A + cc limit.
4. Numerical
term has been with the heatapproach, the it is same as to
results
The recent experimental results of the z-O, 7 ---t 2y decay widths are r(7r” -+ 2-y) = 7.7 & 0.6 eV and T(v + 2y) = 0.510 * 0.026keV [ 191 and the reduced amplitudes are ‘Z~I_.+~~= (2.5f0.1) II Iii27
I
From
I
=
x lo-”
(2.5 f 0.06) x IO-”
Eq.
(24)
and Eq.
(25),
[eV]-‘,
(27)
[eV] -’ .
(28)
we get 7&+2y
(5/3)1+_2,
in the UA( 1) limit. Therefore
to reproduce
the experimental
=
in order
value of ?q_2y,
the
effect of the UA( 1) anomaly should reduce ‘?&+, by a factor 315. In our theoretical calculations, the parameters of the NJL model are the current quark masses m,, = md, m,, the four-quark coupling constant Gs, the coupling constant of the ‘t Hooft instanton induced interaction Gn and the covariant cutoff A. First, we take Go as a free parameter and study the q-meson properties as functions of Go. For the light quark masses, m, = md = 8.0 MeV is taken to reproduce M, = Md N 330 MeV which is the value usually used in the nonrelativistic quark model. Other parameters ms, Go and A are determined so as to reproduce the isospin averaged observed masses, m, = 138.04 MeV, mK = 495.7 MeV and the pion decay constant f,,. The calculated constituent U, d-quark mass is M u,d = 324.9 MeV which is independent of CD. On the other hand the calculated constituent s-quark mass weakly decreases from M, = 556.3 MeV to M, = 503.3 MeV when Gn is changed from Go = 0 to GD = Gz where the observed m,, is reproduced. The fitted result of the current s-quark mass m, is almost independent of Gn and m,T = 192.95 MeV at Go = GL. The ratio of the current s-quark mass to the current u,d-quark mass is m,/m, = 24.1, which agrees wellwithm,/rit=25f2.5 (?%= $(m,+md)) derived from ChBT [20]. The kaon decay constant
M. Tukizawa, M. Oka / Physics Letters B 359 (199s) 210-216
214
600
250
, 20
I
200-
1505 E.
loo-
50-
0
1 0.0
I
I
I
I
I
I
,
0.2
0.4
0.6
0.8
1.0
12
1.4
G
D
’
-60
0 0.0
, 0.2
fK is the prediction and is almost independent of GD. We have obtained f~ = 96.6 MeV at Go = Gz which is about 15% smaller than the observed value. We consider this is the typical predictive power of the N.JL model in the strangeness sector. We next discuss the rr” -+ 2y decay. The calculated result is 7$1_,2~ = 2.50 x 10-i’ [ l/eV] which agrees well with the observed value given in Eq. (27). The current algebra result is I$x._,~~ = a/(rf,,) = 2.514 x lo-” [ l/eV], so the soft pion limit is a good approximation for 7r-O-+ 2y decay. There are two effects of the symmetry breaking on ?$0_+2~. One is the deviation from the GT relation and another is the matrix element of the triangle diagram F(u, 7~~). Our numerical results are g, = 3.44, MU/f,, = 3.52 and F(U,?rO)Mu = 1.015, therefore the deviations from the soft pion limit are very small both in the GT relation and the matrix element of the triangle diagram. Let us now turn to the discussion of the v-meson properties, The calculated results of the r]-meson mass m7] and the mixing angle 0 are shown in Fig. 1, the 17 decay constant f,,is given in Fig. 2 and the 7 -+ 2y decay amplitude &lY is given in Fig. 3 as functions of the non-dimensionalized coupling constant of ‘t Hooft instanton induced interaction c”df = -G~(h/2?r)~f@. As can be seen from Fig. 1, in order to reproduce the observed r]-meson mass, a rather strong instanton
I 0.6
I 0.8 G
Sfl
Fig. 1. Dependence of the calculated q-meson mass tn,, (solid line) and the mixing angle 13(dashed line) on the non-dimensionalized coupling constant of the ‘t Hooft instanton induced interaction G”,“. The observed v-meson mass ~1~= 547.45 MeV is reproduced at G”,” = 1.41.
I 0.4
D
I 1.0
I 1.2
I 1.4
elf
Fig. 2. Dependence of the calculated 7 decay constant fq on the non-dimensionalized coupling constant of the ‘t Hooft instanton induced interaction G’$.
induced interaction is necessary. For example, at Go = Gz, GD(&)/G, = 1.58, it means that the contribution of .Ccgto the dynamical mass of the U, d-quarks is about 60% bigger than that of 1c4. In the previous study of the r] and 7’ mesons in the extended NJL model [ 12161, the strength of the instanton induced interaction has been determined so as to reproduce the observed 7’ mass though the 7’ state has the unphysical decay mode of the q’ + UU, &. The strength determined from m,!, G$ is much smaller than G;),, about l/10 to l/5 of G;),. One of the shortcomings of the NJL model is the lack of the confinement mechanism. It is expected that theconfinement gives rise to an attractive force between quark and antiquark in the 71’meson to prevent the 7’ meson from decaying to the quark and antiquark pair. One of the important results is that the calculated 77 decay constant is very different from the pion decay constant, fv = 206 MeV = 2.23f, at CD = Gz. This suggests that the 7 meson loses the Goldstone boson nature very much. To see whether the 77meson is the Goldstone boson, the GT relation is another important information. For the r] meson, the naive GT relation at the quark level is 2g, f,, = (-&cos 0 -
2 5 sin8]M,
i sinB]M, and at + 1 - ‘jj cost? J$ Go = CL, our numerical results are 2g, fv = 3.202
GeVand/&cosB-2
$sinBIM,+]--&cosB$ 0.892 GeV. Therefore the GT relation
M. Takizawa, M. Oka /Physics Letters B 359 (1995) 210-216
215
able from our analysis. Our calculated result of the mixing angle is 8 = 15.1” which should be compared with 8 N -20” [22]. There are two major differences between our model calculations and the usual analysis. One is that, in the usual analysis, the energy independent mixing is assumed. Another point is that the effects of the U, ( 1) anomaly on the v-meson properties are not taken into account in the usual analysis.
5. Concluding I
00
0.2
I
04
I
06
1
I
I
0.6
1.0
1.2
G De” Fig. 3. Dependence non-dimensionalized induced interaction
remarks
14
of the ~7 -+ 2y decay amplitude ‘&zr on the coupling constant of the ‘t Hooft instanton c”,“.
dose not hold at all. Since the NJL model is known as the model that describes the Goldstone boson properties reasonably well, it is natural to ask whether the present model is applicable to the q meson. For the v’ meson, we expect that the confinement plays an important role since the $-meson pole of the scattering amplitude Eq. (7) appears above the iiu and (ldthreshold. On the other hand, the 17 meson is a tight bound state, so we expect that the NJL model can describe the essential feature of the g meson. In order to confirm it, we have studied the constituent u,dquark mass dependence of the r]-meson properties. By changing m,,d from 7.5 MeV to 8.5 MeV, we have changed Mu,d from about 300 MeV to 360 MeV and other parameters of the model have been chosen so as to reproduce the experimental values of mB, rnK, rn7 and ,fr. The changes of the calculated q decay constant and the mixing angle have been within 2%. We are now in the position to discuss the 77 ---f 2y decay amplitude. Our result is ‘2&zY = 2.73 x lo-” [ 1/eV] at GO = Gz, which is about 10% larger than the experimental value. Therefore the present model reproduces the v-meson mass and the 7 ---f 2-y decay width simultaneously. As for the effects of the symmetry breaking on ?v_~r, our results are F( u, 7) M, = 1.41 andF(u,rl)/F(s,v) = 1.96.Bernardetal. [21] calculated the 71 --t 2y decay width using a similar model. They used a rather weak instanton induced interaction and their result of T(q -+ 27) is about 50% bigger than the experimental value. It is understand-
Using an extended three-flavor Nambu-JonaLasinio model that includes the ‘t Hooft instanton induced interaction, we have studied the Z-O,T,J-+ 2y decays as well as the properties of the pion, the kaon and the v meson. The q-meson mass and the 7 --t 2y decay width have been reproduced simultaneously with a rather strong instanton induced interaction. The calculated 77decay constant is about twice of the pion decay constant. So it is rather hard to consider the r] meson as the Goldstone boson. Because of the above novel picture of the 17 meson, the situation of the mixing angle is also different from the usual analysis. In order to confirm the novel picture of the v meson we have obtained here, certainly further studies of the q-meson properties are necessary. One thing is to study other v-meson decay processes such as 7 ---f 7~O2y, r~ + 3~ in the present framework and such calculations are now in progress. Since the properties of the ~7meson and those of the 7’ meson are closely related, one has to construct the low-energy effective model of QCD which can apply to the 7’ meson. Such an attempt is left to future study. References [ 1] G. ‘t Hooft, Nucl. Phys. B 72 (1974) 461; for a review, see G.A. Christos, Phys. Rep. 116 ( 1984) 251. (21 G. Veneziano, Nucl. Phys. B 159 (1979) 213. [ 3 ] S. Adler, Phys. Rev. 177 ( 1969) 2426; J. Bell and R. Jackiw, Nuovo Cimento A 60 ( 1969) 47. [ 4) J. Wess and B. Zumino, Phys. L&t. B 37 (1971) 95. [51 E. Witten, Nucl. Phys. B 223 (1983) 422. [6] J.E Donoghue, B.R. Holstein and Y.-C.R. Lin, Phys. Rev. Lett. 55 (1985) 2766; 61 (1988) 1527 (E); J. Bijnens, A. Bramon and F. Comet, Phys. Rev. Lett. 61 (1988) 1453; J.F. Donoghue and D. Wyler, Nucl. Phys. B 316 ( 1989) 289.
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[8] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [9] GM. Shore and G. Veneziano, Nucl. Phys. B 381 (1992) 3. [lo] Y. Nambu and G. Jona-I-asinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [ 111 M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. 45 (1971) 1955; G. ‘t Hooft, Phys. Rev. D 14 (1976) 3432. [ 121 T. Kunihiro and T. Hatsuda, Phys. Lett. B 206 (1988) 385; T. Hatsuda and T. Kunihiro, Z. Phys. C 51 ( 1991) 49; Phys. Rep. 247 (1994) 221. [ 131 V. Bernard, R.L. Jaffe and U.-G. Meissner, Nucl. Phys. B 308 (1988) 753. [ 14) Y. Kohyama, K. Kubodera and M. Takizawa, Phys. Lett. B 208 (1988) 165; M. Takizawa, K. Tsushima, Y. Kohyama and K. Kubodera,
Prog. Theor. Phys. 82 ( 1989) 48 1; Nucl. Phys. A 507 ( 1990) 611. [ 151 H. Reinhardt and R. Alkofer, Phys. Len. B 207 (1988) 482; R. Alkofer and H. Reinhardt, Z. Phys. C 45 (1989) 275. 1161 8.. Klimt, M. Lutz, U. Vogl and W. Weise, Nucl. Phys. A 516 (1990) 429; U. Vogl, M. Lutz, S. Klimt and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195. [ 171 D. Ebert and H. Reinhardt, Nucl. Phys. B 271 (1986) 188. ]18] M. Wakamatsu, Ann. of Phys. 193 (1989) 287. [ 191 Panicle Data Group, Phys. Rev. D 50 (1994) 1173. [20] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. [21] V. Bernard, A.H. Blin, B. Hiller, U.-G. Meissner and MC. Ruivo, Phys. Lett. B 305 ( 1993) 163. [22] EJ. Gilman and R. Kauffman, Phys. Rev. D 36 ( 1987) 2761.