Pis'ma v ZhETF, vol.91, iss. 11, pp.611-614
© 2010 June 10
Generalized Hidden Local Symmetry Model confronts the decay
—> 7r+7r_7r_I/T
N. N. Achasov1}, A. A. Kozhevnikov1^+ Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics, 630090 Novosibirsk, Russia
+ Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics, and Novosibirsk State University, 630090 Novosibirsk, Russia
Submitted 15 April 2010 Resubmitted 20 April 2010
Generalized Hidden Local Symmetry (GHLS) model as the chiral model of pseudoscalar, vector, and axial vector mesons, is confronted the ALEPH data on the decay r- —► tt+vt. It is shown that the spectrum of this decay in GHLS falls short of the experimental data. The modifications of GHLS based on inclusion of heavier axial vector mesons are studied. It is shown that the scheme with two additional axial vector isovector mesons with masses m„/ = 1.59 GeV and ma» = 1.88 GeV gives a good description of the ALEPH data.
There is popular chiral model of pseudoscalar, vector, and axial vector mesons and their interactions based on nonlinear realization of chiral symmetry, the so called Generalized Hidden Local Symmetry (GHLS) model [1 -3]. One of its virtue is that the sector of electroweak interactions is introduced in such a way that the low energy relations in the sector of strong interactions are not violated upon inclusion of photons and electroweak gauge bosons. Some interesting two- and three-particle decays as, for example, p° and w 7r+7r_7r°,
were analyzed in the framework of GHLS [1].
Some time ago GHLS with particular choice of free parameters
(a,b,c,d,a5) = (2,2,2,0,1)
(1)
(see Refs. [3-5,7] and (2) for more detail) was applied to the evaluation of the four-pion process p —> 4-k [4-7] and to the comparison with existing data on the reaction e+e- 7r+7r-7r+7r- [8, 9]. It was shown that while the results of calculations do not contradict the data [8] at energies near mp, at higher energies near 1 GeV the cross section of above reaction measured in independent experiments [8, 9], by the factor of about 30 exceeds the values evaluated in GHLS [6, 7]. The contributions of higher resonances p', p" were included to reconcile the data with calculations [6, 7].
Since axial vector meson ai(1260) appears only in the intermediate states of the reaction e+e-7r+7r-7r+7r-, it would be desirable to study the processes where it manifests directly as in the decay r-
which was studied by ALEPH Collaboration [10]. The aim of the present paper is to evaluate
the 7r+7r-7r- spectrum in the decay of r- lepton in the framework of GHLS and compare the results with the ALEPH data. Notice that analogous work in the framework of different chiral model was undertaken, in particular, in Refs. [11, 12].
The basis of the derivation is the lagrangian of the generalized hidden local symmetry model [1, 3] (GHLS). In the sector of strong interactions and in the gauge Cm = 1, Cl = Cfl = C, it looks as
£(GHls) = ^/(0)2^ (d^t + dptf _ ^
\ ¿1
^Tr (Fjy^+F^)--
9 A,
—2m.-,//Tr
2 ig
i Av
(2)
g is the coupling constant to be related to gp7r7r. See (10) below. The notations, assuming the restriction to the sector of the non-strange mesons, are
4V = - duVp - igiVM - igiA^Av],
= d^Av - dvA^ - ig[Vtu, Av] - ig[A,u, Vv], (3)
e-mail: achasov0math.nsc.ru, kozhev0math.nsc.ru
IlHCbMa b ?K3T<J> tom 91 Bbin.11-12 2010
611
Ap = (j ■ A^), £ = expz^jf, where p7r are the vector meson p and pseudoscalar pion fields, respectively, r is the isospin Pauli matrices. The axial vector field AM is not literally the field corresponding to the physical on (1260) meson. The fact is that the lagrangian (2) contains the A — ir mixing term. The latter can be removed upon choosing
b0c.0
g(b7+c0)A(^
where aM is the ai(1260) meson field, and
dp&t - dptf 2 i
(4)
(5)
Free parameters (ao, bo, Co, do), and fl0) oftheGHLS lagrangian with index 0 are bare parameters before renor-malization (see below); [,] stands for commutator. Hereafter the boldface characters, cross (x), and dot (•) stand for vectors, vector product, and scalar product, respectively, in the isotopic space. The terms with free parameters 0:4,5 are necessary for cancelation of momentum dependence in pirir vertex. To provide such cancelation, one should set 04 = 1 + 2q5co/6o [lt 3]. Removing the A — 7T mixing (4) results in non-canonical coefficient in the kinetic term of the pion field. To make it canonical, one should fulfil the following renormalization:
= Z-1'2/*, 7r Z-^n, (00,^,00, do) =
= Z(o,b,c,d), (6)
where
do
b0c0
h
'0 + Co
Z"1 = 1.
See [1,3-5] for more detail.
The amplitude of the decay r- incor-
porates the transition W~ —¥ 7r_7r_7r+. In GHLS, the latter is given by the diagram shown in Fig.l. Necessary terms are obtained from the total GHLS lagrangian which includes electroweak sector [1], and look like
A£Ew = ^zVudW^j. (-fndtlTT±-
+ tH* X t71" X dn*]]± +
•JJtt
bgfla„j_ + agfn[tt x p^ J)
(7)
where stands for the field of a\ meson, Vj_ = (Vi,V2) denotes transverse component of the isotopic vector, W,,i is the W-boson field while g2 and VU!j are the SU(2) electroweak coupling constant and corresponding element of the Kobayashi-Maskawa matrix, respectively. The amplitudes of transitions a\ Sir and ir Sir were
(a)
(c)
. n
(b)
n
■n + W~
-n +
n
n
+W
n
Fig.l. Diagrams describing the transition W~ — 7r_7r_7r+. Shaded circles depict the transition including both the point-like and p-exchange contributions. Permutations of pion momenta are understood
given in Ref. [4, 5] and Ref. [13], respectively. Here we do not fix GHLS parameters to their "canonical" values (1) [1] and allow them to be free. Then the amplitude of the decay a^(q) ir+(qi )ir~ (q2)n~ (<73) should be rewritten as follows: Mai3jr = M[a^(q) 7r+ (qi)n^ (q2)n~ (g3)],
&QV
iMai37r = — [Axq-in + A2q2lx + A3q3lx), (8)
¿J IX
where e^ is the polarization four-vector of a\ meson, and
A n,f, x f P[(l3,qi - ga) - gg3 + m%] - qq3 , -Al —(I + r23) < -—------h
I DP(qi + «2)
4r2(/3 - l)q2q3 +q2 - qqi \
2 m2
A2 =
(9)
0[{q3, gi - Q2) + qq3 - ml] + qq3 Dp{qi + q2) (g2,gi - q3) _ 2r2(0 - l)qiq3 + qqi Dp{qi + q3) m2
Hereafter Pjj interchanges pion momenta g,- and qj, (Qi,Qj) stands for the scalar product, and A3 = P23A2. Note that
9p-k-k = y, m2 = ag2f2, m2ai =(b + c)g2f2, (10)
f„ = 92.4 MeV is the pion decay constant. Parameters r and 0 are the combinations of the GHLS parameters:
r =
b + c r
(11)
The amplitude of the decay W_(g) 7r+ (<7i)7r~ (^2)71"™(93) corresponding to the diagrams Fig.l is
¿Jir
(12)
IIncbMa b JK3T<t Tom 91 Btra. 11-12 2010
a
Generalized Hidden Local Symmetry Model confronts
613
where ^ is the polarization four-vector of W boson and the axial decay current looks like
Jn = ^Qin + X (1 + P23)
%
am„
DAl)
(g2,gi - q3) Dp{qi + g3)
qqi
ar2m2ai
2 Dai(q) 2q»
X \ Mqiv + A2q2tl + A3q3„--fx
x (1-
-P23) [(ml + gig2)(g3,gi - q2)x
0 r2(0^iy
DP(qi am2
+ q2) (1 + p;
mi
0]}
23 y
(qi - q3)n Dp(qi + q3y
(13)
In the above expressions, Dp, D„, and Dai are the inverse propagators of 7r, p, and a\ mesons, respectively. Their expressions are given in Ref. [4]. The terms corresponding to the diagrams (a), (b), (c), and (d) in Fig.l are easily identified by these propagators. Note that the divergence of the axial decay current is
%Jn =
mz
DAq) (g2,gi - q3) DP(qi
q - qqi
q3)
ar
a 2/, " 2777 ,5
2<
23;
■<f
Dai(q)
x (1 + P23)(ml + qiq2)(q3,qi - q2) x 0 r2(0^1)'
Dp(qi + q2)
mi
(14)
One can see that it vanishes at the a\ mass shell in the limit of vanishing pion mass.
The spectrum of the three pion state in the decay t- -k+vt normalized to its branching fraction
is [14]
dB ds
(GFVnd)2(m2^s)
27r(2mT)3rT x [(m2 + 2s)pt(s) + m2pi(s)]
(15)
8 = q2. The transverse and longitudinal spectral functions are, respectively,
Pt(s) = Pi(s) =
2,-Ksfl
J dt>3lr\qj\2
\qJI2
(16)
where d$37r is the element of Lorentz-invariant phase volume of the system 7r-7r-7r+. The numerical integration shows that due to Eq. (14) pi is by at least three
orders of magnitude smaller than pt in all allowed kine-matical range 9m2 < s < m2 and by this reason it is neglected in what follows.
The "canonical" choice (1) of free GHLS parameters [1] with mai = 1.23 GeV results in the spectrum shown with the dot-dashed line in Fig.2. Upon vari-
0.01 r
0.00001
0.001 r
0.0001 .
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
2
s (GeV2)
Fig.2. Spectrum of 7r-7r-7r+ in r decay normalized to the branching fraction BT-_>7r-7r-7r+t,T. The ALEPH data are from Ref. [10]. See the text for more detail
ation of free parameters with the single a\ resonance contribution results in the curve drawn with the dashed line. It corresponds to mai « 1.54 GeV, a « 1.75, r « 1.05, 0 « 0.84 with x2 = 690/112d.o.f. To improve the fit heavier resonances a\, a" were included in a way analogous to ai(1260). Note that there are indications on such resonances both theoretical [15, 16] and experimental [17-19]. The total set of the fitted parameters is first taken to be
(mai, a, r, 0, m.a,, a',r',0', w', ma<<, a", r",0", w"),
where w' parameterizes the gp7r7rw'r'/f„. Compare with Eq.
coupling a^p-K
(8). Analogously for a". The fit chooses w' = 1 and turns out to be insensitive to this parameter leaving \2 = 122/102d.o./. The quality of the fit is considerably improved upon fixing w' = 1 but adding new parameter \j)'-the phase of the a[ contribution. Such phase imitates possible mixing among ai, ai, a" resonances. The results of such type of the fit are
IlHCbMa b ?K3T<1> tom 91 bmu. 11-12 2010
mai = 1.332 ± 0.015 GeV, a = 1.665 ± 0.011, r = 0.332 ± 0.007, /3 = 8.5 ± 0.3, rraai = 1.59 ± 0.01 GeV, a' = 0.99 ± 0.01,
r' = 0.96 ±0.01,/3' = 0.07 ±0.02, (17)
= 28° ±1°, rraa» = 1.88 ± 0.02 GeV,a" = 0.46 ± 0.01, r" = 1.45 ± 0.02,/3" = 0.91 ± 0.05, w" = 1.14 ±0.01,
with x2 = 79/102d.o.f. Corresponding curve is shown in Fig.2 with the solid line. Using (10), (11
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