mixing and decay constants of pseudoscalar mesons: octet-singlet vs. quark flavor basis

EL&ER Nuclear Physics B (Proc. Suppl.) 74 (1999) 15 l-154

PROCEEDINGS SUPPLEMENTS

Mixing and Decay Constants of Pseudoscalar Mesons: Octet-Singlet vs. Quark Flavor Basis Thorsten Feldmann*a

aDepartment of Theoretical Physics, University of Wuppertal, D-42097 Wuppertal, Germany

Although 9-v’ mixing is qualitatively well understood as a consequence of the IT( anomaly in QCD together with a broken SU(3)p flavor symmetry, until recently the values of decay and mixing parameters of the q and q’ were only approximately known, e.g. values for the octet-singlet mixing angle between -20” and -10“ could be found in the literature. New experimental data, especially for the reactions +yr* + 9,~’ and B + $K, together with new theoretical results from higher order corrections in chiial perturbation theory stimulated a phenomenological re-analysis of thii subject, which led to a coherent qualitative and quantitative picture of I] - q’ &b&g and even of q - vi - vC m%ng.

1. 77 - q’ Mixing Schemes

A crucial observation of our analysis (11 is the fact that for a proper treatment of the mixiq one clearly has to distinguish between matrix elements of 9, r,+ states with local currents (e.g. weak decay constants) and overall state mixing. While in the former the Sum symmetry breaking effects, (2m,/(m, + rrzd) II 26) turn out to be essential, in the latter the gluon anomaly plays the important role [2]. Correspondingly, one may think of two possible choices of appropriate basis states as a starting point for the description of 77 - q’ mixing, namely the quark flavor basis (which becomes exact in the limit 712, + 00) and the octet-singlet basis (which becomes exact for m” =md = m,), respectively.

In order to define these bases properly, it is useful to consider a Fock state decomposition of the mesonic states in the parton picture. One then defines the quark flavor basis through

E = cosdlrlp) - sindM = sin (b 1~~) + cos 4 1~~) (1)

with 4 being the mixing-angle and

I:; := ~,(ua+~)/~+~~lgg)+...

* := \k, 133) + !@ 199) + . . . (2)

Here 91 denote (light-cone) wave functions of the corresponding parton states. The effect of

??Supported by Deutsche Forschungsgemeinschaft.

higher Fock states’ (199) I- . . .) is twofold: First, they are necessary for the correct normalization, (%lVj) = &j. s econdly, they reflect the mixing (e.g. through the twist-4 199) component which is present due to the anomaly).

Analogously, in the octet-singlet basis, one obtains

1~ > = coso 1~8) - sin0 lql), 171’) = sin0 Iqs) + case [VI) (3)

with the usual pseudoscalar octet-singlet mixing angle 8 = 4 - arctan A. However, the flavor decomposition in the Fock state expansion looks now more complicated due to the broken sum symmetry

177s) := (e, I’L10 + dd) - 2Q8 Iss)) /&+

(St - fi*:) lgg)/& + . . . 1%) := (e,lua+dd) + q8 Is@) /h+

(fi*: + !q) lgg)/d3 + . . .

(4

Only in the flavor symmetry limit one would have trivial relations between the wave functions, is = Q, = 9, = @I, !!!; = J%!; = Ji-*f/&i,

s = 0, etc. Only in this case one would recover the usually anticipated form of octet and singlet

‘Of course, to construct the wave functions of all Fock states explicitly, one has to solve the QCD bound state problem.

0920-5632/99/S - see front matter Q 1999 Elsevier Science B.V All rights reserved. PI1 SO920-5632(99)00152-S

152 T Feldmann/Nuclear Physics B (PFw. Suppl.) 74 (1999) 151-154

states 17)s) + q8jl,luti + dd - 2s1)/@ + . . . and \q,1) + P&ii + dd+ sz)/& + *:lgg) + . . .

Note that in higher Fock states with increasing number of partons the effect of sum symmetry breaking is washed out (e.g. the ratio of con- stituent quark masses is only 2fi,/(+z, + fid) NN S/3), and thus the octet-singlet basis is still useful for low-energy expansions of QCD like e.g. chiral perturbation theory (ChPT). However, weak decay constants only probe the short-distance properties of the valence Fock states and are thus rather sensitive to sum breaking effects. To see this in more detail, let us define the decay constants2 as (fi7 = 131 MeV)

ow;,If7 = &+d (5) with P = q, 7’; i = q, s (i = 8, l), and the rele- vant flavor combinations of axial-vector currents denoted as J$. Using Eqs. (l-5) one obtains

(;j $) = (k:;;,” -;$

= U(4 diag[f,, Al (6)

with f9(fd) related to the wave function \E,($,) at the origin3, and with U being a usual rotation matrix identical to the one of the state mixing (I).

In the octet-singlet basis one obtains on the

2We stress that occasionally used decay constants “A, A/’ are ill-defined quantities. 3 The decay constants are calculated from the Fock state decomposition as follows (for concreteness we chose the (Q) state as an example)

Here x denotes the usual (light-cone +) momentum frac- tion of the quark and kl its transverse momentum. Note that only the leading quark-antiquark Fock state con- tributes to the decay constant, i.e. Eq. (7) is exact.

other hand

($ ;i) = (;::t -;zss:)

# U(e) diag[fs, Al (8) where we introduced the parametrization of [3]

Note that the decay constants do not simply fol- low the state mixing in the octet-singlet basis; - only in the SU(3)p symmetry limit one has 6s + 6 t 81. Especially the matrix elements of octet/singlet currents with the opposite states do not vanish, (01 J$l~g) = zp, sin(8 - &) fi and (OIJ;,ld = 2pP sin(& - 0) fs. The difference between 6s and 01 following from Eq. (9) is anal- ogous to the one derived within ChPT4 [3].

2. Masses and Decay Constants

The important relation that connects short- distance properties, i.e. decay constants, with long-distance phenomena, i.e. mass-mixing, is provided by the divergences of axial-vector currents including the anomaly (i = u, d, 3, c,. . .)

dp cii ~~75 qi = 2mi ji + z G&‘, (10)

with ji = gi 275 qi. Taking matrix elements (01. . . IP) (for instance (Ol#‘J~,l~) = M: f$ and using the definition of the decay constants (5), the mass matrix in the quark flavor basis is fixed to have the following structure [l]

U(4) diag[Mz, Mifl Ut (4)

(

miq + 2a2 Aa2 = 4ya2 rni, + y2a2 >

(11)

with

2 %9 = 2mq (Wl~q)lfq = MS ds = 2m, (Oljilrls)/fe = 2M$ - Mz (12) 4We like to emphasize that Eq. (8) is not to be read as 1~) = cos~9sl~s) - sin811~~1) etc., i.e. Eq. (3) still holds.

T. Feldmann/Nuclear Physics B (Ptvc. Suppl.) 74 (1999) 151-154 153

and

a2 = & (012 G&J, 9

y = $ (13) s

The mass matrix in the octet-singlet basis can simply be obtained from (11) by a rotation about the ideal mixing angle. Solving for 4, y, a2 and using fq 21 f=, f8 N dm, one obtains the “theoretical” values quoted in Table 1.

Alternatively, the mixing parameters can be determined from phenomenology without using the Sum relations for rnTj and f,?. The mixing angle 4 can be determined by considering appropriate ratios of decay widths/cross sections, in which only the ‘79 or Q component is probed, respectively. The analysis of several independent decay and scattering processes performed in [l] leads to 4 = 39.3” f 1.0’. It is to be stressed that the so-obtained values for the mixing angle 4 (or equivalently for 0 = f$ - arctan Jz) are all con- sistent with each other with a small experimental uncertainty and agree with the “theoretical” ones within 10%.

With this value of the mixing angle the decay constants f, and fs can be estimated from the q, q’ + 77 decay widths5

902M: C, cos+ r(rl +rr] = -

C,sin$ 2 - - -

167r3 [ f9 fs 1 9a2M3 t

r[$ + 771 = - - 16x3’ fq [

C,sin# + C,cos4 2 -

fs 3 04

where C, = 5/9fi and C, = l/9 are the proper charge factors. Combined with the additional information from the structure of the mass matrix, one obtains fq = (1.07 f 0.02) fir and fa = (1.34 f 0.06) fx (see also Table 1). Note that the corresponding difference between Bs, 0, B1 (although formally a higher order sum breaking effect) is enormous!

A prominent example which illustrates the difference between the conventional approach with

5Note that again, the expressions for the two-photon decay widtha take the simple form only in the quark-flavor ha&, in which the decay constant matrix, appearing in the derivation of the anomalous decay, can be inverted in a trivial way.

es = 8 = 81 and the present one is given by the J/lc, + P7 decays. Following [4,5] the decay rates are proportional to the matrix elements I(01 2 GI$ P) I2 which can be calculated using Eqs. (10,ll) and m, 21 rnd ~0, leading to

(15)

from which one obtains by comparison with the experimental value [6] 4 = 39.0” f 1.6” (or 0 = -15.7’ f 1.6’) and 0s = -22.0” f 1.2’.

Direct information on the decay constants fj., can also be obtained from the analysis of the form factors for 7*7 -+ P at large photon virtualities, which are dominated by the valence Fock states in (2,4). Using the modified hard-scattering approach (see [7,8] and references therein), again, the phenomenological parameter set in Table 1 leads to a perfect description of the experimental data [9,10].

3. 77 - r]’ - rlc Mixing

Since the derivation of the pseudoscalar mass matrix via Eq. (10) does not have to make use of flavor symmetry, it can be generalized to r7-$-qc mixing in a straight forward manner [l], leading to a similar msss matrix as in Eq. (11)

(

rni, + 2a2 Aa2 &a2 &a2 rni, + y2a2 yza2 fi.ra2 yza2 m2 + z2a2 cc )

(16)

Of course the mixing between light and heavy pseudoscalars is suppressed by the heavy masses, i.e. a”/mh may be treated as a small parameter, leading to rn& = Mic. The second new parameter is also unambiguously fixed B = fq/fc N fq/fJ/J, = 0.35.

from the phenomenological point of view, namely from the rather large branching ratio for B 3 Kr]’ reported by CLEO [ll], one is mostly interested in the matrix elements of q, 77’ with the charm axial-vector current (0]~7~7sc]P) =

154 T. Feldmann/Nuclear Physics B (Proc. Suppl.) 74 (1999) 151-154

Table 1 Theoretical and phenomenological values of mixing parameters (for details, see [l]).

f&r fslfi? d y a2 WV21 fdfir fl/fr e e8 e1 theory 1.00 1.41 42.4” 0.78 0.281 1.28 1.15 -12.3” -21.0~ -2.70

phenom. 1.07 1.34 39.3” 0.81 0.265 1.26 1.17 -15.40 -21.2” -9.20

2 f; p,. From the diagonahzation of the mass matrix one obtains the following values

f; = - fc ec sines = (-2.4 f 0.2) MeV, f;, = fc ec c0s& = (-6.3 f 0.6) MeV

(17)

where we have defined the mixing angle 0, = --z dwa2/A!ic N -l.O”, which is reasonably small and in accord with Refs. [12-141 and, in particular, with the independent bounds found from the analysis of the rly and $7 transition form factors 171. Obviously, the intrinsic charm in q’ cannot induce a dominant contribution to the B + Kq’ decays (via b + SCE), contrary to what is assumed occasionally [15,16].

An immediate test of the parameter values is provided by a similar ratio of J/Q decay widths as in Eq. (15). Most interestingly, via Eq. (lo), the intrinsic charm picture (i.e. J/T/J -+ c?y, CE + 77’) and the gluon picture of ref. [4] turn out to be equivalent with the result [l]

r[J14 + 77’71 = 02 cose2 k,l 3

r[J/+ + WI ( ) ’ 8 k,c

(18) The values of 8, and 0s found in our approach per- fectly reproduce the experimental value for this ratio [6].

Acknowledgements

I like to express my gratitude to Peter Kroll and Berthold Stech for a fruitful collaboration. I fur- ther enjoyed valuable discussions with Hai-Yang Cheng, Alex Kagan, Alexey Petrov and Vladimir Savinov.

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mixing and decay constants of pseudoscalar mesons: octet-singlet vs. quark flavor basis

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