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Mixing and decay constants of pseudoscalar mesons: the sequel
11 March 1999
Physics Letters B 449 Ž1999. 339–346
Mixing and decay constants of pseudoscalar mesons: the sequel Th. Feldmann a , P. Kroll
a,b
, B. Stech
c
a
Fachbereich Physik, UniÕersitat ¨ Wuppertal, D-42097 Wuppertal, Germany Centre for the Subatomic Structure of Matter, UniÕersity of Adelaide, SA 5005, Australia c Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Heidelberg, D-69120 Heidelberg, Germany
b
Received 21 December 1998; revised 14 January 1999 Editor: P.V. Landshoff
Abstract We present further tests and applications of the new h – hX mixing scheme recently proposed by us. The particle states are decomposed into orthonormal basis vectors in a light-cone Fock representation. Because of flavor symmetry breaking the mixing of the decay constants can be identical to the mixing of particle states at most for a specific choice of this basis. Theoretical and phenomenological considerations show that the quark flavor basis has this property and allows, therefore, for a reduction of the number of mixing parameters. A detailed comparison with other mixing schemes is also presented. q 1999 Published by Elsevier Science B.V. All rights reserved.
In a recent reinvestigation w1x of processes involving h and hX mesons we pointed out that a proper treatment of the h – hX system requires a sharp distinction between the mixing properties of the meson states from the mixing properties of the decay constants. While the particle state mixing involves the global wave functions, the decay constants probe the quark distributions at zero spatial separation. Conventionally, h and hX are expressed as superpositions of an SUŽ3. flavor octet and a flavor singlet state corresponding to an orthogonal transformation with mixing angle u . The decay constants of h and hX defined by their matrix elements with singlet and octet axial vector currents will in general not show the same mixing since flavor symmetry breaking manifests itself differently at small and large distances. Because SUŽ3. breaking is solely caused by the current quark masses a simpler picture can be
expected for properly defined decay constants in the quark-flavor basis. Indeed, a dramatic simplification is achieved by taking two orthogonal basis states 1 which are assumed to have in a Fock state description the parton composition
Ž 1.
Here Ci denote Žlight-cone. wave functions of the corresponding parton states, and the dots stand for
1
In principle, the two-state basis should be extended by states of higher energy, for instance by adding a cc state. Energy considerations indicate, however, that the mixing with these states X is small. The small charm components in the h and h is discussed in Ref. w1x.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 8 5 - 4
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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higher Fock states which also include < gg : components. These higher Fock states play no explicit role in the following discussions where we are mainly interested in mesonic states and decay constants. The physical meson states are related to the basis Ž1. by an orthogonal transformation
ž /
ž /
UŽ f . s
ž
cos f sin f
ysin f , cos f
/
Ž 2.
where f is the mixing angle. Ideal mixing corresponds to the case f s 0. It is to be emphasized that our definition of meson states is given in terms of parton degrees of freedom without introducing model-dependent concepts like constituent quarks. Our central ansatz Ž1,2. has important consequences for the weak decay constants which probe the short-distance properties of the quark-antiquark Fock states. To see this in detail, let us define the decay constants 2 by Ž fp s 131 MeV. ²0 < Jmi 5 < P : ' i f Pi pm , X
H
dx d 2 k H 16p 3
Jmi 5
Ci Ž x ,k H . .
Ž 4.
Here x denotes the usual Žlight-cone q. momentum fraction of the quark and k H its transverse momentum with respect to its parent meson’s momentum. Eq. Ž4. is exact, only the quark-antiquark Fock state contributes to the decay constant, higher Fock states do not contribute. Using Eqs. Ž1. – Ž3., one immediately observes that our ansatz for the Fock decomposition naturally leads to decay constants in the
2
ž
fhq
fhs
fhqX
fhsX
We stress that occasionally used decay constants ‘‘ fh , fh X ’’ are ill-defined quantities.
/
s U Ž f . diag f q , f s .
Ž 5.
The conventional octet-singlet basis states are obtained from the quark-flavor basis states by performing a rotation with the ideal mixing angle. The physical states are then related to the octet-singlet basis states by
ž /
ž /
Ž 6.
with u s f y arctan'2 . The corresponding Fock decompositions of these octet-singlet basis states, following from Eqs. Ž1,2,6., read
Cq q 2Cs < uu q dd y 2 ss :
'6
3 q
Ž 3.
where P s h ,h . Here denotes the axial-vector currents with quark content i s q, s. The decay constants are related to the quark-antiquark wave functions at the origin of configuration space. Because of the fact that light-cone wave functions do not depend on the hadron momentum we can define two basic decay constants f q and f s arising from hq and hs , respectively, f i s 2'6
quark-flavor basis which simply follow the pattern of state mixing:
'2 Ž Cq y Cs .
< uu q dd q ss :
3
'3
'2 Ž Cq y Cs .
< uu q dd y 2 ss :
3
'6
q
2Cq q Cs < uu q dd q ss : 3
'3
q...
q...
Ž 7.
Obviously, it is unavoidable that the so-defined octet Žsinglet. meson state contains an SUŽ3. singlet Žoctet. admixture, except for identical wave functions Cq s Cs , an equality which holds in the flavor symmetry limit only. Only then one would find pure octet and singlet states in Eq. Ž7.. Certainly, these results are based on our central ansatz, namely that h and hX can be decomposed into two orthogonal states where one state has no ss and the other no qq component. One may alternatively start from the assumption that h 8 and h1 defined in Eq. Ž6. have Fock decompositions with only parton octet or singlet combinations in the quark-antiquark sector, respectively. Rotating these states back by the ideal mixing angle, the resulting hq Žhs . state has an ssŽ qq . component, unless the octet and singlet wave functions are equal. However, from both, theoretical and phenomenological considerations performed in Refs. w1–3x, the quark-flavor basis is to be favored.
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
One may also define decay constants through matrix elements of octet and singlet axial-vector currents, analogously to Eq. Ž3.. Using Eqs. Ž1,2., one easily sees that these decay constants cannot be expressed as UŽ u . diagw f 8 , f 1 x. Rather one has fh8
fh1
fh8X
fh1X
f 8 cos u 8 f 8 sin u 8
ž /ž s
yf 1 sin u 1 , f 1 cos u 1
/
Ž 8.
where we use the new and general parametrization introduced in Ref. w2x. The parameters appearing in Eq. Ž8. are related to the basic parameters f , f q and f s , characterizing the quark flavor mixing scheme as follows w1x,
u 8 s f y arctan f 82 s
f q2 q 2 f s2
,
'2 f s fq
,
f 12 s
u 1 s f y arctan
'2 f q fs
,
2 f q2 q f s2
. Ž 9. 3 3 The decay constants f Pi do not follow the pattern of state mixing in the octet-singlet basis; only in the SUŽ3.F symmetry limit one would have u 8 s u s u 1. This is a consequence of the non-trivial Fock decomposition in Eq. Ž7.. The difference between u 8 and u 1 following from Eq. Ž9. leads to the same formula as derived within chiral perturbation theory w2x. In our approach the quantities u 8 and u 1 are parameters determined by the fundamental quantities 3 f , f q and f s . They are not to be used as
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constants Ž3. together with Ž5. one obtains the h – hX mass matrix in the quark flavor basis. Its elements are composed of the gluonic matrix elements ²0 < a s GG˜
fh1
fh8X
fh1X
ž /
s UŽ u8 .
ž
f8
f 1 sin Ž u 8 y u 1 .
0
f 1 cos Ž u 8 y u 1 .
/
.
Ž 11 .
3
We remark at this point, that the flavor singlet axial-vector current is not conserved in QCD. Consequently, the singlet decay constant f 1 , and, hence, f q and f s too, are renormalization scale dependent, although only mildly since the corresponding anomalous dimension is of order a s2 w2x. Varying the scale m between Mh and Mh c, the value of f 1Ž m . changes by 3% only, an effect which we discard.
Then it is tempting to introduce a new basis by
ž /
ž /
.
Ž 12 .
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
342
Table 1 Comparison of different determinations of mixing parameters. The values given in parentheses are not quoted in the original literature but have been evaluated by us from information given therein. Crosses indicate approaches where the difference between u , u 1 and u 8 has been ignored
u
u8
u1
f 8rfp
f 1rfp
method
y 12.38 y15.48
y21.08 y21.28
y2.78 y9.28
1.28 1.26
1.15 1.17
qs–scheme Žtheo.. w1x qs–scheme Žphen.. w1x
– y21.48 y15.58 y19.78 y12.68 yŽ238 y 178. y98
y20.58 x – wy12.28x wy19.58x x wy208x
y48 x – wy30.78x wy5.58x x wy58x
1.28 1.19 – w0.71x w1.27x 1.2 y 1.3 w1.2x
1.25 1.10 – w0.94x w1.17x 1.0 y 1.2 w1.1x
ChPT w2x GMO formula w4x phenomenology w5x model w6x model w7x phenomenology w8–11x UŽ1. A anomaly w12x
The elements of the second matrix on the r.h.s. of Eq. Ž11. can now be viewed as the decay constants of h˜ 8 , h˜ 1 through octet or singlet axial vector currents, respectively. This matrix is still non-diagonal but triangular. The new basis has the special feature that the anomaly contributes to the singlet Žh˜ 1 . mass alone, i.e. one has ²0 <
as 4p
GG˜
Ž 13 .
This property is related to the fact that the ratio ²0 < a s GG˜
4p
It allows to determine a Gell-Mann–Okubo formula for the mass of the h˜ 8 basis state. Transforming the mass matrix found in the quark flavor basis Žsee above. to the new basis Ž12. one finds f 82 m288 ˜˜ s
f q2 Mp2 q 2 f s2 Ž 2 MK2 y Mp2 . 3
.
Ž 14 .
This formula reminds of the suggestion put forward in Ref. w4x Žsee also w13x., namely to use the product f 2 M 2 rather than M 2 to determine the SUŽ3.F breaking effects in the Gell-Mann–Okubo formula 4 .
4 Note however that the decay constants used in the analysis of w4x are not defined as proper matrix elements of weak currents.
Insertion of the relations in Eq. Ž9. into Eq. Ž14. yields
m 288 ˜˜ ,
4 MK2 y Mp2 3
y DGMO
MK2 y Mp2 3
Ž 15 .
with DGM O s 4 Ž f q2 y f s2 .rŽ3 f 82 .. The deviation from the standard Gell-Mann–Okubo relation DGM O can also be derived in chiral perturbation theory w14,8x. The above discussion clearly shows: An analysis which implicitly uses Eq. Ž13. provides for an estimate of the parameter u 8 rather than the angle u . Indeed, previous treatments along these lines obtained mixing angles close to y208, which is consistent with our value of u 8 Žsee Table 1.. As a first test of our mixing approach we analyzed the hg and hXg transition form factors in Ref. w1x. A good description of the experimental data has been found from the phenomenological set of parameters. For details we refer to w1,3x. Let us now turn to further tests and applications of our results which have not been discussed in Ref. w1x. RadiatiÕe decays of S-waÕe quarkonia We define the ratio of decay widths RŽ 3 Sn . s G w 3Sn ™ hXg xrG w 3Sn ™ hg x where 3Sn represents one of the quarkonia Jrc , c X ,F , . . . According to w15x the photon is emitted by the c quarks which then annihilate into lighter quark pairs through the effect of the anomaly. Thus, the creation of the
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
corresponding light mesons is controlled by the matrix element ²0 < a s GG˜ < P :, leading to w1x 4p 3
R Ž Sn . s cot 2u 8
k hX g
3
ž /
Ž 16 .
khg
(
2
where k 12 s Ž M 2 y m12 y m 22 . y 4 m12 m 22 rŽ2 M . denotes the final state’s three-momentum in the rest frame of the decaying particle. The experimental value of R in the Jrc case w16x has already been used in the phenomenological determination of the basic mixing parameters in Ref. w1x. Using the phenomenological value of u 8 quoted in Table 1, we predict RŽ c X . s 5.8 and RŽF . s 6.5. The prediction .4 for RŽ c X . agrees with the experimental value 2.9q5 y1 .8 , which still has uncomfortably large errors however w17x. For the radiative F decays only upper bounds exist at present.
xc J decays into two pseudoscalars Because these are energetic decays the current quarks produced will be in an almost pure SUŽ3. singlet state. However, flavor symmetry violation can occur in the hadronization process. The ratio of xc J decay widths into different pairs of pseudoscalar mesons can be written Ž J s 0,2. G w xc J ™ P1 P2 x G w xc J ™ P3 P4 x
s
C12
2
k 12
ž /ž / C34
k 34
2 Jq1
.
Ž 17 .
For the coefficients Ci j two limiting cases can be considered. If the mesons are formed at hadronic distances the influence of the different decay constants will be a minor one and one expects to a good accuracy Chh s Ch X h X s Cp 0p 0 , Chh X s 0 and, from isospin symmetry, Cpq p ys '2 Cp 0 p 0 Žwhere we included the statistical factor '2 .. If, however, the meson generation starts already at a time at which the inter-quark distances are very small, the decay amplitudes are obtained from the convolution of a hard scattering process with the corresponding wave functions w18x. Assuming equal shapes of the wave functions, an assumption which is not in conflict with present experimental information, differences in the decay amplitudes are then solely due to the different decay constants and the mixing angle. One finds: Chh s f q2 cos 2f q f s2 sin2f s 1.41 Cp 0 p 0 , ChX h X s f q2 sin2f q f s2 cos 2f s 1.53 Cp 0 p 0 and Žwith
343
the statistical factor '2 included. Chh X s '2 Ž f q2 y f s2 . sin f cos f s y0.45 Cp 0 p 0 and Cp q p y s '2 Cp 0p 0 . Numerically we obtain G w xc 0Ž2. ™ hh xrG w xc 0Ž2. ™ p 0p 0 x s 1.9 Ž1.7., G w xc 0Ž2. ™ hXhX xrG w xc0Ž2. ™ p 0p 0 x s 1.9 Ž1.3., G w xc0Ž2. ™ hhX xrG w xc0Ž2. ™ p 0p 0 x s 0.2 Ž0.1. and G w xc0Ž2. ™ pq py xrG w xc0Ž2. ™ p 0p 0 x s 2. From the differences between the two limiting cases it appears that xc J decays are less suited for testing h – hX mixing parameters, but, taking the mixing parameters from other processes, they will provide interesting information on meson formation in these reactions. Experimentally, only the ratio G w x c 0Ž2. ™ hh xr G w x c 0Ž2. ™ p 0p 0 x is known .1 Ž q0 .9 . Ž w16,19x: 0.76q1 for the p 0p 0 branching y0 .5 0.76y0.6 ratio we combined the data with the one for the pq py channel.. At present, the large experimental errors prevent any definite conclusion. Similar relations as given here Žmodified according to the correct charge factors. should hold for two-photon annihilations into pairs of pseudoscalar mesons. g ) g ) ™ h ,h X transition form factors These form factors offer, in principle, a way to measure the angle u 1. Allowing both the gluons to be virtual where at least one of the virtualities q12 and q22 is supposed to be very large, one may easily work out the leading-twist result for these form factors w20x. In this approximation one has Ž i s q, s . Fh i g ) Ž q12 ,q22 . s yCi a s f i dx
H
fi Ž x . x q12 q
Ž 1 y x . q22 Ž 18 .
where f i is the hi distribution amplitude, and Ci a numerical factor Ž C q s '2 , C s s 1.. Combining both the form factors into those for the physical mesons and assuming the equality of the two distribution amplitudes Žwhich at least holds in the formal limit qi2 ™ ` since both the distribution amplitudes evolve into the asymptotic one., one arrives at Fh g ) Ž q12 ,q22 . FhX g ) Ž
q12 ,q22
.
s
'2 f q cos f y f s sin f '2 f q sin f q f s cos f s ytan u 1 . Ž 19 .
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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Of course, at finite values of momentum transfer one may expect corrections from differences between the two distribution amplitudes and from transverse momentum. Such corrections can be worked out following, for instance, Ref. w3x. For a measurement of these form factors one may consider the process pp ™ jet q jet q h ŽhX . where the mesons are supposed to be produced in the central rapidity region w21x. According to Close w22x these form factors may also be of relevance in pp ™ pph ŽhX ., provided the Pomeron couples to quarks A g m andror gg ™ h ŽhX . is the elementary process in the Pomeron-Pomeron interaction. The explicit extraction of the form factors from such measurements may, however, be very difficult. Kilian and Nachtmann w23x discuss g-Odderon-h ŽhX . form factors which appear in diffractive e-p scattering processes. We find that the ratio of these form factors is given by cot u 8 at large momentum transfer. Z ™ h(hX )g decay The treatment of these processes is rather academic since the expected branching ratios are far below the present experimental bounds ŽThe Z ™ pg decay has e.g. been discussed in Ref. w24x.. Nevertheless, they provide an additional example of reactions which are sensitive to the angle u 1. The ratio of the decay widths are given by RŽ Z . s
G w Z ™ hXg x G w Z ™ hg x
s
FhXg Z Fhg Z
2
kh Xg
ž / khg
3
Ž 20 .
where FPg Z is the time-like form factor for Pg transitions mediated by the Z boson. That form factor can be calculated along the same lines as the Pgg ) transition form factor w1,3x. Since the value of MZ2 is very large it suffices to consider the asymptotic limit of the form factor only. In terms of the octet and singlet decay constants, defined in Eq. Ž8., the result reads: FPg Z Ž
MZ2
.s
6 C8g Z f P8 q 6 C1g Z f P1 MZ2
The same transition form factors FPg Z appear in h ŽhX . ™ gmq my decays, but the momentum transfer is very small. In analogy to the P ™ gg decays the amplitudes in this case involve the inverse decay constants, and the h to hX ratio is sensitive to the angle u 8 . To measure these form factors in h ŽhX . ™ gmqmy decays one has to extract the g-Z interference term from suitably chosen asymmetries as discussed in detail in Ref. w25x. RadiatiÕe transitions between light Õector and pseudoscalar mesons The relevant coupling constants are defined by w26x ² P Ž pP . < JmEM < V Ž p V , l . :< q 2s0 s yg V Pg emnrs pPn p Vr ´ s Ž l . .
In Ref. w11x these coupling constants are expressed in terms of meson masses and decay constants by exploiting the chiral anomaly prediction Žat q 2 s 0. and vector meson dominance. However, the difference between u 8 and u 1 has not been considered. Translating the expressions for g P Vg correctly to the quark-flavor scheme, following otherwise Ref. w11x, we arrive at the formulas and values listed in Table 2. The numerical result for the coupling constants depend on the actual values of the vector mixing angle f V , which is expected to amount to only a few degrees Žsee e.g. w5,11x.. The values in Table 2 are calculated for f V s 0. For the vector meson decay Table 2 Various coupling constants g V Pg from theory and experiment w16x. The numerical values are quoted in units of GeVy1 P
h h
X
h
Ž 21 .
where C8g Z s Ž1 y 4sin2u W .rŽ6'6 . and C1g Z s Ž2 y 4sin2u W .rŽ3'3 .. The weak coupling of the flavor octet current is strongly suppressed by Ž1 y 4sin2u W .. Hence, RŽ Z . , cot 2u 1.
Ž 22 .
V
g V Pg Žin units
r
3 cos f 4 fq
r
3 sin f 4 fq
v
cos f cos f V 4 fq
hX
v f
sin f cos f V
hX
f
sin f sin f V 4 fq
y2 sin f sin f V
< g V Pg <Žtheo..
< g V Pg <Žexp..
1.52
1.85"0.34
1.24
1.31"0.12
0.56
0.60"0.15
q2 cos f sin f V
0.46
0.45"0.06
0.78
0.70"0.03
0.95
1.01"0.25
4 fs
cos f sin f V 4 fq
.
4 fs
4 fq
h
mV fV p 2
q2 sin f cos f V 4 fs
y2 cos f cos f V 4 fs
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
constants we take w27x fr s 210 MeV, fv s 195 MeV, ff s 237 MeV. The predictions agree rather well with experiment. Indeed the relations g rh Xg
s
g rhg
g vh Xg
s
g vhg
gfhg gfh Xg
s tan f s 0.82
Ž 23 .
are well confirmed by experiment. In Ref. w11x results of similar quality could only be achieved by using a value of u s u 8 s u 1 s y178 which deviates from the values obtained from other applications substantially.
h and hX admixtures to the pion As is well-known Žsee e.g. w28x. an accurate prescription of the decays of h ŽhX . to three pions can only be achieved by taking isospin violation into account. This effect is usually parametrized in terms of h and hX admixtures to the pion, p 0 s f 3 q e
Ž 24 .
where f 3 denotes the pure isospin-1 state. A straightforward generalization of our mixing scheme yields for the strength of h and hX admixtures in the pion
e s cos f
e X s sin f
m2d d y m2u u 2 Ž Mh2 y Mp2 . m2d d y m 2u u 2 Ž Mh2X y Mp2 .
,
,
Ž 25 .
where the difference m2d d y m 2u u can be estimated from 2Ž MK2 0 y MK2 " q Mp2 " y Mp2 0 . to amount to 0.0104 GeV 2 . A possible difference in u y and d y quark decay constants is ignored in the derivation of Ž25.. The expressions for e and e X look rather simple in the quark flavor scheme and are intimately connected to physical quantities. Inserting our phenomenological number for the mixing angle f we obtain e s 0.014 and e X s 0.0037. By exploiting the properties of the mass matrix w1x the ratio ere X following from Ž25. can also be expressed in terms of u 8 and u
eX e
s ytan u 8
cos u q '2 sin u
ž'
2 cos u y sin u
2
/
.
Ž 26 .
345
The numerical value, following from our phenomenological set of parameters, is 0.26. In contrast, the conventional approach Žsee e.g. w28x., using u s u 8 , y208 gives the much smaller value 0.17. The values of the parameters e and e X have recently been shown to be of importance for the investigation of CP-violation in B ™ pp decays since it breaks the isospin triangle relation for the amplitudes of the three processes Bq™ pqp 0 , B 0 ™ p 0p 0 and B 0 ™ pq py w29x. The value of e X used in Ref. w29x Ž e X s 0.0077. is substantially larger than our value. Summary We discussed the mixing properties of the h and hX meson state vectors and of their decay constants and showed that there is, at most, only one basis where the mixing of the decay constants can follow the pattern of state mixing. Chiral perturbation theory as well as phenomenological analyses favor this proposition for the quark-flavor basis. In general, e.g. in the familiar octet-singlet basis, one needs two angles in order to parametrize the decay constants. However, when using our quark-flavor mixing scheme, these new angles are fixed by the basic parameters f , f q , f s , leading to a number of important consequences for many reactions. The results are quite different from conventional mixing schemes in which the subtleties discussed here are not considered and where, as a consequence of that, the mixing parameters often show a strong process dependence. The improved knowledge of the mixing parameters is also of importance for the analysis of B decays, like B ™ KhX Žsee e.g. w30x. or B ™ pp w29x. Further interesting applications of our approach refer to h and hX production processes in high energy hadron collisions where exotic form factors such as the g ) g )h ŽhX . form factors play an important role.
Acknowledgements T.F. was supported by Deutsche Forschungsgemeinschaft. P.K. thanks the Special Research Centre for the Subatomic Structure of Matter at the University of Adelaide for support and the hospitality extended to him.
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Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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Physics Letters B 449 Ž1999. 339–346
Mixing and decay constants of pseudoscalar mesons: the sequel Th. Feldmann a , P. Kroll
a,b
, B. Stech
c
a
Fachbereich Physik, UniÕersitat ¨ Wuppertal, D-42097 Wuppertal, Germany Centre for the Subatomic Structure of Matter, UniÕersity of Adelaide, SA 5005, Australia c Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Heidelberg, D-69120 Heidelberg, Germany
b
Received 21 December 1998; revised 14 January 1999 Editor: P.V. Landshoff
Abstract We present further tests and applications of the new h – hX mixing scheme recently proposed by us. The particle states are decomposed into orthonormal basis vectors in a light-cone Fock representation. Because of flavor symmetry breaking the mixing of the decay constants can be identical to the mixing of particle states at most for a specific choice of this basis. Theoretical and phenomenological considerations show that the quark flavor basis has this property and allows, therefore, for a reduction of the number of mixing parameters. A detailed comparison with other mixing schemes is also presented. q 1999 Published by Elsevier Science B.V. All rights reserved.
In a recent reinvestigation w1x of processes involving h and hX mesons we pointed out that a proper treatment of the h – hX system requires a sharp distinction between the mixing properties of the meson states from the mixing properties of the decay constants. While the particle state mixing involves the global wave functions, the decay constants probe the quark distributions at zero spatial separation. Conventionally, h and hX are expressed as superpositions of an SUŽ3. flavor octet and a flavor singlet state corresponding to an orthogonal transformation with mixing angle u . The decay constants of h and hX defined by their matrix elements with singlet and octet axial vector currents will in general not show the same mixing since flavor symmetry breaking manifests itself differently at small and large distances. Because SUŽ3. breaking is solely caused by the current quark masses a simpler picture can be
expected for properly defined decay constants in the quark-flavor basis. Indeed, a dramatic simplification is achieved by taking two orthogonal basis states 1 which are assumed to have in a Fock state description the parton composition
Ž 1.
Here Ci denote Žlight-cone. wave functions of the corresponding parton states, and the dots stand for
1
In principle, the two-state basis should be extended by states of higher energy, for instance by adding a cc state. Energy considerations indicate, however, that the mixing with these states X is small. The small charm components in the h and h is discussed in Ref. w1x.
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 0 8 5 - 4
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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higher Fock states which also include < gg : components. These higher Fock states play no explicit role in the following discussions where we are mainly interested in mesonic states and decay constants. The physical meson states are related to the basis Ž1. by an orthogonal transformation
ž /
ž /
UŽ f . s
ž
cos f sin f
ysin f , cos f
/
Ž 2.
where f is the mixing angle. Ideal mixing corresponds to the case f s 0. It is to be emphasized that our definition of meson states is given in terms of parton degrees of freedom without introducing model-dependent concepts like constituent quarks. Our central ansatz Ž1,2. has important consequences for the weak decay constants which probe the short-distance properties of the quark-antiquark Fock states. To see this in detail, let us define the decay constants 2 by Ž fp s 131 MeV. ²0 < Jmi 5 < P : ' i f Pi pm , X
H
dx d 2 k H 16p 3
Jmi 5
Ci Ž x ,k H . .
Ž 4.
Here x denotes the usual Žlight-cone q. momentum fraction of the quark and k H its transverse momentum with respect to its parent meson’s momentum. Eq. Ž4. is exact, only the quark-antiquark Fock state contributes to the decay constant, higher Fock states do not contribute. Using Eqs. Ž1. – Ž3., one immediately observes that our ansatz for the Fock decomposition naturally leads to decay constants in the
2
ž
fhq
fhs
fhqX
fhsX
We stress that occasionally used decay constants ‘‘ fh , fh X ’’ are ill-defined quantities.
/
s U Ž f . diag f q , f s .
Ž 5.
The conventional octet-singlet basis states are obtained from the quark-flavor basis states by performing a rotation with the ideal mixing angle. The physical states are then related to the octet-singlet basis states by
ž /
ž /
Ž 6.
with u s f y arctan'2 . The corresponding Fock decompositions of these octet-singlet basis states, following from Eqs. Ž1,2,6., read
Cq q 2Cs < uu q dd y 2 ss :
'6
3 q
Ž 3.
where P s h ,h . Here denotes the axial-vector currents with quark content i s q, s. The decay constants are related to the quark-antiquark wave functions at the origin of configuration space. Because of the fact that light-cone wave functions do not depend on the hadron momentum we can define two basic decay constants f q and f s arising from hq and hs , respectively, f i s 2'6
quark-flavor basis which simply follow the pattern of state mixing:
'2 Ž Cq y Cs .
< uu q dd q ss :
3
'3
'2 Ž Cq y Cs .
< uu q dd y 2 ss :
3
'6
q
2Cq q Cs < uu q dd q ss : 3
'3
q...
q...
Ž 7.
Obviously, it is unavoidable that the so-defined octet Žsinglet. meson state contains an SUŽ3. singlet Žoctet. admixture, except for identical wave functions Cq s Cs , an equality which holds in the flavor symmetry limit only. Only then one would find pure octet and singlet states in Eq. Ž7.. Certainly, these results are based on our central ansatz, namely that h and hX can be decomposed into two orthogonal states where one state has no ss and the other no qq component. One may alternatively start from the assumption that h 8 and h1 defined in Eq. Ž6. have Fock decompositions with only parton octet or singlet combinations in the quark-antiquark sector, respectively. Rotating these states back by the ideal mixing angle, the resulting hq Žhs . state has an ssŽ qq . component, unless the octet and singlet wave functions are equal. However, from both, theoretical and phenomenological considerations performed in Refs. w1–3x, the quark-flavor basis is to be favored.
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
One may also define decay constants through matrix elements of octet and singlet axial-vector currents, analogously to Eq. Ž3.. Using Eqs. Ž1,2., one easily sees that these decay constants cannot be expressed as UŽ u . diagw f 8 , f 1 x. Rather one has fh8
fh1
fh8X
fh1X
f 8 cos u 8 f 8 sin u 8
ž /ž s
yf 1 sin u 1 , f 1 cos u 1
/
Ž 8.
where we use the new and general parametrization introduced in Ref. w2x. The parameters appearing in Eq. Ž8. are related to the basic parameters f , f q and f s , characterizing the quark flavor mixing scheme as follows w1x,
u 8 s f y arctan f 82 s
f q2 q 2 f s2
,
'2 f s fq
,
f 12 s
u 1 s f y arctan
'2 f q fs
,
2 f q2 q f s2
. Ž 9. 3 3 The decay constants f Pi do not follow the pattern of state mixing in the octet-singlet basis; only in the SUŽ3.F symmetry limit one would have u 8 s u s u 1. This is a consequence of the non-trivial Fock decomposition in Eq. Ž7.. The difference between u 8 and u 1 following from Eq. Ž9. leads to the same formula as derived within chiral perturbation theory w2x. In our approach the quantities u 8 and u 1 are parameters determined by the fundamental quantities 3 f , f q and f s . They are not to be used as
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constants Ž3. together with Ž5. one obtains the h – hX mass matrix in the quark flavor basis. Its elements are composed of the gluonic matrix elements ²0 < a s GG˜
fh1
fh8X
fh1X
ž /
s UŽ u8 .
ž
f8
f 1 sin Ž u 8 y u 1 .
0
f 1 cos Ž u 8 y u 1 .
/
.
Ž 11 .
3
We remark at this point, that the flavor singlet axial-vector current is not conserved in QCD. Consequently, the singlet decay constant f 1 , and, hence, f q and f s too, are renormalization scale dependent, although only mildly since the corresponding anomalous dimension is of order a s2 w2x. Varying the scale m between Mh and Mh c, the value of f 1Ž m . changes by 3% only, an effect which we discard.
Then it is tempting to introduce a new basis by
ž /
ž /
.
Ž 12 .
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
342
Table 1 Comparison of different determinations of mixing parameters. The values given in parentheses are not quoted in the original literature but have been evaluated by us from information given therein. Crosses indicate approaches where the difference between u , u 1 and u 8 has been ignored
u
u8
u1
f 8rfp
f 1rfp
method
y 12.38 y15.48
y21.08 y21.28
y2.78 y9.28
1.28 1.26
1.15 1.17
qs–scheme Žtheo.. w1x qs–scheme Žphen.. w1x
– y21.48 y15.58 y19.78 y12.68 yŽ238 y 178. y98
y20.58 x – wy12.28x wy19.58x x wy208x
y48 x – wy30.78x wy5.58x x wy58x
1.28 1.19 – w0.71x w1.27x 1.2 y 1.3 w1.2x
1.25 1.10 – w0.94x w1.17x 1.0 y 1.2 w1.1x
ChPT w2x GMO formula w4x phenomenology w5x model w6x model w7x phenomenology w8–11x UŽ1. A anomaly w12x
The elements of the second matrix on the r.h.s. of Eq. Ž11. can now be viewed as the decay constants of h˜ 8 , h˜ 1 through octet or singlet axial vector currents, respectively. This matrix is still non-diagonal but triangular. The new basis has the special feature that the anomaly contributes to the singlet Žh˜ 1 . mass alone, i.e. one has ²0 <
as 4p
GG˜
Ž 13 .
This property is related to the fact that the ratio ²0 < a s GG˜
4p
It allows to determine a Gell-Mann–Okubo formula for the mass of the h˜ 8 basis state. Transforming the mass matrix found in the quark flavor basis Žsee above. to the new basis Ž12. one finds f 82 m288 ˜˜ s
f q2 Mp2 q 2 f s2 Ž 2 MK2 y Mp2 . 3
.
Ž 14 .
This formula reminds of the suggestion put forward in Ref. w4x Žsee also w13x., namely to use the product f 2 M 2 rather than M 2 to determine the SUŽ3.F breaking effects in the Gell-Mann–Okubo formula 4 .
4 Note however that the decay constants used in the analysis of w4x are not defined as proper matrix elements of weak currents.
Insertion of the relations in Eq. Ž9. into Eq. Ž14. yields
m 288 ˜˜ ,
4 MK2 y Mp2 3
y DGMO
MK2 y Mp2 3
Ž 15 .
with DGM O s 4 Ž f q2 y f s2 .rŽ3 f 82 .. The deviation from the standard Gell-Mann–Okubo relation DGM O can also be derived in chiral perturbation theory w14,8x. The above discussion clearly shows: An analysis which implicitly uses Eq. Ž13. provides for an estimate of the parameter u 8 rather than the angle u . Indeed, previous treatments along these lines obtained mixing angles close to y208, which is consistent with our value of u 8 Žsee Table 1.. As a first test of our mixing approach we analyzed the hg and hXg transition form factors in Ref. w1x. A good description of the experimental data has been found from the phenomenological set of parameters. For details we refer to w1,3x. Let us now turn to further tests and applications of our results which have not been discussed in Ref. w1x. RadiatiÕe decays of S-waÕe quarkonia We define the ratio of decay widths RŽ 3 Sn . s G w 3Sn ™ hXg xrG w 3Sn ™ hg x where 3Sn represents one of the quarkonia Jrc , c X ,F , . . . According to w15x the photon is emitted by the c quarks which then annihilate into lighter quark pairs through the effect of the anomaly. Thus, the creation of the
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
corresponding light mesons is controlled by the matrix element ²0 < a s GG˜ < P :, leading to w1x 4p 3
R Ž Sn . s cot 2u 8
k hX g
3
ž /
Ž 16 .
khg
(
2
where k 12 s Ž M 2 y m12 y m 22 . y 4 m12 m 22 rŽ2 M . denotes the final state’s three-momentum in the rest frame of the decaying particle. The experimental value of R in the Jrc case w16x has already been used in the phenomenological determination of the basic mixing parameters in Ref. w1x. Using the phenomenological value of u 8 quoted in Table 1, we predict RŽ c X . s 5.8 and RŽF . s 6.5. The prediction .4 for RŽ c X . agrees with the experimental value 2.9q5 y1 .8 , which still has uncomfortably large errors however w17x. For the radiative F decays only upper bounds exist at present.
xc J decays into two pseudoscalars Because these are energetic decays the current quarks produced will be in an almost pure SUŽ3. singlet state. However, flavor symmetry violation can occur in the hadronization process. The ratio of xc J decay widths into different pairs of pseudoscalar mesons can be written Ž J s 0,2. G w xc J ™ P1 P2 x G w xc J ™ P3 P4 x
s
C12
2
k 12
ž /ž / C34
k 34
2 Jq1
.
Ž 17 .
For the coefficients Ci j two limiting cases can be considered. If the mesons are formed at hadronic distances the influence of the different decay constants will be a minor one and one expects to a good accuracy Chh s Ch X h X s Cp 0p 0 , Chh X s 0 and, from isospin symmetry, Cpq p ys '2 Cp 0 p 0 Žwhere we included the statistical factor '2 .. If, however, the meson generation starts already at a time at which the inter-quark distances are very small, the decay amplitudes are obtained from the convolution of a hard scattering process with the corresponding wave functions w18x. Assuming equal shapes of the wave functions, an assumption which is not in conflict with present experimental information, differences in the decay amplitudes are then solely due to the different decay constants and the mixing angle. One finds: Chh s f q2 cos 2f q f s2 sin2f s 1.41 Cp 0 p 0 , ChX h X s f q2 sin2f q f s2 cos 2f s 1.53 Cp 0 p 0 and Žwith
343
the statistical factor '2 included. Chh X s '2 Ž f q2 y f s2 . sin f cos f s y0.45 Cp 0 p 0 and Cp q p y s '2 Cp 0p 0 . Numerically we obtain G w xc 0Ž2. ™ hh xrG w xc 0Ž2. ™ p 0p 0 x s 1.9 Ž1.7., G w xc 0Ž2. ™ hXhX xrG w xc0Ž2. ™ p 0p 0 x s 1.9 Ž1.3., G w xc0Ž2. ™ hhX xrG w xc0Ž2. ™ p 0p 0 x s 0.2 Ž0.1. and G w xc0Ž2. ™ pq py xrG w xc0Ž2. ™ p 0p 0 x s 2. From the differences between the two limiting cases it appears that xc J decays are less suited for testing h – hX mixing parameters, but, taking the mixing parameters from other processes, they will provide interesting information on meson formation in these reactions. Experimentally, only the ratio G w x c 0Ž2. ™ hh xr G w x c 0Ž2. ™ p 0p 0 x is known .1 Ž q0 .9 . Ž w16,19x: 0.76q1 for the p 0p 0 branching y0 .5 0.76y0.6 ratio we combined the data with the one for the pq py channel.. At present, the large experimental errors prevent any definite conclusion. Similar relations as given here Žmodified according to the correct charge factors. should hold for two-photon annihilations into pairs of pseudoscalar mesons. g ) g ) ™ h ,h X transition form factors These form factors offer, in principle, a way to measure the angle u 1. Allowing both the gluons to be virtual where at least one of the virtualities q12 and q22 is supposed to be very large, one may easily work out the leading-twist result for these form factors w20x. In this approximation one has Ž i s q, s . Fh i g ) Ž q12 ,q22 . s yCi a s f i dx
H
fi Ž x . x q12 q
Ž 1 y x . q22 Ž 18 .
where f i is the hi distribution amplitude, and Ci a numerical factor Ž C q s '2 , C s s 1.. Combining both the form factors into those for the physical mesons and assuming the equality of the two distribution amplitudes Žwhich at least holds in the formal limit qi2 ™ ` since both the distribution amplitudes evolve into the asymptotic one., one arrives at Fh g ) Ž q12 ,q22 . FhX g ) Ž
q12 ,q22
.
s
'2 f q cos f y f s sin f '2 f q sin f q f s cos f s ytan u 1 . Ž 19 .
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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Of course, at finite values of momentum transfer one may expect corrections from differences between the two distribution amplitudes and from transverse momentum. Such corrections can be worked out following, for instance, Ref. w3x. For a measurement of these form factors one may consider the process pp ™ jet q jet q h ŽhX . where the mesons are supposed to be produced in the central rapidity region w21x. According to Close w22x these form factors may also be of relevance in pp ™ pph ŽhX ., provided the Pomeron couples to quarks A g m andror gg ™ h ŽhX . is the elementary process in the Pomeron-Pomeron interaction. The explicit extraction of the form factors from such measurements may, however, be very difficult. Kilian and Nachtmann w23x discuss g-Odderon-h ŽhX . form factors which appear in diffractive e-p scattering processes. We find that the ratio of these form factors is given by cot u 8 at large momentum transfer. Z ™ h(hX )g decay The treatment of these processes is rather academic since the expected branching ratios are far below the present experimental bounds ŽThe Z ™ pg decay has e.g. been discussed in Ref. w24x.. Nevertheless, they provide an additional example of reactions which are sensitive to the angle u 1. The ratio of the decay widths are given by RŽ Z . s
G w Z ™ hXg x G w Z ™ hg x
s
FhXg Z Fhg Z
2
kh Xg
ž / khg
3
Ž 20 .
where FPg Z is the time-like form factor for Pg transitions mediated by the Z boson. That form factor can be calculated along the same lines as the Pgg ) transition form factor w1,3x. Since the value of MZ2 is very large it suffices to consider the asymptotic limit of the form factor only. In terms of the octet and singlet decay constants, defined in Eq. Ž8., the result reads: FPg Z Ž
MZ2
.s
6 C8g Z f P8 q 6 C1g Z f P1 MZ2
The same transition form factors FPg Z appear in h ŽhX . ™ gmq my decays, but the momentum transfer is very small. In analogy to the P ™ gg decays the amplitudes in this case involve the inverse decay constants, and the h to hX ratio is sensitive to the angle u 8 . To measure these form factors in h ŽhX . ™ gmqmy decays one has to extract the g-Z interference term from suitably chosen asymmetries as discussed in detail in Ref. w25x. RadiatiÕe transitions between light Õector and pseudoscalar mesons The relevant coupling constants are defined by w26x ² P Ž pP . < JmEM < V Ž p V , l . :< q 2s0 s yg V Pg emnrs pPn p Vr ´ s Ž l . .
In Ref. w11x these coupling constants are expressed in terms of meson masses and decay constants by exploiting the chiral anomaly prediction Žat q 2 s 0. and vector meson dominance. However, the difference between u 8 and u 1 has not been considered. Translating the expressions for g P Vg correctly to the quark-flavor scheme, following otherwise Ref. w11x, we arrive at the formulas and values listed in Table 2. The numerical result for the coupling constants depend on the actual values of the vector mixing angle f V , which is expected to amount to only a few degrees Žsee e.g. w5,11x.. The values in Table 2 are calculated for f V s 0. For the vector meson decay Table 2 Various coupling constants g V Pg from theory and experiment w16x. The numerical values are quoted in units of GeVy1 P
h h
X
h
Ž 21 .
where C8g Z s Ž1 y 4sin2u W .rŽ6'6 . and C1g Z s Ž2 y 4sin2u W .rŽ3'3 .. The weak coupling of the flavor octet current is strongly suppressed by Ž1 y 4sin2u W .. Hence, RŽ Z . , cot 2u 1.
Ž 22 .
V
g V Pg Žin units
r
3 cos f 4 fq
r
3 sin f 4 fq
v
cos f cos f V 4 fq
hX
v f
sin f cos f V
hX
f
sin f sin f V 4 fq
y2 sin f sin f V
< g V Pg <Žtheo..
< g V Pg <Žexp..
1.52
1.85"0.34
1.24
1.31"0.12
0.56
0.60"0.15
q2 cos f sin f V
0.46
0.45"0.06
0.78
0.70"0.03
0.95
1.01"0.25
4 fs
cos f sin f V 4 fq
.
4 fs
4 fq
h
mV fV p 2
q2 sin f cos f V 4 fs
y2 cos f cos f V 4 fs
Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
constants we take w27x fr s 210 MeV, fv s 195 MeV, ff s 237 MeV. The predictions agree rather well with experiment. Indeed the relations g rh Xg
s
g rhg
g vh Xg
s
g vhg
gfhg gfh Xg
s tan f s 0.82
Ž 23 .
are well confirmed by experiment. In Ref. w11x results of similar quality could only be achieved by using a value of u s u 8 s u 1 s y178 which deviates from the values obtained from other applications substantially.
h and hX admixtures to the pion As is well-known Žsee e.g. w28x. an accurate prescription of the decays of h ŽhX . to three pions can only be achieved by taking isospin violation into account. This effect is usually parametrized in terms of h and hX admixtures to the pion, p 0 s f 3 q e
Ž 24 .
where f 3 denotes the pure isospin-1 state. A straightforward generalization of our mixing scheme yields for the strength of h and hX admixtures in the pion
e s cos f
e X s sin f
m2d d y m2u u 2 Ž Mh2 y Mp2 . m2d d y m 2u u 2 Ž Mh2X y Mp2 .
,
,
Ž 25 .
where the difference m2d d y m 2u u can be estimated from 2Ž MK2 0 y MK2 " q Mp2 " y Mp2 0 . to amount to 0.0104 GeV 2 . A possible difference in u y and d y quark decay constants is ignored in the derivation of Ž25.. The expressions for e and e X look rather simple in the quark flavor scheme and are intimately connected to physical quantities. Inserting our phenomenological number for the mixing angle f we obtain e s 0.014 and e X s 0.0037. By exploiting the properties of the mass matrix w1x the ratio ere X following from Ž25. can also be expressed in terms of u 8 and u
eX e
s ytan u 8
cos u q '2 sin u
ž'
2 cos u y sin u
2
/
.
Ž 26 .
345
The numerical value, following from our phenomenological set of parameters, is 0.26. In contrast, the conventional approach Žsee e.g. w28x., using u s u 8 , y208 gives the much smaller value 0.17. The values of the parameters e and e X have recently been shown to be of importance for the investigation of CP-violation in B ™ pp decays since it breaks the isospin triangle relation for the amplitudes of the three processes Bq™ pqp 0 , B 0 ™ p 0p 0 and B 0 ™ pq py w29x. The value of e X used in Ref. w29x Ž e X s 0.0077. is substantially larger than our value. Summary We discussed the mixing properties of the h and hX meson state vectors and of their decay constants and showed that there is, at most, only one basis where the mixing of the decay constants can follow the pattern of state mixing. Chiral perturbation theory as well as phenomenological analyses favor this proposition for the quark-flavor basis. In general, e.g. in the familiar octet-singlet basis, one needs two angles in order to parametrize the decay constants. However, when using our quark-flavor mixing scheme, these new angles are fixed by the basic parameters f , f q , f s , leading to a number of important consequences for many reactions. The results are quite different from conventional mixing schemes in which the subtleties discussed here are not considered and where, as a consequence of that, the mixing parameters often show a strong process dependence. The improved knowledge of the mixing parameters is also of importance for the analysis of B decays, like B ™ KhX Žsee e.g. w30x. or B ™ pp w29x. Further interesting applications of our approach refer to h and hX production processes in high energy hadron collisions where exotic form factors such as the g ) g )h ŽhX . form factors play an important role.
Acknowledgements T.F. was supported by Deutsche Forschungsgemeinschaft. P.K. thanks the Special Research Centre for the Subatomic Structure of Matter at the University of Adelaide for support and the hospitality extended to him.
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Th. Feldmann et al.r Physics Letters B 449 (1999) 339–346
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