Single eta production in heavy quarkonia transitions

Abstract

The $\eta$ production in the $(n,n')$ bottomonium transitions $\Upsilon (n) \to \Upsilon (n') \eta, $ is studied in the method used before for dipion heavy quarkonia transitions. The widths $\Gamma_\eta(n,n')$ are calculated without fitting parameters for $n=2,3,4,5, n'=1$.Resulting $\Gamma_\eta(4,1)$ is found to be large in agreement with recent data.

Full text

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arXiv:0806.2919v1 [hep-ph] 18 Jun 2008
Single eta production in heavy quarkonia
transitions
Yu.A.Simonov, A.I.Veselov
State Research Center
Institute of Theoretical and Experimental Physics,
Moscow, 117218 Russia
June 18, 2008
Abstract
The η production in the (n,n′) bottomonium transitions Υ(n) →
Υ(n′)η, is studied in the method used before for dipion heavy quarko-
nia transitions. The widths Γη(n,n′) are calculated without fitting
parameters for n = 2,3,4,5,n′= 1.Resulting Γη(4,1) is found to be
large in agreement with recent data.
1Introduction
The η and π0production in heavy quarkonia transitions is attracting atten-
tion of experimentalists for a long time [1]. The first result refers to the
ψ(2S) → J/ψ(1S)η process (to be denoted as ψ(2,1)η in what follows, sim-
ilarly for Υ) with
For the Υ(2,1)η and Υ(3,1)η transitions only upper limits B < 2 · 10−3
and B < 2.2 · 10−3were obtained in [2] and [3] correspondingly and prelimi-
nary results appeared recently in [4], B(Υ(2,1)η) = (2.5±0.7±0.5)10−4and
B(Υ(2,1)π0) < 2.1 · 10−4(90% c.l.). On theoretical side in [5] small ratios of
widths
Γη
Γtot= (3.09 ± 0.08)% [1], Γtot= 337 ± 13 keV.
Γ(Υ(2,1)η)
Γ(ψ(2,1)η)
∼= 2.5 · 10−3and
Γ(Υ(3,1)η)
Γ(ψ(2,1)η)= 1.3 · 10−3
(1)
1

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have been predicted, with the model property that the bottomonium yields
of η would be smaller than those of charmonium;specifically in the method
of [6], the width ratio is proportional to O
??mc
mb
?2?
≈ 0.1, for a discussion
see also [6, 7].
However recently [8] new BaBar data have been published on Υ(4,1)η
with the branching ratio
B(Υ(4,1)η) = (1.96 ± 0.06 ± 0.09)10−4
(2)
and
Γ(Υ(4,1)η)
Γ(Υ(4,1)π+π−)= 2.41 ± 0.40 ± 0.12.
This latter result is very large, indeed the corresponding ratio for ψ(2,1)η
transition is ≈ 0.2 and theoretical estimates (1) from [5] for a similar ratio
of Υ(3,1)η/ππ yield 0.015. All this suggests that another mechanism can
be at work in single η production and below we exploit the approach based
on the Field Correlator Method (FCM) [9] recently applied to Υ(n,n′)ππ
transitions with n ≤ 3 in [9, 10], n ≤ 4 in [11] and n = 5 in [12, 13].
The method essentially expoits the mechanism of Internal Loop Radiation
(ILR) with light quark loop inside heavy quarkonium and has two fundamen-
tal parameters – mass vertices in chiral light quark pair q¯ q creation Mbr≈ fπ
and pair creation vertex without pseudoscalars Mω≈ 2ω, where ω(ωs) is the
average energy of the light (strange) quark in the B(Bs) meson. Those are
calculated with relativistic Hamiltonian [14] and considered as fixed for all
types of transitions ω = 0.587 GeV, ωs= 0.639 GeV.
Any process of heavy quarkonium transition with emission of any number
of Nambu-Goldstone (NG) mesons is considered in ILR as proceeding via
intermediate states of B¯B,B¯B∗+ c.c.,Bs¯Bsetc. (or equivalently D¯D etc.)
with NG mesons emitted at vertices.
For one η or π0emission one has diagrams shown in Fig.1, where dashed
line is for the NG meson. As shown in [9, 10, 11], based on the chiral
Lagrangian derived in [15], the meson emission vertex has the structure
(3)
LCDL= −i
?
d4x¯ψ(x)MbrˆU(x)ψ(x) (4)
ˆU = exp
?
iγ5ϕaλa
fπ
?
,ϕaλa=
√2




η
√6+πo
π−,
K−,
√2,π+,K+
Ko
−2η
η
√6−πo
¯K0,
√2,
√6



, (5)
2

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?
&
?
?
%
?
?
&
?
?
%
?
u
u
pu
s
'
$
'
$
( a )
( b )
Fig.1 Single eta production (dashed line) from Υ(n)BB∗vertex (a), and BB∗Υ(n′)
vertex (b).
The lines (1,2,3) in theˆU matrix (2) refer to u,d,s quarks and hence to the
channels B+B−,B0¯B0,B0
of B). Therefore the emission of a single η in heavy quarkonia transitions requires
the flavour SU(3) violation and resides in our approach in the difference of channel
contribution B¯B∗and Bs¯B∗
and B+B−∗channels (with B → D for charmonia).
The paper is devoted to the explicit calculation of single η emission widths in
bottomonium Υ(n,1)η transitions with n = 2,3,4,5. Since theory has no fitting
parameters (the only ones, Mωand Mbrare fixed by dipion transitions) our pre-
dictions depend only on the overlap matrix elements, containing wave functions
of Υ(nS), B,Bs,B∗,B∗
s. The latter have been computed previously in relativistic
Hamiltonian technic in [14] and used extensively in dipion transitions in [11, 12, 13].
The paper is organized as follows. In section 2 general expressions for process
amplitudes are given; in section 3 results of calculations are presented and discussed
and a short summary and prospectives are given.
s¯B0
s(and to the corresponding channels with B∗instead
s, while the π0emission is due the difference of B0¯B0∗
2General formalism
The process of single NG boson emission in bottomonium transition is described
by two diagrams depicted in Fig.1, (a) and (b) which can be written according to
the general formalism of FCM [9, 11, 12] as (we consider η emission)
M = M(1)
η
+ M(2)
η;M(i)
η = M(i)
J(1)
n (p,k)Jn′(p)
E − E(p)
BsB∗
s− M(i)
d3p
(2π)3,
BB∗,i = 1,2(6)
M(1)
η
= γ
?
(7)
3

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where M(2)
of (7). Here γ =
η
has the same form, but without NG boson energy in the denominator
MωMbr2
√2ωηNcfη
functions is (for details see Appendix)
√3and the overlap integral of Υ(nS) and BB∗wave
J(1)
n(p,k) = ¯ zη
123
(0)In,BB∗(p)e
−p2
∆n−k2
4β2
2
(8)
Jn′(p) = ¯ z2
(1)In′,BB∗(p)e
−p2
∆′n
(9)
¯ zη(BB∗) and ¯ z2(BB∗) are Dirac traces of decay matrix elements Υ(nS) → BB∗η
and BB∗→ Υ(n′), respectively they are defined in [9, 11] and below in Appendix.
The special point in our case is that η meson is emitted in P wave, hence one must
extract the corresponding term in the Dirac trace, for details see Appendix.
¯ zη(BB∗) · ¯ z2(BB∗) =
?Mb
2Ω
?24p2unei′il
3ω3
kl.(10)
Here un =
in B(Bs) meson; from Table IV in [9] one finds that Ω = 4.827 GeV, Ωs= 4.830
GeV. In what follows we shall neglect the difference between Ω,Ωsand the mass of
b quark Mb= 4.8 GeV. Note that these large masses cancel in all matrix elements
and final expressions will depend only on energies ω and ωs and differences of
threshold positions: ∆M∗= M(B) + M(B∗) − M(Υ(nS)) and ∆Ms– the same
for Bs,B∗
hence we shall consider only B¯B∗and Bs¯B∗
Indices i′i in ei′ilin (10) refer to the Υ(n′S) and Υ(nS) polarizations respec-
tively. Finally, coefficients β2,β1 and ∆n= 2β2
realistic wave functions of Υ(nS),Υ(n′S) and B,B∗,Bs,B∗
series of oscillator functions and β1,β′
rameters for those functions respectively, see [11] for details.
Finally we define all quantities in the denominator of (7); in M(1)
nominator is
β2
2Ω
∆n(ω+Ω)≈
β2
∆n, and Ω(Ωs) is the average energy of the b quark
2
smasses. Note, that the contribution of the B∗¯B∗,B∗
s¯B∗
schannels vanish
schannels.
1+ β2
2, refer to the expansion of
scomputed in [14] in
1and β2denote the χ2fitted oscillator pa-
BB∗ the de-
E−E(p) = M(Υ(nS))−(ωη+MB+M∗
B+
p2
2MB+(p − k)2
2M∗
B
) ≡ −∆M∗−ωη−E(p,k).
(11)
For M(2)
M(i)
η
one omits ωηand k in (11). Finally one can represent the matrix element
η as follows:
M(1)
η
= γeii′lkl
β2
3∆n
2
?1
ω3
s
L(1)
s
−
1
ω3L(1)
?
e
−k2
4β2
2
(12)
4

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M(2)
η
= γeii′lkl
β2
2
3∆n′
?1
ω3
s
L(2)
s
−
1
ω3L(2)
?
e
−k2
4β2
2
(13)
with
L(1)=
?
d3p p2
(2π)3
(0)In(p)(1)In′(p)e
∆M∗+ ωη+p2
−p2
β2
0
?
2MB+(p−k)2
2M∗
B
?. (14)
L(2)=
?
d3p p2
(2π)3
(1)In(p)(0)In′(p)e
?
sand MB,M∗
−p2
β2
0
∆M∗+
p2
MBB∗
2˜
?
. (15)
For L(1)
β−2
0
=
The width of the Υ(n,n′)η decay is obtained from |M|2averaging over vector
polarizations as
s ,L(2)
∆n+
s
1
one replaces ∆M∗with ∆M∗
Bwith MBs,MB∗
s. Here
1
∆n′.
Γη=1
3
?
i,i′
|M|2dΦ =2k2
27γ2e
−k2
2β2
2dΦ
?????un
?
L(1)
ω3
s
s
−L(1)
ω3
?
+ un′
?
L(2)
ω3
s
s
−L(2)
ω3
??????
2
(16)
where the phase space factor dΦ =
Introducing the average ¯ ω =1
d3k
(2π)32πδ(M(Υ(n))−M(Υ(n′))−ωη−
2(ωs+ ω), one can rewrite (16) as
k2
2M(Υ(n′))).
Γη=
?
Mbr
fη
?2 ?Mω
2¯ ω
?2
ζηk3
¯ ω4e
−k2
2β2
2
?????un
−
??¯ ω
ωs
?3
L(1)
s
−
?¯ ω
ω
?3
L(1)
?
+
un′
??¯ ω
ωs
?3
L(2)
s
?¯ ω
ω
?3
L(2)
??????
2
(17)
with ζη=
One can see from the general structure of Γη, that the main effect comes from
????
16
81πN2
c
∼= 7 · 10−3.
the difference
?
¯ ω
ωs
?3−?¯ ω
ω
?3
????≈ |0.882 −1.139| ≈ 0.257, and from the difference of
|L(i)
s − L(i)| ≤ 0.05.
3Results and discussion
We consider here the single η emission in bottomonium transitions Υ(n,1)η with
n = 2,3,4,5. The corresponding values of ∆M∗,∆M∗
1.
s,ωη,k are given in the Table
5