Glueball plus Pion Production in Photon-Photon Collisions.

27 Pages·0.16 MB·English

by  M. A. ICHOLA

LPC 93 06
hep-ph/9501225
GLUEBALL PLUS PION PRODUCTION
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1 IN HARD EXCLUSIVE TWO-PHOTON PROCESSES

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6 M. A. ICHOLA and J. PARISI

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2 Laboratoire de Physique Corpusculaire, Coll`ege de France
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IN2P3-CNRS
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h 11, Place Marcelin Berthelot, F-75231 Paris Cedex 05, France
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Abstract
We here compute the reaction γ γ G π0 for various glueball candidates
→
G and their assumed quantum states, using a non-relativistic gluon bound-state
model for the glueball.
2
1 Introduction
A model for computing the production and decay of glueballs (G) made
up of two gluons, in any quantum state, has been proposed by Kada et al. [1],
who used it in order to calculate the processes J/Ψ G γ and G γ γ. That
→ ⇔
model was later generalized for more complex reactions by Houra-Yaou et al. [2] ;
the generalized formalism was applied, in particular, to the calculation of glueball
production in high-energy hadron collisions.
The values obtained in Ref. [1] for Γ(G γ γ), considering the main
→
existing glueball candidates, are rather small, and systematically below the exper-
imental limits established till now in measurements of γ γ G . The purpose of
→
this paper is to compute an alternative photon-photon collision process producing
glueballs, i.e. γ γ G π0. The model used in this study is again that of Ref. [2].
→
Let us notice that the pseudoscalar glueball candidate η(1440) was seen in
various measurements of radiative J/Ψ decays [3]. On the other hand , two new
analyses of J/Ψ η(1440)γ performed by the Mark III and the DM2 Collabora-
→
tion [4] both seem to show that, instead of corresponding to a single pseudoscalar
resonance, the η(1440) peak should be due to three different structures (two pseu-
doscalar and one axial) located close to each other in the 1400-1500 MeV range.
However the conclusions of the two analyses are contradictory.
As for the tensor glueball candidate f (1720), it was seen as well by various
2
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experimental groups in radiative J/Ψ decay [5] and also by the WA 76 Collabora-
tion in double-diffractive pp collisions [6]. Here again some doubt has been cast on
the true nature of this resonance by a recent analysis of Mark III [7]. According
to this analysis it should rather be a scalar particle, f (1710), and is unlikely to
0
be a glueball. Thus there is, here again, an obvious disagreement between various
experimental groups (see [8]).
Finally, as regards the X(2220), it was also found by the Mark III Collab-
oration in radiative J/Ψ decay, and was assigned the quantum numbers : J even
(without further precision), P = + [9]. On the other hand, it was recently seen
by two experimental groups in hadronic reactions, but they disagreed on its spin
value (JP = 2+ vs. JP = 4+) [10] ; specialists now tend to believe that the X is
rather a ss¯ quarkonium state belonging to the nonet 3F.
It is obvious that, in all three cases considered, no firm conclusion has yet
been reached as to the nature of these resonances. Therefore we consider that none
of them has been discarded as a glueball candidate, and we continue treating them
as in previous papers [1, 2].
4
2 Details of calculation
Assuming, here again, the glueball to be a weakly bound state of two non-
relativistic gluons, we are led (see Sec. II of Ref. [2]) to the formula
λλ′,Λ (s,Θ) = f lim 1 d(cosθ) dφ
Mγγ′→Gπ0 L β→0 βL 4 π ×
Z
ζLSJ Λ(θ,φ) λλ′,λ1λ2 (s,Θ,θ,φ) (1)
× λ1λ2 Mγγ′→g1g2π0
λX1,λ2
connecting the helicity amplitudes of the process γ γ′ G π0 with those of the
→
subprocess γ γ′ g g π0. In formula (1), s and Θ are, respectively, the total
1 2
→
′
energy squared and the pion emission angle in the γ γ c.m. frame, while θ (φ)
is the orbital (azimuthal) emission angle of either gluon in their c.m. frame (the
glueball rest frame) ; see Fig. 1. We call J, L, S, respectively, the total spin, the
orbital angular momentum and the intrinsic spin of the glueball, while Λ is its spin
′
component along the z axis of Fig. 1. In addition we call λ, λ the helicities of
′
the photons γ, γ , while λ , λ are those of the gluons g , g (all helicities being
1 2 1 2
defined in the glueball rest frame). The angular projection function ζLSJΛ(θ,φ) is
λ1λ2
defined as :
ζLSJ Λ(θ,φ) = dJ e−iΛφ L S 0 Λ¯ L S J Λ¯ 1 1 λ λ 1 1 S Λ¯ (2)
λ1λ2 ΛΛ¯ h | ih 1 − 2 | i
where Λ¯ = λ λ . Finally, β is the velocity of either gluon in the glueball rest
1 2
−
frame, while f is given by
L
2 L + 1 2 i L (2 L + 1)!! d L
f = R (r) (3)
L L
s 2 π M (cid:18)−M(cid:19) L!  dr! r→0
 
5
where M is the glueball mass, and R (r) its radial wave function in configuration
L
space.
′
As in Ref. [2], we assume the glueball to be relativistic in the γ γ c.m.
frame : M/√s 0. In that approximation, the gluons g , g are also treated
1 2
→
as massless in the subprocess γ γ′ g g π0. The latter subprocess involves 16
1 2
→
helicity amplitudes. Noticing that, due to angular momentum and parity conser-
vation, those amplitudes remain unchanged when all photon and gluon helicities
are reversed, only eight of them are needed.
For the calculation of those amplitudes, we use the Brodsky-Lepage model
[11], i.e. :
λλ′,λ1λ2 (s,Θ,θ,φ) = dx Φ (x) λλ′,λ1λ2 (x,s,Θ,θ,φ) (4)
Mγγ′→g1g2π0 π Mγγ′→g1g2(qq¯)PS
Z
where Φ is the pion distribution amplitude , while in the calculation of
π
λλ′,λ1λ2 one makes the substitution : v u¯ γ p /√2, taking into ac-
Mγγ′→g1g2(qq¯)PS q¯ q → 5 π
count the fact that the (q q¯) system is in a pseudoscalar state. The helicity ampli-
tudes λλ′,λ1λ2 needed are obtained, at lowest order in QCD, by summing
Mγγ′→g1g2(qq¯)PS
over the contributions of all diagrams of Fig. 2.
We then remain with the task of applying formulas (4) and (1). Actually,
we find more convenient to integrate over θ,φ first, applying formula (1), and to
leave the convolution over φ (x) for the following stage of our calculation.
π
Formula (1), where we substitute (q q¯) for π0, leads us, for the var-
PS
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ious quantum states considered (for the main glueball candidates) to the ex-
pressions λλ′,Λ shown hereafter. We here fixed λ = +1, since ampli-
Mγγ′→G(qq¯)PS
tudes with λ = 1 are derived therefrom by applying the relation −λ−λ′,−Λ =
− M
( 1)J+L−Λ−1 λλ′,Λ.
− M
All other helicity amplitudes are vanishing at that order.
Let cosΘ = u, N = x (1 x) s, y = 2 (1 x) (1+u),
− − −
y = 1+u (1 2 x) and z = 1+u 2 x.
1
− −
In addition we set :
y
A = 2(1 u)(1 2x) u
1
− − − − z
2u
B =
1
− z
y
C = 4x2(1+u)+2(u+3)(1 2x)+2(1 x)(1 u2)+(4(x 1)2 1 u2)
1
− − − − − − z
D = 4(1 x)(1+u)
1
− −
y
A = 4(1 x)
2
− − z
(1 x)
B = 4 −
2
z
y
C = 2(1+u)(1 x)( 2x+1 u)+(4(x 1)2 +1 u2)
2
− − − − − z
D = 4(1 x(1 u))
2
− −
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We get :
J = 0, L = S = 0
B y A
++,0 = 8 π 1 1 − 1 (5)
Mγγ′→G(qq¯)PS − y N
1
B y A
+−,0 = 8 π 2 1 − 2 (6)
Mγγ′→G(qq¯)PS − y N
1
J = 0, L = S = 1
C z
++,0 = 8 π 1 (7)
Mγγ′→G(qq¯)PS 3 y2 N
1
C z
+−,0 = 8 π 2 (8)
Mγγ′→G(qq¯)PS − 3y2 N
1
J = 2, L = 0, S = 2
B y A
++,0 = 16 π 1 1 − 1 (9)
Mγγ′→G(qq¯)PS − y N
1
D
++,2 = 16 π 1 (10)
Mγγ′→G(qq¯)PS − y N
1
B y A
+−,0 = 16 π 2 1 − 2 (11)
Mγγ′→G(qq¯)PS − y N
1
D
+−,−2 = 16 π 2 (12)
Mγγ′→G(qq¯)PS − y N
1
J = 2, L = 2, S = 0
B y3 A (z2 +y2)
++,0 = 32 π 1 1 − 1 1 (13)
Mγγ′→G(qq¯)PS 15 y3 N
1
8
D
++,2 = 64 π 1 (14)
Mγγ′→G(qq¯)PS − 15 y N
1
B y3 A (z2 +y2)
+−,0 = 32 π 2 1 − 2 1 (15)
Mγγ′→G(qq¯)PS 15 y3 N
1
D
+−,−2 = 64 π 2 (16)
Mγγ′→G(qq¯)PS − 15 y N
1
J = 2, L = S = 2
17 B y3 +A (4z2 17y2)
++,0 = 8 π 1 1 1 − 1 (17)
Mγγ′→G(qq¯)PS 15 y3 N
1
D (4z2 3y2)
++,2 = 16 π 1 − 1 (18)
Mγγ′→G(qq¯)PS 15 y3 N
1
17 B y3 +A (4z2 17y2)
+−,0 = 8 π 2 1 2 − 1 (19)
Mγγ′→G(qq¯)PS 15 y3 N
1
D (4z2 3y2)
+−,−2 = 16 π 1 − 1 (20)
Mγγ′→G(qq¯)PS 15 y N
1
J = 4, L = S = 2
B y3 A (z2 +y2)
++,0 = 128 π 1 1 − 1 1 (21)
Mγγ′→G(qq¯)PS 75 y3 N
1
D (z2 +y2)
++,2 = 64 π 1 1 (22)
Mγγ′→G(qq¯)PS 15 y3 N
n1
B y3 A (z2 +y2)
+−,0 = 128 π 2 1 − 2 1 (23)
Mγγ′→G(qq¯)PS 75 y3 N
1
D (z2 +y2)
+−,−2 = 64 π 2 1 (24)
Mγγ′→G(qq¯)PS 15 y3 N
1
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Intheaboveformulas, coupling constants, aswell aschargeandcolorfactors,
were left aside.
For Φ we here choose two different expressions given in the literature,
π
namely that of Chernyak and Zhitnitsky [12]
ΦCZ(x) = 5 √3 f x (1 x) (2x 1)2 (25)
π π − −
and the so-called asymptotic distribution amplitude [11]
Φas(x) = √3 f x (1 x) (26)
π π −
where f is the pion leptonic decay constant.
π
Convolution of Φ with formulas (5) - - (24) then leads us to the amplitudes
π
λλ′,Λ needed.
Mγγ′→Gπ0
Therefrom one obtains the corresponding differential cross section :
dσγγ′→Gπ0 = 4 π3α2αs2 f2 f2 λλ′,Λ 2 (27)
dt s2 ch c Mγγ′→Gπ0
λXλ′,Λ (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where the charge and color factors are easily obtained as : f2 = 1/18, f2 = 2/3.
ch c
This expression of the cross section still contains an undetermined constant,
f2 (see formula (1)). In order to eliminate it, we use the same procedure as in
L
Refs. [1] and [2], i.e. we write :
dσγγ′→Gπ0 dσγγ′→Gπ0 Γ(J/Ψ G γ) B( G x y)
B( G x y) → → (28)
dt → ≡ dt Γ(J/Ψ G γ)
→
where B( G x y) is the branching ratio for glueball decay in a given channel
→
(actually we shall only consider the main decay channel for each glueball candi-
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