ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 9, с. 1234-1238
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
ISOSCALAR AMPLITUDE DOMINANCE IN e+e- ANNIHILATION TO NN PAIR CLOSE TO THE THRESHOLD
©2014 V. F. Dmitriev*, A. I. Milstein**, S. G. Salnikov***
Budker Institute of Nuclear Physics, Novosibirsk; Novosibirsk State University, Russia Received June 8, 2013
We use the Paris nucleon-antinucleon optical potential for explanation of experimental data in the process e+e- ^ pp near threshold. It turns out that final-state interaction due to Paris optical potential allows us to reproduce available experimental data. It follows from our consideration that the isoscalar form factor is much larger than the isovector one.
DOI: 10.7868/S0044002714080042
1. INTRODUCTION
At present, QCD cannot describe quantitatively the low-energy nucleon-antinucleon interaction, and various phenomenological approaches have been suggested in order to explain numerous experimental data, see, e.g., [1-6] and recent reviews [7, 8]. However, parameters of the models still cannot be extracted with a good accuracy from the experimental data [9].
Very recently, renewed interest in low-energy nucleon-antinucleon physics has been stimulated by the experimental observation of a strong enhancement of decay probability at low invariant mass of pp in the processes J— YPP [10], B + — K+pp and B0 — D0pp [11-13], B+ — n+pp and B+ — — K°pp [14], Y — ypp [15]. One of the most natural explanations of this enhancement is final-state interaction of the proton and antiproton [16-21].
A similar phenomenon was observed in the investigation of the proton (antiproton) electric, GE(Q2), and magnetic, GM(Q2), form factors in the process e+e- — pp [22-24]. Namely, it was found that the ratio \Ge(Q2)/Gm(Q2)| strongly depends on Q2 = = 4E2 (in the center-of-mass frame) in the narrow region of the energy E near the threshold of pp) production. Such strong dependence at small E is related to the large-scale interaction of proton and antiproton. Therefore, it is possible to apply the approaches of [1 -6] for an explanation of experimental data in the process e+e- — pp. In the present paper, we use
* E-mail: V.F.Dmitriev@inp.nsk.su
E-mail: A.I.Milstein@inp.nsk.su
E-mail: salsergey@gmail.com
the Paris nucleon-antinucleon optical potential Vnn which has the form [4, 5]:
VNN = UNN — iWNN, (1)
where the real part UN^ is the G-parity transform of the well established Paris NN potential for the long-and medium-range distances (r > 1fm), and some phenomenological part for the short distances. The absorptive part Wnn of the optical potential takes into account the inelastic channels of NN interaction, i.e. annihilation into mesons. It is essential at short distances and depends on the kinetic energy of the particles. We perform calculations in the non-relativistic approximation. The Coulomb interaction between proton and antiproton is important only for the kinetic energy T < (na)2M ~ 1 MeV, where a is the fine-structure constant, and M is the proton mass. Here we consider the process for kinetic energies T » 1 MeV, therefore we neglect the Coulomb interaction.
Taking into account that the difference of the cross sections e+e- — pp and e+e- — nn is small [24, 25], we calculate the cross sections at a given isospin final states and compare them with the experimental data for e+e- — pp. As a result, we found that the amplitude with the isospin I = 0 strongly dominates. Besides, our prediction for the ratio \GE(Q2)/GM(Q2)\ depends on the parameters of the Paris potential but is independent of the form factor at threshold. The ratio is also in a qualitative agreement with experimental data.
2. AMPLITUDE OF THE PROCESS
In the nonrelativistic approximation, the amplitude of NN pair production in a certain isospin channel
I = 0,1 near threshold can be presented as follows (in units 4na/Q2):
Gl e
I pi ^ ^
ß2
k2eß - 3(k • eß)k
6M 2
(2)
Si = n m + H (Q ) + Y [H (Q ) - H (Q)],
GI = (Q2) -F(Q2), where /3 = k/M < 1, is a virtual photon polarization vector, corresponding to the projection of spin Jz = / = ±1, and e\ is the spin-1 function of NN pair, A = ±1,0 is the projection of spin on the vector k. Two tensor structures in Eq. (2) correspond to the s-wave and d-wave production amplitudes. The total angular momentum of the NN pair is fixed by a production mechanism. The functions F[(Q2) and F2> (Q2) are the Dirac form factors of the NN pair which include the effects of final-state interaction and have a pronounced Q2 behavior near the threshold. Summation over the polarization of nucleon pair and averaging over the polarization of virtual photon is performed using the equations,
(3)
E
E
A=1,2,3
i* j _ ^ _
& 11 & 11 — ~ 0 I —
m m 2 ^
e^ei = öij,
1
(öij - PiPj/P2 ),
1,2
where P is the electron momentum.
Our aim is to single out the effects of finalstate interaction. In order to do that, we write the amplitude (2) in the form
d3p
T
x m
_${(-)*
(2n)3 kx
(P) x
(4)
Gs eM + Gd
j p2eM - 3(p • eM)P
6M 2
where ^(p) is the Fourier transform of the function ^k(")(r), the wave function of the NN pair in coordinate space. This wave function is the solution of the Schrodinger equation
J (—> k
(r )H
MßWkx^*
(r),
(5)
P2
H=M+ VNN,
where V
*JA—)* (r)
Nn is the optical potential. Note that
:k A Vr) is the left eigenfunction of the biorthogonal set of eigenfunctions of the non-Hermitian operator H. The asymptotic form of the wave function at large distances reads
In Eq. (4) the form factors Gs and Gd are
32
Gl = F{ + Fl + ^(Fl-F{), (7)
Gd = F{ - Fi,
where the "bare" Dirac form factors F1 and F2 do not account for the effect of final-state interaction. Near threshold these form factors are smooth functions of Q2 and can be treated as phenomenological constants.
3. WAVE FUNCTION Let us introduce the vector spherical functions Jn) as
Y5» = £ cl^-mYLUn^-m (8)
m
where YLm(n) are spherical harmonics, CJ^li1^-m are Clebsch—Gordan coefficients, and n = r/r. In Eq. (8) the quantization axes is directed along the
vector k. Then the wave function can be
written in the form
<i™)W = EVM5j+I)Cj0Aiy/(r)x (9)
J
x YjA(n) + V V4tt(2J - 1 )C£ 10 1A x
J
+
uJ (r)YjA-1(n)+w{} (r)Yj+1(n) + ^VM2J + 3)C^10)1AX
J
uJ (r)YJj-1(n)+wIj} (r)Yj+1(n)
Here, the functions vJ(r), u—(r), and w—(r) have the asymptotic form at large distances
vj(r) = 7T7
1
2ikr
nJJ A[kr—Jn/2] —i[kr — Jn/2] ^n e e
, (10)
ul j(r) = 7T7
1
2ikr
sJJ i[kr — (J—1)n/2] _ S11 e
e
—i[kr—( J— 1)n/2]
1
wIj{r) = 1)-/2],
2ikr
yi2jir) = A-sllé^-v-
2JV ' 2ikr 21
1 \ nlJ A\kr-(J+l)ii/2} S22 e —
W2 j(r) = —
2ikr
e
—i[kr—(J+1)n/2]
J (—), kx
ikr
-„k, where S^-7 and S1- are some functions of energy with
(r) « eAetk'r + /aa' ~ £ A' • (6) \siJ\ <; l and \S(/\ < 1. Due to angular momentum
^^EPHA^ OH3HKA TOM 77 № 9 2014
x
X
X
8
conservation, only the terms with J = 1 and L = = J ± 1 (i.e., L = 0,2) contribute to the matrix element (4). Then the amplitude (4) can be written as
T{ß = 72 lim
^ r—t 0
GS eß -
(11
jeß A-3(V-eM)V d 6M2
4>U r) = [u{Kr)ex + wI1*1(r)V^Y2lx(n)} + Finally we have
GÏ
^¡2M2 \ r2
Thus, in the non-relativistic approximation the ratio GE/GM is independent of the constant G[,
GM
u
11
(0) - ^i(O)
(16)
Note that the electromagnetic interaction is important only in the narrow region where 3 ~ na and the nucleon energy is E = Mj32/2 ~ 0.3 MeV. In this paper we will not consider this narrow region and neglect the electromagnetic interaction in the potential. Then, the amplitude of pp pair production,
T^pJ, and the amplitude of nn pair production, T^J,
(12) have the form
rpl I 7^0
T(p) _ A/i ' \ß A/i
t1 _ t0
_ A/i A/i A/i —
The first term in Gd contains the large factor 6/32, while the second term in Gl is small due to the proton mass M in denominator. Thus, in the non-relativistic approximation the amplitude reads
TI^ = GIs{V2uInm^-e*x)+ (13)
+ u2i(0)[(eM • e*x) - 3(k • e^)(k • e*x)]
where k = k/k. The interpretation of this equation is the following. As a result of re-scattering due to the tensor forces, the pair produced at the origin in s wave has a non-zero amplitude to transfer to d wave.
4. CROSS SECTION AND SACHS FORM FACTORS
The cross section corresponding to the amplitude (4) has the form in the center-of-mass frame (see, e.g., [26])
pa2
V2 ' A/i V2 '
The contribution of the isospin I to the total cross section of the nucleon pair production reads
2n3a2
a
Q2
G |2 [lull (0)|2 + |u2i (0)|2]. (17)
Thus, to describe the energy dependence of the ratio GE/Gim and the cross section a1 in the non-relativistic approximation, it is necessary to know the functions «{1(0) and u21 (0).
Let us write the hamiltonian H1 for the isospin I as follows:
- P2
m+ v° (-r)ÔL0 + V2 (-r)ÔL2 + V^Sl2'
(18)
S12 = 6(S • n)2 - 4,
da dû
4Q2
where S is the spin operator for the spin-one system of produced pair, (—p2) is the radial part of the Laplace operator, and L denotes the orbital angular momentum. Then the radial wave functions uIn1 and
Gm(Q2)|2(1 + cos2 9) + (14) wn 1, n = 12, satisfy the equations
+
4M2
Ge (Q2)|2 sin2 9
M
X + Vx = 2Ex,
(19)
Here, 9 is the angle between the electron (positron) momentum P and the momentum of the final particle k. In terms of the "dressed" form factors Gl and Gd the electromagnetic Sachs form factors, corresponding to the contribution of the amplitude with the isospin I, have the form
V
Vl
-2V2 Vi \
-2^2Vi Vi-2Vi
X =
u
nl
w
n1
The asymptotic form of the solutions at large distances is, Eq. (10),
GIM = G1S+ = G>Î!(0) + -^21(0)], (15)
2 M R2
-qGE = Gl - ^Gl = GM.i0) - 0)].
ß
2
1
u1i(r) =
1
2ikr
all ikr -ikr S11 e — e
1
2ikr'
(20)
,1 /m\ _ ^ qII „ikr
2ikr 21 '
u21 (r) =
HŒPHAfl OH3HKA TOM 77 № 9 2014
l
a, pb 1000 |-
800
600
400
200
0
1.90 1.95 2.00 2.05 2.10 2.15 2.20
2E, GeV
Fig. 1. Calculated isoscalar cross section normalized to the data at the third point.
\geЮМ \ 2.0
a, pb
8
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