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The production of the exotic atoms π+π−, K+π− and K+K−
NUCLEAR PHYSICS A
Nuclear Physics A562 (1993) 446-460 North-Holland
The production of the exotic atoms X+X-, K+ z- and K +KS. Wycech’ Soltan Institute for Nuclear Studies, Warsaw, Poland
A.M. Green 2 Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland Received 23 February 1993 (Revised 4 May 1993) Abstract: Estimates are made of the signals to be expected in the production of the exotic atoms Pionium (n+n-), Kaonium (K+K-) and also K+n- in pp, pd and e+e- reactions. Such experiments are now being undertaken or contemplated at CELSIUS, CERN, COSY, Indiana and SATURNE.
1. Introduction
Two-meson systems (M1M2) are of great theoretical interest since they are the simplest involving the interaction between two hadrons. For example, at a microscopic level, MI M2 is thought to be well described in terms of quarks as (qq) (qq) - the next degree of quark complication after that of a single meson (44) or a baryon (qqq ). Also at low energies some two-meson systems exhibit special symmetries such as chiral symmetry in the case of two pions. In spite of this theoretical interest, the amount of available experimental data is limited to the following. (a) In the nn system, phase shifts are known upto an energy of ~1.5 GeV in the centre-of-mass as
- extracted from the final state interaction
in, for example, reactions such
[II aN + (na)N.
In addition, the low energy parameters (i.e. scattering lengths and effective ranges) can also be studied by the Ke4 decays e.g. K- + a+ a-ei?. The standard compilation is that of ref. [ 21 with a more recent review being ref. [ 31. (b) In the Kn system, phase shifts are known upto M 1.SGeV [ 41. For a recent review of the theoretical and experimental status see ref. [ 5 1. (c) In the KR system very little is known experimentally, the main source of infor+ KK see refs. [ 5-71 for reviews of the current mation being the indirect process 717~ theoretical and experimental status. ’ E-mail address: [email protected] 2 E-mail address: greenefinuhcb 0375-9474/93/$06.00
@ 1993-Elsevier
Science Publishers B.V. All rights reserved
447
S. Wycech, A.M. Green /Exotic atoms So far the study of two-meson
atomic states has not been feasible experimentally.
How-
ever, there is a hint that they may be formed in high-energy proton-nucleus collisions [ 81. An improved version of this particular experiment is now being proposed at CERN [ 91. In principle, dominated
the energy levels and, in particular, by the Coulomb
interaction,
the lifetimes of such systems, even though
could give valuable information
about the much
shorter range strong interaction. For anti-protonic atoms @A), this field has, for a long time, been the object of much research both theoretically and experimentally [ IO,111, and has successfully produced constraints on the basic F-nucleon interaction. Recently, proposals have been made [ 12,151 to produce pionium tion p+d-
3He + (~+a-
horn.
(n+ rr- ) in the reac(1.1)
It is mainly this experiment that will be discussed in some detail in this article. However, the model is easily extended to other similar reactions - for example - the process pp + ppn+n(at threshold) recently proposed at the COSY-Julich synchroton ring [ 131. In addition, the formalism developed for these reactions is also applicable to other exotic atoms such as K+K- or K+a-. These atoms may also be accessible in the near future-for example, there are indications of a signal at the KR threshold in the recent pd + 3HeX at Saturne [ 141. In section 2 some basic features of exotic atoms are given. In section 3, the coalescence model is introduced to estimate the relative rate of atom (A4+M-) production compared with the uncorrelated pair production MiA42. For the reaction in eq. (l.l), this is basically the ratio of the phase space factors L2/L3, where L* is the two-body phase space L*[ 3He, (X+X-) atom] and L3 is the three body-phase space L3 [ 3He, an]. Each of these is also weighted by an appropriate production amplitude for the (IC+X-)~~~,,, and uncorrelated XX. These two amplitudes are derived in section 4 and the atomic signal in the cross-section for the recoil ‘He energy distribution is calculated for that kinematics which probes the energy of the expected atom. In section 5 the formation of pionium and kaonium in 2p-states is discussed and some conclusions are given in section 6.
2. Basic features of strange atoms Someofthebasicpropertiesof (rr+lr-), (K-n+) and (K-K+) atomsaresummarised in Table 1. Pionium has been discussed earlier in a number of papers [ 161 and experimental searches have either been attempted [8] or proposed [ 121. In this paper some additional properties of the simplest strange atoms are also calculated. Table 1 collects together the Bohr radii ( 1/,~a) and some binding energies [pa*/2 (n + I + I)* ] of these atoms, where p is the appropriate reduced mass. However, the quantity of special interest is the lifetime, since it is connected to meson-meson scattering lengths by the relation given in eq. (A.7) as r = -4p2a31mA,,.
(2.1)
448
S. Wycech, A.M. Green /Exotic atoms TABLET
Some properties of the exotic atoms 71+n - , K + 71- and K + K -.
n+n-
K+n-
K+K-
Bohr radius/fm
387.46
248.51
109.55
El,lkeV -%,lkeV Basic decay from 1s state Width (ev) Lifetime (1O-15 s)
-1.86
-2.90
-6.57
-0.46
-0.72
-1.64
RwJ
KY
7171and t/71
0.24 2.72
0.18 3.59
639.4 0.0010
In the rr+rr- and K+n- cases, Im A. is generated by isospin symmetry breaking through the meson mass differences within the rr and K multiplets. It may be expressed in terms of the atomic decay momentum and the combinations of the isospin scattering lengths given in Table A.1 and eq. (A.2) in appendix 1. The lifetimes given in Table 1 are calculated using ac, = -0.26 a2 = 0.02 for the na-case, al = -0.22 a3 = 0.06 for the Kn-case. The units are l/m, and the signs of the a’s are opposite to those used in ref. [2], the source of these numbers. At this stage there is no direct data on the K+K- scattering lengths. This system is half isospin 0 and half isospin 1. The I = 0 component is strongly coupled to the fo (975 ) meson and decays into pions [ 6 1. Simple estimates based on the assumption that fs is a KK quasibound state yield Im A0 FZ- 1 fm and similar estimates follow from phenomenological analyses [ 71. The Al length is not known, but it could well be equally large due to nq coupling via the a0 (983) resonance. For illustration, the width in Table 1 is, therefore, calculated using K,, = 0.328, Kc, = 2.53 and Kc0 = 0.948 all in fm. - reasonable values suggested in ref. [ 71. These are obtained following ref. [ 71 by fixing the 71-a resonance
at the fo energy (which yields Koo) and, from eq. (A.2), result
in Arc = 1.48 - il.33 fm. Lifetimes of the atomic Is-states are too short to be measured directly by electronic methods. Another way to possibly measure such lifetimes is to observe mesonic decay modes and compare it with yy decays, which are exactly calculable. The main problem with this method is the large background due to the direct production of the decay mesons. This problem is discussed in the next section. Another method would be to create relativistic atoms or atoms in 2p-states. This latter possibility is discussed in section 5.
3. The coalescence model The production of atoms in the coalescence model is assumed to be given essentially by the atomic wave function t,u. Let the M+Mmeson-pair production amplitude be
449
S. Wjtcech,A.M. Green /Exotic atoms F(P,q,pi,p,),
where P and g are the total and relative pair momenta,
p/ are the momenta
of other particles
produce atoms in a state y is then
F(P,Pi,Pf) =
while pi and
in the initial and final states. The amplitude
s
F(4)
F(P,q>p,,pf) (2ndc
to
(3.1)
In general, the momenta involved in w (a ) are small on the usual short range scale of meson production mechanisms. Hence for s-states
and so the amplitude normalisation
is related to the atomic wave function
constant.
The relative
at zero range i.e. with the
rate of atom production/pair
production
is then
given by phase space as W(atom) W(pair)
1 IF(q = 0)12dL”+‘I~(0)12/(2~~+~=
s IF(q)12dL”+2
1
3
(3.3)
where, for example, L ‘+’ denotes the combined phase space for the atom and the other (~1) final state particles. The reduced mass factor pM+ M- is a remnant of the relativistic phase space element, which for a single particle is dp/ [2E ( 2n 131. However, as shown in appendix 2, eq. (3.3) has its limitations, since the atom is unstable. Using the above expressions for the reaction in eq. ( 1. I f , the coalescence model yields relative pionium production rates of 2 x 10P3 at E - J&&old M IMeV dropping to 8 x 10w5 at 10 MeV. Relative rates for KK atom production are higher, mainly because of the (mK/mn)2 M IO factor that enters from the 1t,u(0) 1*/p term in eq. (3.3). The rates are small due to the large atomic radii and are clearly unmeasurable. However, some improvement may be obtained with subthreshold experiments - as described in the next section. Equation (3.3) is useful also in calculations of the atomic production via decay processes. An example of such a reaction is the decay Ki -+ rc’n+ndiscussed in the second of references [ 161. The rate 5.1 x IO-’ given there has been obtained with some additional factors : (a) a 5.59% branching ratio for the three charged pion decay channel, (bf eq. (3.3) is used with iz = I, (c) an additional factor of 2 comes from the two rr+‘s in the final state, (d) a summation over all nS states is done . With the rate of K+K- pairs of 2 x lO”/year expected at DAQNE, this estimate yields 2 x lo4 pionia produced per year. Similar calculations give a rate 3.9 x lo-’ for the pionium atoms produced per decay of the eta meson. For the three body decays of D-mesons a branching ratio of (pionium / K-x+x) equal to 5.4 x lo-* is obtained. With a 10% branching to this K-n’nchannel and a rate of 2 x 10’ D-mesons per year expected at Tau-factories [ 17 1, one obtains less than one event per year. The number of K-z+ atoms would be about three times larger. Four body decays offer a higher chance of 10M6 to form the K-K+ meson per K-z+~+n~ decay mode.
450
S. Wyceeh, A.M. Green /Exotic atoms
0 0
t
+ k=
Cl dl Fig. I. The basic amplitudes for atom formation and background above and below the R+~Lthreshold: (a) Direct atom; (b) Fano resonance; (c) Direct zone background; (d) Charged background.
4. Subthreshold production In this section is discussed the atomic signal expected to occur on the 3He recoil energy distribution in the p + d reaction of eq. ( 1.1). The notation is adapted to the pionium case, but the formalism is applicable to kaonium and to other cases including heavier targets. To be more specific an experiment is discussed for creating pionium below the X+X- production
threshold p+d
[ 121 by means of the process -+
“He+
(7~+n-),,
+
3He -I- n’l~’
---t
3He + YV
The pionium decays predominantly into Z’X’ and these decaying pions would be observed on the background of neutral pions from the direct n”zo production. The aim of this section is to calculate the atomic signal in the non0 channel as reflected in the 3He recoil. In a model calculation this signal, relative to the overall rate, is evaluated. The basic atom fo~ation amplitudes are given in Figs. la and lb. These will interfere with the large background of non0 production from the direct process of Fig. Ic. In the above type of recoil experiment, if performed above the X+X- threshold, some background will be due also to the non-interfering processes of fig. Id. Magnitudes of the signal and the background are discussed below. These are also used to justify the coalescence model and to find its limitations. At first the two-meson subsystem in Fig. 1 is discussed. Later on, this is extended to the three body situation. Details of the calculations as well as some definitions are collected in appendix 1. As will be shown below, the atom is formed predominantly via the charged intermediate state of fig la i.e. as an intermediate state in the reaction niz-+ n”xo. The relevant amplitude is the product of three factors Fc.G:. ffCT,,, where FE
S. Wycech, A.M. Green /Exotic
is an amplitude
for the production
451
atoms
of charged mesons and G,’ is a coulomb propagator
for
the closed channel (c) that - close to the 1s-atomic state - is dominated by 1t,u)( WI/ (E ~0). The last factor ( fc,,) is a transition matrix, which for reasons of normalisation, is put into a pseudopotential form ? = (2n/,~)6 (r) T. The detailed properties and the parametrisation of T are given in appendix 1, where it is shown that close to the atomic state pole T,, is T where R = - (2n/p)j~ production amplitude F,, = F,.G,c&,
A CoNN 1 + A,,&
(O)]*. Collecting
together the three factors results in the atomic
drF,(r)v(r).
=
(4.1)
- EC))’
2. P
I
Ac0 dr’ w(r’)6(+), E - EO+ RA,, s
where A,, is the scattering length in the rr+rr- channel responsible for the atomic decay into the 7r”rro channel.
and AC0 is the transition The distances
involved
(4.2) length in F(r)
are much shorter than the atomic radii and so the first integral reduces to ~(0) . F,, where FC = SdrF,(r) is now the n+a- production amplitude. The atomic formation amplitude becomes F,, = -FC.
and displays the standard Breit-Wigner case of the other production amplitude an analogous calculation one obtains F,,
= Fo
R&o ( E - EO + R&c >
(4.3)
shape for the line. This does not happen in the due to the rr”7co --t rc”7roreaction of Fig. lb. By
G, . T;,, eiq,.r
=I
drF,(r)-
x”[
1 +%K,,
r
E - 60 + R(K,, (
- K&/K,,) E - 60 + RA,,
I
’
(4.4)
where now F,, (I) is an amplitude for noa production to be integrated over the intermediate state propagator. The terms in brackets come from TO, expressed by eq. (A.21 and expanded around the atomic pole with the help of eq. (A.61 and some simple but lengthy algebra. The terms in the round brackets constitute another atomic signal that now displays a typical Fano resonance consisting of a peak and a zero in the amplitude [21]. The position of the peak is given by the atomic level energy and the separation between the zero and the peak is a measure of the K,, coupling strength relative to the K,, and KC, strengths. The pionium signals in these two channels are plotted in Fig. 2a. They are clearly too narrow to be measured directly at the present time. A similar structure is also predicted for the (K+n-)-atom. However, in the K?? system - see Table 1 and Fig. 2b - the peaks are expected to have widths of the order of keV and may be even broader in higher Z systems. Such reactions with an initial proton beam of 3 keV resolution could possibly permit a direct measurement of this shape [ 121. The contribution of the elastic rr”rro scattering to pionium production from eq. (4.4) is small. One reason is the (K,,/r) factor, which is z 0.1. Another is that the rr + 7c- produc-
S. Wycech, A.M. Green /Exotic atoms
452
0.1
/
I -LO
-3.0 -2.0 -1.0
0
10
2.0 3.0 E-E.
LO > (eV)
A 3
-
5, 3’ ,
‘\
XI0 ‘\ ._
-05
0
0.5
10
__-__ 1.5
z-
Fig. 2. Shape of the Is-atomic state contribution to the background amplitude. The moduli squared of the round bracket terms (...) in eqs. (4.3,4.4) are plotted - the dashed line corresponds to eq. (4.3) and the solid line to eq. (4.4). (a) The effect of pionium. The aok contribution of eq. (4.4) exhibits a Fano resonance with its characteristic zero. Here the values of a0 and a2 quoted in section 2 are used. N.B. The semi-log scale . (b) The effect of kaonium. For illustration, reasonable values of K,, = 0.328, KC, = 2.53 and KC0 = 0.948 all in fm are used - see ref. [7]. For clarity, the dashed line has been multiplied by a factor of 10.
S. Wycech, A.M. Green /Exotic atoms tion cross-section pion production
453
is larger by a factor of about 3-4. The lower estimate comes from two cross-sections
at pl&, = 2.23 GeV/c [ 201. In fact aspI& decreases this fac-
tor increases. The upper estimate comes from the coupling ratio G(pnn+)/G(ppn’) = fi [2]. This is a fortunate conclusion, since the F,, amplitude depends on the off-shell structure in F. (r). However, from Fig. lc the direct production of n”zo described by simply F. contributes both background and interference terms to the atomic amplitude F,,. To study these effects one needs 3-body kinematics appropriate for the final states in Fig. 1. Let the energy be measured final state in the cm system is
relative to the the 7t+7c- threshold,
E, = q,2/2p -A
which for the
+ ER(~),
(4.5)
where E,(P) = P2/2pR is the ‘He recoil energy and A = 2 (m, - mo) is the threshold mass difference. Neglecting the F,, contribution from eq. (4.4), since it is expected to be only a 5% correction to Fco, the formation probabilities are given by the integration IFco + F,12dL3
w= =
J{lFo12+
[I&o +
El2 -
IFo121}dL3
= w, + w,
(4.6)
over the three body final phase space. This is done in appendix 2. The second term on the RHS gives the probability of atom formation WA, whereas the first term is the z”zo background. Integration over the atomic line and the phase space yields the total atom formation probability WA =
w
lFc12 L2 . (1 + 2q01m;;;co))
where L2 is the phase space element
L2 = P~/[4~(2m
,
(4.7)
+ A43)] with P --) PA =
EC) being the recoil momentum at the atom production energy and M3 is the mass of 3He. The last factor in eq. (4.7), where 1 = Fo/Fc, is due to the atomic $pR
(1~1 +
background interference. This term is an improvement over the coalescence model estimate given by eq. (3.3)) but the correction is negligible (< 1Oe3 ) in the pionium case. Similarly in the K-K+ case ImA,, z low4 - again a negligible correction. However, it may be of the order 1 in the K?? case. In general, the experimental
resolution
will be
such that many atomic states will fall inside this resolution.
In this case the above factor
lw(0)12 is replaced by C,, Iw,(0)12, where n is the principal lead to an enhancement that is, at most, 20%.
quantum
The recoil energy distribution less and is given by &(ER)
due to direct n”zo production
number.
This can
requires one integration
= [FoI~P(ER)J~~U((EC + A) - ER)/[16a3(2m + M3)1.
(4.8)
A similar expression holds for the recoil due to n+n- production, if experiments are performed above that threshold. The necessary change is F. + F,, EC + EC - A. The latter effect of the II+W- channel may be minimised by making the experiment close to
454 the threshold
S. Wycech, A.M. Green /Exotic atoms [ 121. However, in the energy range of interest, the rr”zo production
a constant but large background. WA/ [ %(ER)dER] [ 121. As discussed
With the energy fixed at the n+z-
makes
threshold one obtains
x IFc/Fo12/200, where a recoil energy resolution of 70 keV was used above, the enhancement due to IFc/Fo12 is expected to be at least a
factor of 3. In spite of this, one should expect a sizeable background
in the subthreshold
pionium production. Only if the resolution can be reduced substantially value of 70 keV will the atomic state signal become significant.
from the stated
Kaonium production is likely to be dominated by the fo (975) and a0 (983) resonances. Indeed, a peak is observed at Saturne in the reaction p (d, 3He)X around the Kx threshold [ 141. The K+ K- atom production rate and the background recoil energy distribution due to rcrc or ?~a are given by equations analogous to eqs. (4.7), (4.8), but with relativistic phase space. Now one obtains WAD[ WB(ER)AER] z lFc/Fo12/225, where Fc/Fo is the ratio of resonant decays into KK versus nrr or r]n. This factor is available in the f0(975), Z = 0 case and IFc/Foj2 = gK/g= = 2 - see ref. [6]. Unfortunately, there is no experimental estimate of the a0 (983), Z = 1 coupling to KK relative to the r~rccoupling. However, in some theoretical models, a ratio similar to that in the fs case is suggested [ 231. In view of the strong KK, m coupling to the resonant states, the resultant scattering lengths and K-matrix elements could be as large as l-2 fm. This generates structure in the atomic line with a width of the order keV. A full set of K-matrix elements is not yet available and so a more detailed structure cannot be calculated. In particular, one cannot estimate with reliable parameters the magnitude of the Fano type of structure already predicted for pionium in Fig. 2a. For illustration, in Fig. 2b reasonable values for the K-matrix suggested in ref. [7] are used. However, it should be emphasised that the detailed structure seen in this figure is very dependent on the actual numbers used in the K-matrix . The interference effect in eq. (4.7) may be rather large, since 12qoAcoI x 4 is expected. The K+K- case is rather similar to that of pionium in the sense that the threshold of the open channel K”no is close to the energy of the atom. However, the major difference is that a suitable mechanism for producing K+n- or K”no is not known - unlike: (a) Pionium,
which is part of the well known two-pion
production
present in many
reactions e.g. as in eq. ( 1.1)) or (b) Kaonium, which can be produced via the fo (975) and uo (983) resonances. One possibility is to produce Ka in the reaction pp + pC+n-K+ - but this has a very small cross-section
e.g. 0.019mb at 3.47 GeV/c
[20].
5. The production of exotic atoms in Zp-states The formation of pionium in a 2p-state would make its detection much easier. The main decay mode is the X-ray transition to the 1s-state with a rate of 0.86 x 10”s-’ [ 241, This time is long enough for the atom to leave the target and so facilitate measurements of its decay modes. However, its production rate is very slow, since it involves an additional factor of cy2. Consider that a vector meson, e.g. p or 4, produced by a reaction such as
455
S. Wycech, A.M. Green /Exotic atoms
pd +
P 3He decays into pionium
or kaonium.
Let the meson production
amplitude
FE +q, where Q is a unit vector related to the vector particle and q is the momentum meson-meson
centre-of-mass.
2 yields an atom formation
Employing
a procedure
rate in a coalescence
analogous
be
in the
to that used in section
model of the form
(5.1) Here, w’ is the radial derivative of the p-wave atomic function at the origin - ,/w for a 2p-state - and (...) denotes the average of q2 over the final phase space. Close to the rr+z- production threshold this ratio follows ETsi2 and, in principle, may be quite large. However, in practice, vector meson production may be minute. Estimates suggest that, at EC = 10 MeV, the ratio WA/ WB(ER)AER is of the order lo-‘. Similar calculations may be carried out for estimating the chances to produce kaonium in the reaction e+e- --f 4 + (K+K-)2,. Right at the (K+K-) atomic energy, one has WA/[ WB(E)AE] M 0.4 x 10e6 using the DA@NE resolution of 4 x 10m4. This reaction is 32 MeV away from the 4 energy, which gives another l/500 reduction factor. With the expected rate of 1044/s, this would produce about one kaoniumlday.
6. Conclusion In this article, signals indicating the presence of the exotic atoms pionium (~+a-), kaonium (K+K- ) and K+n- are estimated in various reactions. The emphasis has been on the production of pionium in the reaction p + d + ‘He + (M+M-)atom, but the formalism proposed applies equally well to kaonium and the K+n- atom. Furthermore, following the suggestion in ref. [ 121, the signal in the 3He recoil energy distribution is expressed as the ratio of probabilities WA (AtOm)/Wg (Background)dEa, where AER is the recoil energy resolution. In the pionium case, since this resolution is 70 keV compared with the atomic line width of a few eV, the signal is only z 1% of the n”rro background. Only if the resolution can be decreased by an order of magnitude could such an atomic signal become measureable. The situation for seeing the K+n- atom is somewhat similar to that of pionium with the additional complication that the basic Ka states are more difficult to generate i.e. in eqs. (4.3,4.4), the FC,O(Kz) factors are much smaller than the corresponding FC,O(m) factors. Possibly it is kaonium that will be most easy to observe, since this has a width that is of the order 1 keV compared with fractions of an eV for pionium and the K’natom - see Table A. 1. In addition, the basic mechanism for creating this atom is enhanced, since it appears via the fo (975 1 and a0 (983) resonances. In comparison, the production of, for example, the p-meson would generate pionium only in a Zp-wave and this would be expected to have a very small probability. It is possible that the peaks seen at the Ki? threshold in the p (d, ‘He)X reaction at Saturne are already partly due to kaonium formation. Unfortunately, unlike the pionium and K+n- systems, the theoretical esti-
456
S.
Wycech, A.M. Green /Exotic atoms
mates based on eqs. (4.7,4.8) can not yet be made, since they require a knowledge of the K-matrix elements KC,(K+K- -+ K+K-),Koo(wa -+ mr) and Kco(K+K- --+m). For pionium
and K+K-,
the corresponding
K-matrix
elements are sufficiently
show that their effect - reflected by the interference
well known to
term in eq. (4.7) - represents
only a
low3 correction to the basic coalescence model. On the other hand, for the Ki? system, all that can be said at present is that ImA,, z - 1 fm, which is sufficient to indicate that the interference term in eq. (4.7) could be significant. Similarly, the effect of the K+Katom in the background production of Fig. lb [i.e. eq. (4.4) ] could also be significant, if Re A,, and also the K-matrix elements are sufficiently large. However, it is possible that these parameters could conspire to give a Fano zero that greatly reduces the overall effect. As outlined in section 5, the production of atoms in the 2p-state, although desirable for observing such atoms when once generated, is very unlikely. Similarly, the direct production via e+e- + 4 + (K+ K- )zP seems also to be impractical. So far the discussion has concentrated on p + d reactions with the conclusion that the signals for atom production will be very difficult to detect compared with the ever present background. In pp + ppn+nreactions this problem is further complicated by the need to consider 3 and 4 phase space factors i.e. n = 2 in eq. (3.3). The relative atomic production rates W (atom)/ W (pair) in the pp ---f ppn+nand KK or in the pp --t K-n+pC+ reactions are now higher . Eq. (3.3) produces a rate of 1.26 x 10-5[2~,,,,+M-/(E - E(threshold))3/2] that is more than lo-* at 1 MeV or 10-4-10-3 at 10 MeV above the thresholds . However, it would then be necessary to measure the energies (El, E2) of both protons and to concentrate on the events with Ei + E2 tuned to pick up the atomic signal. The ratio of the atomic formation probability to that of uncorrelated mesons would now have the form WA/ WB (E,, , EP2) AE,, AED i.e. with the resolution appearing twice. This is clearly a much more demanding experiment than the p + d proposal.
One of the authors
(S.W.) wishes to acknowledge
the hospitality
of the Research In-
stitute for Theoretical Physics, Helsinki, where part of this work was carried out. In addition, he wishes to acknowledge the receipt of the KBN grant number 2p 302 14004, which partially covered the expenses incurred in this collaboration. Also the authors wish to thank Dr. B. Hoistad , Dr. H. Nann, Dr. W. Oelert and Dr. J. Stepaniak for useful communications
and discussions.
Appendix A
SCATTERING PARAMETERS As there exists conflicting sign conventions for the low energy expansions a few basic definitions seems appropriate. The scattering matrix is normalised according to the Lippman-Schwinger equation ? = ^v + v (EC - go)-’ ? that leads to a low energy
451
S. Wycech, A.M. Green /Exotic atoms TABLE
A.1
The isospin structure of the K-matrix Channel c
n+7l-
K+Z-
K+K-
Channel o
7rW
K%O
nn, rln
KC,
(a2 + 2a0)/3
(a3 + %)/3
00 + AI )/2
KM
JZ(a2
vQ(a3 -al
K 00
(2a2
+
a0)/3
@a3 + 4
a0)/3
)/3
J/3
parametrization 1/T = 1/a + iq, where a is a scattering length and q is a relative momentum. Phenomenological scattering lengths are used and an off-shell extension of this model is obtained by a pseudopotential ? = (27r/p )6 (r) T that may generate a multiple scattering expansion. Scattering lengths defined in this way a = - tan S/q differ in sign from those defined in the compilation of ref. [ 21. Consider s-wave scattering in a system consisting of two channels c and o. The Kmatrix, i.e. a generalisation of the scattering length, and the T-matrix that follows from it are denoted by
z=
(:::I)
and
T=
(
Aoo/ (1 + iqOAOO)AC,/ ( 1 + iqcAcc)
(1 + iwL)
A,,/
A,,/ ( 1 + iq,A,,)
64.1) ’
where qO,Care the centre-of-mass momenta of the two mesons in the two channels o, c. The channel scattering lengths Aii are expressed in terms of the K-matrix elements, via the solution of T = K - iKqT, by [ 181 A,, = Kcc - iK~OqO/(l + iMGO), &
= K,,l(
A,, = &,
1 + iqJGo ),
- iKkk/(l
+ iqcKcc).
64.2)
These equations form a basis in which to describe two channel scattering in terms of the three parameters of the K-matrix. The systems of interest, like nn or Kn, are believed to display a high degree of isospin symmetry in their interactions. The K-matrix is given by two real parameters referring to isospin 0, 2 in the X+X- case and l/2, 312 in the K+n- case. The isospin structure of the K-matrix is given in Table A. 1. The case of K +K- is more complicated due to the mass difference of the K + K- and ??K” and also the presence of the two decay channels 7~71(I = 0) and ~7c (I = 1). Also notice the unitarity condition, which follows from eqs. (A.2), Im-4,
= -I&I*%.
64.3)
458
S. Wpxch, A.M. GreenI Exotic atoms
Isospin symmetry
is broken by coulomb interactions
and meson mass differences.
The
first allow for atomic binding and the second induce decays of these atomic systems. The channels
in Table A. 1 are, accordingly,
named closed (c) and open (0).
To describe atomic systems produced eqs. (A. 1), (A.2). The standard
in collisions,
these effects.have
to be built into
way is to separate long range effects that enter coulomb
propagators G” and modifications at short ranges entering “coulomb corrected” scattering lengths. The propagators that describe coulomb and short ranged interactions are given by Gc + GCTGc. As a consequence,
the scattering
fii = f”a,
is described by amplitudes
+ e’@icTEjci
of the form
eioj,
(A.4)
where f”, CTand c are the coulomb scattering amplitudes, phases and penetration factors. The latter arise in asymptotic solutions only and are of no concern in the present problem. The coulomb corrections to T’ arise in terms of known coulomb functions that should replace iq in eqs. (A.l),
(A.2): iq -+ f = 2yh + iqc2,
wherec2
= 2aq/[exp(2nq)
- i]
and
h = $[y(is)
(A.5) + v(-iv)]
- $lnq!
Also y = ZZ’ap and v = y/q - see for example ref. [22]. All these functions are diagonal in channel indices, which were dropped, however. It is easy to check that for neutral particles (2 = 2’ = 0) the f = iq. Now, the atomic states created in the intermediate states of nuclear reactions are generated by poies in f of eq. (AS). The expansion off around this pole yields [ 191 f=
R
whereR = -$]v(0)12
ETq
= -2h203,
co is the pure coulomb energy of an s-level and ECis the energy distance from the M+Mthreshold. As a first application of eq. (A.S), the position of the atomic pole in T,, of eq. (A. 1) is found. One obtains immediately the atomic energy shifted by strong interactions
E -
p
=
Eo
+
~:W(0)12Acc = 21U2ar3&, Eo +
(A.7)
which is a well known result relating level shifts and widths to the scattering length. The width expressed by K-matrix
elements
is
(A.81
Appendix B B. 1. PHASE SPACE FACTORS AND INTEGRALS
In this appendix are calculated phase space factors and integrals over the atomic line. Non-relativistic energies are used for the 3Herr%o final state. For a final state containing
4.59
S. Wycech, A.M. Green /Exotic atoms three particles
(B.1) Going over to energy variables recoil energy ER(P)
E4 = q2/2p in the rr7z centre-of-mass
= P2/2pR with ,UR = 2mM3/(2m
system and the
+ M3), eq. (B.l) becomes
dL3 = 6 (Eo - Eg - ER ) Pq dEsdEq/C, where c = 16a3 (2m + M3). Thus the background energy distribution %(ER)
=
rr”7coemission
(B.2) generates
the recoil
given by
l&l2
6(Eo - E4 - ER)PqdEq/c
= IF,j2P~~lc.
(B.3)
J
Here EO is the energy related to the x0x0 threshold. Now the atomic formation probability the energy related to the rc+rcis calculated and EC = E. - 2 (m+ - mo) = EO - A t.e. . threshold is more appropriate for use. The atomic signal from eq. (4.6) using eq. (4.3) becomes %
IRAeo - 112-1 E4 - A - e. + RAcc
=
If’12
=
lfil*J [IEo _
Jr
I
ER
yf$, +
R&
-
1s(Eo-E,-En)PqdERdEq/C II2
-
11f’qdER/C,
U-3.4)
where 1 = F,/F, and q = J2,u ( E. - ER). Since the atomic line is so narrow, it is assumed that q is constant over the energy range of interest. The integration over the atomic line is now performed using the relation SdEs/[ (ER - E)~ + (r/2j2] = 27c/T and also by taking into account the unitarity condition of eq. (A.3) and replacing R by eq. (A.6). This results in
w, = IFcl’w
(1+ 2qo(11$Aco)) [4n(22+M )I,
(B.5)
3
where the first term comes from the line and the second represents background.
interference
with the
The square bracket term is the L2 phase space factor for the atom and the
‘He system, PA = ,/m and qO is the relative momentum
is the recoil momentum
at the atom production
point
of the won0 emitted in the atomic decays.
References [ 1] B.R. Martin, D. Morgan and G. Shaw, Pion-pion interactions [2] [ 31 [4] [5] [6]
in particle physics (Academic Press, 1976) 0. Dumbrajs et al., Nucl. Phys. 216 (1983) 277 W. Ochs, Results on Low energy ax Scattering, MPI-Ph/Ph 9 1-35 a contribution to nN News Letter (eds. G. Hohler, W. Kluge and B.M.K. Nefkens) D. Aston et al., Nucl. Phys. B296 (1988) 293 D. Lohse, J.W. Durso, K. Holinde and J. Speth, Nucl. Phys. A516 (1990) 513 T.A. Armstrong et al., Z. Phys. C51 (1991) 351
460
S. Wycech, A.M. Green /Exotic atoms [7] F. Cannata, J.P. Dedonder [8] [9]
[lo] [ 111 [ 121 [ 131 [ 141 [ 151 [ 161 [ 171 [ 181 [ 191 [20] [21] [22] [23] [24]
and L. Lesniak, Phys. Lett. B207 (1988) 115; Z. Physik A343 (1992) 451 L.L. Nemenov, Sov.J. Nucl. Phys. 41 (1985) 629; L.G. Afanasyev et al., Phys. Lett. B255 ( 199 1) 146 Bern-Dubna Collab., Lifetime measurement of x+x-atoms to test low energy QCD predictions, Letter of Intent SPSLC92-44/I 191. A.M. Green and S. Wycech, Nucl. Phys. A467 (1987) 744 C. Batty, Nucl. Phys. A508 ( 1990) 89c H. Nann, in Proc. Workshop on Meson Production, interaction and decay (Cracow, Poland, May 1991), Eds. A. Magiera, W. Oelert and E. Grosse (World Scientific 1991), p. 100 W. Oelert, in Proc. Workshop on Meson Production, interaction and decay (Cracow, Poland, May 1991), Eds. A. Magiera, W. Oelert and E. Grosse (World Scientific 1991), p. 199 R. Siebert, in Proc. Workshop on Meson Production, interaction and decay (Cracow, Poland, May 199 1), Eds. A. Magiera, W. Oelert and E. Grosse (World Scientific 199 1 ), p. 166 CELSIUS proposal, Meson production in light ion collisions at CELSIUS - spokesmen: B. Hoistad and T. Johansson S.M. Bilenkij et al., Yad. Phys. 10 (1969) 812 H. Pilkuhn and S. Wycech, Phys. L&t. 76B (1978) 29 J. Kirkby, Physics and design of a tau-charm factory in Spain presented at Workshop in Valencia, Spain, October 199 1 and CERN-PPE/92-30 13 February 1992 H. Pilkuhn, The interaction of hadrons (North Holland, 1967) p.207 J. Carbonell, J.-M. Richard and S. Wycech, Z. Phys. A343 (1992) 325 V. Flaminio, W.G. Moorhead, D.R.O. Morrison and N. Rivoire, HERA-CERN 1984 compilation III U. Fano, Phys. Rev. 121 (1961) 1866 R.G. Newton, Scattering theory of waves and particles (Springer, New York, 1982) J. Weinstein and N. Isgur, Phys. Rev. D43 (1990) 95; A. Bramon and S. Narison, Mod. Phys. Lett. A4 (1989) 1113 0. Dumbrais, Z. Phys. A321 (1985) 297
Nuclear Physics A562 (1993) 446-460 North-Holland
The production of the exotic atoms X+X-, K+ z- and K +KS. Wycech’ Soltan Institute for Nuclear Studies, Warsaw, Poland
A.M. Green 2 Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland Received 23 February 1993 (Revised 4 May 1993) Abstract: Estimates are made of the signals to be expected in the production of the exotic atoms Pionium (n+n-), Kaonium (K+K-) and also K+n- in pp, pd and e+e- reactions. Such experiments are now being undertaken or contemplated at CELSIUS, CERN, COSY, Indiana and SATURNE.
1. Introduction
Two-meson systems (M1M2) are of great theoretical interest since they are the simplest involving the interaction between two hadrons. For example, at a microscopic level, MI M2 is thought to be well described in terms of quarks as (qq) (qq) - the next degree of quark complication after that of a single meson (44) or a baryon (qqq ). Also at low energies some two-meson systems exhibit special symmetries such as chiral symmetry in the case of two pions. In spite of this theoretical interest, the amount of available experimental data is limited to the following. (a) In the nn system, phase shifts are known upto an energy of ~1.5 GeV in the centre-of-mass as
- extracted from the final state interaction
in, for example, reactions such
[II aN + (na)N.
In addition, the low energy parameters (i.e. scattering lengths and effective ranges) can also be studied by the Ke4 decays e.g. K- + a+ a-ei?. The standard compilation is that of ref. [ 21 with a more recent review being ref. [ 31. (b) In the Kn system, phase shifts are known upto M 1.SGeV [ 41. For a recent review of the theoretical and experimental status see ref. [ 5 1. (c) In the KR system very little is known experimentally, the main source of infor+ KK see refs. [ 5-71 for reviews of the current mation being the indirect process 717~ theoretical and experimental status. ’ E-mail address: [email protected] 2 E-mail address: greenefinuhcb 0375-9474/93/$06.00
@ 1993-Elsevier
Science Publishers B.V. All rights reserved
447
S. Wycech, A.M. Green /Exotic atoms So far the study of two-meson
atomic states has not been feasible experimentally.
How-
ever, there is a hint that they may be formed in high-energy proton-nucleus collisions [ 81. An improved version of this particular experiment is now being proposed at CERN [ 91. In principle, dominated
the energy levels and, in particular, by the Coulomb
interaction,
the lifetimes of such systems, even though
could give valuable information
about the much
shorter range strong interaction. For anti-protonic atoms @A), this field has, for a long time, been the object of much research both theoretically and experimentally [ IO,111, and has successfully produced constraints on the basic F-nucleon interaction. Recently, proposals have been made [ 12,151 to produce pionium tion p+d-
3He + (~+a-
horn.
(n+ rr- ) in the reac(1.1)
It is mainly this experiment that will be discussed in some detail in this article. However, the model is easily extended to other similar reactions - for example - the process pp + ppn+n(at threshold) recently proposed at the COSY-Julich synchroton ring [ 131. In addition, the formalism developed for these reactions is also applicable to other exotic atoms such as K+K- or K+a-. These atoms may also be accessible in the near future-for example, there are indications of a signal at the KR threshold in the recent pd + 3HeX at Saturne [ 141. In section 2 some basic features of exotic atoms are given. In section 3, the coalescence model is introduced to estimate the relative rate of atom (A4+M-) production compared with the uncorrelated pair production MiA42. For the reaction in eq. (l.l), this is basically the ratio of the phase space factors L2/L3, where L* is the two-body phase space L*[ 3He, (X+X-) atom] and L3 is the three body-phase space L3 [ 3He, an]. Each of these is also weighted by an appropriate production amplitude for the (IC+X-)~~~,,, and uncorrelated XX. These two amplitudes are derived in section 4 and the atomic signal in the cross-section for the recoil ‘He energy distribution is calculated for that kinematics which probes the energy of the expected atom. In section 5 the formation of pionium and kaonium in 2p-states is discussed and some conclusions are given in section 6.
2. Basic features of strange atoms Someofthebasicpropertiesof (rr+lr-), (K-n+) and (K-K+) atomsaresummarised in Table 1. Pionium has been discussed earlier in a number of papers [ 161 and experimental searches have either been attempted [8] or proposed [ 121. In this paper some additional properties of the simplest strange atoms are also calculated. Table 1 collects together the Bohr radii ( 1/,~a) and some binding energies [pa*/2 (n + I + I)* ] of these atoms, where p is the appropriate reduced mass. However, the quantity of special interest is the lifetime, since it is connected to meson-meson scattering lengths by the relation given in eq. (A.7) as r = -4p2a31mA,,.
(2.1)
448
S. Wycech, A.M. Green /Exotic atoms TABLET
Some properties of the exotic atoms 71+n - , K + 71- and K + K -.
n+n-
K+n-
K+K-
Bohr radius/fm
387.46
248.51
109.55
El,lkeV -%,lkeV Basic decay from 1s state Width (ev) Lifetime (1O-15 s)
-1.86
-2.90
-6.57
-0.46
-0.72
-1.64
RwJ
KY
7171and t/71
0.24 2.72
0.18 3.59
639.4 0.0010
In the rr+rr- and K+n- cases, Im A. is generated by isospin symmetry breaking through the meson mass differences within the rr and K multiplets. It may be expressed in terms of the atomic decay momentum and the combinations of the isospin scattering lengths given in Table A.1 and eq. (A.2) in appendix 1. The lifetimes given in Table 1 are calculated using ac, = -0.26 a2 = 0.02 for the na-case, al = -0.22 a3 = 0.06 for the Kn-case. The units are l/m, and the signs of the a’s are opposite to those used in ref. [2], the source of these numbers. At this stage there is no direct data on the K+K- scattering lengths. This system is half isospin 0 and half isospin 1. The I = 0 component is strongly coupled to the fo (975 ) meson and decays into pions [ 6 1. Simple estimates based on the assumption that fs is a KK quasibound state yield Im A0 FZ- 1 fm and similar estimates follow from phenomenological analyses [ 71. The Al length is not known, but it could well be equally large due to nq coupling via the a0 (983) resonance. For illustration, the width in Table 1 is, therefore, calculated using K,, = 0.328, Kc, = 2.53 and Kc0 = 0.948 all in fm. - reasonable values suggested in ref. [ 71. These are obtained following ref. [ 71 by fixing the 71-a resonance
at the fo energy (which yields Koo) and, from eq. (A.2), result
in Arc = 1.48 - il.33 fm. Lifetimes of the atomic Is-states are too short to be measured directly by electronic methods. Another way to possibly measure such lifetimes is to observe mesonic decay modes and compare it with yy decays, which are exactly calculable. The main problem with this method is the large background due to the direct production of the decay mesons. This problem is discussed in the next section. Another method would be to create relativistic atoms or atoms in 2p-states. This latter possibility is discussed in section 5.
3. The coalescence model The production of atoms in the coalescence model is assumed to be given essentially by the atomic wave function t,u. Let the M+Mmeson-pair production amplitude be
449
S. Wjtcech,A.M. Green /Exotic atoms F(P,q,pi,p,),
where P and g are the total and relative pair momenta,
p/ are the momenta
of other particles
produce atoms in a state y is then
F(P,Pi,Pf) =
while pi and
in the initial and final states. The amplitude
s
F(4)
F(P,q>p,,pf) (2ndc
to
(3.1)
In general, the momenta involved in w (a ) are small on the usual short range scale of meson production mechanisms. Hence for s-states
and so the amplitude normalisation
is related to the atomic wave function
constant.
The relative
at zero range i.e. with the
rate of atom production/pair
production
is then
given by phase space as W(atom) W(pair)
1 IF(q = 0)12dL”+‘I~(0)12/(2~~+~=
s IF(q)12dL”+2
1
3
(3.3)
where, for example, L ‘+’ denotes the combined phase space for the atom and the other (~1) final state particles. The reduced mass factor pM+ M- is a remnant of the relativistic phase space element, which for a single particle is dp/ [2E ( 2n 131. However, as shown in appendix 2, eq. (3.3) has its limitations, since the atom is unstable. Using the above expressions for the reaction in eq. ( 1. I f , the coalescence model yields relative pionium production rates of 2 x 10P3 at E - J&&old M IMeV dropping to 8 x 10w5 at 10 MeV. Relative rates for KK atom production are higher, mainly because of the (mK/mn)2 M IO factor that enters from the 1t,u(0) 1*/p term in eq. (3.3). The rates are small due to the large atomic radii and are clearly unmeasurable. However, some improvement may be obtained with subthreshold experiments - as described in the next section. Equation (3.3) is useful also in calculations of the atomic production via decay processes. An example of such a reaction is the decay Ki -+ rc’n+ndiscussed in the second of references [ 161. The rate 5.1 x IO-’ given there has been obtained with some additional factors : (a) a 5.59% branching ratio for the three charged pion decay channel, (bf eq. (3.3) is used with iz = I, (c) an additional factor of 2 comes from the two rr+‘s in the final state, (d) a summation over all nS states is done . With the rate of K+K- pairs of 2 x lO”/year expected at DAQNE, this estimate yields 2 x lo4 pionia produced per year. Similar calculations give a rate 3.9 x lo-’ for the pionium atoms produced per decay of the eta meson. For the three body decays of D-mesons a branching ratio of (pionium / K-x+x) equal to 5.4 x lo-* is obtained. With a 10% branching to this K-n’nchannel and a rate of 2 x 10’ D-mesons per year expected at Tau-factories [ 17 1, one obtains less than one event per year. The number of K-z+ atoms would be about three times larger. Four body decays offer a higher chance of 10M6 to form the K-K+ meson per K-z+~+n~ decay mode.
450
S. Wyceeh, A.M. Green /Exotic atoms
0 0
t
+ k=
Cl dl Fig. I. The basic amplitudes for atom formation and background above and below the R+~Lthreshold: (a) Direct atom; (b) Fano resonance; (c) Direct zone background; (d) Charged background.
4. Subthreshold production In this section is discussed the atomic signal expected to occur on the 3He recoil energy distribution in the p + d reaction of eq. ( 1.1). The notation is adapted to the pionium case, but the formalism is applicable to kaonium and to other cases including heavier targets. To be more specific an experiment is discussed for creating pionium below the X+X- production
threshold p+d
[ 121 by means of the process -+
“He+
(7~+n-),,
+
3He -I- n’l~’
---t
3He + YV
The pionium decays predominantly into Z’X’ and these decaying pions would be observed on the background of neutral pions from the direct n”zo production. The aim of this section is to calculate the atomic signal in the non0 channel as reflected in the 3He recoil. In a model calculation this signal, relative to the overall rate, is evaluated. The basic atom fo~ation amplitudes are given in Figs. la and lb. These will interfere with the large background of non0 production from the direct process of Fig. Ic. In the above type of recoil experiment, if performed above the X+X- threshold, some background will be due also to the non-interfering processes of fig. Id. Magnitudes of the signal and the background are discussed below. These are also used to justify the coalescence model and to find its limitations. At first the two-meson subsystem in Fig. 1 is discussed. Later on, this is extended to the three body situation. Details of the calculations as well as some definitions are collected in appendix 1. As will be shown below, the atom is formed predominantly via the charged intermediate state of fig la i.e. as an intermediate state in the reaction niz-+ n”xo. The relevant amplitude is the product of three factors Fc.G:. ffCT,,, where FE
S. Wycech, A.M. Green /Exotic
is an amplitude
for the production
451
atoms
of charged mesons and G,’ is a coulomb propagator
for
the closed channel (c) that - close to the 1s-atomic state - is dominated by 1t,u)( WI/ (E ~0). The last factor ( fc,,) is a transition matrix, which for reasons of normalisation, is put into a pseudopotential form ? = (2n/,~)6 (r) T. The detailed properties and the parametrisation of T are given in appendix 1, where it is shown that close to the atomic state pole T,, is T where R = - (2n/p)j~ production amplitude F,, = F,.G,c&,
A CoNN 1 + A,,&
(O)]*. Collecting
together the three factors results in the atomic
drF,(r)v(r).
=
(4.1)
- EC))’
2. P
I
Ac0 dr’ w(r’)6(+), E - EO+ RA,, s
where A,, is the scattering length in the rr+rr- channel responsible for the atomic decay into the 7r”rro channel.
and AC0 is the transition The distances
involved
(4.2) length in F(r)
are much shorter than the atomic radii and so the first integral reduces to ~(0) . F,, where FC = SdrF,(r) is now the n+a- production amplitude. The atomic formation amplitude becomes F,, = -FC.
and displays the standard Breit-Wigner case of the other production amplitude an analogous calculation one obtains F,,
= Fo
R&o ( E - EO + R&c >
(4.3)
shape for the line. This does not happen in the due to the rr”7co --t rc”7roreaction of Fig. lb. By
G, . T;,, eiq,.r
=I
drF,(r)-
x”[
1 +%K,,
r
E - 60 + R(K,, (
- K&/K,,) E - 60 + RA,,
I
’
(4.4)
where now F,, (I) is an amplitude for noa production to be integrated over the intermediate state propagator. The terms in brackets come from TO, expressed by eq. (A.21 and expanded around the atomic pole with the help of eq. (A.61 and some simple but lengthy algebra. The terms in the round brackets constitute another atomic signal that now displays a typical Fano resonance consisting of a peak and a zero in the amplitude [21]. The position of the peak is given by the atomic level energy and the separation between the zero and the peak is a measure of the K,, coupling strength relative to the K,, and KC, strengths. The pionium signals in these two channels are plotted in Fig. 2a. They are clearly too narrow to be measured directly at the present time. A similar structure is also predicted for the (K+n-)-atom. However, in the K?? system - see Table 1 and Fig. 2b - the peaks are expected to have widths of the order of keV and may be even broader in higher Z systems. Such reactions with an initial proton beam of 3 keV resolution could possibly permit a direct measurement of this shape [ 121. The contribution of the elastic rr”rro scattering to pionium production from eq. (4.4) is small. One reason is the (K,,/r) factor, which is z 0.1. Another is that the rr + 7c- produc-
S. Wycech, A.M. Green /Exotic atoms
452
0.1
/
I -LO
-3.0 -2.0 -1.0
0
10
2.0 3.0 E-E.
LO > (eV)
A 3
-
5, 3’ ,
‘\
XI0 ‘\ ._
-05
0
0.5
10
__-__ 1.5
z-
Fig. 2. Shape of the Is-atomic state contribution to the background amplitude. The moduli squared of the round bracket terms (...) in eqs. (4.3,4.4) are plotted - the dashed line corresponds to eq. (4.3) and the solid line to eq. (4.4). (a) The effect of pionium. The aok contribution of eq. (4.4) exhibits a Fano resonance with its characteristic zero. Here the values of a0 and a2 quoted in section 2 are used. N.B. The semi-log scale . (b) The effect of kaonium. For illustration, reasonable values of K,, = 0.328, KC, = 2.53 and KC0 = 0.948 all in fm are used - see ref. [7]. For clarity, the dashed line has been multiplied by a factor of 10.
S. Wycech, A.M. Green /Exotic atoms tion cross-section pion production
453
is larger by a factor of about 3-4. The lower estimate comes from two cross-sections
at pl&, = 2.23 GeV/c [ 201. In fact aspI& decreases this fac-
tor increases. The upper estimate comes from the coupling ratio G(pnn+)/G(ppn’) = fi [2]. This is a fortunate conclusion, since the F,, amplitude depends on the off-shell structure in F. (r). However, from Fig. lc the direct production of n”zo described by simply F. contributes both background and interference terms to the atomic amplitude F,,. To study these effects one needs 3-body kinematics appropriate for the final states in Fig. 1. Let the energy be measured final state in the cm system is
relative to the the 7t+7c- threshold,
E, = q,2/2p -A
which for the
+ ER(~),
(4.5)
where E,(P) = P2/2pR is the ‘He recoil energy and A = 2 (m, - mo) is the threshold mass difference. Neglecting the F,, contribution from eq. (4.4), since it is expected to be only a 5% correction to Fco, the formation probabilities are given by the integration IFco + F,12dL3
w= =
J{lFo12+
[I&o +
El2 -
IFo121}dL3
= w, + w,
(4.6)
over the three body final phase space. This is done in appendix 2. The second term on the RHS gives the probability of atom formation WA, whereas the first term is the z”zo background. Integration over the atomic line and the phase space yields the total atom formation probability WA =
w
lFc12 L2 . (1 + 2q01m;;;co))
where L2 is the phase space element
L2 = P~/[4~(2m
,
(4.7)
+ A43)] with P --) PA =
EC) being the recoil momentum at the atom production energy and M3 is the mass of 3He. The last factor in eq. (4.7), where 1 = Fo/Fc, is due to the atomic $pR
(1~1 +
background interference. This term is an improvement over the coalescence model estimate given by eq. (3.3)) but the correction is negligible (< 1Oe3 ) in the pionium case. Similarly in the K-K+ case ImA,, z low4 - again a negligible correction. However, it may be of the order 1 in the K?? case. In general, the experimental
resolution
will be
such that many atomic states will fall inside this resolution.
In this case the above factor
lw(0)12 is replaced by C,, Iw,(0)12, where n is the principal lead to an enhancement that is, at most, 20%.
quantum
The recoil energy distribution less and is given by &(ER)
due to direct n”zo production
number.
This can
requires one integration
= [FoI~P(ER)J~~U((EC + A) - ER)/[16a3(2m + M3)1.
(4.8)
A similar expression holds for the recoil due to n+n- production, if experiments are performed above that threshold. The necessary change is F. + F,, EC + EC - A. The latter effect of the II+W- channel may be minimised by making the experiment close to
454 the threshold
S. Wycech, A.M. Green /Exotic atoms [ 121. However, in the energy range of interest, the rr”zo production
a constant but large background. WA/ [ %(ER)dER] [ 121. As discussed
With the energy fixed at the n+z-
makes
threshold one obtains
x IFc/Fo12/200, where a recoil energy resolution of 70 keV was used above, the enhancement due to IFc/Fo12 is expected to be at least a
factor of 3. In spite of this, one should expect a sizeable background
in the subthreshold
pionium production. Only if the resolution can be reduced substantially value of 70 keV will the atomic state signal become significant.
from the stated
Kaonium production is likely to be dominated by the fo (975) and a0 (983) resonances. Indeed, a peak is observed at Saturne in the reaction p (d, 3He)X around the Kx threshold [ 141. The K+ K- atom production rate and the background recoil energy distribution due to rcrc or ?~a are given by equations analogous to eqs. (4.7), (4.8), but with relativistic phase space. Now one obtains WAD[ WB(ER)AER] z lFc/Fo12/225, where Fc/Fo is the ratio of resonant decays into KK versus nrr or r]n. This factor is available in the f0(975), Z = 0 case and IFc/Foj2 = gK/g= = 2 - see ref. [6]. Unfortunately, there is no experimental estimate of the a0 (983), Z = 1 coupling to KK relative to the r~rccoupling. However, in some theoretical models, a ratio similar to that in the fs case is suggested [ 231. In view of the strong KK, m coupling to the resonant states, the resultant scattering lengths and K-matrix elements could be as large as l-2 fm. This generates structure in the atomic line with a width of the order keV. A full set of K-matrix elements is not yet available and so a more detailed structure cannot be calculated. In particular, one cannot estimate with reliable parameters the magnitude of the Fano type of structure already predicted for pionium in Fig. 2a. For illustration, in Fig. 2b reasonable values for the K-matrix suggested in ref. [7] are used. However, it should be emphasised that the detailed structure seen in this figure is very dependent on the actual numbers used in the K-matrix . The interference effect in eq. (4.7) may be rather large, since 12qoAcoI x 4 is expected. The K+K- case is rather similar to that of pionium in the sense that the threshold of the open channel K”no is close to the energy of the atom. However, the major difference is that a suitable mechanism for producing K+n- or K”no is not known - unlike: (a) Pionium,
which is part of the well known two-pion
production
present in many
reactions e.g. as in eq. ( 1.1)) or (b) Kaonium, which can be produced via the fo (975) and uo (983) resonances. One possibility is to produce Ka in the reaction pp + pC+n-K+ - but this has a very small cross-section
e.g. 0.019mb at 3.47 GeV/c
[20].
5. The production of exotic atoms in Zp-states The formation of pionium in a 2p-state would make its detection much easier. The main decay mode is the X-ray transition to the 1s-state with a rate of 0.86 x 10”s-’ [ 241, This time is long enough for the atom to leave the target and so facilitate measurements of its decay modes. However, its production rate is very slow, since it involves an additional factor of cy2. Consider that a vector meson, e.g. p or 4, produced by a reaction such as
455
S. Wycech, A.M. Green /Exotic atoms
pd +
P 3He decays into pionium
or kaonium.
Let the meson production
amplitude
FE +q, where Q is a unit vector related to the vector particle and q is the momentum meson-meson
centre-of-mass.
2 yields an atom formation
Employing
a procedure
rate in a coalescence
analogous
be
in the
to that used in section
model of the form
(5.1) Here, w’ is the radial derivative of the p-wave atomic function at the origin - ,/w for a 2p-state - and (...) denotes the average of q2 over the final phase space. Close to the rr+z- production threshold this ratio follows ETsi2 and, in principle, may be quite large. However, in practice, vector meson production may be minute. Estimates suggest that, at EC = 10 MeV, the ratio WA/ WB(ER)AER is of the order lo-‘. Similar calculations may be carried out for estimating the chances to produce kaonium in the reaction e+e- --f 4 + (K+K-)2,. Right at the (K+K-) atomic energy, one has WA/[ WB(E)AE] M 0.4 x 10e6 using the DA@NE resolution of 4 x 10m4. This reaction is 32 MeV away from the 4 energy, which gives another l/500 reduction factor. With the expected rate of 1044/s, this would produce about one kaoniumlday.
6. Conclusion In this article, signals indicating the presence of the exotic atoms pionium (~+a-), kaonium (K+K- ) and K+n- are estimated in various reactions. The emphasis has been on the production of pionium in the reaction p + d + ‘He + (M+M-)atom, but the formalism proposed applies equally well to kaonium and the K+n- atom. Furthermore, following the suggestion in ref. [ 121, the signal in the 3He recoil energy distribution is expressed as the ratio of probabilities WA (AtOm)/Wg (Background)dEa, where AER is the recoil energy resolution. In the pionium case, since this resolution is 70 keV compared with the atomic line width of a few eV, the signal is only z 1% of the n”rro background. Only if the resolution can be decreased by an order of magnitude could such an atomic signal become measureable. The situation for seeing the K+n- atom is somewhat similar to that of pionium with the additional complication that the basic Ka states are more difficult to generate i.e. in eqs. (4.3,4.4), the FC,O(Kz) factors are much smaller than the corresponding FC,O(m) factors. Possibly it is kaonium that will be most easy to observe, since this has a width that is of the order 1 keV compared with fractions of an eV for pionium and the K’natom - see Table A. 1. In addition, the basic mechanism for creating this atom is enhanced, since it appears via the fo (975 1 and a0 (983) resonances. In comparison, the production of, for example, the p-meson would generate pionium only in a Zp-wave and this would be expected to have a very small probability. It is possible that the peaks seen at the Ki? threshold in the p (d, ‘He)X reaction at Saturne are already partly due to kaonium formation. Unfortunately, unlike the pionium and K+n- systems, the theoretical esti-
456
S.
Wycech, A.M. Green /Exotic atoms
mates based on eqs. (4.7,4.8) can not yet be made, since they require a knowledge of the K-matrix elements KC,(K+K- -+ K+K-),Koo(wa -+ mr) and Kco(K+K- --+m). For pionium
and K+K-,
the corresponding
K-matrix
elements are sufficiently
show that their effect - reflected by the interference
well known to
term in eq. (4.7) - represents
only a
low3 correction to the basic coalescence model. On the other hand, for the Ki? system, all that can be said at present is that ImA,, z - 1 fm, which is sufficient to indicate that the interference term in eq. (4.7) could be significant. Similarly, the effect of the K+Katom in the background production of Fig. lb [i.e. eq. (4.4) ] could also be significant, if Re A,, and also the K-matrix elements are sufficiently large. However, it is possible that these parameters could conspire to give a Fano zero that greatly reduces the overall effect. As outlined in section 5, the production of atoms in the 2p-state, although desirable for observing such atoms when once generated, is very unlikely. Similarly, the direct production via e+e- + 4 + (K+ K- )zP seems also to be impractical. So far the discussion has concentrated on p + d reactions with the conclusion that the signals for atom production will be very difficult to detect compared with the ever present background. In pp + ppn+nreactions this problem is further complicated by the need to consider 3 and 4 phase space factors i.e. n = 2 in eq. (3.3). The relative atomic production rates W (atom)/ W (pair) in the pp ---f ppn+nand KK or in the pp --t K-n+pC+ reactions are now higher . Eq. (3.3) produces a rate of 1.26 x 10-5[2~,,,,+M-/(E - E(threshold))3/2] that is more than lo-* at 1 MeV or 10-4-10-3 at 10 MeV above the thresholds . However, it would then be necessary to measure the energies (El, E2) of both protons and to concentrate on the events with Ei + E2 tuned to pick up the atomic signal. The ratio of the atomic formation probability to that of uncorrelated mesons would now have the form WA/ WB (E,, , EP2) AE,, AED i.e. with the resolution appearing twice. This is clearly a much more demanding experiment than the p + d proposal.
One of the authors
(S.W.) wishes to acknowledge
the hospitality
of the Research In-
stitute for Theoretical Physics, Helsinki, where part of this work was carried out. In addition, he wishes to acknowledge the receipt of the KBN grant number 2p 302 14004, which partially covered the expenses incurred in this collaboration. Also the authors wish to thank Dr. B. Hoistad , Dr. H. Nann, Dr. W. Oelert and Dr. J. Stepaniak for useful communications
and discussions.
Appendix A
SCATTERING PARAMETERS As there exists conflicting sign conventions for the low energy expansions a few basic definitions seems appropriate. The scattering matrix is normalised according to the Lippman-Schwinger equation ? = ^v + v (EC - go)-’ ? that leads to a low energy
451
S. Wycech, A.M. Green /Exotic atoms TABLE
A.1
The isospin structure of the K-matrix Channel c
n+7l-
K+Z-
K+K-
Channel o
7rW
K%O
nn, rln
KC,
(a2 + 2a0)/3
(a3 + %)/3
00 + AI )/2
KM
JZ(a2
vQ(a3 -al
K 00
(2a2
+
a0)/3
@a3 + 4
a0)/3
)/3
J/3
parametrization 1/T = 1/a + iq, where a is a scattering length and q is a relative momentum. Phenomenological scattering lengths are used and an off-shell extension of this model is obtained by a pseudopotential ? = (27r/p )6 (r) T that may generate a multiple scattering expansion. Scattering lengths defined in this way a = - tan S/q differ in sign from those defined in the compilation of ref. [ 21. Consider s-wave scattering in a system consisting of two channels c and o. The Kmatrix, i.e. a generalisation of the scattering length, and the T-matrix that follows from it are denoted by
z=
(:::I)
and
T=
(
Aoo/ (1 + iqOAOO)AC,/ ( 1 + iqcAcc)
(1 + iwL)
A,,/
A,,/ ( 1 + iq,A,,)
64.1) ’
where qO,Care the centre-of-mass momenta of the two mesons in the two channels o, c. The channel scattering lengths Aii are expressed in terms of the K-matrix elements, via the solution of T = K - iKqT, by [ 181 A,, = Kcc - iK~OqO/(l + iMGO), &
= K,,l(
A,, = &,
1 + iqJGo ),
- iKkk/(l
+ iqcKcc).
64.2)
These equations form a basis in which to describe two channel scattering in terms of the three parameters of the K-matrix. The systems of interest, like nn or Kn, are believed to display a high degree of isospin symmetry in their interactions. The K-matrix is given by two real parameters referring to isospin 0, 2 in the X+X- case and l/2, 312 in the K+n- case. The isospin structure of the K-matrix is given in Table A. 1. The case of K +K- is more complicated due to the mass difference of the K + K- and ??K” and also the presence of the two decay channels 7~71(I = 0) and ~7c (I = 1). Also notice the unitarity condition, which follows from eqs. (A.2), Im-4,
= -I&I*%.
64.3)
458
S. Wpxch, A.M. GreenI Exotic atoms
Isospin symmetry
is broken by coulomb interactions
and meson mass differences.
The
first allow for atomic binding and the second induce decays of these atomic systems. The channels
in Table A. 1 are, accordingly,
named closed (c) and open (0).
To describe atomic systems produced eqs. (A. 1), (A.2). The standard
in collisions,
these effects.have
to be built into
way is to separate long range effects that enter coulomb
propagators G” and modifications at short ranges entering “coulomb corrected” scattering lengths. The propagators that describe coulomb and short ranged interactions are given by Gc + GCTGc. As a consequence,
the scattering
fii = f”a,
is described by amplitudes
+ e’@icTEjci
of the form
eioj,
(A.4)
where f”, CTand c are the coulomb scattering amplitudes, phases and penetration factors. The latter arise in asymptotic solutions only and are of no concern in the present problem. The coulomb corrections to T’ arise in terms of known coulomb functions that should replace iq in eqs. (A.l),
(A.2): iq -+ f = 2yh + iqc2,
wherec2
= 2aq/[exp(2nq)
- i]
and
h = $[y(is)
(A.5) + v(-iv)]
- $lnq!
Also y = ZZ’ap and v = y/q - see for example ref. [22]. All these functions are diagonal in channel indices, which were dropped, however. It is easy to check that for neutral particles (2 = 2’ = 0) the f = iq. Now, the atomic states created in the intermediate states of nuclear reactions are generated by poies in f of eq. (AS). The expansion off around this pole yields [ 191 f=
R
whereR = -$]v(0)12
ETq
= -2h203,
co is the pure coulomb energy of an s-level and ECis the energy distance from the M+Mthreshold. As a first application of eq. (A.S), the position of the atomic pole in T,, of eq. (A. 1) is found. One obtains immediately the atomic energy shifted by strong interactions
E -
p
=
Eo
+
~:W(0)12Acc = 21U2ar3&, Eo +
(A.7)
which is a well known result relating level shifts and widths to the scattering length. The width expressed by K-matrix
elements
is
(A.81
Appendix B B. 1. PHASE SPACE FACTORS AND INTEGRALS
In this appendix are calculated phase space factors and integrals over the atomic line. Non-relativistic energies are used for the 3Herr%o final state. For a final state containing
4.59
S. Wycech, A.M. Green /Exotic atoms three particles
(B.1) Going over to energy variables recoil energy ER(P)
E4 = q2/2p in the rr7z centre-of-mass
= P2/2pR with ,UR = 2mM3/(2m
system and the
+ M3), eq. (B.l) becomes
dL3 = 6 (Eo - Eg - ER ) Pq dEsdEq/C, where c = 16a3 (2m + M3). Thus the background energy distribution %(ER)
=
rr”7coemission
(B.2) generates
the recoil
given by
l&l2
6(Eo - E4 - ER)PqdEq/c
= IF,j2P~~lc.
(B.3)
J
Here EO is the energy related to the x0x0 threshold. Now the atomic formation probability the energy related to the rc+rcis calculated and EC = E. - 2 (m+ - mo) = EO - A t.e. . threshold is more appropriate for use. The atomic signal from eq. (4.6) using eq. (4.3) becomes %
IRAeo - 112-1 E4 - A - e. + RAcc
=
If’12
=
lfil*J [IEo _
Jr
I
ER
yf$, +
R&
-
1s(Eo-E,-En)PqdERdEq/C II2
-
11f’qdER/C,
U-3.4)
where 1 = F,/F, and q = J2,u ( E. - ER). Since the atomic line is so narrow, it is assumed that q is constant over the energy range of interest. The integration over the atomic line is now performed using the relation SdEs/[ (ER - E)~ + (r/2j2] = 27c/T and also by taking into account the unitarity condition of eq. (A.3) and replacing R by eq. (A.6). This results in
w, = IFcl’w
(1+ 2qo(11$Aco)) [4n(22+M )I,
(B.5)
3
where the first term comes from the line and the second represents background.
interference
with the
The square bracket term is the L2 phase space factor for the atom and the
‘He system, PA = ,/m and qO is the relative momentum
is the recoil momentum
at the atom production
point
of the won0 emitted in the atomic decays.
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