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Volume 116B, number 4 PHYSICS LETTERS 14 October 1982 ROLE OF INTERNAL SYMMETRY IN p~ ANNIHILATION Berndt MiJLLER and Johann RAFELSKI Institut ffir'Theoretische Physik der W. Goethe-Universitdt, Postfach 11 19 32, D-6000 Frankfurt am Main, Germany Received I4 April 1982 Revised manuscript received 30 June 1982 Internal symmetry influences the form of quantum spectra as well as the abundance of particles emitted by a thermal source. Isospin conservation in proton-antiproton annihilation is used to quantitatively illustrate the influence of these phenomena on mean pion multiplicities and the apparent primary pion temperature parameter. Our aim is to study the influence of an internal (ex- act) symmetry on quantum statistics. To illustrate the problem we may consider an assembly of isospin carry- ing particles in contact with a heat bath. The statistical quantum distribution of these particles will be considei- ed and it will be shown that the requirement of sym- metry conservation alters the form of the statistical distribution. Such a thermodynamic system may be realized in proton-antiproton annihilation at moder- ate energies, with isospin being the conserved quantum number [1]. While our theory leads to qualitative and quantitative predictions which can be tested experimen- tally, a practical difficulty resides in the fact that one cannot easily distinguish primary emitted particles, e.g. pions, from the secondary pions that arise due to the decay of resonances outside the annihilation region We will, however, study here the apparent slope param- eter ("temperature") as function of primary pion en- ergy and the average (primary and secondary) pion multiplicities as function of the volume of the hadronic fireball. The influence of symmetry conservation will then be seen in a comparison with analogous results that follow when the isospin quantum number is ignored. Our choice of the proton-antiproton system, and its annihilation channel in particular, is motivated by the observation that in this channel an automatic selec- tion is made of those reactions in which all three quarks and all three antiquarks of projectile and target must participate. Hence the annihilation provides a "cen- tral" trigger for the p~-reaction, and we may assume that all primary particles originate in a finite common volume. Furthermore, the known exponential be- haviour of the particle spectra from p~-annihllation suggests the use of the canonical ensemble for the de- scription of the central fireball. In this spirit we will implement total energy and momentum conservation only on average while isospin will be conserved exact- ly. Another (practical) reason to consider p~-annihila- tion as an example for the influence of internal sym- metry on quantum statistics is the availability of a low energy antiproton ring (LEAR) at CERN in the near future [2], which will deliver the precise data necessary for a detailed comparison between theory and experiment. As both the I = 0 and the I = 1 channels are popu- lated in proton-antiproton annihilations, we will take a weighted average of both in this discussion. More ideal, but not yet practical, would be the study of deuteron-antideuteron or ~-f i annihilations which populate only/= 0 channels. A further approximation (at the level of 1% in the multiplicities) we make is to explicitly exclude (besides one particle annihilations which are possible in the naive canonical approach) also annihilation into two pions. This channel is easily seen tO be strongly suppressed with respect to other channels when exact energy and momentum conser- vation is implemented. 274 0 031-9163/82/0000-0000/$02.75 1982 North-Holland Volume 116B, number 4 PHYSICS LETTERS 14 October 1982 In order to correlate the reaction volume of a hy- pothetical annihilation fireball with the centre-of-mass energy, q~, we assume that the volume grows like in the spirit of the MIT bag model [3]. By comparing the observed pion multiplicity at threshold with our results we can fix the proportionality constant v@V ---- b = 170 MeV/fm 3 < 4B, where B = 60 MeV/fm 3 is the bag constant. This indicates that hadronization occurs, as expected above the energy density of nu- clear matter, but below that of individual nucleons. We assume that the transition proceeds at the fixed limiting temperature T O = 160 MeV. Would it not be for quantum and symmetry effects, particle spectra in p~-annihilation would reflect a universal slope 1/T O = 6.25 (GeV)-1 in the transverse mass distribution. The primary multiplicities of the mesonic resonanes are dependent on their masses and degeneracies. In this context the role played by the broad resonances must be carefully considered. Aside from the narrow zr, r/, co mesons we include the following wide reso- nances: p, A1, B, A 2 and f. All these resonances still contribute significantly to the total pion multiplicity. In the formal expressions below we include the influ- ence of the resonance widths by explicitly distribut- ing these resonances over a wide mass interval with a suitable weight function. The emphasis of our present work is put on the exact conservation of isospin among all particles con- tained in the fireball. Having a long history [1], this problem has only recently been approached with the mathematical apparatus of group theory that allows for a projection onto a particular irreducible repre- sentation of a compact semi-simple Lie group [4]. The formal procedure is based on the standard grand canonical approach to thermodynamics: Given ZN(T , V), the canonical partition function for a fLxed con- served particle number N, one considers the generating function Z(X, T, V) = NNXNZN, which usually can be calculated much more easily. The inversion by contour integration ZN= ~ dX Z/X N+I has, for example, been discussed in the context of the statistical bootstrap model with finite baryon number [5]. Substituting exp (i) for X leads to formulas similar to those of isospin problem, the difference being that isospin is described by a nonabelian symmetry group. Therefore we must define the generating function of the microcanonical partition functions by summation over the irreducible representations of the isospin group SU(2): r, v) = a/-lx(o)zz, (1) where XI() is the character function of the given re- presentation of total isospin I and d I = 2/+ 1 is the dimension of the representation. It is important to remember that the generating function Z(~), T, V) has no immediate physical meaning but~ once it is known, Z I can be easily constructed by making use of orthog- onality and completeness of the characters of a com- pact semi-simple Lie group [6] : Z I = d I fdu(~) X~(~) Z(~, T, V) (2) where/~() is the group Haar measure. For the isospin group SU(2) we have in particular: dg(~) = (d~/27r)sin2(q~/2), q~E [0,470 XI() = X~() = sin[(/+ 1/2)~]/sin(C/2). (3) In order to compute Z we recall the definition of the group characters XI(~)) =~ (I, 13 [ exp(--iq~]3)[I, I3) , (4) ~3 where the sum runs over the entire multiplet of the given irreducible representation. (The generalization of this formula to other groups is obvious.) The micro- canonical partition function is defined as trace over all states transforming according to a particular repre- sentation I (indicated by the projector/3/): zz= tr[P)exp(- >b] = %lexp(- Jq)lvz>, (5) vI where Iv I) is a state of fixed isospin but arbitrary par- ticle content: Noting that strong interactions conserve isospin, i.e. H and ]3 commute , we find [4] by insert- ing eqs. (4), (5) into eq. (1): Z(~b, T, V) = tr exp( - /~ r - i~bi3) = exp( /~ In Zi(~b, T, V,m=mi) ) . (6) We have indicated here that the generating function 275 Volume 116B, number 4 PHYSICS LETTERS 14 October 1982 factorizes into contributions from different particles when we assume that the effect of the strong int,erac- tions is approximately described by the multitude of non-interacting resonances [5] To account for the width of the p, A1, B, A2, fmeso n resonances, we re- place the discrete sum over i in eq. (6) by an integral over a mass distribution function composed of lorentzians at resonant mass and with experimental width but cut off at the mass threshold [7] * 1 For ideal Bose or Fermi gases Z is easily evaluated: In Zi(4, T, V, m) = ~ ~ (+-)ln{1 + exp[--~6ok(m) -- i413]} , (7) k h where 6ok(m ) are the eigenmodes of particles with mass m in the volume V. The upper sign refers to fer- mions, the lower sign to bosons. We now introduce the approximate constraint of energy-momentum conser- vation by recalling that the trace in eq. (6) cannot in- clude states with zero or one particle. We furthermore omit, as already mentioned, the states containing only two pions. These corrections lead us to consider the modified generating function ~E corrected for energy- momentum conservation: ~ 1 ~1 2 E(4, T, V)-- I17i-l-EZ] 1, l l where w ~ Z~ k~' [1 + 2cos(n4)] exp(--n/36o~:) --/t/ (i = rr, p, A1, B, A2), zn~ =g-'/n Ek exp (-n~6o~) (i = ~/, 6o, f) (s) and gi the degeneracy due to spin eq. (2) we can ex- tract the modified partition function Z~ for fixed isospin I that omits zero-, one-, and two-pion contri- butions: ,1 This procedure gives a sufficiently exact approximation of the influence of multi-pion phase space on the resonance shape. 7r Z E = 2(2+ 1) f d(4fir) sin(4/2) 0 X sin[(/+ 1/2)4] zE(4, T, V). (9) The particle spectra follow from the expressions given above, We consider the quantity n~(, T, V, m)=[tr exp( -~/1) ] - l t r [~exp(-/gar)] I.J,E I ,E (10) to be the average number of particle of sort i, energy 6ok(mi) in a system coupled to total isospin I and omit- ring the states as discussed above (indicated by "E"). Recalling that H = E nk6ok, , we easily find n}c=--[~/~(fl6o~c)] ln[i./,Etr exp(--/3/~)] = --[0/a (/~6o})1 In@) . (11) In the actual calculation the frequency differentiation must be carried out before the isospin projection. The result, e.g., for the primary pion multiplicity is 2 f dsin(4]2)sin[(/+ 112)4] n~r(W) = zE 0 "ff- X [( Ni "Zi ) ~n exp(-n~6o) [ l + 2 cos(n4'] - exp(-/36o) (1 + 2 cos 4) (1 +~1) exp(-2~6o) (1 + 2 cos(24)]] . (12) .d The particle multiplicities are bound by (analytic) inte- gration over the corresponding spectra. We notice that this method also provides us with the actual shape of the various wide resonances as determined by invariant mass analysis. The actual numerical calculations are straightfor- ward. We go to the continuum limit: ~-+ v f d3p co~c _+(p2 + m})1/2 (13) k (2703 ' which turns out to be satisfactory already for a rather small hadronic volume. In a first step the function ~'E, eq. (8) is calculated. After expanding the logarithm in 276 Volume 116B, number 4 PHYSICS LETTERS 14 October 1982 eq. (7) into a Taylor series, the momentum integrals can be carried out analytically and the resulting series of Bessel functions are summed numerically. The de- rivative o fZ E with respect to frequency is then pro- jected onto isospin I = 0 and I = 1, of eq. (12), and the weighted average of both results is taken (i.e. Z 0 + lZx) . In fig. 1 we show the effective primary pion tem- perature (i.e. the slope parameter from a local fit to the spectrum) as a function of the pion energy. With- out isospin projection, the Bose-Einstein (B-E) dis- tribution function fits almost exactly and would lead to a slope of 160 MeV, the input temperature at which the pions are produced. After isospin projection, the spectra are found to deviate sfrongly from the Bose distribution. In particular, we do not find the "con- densation" enhancement at low energies. This leads to a high apparent temperature at low pion energies. This observation is in agreement with experimental findings in several exclusive channels * 2. By taking a Maxwell-Boltzmann (M-B) distribu- tion function for the temperature fit (i.e. taking the logarithmic derivative of the spectrum) a systematical- ly lower temperature is found when isospin is con- served. This indicates that some contribution from the condensation terms survives the isospin projection. ,2 The most comprehensive recent study of pp-annihilations is contained in ref. [8]. ZOO 180 :> "-~ 160 _ ~ t I 7 - bin"- Fit E IZO I I I I I ZOO 300 400 500 600 E ( MeW Fig. 1. Slope parameter (temperature) of primary pions as function of pion energy assuming Bose-:Einstein (B-E) or Maxwell-Boltzman (B-M) spectral forms for isospin project- ed spectra. Without projection the slope parameter is 160 MeV for the Bose-Einstein form. All curves for Vrs = 4.9 GeV. The average temperature, after weigthing the tempera- ture function with the spectrum, turns out to be about 145 MeV. This is again in qualitative agreement with experiments that single out the annihilation channel in p~-reactions [8]. The slope parameter is found to depend relatively little on the size of the reaction volume, i.e. on the c.m. energy. What changes most with volume is the normalization of the spectra, and hence multiplicities. The pertinent results are shown in fig. 2. To start with, fig. 2a gives the total pion multiplicity as function of V or x/~, respectively: tttot zr = n~r + 2np + 2.8nto + 1.62n n + 3hA1 + 3.8n B + 3hA2 + 2.2nf, (14) l I I / I 10 nr~ / - // k 9 -- o Amsterdam / ,ivorpool / T T /T - *CERN / / ,L, l / I - ,~,~/ 7- ~/ / (o) 5 6 / - ~ p~-onnihitotion [ , , , $I/2 [GeV 2 3 4 I I I ~0 25 V[frn3 ] 30 15 20 1.0 0.8 0.6 0.4 0.2 0 F J tb) , 1 , l s,1/2[Ge, v] Z 3 4 5 Fig. 2. (a) Final pion multiplicity in p~-annihilation. The solid line includes isospin projection (I = 0, 1), while the dashed line is obtained without isospin projection. (b) Primary meson mul- tiplicities are functions of the reaction volume V, the relation- ship with x/~ is established empirically by fitting the total pion multiplicity at threshold. 277 Volume 116B, number 4 PHYSICS LETTERS 14 October 1982 Still higher resonances are here irrelevant, but the f meson contributes on average 0.2 pions per annihila- tion event. The full line shows our results with isospin projection. The dashed line above was obtained with- out isospin projection. We also show several experimen- tal results in which the analysis includes missing mass fits [8,9]. We observe a quite surprising agreement with our predictions, in the sense that the rise in mul- tiplicity with ,v/)-iS well reproduced if the relation be- tween X/s-and V is calibrated at threshold. In fig. 2b we show the primary meson multiplici- ties from which the total pion multiplicity has been derived. It is interesting to note that the total p yield, which originates from p, A 1 and A 2 mesons, leads to a p/rr ratio between 0.34 and 0.36 within the energy range covered by our calculations. This result must be com- pared with that of Whyman et al. [10] who find pOfir- = 0.14 + 0.01. On the other hand we t'md that the 6o meson, which also is a decay product of the B meson, is as abundant as the p meson and seven times more abundant than the f meson. While this last obser- vation agrees well with experimental data [11], our p abundance is too large as compared to the pion yield u 0 ,,o (ass ruing p /rr- _ pfir) and too small as compared to the 6o abundance (assuming pO[6o ~ ~ p/co) where ex- periment [11] gives pO/6o ~ 1. This apparent discrep- ancy may be resolved in the following way: Investigat- ing the influence of statistical population of states on the resonance shape w~ find that a substantial part of the resonance yield (up to 40% of p and 70% of A1) would be attributed to the non-resonant phase space in the course of a normal invariant mass fit analysis. Only resonances with width F