Quantum statistics and isospin invariance

Nuclear Physics BI00 (1975) 302-312 © North-ftolland Publishing Company

QUANTUM STATISTICS AND ISOSPIN INVARIANCE J, KRIPFGANZ

Sektion Physik, Karl-Marx-Universiti~t Leipzig, DDR Received 4 August 1975 The level density of an ideal Bose or Fermi gas is written in terms of usual phase-space integrals taking isospin conservation into account. This cluster decomposition includes quantum statistics corrections between equal as well as unequal charge states of the considered particles. The isospin weights are given. These results are used to formulate a simple isospin-invariant statistical bootstrap model with Bose statistics. In the framework of this model the production of neutral pions in e+e- and NN annihilation is investigated. 1. Introduction The effects of Bose and Fermi statistics are well understood in macroscopic statistical mechanics. This is not the case, however, for the application of statistical ideas to elementary particle physics. In general, statistical models are based on the usual phase-space integrals corresponding to Boltzmann statistics. In a recent paper Chaichian, ttagedorn and Hayashi [1] investigate the density of states of an ideal relativistic gas of isoscalar bosons or fermions. The level density is written as a cluster decomposition over usual phase-space integrals. The authors point out that the effects of Bose or Fermi statistics may be very significant and cannot be considered as small corrections to the Boltzmann term. There are also some experimental indications of the presence of Bose statistics effects. In the framework of an independent cluster model it has been shown that the short range correlations in the azimuthal angle can only be understood if the effect of Bose statistics is taken into account [2]. These results encouraged us in studying the consequences of quantum statistics further. In the present paper the analysis of ref. [1] is generalized to include isospin invariance. The incorporation of non-Abelian internal symmetries leads to some nontrivial effects justifying this treatment. We perform a cluster decomposition of the density of states of an ideal Bose or Fermi gas and obtain clustering effects not only for equal charge states (e.g. n+n +) but also between different ones (n*n 0 etc.). This has to be so in an isospin-invariant scheme. This cluster decomposition is investigated in sect. 2. In particular, the corresponding isospin weights are calculated. In sect. 3 these results are used to formulate and solve an isospin-invariant statistical bootstrap model with Bose statistics. The ratio of neutral and charged pions is 302

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computed for isoscalar and isovector fireballs. This analysis is motivated by a recent suggestion [3] that the neutral excess found in e+e - annihilation [4] (and p~ annihilation at rest [5]) could be understood as an excess of neutral pions caused by Bose statistics. However, in the framework of the model studied in sect. 3 no significant excess of neutral pions is found. This is a consequence of isospin invariance. Our results are summarized and discussed in sect. 4.

2. Isospin-invariant cluster decomposition of the density of states of an ideal Bose or Fermi gas We consider an ideal gas o f { N T, N T+I .... ,NT) bosons or fermions of isospin T The phase space available for such a system is given by

f2 (N t } (Q2, I) = Tr (Nt} P QP[,

(1)

where Q is the total four-momentum and (I, i) the total isospin. PQ and PIi are the projection operators on the irreducible representations of the corresponding symmetry group

pQ _

1 f d 4 a e iaQ U(a), (2re) 4

(2)

p f _ 2 / + 1 f d R D/. ( R - l ) U(R). 8rr 2

(3)

U(a) and U(R) describe 4-dimensional translations and rotations in isospace, respectively. The D / (R) are rotation matrices. The measure dR is given by 27r ~r 27r fdR: f fsin /3 d/3 f dT, 0 0 0 where a,/3, 7 are the Euler angles. The trace Tr{Nt} in eq. (1) refers to the subspace of given particle number T Tr (Nt} A = (npt~} t = -

5(N - ~P npt) ((npt}lA[

(4)

with

I(,,_.)> : FI ~(a t)npt I0). pt

(5)

pt . pt

The occupation number npt indicates how many particles have momentum p and third isospin component t. The creation and annihilation operators obey commutation or anti-commutation relations, respectively +

Bose statistics (B)

[apt, ap, t, ] = ~pp, ~ tt"

(6)

Fermi statistics (F)

(apt , ap+t,} = (Spp,(~tt,.

(7)

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304

In order to calculate the trace (1) we use the transformation properties of the creation operators

U(a) a;t U- 1(a) = e lap a pt, + 1

_

T

(8) +

U(R) a+pt U- (R) - Dtt,(R ) apt, ,

(9)

and the invariance of the vacuum. It is this mixing of creation operators of different charge (eq. (9)) from which the particular effects of isospin invariance result. We represent now the/5 functions in eq. (4) by Fourier integrals and obtain

a{xt;(o2,~)-~ ~13 et 2~+1 (e~D~.fR_t) (2n-i)2T+l J t zNt +1 8rr2 0 Xl /'d4 a e_ia Q Z(a, R, (zt}), (2zr)4 a

(10)

where the grand canonical partition function Z(a, R,

(zt})

is given by

Z(a, R, {zt} ) = r~l ({nptil U (ZtXpDT'(R) apt')nPt IO) Wpt 5

pt

(1 l)

x/n~pt [

with

Xp = e iap.

(12)

It will be useful to begin with the consideration of a single momentmn state. We introduce the notation

z0((~,t,))

<(n t) I H (~

-:

+

"

Io).

03)

Nil1 t •

In terms ofZo((Dtt,}), Z(a, R, (zt}) is given by

Z(a, R, {zt)) = ~Zo((Z t Xp DT,(R)}). p

(14)

We start with the calculation of Zo((Dtt,}) in the case of Bose statistics. Using the commutation relations (6) the scalar product in eq. (13) can be performed, and we obtain I-[ nt!

zg({~,,,}): C

C

'

I-l~ ot,'

(nt) (qtt,} l-Iqtt,! tt' --tt' tt' X~(,ll--~t,

qlt, ) ~t ~(,,t .- ~t qt,t).

(15)

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305

Representing the second set of 6 functions by Fourier integrals we find the expression (15) to be equivalent to

zB({~tt,}) -

1

qfI-[dYt

(27r/) 2T+1 ~ ' t

YU

(1--~t

Dtt, ytYtT1)-1

.

(16)

'

These integrals can be performed, and we finally obtain

Z~((Dtt,} ) : (det (S tt, - Dtt,)) 1 . (17) In the case of Fermi statistics the calculation ofZo({Dtt,} ) is trivial and yields zF({Dtt,}) = det (6tt, + Dtt, ) (18) According to eq. (14) the grand canonical partition function Z(a, R, {zt}) can be expressed as

zB'F(a,R,

logZB'l:((ztxpDT,(R)})}

(zt})=exp{ F

9'~

-= exp { F

t

(19)

II (Ztxp)nt}

B F ({D T, (R)}) are given by where the expansion coefficients F {n't} FB'F{nt}({DT' (R)}) - I-I-Lit nt!

az't't an-~-t log zB'F({ztDT,(R)})I{zt=O},

Expanding the exponential in eq. (19),

Z(a, R, {zt)) takes the

{L{nt)} (n t} L(nt}!

(20)

form

(n't}

\ p t "- - /

(21) A parameter L{n. } occurs for each particular set (nt}. The sum goes over the possible values O, 1,2 .... ¢Fhe physical meaning of L{nt} will become clear soon. The representation (21) of the grand canonical partition function is convenient to go back to the microcanonical formulation via eq. (10). In this way we obtain the following cluster decomposition of the density of states

~2B t" (Q2,1) = ~ G B'I: H (~it} {L(,~}} ;i,{c(,,)} t 8(Nt - {~nt}n eL(,tt}) ) L(nt}

xfa(4)(O - ~; Z

{nt} J{nt}

1 L{nt) Bd4pi{nt }

e

j{.t}){,,,} Z{nt)" T~ H J{nt} H

n2

6+(,P2{n,}-- .2m2 ) (22)

with 1l : ~ t

Ht.

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306

In deriving eq. (22) we approximate the discrete momentum spectrum for simplicity by a continuum ~[] - > / B d 4 p 8+(p 2 - m2). p We note that the level density (22) contains terms with multiple masses (nm) of the constituents of the gas. Such terms are called (Bose or Fermi) clusters. L{n } re. , t presents the number of such clusters contalmng { n T, n T+ 1..... n T} particles of a third isospin component - T , ..., t, ..., T in one and the same m o m e n t u m state. Our main result is the determination of the isospin weights of configurations containing Bose or Fermi clusters B,F

GIi,{L{nt }

_2•+ 1/"

~ 2 J dR B,I"

/~ii(R - 1 ) I I

~nt)

(FynI~({DT,(R)}))L{nt} L tJ "

(23)

T

The contributions F~nt)({Dtt,(R)} ) are defined by eq, (20) together with eq. (17) or eq. (18), respectively. An example of particular interest is an ideal gas of isovector bosons (pions). In this case we obtain

F l,0,0 =

ll(S),

e 2,0,0 ) = (D{ t(m) 2,

F 1,1,0 = DI2(R) D I(R), F ,o,o} = 1

1

1

F~32,1,0} = D 11 (R) D12(R ) D21(R), F~I,I,1 }

=DI2(R) D~3(R)D~I(R)+DI3(R)D~2(R)D{I(R),

(24)

etc. In the Boltzmann limit no Bose or Fermi clusters are present. In this case the isospin weights (23) are the well-known Cerulus coefficients [6]. It should be noted that the isospin weights (23) may be negative for some configurations, i.e. the equivalent interaction simulated by the cluster decomposition may be repulsive. This is not at all an unphysical feature, since these Bose or Fermi clusters are by no means real physical objects like resonances. In particular, a Fermi cluster may contain more than one fermion of the same kind in the same m o m e n t u m state. Such pure mathematical objects may have negative probabilities. Finally we emphasize once more that isospin invariance causes statistics effects also between different charge states of an isomultiplet. This effect vanishes if only charge conservation is taken into account. Formally this is achieved by putting the

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non-diagonal elements of the D functions zero (see, e.g., eqs. (24)). In the case of isoscalar particles we reproduce the results of ref. [1].

3. An application to the neutral-charged ratio Favoured reactions for the application of statistical ideas in elementary particle physics are e+e - annihilation into hadrons and NNt annihilation at rest. This is because these processes are ahnost isotropic in momentum space. In both reactions it has been found [4,5] that the charged secondaries take off an energy definitely smaller than } of the total available energy. It is well-known that statistical models in the Boltzmann limit cannot account for this neutral excess (see also fig. 1). We will use now the results of sect. 2 to investigate possible effects of Bose statistics on the neutral-charged ratio. The modern version of the statistical model is the statistical bootstrap model [7,8]. It incorporates the hadronic mass spectrum and predicts its asymptotic behaviour to be exponential in the mass. Fireballs are introduced as typical average resonances. The decay of such a fireball is described statistically, i.e. the weight of a certain decay configuration is given by the available phase space. As decay products all possible hadronic states have to be counted. This includes the predicted heavy resonances which decay further and lead to a cascade decay. We formulate and solve now a statistical bootstrap model with isospin conservation and Bose statistics. So far either versions with the correct statistics but without internal symmetries [9] or isospin invariant versions in the Boltzmann limit [10] have been studied. It has been shown [8] that a fireball mainly decays into some pions and one h e a w restfireball. The dominant decay mode is that into one pion and a restfireball. In this approximation coherence effects are not present. They are destroyed by the time sequence due to the decay chain. Bose corrections arise if the considered pions are emitted in one and the same decay configuration. We estimate these coherence effects in the first non-vanishing order, i.e. we take into account the emission o f one or two pions per decay step. The emission of more pions is strongly suppressed by the dynamics of the decay cascade governed by the hadronic mass spectrum. In order to incorporate Bose statistics the emission of a two-pion Bose cluster has to be considered besides that of one or two pions. The corresponding probability will be negative for some charge combinations, as discussed in sect. 2. This is not an unphysical feature, since these Bose clusters are pure mathematical objects. We exclude exotic states, in the sense of a simple quark model. Only non-exotic quantum number ( / = 0,1) are allowed for the decaying as well as the remaining fireball. In this way, using the isospin weights (23), the following bootstrap equations in configuration space are obtained

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308

j+l = g+ _ g+o + ({go + 0"4go0 + 0.2g+_ ) f l + (½g+ _ 0.2g+o ) j4 + 0.6g++ j.1 + (g+ _ g+o ) ./CO)+ O(gig/) ' f~ = go - g+- + (½g- -- 0'2g-o) f+l + (0.6go0 + 0.8g+ ).t~ + (½g+ _ 0.2g+o ) fl. + (go - g+ )'[~ + O(gigj)' •Co0

=13go0 + 2g+ -- "t- ( l g

- 3 g'

1 O)J+"l +(3gO -}g+-)f(l/

+(½g+- - -~g+0 )J'l-- + (~go0 +2.,32g+ _ )'"0.t 0 +O(gigj)"

(25)

f/denotes the Laplace transformed product of the hadronic mass spectrum p(Q2, I) and the generating function ~(Q2,/, i I {ht)) of a fireball of four-momentum Q and isospin (I, i)

.ti( =f d4Q e

xO p(Q2, I) B #(Q2, I, i l {h,}).

(26)

The gi and gij describe the emission of a single pion or a two-pion Bose cluster, respectively

gi = hif daQe--rOB 8+ (Q2 " m2n) '

(27)

~! =hihif d4Q e- XQ B

(28)

8+(Q 2 - 4m2).

For simplicity, the Boltzmann terms O(gigj) corresponding to the emission of two pions are not written down explicitly. This model, described by the bootstrap equations (25), should not be applied to exclusive channels near their threshold. In this limit the continuous counting of momentum states used in eq. (22) is not justified, and, combined with some negative probabilities for Bose clusters, may lead to negative probabilities for physical states. Similar effects have been discussed in ref. [1 ]. The solution of the bootstrap equations (25) is trivial. It is useful to write it as a series expansion with respect to the input spectra & and gij (29) 4-1(1) •(2)) ~'i ' ij J

\t

/ \i~<]"

The expansion coefficients w(I, i I {i}1), lb.2))) can be calculated recursively. Going back to nmmentum space, i.e. inverting the Laplace transformation, we obtain front eq. (29)

Y. Kripfganz /Quantum statistics

309

Bp(Q 2, I) ~(O21i l{hi)) =

W(l,i[{l"i(1)' l(2)}}g2(Q2il ij" ~1(1) /(2)) i • ij ~

(1) (v)

,l-)B

1(l)+l(2)[rj 1}1),~/'1-1 ./(2)~ [llh i | [ l l ( h i h i I i ] I \i / \i<~j " !

(30) The ~(Q2 [/(1),/(2)) are the phase space integrals for/(1) = ]~i/~1) single pions and /(2) = v' •(2) two-pion Bose clusters (of mass 2riG). Now we are going to study the physical properties of the solution (30). To that end, it is often convenient to use the maximum temperature T 0 instead of the coupling constant B as the free parameter of the model. The maximum temperature is given by the rightmost singularity o f f i / in the inverse temperature plane (3 -- x/x2), which corresponds to the zero of the coefficient determinant of the system (25). In this way, T O is related to the coupling constant B, and we obtain 0 : (L,(I)(B, TO) +g(2)(B, TO) - 1)(g(2)(B, TO) - 1) - (g(l)(B, TO) g(2)(B, TO)) 2 (31) where g(l)(B, TO) and g(2)(B, TO) are given by

gO)(B, TO) - g / h i = 2~rm B T 0 K~(m r/TO),

(32)

g(Z)(B, TO) ~ gij/hi hj = ¼ 27r mrr a T OK 1(2turf/T0).

(33)

In particular, we verify the known result [9] that for given coupling constant B, T 0 is smaller if Bose statistics effects are taken into account TBOSe <5 T B°ltzmann 0 ----0 The Boltzmann limit corresponds to g(2J(B, TO) = O. We turn now to the discussion of the neutral-charged ratio. This analysis is based on the solution (30) where the expansion coefficients w(/, i [{l!1), l~2))) are calculated numerically. We obtain the following results: (i) For isoscalar fireballs we find (n,To). --= 0.5, (n~ch),

I = 0,

(34)

independent of the total pion number n, tile fireball mass M and the choice of the coupling constant. This is required by general isospin bounds [11] and in dicates only that our calculations should be correct. These bounds are not respected be the results of ref. [3]. The reason is that in this case only Bose clusters of 7r0 are considered. Such a procedure does not treat the charge states of the pion on an equal footing and violates isospin invariance. (ii) Without any further calculation, we can conclude that also for isovector fireballs the average number of rr +, lr0, rr- is equal (n ,) = (nno} = (n _),

I = 0, 1,

M~'~',

(35)

310

J. Kripfganz /Quantum statistics



52

.51

.50

2.

i.

~.

~.

10,

N iGeV/c~

I:ig. 1. Neutral-charged ratio /~as a function of the fireball mass M. Full curve: statistical bootstrap model witb Bose statistics, defined in sect. 3; dashed curve: Boltzmann limit. T o = 190 MeV in both cases.

.51

.50

2.

i.

ME&/cq

Fig. 2. (nno>lw'rsus M; Bose statistics, for several values of the maximum temperature: T O = 160,190,220 MeV. if the fireball mass M is large enough, i.e. if the decay chain is very long. In this case, in leading order the decay spectra do not d e p e n d on the isospin of the primary fireball. (iii) F o r isovector fireballs at n o n - a s y m p t o t i c masses we obtain a neutral-charged ratio shown in fig. 1. We notice a small increase c o m p a r e d with the B o l t z m a n n limit. However, the resulting value is still very near to 0.5, and no remarkable excess of neutral pions is found. This result is not very sensitive to the value o f the m a x i m u m temperature, as indicated in fig. 2.

4. S u m m a r y and discussion We have seen that the density o f states o f an ideal q u a n t u m gas can be expressed as a cluster d e c o m p o s i t i o n , i.e. as an expansion over usual phase-space integrals. Be-

J. Kripfganz / Quantum statistics

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sides the ground state particles of mass m, virtual states of multiple masses n m (clusters) occur. This result is not new. In the present paper we have incorporated nonAbelian internal symmetries, actually isospin invariance. In this case there are coherence effects not only between equally charged particles (as one would expect naively) but also between different charge states of a given isomultiplet. In other words, in the cluster expansion there occur not only rr+~r+, 7r0rr°, 7r+rr+rr+.... Bose clusters, but also rr+rr0, 7r+rr- clusters etc. One of our main results is the calculation of the isospin weights of configurations containing such Bose or Fermi clusters. These isospin weights are given by eq. (23). In order to estimate the effects of such quantum statistics corrections, an isospin invariant statistical bootstrap model with Bose statistics has been formulated and solved. In particular, the question has been studied wether the neutral excess, found in annihilation reactions, could be understood as an excess of neutral pions caused by Bose statistics, or not. As discussed in sect. 3, the answer is negative. One could make the objection that the simple statistical bootstrap model studied in sect. 3 is not representative for statistical models at all. In fact, the decay chain suppresses coherence effects. They are present only for particles emitted in one and the same decay generation. Since for isovector fireballs the neutral-charged ratio slightly increases, compared with Boltzmann statistics (fig. 1), one should expect a further slight increase for pure phase-space models. However, the numerical values shown in fig. 1 did not encouraged us to perform the corresponding calculations for pure phase space. We do not expect any significant effect. In any case, for pion systems of total isospin I = 0 the average numbers of n+nOn - must strictly be equal. The fact that statistical models (and in particular tile model discussed here) are not able to explain the excess of neutral energy may not be such a serious failure as it looks. The origin of the neutral excess is experimentally and theoretically - not well understood. We do not try to give a detailed discussion here [12]. What one should emphasize is that it is by no means clear that the neutral excess is due to neutral pions, that strong interaction processes are responsible for it. In any case, there are contributions from electromagnetic decays which do not have to respect isospin invariance. When we do not find a significant effect of Bose statistics on the neutral-charged ratio this does not mean that the quantum statistics effects are always small in a statistical model. In particular, the study of the short range correlations in the azimuthal angle deserves further attention. I am indebted to many colleagues for discussions, comments and criticism, particularly R. Hagedorn, M. Chaichian, G. and J. Ranft and U. Wambach. I would like to thank D. Amati for the kind hospitality of the CERN Theory Division where this work was completed.

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References [ 1] [2J [3] [4] [5]

M. Chaichian, R. Hagedorn and M. Hayashi, CERN preprint TH. 1975 (1975). J. Ranft and G. Ranft, preprint KMU-ItEP-7501 (1975), and references therein. U.M. Wambaeh, Dortmund preprint (November, 1974). B. Richter, Proc. 17th Int. Conf. on high-energy physics, London, 1974, p. IV-37. C. Ghesquiere, Proc. Syrup. on antinucleon-nucleon interactions, Liblice-Prague, 1974, p. 436. [61 F. Cerulus, Nuovo Cimento 19 (1961) 528. [7] R. ttagedorn, Nuovo Cimento Suppl. 3 (1965) 147. [8] S, Frautschi, Phys. Rev. D3 (1971) 2821. [9] K. Fabricius and U.M. Wambach, Nucl. Phys. B68 (1974) 349. 1101 J. Kripfganz, Proc. 5th Int. Syrup. on many-particle hadrodynamics, Eisenach-keipzig, 1974, p. 643. i l l 1 C.11. Llewellyn Smith and A. Pais, Phys. Rev. D6 (1972) 2625; M. Moshinsky et al., Phys. Rev. DIO (1974) 1587. [12] tl.J. Mohring, preprint KMU-ttEP-7506 (1975); F. Csikor, Budapest preprint No. 346 (1975).