Hydrodynamical approach to the electron-positron annihilation into hadrons

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PHYSICS LETTERS

30 September 1974

HYDRODYNAMICAL APPROACH TO THE ELECTRON-POSITRON ANNIHILATION INTO HADRONS E.L. FEINBERG P..N. Lebedev PhisicalInstitute, 117924 Moscow, USSR Received 17 July 1974 The problem is reconsidered and it stressed that viscosity must be taken into account. This makes superfluous introduction of an "initial hadronic volume" Vo - m~rI which in the e+e- case can hardly be physically motivated.The viscosity becomes the single source of dissipation. Rough estimate of multiplicity is presented. Further, black body electromagnetic radiation from the hot hadronic clump turns out to be essential, It leads to an increase of energy spent on neutral component.

Introduction. Peculiar features ofe+e - ~ hadrons annihilation at the total CMS energy E--- 3 - 5 GeV [1] have stimulated many attempts ofhydrodynamical treatment of this process [2]. The features important in this respect are: 1) The average energy ofhadrons is almost E-independent (maybe somewhat increases with E ) and is close to the energy of Bose particles at the temperature T ~ m~r (pion mass). 2) The energy spectra of produced 7r, K and/~ are o f e x p ( - E i / T ) type with the identical T ~ 0.16 GeV ~ rnTr. 3) Relative abundances ofrr, K and ~ do not contradict the thermodynamical predictions for the same T. However, 4) the energy of produced neutral particles exceeds the value given by isotopic invariance. This contradicts the simple hydrodynamically-thermodynamical picture as well. 5) Charged multiplicity seems to be (nearly?) constant which never has been met in calculations on hydrodynamics of hadronic collisions. Although they are preliminary, these data deserve attention. All hydrodynamical treatments of this process [2] used directly the classical Landau theory developed for the case of the central hadron-hadron collision. We wish to point to the necessity of going beyond the frames of the Landau basic assumptions. We also give (rather rough) estimates of possible influence of two phenomena not taken into account by Landau. They are: the viscosity as a source of the entropy increase leading to multiple production, and the direct production of 3,-quanta (possibly also ofe+e - pairs etc.) in the process of expansion of the hot hadronic "plasma" (this is essentially the black body radiation which may be called also "fluctuation radiation" since it is due to thermal fluctuations

of charge density). Both these phenomena, as will be seen, can be essential in the process under consideration. The process differs from the one treated by Landau in two respects: 1) Here the colliding particles do not have intrinsic natural length parameter (at least the one of the order of m~-1), the "initial size" of the hydrodynamical system may be arbitrarily small and the initial entropy may be small correspondingly; shock waves cannot secure its increase. 2) The system is spherically symmetrical (at least approximately), instead of being the single dimensional one as in hadronic collisions, and the early stage of expansion with high temperature plays quantitatively different part. The most puzzling experimentally found property of the process is the constancy of annihilation cross section, o ~ 20 nb ~ rt(lO-16cm) 2. This problem is beyond the scope of the hydrodynamics. It seems remarkable that such a o corresponds, on one hand, to the critical weak interaction length, r o ~ lO-16cm; on the other hand to the electrostatic energy e2/ro~ 1GeV which is of the order ofhadronic masses. Thus the problem of the e value might pertain to the still nonexisting universal theory of strong, electromagnetic and weak interactions. State equation o f the hadronic plasma. We assume state equation p = c2e, p being the pressure, e the energy density, c 2 is the sound velocity. Landau used c 2 = 1/3 which holds for the extremely relativistic electronphoton gas. For the hadronic plasma this may hold if T ~ m n, rn N and the theory is renormalizable [3]. Generally speaking any c 2 ~< 1 is possible. In our case initial temperatures are actually high and c 2 = 1/3 may be

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reasonable. In general case e ~ T I/c2 +1 and the entropy density s ~ e/T. Viscosity. Its importance in any hydrodynamical process is measured by the value of the Reynolds number which is the ratio of kinematic terms Tiok = (e + p ) u iu k + p 5 ik (u i is four velocity) to dissipative ones r ik in the full expression for the energy momentum tensor/n'k= ~ k +rik ' i.e. Re ~ IZiokl/Irikl. Viscous terms are present in r combinations like V Out/ axluiu k and rlaui/Ou k, rl being the viscosity coefficient. Since p ~< e and Oui/~x k ~ (u ° - 1 ) [ L , L being the characteristic macroscopic length parameter, then Re ~ e L / [ ( u o - 1 ) n ] . The value of ~7is of fundamental importance. If the smallest microscopic length parameter is T -1 , which is true when T>> m~, m y and all masses fall away, then both the dimensionality considerations [4] and the approximate field theoretical calculations [5] lead to r / ~ T 3. This value oft/will be used below although for c 2 :~ 1/3 it is open to criticism (see below). Further, u ° - 1 usually is not large u ° - 1 ~ 1 . We define correspondingly: Re = e L / ~ 3 .

(1)

In the one dimensional case L is thickness of the system in the given stage of expansion and e L m ~ 2 is the total CMS energy E. We shall put m~ 1 to be unit o f length, m , the unit o f energy. Therefore, when E = 5 GeV we have E ~ 34. Thus Re = E / T 3 .

(1 a)

In the spherical case L is the radius o f the system, R, a n d E = e R 3, Re = E / T 3 R 2 .

(lb)

Combining these equations with the formula e ~ T 1/e2+1 we obtain for the one dimensional case: Re = (L/Li)3c2/(l+c2),

T = (E/L)c:l(l+ea) ,

(2) L 1 = E-(1-2c2)/3c2 . For the spherical case: Re = ( R / R 1 ) (7c2 -2)/(1+c2), R 1 = E-(t-2c2)/(Tca-2).

T = ( E / R 3 ) c2/(1+c2) , (3)

Decay into final particles occurs at T = Tk "" rn~r = 1, i.e. when, in the spherical case, 204

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R = R k ~ E I/3.

(3a)

For the linear case the same procedure gives L k ~ E, however this value is fictitious: long before L becomes E, at L ~ I the one dimensional expansion goes over into a three dimensional one. The true size at the moment of decay is L ~ In E. When Re <~ 1 viscosity secures dissipation. When Re becomes large expansion becomes isentropic. We shall estimate the influence o f viscosity on the final entropy and thus on multiplicity (n) substituting for all characteristics their values averaged over the system volume. Therefore, we assume that they are distributed rather uniformly over the volume. In the linear case (n) ~ S 1 = s (L = L 1)L 1 and we have [4]: (n) "-E2/3

(one dimensional)

(4)

for any c 2 (while when viscosity is neglected (n) ~ E 1/2 at c 2 = 1[3; it is even smaller for higher c2). In the spherical case the result ,to a great extent depends on c 2. I f c 2 < 2 / 7 , then for s m a l l R , R < R 1 , we have Re > 1 and dissipation is quite low. Decay into separate hadrons takes place at R < R 1. However i f c 2 > 2/7, then S 1 = s ( R = R 1 ) - R ~ gives: ( n > ~ E (6c~-2)/(7c2-2)

(spherical, c2 > 2/7) . (5)

Thus (n) = const, at c 2 = 1 / 3 ,

(5 a)

( ( n > ~ E 2/3 at c 2 = 1/2). With the present accuracy (or, better to say, inaccuracy) of estimates, from eqs. (3a) and (5a) we have the average energy o f the produced particle ( E i) "" E l ( n ) increasing as the overall CMS energy E. However, this~holds only if we neglect the energy going over to other possible degrees of freedom and we are presently going to discuss this effect. Electromagnetic radiation. It was shown in refs. [6] that in the process ofhydrodynamical expansion the hot hadronic plasma must emit 3,-quanta, (and in the next order in e 2 also electronic and muonic pairs). If E is not very large 3,-quanta freely go out and are not reabsorbed in the plasma. Their number (n,r), relative to the number of pions, (n~r), slowly increases with energy. For the central collision in the Landau theory, it was found as (n,r)/(n ~> ~ const .e2( n~>1/3 , const • ~20, (n~) ~ E 1/2. Let us transfer calculations o f the

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number of photons performed for a linear case [6] to the spherical one. Here, as already has been mentioned, we have no intrinsic length parameter and the initial radius o f the hadronic system can be arbitrarily small, maybe even Rmi n ~ V ~ ' ~ 10-16cm ~ 1 0 - 3 m ~ 1. In this case the initial temperature is high, T O (E/lO-9) c2/(1+c2) i.e. for E = 5 GeV ~ 34 and c 2 = 1/3 To ~ 400m~. The 7-quantum energy radiated within the element d4x according to dimensionality arguments is dW~t = Ae 2 T5d4x, the number of quanta (of the average energy ~ T) is dn,y = Be2T4d4x (A, B unknown constants o f the order o f one) n.r is an invariant and may be calculated in the overall CMS:

(n v >= Be 2 f T4d Vdt.

(6)

Assuming that T is the volume averaged quantity (3), a n d R = R(t), d t = (dt/dR)dR ~ dR, we have for the number o f quanta radiated during the expansion ( V = R 3)

Rk dR (n.r) = Be2E4e2/(l+c 2) f R(9e2-3)/(l+c 2)

.

(7)

Rmin Of course, the energy of quanta emitted at different stages of expansion varies greatly. Using (3a), we have, assuming R k ~ R m i n :

( ER k ~ E 4/3 ,

c 2 = 1/3;

] 4'~ Rk El/3 c 2 (n~) ~ Be z X 1 E t~ln Rmi----~ ~ E4/31n Rmi----n,

~E4C2/(1+c2) l + c 2 8c 2 - 4

=

1/2;

1 , c2>i/2. (8e~-4)/(1+c2)

Rmin

Since (n~) for c 2 = 1/3 is constant, we have, (n 7)[(nn) = Be2E4/3/(nn),

(8a, 9)

which even at E = 5 GeV ~ 34 can be markedly large (although (n~r) ~ 10). Since in this case the integral (7) is determined by the upper limit, where T ~ ran, the dominating energy o f quanta is the same as for pions. This is ~ 0.5 GeV if we assume that hydrodynamical flow does not influence very essentially their energy at least at E <~ 10GeV. If c 2 exceeds 1/3 and Rmi n is small, then (n. r) is even greater and for c 2 > 1/2 the emission at earlier stages of expansion, R ~ Rmin, T >> m,r, predominates.

30 September 1974

The effect is essentially very simple. Charge density fluctuations in the hot plasma at T~" m~r have the linear size equal to the smallest correlation length, l ~ T -1 , volume "- T - 3 , their time variation is determined by time correlation length r which is again ~ T - 1 . Thus in the volume ~ R 3 we have at any moment v ~ R 3 T 3 "elementary charged fluctuations" which emit "y-quanta with some effective squared charge Be 2, the frequency being ~1 --1 ~ T*. During the life o f this clump with the radius R this happens ~ R/r ~ R T times. If the f'mal stage dominates, c 2 = 1/3, R ~ R k, then the number o f emitted quanta is (see eq. (3a)) (n.t >~e2BR 4 T4, just as in eq. (8). It is probably worth reminding that usual radiation of n~r charged particles emitted when they are generated gives only n~ "~ e2n~ quanta. The radiation has an upper bound if the equilibrium with the pionic degrees o f freedom is established. In this case the energy is equally distributed among 5 degrees o f freedom (n +, lr-, n ° and two 7-polarizations, YL and "YR)" Thus the "t's, arising also from ~ro, carry away (3/5)E, while in the absence o f direct, black body production this figure is only (1/3)E. The threshold energy Eth for appearance of such an equilibrium may be estimated e.g. from the condition that the energy of 7-radiation becomes of the order of E. For the linear case it was found in ref. [6], Eth 300 GeV. On the other hand we may take as a basis the reabsorbtion o f radiation within the lump (as suggested by E.V. Shuryak). The corresponding condition may be written as a w R = 1 where a w is the extinction coefficient o f a plane wave of frequency co within the plasma. According to eq. (4.6) in ref. [6], for co ~ T, a w ~ e2T. Thus Ra w = Ce2E1/4R 1/4 (for any c2), Cbeing some constant of the order o f one. At R = R k Oa) this means Ce2E~ 3 = 1, Eth = (Ce2) -3. This condition is rather severe. In ref. [6] various estimates *Here from follows an additional condition on integration in (6), (7): the system should be large enough to contain z, > 1 "elementary fluctuations", otherwise the charges are distributed uniformly and may not radiate. According to f3) RT = Ee2/(I+c2)R(I-2e2)/(I+c2). It is seen that v > 1 is satisfied fore 2 l/2thecondition becomes R(2c~ -1)1(1+c2-) < EC2l(l+c2), which also holds, since here the lower limit is important and Rmin~E7 Where "r is any positive power. A similar condition (with somewhat different argumentation) was imposed in ref. [6 ] where it actually should be neglected, since in the transversedirection (in the linear case) always u ~ L T~> 1 holds since L ~ L.L ~ 1, while it was taken for L =LII~L±,which is tmnecessa~. 205

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showed that under e 2 we should understand e 2 = 4n/ 137 -~ 0.1. Even in this case Eth seems to be very large and exceeding actually used energies. Thus it seems quite possible that all ~/-radjation goes freely out of the clump like neutrinos go from the interior of the Sun. Discussion of results. Thus, viscosity secures necessary dissipation even in absence o f other phenomena (shock waves in the Landau theory etc.). The spherical case pertains not only to the e+e - -> hadrons annihilation, but also to the decay of a point-like (or almost point like, r o < m ~ 1) heavy (mass ~ 5 GeV) parton. If can be treated as a "point-like explosion". At c 2 = 1/3 multiplicity should be close to the constant one. Second, direct production of'/-quanta (at essentially higher E also of electronic and muonic pairs, the mass is not very essential [6]) can be important and can explain the increase of energy going over into neutral component. The experiment must show whether this excess of neutral particles actually refers to T-quanta. The weak points in the above reasonings are: 1) The choice 17 ~ T 3 is well motivated i f c 2 = 1/3, but is dubious for other c 2, since masses or other dimensional quantities besides T can appear in ~. This can influence estimates of (nn). It may indirectly influence the estimate of (n, r) as well but apparently only slightly. 2) In all estimates the averages over the volume were used. Meanwhile, it was shown (in the viscosity free case) [2] that in the spherical case the entropy tends to be concentrated closer to the surface. However, for not too large E the volume of this region does not differ drastically from the total volume. We may expect that viscosity additionally helps to makes the system more uniform.

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3) In estimating 1/2 situation is muc h better). I am very grateful to D.S. Chemavsky for stimulatin~ discussions, to J.D.B. Bjorken for kindly communicatin~ to me the SLAC experimental data and for the interest shown to the discussed ideas, and to E.V. Shuryak for a helpful remark.

References

[1] J.D. Bjorken, private communication; C.H. Liewelynn-Smith CERN preprint TH.1849, 28 March 1974. [2] E.V. Shuryak, Phys. Lett. 34B (1971) 509; also the paper presented to the Vth Intern. Symp. on Multiparticle hadrodynamics, June, 1974, Eisenach-Leipzig; F. Cooper, G. Fry, E. Sehonberg, Phys. Rev. Lett. 31 (1974) 375; M. Chaichian, E. Suchonen, Bielefeld preprint, Bi-74/04, January, 1974. [3] H. Ezawa, Y. Tomozawa, H. Umezawa, Nuovo Cim. 5 (1957) 810. [4] E.L. Feinberg in Quantum field theory and hydrodynamics, Proc. Lebedev Physical Institute, 39 (1966) 165 (Consultant Bureau, New York). [5] C. lso, K. Mori, M. Namiki, Progr. Theor. Phys. 22 (1959) 403. [6] E.L. Feinberg, Izvestia Akademii Nauk, Ser. Fyzicheskaya, 26 (1962) 622; 34 (1970) 1987 (both in Russian). Short communication: Proc. IX Conf. High Energy Physics, Kiev, 1959, vol. 1, p. 220, Moscow, 1960.