Reggeon field theory and markov processes

Volume 77B, n u m b e r 2

PHYSICS LETTERS

31 July 1978

REGGEON FIELD THEORY AND MARKOV PROCESSES P. GRASSBERGER and K. SUNDERMEYER Faculty o f Physics, University o f Wuppertal, Germany Received 24 April 1978

Reggeon field theory with a quartic coupling in addition to the standard cubic one is shown to be mathematically equivalent to a chemical process where a radical can undergo diffusion, absorption, recombination, and autocatalytic production. Physically, these "radicals" are wee partons.

It has often been observed that Reggeon field theory (RFT) has many features which are more typical for a stochastic process than for a quantum theory. This starts with the fact that the evolution parameter is X/Z-i" × rapidity, so that the equation of motion of the free theory is formally a diffusion equation rather than a Schr6dinger equation. Later [1] it was shown that a bare pomeron corresponds also physically to a random walk in impact parameter. Finally, Alessandrini et al. [2] have observed that the classical approximation to RFT corresponds mathematically to a non-linear diffusion process. The physical basis for these analogies rests in the non-covariant parton model [3,4]. It was shown in ref. [5] that essential features of RFT can be understood if one makes the ad hoc assumption that only wee partons interact both in the collision between two hadrons and during a boost of a single hadron *1. Wee partons in a given Lorentz frame are those with momenta of order unity. The "Schr6dinger equation" of RFT then describes the evolution of n-wee-parton densities during a boost. During a boost, four basic things can happen to wee partons: (i) they diffuse in the impact parameter plane and stay wee; (ii) they become hard, and thus get effectively lost; (iii) they split into two and thus stay wee; (iv) two of them combine to form a parton which ,1 For a critique of this assumption, see ref. [6].

220

can be either wee or hard. In the following, we shall neglect for simplicity diffusion, i.e. we consider reggeon quantum mechanics in zero transverse dimensions. It will be restored at the end. Analogous processes occur frequently - and have been studied in extenso [7] - in chemical reactions. Consider a mixture which contains two types of molecules M, N, the numbers of which are kept constant, and a radical R which can undergo the following reactions: kl

R+M~A, R+N~ R+R~

k2

v

(1) 2R,

(2)

B,

(3)

where A and B are chemically inert, and where kl, k2, p and v are the respective rate constants. Eqs. ( 1 ) - ( 3 ) correspond precisely to the basic processes (ii)-(iv). The distribution Pn of the number of radicals of type R satisfies the master equation [7] ( d / d t ) P n ( t ) = p [(n + 1)Pn * 1 - nPn ] + o[(n -

1 ) P n _ 1 -- n P n ]

(4) + lp[(n

+ 1)nPn + 1 - n ( n -

1)Pn]

+½v[(n + 2)(n + 1)Pn+ 2 - n ( n - 1)Pn] , where (n M and n N are the numbers of molecules of

Volume 77B, number 2

PHYSICS LETTERS

type M,N)

31 July 1978

( d/dy)tpk(y) = -

19 = kln M ,

(5,6)

o = k2n N .

For the generating function of this distribution,

:-(1 -

(1/~.)(Ola k HI ~#>

ap)k¢k --½ir{

(k- 1)~/~k_ 1

(15)

+ k x / k + 1 ~k+ 1 ) - ¼ X k ( k - 1 ) ~ .

oo

a(z, t) : ~

n=0

Pn(t)z n ,

(7)

This last equation is equivalent to eq. (I 1) provided we associate

y +-+ t,

we then find

V~.~Ok/~O0 +~ ( - ( 2 v + p)/2o)k/2nk, (O/Ot)G(z, t) = { p(1 - z)O/Oz + oz(z - 1)b/Oz

(8) +/.t]'Z(1 --z)O2/Oz 2 + v~-(1 -- z2)O2/az 2} G(z, t) ' The binomial moments, (9)

are related to G by

t)tz:

(10)

1,

and thus satisfy

(d/dt)nk(t) = (o - o)kn k + ok(k - 1)nk _ 1 1

--(Tp

+

v)knk+ 1 - ½ ( p + v ) k ( k - 1 ) n

(11)

(16)

p--o,

+--+ 2(p + v),

amplitude. In order to see this, let us consider the RFT hamiltonian H = ( 1 - a p ) a t a + ~ lir .a t ( d f +a)a+-gLa , 2a2.

(12)

As we had already mentioned, we neglect for the moment the pomeron slope, whence we have replaced the pomeron field by annihilation and creation operators a, at, which satisfy al0) = 0 .

(13)

Also, we have included a quartic pomeron coupling in addition to the standard cubic one. The evolution of the k-pomeron amplitude, ~akO') = (1/V~-!)
?

True RFT with C~p 4 : 0 can be discussed in complete analogy, by replacing the generating function G(z, t) by a functional G ( [ z ] , t), and including a diffusion term. There is just one problem: a strictly local interaction is incompatible with positivity of probabilities (see also ref. [5] ), although it can be arbitrarily closely approximated. Neglecting this mathematical subtlety *2, we replace eq. (8) by ,3

k.

Our crucial observation is that this equation for the kth binomial moment of the parton distribution is essentially the same as the equation for the k-pomeron

[a, at] = 1,

+-+

r +-+ -2V{(½la + v)o.

nk = ,

nk(t) =(~/Oz) kG(z,

l--Otp

(14)

during a boost of the arbitrary state I~O, )) is given by

~ G ( [ z ] , t ) = f d2b {~'pV2z(b) s A ) 6 + p(1 - z ( b ) ) ~zz(~ + oz(b)(z(b) - 1) 82 + u½z(b)(1 - z(b)) - 8z(b) 2

+ p-~(1 - z2(b)) ~

6

8z(b) (17)

t G ( [ z ] , t).

5z(b) z J

All further details are left as an exercise for the reader. We just mention that k-pomeron wave functions become equivalent to k-wee-parton densities, as claimed in ref. [5]. Let us end with some remarks, first about RFT and then about chemical reactions. ,2 Which corresponds to the need for infinite renormalization in the standard formulation of RFT. ,a Another - mathematically rigorous - possibility would consist in putting everything on a lattice. 221

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PHYSICS LETTERS

For eq. (4) to be a true master equation, O, o,/a and v must all be non-negative. This implies k>0

i f r q: 0 ,

(18)

thus standard RFT, with a cubic coupling only, is not equivalent to a Markov process in the above sense. Another consequence o f the non-negativeness of the transition rates is X(1 - ap)/r 2/> - 1.

(19)

The equality sign holds only when p = v = 0,

(20)

and has been studied in detail by Bronzan and Sugar [8]. If and only if eq. (20) holds, there exists a stationary Poisson distribution (for the problem without diffusion) with (n) = 2o/~t which corresponds to the coherent eigenstate o f H found in ref. [8]. This is easily understood, as eq. (20) means that partons during a boost always stay wee in the sense that they can interact even when their momenta become arbitrarily large. This is o f course in conflict with the general parton philosophy and might explain the problems with t-channel unitarity found in ref. [8]. The present note should clarify certain aspects o f R F T which are very obscure in the standard formulation, but it poses also some problems for RFT. In ref. [5], it was pointed out that RFT corresponds to a p a r t o n - p a r t o n S-matrix which is non-unitary in the s-channel - provided partons and pomerons are related as above. In that paper, the relationship was however only conjectured, and therefore this problem was not taken very seriously. As we now have found this relation to be mathematically exact, we have to take the unitarity problem more seriously - although, we admit, it is seen only at the partonic and not at the hadronic level. Another intriguing question is why one should be allowed to treat wee partons like classical particles, as we did in this paper. The last and most crucial question is whether it is justified to neglect the interaction between hard partons. Indeed, all known theories (and data as well!) show some hard scattering effects which make the wee parton evolution non-Markovian. In that case, if one can derive anything similar to RFT, one should find a reggeon lagrangian which is nonlocal in rapidity. The critical behaviour of that theory might be '~ery dif222

31 July 1978

ferent from that o f standard RFT. Non-trivial consequences of our observation for chemical reactions follow from the fact that RFT exhibits for certain parameter values a second-order critical phenomenon. Due to universality o f such phenomena, the critical indices should not be altered by adding a quartic reggeon coupling [9]. Thus, results from RFT can immediately be taken over to chemical systems with reactions ( 1 ) - ( 3 ) . The only difference is that the physical number o f dimensions becomes D = 3 instead o f D = 2 in RFT. Assume that, at time t -- 0, a certain number of radicals is implanted at the origin x = 0. At the critical point, the density o f radicals becomes then asymptotically

O(x,t)

~

tUF(x2/tv),

(21)

t--+ ~

where F is a calculable function, and/~ and v are critical indices. Expansion in e = (4 - D ) gives for e = 1 [10] : kt = - 2 + 1~ e + ( ~ In ~4 *. 5X, ,),t ~ e ) 2 + O ( e 3 ) ~ _ l . 4 7 (22) v = l + ~ 4 e + ( ~ l n5 9~ + ~4) ~ e 7) 9 x / 1 "2 + O ( e 3 ) ~ 1 . 0 6 . (23) Although these expansions are only asymptotic, the numerical values should be roughly correct. One of us (P.G.) wishes to thank F. Gu6rin for her collaboration during early stages o f this work, and F. Henyey for very instructive discussions. [ 1 ] V.N. Gribov, Proc. VIIIth Winter LNPI School (Leningrad, 1973) Vol. II, p. 5; F.S. Henyey, Phys. Lett. 45B (1973) 363,469. [2] V. Allessandrini, D. Amati and M. Ciafaloni, Nucl. Phys. B130 (1977) 429. [3] R.P. Feynman, Photon-hadron interactions (Benjamin, New York, 1972); 5th Hawaii Topical Conf. on Particles, eds. P.N. Dobson et al. (Univ. Press of Hawaii, 1974) X. [4] J. Koplik and A.H. Mueller, Phys. Rev. D12 (1975) 3638. [5] P. Grassberger, Nucl. Phys. B125 (1977) 83. [6] L. Caneschi, I. HaUiday and A. Schwimmer, preprint CERN-TH-2389 (1977). 17] K.J. McNeil and D.F. Walls, J.Stat. Phys. 10 (1973) 439; H. Haken, Rev. Mod. Phys. 47 (1975) 67. [8] J.B. Bronzan and R.L. Sugar, Phys. Rev, D16 (1977) 466. [9] H.D.I. Abarbanel et al., Phys. Rep. 21C (1975) 119. [10] J.B. Bronzan and J.W. Dash, Phys. Lett. 51B (1974) 496; erratum,Phys. Rev. D12 (1975) 1850.