Time-Longitudinal Dilations and Scaling Behavior of High-Energy Collisions

The assumption that a dilation transformation in time and one space direction is implementable for matrix elements of certain operators between states with constituent particles moving mainly along that direction with large momenta, leads to scaling of inclusive processes in the high-energy limit. Without further restriction, certain exclusive processes likewise survive this limit. A nonzero limit for some of these processes, namely, those involving a transfer of quantum numbers, is unacceptable. This difficulty is overcome by postulating that conserved quantities may be broken up into right-and left-moving ones, and each is separately conserved.

Time-Longitudinal Dilations and Scaling Behavior of High-Energy CollisionsM yron Bander DePartment of Physics, University of California, ovine, Iruine, Califorriia 92/64   (Received 10 February 1971)   The assumption that a dilation transformation in time and one space direction is implementable for matrix elements of certain operators between states with constituent particles mov- ing mainly along that direction with large momenta, leads to scaling of inclusive processes in the high-energy limit.Without further restriction, certain exclusive processes likewise survive this limit.A nonzero limit for some of these processes, namely, those involving a transfer of quantum numbers, is unacceptable.This difficulty is overcome by postulating that conserved quantities may be broken up into right-and left-moving ones, and each is separately conserved.I. INTRODUCTION   It has been recently noted that rates for many high-energy processes have certain scaling prop- erties; they are functions of dimensionless ratios of variables describing these processes.It was conjectured by Bjorken' and experimentally confirmed' that the structure functions of inelastic electron scattering depend only on the ratio of the energy transfer and momentum transfer squared.This relation holds true only in the region where both variables are large.Likewise, for inclusive processes, Feynman' suggested that the spectrum of a definite particle emerging from a high-energy two-body collision depends only on the ratio of longitudinal momentum to center-of-mass energy, q, ~/vs, and on the transverse momentum q~.
Such dependences on ratios of dynamical quantities are suggestive of the invariance of the forces causing these processes under scale transfor- mations, ' x-xx, p-p/x.(&) Invariance under such transformations implies the nonexistence of a scale of length and thus requires that all physical quantities depend only on dimensionless ratios.It is clear that the transforma- tions generated by Eq. ( 1) are, at best, approxi- mate symmetries valid at high energies.In this paper we shall be interested in processes initiated by two particles with large total center-of-mass energy.The experimental observation that transverse momenta' are bounded indicates that the complete four-dimensional dilation invariance is too strong.With this in mind, we shall explore invariance under time-longitudinal dilations (TLD), formations imPli ed by Eq. ( 2) are valid for matrix elements of local oPerators, or Products of local operators, taken between states of large total mass and zoith all constituent Particles having large longitudinal and small transverse momenta.
We shall refer to such states as longitudinal states (LS).We note that in the center-of-mass system, the initial state of a high-energy two-body collision is a LS.
In the subsequent sections w'e shall examine the consequences of the assumption that the above transformation may be implemented.Among these are Feynman's conjecture' on inclusive processes, as well as the survival of certain exclusive processes at high energies, namely, those dominated by the exchange of a Pomeranchuk trajectory.Without further assumptions, we find that exclusive processes with an exchange of quantum num- bers likewise survive at high energies.As this is unacceptable, there must be a scheme which en- sures that the matrix elements for these reactions do vanish.We propose a scheme where any con- served operator may be decomposed into a part acting only on leftor on right-moving constituents.
The assumption that each part, separately, is a symmetry eliminates the nonzero limit of the un- wanted processes.
Before closing this section, it may be worth- while to present a different way of understanding the transformations of Eq. ( 2) and Eq.(2).Bather than considering them to be a symmetry of some truncated Hamiltonian, they may be considered a symmetry of the equations describing some approximate model, as for instance, the multiperipheral model.Scaling properties of exclusive reactions have been obtained from these models.' II. ASSUMED SYMMETRY Again we expect this to be valid under very limited circumstances.

III. DIMENSIONS OF SOURCE CURRENTS
A. Scalar Particles If IP(x) is some local operator normalized through the condition (2&)'a2p, &0 ly(O) I p& =1, where lp& denotes a state of the particle of interest, the source current j(x) is defined by (s'+ m') Ip(x) = j(x) .
We wish to determine the dimensionality of j(x) under TLD.Let In&, IP& be any multiparticle longitudinal states and let I p, n& be the state formed by adding the particle of interest to the state In&.Let lp, n& likewise be a LS.Using the standard reduction technique, we obtain (27i)"'V'2po {0I p(0) I p, )I& =u(p, )I. ), with u(p, )I) a Dirae spinor for a particle of momen- tum p and helicity X.The source current for the particle under consideration is obtained from (ty'm))t)(x) =)((x) .We again use the reduction formulas on states formed analogously to those of the previous section: {p; out I p)I., n, in) ~-ip ' g =i ""~2 {p, out I x(x)u(p, )I. ) I n, in) .( 12) In this limit, we may consider the spinors to be of the form vpp (1 k(TII )/2 for helicities +-, ', respec- tively.Performing a TLD on both sides of E(I.( 12), we obtain (l*o )U()))((x)U'(~)=)'i'(1+o"))((~t, ~x, x,).( 12) Repeating this argument for the reduction of an antiparticle state, we get the unrestricted relation o"(p;outljt(x) In; in) .
Thus we find that the d of Eq. (4}, for scalar (or pseudoscalar) fields is d= -2.
We now concentrate on high incident energies and assume that i a, t)) is a LS.With the dimensions obtained in the previous section the following scaling properties hold: W (pll, pir s) =)t.'W (p /)l.r pi, s/)l.') W(/So)(pll, p"S) = )l.' W(""()@(pl, /)l., P"S/)l.'),W (p p S) =) (o(p)+ a(o))W(1) (p /) p, S/)l. ) with a(y. ) = 0 for p, = 0 and parallel, and a(t), ) = 1 for p. transverse.Implementing these scaling laws, we note that the above structure functions may be A more complicated inclusive reaction is one in which two particles are detected: a+b c+d+X.This may be generalized to multiparticle inclusive spectra.

V. EXCLUSIVE PROCESSES
Pursuing the ideas of time-longitudinal dilations, we may obtain scaling properties for exclusive reactions which are not so welcome as those for inclusive reactions.There are two typical exclu- sive reactions for which we may obtain such results: In this reaction, c and d are detected and all other particles are summed over.The cross section in this case is proportional to (a, t)IT*(j,(x) j~(y)}T*(j,(z) j~(0)}ia, b), (24) where T*(j,(x)j, (y)) is related to the ordinary time-ordered product by T *(j,(x) j (y) } = T(j,(x)j (y)) -&(x-y)e(x) -e 5(x-y)y (x) -"( 25) where lP(x), lP, (x) are local operators, and the subtractions ensure that T* is Lorentz-invariant.' Utilization of the reduction techniques permits us to establish the dimensions of these extra operators.Concentrating for definiteness on the case where c and d are spinless, we find U(~) T*( j,( )j, (y)) U'()l, ) =)l.T*( j ()lx, )l xll, xi)ju ()l yo )lxll )l ~)) (26) From the above, we obtain the scaling property for the cross section: