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Classical kinks and their quantization in supercritical Reggeon field theory
Nuclear Physics B130 (1977) 429-485 0 North-Holland Publishing Company
CLASSICAL KINKS AND THEIR QUANTIZATION IN SUPERCRITICAL REGGEON FIELD THEORY V. ALESSANDRINI Laboratoire de Physique Thbrique et Hautes Energies, Orsay *
D. AMATI CERN, Geneva
M. CIAFALONI Scuola Not-male Superiore, Pisa and INFN, Sezione di Piss Received 19 July 1977
The existence of kink states in supercritical reggeon field theory ((~(0) > arc) is related to the expanding disc behaviour previously found in a lattice approximation. The classical solitary waves of RFT with o(O) - 1 = ~1> 0 are classified, for the case of a one-dimensional impact parameter space (D = 1). We find kink-like solutions corresponding to a range of velocities up to a critical one for which the solution has a phonon-like dispersion law (zero mass). The general methods of kink quantization are adapted to RFT. By then quantizing the theory around the solution with critical velocity we are able to give the low-lying spectrum and to construct the quantum kink states, which in RFT are connected to the vacuum by field operators. Their contribution to the Green function is found to restore the cluster property by cancelling, at large impact parameter, the constant term provided by the second ground state present in the ordered phase. For the impact parameter IBI < 2& logs the Green function tends to a constant proving therefore for D = 1, the expanding disc behaviour which is consistent with s-channel unitarity.
1. Introduction The peculiar nature of the critical phenomenon characteristic of the reggeon field theory (RFT) when the reggeon intercept goes through one, has been recently unraveled by using field quantization techniques apt to identify the phase transition mechanism [ 1,2]. A lattice is introduced in impact parameter space, the RFT Hamiltonian is diagonalized at each lattice point and finally an analog spin model is ob-
l
Laboratoire associC au CNRS. 429
430
V. Alessandrini et al. / Kink states in supercritical RFT
tamed by keeping the two lowest-lying states at each site plus the intersite interactions arising from the kinetic terms in the Lagrangian [1,3]. It was recognized [ 11, by using this approach, that contrarily to what happens in more orthodox field theories, the vacuum state does not change structure as the reggeon intercept goes through the critical value. There is however a drastic change in the spectrum of the theory, in the sense that a new energy eigenstate shows up that is degenerate with the old perturbative vacuum in the thermodynamic limit. Even more important, field operators have finite matrix elements between these two degenerate states in this limit, which means that these two degenerate vacua belong to the same Hilbert space. This peculiar feature of RFT can be traced, as we shall see later on, to the non-hermiticity of the Hamiltonian. The two degenerate vacua of the spin model can be thought of as spatially homogeneous states where all spins are up or down, respectively. It has also been recognized [2], always in the context of the spin model, the important role played by spatially inhomogeneous states of a kink-like type in which a dislocation - frontier between spins up and down - propagates with a definite speed. These states give rise to a low-lying continuum with zero energy gap with respect to the ground states, and they are therefore responsible for the peculiar asymptotic behavior of the Green functions, characteristic of a disk expanding with rapidity in impact-parameter space. Nevertheless the spin model, explicitly tailored to put in evidence the collective modes responsible for the phase transition, mistreats badly other degrees of freedom like, for example, higher single site excitations which cannot be neglected far away from the critical point. This is therefore reflected in an unphysical dependence of the disk expansion speed on the infrared cutoff-intersite distance a in the ordered phase of the theory, when a(O) >ocrtt(O). Strictly speaking the disk expansion speed goes to zero as a goes to zero, so the continuum limit cannot be taken in the spin model, without introducing further degrees of freedom. A possible way out, proposed by Bronzan and Sugar [4], is to introduce a quartic interaction with a coupling X’ = h2/p, where h is the triple-pomeron coupling and p= a(O) - 1. At this “magic” value the lowest states at each site become exactly degenerate and the limit a -+ 0 is harmless. The authors then construct the box states in the continuum limit, by paralleling the treatment of ref. [2], and get a physical expansion velocity. The purpose of the present paper is to propose a more general quantization procedure of RFT based on the observation that the non-perturbative effects generated by degrees of freedom neglected in the spin model are already present at the classical level. In fact, we will be able to identify in the continuum limit (and for one transverse space dimension, D = 1) the classical analogues of the kink-like states of the spin model. The fact that the non-perturbative effects can be treated in the semiclassical limit follows from the property that the RFT Lagrangian scales with X as (l/h’) f?(x$, h$). Therefore, as in ?Q4, X plays the role of a semi-classical expansion parameter, and we expect that as h + 0 the non-perturbative effects are exhausted by the classical solutions. Our method is then similar to the ones recently applied to hermitian field theo-
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431
ries - as Q4 in two dimensions - but there are some peculiar aspects stemming from the non-hermiticity of RFT and/or its lack of Galilean invariance. In fact, as learnt from the spin model, there is just one Hilbert space in the ordered phase, and the kink states are connected to the vacuum by field operators. Moreover, we find that classical kink solutions exist for any speed u, and that for a critical value of u they have a phonon-like dispersion law (zero mass). The quantum kink states have therefore small energy gap for small momenta and contribute in an essential way to the disc’s expansion. The critical speed is just the expansion velocity of the disc - actually a box for D = 1 - and turns out to be the same as in the simple eikonalization of a pomeron above one. This paper is organized as follows: in sect. 2, we rephrase some results of the spin model and recall mainly the properties of the two degenerate ground states. We also present a heuristic argument leading in an intuitive way to the expanding disk behavior of hadronic scattering, based on a generalized coherent state formalism. The details of these calculations are relegated to appendix A because they are not essential for the rest of the paper. In sect. 3, we discuss the classical kink solutions of RFT. We prove the existence of a large class of kink-like solutions and explore its main properties, like dispersion relation and dependence on the propagation speed. Sect. 4 is devoted to a general discussion of the kink quantization of RFT. We have adapted to our case the well-known methods of kink quantization [S-7], adding some further understanding to the treatment of the zero-mode problem in a case in which one is forced - as we are - to use the Hamiltonian path integral formulation instead of the Lagrangian one. In sect. 5 we discuss the dependence of the procedure on the classical solution around which we expand and we show how one is driven to a specific choice in order to avoid linear (tadpole) terms in the quantum fluctuations. This fixes the disk expansion speed. The quantization of the theory is then performed up to order zero in the triple-pomeron coupling, and the RFT spectrum in the ordered phase obtained and discussed. Technical details related to the completeness of our eigenfunctions expansions and the calculation of zero-mode states are discussed in appendix B. Finally in sect. 6 we compute the two-point Green function and S-matrix for high energies finding the disk behavior. We then conclude by a discussion of our results, emphasizing the peculiarities of the field theory we considered.
2. Preliminary
results and semiclassical approximation
We shall adopt in this paper the notations and conventions used before in connection with the spin model [ 11. The Gribov fields J/Q, u) and $(b, JJ) are defined in a D-dimensional impact parameter space b, the rapidity y playing the role of imaginary time. By resealing the Gribov fields according to q=i$,
p=iJ/,
(2-l)
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432
the triple-pomeron as
coupling term becomes real, and the Lagrangian can be written
2 =44&P- WP
+
O’vbq.VbP + hq(q+P>P>
(23
where the mass parameter /Ais related to the pomeron intercept by /.l=cu(O)-
1.
(2.3)
It must be kept in mind that q and p are creation and destruction pectively, satisfying the equal-time commutation relations [P(bt Y), 4@‘,
VII= -6@)(b- b’)9
operators,
res-
(2.4)
the wrong sign of the commutator being of course a reflection of the imaginary nature of the triple-pomeron coupling, that is to say, of the non-hermiticity of the RFT Hamiltonian. The intercept parameter p in eq. (2.2) is in principle the bare one. It can be transformed into the renormalized intercept by adding a suitable mass renormalization counterterm, which is perturbative in A. Since we shall not be concerned in this paper with renormalization effects, we shall ignore counterterms, and accept that the critical intercept is given by ~,.~r = 0. The RFT Hamiltonian reads, in our notation:
H=
Jdb[iw -
a’vq- VP - Wq +p)Pl
(2.5)
and the two states which play in the spin model [ 1,2] the role of degenerate vacua in the ordered phase /J > 0 can easily be identified. These states are in fact in one to one correspondence with the stationary points of the classical action corresponding to constant field configuration, i.e., (q, p) = (0,O); (p/h, 0) and (0, p/X). They are obtained by remarking that constant field configurations correspond to zero-momentum components of the Gribov fields, and that the zero-momentum part of H is just the RFT Hamiltonian in zero transverse dimensions, whose spectrum is well known [8,9], with h replaced by A/&‘*. The lowest-lying state is the perturbative vacuum I&), defined by PlGfkJ= 0,
(4014 = 0 ,
(2.6)
which, because of the ordering properties of H, is always an exact zero-energy eigenstate. This state does not change its structure at the critical point but, as 1-1becomes positive, a new state which we shall call IGo) shows up on top of it with an energy gap
Av=exp[-f (3’v], which vanishes exponentially
l
in the thermodynamic
V denotes the impact parameter space volume.
(2.7) limit V+ 00. This second state
V. Alessandrini et al. /Kink states in supercriticalRFT
is associated with the “shifted pomeron”
stationary
points, and it is given by
Itie) = I&) - 141)) where I&) approaches,
433
(2.8)
as p/h + 00, a coherent state
141>= exp -$Jq@,
0) db]l@cr) )
(2.9)
such that
P(b,0) l91)=!
l$Jl).
Even though the Hamiltonian m = (_ l)lS@)PW db ,
(2.10) His not hermitian,
there exists a metric operator
[l] (2.11)
such that ‘VZH+CYl =H.
(2.12)
Therefore, right and left eigenstates of H are connected by the metric operator 311, It follows then that the perturbative vacuum is self-conjugate, while the dual state of I&) is (2.13) such that (2.14) These right and left eigenstates of H corresponding to the same energy level are therefore associated with the @/A, 0) and (0, p/A) fixed points of the classical action. The fact that a coherent state shows up as a new ground state after the phase transition is a characteristic feature of non-relativistic microscopic theories of Bose-Einstein condensation, such as the theory of superfluidity [lo]. But in hermitial theories this state has an exponentially vanishing overlap with the old ground state in the thermodynamic limit, so it really corresponds to a different phase of the system. This is precisely what does not happen in RFT, due to the non-hermiticity of H. If we compute the norm of the coherent state l@r>, taking care of the q, p commutation rule (2.4), we get (#I~ lf#q) N e-(r/h)2
v,
(2.15)
while in a hermitian theory one would obtain the opposite sign in the exponential, thereby giving for a normalized coherent state an exponentially vanishing overlap with the perturbative vacuum lq@. In our case, however, l$r) is a zero norm state in
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V. Alessandrini et al. /Kink states in supercriticalRFT
the thermodynamic limit while I$,-,) has negative norm, l@r) having a finite overlap with the perturbative vacuum I&,). Moreover, from the explicit representation of the states one can immediately get the transition matrix elements of the Gribov fields between the two vacua, given by (2.16) These peculiar features of RFT were already recognized in the spin model. In A#4 theory, different shifts of the quantum fields define different disconnected Hilbert spaces, corresponding to different phases of the system. Because of the existence of topological conservation laws, the kink sector is also different from the vacuum sector. The old perturbative vacuum disappears because it becomes unstable. In RFT there are no different disconnected phases: all would-be phases are connected by transition matrix elements of the quantum fields that remain finite in the thermodynamic limit. Even the kink states that we shall discuss later on, which interpolate in impact parameter space between the two vacua ($I,,) and l$r), will have finite overlap with them. Therefore shifting the fields will be in RFT a matter of convenience and not of absolute necessity because the kink sector will be the same as the vacuum sector. Let us now discuss the coupling to the external particles. Under the hypothesis of eikonal couplings, the RFT scattering amplitude can be written in the Schrbdinger picture as the following matrix element of the evolution operator between coherent states (2.17) where B is the distance in impact parameter space between the colliding hadrons, which act in this theory as external sources of pomerons. The source distributions g(b) and f(b) are taken to be strongly peaked functions near b = 0. Independently of the assumption on pomeron coupling S, as given by (2.17), is in general the generating functional for Green functions, which are obtained by functional differentiation with respect to the sources. It was shown in ref. [2] that, in the context of the spin model, the Green functions have an expanding disk behavior, which means S = 1 for 1B I- uY > 0. We shall restrict ourselves from now on to the case of a one-dimensional impact parameter space. In the spin model the expansion velocity u depends on P - p,,.rt and on the intersite distance a [2]. It is reasonable that, away from the critical point, the dispersion law which is related to the expansion speed IJ cannot be correctly obtained with a method that has neglected other excitations not responsible for the critical phenomenon. We shall present in the next sections a quantization technique which is better tailored to give the complete low-energy spectrum in the ordered phase. We would like to sketch, however, in the rest of this section, a heuristic argument which shows intuitively the emergence of the expanding disk. The argument is heu-
V. Alessandrini et al. /Kink states in supercriticalRFT
43.5
ristic in the sense that some steps are justified a posterion’ by the results of the following sections, and all the details of the calculations are given in appendix A. From eq. (2.17) it is clear that the S-matrix can be computed by solving the Schrodinger equation of RFT
HI*(y)) = -
$ Iwj))
(2.18)
with the initial condition that I*(O)) is the initial coherent state, the S-matrix being then the overlap of I*(Y)) with the final state. We then write the most general state vector of RFT as a generalized coherent state, in terms of an infinite set of correlation functions, and transform the Schrodinger equation into a set of differential equations for these correlation functions. Needless to say, the problem still contains all the intricacies of the theory, and the next step is to calculate a solution in the semiclassical approximation. The fact that this approximation is extremely relevant in RFT in the ordered phase is indicated by the fact that the RFT Lagrangian has the following scaling property with respect to the triple-pomeron coupling (2.19)
~(M;h)=&V?4~~;1),
which means that the weak coupling limit will be correctly described by the semiclassical approximation. Of course the triple-pomeron coupling is dimensionful, so h = 1 is to be understood in terms of the unit of length, which is fixed by CK’. We then find that, in the semiclassical approximation, the state vector is given ~,J,(Y))
= e-jdbg(b,
Y)q(b)+
correlations [$0,
(2.19)
where the correlations are higher order polynomials in 4, and g(b, JJ) satisfies a nonlinear diffusion equation with initial condition g(b, 0) = g(b), the initial source function. This diffusion equation is in fact the classical equation of motion for the field p,,(b, y) when qcl(b, y) is set equal to zero. We discuss then its solutions in appendix A, where it is shown that g(b, y) propagates in y by expanding from g(b, 0) and reaching, for large y, a box-like form that expands with a well defined velocity u,, = 2&, i.e. g(b,y)+(u,,y+ti-
lbi),
(2.20)
where 6 is a “phase shift” that depends explicitly on the initial condition. To the extent that correlations can be neglected, the state vector I I)(Y)) is asymptotically in Y a coherent state, and then it is trivial to see that the S-matrix is given by -!ldbf(b-B)B(uoYtS-IbI)
1 ,
(2.21)
V. Alessandrini et al. /Kink
436
states in supercritical RFT
which, due to the fact that f(b - B) is localized around b = B, is 1 if 1B I> u. Y + 6. This is just the expanding disk behavior of the scattering matrix. Of course the evolution of the state vector in terms of a single coherent state is only approximate, because the classical solutions that dominate in principle the semi-classical limit should be those of the coupled field equations. This effect is taken care of by the correlation functions in the state vector, that we have neglected in eq. (2.21). However, from the more careful analysis that we shall perform in the next sections it will turn out that the low-energy part of the spectrum of RFT - and therefore the large Y behavior - will be really controlled by uncoupled field configurations. We shall therefore justify a posterion’ the validity of the simple picture presented here. 3. Classical kink solutions of RFT We shall investigate in this section the existence and properties of a large family of kink solutions to the classical equations of motion of RFT, for the case of a onedimensional impact parameter space. The equations of motion that follow from the Lagrangian (2.2) are
ap
2
__pp+hgZ+2Xpq-a’~=Q, aY
(3.la)
a% -a4_w+hq”+2xpq-u’--$=o, (3.lb) ay and we are particularly interested in kink-like solutions which depend only on a linear combination of y and b which we define as P’PUY
-
(3.2)
where the parameter IJ is the kink propagation ables we write then the classical fields as
Pcdb,Y) =;
4~)
3
qc&
Y)
velocity *. With this choice of vari-
=; g@)
(3.3)
and obtain for the fields q(p), q(p) the coupled differential
equations
+u3,tcp-(p2-2l@=o,
(3.4a)
~t~t+~2_2$@Y=o.
(3.4b)
The fixed points of these equations tions) are given by (&a=
(O,O) (091) ( (1,O)
corresponding
to constant (p-independent
.
* We notice that u as defined here, and in the following, is adimensional. tion velocity
measured in units of &&.
solu-
(35) It is the kink propaga-
V. Alessandrini et al. /Kink states in supercriticalRFT
431
The first one obviously corresponds to the perturbative vacuum associated with unshifted pomeron states, while the two others correspond to the new vacuum associated with shifted pomeron states, as discussed in the previous section. As it is well known, these fmed points are unstable. They are actually saddle points in the sense that in the (q, $) plane corresponding to phase space they are attractive in some directions and repulsive in others. From the general structure of eqs. (3.4) we can see that 3
V%co)= V-,(-P)
?“@I = Y-,(-P)
9
(3.6)
which means that we can restrict our attention only to positive propagation velocities; solutions with negative u being then trivially obtained by means of these reflection properties. Moreover, it also follows from eqs. (3.4) that if {q,(p), P&J)} is a a solution, then {a(-p), cp,(-p)} is also a solution. For all the cases we shall be interested in these two solutions coincide, so we shall have CPU(P)= &(-P)
7
(3.7)
once the point p = 0 has been fixed by using the translational invariance of the equations of motion. Moreover, a class of solutions of the single-field equation (q = 0, cp= &I) or vice versa) can be proved to exist for 0 < u < 00. They are discussed in appendix A. Some particular kink solutions of eqs. (3.4) are already known. Cardy and Sugar have found exact analytic solutions for two particular values of the propagation speed. They are given by
cp=
i [ 1 7 tanh &
- i(cosh &I)-~]
foru=d,
q
.i [ 1 T tanh &J
- $(cosh x&)-~]
foru=G.
(I(
(3.8)
Each of these solutions can be represented in the (q, q) plane by a phase-space trajectory starting at the (0, 1) fured point when p = -00, and ending at the (1,O) one when p = 00, as shown in fig. 1, The first solution (3.8a) has both fields always positive and smaller than 1 and is represented therefore by a trajectory internal to
Fig. 1. The phase-space trajectories of the Cardy-Sugar kink solutions.
V. Alessandrini et al. / Kink states in supercritical RFT
438
the triangle defined by the three fixed points given by eq. (3.5). The second solution is represented by a trajectory that lies outside the triangle. Due to the reflection property (3.6), one finds for u = -J$ and u = -4 the same trajectories but evolving in the opposite sense, from (1,O) to (0, 1). We are interested in finding out about the existence of some further solutions of the same kind, and in particular the way they may depend on the propagation velocity u. This is a non-trivial problem in RFT due to the lack of Galilean invariance, which means that kink solutions with different velocities are not related by a simple change of variables, and consequently that the functional form of the kink solutions change in a non-trivial way as the propagation velocity u is changed. We shall heavily rely on the existence of a constant of motion for the system (3.4), which is easily obtained by multiplying the first equation by $, the second by (d and adding to obtain a total derivative which can be integrated to give ~tcp~(l-cp-~=c,
(3.9)
where C is a constant. This constant of the motion plays the same role of energy conservation in an ordinary mechanical system. It severely restricts the shape of the phase-space trajectories in the (q, @) plane. The kink solutions will reach (or leave) asymptotically one of the fixed points (3.5). It then follows that C = 0, and then &j
=-pq(l
_I+-cp)
(3.10)
for all p. Therefore, in all of the (~,a plane the sign of @is determined, it is exhibited in fig. 2. One can immediately see, for example, that inside the triangle determined by the three fixed points L$ is negative, so if one of the fields decreases as a function of p, the other must increase and vice versa. Moreover, the phase-space trajectory can never be parallel to one of the coordinate axes, except when it crosses one of the three lines of fig. 2, which are the coordinates axe themselves cp= 0 and Cp=0, and the straight line p + Cp=1. It is easy to deduce in this way that the origin (0,O) can only be reached by infinite looping around it or from one of the coordinate axes which implies, however, that one of the fields vanishes identically for all p. We are left in this case with the uncoupled field situation discussed at length in the appendix A.
Fig. 2. The sign of $
in the (9, F) plane.
V. Alessandrini et al. /Kink states in supercriticalRFT
439
3.1. Linearized equations and numerical results Let us now study the behavior of the classical solutions near the (9,9) = (0, 1) fixed point. We then introduce x=1-9
(3.11)
and linearize eqs. (3.4) in 9 and x. This gives cp-t$-9=0,
(3.12a)
ji-vg-x+29=0.
(3.12b)
We shall restrict ourselves to v > 0; for v < 0 the solutions are obtained by the reflection properties expressed by eq. (3.6). The eq. (3.12a) implies 9@) = A eVkp with A an arbitrary
(3.13) constant,
and
k=-&v&d-.
(3.14)
In order to solve eq. (3.12b) we have to distinguish two possibilities, according to whether 9 is of the same order or much smaller than x. We therefore consider the two cases
(a) f -O(l),
(b) ;
<
For case (a), x is driven by the solution of the 9 equation, x=Beekp,
with
x=BeVKP
ri=$vfJW.
and we find
A -=vk, B
while for case (b) x is decoupled
with B an arbitrary
(3.15)
(3.16)
of 9 and its behavior is then of the form (3.17)
constant
and (3.18)
It is clear that the major difference between cases (a) and (b) is that in the first one the phase-space trajectory will reach (or leave) the fixed point (0, 1) with a finite slope, while in the second case it will do so with a zero slope, along the 9 axis. We still have to consider, however, the various sign possibilities in eqs. (3.14) and (3.18). Let us first consider trajectories of the class (a), in which case we only have to worry about the two sign possibilities in eq. (3.14). For the positive sign, k > 0 and the linearized solution is relevant for p + +oo. Since we are restricting ourselves to v > 0, it follows that A/B > 0, and the solution therefore tends towards the (0, 1) fixed point as p + = from inside the triangle of fig. 2. Indeed, the slope with which
440
V. Alessandrini et al. /Kink states in supercriticalRFT
it reaches the fixed point in the (p, q) plane is given by Sr = lim Z- = lim Z-_ = -$u(m p-+- ( -x 1 p-+-a$- 1
- u) ,
(3.19)
which shows that Sr is always negative, varying monotonically between 0 and (-1) as u varies from 0 to infinity. In this limiting case the phase-space trajectory reaches the fixed point along the hypothenusa of the triangle of fig. 2. If instead k < 0 (always in the case (a)), the linearized solution is relevant for p + -00, and since A/B < 0 we conclude that the phase-space trajectory leaves the fixed point by going outside the triangle. Indeed, the slope in the (p, $) plane is given by Sz=
lim Z- =+u($GZtu)>O. p+_m ( -x 1
(3.20)
Let us now turn to case (b). It is a simple matter to realize that the only sign combination of eqs. (3.14) and (3.18) compatible with cpbeing exponentially smaller than x is
K=~“-&&zi,
k=-$v-&%i,
(3.21)
which means that the solution of the linearized problem is relevant when p + -00. Since A and B are uncorrelated, the possible solutions leave the tixed point (0, 1) with vanishing slope but may end up either from inside or outside the triangle. We should also mention that for this class (b) of solutions tp/x vanishes exponentially in p except when IJ = 0, in which case cp/x goes to zero as l/p. In this case the solution of the linearized equations (3.12) is (P=A e’P
x=Ap
efp
(3.22)
with both signs in the exponential being the same. To sum up, the behavior of the kink solutions of RFT near the (0, 1) fixed point for IJ > 0 can be summarized as follows (i) case (a), k > 0, the solutions arrive to the fixed point from inside the triangle with a finite slope ranging from Sr = 0 for u = 0 to St = -1 for u = 00. (ii) case (a), k < 0, kink solutions leave the fixed point with a finite positive slope Sa going outside of the triangle. (iii) case (b), there is a unique sign choice for k and K, given by eq. (3.2 l), and the kink solution leaves the fixed point with zero slope. The behavior of the classical solutions near the other non-perturbative futed point, namely, (p, $) = (1,O) can be obtained at once from the previous results, due to the symmetry of the equations of motion, by interchanging cpand $ and changing p into (-p). We can also locate the Cardy-Sugar solutions within this classification. The first one, given by eq. (3.8a) is of type (b), leaving (0, 1) with zero slope, while the
441
V. Alessandrini et al. /Kink states in supercritical RFT
second one corresponds to case (ii), leaving the fured point and going outside of the triangle with slope Sz = tf. We have investigated numerically the existence of kink solutions that interpolate between the (0, 1) and (1,O) fHed points. These solutions will satisfy the conservation law provided by eq. (3.10), and, as discussed before, they will not reach the origin (0,O) unless one of the fields vanishes identically. Let us define p = 0 by the condition cp(O)= fi0) .
(3.23)
It then follows from eqs. (3.10) and (3.7) that (cl(O)= -l&O) = *do)
d_
.
(3.24)
Therefore, for every given p(O), these relations specify a complete set of initial conditions for the integration of the coupled system of equations of motion. We have performed this integration numerically by searching, for every given cp(O), the value of the propagation speed u such that the solution approaches steadily the fixed points. We obtained in that way a large class of kink solutions, whose phase-space trajectories are either inside or outside the triangle determined by the three fixed points, corresponding to the two different signs of eq. (3.24). We shall discuss their properties separately. We present first our results for the classical solutions whose phase-space trajectory are inside the triangle. We find in this case kink solutions belonging to the class (a) for all propagation velocities in the range 0 < IJ< 00, solutions belonging to the class (b) for velocities in the range 0 < u < 2, and the solution having the behavior of eq. (3.22) for u = 0. Needless to say these numerical solutions match the known behavior near the fixed points. In order to make our results more clear and to make more evident the continuity properties of these solutions as the propagation velocity changes, we turn our class (a) solutions for positive u into solutions of negative u (-< u < 0) by means of the reflection property (3.6). We are therefore left with
Fig. 3. The phase-space trajectories of the family of kink solutions inside the triangle, with --
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V. Alessandrini et al. /Kink
states in supercritical RFT
a family of kink solutions with propagation velocity in the range -00 < u < 2, whose properties we now discuss. The phase-space trajectories of this family of solutions are sketched in fig. 3. We discuss the evolution of the phase-space trajectory as a function ofy because in this case the sense of the evolution, indicated by an arrow in fig. 3, does not change when IJ is changed to (-IJ). This is not the case for the evolution in p, as follows from eq. (3.6). As u tends to (--), the trajectory tends to the hypotenusa of the triangle, evolving in y from the (1,O) to the (0, 1) fixed point. As u begins to increase through negative values, the phase-space trajectory begins to sweep the interior of the triangle, always propagating in the same direction and leaving and reaching the fixed points with a non-zero slope. When u = 0, we find a static solution exhibited in fig. 3, whose behavior near the fixed points is given by eqs. (3.22) and for which obviously there is no arrow corresponding to time evolution. When the velocity u becomes positive, the solutions switch to class (b). They keep sweeping the interior of the triangle as u increases, but they switch their sense of evolution in the sense that they evolve from the (0,l) to the (1,O) fixed points. Since one of the fields is exponentially smaller than the other near the futed points, their phase-space trajectory is in those regions very close to the cpor p axis. The emergence of the critical propagation velocity u = 2, which will play a central role in the rest of the paper, can be seen as follows. As u approaches 2 from below, the phase-space trajectories become more and more close to the sides of the triangle (the cp,9 axis) thereby spending a large amount of “time” near the origin, that is to say, the perturbative vacuum. In this limit the coupling between the two field equations becomes less relevant, and the overlap between cpand q tends to zero. In the
-1
0
w
1
2
”
Fig. 4. The value of q(O) as a function of v for the kink solutions inside the triangle.
V. Alessandrini et al. /Kink
states in supercritical RFT
443
limit u = 2, the classical fields are solutions of uncoupled field equations, and the phase-space trajectory spends an infinite time in the perturbative vacuum. We stress again the perfect continuity of this family of solutions that we have found. This is also seen in fig. 4, where we plot the value of cp(O)as a function of u. The Cardy-Sugar solution (3.8a) is a particular member of this family without any further relevance besides the fact that it is given by a simple analytic expression. We have also found kink solutions outside of the triangle, corresponding to a choice of initial conditions such that ~(0) < 0 and d(O) > 0. They look like the corresponding Cardy-Sugar solution represented in fig. 1. In agreement with our expectations from the analysis of the linealized equations we find again a family of class (b) solutions with velocities in the range 0 < IJ < 2, such that when u + 2 they become very close to the sides of the triangle and eventually merge at u = 2 with the corresponding solution obtained from the interior of the triangle. Notice, however, that while the classical solutions {&), aO)} whose phase-space trajectory is inside the triangle are monotonic functions of p, those which are outside it necessarily have an oscillation, as functions of p. 3.2. Energy-momentum relationship We shall anyhow concentrate our attention on the solutions inside the triangle, which are the ones which are relevant for our quantization procedure. We turn now to the calculation of the energy, the momentum and the action associated with them. From eq. (2.5) and (3.3) we obtain, after changing variables from b to p: (3.24) where the dot means as usual the derivative with respect to p. The classical momentum is given by (-i Ycpcl), where
(3.25) and the classical action corresponding
to a kink of speed u is (3.26)
It is worth commenting at this point about the imaginary values of the classical momentum, (-i 3’cl). The corresponding quantum operator defined in terms of the p and 4 fields is the generator of real translations in impact parameter space and con-
444
V. Alessandrini et al. /Kink
states in supercritical RFT
tains an i because p and 4 satisfy commutation relations adequate to creation and destruction operators. Its classical values are not expectation values but rather transition matrix elements of P between a right coherent eigenstate of p and a left coherent eigenstate of 4. Since for our classical solutions p and 4 have real values, the transition matrix element turns out to be imaginary. It follows immediately from eq. (3.25) that rS,r is just (n/X)* times the area under the phase space trajectory of the classical solution. Therefore, for the family of solutions under consideration, PC1starts from :(J.L/X)~ when LJ+ -00 and decreases monotonically to zero as v-, 2-. By using eq. (3.10) the classical energy can be written as:
Ple1
0--
-
=-2
j dp& _m
cc&G”
=2
s
dp@(l
-LQ-~)~O.
(3.27)
_m
Therefore, eCl is always positive definite, starting from zero when u = 2, then increas ing and finally decreasing to zero when IJ+ -00. Moreover, by manipulating the equations of motion one can prove that AC, =
EC1 -
l&i
3, =;
0x *fi
j- dp [cpq2+ q2g] . _m
Therefore, for the class of solutions we are considering the action density LA,.,is always positive definite. It goes to zero as LJ-+ 2-, and increases monotonically to (+-) as u tends to (--). We are interested in studying in more detail the limit u + 2-, which will be the relevant one in our quantization procedure. We have seen that in this limit ~(0) tends to zero. Moreover, the overlap between cpand g is also very small; in fact q(p) + 0 for p < 0 while q(p) + 0 for p > 0, so the field equations are effectively decoupled. We shall therefore study the properties of the kink solutions by using cp(O)as a small expansion parameter. In this case the initial conditions (3.24) become l&(0)=$(0)2:$(0)=--0).
(3.29)
Moreover, for p > 0 $$I)is very small, and therefore in this region we can neglect quadratic terms in @-@).We are therefore led to the following field equations for P>O +uct)+q-\o*=o,
(3.30)
++$(1-21p)=o.
(3.31)
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states in supercritical RFT
445
Eq. (3.31) can be solved by using (3.30). The solution is (3.32)
@P) = G(P) emuP ,
which satisfies the initial conditions (3.29) and gives g(p) in terms of the solution of the single field equation for cp, eq. (3.30). Let us now show that this equation admits a unique solution with ~(0) = 0(2 - u). In fact, as it is done also in the appendix, the single field equation can be recast in the form (3.33) where the velocity field W(p)= cp has a unique phase-space trajectory with boundary condition w’( 1) = (1 t 42x1 - cp). As remarked in the appendix A, cp= 0 is a spiral point for the trajectory if u < 2 (damped oscillations) and becomes a regular end point when u = 2 (critical damping). We show in fig. 5 the form of W(q). It is then clear that the boundary condition cp(O)= (p(O) is satisfied for u < 2 at the intersection I of W = W(q)and W = cp; in such a way that cp(O)+ 0 as u + 2- (see fig. 5). The same remarks prove that there are solutions with u < 2 outside the triangle, satisfying cp(O)= -d(O), given by the intersection II of W = W(p)with W = -q. We are now prepared to discuss the order of magnitude, in the small parameter and s4,. First of all we notice that 3’Cris the area under the cpo = cp(O)of %I 9 ~Ccl> phase-space trajectory in the cp,$ plane, and that it is of O(C&. Indeed, by using the solution (2.32) for g(p), we can compute Yer as follows:
=2 f
G2 e-“P dp t & .
(3.34)
0
For small values of p, the solution I&) of eq. (3.30) grows like cpo exp(& up) for
L
1
Q
Fig. 5. The phase-space (W, 9) trajectory for the kink solutions of the single-field equation. Full line: v < 2. Dotted line: v = 2.
V. Alessandrini et al. /Kink
446
states in supercritical RFT
u = 2-. Therefore, the integrand in eq. (3.34) is of order (pg over a length of order log( 1/go), and outside of this region it is exponentially small. It then follows that PC, is of order O(& ln( l/qe)). We shall neglect logarithms with respect to powers of (p. in our estimates. In what concerns the classical energy eC1,we can write
0x2
-
1
-+$dp
C-1= -4
s
r-la
0
= 4~ 7
$* e-“P dp _ 4 J
0
and, by performing
*__&
=
2u
i
(3.35)
0
a partial integration
e,,
$4 e-“p dp
in the last term, we obtain
$* ePp dp +
2$(o)*.
(3.36)
0 Therefore, using the initial condition
0
; * [f,, - uJ&
;p,,] = (2 - u)l& .
so we conclude that, when u + 2-, sion relation has the form f,r = 2fi
(3.30) it follows that (3.37)
ecl and Y,_tboth go to zero, and that the disper-
YC,I+ 0(‘9,](2 - u)).
(3.39)
The meaning of this result is clear; it is just the Taylor expansion of the dispersion law e,r = ~(‘3’~~) of our family of kink solutions. This dispersion law is not simple because of the lack of Galilean invariance of RFT. In A@4 the action, being invariant under Lorentz transformations, does not depend on the propagation speed u of the
Fig. 6. The action per unit rapidity for the various families of classical kink solutions, as a function of (A) solutions inside the triangle: full curve; (B) solutions outside the triangle: dashed curve; (C) single-field solutions: dot-and-dash curve.
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states in supercritical RFT
441
moving kinks. The dispersion law for a moving kink is then a direct consequence of Lorentz invariance. For our family of kinks solutions with -m < IJ < 2, the action does depend on u, the u dependence of the solution is non-trivial and the general dispersion law cannot be obtained in analytic form. However, as u + 2-, or, what is the same, for small classical momenta, the dispersion law of the classical solution becomes a phonon-type dispersion law, with a zero energy gap. Besides the family of classical solutions we have just discussed, we must recall the existence of kink solutions outside the triangle, which have negative action. Moreover, as discussed in appendix A, there are also classical kink solutions of the uncoupled field equations - with one of the fields identically zero - which connect one of the (0, 1) or (1,O) fixed points to the perturbative vacuum. The action, energy and momentum of these kink solutions is obviously zero, and their propagation speed is bigger than 2 *. The critical propagation speed is then reached from above when one considers the uncoupled field equations, and from below, as we have discussed at length, when one starts from the coupled case. We summarize in fig. 6 the action per unit rapidity interval of the various families of classical solutions discussed in this paper. As we have done in fig. 3, we exhibit the perfect continuity in u by choosing the negative u branch when convenient.
4. Quantization
of kink states
The purpose of this section is to set up a quantization scheme in RFT by expanding around one of the classical solutions discussed in the previous section. We shall therefore adapt to RFT the methods introduced to quantize X$4 theory in 1 + 1 space-time dimensions around a moving kink. The different methods used - canonical quantization or path-integral formulation - are of course all equivalent, and their main task is to isolate the collective coordinates associated with translational invariance and, to integrate over the translational mode. We shall adopt in this paper the path-integral formulation of Gervais and Sakita [6]. In this formalism the global translational invariance is transformed into a local gauge invariance by introducing an extra degree of freedom and a constraint. The collective coordinate degree of freedom can then be easily integrated away when one considers matrix elements of the evolution operator between eigenstates of total momentum. In our case, after projecting total momentum in the initial and final states of eq. (2.17) we get the following representation for the matrix element
(4.1) l
The ones with 0 < v < 2 have an oscillating
field.
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V. Alessandrini et al. /Kink
states in supercritical RFT
In this equation +r,k and qr,k are the reduced wave functionals for the initial and final state, the plane wave factors of the collective coordinate part of the wave functional have already been used in the collective coordinate integration. The reggeon fields have already been transformed to the body-fixed system, namely, p = p@,y) and 4 = q@,y). The first delta functional represents the constraint, and fixes the quantum momentum operator (4.2) equal to the total momentum eigenvalue k. Moreover, Q@, 4) = 0 is the gauge condition, to be chosen so as to eliminate the zero mode, and J is the corresponding Jacobian. Taking advantage of the presence of the constraint in the functional integration, we can rewrite the S-matrix as:
X exp
fdy[-i&k
+6j(qp
1%
eff> dpl
,
(4.3)
0
where the effective Hamiltonian
density 91,rr is given by: (4.4)
Since p is the transverse coordinate in the body-fixed system, the classical static fields that make stationary the effective action of eq. (4.3) satisfy the equations of motion
691eff --0,
MP)
_
&%ff
_
&P(P) -OF
(4.5)
which, given the form (4.4) of BI,n, are precisely the classical equations of motion (3.la,b) discussed in the previous section. Their solutions are then the moving kinks propagating with velocity u. Notice, however, that these moving kinks are local minima of the unconstrained action. One could still try to fix u, which plays the role of a Lagrange multiplier, in such a way that the constraint k = P is satisfied, but this will not be possible in general given the imaginary nature of the classical momentum of our families of kinks. We shall come back to this problem in the next section, where we shall discuss at length the choice of the classical solution used to shift the reggeon fields. The classical kink solutions contain both fields qCi and pcl, and it will prove convenient to introduce a spinorial formalism in order to simplify the subsequent expressions. Moreover, we want to treat creation and destruction operators in a sym-
V. Alessandrini et al. /Kink states in supercritical RFT
metrical way. We then introduce
f=
449
a spinor field E by
0 ;
(4.6)
and a metric tensor 0
u=
-1
( ) 1
(4.7)
0
in such a way that
This is typical of “first order” formalisms even for Hermitian theories [S]. Notice that c[ = qp - p4 is zero at the classical level, but is of course the commutator of p and 4 at the quantum level. If the scalar product is defined as (r, 8 = /-dp (Q’P - ~‘4) = -6
F’) 2
(4.9)
then the transformations that leave this scalar product invariant at the Lagrangian level are equivalent to transformations that leave the commutation relations invariant at the Hamiltonian level, which are of course Bogoliubov transformations. Moreover, from eq. (4.9) it follows that for any c-number spinor [, (4.10) and therefore, in order to define a non-vanishing norm we shall be led to introduce a spinor dual to .& which is not gn. Let us now perform a shift of the reggeon fields, by defining the shifted fields according to (4.11) where (4.12)
(do) and a) being the solutions action density SQ0) becomes:
of eqs. (3.4a,b). In terms of the shifted fields the
(4.13)
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V. Alessandrimi et al. /Kink states in supercritical RFT
where Syz is a differential
Bc2
operator of the form
(4.14)
=lJ
where
B = 24~) >
B=2&p),
O(u)=u$d2
+2(cpt$)-
-dp2
1,
DT (u) = D(-u) .
(4.15)
In order to properly incorporate in the shifted action density (4.13) the gauge condition and the constraint, let us investigate separately the degrees of freedom associated with the translational invariance of the RFT action. It follows, as usual, from translational invariance that
lo@) =No %
(4.16)
is a zero-energy eigenvector of 5X2, i.e. g,to
&I~2
= 0,
(4.17)
=o,
No being a normalization factor to be discussed later on. Moreover, differentiating with respect to u the classical equations of motion (3.4a,b) we get
j&l
8
2
,/pEcl --
a~
-0
ap
(4.18)
3
which implies that the spinor x0 defined as
Lo being a new normalization 9f,Xo(P)
= -P
5 No
(o(P)
factor, satisfies ’
(4.20)
From eqs. (3.17) and (3.20) it follows at once that both to and x0 are orthogonal to any eigenvector of 91, with non-zero eigenvalue. Notice that the vector x0 just defined is not a zero-frequency eigenvector. It will play however an important role because it is the conjugate vector to the zero mode eigenvector to, and it will have to be kept in order to have a complete set of eigenfunctions to expand the shifted field. The arbitrary normalization constant Lo will be fixed by the condition (X0> Eo) = 1.
(4.2 1)
V. Alessandrini et al. /Kink states in supercritical RFT
451
We want to emphasize that the formulation we are describing here is by no means specific to RFT. The same pattern emerges when one performs a kink quantization of Lp4 theory, for example, using a first-order formalism. We can now discuss the choice of the gauge condition Q = 0. As is usually done, we impose a gauge condition that ensures that the spinor field [ has no component on the zero mode. Our gauge condition is then, as suggested by eq. (4.2 1)
Q=(f?o,t)=O.
(4.22)
The classical part of ,!jin eq. (4.11) will satisfy by itself the gauge condition provided that the over-all translational parameter a left undetermined by the equations of motion in ,&r(p t a) is adequately chosen. One can easily check that for the class of kink solutions discussed in the previous section our gauge condition fixes a = 0. Therefore, eq. (4.22) reduces to a condition for the shifted field l’, and the most general solution is 114
E =;
EC1 +
(a 1 !-
Pox0
+
al 3
(4.23)
where v is the genuine quantum fluctuation, namely, the part of the shifted field having components on the remaining non-zero modes:
(xo,17)=(~oo,77)=o. Once the gauge condition
(4.24)
is chosen, the Jacobian J of eq. (4.3) is given by (4.25)
and, by using the expansion
J=
(4.23) we obtain
v[;$ +($y”(a~~,xo)f(~)1'4~o(a,w0.XO)].
We can next use the constraint in eq. (4.3) to fix the x0 component field. The constraint equation reads, in our spinor formalism,
(4.26)
of the shifted (4.27)
i(P-k)=~(~,a,U-ik=O, which in terms of the shifted fields become of the form R-F[A,]
=o,
where we have introduced
(4.28) the notation: (4.29)
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V. Alessandrini et al. /Kink states in supercriticalRFT
Wol =-A,
(”y4[:j$-) + y4 @,ii, x0) 2
(4.30) If the constraint delta functional 6 [R - F(Ao)] is used to perform the A0 integration, one gets a new Jacobian factor of the form [lI,F’(Ao)]-’ , evaluated at the value of A0 that satisfies the constraint, namely 2&7&Y) II2 a,(v)=-fW[f2W+ @o,apxo)] 1
g 1’4 X +
fW=
0
p
t
(4.3 1)
(a,q, x0)
~~0,~,x0~
(4.32) *
We have chosen the root of eq. (4.28) that guarantees that A0 vanishes when R = 0, that is to say, when the constraint is saturated by the classical momentum and the quantum fluctuation 1) is zero. By computing F’[A,] from eq. (4.30) and using (4.3 1,32) we then find that this new Jacobian exactly cancels - up to a constant the first one given by eq. (4.26), i.e. CX’114 (4.33) ~ FfrAol = y 1 = constant . l--Y 1 This result confirms the internal consistency of our procedure. In the general formulation of the collective coordinate method the over-all Jacobian one gets in the path integral can be written as the commutator of the gauge condition with the constraint, Calling A0 the to component of g, we have written the constraint and the gauge condition as:
+l,(v)+f
- (f2
+;;$Qg’]
~LJOWI
(4.34)
and the fact that the over-all Jacobian is a constant means that A0 and Ao, considered as operators, satisfy canonical commutation relations. Therefore the identification of x0 as the conjugate vector to to is fully justified. We then conclude that the constraint and the gauge condition can be integrated, without adding any extra terms to the action density in eq. (4.13). All what one has to do is to shift the fields according to eq. (4.23), with A0 fured by eqs. (4.3 1,32). The n field is to be integrated in the subspace orthogonal to the zero mode. Let us now turn our attention to the normal modes which diagonalize the quadratic part of the action, and which will provide us with a basis to expand the n field. They are solutions of the eigenvahre equation %r;,
= %En,
(n # 0) .
(4.35)
V. Alessandrini et al. /Kink
We next define the components
states in supercritical RFT
453
of &, by
(4.36) and, using the explicit form of 91z given by eqs. (4.14, 15) the eigenvalue equation reads (4.37a)
1
-2qF,,+ u$+-$+l-2(qtlp)
Fn=T
F,,.
(4.3713)
By inspection of this system of equations, and using also the fact that the classical solutions we are considering have the property that cp(p) = a-p), it follows that if .$, is an eigenfunction of 91 2 with eigenvalue o,, then ES-P) E-“@)=(F is an eigenfunction RZt-n@)
n
(4.38)
(_J of g2 with eigenvalue (--0,)
= -%t-n(P)
(n # 0)
.
(4.39)
This doubling of eigenfrequencies is typical of first-order formalisms, which deal with creation and destruction operators (or coordinate and momenta) as independent variables. We still have to find the left eigenfunctions of $X2. It also follows by inspection that t*n 9!2
= Q+ztkn
9
(4.40)
which means that the spinor dual to .$, is g_n. It is obvious then that the correct normalization conditions to be imposed on our eigenfunctions is (L,~,)=-(~n,~~)=6n,-mun,
(4.4 1)
where on is the sign of n, and we adopt the convention that positive n corresponds to positive 0,. The .diagonalization of the quadratic part of the RFT effective Hamiltonian proceeds then as usual with Bogoliubov transformations. From the completeness relation (cf. appendix B)
55
n>O
=
[tn@) E-n@‘) - t-n@) inCP’>l + EO@)XO(P’)- XO@)tO@')
1 %J -P’) ,
(4.42)
4.54
V. Alessandrini et al. /Kink
it follows from eqs. (4.20,35 tor $X2 is 812
=
c n>O
States in supercritical RFT
and 38) that the spectral decomposition
un[tn(P)E-nCP’)
+ E-n(P)&@‘)1
+ dOSO@)
tO:o@‘)
of the opera-
.
(4.43)
If we now define, for n > 0 ‘4, = C_, 3771, then the expansion
n(P,.Y)=n$i
An
= (En
977)t
(4.44)
(n >o> 3
for the spinor field 77becomes
(4.45)
[A,01)~,(p)-A,01)5-n(p)l
and therefore the eigenfunction
HPJ) =f Ed(P)+
expansion
for the full reggeon field [ reads
0$ 1’4
(4.46)
The coefficients A, and 2, must be considered as the integration variables in the path-integral formulation, while, as we emphasized before,do is given by the constraint equation (4.32) in terms of the integration variable n, and the gauge con. dition has fured Ae = 0 in an extra term of the form Aoto that could have been added in eq. (4.46). In a canonical formalism x, and A, become, as we shall soon confirm, creation and destruction operators of a quantum of excitation energy w, corresponding to the small oscillations around the classical solution. Indeed, they satisfy the commutation relations [An9 Aml
= 6n.m
(4.47)
7
which are a consequence of the original commutation relations of the reggeon fields and of the completeness of our set of eigenfunctions (cf. appendix B). One has to be careful, however, with the ordering of operators, which is unambiguously fixed by the ordering of operators in the original Hamiltonian. In terms of the shifted fields, the action density (4.13) becomes (4.48) where the energy E is given, in the quadratic approximation E = dcl t ivfi
k -
$sdp
by
[A020 + fj] 5& [AOXO+ VI+ MT,
where 6E are ordering terms that we shall discuss later. By using eqs. (4.20,2
(4.49) 1 and
states in supercritical RFT
V. Alessandrini et al. /Kid
455
24) one obtains E=sQcItivflpkt~A$--and from eqs. (4.43,44
Lo No
-i
s
dpijG&qtfiE
and 47)
E=s&+ivfik+$Aa$+
c
o,(~,A,+~)+cSE.
(4.5 1)
n=l
The ordering term 6E is fixed by remembering that the unshifted Hamiltonian is ordered in such a way that the perturbative vacuum is an exact eigenstate. If one shifts the reggeon fields by keeping the ordering of the operators, one gets the quadratic form $‘D(v) $’ •t&j/l* + B$‘* ,
(4.52)
where +’ and 5’ are the components of the shifted spinor [‘, defined in (4.11). There fore, by comparing this quadratic form with the one we have actually diagonalized, one can see immediately that the ordering terms are what is needed in order to replace $‘D(v)$’ by a symmetrized form. Then, (4.53)
the w:(v) being the eigenvalues of the same Schriidinger problem as before, but without off-diagonal coupling. The final result for the energy is, then OD
E=dpQd tivfik+$.d$$t
c
n=l
u,(v).&An
+; 5
n=l
[tin(v) -w:(v)] (4.54)
5. Discussion of the spectrum 5.1. Choice of the parameter v In order to calculate the energy spectrum from eq. (4.54) we must calculate the eigenfrequencies w, by solving the eigenvalue equations (4.37a, b) which depend explicitly on the classical solution {q”(p), &&I)} used to shift the reggeon fields. Also PQ,_toccurring in (4.54) depends on the classical solution in a rather non-trivial way, as we discussed at length in sect. 3. We must therefore face the problem of the actual choice of the classical solution. In principle any one is equally good: quantum fluctuations will span the whole path integral. Our purpose is however to construct eigenstates of the Hamiltonian by treating all nonperturbative effects in the semiclassical limit. We have therefore to
.
456
V. Aiessandrini et al. /Kink
states in supercritical RFT
require after shifting around the solutions chosen, that the Hamiltonian operator contains the quantum fluctuations in a positive definite quadratic form. This restricts our choice, because for some of them the quadratic form is not positive, and for others linear terms in the quantum fluctuations (tadpoles) can be present due to the momentum conservation constraint. A problem which is similar in some aspects to ours, arises in Xp4 theory, related to the arbitrary choice of the kink propagation velocity. Due to Lorentz invariance the classical action in Xv4 theory is independent of u (u can be changed by a Lorentz transformation). However, the classical energy and momentum depend on u, the different terms in the expansion of the Hamiltonian are then u-dependent and to check the independence of u of the sum is highly non-trivial. Since u is introduced essentially as a Lagrange multiplier (see eq. (4.31)), the quantization is usually done by choosing u in such a way that the constraint is saturated at the classical level, i.e. 9et(u) = k. Only in this case the moving kink is a minimum of the constrained action and the expansion of the Hamiltonian does not contain linear terms in the shifted fields. If u is not chosen in this way the constraint equation introduces tadpole terms which are perturbative in the sense that they are proportional to the small coupling constant h, and which must be explicitly summed. The result of this summation has a u dependence that compensates the u dependence of the classical energy. The previous remarks may be intuitively understood as follows: in X(p4 theory, the kink energy is given by
Ekink(k) = Jk* + (MO + AM)*+ O(h)
,
where Me = 0( l/X) and &l4 is a renormalization the constraint, with the result
(5.1) mass * . If u is chosen to saturate
then the result for the Hamiltonian that is obtained is equivalent to the expansion in AM of eq. (5.1). For other values of u, the form of the Hamiltonian is equivalent to an expansion of Ekink(k) in powers of a combination of AM and the fraction of k not saturated by PC,. In our case, the situation is complicated by two facts. In the first place the theory is not Galilean invariant, and therefore the classical action depends on u. As a consequence, we do not expect a simple particle-like dispersion law to hold for an arbitrary value of u, as it is the case in eq. (5.1) for hp4 theory. On the other hand, the classical momentum is imaginary, and consequently we cannot hope to saturate the constraint k = iTc;p,lwith our classical solutions for physical (real) values of k. This implies that in general we are bound to have tadpoles coming from the momentum conservation constraint. Fortunately, it turns out that we can dispose of both difficulties at once by choosl
See, e.g., eqs. (4.15) and (4.24)
of Baacke
and Rothe,
ref. 171.
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states in supercritical RFT
457
ing to quantize the theory around solutions of class (A) exhibited in fig. 3 - namely, the ones inside the triangle - having a propagation velocity u such that u + 2-. That is to say, we pick the classical solutions of class (A) having a vanishing classical momentum 5, = 0((2 - u)~), according to our estimates of sect. 3. We shall then show that no tadpole occurs in this case, for any value of the quantization momentum k, and that the phonon-like dispersion formula e,r - V&
PC, = SQ,, = 0 [9e,(2 - u)] + 0
-
(5.3)
is not destroyed by the quantum fluctuation, in the sense that they do not introduce, for example, an energy gap. A first intuitive justification for the absence of tadpoles can be given by the following argument, Going back again to the general form of the energy given by eq. (4.54), we can see that tadpoles will in general show up through the Ai term, which is fixed by the momentum conservation constraint and given by eq. (4.3 1). Indeed, it is easy to see that A; contains linear terms in the quantum fluctuation n. Let us first discuss the role played by the Ai term at the purely classical level, by neglecting the quantum fluctuation n. Let us then fix for simplicity k = 0 and compare the energy gap J!?~~@ = 0) predicted by eq. (4.54) for two different values of the velocity u, i.e. u = 2 and u < 2. We expect them to be the same, at the expense of the fluctuation in the direction of x0, that is to say, A o, which corresponds to changing 3’er in the classical action. This is indeed what happens, because for u = 2 both 34,r and Ai are identically zero (if n = 0) and E,,,i,,(k = 0) = 0. If, instead, u < 2, then A0 as given by (4.31) turns out to be of order dYCr, and the contribution of the A; is of the form (5.4) which corresponds to rewriting the previous result for u = 2, Emin (k = 0) N &(P = 0), by expanding 9QCl(3 = 0) around ;P= PCt # 0. This shows then that the A0 contribution is minimized at the classical level if sQ,t is stationary with respect to Y.-t, which happens only for u + 2-. We shall also show later in this section that also the quantum corrections to A o vanish in this limit. The aim of this rather lengthy discussion was to motivate and partially justify our choice of the u + 2- kink solutions inside the triangle to shift the reggeon fields. A full justification of our choice will emerge as we proceed along this section. We shall not consider the class outside the triangle. They correspond to classical fields which are not monotonous functions’of p, implying that the zero-mode wave function has nodes. At the quantum level this means that the zero mode is not the ground state, and therefore they are not stable. Moreover, these solutions do not yield for Ai the result (5.4), not even for u --f 2 - , implying that one has also tadpoles in this case. As for the family of solutions of the single-field equations, the action, as well as the energy and the momentum, vanish for any u. For u < 2, the zero mode wave function again has nodes, and we expect them to be unstable. The case u > 2 and
458
V. Alessandrini
et al. /Kink
states in supercritical
RFT
especially u -+ 2+ is the one closest to our choice (u + 2-). The main drawback of this possibility is that it does not satisfy explicitly the p * q+ symmetry of RFT. Therefore, the quantum states dual to the ones found by shifting around, say, (JJ, 4) = (C&A, 0) are to be obtained by performing a different shift around (0, &~/l/h).This poses technical problems and we prefer to use an explicitly symmetric procedure. We would like to mention at this point the entirely different choice of classical solution made by White [ 111, to discuss the same problem we are concerned with. In our language, he chooses to quantize around the limiting solution u + -m of the class (a), which is of the form
cp@)-- ;<1- t&y), aP)--
;<1+t&Y),
(5.5)
i.e. reduces to the “instanton” along the hypothenusa of the triangle in fig. 3. As we have already discussed in sect. 3, such a solution has the opposite time evolution to the ones we consider, that is to say, it satisfies &=-oo)=O,
cpo,=+=)=O,
which are to be compared with the perturbative our kink solutions cp(j=-w)=O,
&V=+-)=O.
(5.6) boundary
conditions
satisfied by
(5.7)
White claims that, in order to have analyticity in the parameter /J for an infinite volume V in impact parameter space the boundary conditions (5.7) have to be changed, and that (5.6) ends up to be the only possible choice. Our approach in this and in the preceding papers [ 1,2] has been, on the contrary, to keep the boundary conditions fixed and demand analyticity in p only at a finite volume I’ of impact parameter space, and fixed rapidity Y. Analyticity in w can be proved in this case [8,9], and the matching with ordinary perturbation theory can be argued [9]. The S-matrix (2.17) is in this case stable against small, p-independent variations of the boundary condition (5.7) because the vacuum l&J is the only ground state at finite V. When I/ + 00 a phase transition has been found [l] due to the ground state degeneracy, and a non-analytic behavior in /J shows up at some critical value /.Icrit. We therefore consider White’s approach as basically different from ours, and from our point of view is open to criticism. Notice for example that the IJ-+ -00 kink has an infinite action per unit time that White subtracts away. This infinite action per unit time corresponds, in the language of sect. 2, to the exponentially vanishing normalization factor of the new ground state l$t), given by eq. (2.15). In subtracting it away, White keeps a finite norm for states which for us are of zero norm. It is then easily understandable that, if his shifting procedure is convergent, he will get a spectrum completely different from ours.
V. Alessandrini et al. /Kink states in supercritical RFT
459
5.2. Eigenfrequencies and eigenfunctions for v --,2We now discuss the calculation of the eigenfrequencies and eigenfunctions corresponding to quantum fluctuations around a classical kink solution with velocity v -+ 2-. In order to deal with the eigenvalue problem defined by eqs. (4.37a,b) we shall rewrite them in a more suitable form by eliminating the first derivative from the differential operators involved. Let us then define the functions f,(p) andfn(p)
(5.8) where C, is a normalization factor to be defined later on. Then, the relation (4.38) that gives the components of &n(p) in terms of those of&(p) implies the obvious relation f-,(p)
= U-p)
(5.9)
between the (+c+) and the (-w,) solutions. Introducing (5.8) in (4.37a,b), tem reduces to a coupled Schrijdinger problem of the form
1-$ + z 1 &XP) cup
f,(p)
W(p) -
f,(p)
where the potential D,--
2
+ - $ [
, 7
the sys-
+ Q(p) emuPfn(p) = 0
(S.lOa)
1
(5.10b)
+ W(p) + F
s,(p) = 0
W(p), defined in terms of the operator D(v) of eq. (4.15) by
d2 w(p) +
=
emupI D(v) evp12 ,
(5.11)
dp is given explicitly W(p)=+?-
in terms of the classical solution 1 t2[&)+&)].
{cp,(p} by (5.12)
For small values of (2 - v), the shape of the potential is sketched in fig. 7. It looks roughly speaking like a well centered at the origin p = 0 of height 2 and range 2L. Due to the fact that 40) = p(O) 2: 0(2 - v) and that cp(p) behaves as ~(0) exp(ivp) in the perturbative region cp(p) << 1, we can estimate the range L to be L = log(2 - v) .
(5.13)
Moreover, at the bottom of the well, near p = 0, IV@) takes a very small value of the order of (2 - v). We notice next that the coupled system of eqs. (5.10a,b) can be decoupled as v N 2- because in this case one of the non-diagonal potentials is always small. Let us consider for example the region p > 0. From our previous discussions in sect. 3,
V. Alessandrini et al. /Kink states in supercriticalRFT
460
Fig. 7. Shape of the potential W of eq. (5 .l 1) and of the classical fields. G is represented by a dashed line, and q by a dotted-dashed one.
we know that in this region cp@) grows first like ~(0) exp(iup) and then saturates at the value cp= 1. Since the exponential growth of cp@) is overcompensated by the factor exp(-up), it follows that when p > 0 the nondiagonal potential cp(p) exp(--up) is of order cp(O),that is to say, (2 - u). Always in the region p > 0, we can easily see that the other nondiagonal potential, -a) exp(up), can be of order 1. Indeed, from the known approximate solution for-&) given by eq. (3.32), it follows that m)
cup N $@)
(5.14)
and in the region p - L (b(p) is of order 1. The opposite results about the magnitude of the nondiagonal potentials hold in the region p < 0. Let us start by considering positive energy solutions, on > 0. Because of the previous discussion, in the region p > 0 eq. (5.10a) reduces to &fXJ)
(5.15)
= %LMP)
whereDs is the Schrodinger operator of eq. (5.11). Eq. (5.10b) becomes an inhomogeneous equation that determines.& in terms off,, . Indeed, DS being a positive semidefinite operator, there is no non-trivial solution of the homogeneous equation (Ds t w,)f = 0 when o, > 0. Also because of this fact, when we consider the region p < 0 where the inhomogeneous term - proportional to q exp(up) - can be neglected, we conclude that the only solution of (5.10b) is &l(P) = 0,
p
n>O.
(5.16)
Plugging this result back into eq. (5.10a), we obtain again eq. (5.15) for p < 0. There fore, eq. (5.15) is valid everywhere, i.e. in the whole region - < p < 00. We are therefore left with an ordinary Schriidinger equation involving a potential IV(p) which is approximately a well of height 2 and range L. When (2 - u) is small, L is very large [cf. eq. (5.13)] and the low-lying eigenvalues will be approximately given,forn >O,by n=2m-1 (5.17) n=2m. 1)n/L >
\(2m+
461
On the other hand the ground state w. should be of order 0(2 -- u), because we know that the exact coupled equations have an exact zero mode arising from translational invariance, and they differ from the present approximate ones by 0(2 - u). In deriving the zero-mode eigenfunctions from translational invariance one realizes that the shape of the potential is extremely important in order to have the translational mode correctly placed at o = 0, so the estimate (5.17) cannot be valid in this case. The fact that the ground state o. of our Schrijdinger problem is related to the zero mode arises of course from the observation that the zero-mode eigenfunctions, given by (4.16), have no nodes. The stability of our classical solutions - and the fact that Ds is positive semidefinite is then guaranteed. Since the Schrijdinger potential IV(p) is an even function of p, the Schrijdinger eigenfunctions f,(p) will satisfy the parity condition fn(-P)
=
(-l)"fn@)
(n = 0, l> ..*) .
,
They represent a complete orthonormal
5
n=O
fn(P)fn(P’)
= WP -P’)
(5.18)
set of functions
satisfying
(5.19)
3
(f.,fm)=SdPf.@)f,@')=Sn,m _ca
(5.20)
.
In what concerns f,&), we have already established that it is determined unique solution of eq. (5. lob)
by the
fn@)= [Ds + ~nl-12ti>e”Pfn@) =
5
fm (P)(fm
m=O
,2@ euP’fn)
where we have used the completeness
(fm
,V
e”“’f,)
= J
d/J’ fm b’)
(5.21)
>
%+~rn
relation (5.20) and the obvious notation
V(P’)
_m
euP’ fn b’)
(5.22)
.
We notice that, as expected, eq. (5.22) continues smoothly into eq. (5.16) for p < 0. Indeed, the support of&r’) exp(up’) is p’ large and positive. Having discussed the solutions of eqs. (5.10a,b) for on > 0, we can use eqs. (4.38,39) to derive at once the eigenfunctions for the case w, < 0:
C;_n@) = C_“( i
YEi)
= C-n( ::I::,
n
?I
= C-n(iI:
r:II,2)
.
(5.23)
462
Using the reflection properties of f,(p), = (-l)“C,, we can rewrite l_,Cp) as
~-,co)
given by eq. (4.18)
and fixing C_,
(5.24)
(n > 0) ,
= G(
where g,@) is now given by g
(p)
=
n
5 fm(PI Cr,,2~ cup’fn> m=O
=
%+Wm
(-1Y L-P)
(5.25)
and, of course, &@) = 0 for p > 0. We can now use the normalization (4.41) to determine C,, :
=c;s,.,
‘C,
= 1)
condition
(5.26)
where use has been made of the orthogonality relation (5.20) and of the fact that the functions&@) and &@) have everywhere a negligible overlap, namely, Xk%Z@) N 0. We now turn to a discussion of the zero-mode. We have to identify the zero-mode wave functions to and its conjugate vector x0, defined in terms of the classical sohrtions by eqs. (4.16) and (4.19), with the eigenfunctions we obtained for n = 0. Notice that we have two eigenfunctions available, by setting n = 0 in .$ or ,$_, . We discuss this problem at length in appendix B, where the approximate forms of lo and x0 are given for (2 - u) small. We are more concerned now with an estimate of the normalization factors No and Lo as IJ+ 2-, given the role they play as coefficients of the ,4; term in the general form of the energy given by eq. (4.54). In order to compute No we recall the fact (see appendix B) that, for (2 - IJ) small, the correct identification of E. is f. (-p)
evp12
fo(-p)
emuP”
Q-P>
Therefore, by identifying (4.16) we obtain foc0)
= NOW)
for
eMupI
p >
,
0 the upper component
(P>O)
.
(5.27)
of to with the exact result
(5.28)
V. Alessandrini et al. /Kink states in supercriticalRFT
463
and, since fe@) is an even function
l= [ dp fi@) -00
= 2Nz s
dp (b2 eevp .
(5.29)
0
This integral has already been discussed in sect. 3 [cf. eq. (3.34)] and proved to be of order (pg up to irrelevant logarithms. Therefore, No = O(&)
.
(5.30) constant, Lo, is determined
The remaining normalization condition (4.2 1)
_m
1 =(~o,~o)=LoNo
fdp
g$
-;
1
m-a$
s
ap - dp. _m au ap
= 2LoNo
(5.31)
We have made use of the fact that &I) = -$(-+I), acp(p) av=--’
$
from the normalization
and that
a$(+)
(5.32)
au
This last statement is just a consequence of the reflection back to eq. (5.31), we observe next that, for p > 0
aim am
_vpl2
aU=ave
_ 4 u&O)
e-vP/2 .
property (3.7). Going
(5.33)
The second term of the right-hand side is of order cpo, but the first is of order cpo/(2 -II). Since we know also that &J) is of 0(2 - V) for p > 0, we conclude that 1
_=o &No and therefore, Lo ,-0(2-u). 1”o
(
$u
(5.34) 1
from eq. (5.29) we obtain (5.35)
We have now all the information we need in order to evaluate the energy spectrum (4.53) in the limit IJ+ 2-. We note that: (i) The classical action density Act = 0 [ PCl(2 - u)] vanishes as a consequence of the phonon-like dispersion law. (ii) The constraint term (pi~Lo/No) also vanishes, because of (5.35). Notice that, away from u = 2, A: contains a tadpole which by eqs. (4.3 1,32) is proportional to
464
V. Alessandrini et al. 1 Kink states in supercritical RFT
N;‘($, aPxo). It is only for IJ + 2- that, thanks to (5.30),,4; is quadratic in n, and moreover gives a vanishing contribution to the energy because of the behavior of (Lo/No) given by (5.35). (iii) The zero-point energy
(5.36) vanishes as well, because Ok + o,(u) for u + 2-. We have in fact previously shown that u -+ 2- the eigenvalue problem for w, reduces to (5.15) whose eigenfrequencies, by (5.1 l), are 0:. The correction (0, - WE) can be computed to first order of perturbation theory, and the result for AEo is mo
= c’
un >2v e-uofk>(fky
Gfn)
wn + ak
n,k
= o(2
_ u>
9
(5.37)
where C’ excludes the term n = k = 0 in the sum. By the results above, the final formula for the energy becomes E=2ifik+p
c
w,A,A,+O(h).
(5.38)
n=1
Note that the quantum fluctuations have a small gap wr cx 7r/2L and for u + 2(L 51 iln(2 - u)l + 00) give rise to a continuum headed by the kink state. The above formula for the spectrum of RFT represents the main result of this section. Let us analyze the procedure that allowed us to establish it. We have seen the importance of keeping the triple pomeron coupling h and (2 - u) as independent parameters. As we discussed at the beginning, the quantization of RFT should be independent of u and therefore independent of the classical solution chosen to shift the reggeon fields. However, it is only for u + 2- that we succeeded in having, after shifting, a bona fide quadratic form, and the reason for this to happen is easily understood. For every u < 2 there is a classical kink solution that represents an unconstrained minimum of the action corresponding to some classical momentum ;p,l. If the theory could be quantized in such a way that k = P = i f&, then of course the expansion around the classical solution would involve only quadratic terms. This is not possible in our case, but the results of this section show that at the optimal value u = 2 the term in the Hamiltonian arising from the constraint not only is quadratic in the quantum fluctuation but also gives a vanishing contribution. Truly enough, this is achieved at the expense of having a continuum spectrum for the quantum fluctuations. The physical meaning of u = 2 is that the classical action density has an extremum at YCt= 0, where the phonon-like dispersion formula eC1= 2fl@ 3,t + O(@‘) holds. This dispersion law is preserved in the quantum case because there is no mass renormalization of the kink arising from zero-point fluctuations. We therefore find
V. Alessandrini et al. /Kink states in supercriticalRFT
465
that kink states with u = 2 give rise to a zero energy gap for k + 0. It must be remarked that, if our shifting method is properly convergent, the kink states and its quantum fluctuations are bona fide eigenstates of the non-hermitian RFT Hamiltonian. The same conclusion was reached in the spin model [2] where, only in the infinite volume limit, the kink states were found to have the dispersion law E(k) 0: [ 1 - exp(-ika)] = ika, a being the interlattice distance. However the continuous spectrum arises in that case in a discontinuous way, the finite volume spectrum being highly degenerate, but with a finite k = 0 gap. The same situation seems to occur in ref. [4]. Therefore, the occurrence of the phonon-like dispersion law appears more neatly in the present method than in the spin model.
6. Quantum
states and the expanding disk
Let us summarize the structure of the lower-lying spectrum. For y > pC we have the two degenerate ground states discussed in sect. 1 (the vacuum IGo> and its companion 1I)~> of eq. (2.8)) and, furthermore, two bands of states whose energy gap E = 52ikfivanishes for k + 0. Both ( I)~) and the kink states can be thought of as the lowest members of families of states obtained by a quadratic expansion of the Hamiltonian around the corresponding classical solutions. Let us stress again the difference between our situation and the similar one occurring, e.g., in hp4 theory. In an hermitian field theory the expansions around the space independent fixed points and the kinks define superselected Hilbert spaces. The existence of the topological charge VP(-) - cp(--00) is just a sign of that. In RFT we shall find instead that also the kink states, as the ground state /Go), are connected to the perturbative vacuum I@,,)by field operators. This is due to the peculiar role of the metric operator % of eq. (2.1 l), and implies that the vacuum I&,), l$e) and the kink states belong to the same (metric) space. Two logical consequences follow from the previous remark. On the one hand all states found so far are to be considered together in the time evolution operator and in the completeness relation. This is what we shall do in the following, and we shall make sure that we do not double-count by checking that energy eigenstates are properly orthogonalized. On the other hand it is not possible, in our case, that the semiclassical expansion we have used be equivalent to a “bona-fide shift”. By this we mean that some quantum fluctuations must be large when we compute matrix elements of ageneral operator. This also means that, in principle, one should be able to find, say, the kink state, starting from vacuum excitations if large fluctuations are allowed (in a sense, this has been done in the spin model). Nevertheless, the basis of our approach is that some classical solutions exhaust all the non-perturbative content of the quantum states, and that therefore a quadratic expansion is acceptable - at least locally - for the Hamiltonian operator. This is enough in order to identify the lowest lying states mentioned before, which in this respect are analogous to the breather modes found in the vacuum sector of Xv4 theory. A different question is how to take into account excited states around the
466
V. Alessandrini et al. / Kink states in supercriticalRFT
classical solutions. We shall discuss this point briefly later on, and we shall first concentrate on finding the contribution of kink states in the completeness relation. 6.1. The quantum kink state The quantum energy operator
kink states correspond to the ground state of momentum of eq. (5.38). They are defined by
k of the
l
Elk) = 2ikfilk)
,.
The second condition
Plk)=klk).
is automatically
implemented
(6.1) by writing the kink states as
_m
Using eq. (5.38) the kink state I&r>is then determined
&l&,)=0,
up to O(h) corrections
(n+O) *
by (6.3)
We remind that the gauge condition 2 o = (j&, , l) = 0 has fixed the arbitrary translation parameter a that occurs in the classical solution to, say, a = 0. Notice that eq. (6.3) assumes that Ec~,A,A, commutes with P, which is true approximately to the order XJp to which we are working. Let us recall that, from eqs. (4.44) and (4.36) &I =(C,,~-ed)=(~-,,p-~~)=(F_,,q-~~),
(6.4)
where F_, and F_, are given explicitly in eqs. (5.23) and (5.24,25) for the u + 2solutions we are using. By representing the operator p(p) by -a/aq(p), we can look for solutions of eq. (6.3) of the form /J I&I)=
ew[--(a,q)+
1 2@7,
h
p4)11@0),
(6.5)
where (Y@)is an unknown function and /3 an operator with symmetric kernel /3(p, p’). Then, by using eq. (6.4) we find that eq. (6.3) reduces to (F-n,o-@+(F-n,@)=O,
(6.6) nf0
(%,
p4) - (L,,
q) = 0 .
(6.7)
By using the explicit expressions (5.24,25) for F_, and F_n, as well as the fact that &I) exp(ivp) is of order 0(2 - u), it follows that the term (F_, , @) is of order 0(2 - u). We then find o(p) = cp(p) t constant &I) t 0(2 - u) , where the (b@) term comes from the fact that eq. (6.6) does not determine l
We use different
the zero-
symbols for the (same) Hamiltonian operator in order to distinguish its form in terms of the unshifted fields 4 and p from its form E of eq. (4.50) in terms of the shifted ones II and fi.
H of eq. (2.5) written written
(6.8)
V. Alessandrini et al. /Kink
states in supercritical RFT
467
mode components of both Q@) and p(p, p’). These components correspond to an infinitesimal translation in (6.5). Since we have to perform this translation anyway in order to obtain momentum eigenstates, we can assume that CYand /3have no zeromode components. With this specification al(P) = cp(P) + O(2 - u) ’
(6.9)
In the same way we obtain from eq. (6.7)
Ph P’>=n
mfi, P,, euP/*fn(p) euP/*fm(p’) ,
pnm = pm, = _
‘-,‘yyfn n
.
(6.10)
(6.11)
m
By putting eqs. (6.2), (6.5) and (6.9) together, we finally get
Ik)=JdaeCika exP (Pa =
[
- t (Vap 4) + i(4, Pa4)
cP(P - a>, ba =NP - a, P’-
The field cpis approximately, forp>O: $=cp-cp*
t@,
)3 1I@0
a>.
for LJ-, 2-, a solution of the single-field equation
= -2pco)
(6.13)
(b(O) = V(O)
and the kernel fl(p, p’) is a solution of the differential [ua, - a; + 2V@) -
(6.12)
equation
11&J,P’)+ [$I - $3 + MP’) - 11P(PP’>
6(p - P’) + (2V)oo
fo@uafJ') .
(6.14)
It is interesting to notice that - aside from the elimination of the zero mode in eq. (6.14) - both eqs. (6.13) and (6.14) can be derived from the generalized coherent state approach developed in appendix A. Indeed, they are essentially the first two correlation functions, in the particular case in which these correlation functions are kink-like, i.e., they depend on y only through the body-fixed variable p. As discussed in appendix A, these equations are the semiclassical approximations to the exact equations for the correlation functions, and therefore they do not include renormalization effects *. This means that, provided we limit ourselves to the null gap states with u = 2, the approximation of expanding the equations of motion in powers of 4 is essentially correct for the calculation of right eigenstates (cf. sect. 2). * This indicates a method for including renormalization effects in the construction of the states. For instance, the addition of the term p(p, p’) in (A.4) to (6.12) would give a mass renormalias already found in the spin zation. The ordered phase would occur for p > ccc = 0(,X*/m), model.
468
V. Alessandrini et al. /Kink states in supercriticalRFT
6.2. Approximate
evolution operator and S-matrix
We want now to investigate the normalization properties of kink states. We first notice that for each kink state with speed v = 2 and energy 2ikG, which we shall denote by Ik), , there is an antikink state Ik)_ with speed -2 and energy 2ikd&, due to the reflection symmetry (3.6) and (3.7). This antikink state Ik)_ is simply obtained by the replacement p + -p in cp@) and p. The states Ik>+and Ik)_ propagate in space in opposite directions: left for Ik)+ and right for IkL, corresponding to a single direction in time, namely, the one for which p -+ p/h asy + 00. This is exhibited in fig. 8.
a)
b)
Fig. 8. Time evolution of (a) the kink and (b) the antikink states.
For the purpose of this section we shall neglect the correlation (PC(/~,and we shall write Ik)+=Jdue-‘kaexp Ik)_ = rda
-; [
emika exp
c
-:
1 1
dpq(P-a)&) dp 4-p
(6.15a)
I&o>,
+ a> 4(p)
Ido)
0 with respect to
.
(6.15b)
They satisfy: Elk),
= +iv,kl k>+_,
where we have introduced
Plk), = klk),
,
(6.16)
the critical speed (6.17)
ve=24&.
We want now to find the left eigenstates corresponding to (6.16). By introducing the parity operator I, which obviously commutes with E and anticommutes with P, we note that from (6.16) and the definition of the metric operator %! (see eq. (2.11)) it follows that _(-klZE = _GkIEI= _(-klZP=k
_(-kll.
ivok _(-kll,
(6.18) (6.19)
469
V. Alessandrini et al. f Kink states in supercritical RFT
Moreover, one can check that: _Ckll=
+(-klI=
+&I ,
(6.20)
_(kl ,
which means that in the left eigenstates t and - exchange their roles. This fact is confirmed by the evaluation of the scalar products according to the metric operator (2.11). While I k)+ and Ik)_ are themselves null states in the thermodynamic limit +Uclk)+ = _Uclk)_ = exp[-
~(p/A>2Vl s
0
(6.2 1)
the scalar product _(kl k>+ is not trivial, because it contains only a partial overlap of cp@) and cp(--p). Indeed, one easily obtains _&‘)+
(6.22)
= 2n6(k - k’) N(k) ,
where
(e --f o+)
(6.23)
and we have roughly approximated the integral in the exponent by @(-a) for A<, with the ground states i$e> and I I),,) = I #,,) - I c#q) discussed in sect. 2. Of course the kink states are orthogonal to the ground states for k # 0, but in order to keep this property for k = 0 we have to project away the vacuum component by subtracting a 6 function part. Then, we define Ik): = Ik>+ - 2d(k)
lG1> , (6.24)
Ikf_ = Ik)_ - 2n8(k)l@t), in such a way that the new states are automatically
orthogonal
to both I& > and
($0). By taking into account the normalization factors (6.21) and (6.22) we are now able to write the contribution of ground states and kinks to the completeness relation. We get:
t
s-=dk -
0
{-ik e
ivOkY
Ikf_ +(kl’ t ik emivokY Ik): _(kl’}
(6.25)
2n
+ quantum
excitations
.
* This is true if q(p) is approximated by a step function centered at p = 0. Strictly speaking q(p) should be centered at the transition region p = L. This would give rise to an extra factor exp(2ilk) iniV(k), which, however, cancels against similar factors in the matrix elements.
V. Alessandrini et al. /Kink
470
states in supercritical RFT
From this completeness relation we can calculate the pomeron Green functions. for /AY-+ =. We shall define the propagator as D(j3, Y) = -(@cl p(O,O) eiPB eCEYq(O, O)l&> .
(6.26)
This definition is stable against addition of time-dependent g@e) spread over a volume V, provided we put
@‘“‘(B, Y)p,.
= j_im_ ,lir,
[analytically
continued
external sourcesf(-vo),
DgLyoj,
gtyoj(B, Y)]
(6.27) where the analytic continuation from 1_1<0 to P > 0 is unambiguously made at finite space volume. In fact, I q?~,, ) is the true ground state at finite V and is automatically picked up by the y. + m limit l . With this proviso, the ground state contribution to (6.26) is, by (6.25) and (2.16) and always neglecting relative corrections of O(h/p) 2 DC4
-~~~lpI~~~~-~~~~,l4l~o~
r> z B
=
$
(6.28)
.
0
fixed
The limit (6.28) is asymptotically reached at finite B. However, one should also have that D(B, Y) + 0 for I B I + m at finite Y, by the cluster decomposition of the correlation function. Therefore, we need additional states of small momentum which cancel the contribution (6.28) at large impact parameters. We shall now show that in fact the kink states provide their cancellation l * . By neglecting for the time being quantum excitations, we get ,from (6.25) and (6.26): 2
dk
0 J
D(B, r> = z
-
2n ecikB [x+(k) x_(k)
kuokY + c.c.]
(6.28)
,
where xi (7~)= Q. Ip(0) Ih’>*. Using (6.15) and (6.24) we find x+(k) =x(k)
=f
x_(k)=X,(k)=x
[Jda evikaO(a)[~dae-ikaO(-a)-
2n6(k)] 2n6(k)]
= -$ =-!
,
& &
.
(6.30)
Therefore, by using the inverse transforms: O(z) =
&j&
eikz
hk =&jdk ewikz &
,
(6.3 1)
* Ambiguities both in the analytic continuation and in the ground state may arise only if the V-t - limit is taken first. l * The same cancellation mechanism was independently exhibited in the particular case of RFT with “magic” quartic couplings studied in ref. [4].
V. Alessandrini et al. /Kink states in supercritical RFT
471
we finally get: [i -O(B-LJ,Y)-C~(-B-U,Y)]
D@, Y)=
020(voY-
= f
,B,),
(/Jr>>
1,
h<
(6.32)
This is the final result of our calculation. It clearly shows the role of the kink state in cancelling the ground state contribution at large 1B I. The pomeron propagator of eq. (6.32) is a grey disc of opacity (P/A)~ and radius R(Y)=voY=2fiY.
(6.33)
The J-plane structure of this result, in which kink excitations are neglected, can be analyzed as follows. The contribution of ]rJo ) is a fixed pole at J = 1, (J = 1 - E), while the kink and antikink are complex conjugate poles with residues which are singular for k + 0. They combine together to give
A2 0;
W k) =
J+l27r6(k) +
1 ik(J-
1
1 + ikuo)
+ C.C.
2
= (J - l)(J - 1 t iv&) which is the J-plane structure kink excitations, as discussed kink poles are really speaking The S-matrix corresponding evaluated by the same method
+ Cc. = (J-
1)2 t u;k2
’
(6.34)
of the expanding disk for D = 1. The introduction of later on, gives rise to J-plane cuts and the kink and antithe edge of those cuts. to eikonal-type couplings, given by eq. (2.17), can be and, for point-like sources is asymptotically given by *
Scf, g; B, Y) - 1 - [ 1 - eVgW/‘] [I - e-fP/h]
O(ue Y - iBI) ,
(6.35)
as in the quantum spin model. This provides inside the disk a factorized opacity bounded by one. Notice also that this result for the S-matrix is really valid in the weak coupling limit, namely X << p. For larger values of h/p, renormalization effects neglected here would eventually replace p/X by up/h, where u is an order parameter which vanishes at p = po. Moreover, the precise form of the asymptotic S-matrix changes due to correlation effects in IrJo> (see Cardy, ref. [3]).
* This simply follows from the approximate representation exp(-b/h)
gdd)
h~cP
1 - [ 1 - exp(-gcl/Ql O(P).
412
V. Alessandrini et al. /Kink states in supercriticalRFT
6.3. Excited states Taking into account excited states is not easy to for several reasons. The first question of principle that arises is whether or not to count both the excitations around I Go>(shifted pomerons) and those around the kink states. This difficulty is shared with other theories which have in the same superselection sector several minima of the action which are not perturbatively connected. We can advance some arguments which indicate that the expansion around the kinks could provide, at least approximately, all the elementary excitations of the theory. Let us in fact try to identify the excited modes around the kink. We have in the first place the lower-lying states inside the well (see fig. 7) with dispersion law w, = /Ai, (ki << 1). These are a band of states with a small gap or , which collapse into a continuum in the limit u + 2- (L + 00). The intuitive picture that we have of these states is that they correspond to unshifted pomerons excitations (around the perturbative vacuum I&)) in the presence of the kink, and viewed from a reference frame moving at the critical speed uo. First of all, notice that the bottom of the well corresponds to the perturbative vacuum and that, as IJ approaches 2, our kink solutions spend a very long “time” 2L in this state. Secondly, let us see how an unshifted pomeron looks like in the kink rest-frame. The transformation b + b + uoy is a Galilean transformatinn with imaginary velocity iv0 = 2ifi, because rapidity y corresponds to imaginary time. Correspondingly, the pomeron dispersion law can be rewritten in terms of momentum k’ in the moving frame as follows 2
= iuok + cx’k” , where iuo/2cw’is the center of mass momentum corresponding to a pomeron “mass” 1/2cX’. Formally, this is exactly the spectrum that we get by quantizing around the kink if k’ is identified with &? k,. However, since k’ = k - iuo/2cu’ - as it must be for free pomerons - eq. (6.36) is just a trivial restatement of the original dispersion law. What happens in our case is that unshifted pomerons are in interaction with the kink, which eats up their momentum in such a way that k’ is not fixed by the momentum conservation constraint, and can be interpreted in fact as &? k,. Momentum is taken away by the zero mode A0 which, according to the results of sect. 5, does not contribute to w(7c) for u + 2-. This is the mechanism by means of which the kink stabilizes the unshifted pomerons. It is also easy to understand the higher modes o, > 2~1,which correspond to a true continuum above the kink well. They are essentially fluctuations around the value cp= p/X of the fields, and can therefore be interpreted as shifted pomerons in interaction with the kink, viewed also from the kink rest frame. Their characteristic frequency is-shifted to 2~ by the same mechanism that pushes the unshifted pomerons up to zero frequencies, namely, the transformation of energies under a Galilean transformation. To sum up, since we have around the kink both shifted and unshifted
V. Alessandrini et al. /Kink states in supercriticalRFT
473
pomerons [with dispersion law (6.36)] we feel that these excitations can provide all relevant corrections. Another question of principle that we have to face is the evaluation of corrections introduced by the excited states to the expanding disk just found. The most important are clearly those arising from small w values (distorted pomerons). From the point of view of the J plane they give rise, in the limit u + 2-, to cut singularities at J(w) = 1 + 2ikue - w,
(6.37)
whose branch points (w = 0) coincide with the kink singularities. These contributions will not spoil the disk result provided the cut discontinuity (due to excited states) is less singular than o -iI2 for any k, including k = 0. We have performed a rough estimate of the contribution of the nth mode to the evolution operator by approximatingf,@) of (5.15) by eigenfunctions of a square well of range 2L. The result indicates that the cut discontinuities vanish at the branch points for any real k. Therefore we expect our results to be valid both for Y >> max(uu IB I, 1/p) and for lB I >> Y/ue. The excitations contribute mostly in the transition region IBI - ue Y. More precisely, in our rough calculations, they are most important outside the disk where they quantitatively reproduce - for IB I > u. Y >> uo/p - the unshifted pomeron tail, thus correcting the disk result utot a Y (forD= 1) by an inverse power of Y = log s. A proof of this indication would show the matching with ordinary perturbation theory, which is expected to be correct outside the disk. We are still unable to understand whether these distorted pomerons which in our approach correspond to bona fide eigenstates of the Hamiltonian, are relevant to the fulfillment of t-channel unitarity for t > 0 (k2 < 0). A detailed analysis of the role of the excited states is left to future investigation.
7. Conclusions In order to find the lower-lying spectrum of RFT, which determines the asymptotic properties of the amplitudes, we have introduced a quantization procedure around classical solutions, suitable for the supercritical phase (o(O) > ar,). We have been able to identify the relevant solutions in the case of a one-dimensional impact parameter space. We have shown that kink-like classical solutions exist for a range of values of the speed u up to a critical value 2, for which the dispersion law is phonon like. This particular value is the best suited for our quantization procedure, even if track must be kept of (2 - u), which acts as an infrared regulator parameter and can be sent to zero only at the end of the calculation. We have then obtained the kink-like quantum states and the quantum excitations around them. The spectrum of the kink states is E = *2ifi k, while the excitations give rise to a (real) continuum starting from there.
V. Alessandrini et al. /Kink states in supercriticalRFT
414
We have finally computed the contribution to the Green functions of the lower lying states, i.e., the vacuum, its degenerate companion and the kink states. The contribution of the latter was found to cancel the one of the degenerate ground states for lB1 > 2fi Y, while for 2fi Y > IBl, the Green function was still asymptotically constant. This is just the disk behavior typical of RFT, with the expected expansion velocity. We argued that the lower lying excited states can be interpreted as perburbative pomeron excitations viewed in the moving kink frame, and that their contributions do not substantially modify the expanding disk behavior. On the light of our results we think that a basic characteristic of thistheory is the simultaneous interplay of different states which cannot be obtained perturbatively from each other. The quantization around classical solutions is well suited to isolate those nonperturbative effects. In this way we were able to spot the contribution to the evolution operator of various quantum states corresponding to different classical solutions. This implies that the theory cannot be interpreted as a perturbative expansion around a single, suitably chosen ground state (“shifted” vacuum). We wish to thank E. Edberg and K. Kijlbig for their help in numerical
calculations,
Appendix A We shall first discuss in this appendix a generalized coherent state approach to the solution of the Schrodinger equation (2.18) of RFT. After having shown the relevance of the single field equation of motion in the semiclassical limit, we shall give a detailed discussion of its solutions. A. Generalized coherent states The most general state vector leti)) HI \k(Y)) = -
$
solution of the RFT Schrijdinger
equation
(A.1)
I\k@))
can, without any loss of generality, be written in the form I@@)) = epAcy) !@) , where l&,> is the perturbative A(y) =Nfj
+
Jdbi
(A-2) vacuum, and A@) an operator given by .. . dbN s(bi)
.. . &N)
G&i
.. . bN; u)
(A.3)
in terms of an infinite number of correlation functions G,. The Schrijdinger equation (A.l) can now be transformed into an infinite set of coupled differential equa-
V. Alessandrini et al. /Kink states in supercriticalRFT
475
tions of first order in “time” y for these correlation functions. These equations are obtained by computing both sides of eq. (A.l), using the RFT Hamiltonian given by eq. (2.5) and the commutation relations (2.4) and by identifying powers of creation operators. Although the general equation for a+~ is not difficult to obtain, we write below only the first two equations for simplicity
- aG,(b,y) =-~tol'vf>G,(b,y)tXG:(b,y)+~Gz(b,b,~),
(A.4)
aY
-
aGz(b,
b’;y) =-6~ t a’v;)G2
ay
+ 2X[G1 (b, y) + Cl
- (j,.i t cth$)Gz
+ 2M(b
(b’,y)lGz + 2%@, b, b’;y)
-
b’)G1@A (A-5)
and so on. In general, a,,G, depends on G,+, and GN_t . For an initial coherent state of the form considered in sect. 2, these equations are to be solved with the initial condition
G,@,0) =g@), GN(bl,
.. . . bN; 0) = 0
forN>
2,
(A-6)
g(b) being the initial source distribution. We are still at the starting point in the sense that we have not yet made any approximation that allows us to handle the problem under consideration. We now proceed to compute a solution in the semiclassical approximation which, as discussed in the text, should be good in the weak coupling limit for o(O) > 1. The semi-classical limit for the coupled equations for the correlation functions is easily obtained, because in this case A(V) is just the classical action and, considered as a functional of the dynamical variable q(b) it satisfies the Hamilton-Jacobi equation
(A-7) Replacing the expansion (A.3) for A(y) and identifying again powers of q(b) one can easily obtain the semi-classical limit for a,G,. The result is neatly phrased by saying that the semi-classical limit corresponds to dropping the “feedback” terms in the exact equations, that is to say, terms proportional to GN+~ in the equation for a,G,. What is left is then a set of uncoupled equations for the correlation functions of the form
a(%_(p + dV;)G1 --
ay
- XG: ,
(A.@
V. Alessandrini et al. /Kink states in supercritica~RFT
416 =2 -=
[@ + dv;,
+ 01 •t ~‘v3 + 2VG,(b,y)
+GI(~‘,YNIG~
au
+-2AS(b-b’)Gl(b,y).
(A.91
The only non-linear equation is that of G r ; G, satisfying a linear differential equation that has as an input the lower correlation functions. By means of an adequate resealing of the correlation functions one can check that the semi-classical equations are independent of X, and that the feedback term (radiative corrections) are perturbative in A. We are therefore naturally led to study the equation for Gr , which is exactly the single-field equation for p,t(b, y). Higher correlations take into account the fact that the classical field qCt@, y) is non-zero, and they can in principle be obtained by solving linear equations once Gr is known. A.2. The single-field equation We shall henceforth restrict to a one-dimensional impact parameter space, and study the single-field equation (A.8). We want to illustrate how a wave generated by a source approaches asymptotically solitons with a well-defined (critical) speed. Let us consider the well-defined boundary value problem for the single-field equation
(A.lO)
By introducing t = py, x = @b and up= (p/A) pCl we eliminate all dimensional constants, and the problem reduces to + = cp- cpa t cp” )
dO,x) =p
fhmix)= %@c).
(A. 11)
Non-linear diffusion equations of this kind have been widely studied in the mathematical literature [ 121, and the qualitative features of (A.1 1) have already been given in the context of RFT [13]. They are: (i) The only fixed points, i.e. static solutions of (A.1 l), are cp= 0 and cp= 1. (ii) If the source f(b) is everywhere smaller than p/h, namely, 0
V. Alessandrini et al. /Kink states in supercriticalRFT
417
the negative (p2 term in the differential equation, provided that the source ue (x) is everywhere smaller than the saturation value of cp. We want to clarify how cp= 1 is approached for large times. We shall see that this approach occurs with a well-defined speed, which picks up one among many soliton solutions of (A.1 1). Let us therefore look for homogeneous waves that interpolate between the two fixed points, of the form * (A.13)
cp(P) = cp(ut *x) , such that d-=)=0,
cp(+-)=l.
They satisfy the differential
(A.14) equation
Ip=-cptcpatu+
(A.15)
and solutions to this equation with boundary conditions (A.14) exist for any positive speed 0 < u < 00 l’ . In fact, d-p) can be interpreted as the coordinate of a particle of unit mass moving in a potential (A.16)
V(p) = &a - $3
and subject also to a friction force with viscosity coefficient u. Indeed, by reversing the sign of “time” p the term u$ in (A.15) changes sign and becomes then a dissipative force. The shape of the potential V(q) is exhibited in fig. 9. Since for the timereversed motion cphas to start at cp= 1 and end up at ‘p = 0, it is obvious from the mechanical analogy that this will happen for any value of u. The time-reversed motion will correspond to the particle moving downhill and performing damped oscillations around the bottom of the well for small values of the viscosity coefficient u, or reaching very slowly cp= 0 without oscillating for large values of u. The borderline between these two regimes corresponds to a critical velocity u, that we discuss next. But it is quite clear that solutions of (A. 15) exist for any positive u. When cpbecomes small, the potential V(p) approaches a harmonic one, and the problem becomes that of a damped oscillator. Linearizing the equation and looking for solutions of the form cp(p) = exp(k,p) we get k; -uko
+ 1 =0,
:.
j+$u+@T.
(A.17)
We have therefore damped oscillations for d-p) around cp= 0 for u < 2, and at most one oscillation for u > 2. The phase-space trajectory corresponding to the solutions of eq. (A. 15) can be
*Cf. footnote after eq. (3.2). ** We are indebted to G. Parisi for the argument that follows. More rigorous derivations can be found in the mathematical literature [ 121.
V. Alessandrini et al. /Kink
478
states in supercritical RFT
Fig. 9. The potential V(q) of the single field equation (A.15).
found by introducing
the velocity field W(q) = Cp,which satisfies the equation (A.18)
and in order to define W(p)uniquely we shall give a boundary condition at cp= 1. From the linearized form of eq. (A.15) near cp= 1, it follows that q is of the form (A.19) but k_ has to be discarded because from (A.14) Cpco)must reach 1 as p +m. Therefore, it follows that
W((P) -=k+=&@t1, 1-q
asp-+1
(A.20)
Eqs. (A.18) and (A.20) define a unique trajectory in the (W,cp) plane that is sketched in figs. 10a and b for u < 2 and u > 2 respectively. For u < 2, fig. 10a shows the existence of an infinite number of damped oscillations for the time-reversed motion. The origin is then a spiral point for W(q). On the contrary, for u > 2 we observe that no damped oscillation can exist. In fact, in such a case the phase space trajectory would cross the cp= 0 axis in fig. lob at some point W,,.But then the corresponding unique trajectory would lie necessarily above the line W = k&o and could not possibly end up in q = 1. a)
b)
Fig. 10. Phase-space (IV, 9) trajectories for the kink solutions of the single-field equation. (a) u<2,(b)u>2.
V. Alessandrini et al. /Kink
states in supercritical RFT
479
This last statement can also be easily understood by means of the mechanical analogy. Let us think of cp(p) as the coordinate of a particle. After the particle crosses cp= 0 with a finite speed to the right, it will be subject to a restoring force (-cp), an anti-dissipative force u$ that feeds energy and repels it to the right, and the nonlinear force cp2 that further repels it to the right. Therefore, the velocity corresponding to the linear problem where cp2 is neglected gives a lower bound to the actual velocity of the particle. Going back to fig. lob, we remark that for u > 2 the phase space trajectory is asymptotically tangent to W = k&, kg being the smaller exponent of eq. (A. 17), which is asymptotically dominant for p -+ -00, Therefore, we conclude that the soliton for IJ > 2 lies always in 0 < cp< 1. It is also clear that its steepness increases when u decreases from large values to the critical value LJ= 2. Indeed, by identifying the work performed by the non-conservative force u$ between cp= 0 and cp= 1 with the variation of the mechanical energy of the particle, one gets
U
s
‘b2dp=V(1)-V(O)=;,
(A.2 1)
-m
which shows that, as u decreases the steepness of p(p) increases if cpis a monotonous function of p. The different shapes of solitons of u > 2 are exhibited in fig. 11. We are now in a position to discuss the asymptotic propagation speed in the original initial value problem (A. 11). The basic idea is to rewrite the differential equation (A.1 1) in a frame moving with speed u in the positive x-direction, for example. Then we have cp(x, t) = u(ut -x,
t) = U(P, t) ,
(A.22)
au=u-uutu-u2, .. .
at
u(-x,
0) = 240(x).
(A.23)
The solitary waves we discussed before are now static solutions of (A.23), and it is clear that they are possible asymptotic limits for u@, t) as t + 00 at fixed p. The mechanism that chooses precisely the u = 2 soliton for the asymptotic limit of u@, t) is given by the following results which are proved in ref. [ 121. (a) For any u > 2, if the initial condition ue (xx>vanishes more rapidly than any
Fig. 11. Different shapes of kink solutions of the smgie-ileid equation, for v > 2.
V. Alessandrini et al. /Kink
480
states in supercritical RFT
Fig. 12. The way in which the solution of the diffusion equation (A.ll)
gets asymptotically
trapped between kinks with u > 2 and v < 2.
exponential for x + 00, it is possible to find a soliton solution $>(p) such that u(p, t) < @,@). Moreover, a&_~, t) < 0. (b) For any u < 2, there is an (oscillating) soliton solution with @<@e) = 0 such that U(JJ, t) > u<(p) -@ > pe) - and sufficiently large time. Moreover, atub, t) >O. The situation for cp(x, t) is therefore the one depicted in fig. 12. What points (a) and (b) are telling us is that, in a frame moving with velocity u > 2 one would see z&, t) - bounded from above by &,@) - decreasing with time t, while in a frame moving with velocity u < 2 one would see u(p, t) - bounded from below by #c(p) for p >pe - increasing with time. From (a) and (b) and known theorems, it follows that
cp(ut- p, t) = u(p, t) --f 0
if
u > 2,
cp(ut-p, t) = u@, t) --f 1
if
u<2.
(A.24)
In other words, rp(x, t) gets trapped between solitons with u > 2 and u < 2, and has to propagate with the asymptotic speed u = 2. Without attempting a proof of these statements here, let us notice that the properties of u@, t) discussed in the preceeding paragraph are indeed shared by the perturbative solution (A.12). For the special case uo@) Q:S(x) we have in fact lpo(ut-P7
t)=& W-J t[
=&
exp[t(l
(ut 4t
- $3)
PI2 1 t iu2p - p2/4t] .
(A.29
Completely similar arguments hold for x < 0. For simplicity we shall then summarize the results described above by the statement (not strictly proved in the mathematical literature) that cp(x, t) approaches asymptotically the soliton with u=2. In other words ~x,t)~[@2(2t-X-~)t$2(2t+X+b)-1]
(A.26)
for some values of the translational parameters a and b which depend on the specific form of the initial condition u. (xx>.This is essentially equivalent to the equation (2.20) of the text.
V. Alessandrini et al. /Kink states in supercritical RFT
Appendix
481
B
Zero modes and completeness The purpose of this appendix is to relate the zero-mode wave function .&@) and its conjugate vector x,,(p) to the lowest eigenfunctionfe(P) of the SchrSdinger operator DS of equation (5.11). We are particularly interested in understanding their role in the completeness relation, in order to prove the spectral decomposition of the identity given by eq. (4.42) as well as the spectral decomposition of the Hamiltonian % s given by eq. (4.43). Our strategy will be to prove these results from the known completeness of the eigenfunctions of DS. Let us recall that [&I) and X,-&I) are, by eqs. (4.16) and (4.19)
(B.1)
03.2) From the known properties of our classical solutions as functions of p and u (see eq. (5.32) as well as the comment preceding it) we deduce that the components of [e and x0 have the symmetry property
FobI = -G-P)
(B-3)
3
.
Go(P) = Go(-P)
(B-4)
Therefore, in order to satisfy the symmetry property (B.3) between the upper and lower components of go, we identify it with the eigenfunctions introduced in sect. 5 as follows: fob) to(P)
evp12
=(
fo@) emupI
1
d(P)
-
(
J;b(--p)
eup12
fo(--p)
emupI
1
0(-P)
*
(B-5)
Likewise, in order to satisfy (B.4) we write x0@) as go (-p)
go@) evpt2 xocp) =
( go (J.I) e-vp/2 1
B(P) +
where go@), go(p) are functions read Dsfo +2~e-VP~o 2g evPf.
+
( go(-p)
evp12 emvp12
yet to be determined.
1
0(-P)
(B.6)
Eqs. (4.17) and (4.20) now
=0,
Dsfo = 0 ,
(B-7)
V. Alessnndrini et al. / Kink states in supercriticalRFT
482
Dsg,-, + 29 ewvp go = Szf,, , 2@ evPgo t Dsgo =
(a=-&O/NO)>
-CZfo,
w3)
Let us now try to express, for (2 - u) << 1, all the components terms of
fo(p) = N0 cb(p) e-vp/2 @>O) .
of go and x0 in
(B.9)
The function so defined, which is the exact expression for the upper component of the zero mode for p > 0, is also approximately equal to the ground-state wave functions of Ds introduced in sect. 5, corresponding to the eigenvalue w. = 0(2 - u), and for which we have used the same notation. In fact, both can be identified as u + 2-. Since the ground-state wave function of DS is an even function, its derivative at the origin vanishes. We check that this is approximately true for fob). By using (3.29) and the equation of motion at p = 0, we obtain fO(0)=fvO[~O)-$0e]
= N&u-
We therefore extend the definition function
fo(-P) = fo(Ph
(P> 0)
of
l)%
t&j]
=0(2-u).
(B.10)
fo@) to p < 0 by demanding it to be an even
3
(B.11)
having then the property that Dsfo = 0(2 - u),
fo(0) = 0(2 - u) .
(B.12)
We next go back to eqs. (B.7) which reduce, to leading order in (2 - u), and for p >o,to
Dsfo= 0, (P >Oh
2$fc, + Dsfo = 0 ,
where use has been made of the fact that, for p > 0, cpexp(-up) that (p = (b exp(-up). Due to the fact that, for p > 0
O=$[Dsfol the two independent
=Dsfo
is of 0(2 - u), and
(B.14)
solutions of (B.13) are
0
fo
(B.15)
a
4
+2;fo,
(B.13)
fo
IT
(b)
( .I’ fo
V. Alessandrini et al. /Kink
states in supercritical RFT
483
and the zero mode &-, must be a linear combination of the two, which is determined by using eq. (B.3). This equation implies that, at the origin,fe(O) = -fe(O). Therefore in order to satisfy this relation at p = 0 between the upper and lower components of to, we must have the antisymmetric combination, i.e.: f. evp12 &)(p)=No
fg
_i cf, _ fe) e-VP/2
For p < 0, .$e is automatically
1
(p >o;
u-2_).
(B.16)
’
given by (BS), i.e.
=(
[o(P)
(p CO;
u+2_)
(B.17)
Notice that the upper component of .$,@I) in p < 0 contains also f,-,, and therefore it is neither even nor odd under p * -p as expected from the known shape of p(p). A similar situation occurs for x0(p), in the limit u -+ 2-. In fact, by eq. (5 35) (B.18) and therefore the inhomogeneous term in the right-hand side of (B.8) vanishes to leading order in (2 - u). The equations determining the components of x0(p) reduce then in this limit to eq. (B.13). In other words, x0(p) becomes, for u + 2-, a second zero-mode solution. This is not so surprising if one remembers that the two components of the classical solution, &) and $@), are less and less correlated as u + 2-, and that in the limit u = 2 they are independent solutions of single field equations. However, this time the boundary condition at p = 0 is given by eq. (B.4), which implies g(O) = g(O), and therefore x0 is proportional to the symmetric combination of the two independent solutions (B.15):
%I
X0(P)= Lo z
f. evpi2 = :
(jJ >o; Q. t fo) e-vp/2
where the factor 1 is determined
(x07 to) = 2 s
by the normalization
[Fo(n)Go(~)
-
u+2_),
(B.19)
’ condition
~oW~oW1 c-b
0
=2s 0
f;(p) dp = 1.
(B.20)
484
V. Alessandrini et al. / Kink states in supercritical RFT
Finally, the form of x0 for p < 0 is obtained
from eq. (B.6)
v. U-+2_).
(PC%
X0(P) 21 ;
(B.2 1)
Let us remark that the expression for x0 differs from the one for to essentially because of the replacement f. + -f.in the lower component. It must also be noticed that, since fo(0)= 0 and fo(p) is of order 1 for p z L, the term proportional to f. in the lower component of (B.19) is negligible compared to the other term for moderate values of p. On the other side, in this region the lower component of x0 can be estimated as follows (for 0 < p
e
-up/*
-L -
z Oau’
Lo
g
-2f-Lo-
v-2
cbe
--VP
.
(B.22)
2-v
Therefore, by comparing it with f. givenby (B.9) we get --
Lo --=;(2-u) _a No I-1
(B.23)
in agreement with the argument in the text. The explicit expressions we have obtained for the vectors E. and x0, together with the similar ones for .&, (n # 0) discussed in the text, allows us to prove the completeness relation:
njjl[MP)L(P')
- L(P)~n(P')l
(B.24)
+ 5o(P)jio(P') - xo(P)5_dP')= IS(P -PI).
In fact, by using the eqs. (5.8) and (5.9) of the text, we get for the left-hand side of (B.24) a matrixM@, p')which, for the case p, p' > 0 is given by M,, =Mz2 = 2 e"P~2fn(p)e-"P~2fn(p'), n=O Ml*=09
M2r = {C n=l
e--vD/2 g,(P) e-"P"*fn(P')
t e-"P/2fo(p)e-vP"2fo(p')- (p * P’)} It is clear that, by the completeness
.
relation for the hermitian
(B.25) operator Ds given by
V. Alessandrini et al. /Kink states in supercritical RFT
485
eq. (5.19) in the text, we get Ml1
‘M22
= SC0 -P’)
(B.26)
9
because the similarity transformation exp(iup) the other hand, from eq. (B.14) we get fo@) =-
c f,(p)(2G),a nf0
+ constant
leaves the h-function
.fa(p) .
unchanged.
On
(B.27)
an
However, only the first term contributes to M2 r , the second one being canceled when the expression is antisymmetrized in p and p’. Then, using (B.27) and the expression (5.25) for g,(p), we get
M21
= fIl
Sk
f&3)&@‘)
e-*+p’)/2
-(p
*P’)
s 0.
(B.28)
because the first term is symmetric in (p, p’). This completes the proof of eq. (B.24). From the completeness relation we also get the spectral decomposition of the operator SK2 on both sides of (B.24), it follows that 81’2x0 = ato
3
912to
= 0
(B.29)
and acting with $X2 on both sides of (B.24), it follows that
References [l] D. Amati, M. Le Bellac, G. Marchesini, M. Ciafaloni, Nuclear Phys. 8112 (1976) 107. [2] D. Amati, G. Marchesini, M. Ciafaloni, G. Parisi, Nucl. Phys. B114 (1976) 483. [3] J.L. Cardy, Nucl. Phys. B115 (1976) 141; R.C. Brower, M.A. Furman and K. Subbarao, UCSC preprint, 1976. [4] J.B. Bronzan and R.L. Sugar, U.C. Santa Barbara preprint TH 77-l. [5] N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606. [6] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DlO (1974) 4114,4130, Dll (1975) 3424; J.L. Gervais and B. Sakita, Phys. Rev. Dll (1975) 2943. [7] J. Baacke and H.J. Rothe, Nucl. Phys. B125 (1977) 108. [8] V. Alessandrini, D. Amati and R. Jengo, Nucl. Phys. B108 (1976) 425; R. Jengo, Nucl. Phys. B108 (1976) 447; J. Bronzan, J. Shapiro and R. Sugar, Phys. Rev. D14 (1976) 618. [9] M. Ciafaloni, M. Le Bellac and G.C. Rossi, Nucl. Phys. B130 (1977) 388. [lo] P.W. Anderson, Rev. Mod. Phys. 38 (1966) 298. [ll] A.R. White, CERN preprint TH. 2259 (1977), Nucl. Phys. B, submitted. [ 121 D.G. Aronson and H.F. Weinberger, Non-linear diffusion etc., in Lecture Notes in Math., 446 (Springer Verlag, 1975) and references therein. [13] M. Ciafaloni and G. Marchesini, Nucl. Phys. B109 (1976) 261.
CLASSICAL KINKS AND THEIR QUANTIZATION IN SUPERCRITICAL REGGEON FIELD THEORY V. ALESSANDRINI Laboratoire de Physique Thbrique et Hautes Energies, Orsay *
D. AMATI CERN, Geneva
M. CIAFALONI Scuola Not-male Superiore, Pisa and INFN, Sezione di Piss Received 19 July 1977
The existence of kink states in supercritical reggeon field theory ((~(0) > arc) is related to the expanding disc behaviour previously found in a lattice approximation. The classical solitary waves of RFT with o(O) - 1 = ~1> 0 are classified, for the case of a one-dimensional impact parameter space (D = 1). We find kink-like solutions corresponding to a range of velocities up to a critical one for which the solution has a phonon-like dispersion law (zero mass). The general methods of kink quantization are adapted to RFT. By then quantizing the theory around the solution with critical velocity we are able to give the low-lying spectrum and to construct the quantum kink states, which in RFT are connected to the vacuum by field operators. Their contribution to the Green function is found to restore the cluster property by cancelling, at large impact parameter, the constant term provided by the second ground state present in the ordered phase. For the impact parameter IBI < 2& logs the Green function tends to a constant proving therefore for D = 1, the expanding disc behaviour which is consistent with s-channel unitarity.
1. Introduction The peculiar nature of the critical phenomenon characteristic of the reggeon field theory (RFT) when the reggeon intercept goes through one, has been recently unraveled by using field quantization techniques apt to identify the phase transition mechanism [ 1,2]. A lattice is introduced in impact parameter space, the RFT Hamiltonian is diagonalized at each lattice point and finally an analog spin model is ob-
l
Laboratoire associC au CNRS. 429
430
V. Alessandrini et al. / Kink states in supercritical RFT
tamed by keeping the two lowest-lying states at each site plus the intersite interactions arising from the kinetic terms in the Lagrangian [1,3]. It was recognized [ 11, by using this approach, that contrarily to what happens in more orthodox field theories, the vacuum state does not change structure as the reggeon intercept goes through the critical value. There is however a drastic change in the spectrum of the theory, in the sense that a new energy eigenstate shows up that is degenerate with the old perturbative vacuum in the thermodynamic limit. Even more important, field operators have finite matrix elements between these two degenerate states in this limit, which means that these two degenerate vacua belong to the same Hilbert space. This peculiar feature of RFT can be traced, as we shall see later on, to the non-hermiticity of the Hamiltonian. The two degenerate vacua of the spin model can be thought of as spatially homogeneous states where all spins are up or down, respectively. It has also been recognized [2], always in the context of the spin model, the important role played by spatially inhomogeneous states of a kink-like type in which a dislocation - frontier between spins up and down - propagates with a definite speed. These states give rise to a low-lying continuum with zero energy gap with respect to the ground states, and they are therefore responsible for the peculiar asymptotic behavior of the Green functions, characteristic of a disk expanding with rapidity in impact-parameter space. Nevertheless the spin model, explicitly tailored to put in evidence the collective modes responsible for the phase transition, mistreats badly other degrees of freedom like, for example, higher single site excitations which cannot be neglected far away from the critical point. This is therefore reflected in an unphysical dependence of the disk expansion speed on the infrared cutoff-intersite distance a in the ordered phase of the theory, when a(O) >ocrtt(O). Strictly speaking the disk expansion speed goes to zero as a goes to zero, so the continuum limit cannot be taken in the spin model, without introducing further degrees of freedom. A possible way out, proposed by Bronzan and Sugar [4], is to introduce a quartic interaction with a coupling X’ = h2/p, where h is the triple-pomeron coupling and p= a(O) - 1. At this “magic” value the lowest states at each site become exactly degenerate and the limit a -+ 0 is harmless. The authors then construct the box states in the continuum limit, by paralleling the treatment of ref. [2], and get a physical expansion velocity. The purpose of the present paper is to propose a more general quantization procedure of RFT based on the observation that the non-perturbative effects generated by degrees of freedom neglected in the spin model are already present at the classical level. In fact, we will be able to identify in the continuum limit (and for one transverse space dimension, D = 1) the classical analogues of the kink-like states of the spin model. The fact that the non-perturbative effects can be treated in the semiclassical limit follows from the property that the RFT Lagrangian scales with X as (l/h’) f?(x$, h$). Therefore, as in ?Q4, X plays the role of a semi-classical expansion parameter, and we expect that as h + 0 the non-perturbative effects are exhausted by the classical solutions. Our method is then similar to the ones recently applied to hermitian field theo-
V. Alessandrini et al. /Kink states in supercritical RFT
431
ries - as Q4 in two dimensions - but there are some peculiar aspects stemming from the non-hermiticity of RFT and/or its lack of Galilean invariance. In fact, as learnt from the spin model, there is just one Hilbert space in the ordered phase, and the kink states are connected to the vacuum by field operators. Moreover, we find that classical kink solutions exist for any speed u, and that for a critical value of u they have a phonon-like dispersion law (zero mass). The quantum kink states have therefore small energy gap for small momenta and contribute in an essential way to the disc’s expansion. The critical speed is just the expansion velocity of the disc - actually a box for D = 1 - and turns out to be the same as in the simple eikonalization of a pomeron above one. This paper is organized as follows: in sect. 2, we rephrase some results of the spin model and recall mainly the properties of the two degenerate ground states. We also present a heuristic argument leading in an intuitive way to the expanding disk behavior of hadronic scattering, based on a generalized coherent state formalism. The details of these calculations are relegated to appendix A because they are not essential for the rest of the paper. In sect. 3, we discuss the classical kink solutions of RFT. We prove the existence of a large class of kink-like solutions and explore its main properties, like dispersion relation and dependence on the propagation speed. Sect. 4 is devoted to a general discussion of the kink quantization of RFT. We have adapted to our case the well-known methods of kink quantization [S-7], adding some further understanding to the treatment of the zero-mode problem in a case in which one is forced - as we are - to use the Hamiltonian path integral formulation instead of the Lagrangian one. In sect. 5 we discuss the dependence of the procedure on the classical solution around which we expand and we show how one is driven to a specific choice in order to avoid linear (tadpole) terms in the quantum fluctuations. This fixes the disk expansion speed. The quantization of the theory is then performed up to order zero in the triple-pomeron coupling, and the RFT spectrum in the ordered phase obtained and discussed. Technical details related to the completeness of our eigenfunctions expansions and the calculation of zero-mode states are discussed in appendix B. Finally in sect. 6 we compute the two-point Green function and S-matrix for high energies finding the disk behavior. We then conclude by a discussion of our results, emphasizing the peculiarities of the field theory we considered.
2. Preliminary
results and semiclassical approximation
We shall adopt in this paper the notations and conventions used before in connection with the spin model [ 11. The Gribov fields J/Q, u) and $(b, JJ) are defined in a D-dimensional impact parameter space b, the rapidity y playing the role of imaginary time. By resealing the Gribov fields according to q=i$,
p=iJ/,
(2-l)
V. Alessandrini et al. / Kink states in supercritical RFT
432
the triple-pomeron as
coupling term becomes real, and the Lagrangian can be written
2 =44&P- WP
+
O’vbq.VbP + hq(q+P>P>
(23
where the mass parameter /Ais related to the pomeron intercept by /.l=cu(O)-
1.
(2.3)
It must be kept in mind that q and p are creation and destruction pectively, satisfying the equal-time commutation relations [P(bt Y), 4@‘,
VII= -6@)(b- b’)9
operators,
res-
(2.4)
the wrong sign of the commutator being of course a reflection of the imaginary nature of the triple-pomeron coupling, that is to say, of the non-hermiticity of the RFT Hamiltonian. The intercept parameter p in eq. (2.2) is in principle the bare one. It can be transformed into the renormalized intercept by adding a suitable mass renormalization counterterm, which is perturbative in A. Since we shall not be concerned in this paper with renormalization effects, we shall ignore counterterms, and accept that the critical intercept is given by ~,.~r = 0. The RFT Hamiltonian reads, in our notation:
H=
Jdb[iw -
a’vq- VP - Wq +p)Pl
(2.5)
and the two states which play in the spin model [ 1,2] the role of degenerate vacua in the ordered phase /J > 0 can easily be identified. These states are in fact in one to one correspondence with the stationary points of the classical action corresponding to constant field configuration, i.e., (q, p) = (0,O); (p/h, 0) and (0, p/X). They are obtained by remarking that constant field configurations correspond to zero-momentum components of the Gribov fields, and that the zero-momentum part of H is just the RFT Hamiltonian in zero transverse dimensions, whose spectrum is well known [8,9], with h replaced by A/&‘*. The lowest-lying state is the perturbative vacuum I&), defined by PlGfkJ= 0,
(4014 = 0 ,
(2.6)
which, because of the ordering properties of H, is always an exact zero-energy eigenstate. This state does not change its structure at the critical point but, as 1-1becomes positive, a new state which we shall call IGo) shows up on top of it with an energy gap
Av=exp[-f (3’v], which vanishes exponentially
l
in the thermodynamic
V denotes the impact parameter space volume.
(2.7) limit V+ 00. This second state
V. Alessandrini et al. /Kink states in supercriticalRFT
is associated with the “shifted pomeron”
stationary
points, and it is given by
Itie) = I&) - 141)) where I&) approaches,
433
(2.8)
as p/h + 00, a coherent state
141>= exp -$Jq@,
0) db]l@cr) )
(2.9)
such that
P(b,0) l91)=!
l$Jl).
Even though the Hamiltonian m = (_ l)lS@)PW db ,
(2.10) His not hermitian,
there exists a metric operator
[l] (2.11)
such that ‘VZH+CYl =H.
(2.12)
Therefore, right and left eigenstates of H are connected by the metric operator 311, It follows then that the perturbative vacuum is self-conjugate, while the dual state of I&) is (2.13) such that (2.14) These right and left eigenstates of H corresponding to the same energy level are therefore associated with the @/A, 0) and (0, p/A) fixed points of the classical action. The fact that a coherent state shows up as a new ground state after the phase transition is a characteristic feature of non-relativistic microscopic theories of Bose-Einstein condensation, such as the theory of superfluidity [lo]. But in hermitial theories this state has an exponentially vanishing overlap with the old ground state in the thermodynamic limit, so it really corresponds to a different phase of the system. This is precisely what does not happen in RFT, due to the non-hermiticity of H. If we compute the norm of the coherent state l@r>, taking care of the q, p commutation rule (2.4), we get (#I~ lf#q) N e-(r/h)2
v,
(2.15)
while in a hermitian theory one would obtain the opposite sign in the exponential, thereby giving for a normalized coherent state an exponentially vanishing overlap with the perturbative vacuum lq@. In our case, however, l$r) is a zero norm state in
434
V. Alessandrini et al. /Kink states in supercriticalRFT
the thermodynamic limit while I$,-,) has negative norm, l@r) having a finite overlap with the perturbative vacuum I&,). Moreover, from the explicit representation of the states one can immediately get the transition matrix elements of the Gribov fields between the two vacua, given by (2.16) These peculiar features of RFT were already recognized in the spin model. In A#4 theory, different shifts of the quantum fields define different disconnected Hilbert spaces, corresponding to different phases of the system. Because of the existence of topological conservation laws, the kink sector is also different from the vacuum sector. The old perturbative vacuum disappears because it becomes unstable. In RFT there are no different disconnected phases: all would-be phases are connected by transition matrix elements of the quantum fields that remain finite in the thermodynamic limit. Even the kink states that we shall discuss later on, which interpolate in impact parameter space between the two vacua ($I,,) and l$r), will have finite overlap with them. Therefore shifting the fields will be in RFT a matter of convenience and not of absolute necessity because the kink sector will be the same as the vacuum sector. Let us now discuss the coupling to the external particles. Under the hypothesis of eikonal couplings, the RFT scattering amplitude can be written in the Schrbdinger picture as the following matrix element of the evolution operator between coherent states (2.17) where B is the distance in impact parameter space between the colliding hadrons, which act in this theory as external sources of pomerons. The source distributions g(b) and f(b) are taken to be strongly peaked functions near b = 0. Independently of the assumption on pomeron coupling S, as given by (2.17), is in general the generating functional for Green functions, which are obtained by functional differentiation with respect to the sources. It was shown in ref. [2] that, in the context of the spin model, the Green functions have an expanding disk behavior, which means S = 1 for 1B I- uY > 0. We shall restrict ourselves from now on to the case of a one-dimensional impact parameter space. In the spin model the expansion velocity u depends on P - p,,.rt and on the intersite distance a [2]. It is reasonable that, away from the critical point, the dispersion law which is related to the expansion speed IJ cannot be correctly obtained with a method that has neglected other excitations not responsible for the critical phenomenon. We shall present in the next sections a quantization technique which is better tailored to give the complete low-energy spectrum in the ordered phase. We would like to sketch, however, in the rest of this section, a heuristic argument which shows intuitively the emergence of the expanding disk. The argument is heu-
V. Alessandrini et al. /Kink states in supercriticalRFT
43.5
ristic in the sense that some steps are justified a posterion’ by the results of the following sections, and all the details of the calculations are given in appendix A. From eq. (2.17) it is clear that the S-matrix can be computed by solving the Schrodinger equation of RFT
HI*(y)) = -
$ Iwj))
(2.18)
with the initial condition that I*(O)) is the initial coherent state, the S-matrix being then the overlap of I*(Y)) with the final state. We then write the most general state vector of RFT as a generalized coherent state, in terms of an infinite set of correlation functions, and transform the Schrodinger equation into a set of differential equations for these correlation functions. Needless to say, the problem still contains all the intricacies of the theory, and the next step is to calculate a solution in the semiclassical approximation. The fact that this approximation is extremely relevant in RFT in the ordered phase is indicated by the fact that the RFT Lagrangian has the following scaling property with respect to the triple-pomeron coupling (2.19)
~(M;h)=&V?4~~;1),
which means that the weak coupling limit will be correctly described by the semiclassical approximation. Of course the triple-pomeron coupling is dimensionful, so h = 1 is to be understood in terms of the unit of length, which is fixed by CK’. We then find that, in the semiclassical approximation, the state vector is given ~,J,(Y))
= e-jdbg(b,
Y)q(b)+
correlations [$0,
(2.19)
where the correlations are higher order polynomials in 4, and g(b, JJ) satisfies a nonlinear diffusion equation with initial condition g(b, 0) = g(b), the initial source function. This diffusion equation is in fact the classical equation of motion for the field p,,(b, y) when qcl(b, y) is set equal to zero. We discuss then its solutions in appendix A, where it is shown that g(b, y) propagates in y by expanding from g(b, 0) and reaching, for large y, a box-like form that expands with a well defined velocity u,, = 2&, i.e. g(b,y)+(u,,y+ti-
lbi),
(2.20)
where 6 is a “phase shift” that depends explicitly on the initial condition. To the extent that correlations can be neglected, the state vector I I)(Y)) is asymptotically in Y a coherent state, and then it is trivial to see that the S-matrix is given by -!ldbf(b-B)B(uoYtS-IbI)
1 ,
(2.21)
V. Alessandrini et al. /Kink
436
states in supercritical RFT
which, due to the fact that f(b - B) is localized around b = B, is 1 if 1B I> u. Y + 6. This is just the expanding disk behavior of the scattering matrix. Of course the evolution of the state vector in terms of a single coherent state is only approximate, because the classical solutions that dominate in principle the semi-classical limit should be those of the coupled field equations. This effect is taken care of by the correlation functions in the state vector, that we have neglected in eq. (2.21). However, from the more careful analysis that we shall perform in the next sections it will turn out that the low-energy part of the spectrum of RFT - and therefore the large Y behavior - will be really controlled by uncoupled field configurations. We shall therefore justify a posterion’ the validity of the simple picture presented here. 3. Classical kink solutions of RFT We shall investigate in this section the existence and properties of a large family of kink solutions to the classical equations of motion of RFT, for the case of a onedimensional impact parameter space. The equations of motion that follow from the Lagrangian (2.2) are
ap
2
__pp+hgZ+2Xpq-a’~=Q, aY
(3.la)
a% -a4_w+hq”+2xpq-u’--$=o, (3.lb) ay and we are particularly interested in kink-like solutions which depend only on a linear combination of y and b which we define as P’PUY
-
(3.2)
where the parameter IJ is the kink propagation ables we write then the classical fields as
Pcdb,Y) =;
4~)
3
qc&
Y)
velocity *. With this choice of vari-
=; g@)
(3.3)
and obtain for the fields q(p), q(p) the coupled differential
equations
+u3,tcp-(p2-2l@=o,
(3.4a)
~t~t+~2_2$@Y=o.
(3.4b)
The fixed points of these equations tions) are given by (&a=
(O,O) (091) ( (1,O)
corresponding
to constant (p-independent
.
* We notice that u as defined here, and in the following, is adimensional. tion velocity
measured in units of &&.
solu-
(35) It is the kink propaga-
V. Alessandrini et al. /Kink states in supercriticalRFT
431
The first one obviously corresponds to the perturbative vacuum associated with unshifted pomeron states, while the two others correspond to the new vacuum associated with shifted pomeron states, as discussed in the previous section. As it is well known, these fmed points are unstable. They are actually saddle points in the sense that in the (q, $) plane corresponding to phase space they are attractive in some directions and repulsive in others. From the general structure of eqs. (3.4) we can see that 3
V%co)= V-,(-P)
?“@I = Y-,(-P)
9
(3.6)
which means that we can restrict our attention only to positive propagation velocities; solutions with negative u being then trivially obtained by means of these reflection properties. Moreover, it also follows from eqs. (3.4) that if {q,(p), P&J)} is a a solution, then {a(-p), cp,(-p)} is also a solution. For all the cases we shall be interested in these two solutions coincide, so we shall have CPU(P)= &(-P)
7
(3.7)
once the point p = 0 has been fixed by using the translational invariance of the equations of motion. Moreover, a class of solutions of the single-field equation (q = 0, cp= &I) or vice versa) can be proved to exist for 0 < u < 00. They are discussed in appendix A. Some particular kink solutions of eqs. (3.4) are already known. Cardy and Sugar have found exact analytic solutions for two particular values of the propagation speed. They are given by
cp=
i [ 1 7 tanh &
- i(cosh &I)-~]
foru=d,
q
.i [ 1 T tanh &J
- $(cosh x&)-~]
foru=G.
(I(
(3.8)
Each of these solutions can be represented in the (q, q) plane by a phase-space trajectory starting at the (0, 1) fured point when p = -00, and ending at the (1,O) one when p = 00, as shown in fig. 1, The first solution (3.8a) has both fields always positive and smaller than 1 and is represented therefore by a trajectory internal to
Fig. 1. The phase-space trajectories of the Cardy-Sugar kink solutions.
V. Alessandrini et al. / Kink states in supercritical RFT
438
the triangle defined by the three fixed points given by eq. (3.5). The second solution is represented by a trajectory that lies outside the triangle. Due to the reflection property (3.6), one finds for u = -J$ and u = -4 the same trajectories but evolving in the opposite sense, from (1,O) to (0, 1). We are interested in finding out about the existence of some further solutions of the same kind, and in particular the way they may depend on the propagation velocity u. This is a non-trivial problem in RFT due to the lack of Galilean invariance, which means that kink solutions with different velocities are not related by a simple change of variables, and consequently that the functional form of the kink solutions change in a non-trivial way as the propagation velocity u is changed. We shall heavily rely on the existence of a constant of motion for the system (3.4), which is easily obtained by multiplying the first equation by $, the second by (d and adding to obtain a total derivative which can be integrated to give ~tcp~(l-cp-~=c,
(3.9)
where C is a constant. This constant of the motion plays the same role of energy conservation in an ordinary mechanical system. It severely restricts the shape of the phase-space trajectories in the (q, @) plane. The kink solutions will reach (or leave) asymptotically one of the fixed points (3.5). It then follows that C = 0, and then &j
=-pq(l
_I+-cp)
(3.10)
for all p. Therefore, in all of the (~,a plane the sign of @is determined, it is exhibited in fig. 2. One can immediately see, for example, that inside the triangle determined by the three fixed points L$ is negative, so if one of the fields decreases as a function of p, the other must increase and vice versa. Moreover, the phase-space trajectory can never be parallel to one of the coordinate axes, except when it crosses one of the three lines of fig. 2, which are the coordinates axe themselves cp= 0 and Cp=0, and the straight line p + Cp=1. It is easy to deduce in this way that the origin (0,O) can only be reached by infinite looping around it or from one of the coordinate axes which implies, however, that one of the fields vanishes identically for all p. We are left in this case with the uncoupled field situation discussed at length in the appendix A.
Fig. 2. The sign of $
in the (9, F) plane.
V. Alessandrini et al. /Kink states in supercriticalRFT
439
3.1. Linearized equations and numerical results Let us now study the behavior of the classical solutions near the (9,9) = (0, 1) fixed point. We then introduce x=1-9
(3.11)
and linearize eqs. (3.4) in 9 and x. This gives cp-t$-9=0,
(3.12a)
ji-vg-x+29=0.
(3.12b)
We shall restrict ourselves to v > 0; for v < 0 the solutions are obtained by the reflection properties expressed by eq. (3.6). The eq. (3.12a) implies 9@) = A eVkp with A an arbitrary
(3.13) constant,
and
k=-&v&d-.
(3.14)
In order to solve eq. (3.12b) we have to distinguish two possibilities, according to whether 9 is of the same order or much smaller than x. We therefore consider the two cases
(a) f -O(l),
(b) ;
<
For case (a), x is driven by the solution of the 9 equation, x=Beekp,
with
x=BeVKP
ri=$vfJW.
and we find
A -=vk, B
while for case (b) x is decoupled
with B an arbitrary
(3.15)
(3.16)
of 9 and its behavior is then of the form (3.17)
constant
and (3.18)
It is clear that the major difference between cases (a) and (b) is that in the first one the phase-space trajectory will reach (or leave) the fixed point (0, 1) with a finite slope, while in the second case it will do so with a zero slope, along the 9 axis. We still have to consider, however, the various sign possibilities in eqs. (3.14) and (3.18). Let us first consider trajectories of the class (a), in which case we only have to worry about the two sign possibilities in eq. (3.14). For the positive sign, k > 0 and the linearized solution is relevant for p + +oo. Since we are restricting ourselves to v > 0, it follows that A/B > 0, and the solution therefore tends towards the (0, 1) fixed point as p + = from inside the triangle of fig. 2. Indeed, the slope with which
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V. Alessandrini et al. /Kink states in supercriticalRFT
it reaches the fixed point in the (p, q) plane is given by Sr = lim Z- = lim Z-_ = -$u(m p-+- ( -x 1 p-+-a$- 1
- u) ,
(3.19)
which shows that Sr is always negative, varying monotonically between 0 and (-1) as u varies from 0 to infinity. In this limiting case the phase-space trajectory reaches the fixed point along the hypothenusa of the triangle of fig. 2. If instead k < 0 (always in the case (a)), the linearized solution is relevant for p + -00, and since A/B < 0 we conclude that the phase-space trajectory leaves the fixed point by going outside the triangle. Indeed, the slope in the (p, $) plane is given by Sz=
lim Z- =+u($GZtu)>O. p+_m ( -x 1
(3.20)
Let us now turn to case (b). It is a simple matter to realize that the only sign combination of eqs. (3.14) and (3.18) compatible with cpbeing exponentially smaller than x is
K=~“-&&zi,
k=-$v-&%i,
(3.21)
which means that the solution of the linearized problem is relevant when p + -00. Since A and B are uncorrelated, the possible solutions leave the tixed point (0, 1) with vanishing slope but may end up either from inside or outside the triangle. We should also mention that for this class (b) of solutions tp/x vanishes exponentially in p except when IJ = 0, in which case cp/x goes to zero as l/p. In this case the solution of the linearized equations (3.12) is (P=A e’P
x=Ap
efp
(3.22)
with both signs in the exponential being the same. To sum up, the behavior of the kink solutions of RFT near the (0, 1) fixed point for IJ > 0 can be summarized as follows (i) case (a), k > 0, the solutions arrive to the fixed point from inside the triangle with a finite slope ranging from Sr = 0 for u = 0 to St = -1 for u = 00. (ii) case (a), k < 0, kink solutions leave the fixed point with a finite positive slope Sa going outside of the triangle. (iii) case (b), there is a unique sign choice for k and K, given by eq. (3.2 l), and the kink solution leaves the fixed point with zero slope. The behavior of the classical solutions near the other non-perturbative futed point, namely, (p, $) = (1,O) can be obtained at once from the previous results, due to the symmetry of the equations of motion, by interchanging cpand $ and changing p into (-p). We can also locate the Cardy-Sugar solutions within this classification. The first one, given by eq. (3.8a) is of type (b), leaving (0, 1) with zero slope, while the
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V. Alessandrini et al. /Kink states in supercritical RFT
second one corresponds to case (ii), leaving the fured point and going outside of the triangle with slope Sz = tf. We have investigated numerically the existence of kink solutions that interpolate between the (0, 1) and (1,O) fHed points. These solutions will satisfy the conservation law provided by eq. (3.10), and, as discussed before, they will not reach the origin (0,O) unless one of the fields vanishes identically. Let us define p = 0 by the condition cp(O)= fi0) .
(3.23)
It then follows from eqs. (3.10) and (3.7) that (cl(O)= -l&O) = *do)
d_
.
(3.24)
Therefore, for every given p(O), these relations specify a complete set of initial conditions for the integration of the coupled system of equations of motion. We have performed this integration numerically by searching, for every given cp(O), the value of the propagation speed u such that the solution approaches steadily the fixed points. We obtained in that way a large class of kink solutions, whose phase-space trajectories are either inside or outside the triangle determined by the three fixed points, corresponding to the two different signs of eq. (3.24). We shall discuss their properties separately. We present first our results for the classical solutions whose phase-space trajectory are inside the triangle. We find in this case kink solutions belonging to the class (a) for all propagation velocities in the range 0 < IJ< 00, solutions belonging to the class (b) for velocities in the range 0 < u < 2, and the solution having the behavior of eq. (3.22) for u = 0. Needless to say these numerical solutions match the known behavior near the fixed points. In order to make our results more clear and to make more evident the continuity properties of these solutions as the propagation velocity changes, we turn our class (a) solutions for positive u into solutions of negative u (-< u < 0) by means of the reflection property (3.6). We are therefore left with
Fig. 3. The phase-space trajectories of the family of kink solutions inside the triangle, with --
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V. Alessandrini et al. /Kink
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a family of kink solutions with propagation velocity in the range -00 < u < 2, whose properties we now discuss. The phase-space trajectories of this family of solutions are sketched in fig. 3. We discuss the evolution of the phase-space trajectory as a function ofy because in this case the sense of the evolution, indicated by an arrow in fig. 3, does not change when IJ is changed to (-IJ). This is not the case for the evolution in p, as follows from eq. (3.6). As u tends to (--), the trajectory tends to the hypotenusa of the triangle, evolving in y from the (1,O) to the (0, 1) fixed point. As u begins to increase through negative values, the phase-space trajectory begins to sweep the interior of the triangle, always propagating in the same direction and leaving and reaching the fixed points with a non-zero slope. When u = 0, we find a static solution exhibited in fig. 3, whose behavior near the fixed points is given by eqs. (3.22) and for which obviously there is no arrow corresponding to time evolution. When the velocity u becomes positive, the solutions switch to class (b). They keep sweeping the interior of the triangle as u increases, but they switch their sense of evolution in the sense that they evolve from the (0,l) to the (1,O) fixed points. Since one of the fields is exponentially smaller than the other near the futed points, their phase-space trajectory is in those regions very close to the cpor p axis. The emergence of the critical propagation velocity u = 2, which will play a central role in the rest of the paper, can be seen as follows. As u approaches 2 from below, the phase-space trajectories become more and more close to the sides of the triangle (the cp,9 axis) thereby spending a large amount of “time” near the origin, that is to say, the perturbative vacuum. In this limit the coupling between the two field equations becomes less relevant, and the overlap between cpand q tends to zero. In the
-1
0
w
1
2
”
Fig. 4. The value of q(O) as a function of v for the kink solutions inside the triangle.
V. Alessandrini et al. /Kink
states in supercritical RFT
443
limit u = 2, the classical fields are solutions of uncoupled field equations, and the phase-space trajectory spends an infinite time in the perturbative vacuum. We stress again the perfect continuity of this family of solutions that we have found. This is also seen in fig. 4, where we plot the value of cp(O)as a function of u. The Cardy-Sugar solution (3.8a) is a particular member of this family without any further relevance besides the fact that it is given by a simple analytic expression. We have also found kink solutions outside of the triangle, corresponding to a choice of initial conditions such that ~(0) < 0 and d(O) > 0. They look like the corresponding Cardy-Sugar solution represented in fig. 1. In agreement with our expectations from the analysis of the linealized equations we find again a family of class (b) solutions with velocities in the range 0 < IJ < 2, such that when u + 2 they become very close to the sides of the triangle and eventually merge at u = 2 with the corresponding solution obtained from the interior of the triangle. Notice, however, that while the classical solutions {&), aO)} whose phase-space trajectory is inside the triangle are monotonic functions of p, those which are outside it necessarily have an oscillation, as functions of p. 3.2. Energy-momentum relationship We shall anyhow concentrate our attention on the solutions inside the triangle, which are the ones which are relevant for our quantization procedure. We turn now to the calculation of the energy, the momentum and the action associated with them. From eq. (2.5) and (3.3) we obtain, after changing variables from b to p: (3.24) where the dot means as usual the derivative with respect to p. The classical momentum is given by (-i Ycpcl), where
(3.25) and the classical action corresponding
to a kink of speed u is (3.26)
It is worth commenting at this point about the imaginary values of the classical momentum, (-i 3’cl). The corresponding quantum operator defined in terms of the p and 4 fields is the generator of real translations in impact parameter space and con-
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V. Alessandrini et al. /Kink
states in supercritical RFT
tains an i because p and 4 satisfy commutation relations adequate to creation and destruction operators. Its classical values are not expectation values but rather transition matrix elements of P between a right coherent eigenstate of p and a left coherent eigenstate of 4. Since for our classical solutions p and 4 have real values, the transition matrix element turns out to be imaginary. It follows immediately from eq. (3.25) that rS,r is just (n/X)* times the area under the phase space trajectory of the classical solution. Therefore, for the family of solutions under consideration, PC1starts from :(J.L/X)~ when LJ+ -00 and decreases monotonically to zero as v-, 2-. By using eq. (3.10) the classical energy can be written as:
Ple1
0--
-
=-2
j dp& _m
cc&G”
=2
s
dp@(l
-LQ-~)~O.
(3.27)
_m
Therefore, eCl is always positive definite, starting from zero when u = 2, then increas ing and finally decreasing to zero when IJ+ -00. Moreover, by manipulating the equations of motion one can prove that AC, =
EC1 -
l&i
3, =;
0x *fi
j- dp [cpq2+ q2g] . _m
Therefore, for the class of solutions we are considering the action density LA,.,is always positive definite. It goes to zero as LJ-+ 2-, and increases monotonically to (+-) as u tends to (--). We are interested in studying in more detail the limit u + 2-, which will be the relevant one in our quantization procedure. We have seen that in this limit ~(0) tends to zero. Moreover, the overlap between cpand g is also very small; in fact q(p) + 0 for p < 0 while q(p) + 0 for p > 0, so the field equations are effectively decoupled. We shall therefore study the properties of the kink solutions by using cp(O)as a small expansion parameter. In this case the initial conditions (3.24) become l&(0)=$(0)2:$(0)=--0).
(3.29)
Moreover, for p > 0 $$I)is very small, and therefore in this region we can neglect quadratic terms in @-@).We are therefore led to the following field equations for P>O +uct)+q-\o*=o,
(3.30)
++$(1-21p)=o.
(3.31)
V. Alessandrini et al. /Kink
states in supercritical RFT
445
Eq. (3.31) can be solved by using (3.30). The solution is (3.32)
@P) = G(P) emuP ,
which satisfies the initial conditions (3.29) and gives g(p) in terms of the solution of the single field equation for cp, eq. (3.30). Let us now show that this equation admits a unique solution with ~(0) = 0(2 - u). In fact, as it is done also in the appendix, the single field equation can be recast in the form (3.33) where the velocity field W(p)= cp has a unique phase-space trajectory with boundary condition w’( 1) = (1 t 42x1 - cp). As remarked in the appendix A, cp= 0 is a spiral point for the trajectory if u < 2 (damped oscillations) and becomes a regular end point when u = 2 (critical damping). We show in fig. 5 the form of W(q). It is then clear that the boundary condition cp(O)= (p(O) is satisfied for u < 2 at the intersection I of W = W(q)and W = cp; in such a way that cp(O)+ 0 as u + 2- (see fig. 5). The same remarks prove that there are solutions with u < 2 outside the triangle, satisfying cp(O)= -d(O), given by the intersection II of W = W(p)with W = -q. We are now prepared to discuss the order of magnitude, in the small parameter and s4,. First of all we notice that 3’Cris the area under the cpo = cp(O)of %I 9 ~Ccl> phase-space trajectory in the cp,$ plane, and that it is of O(C&. Indeed, by using the solution (2.32) for g(p), we can compute Yer as follows:
=2 f
G2 e-“P dp t & .
(3.34)
0
For small values of p, the solution I&) of eq. (3.30) grows like cpo exp(& up) for
L
1
Q
Fig. 5. The phase-space (W, 9) trajectory for the kink solutions of the single-field equation. Full line: v < 2. Dotted line: v = 2.
V. Alessandrini et al. /Kink
446
states in supercritical RFT
u = 2-. Therefore, the integrand in eq. (3.34) is of order (pg over a length of order log( 1/go), and outside of this region it is exponentially small. It then follows that PC, is of order O(& ln( l/qe)). We shall neglect logarithms with respect to powers of (p. in our estimates. In what concerns the classical energy eC1,we can write
0x2
-
1
-+$dp
C-1= -4
s
r-la
0
= 4~ 7
$* e-“P dp _ 4 J
0
and, by performing
*__&
=
2u
i
(3.35)
0
a partial integration
e,,
$4 e-“p dp
in the last term, we obtain
$* ePp dp +
2$(o)*.
(3.36)
0 Therefore, using the initial condition
0
; * [f,, - uJ&
;p,,] = (2 - u)l& .
so we conclude that, when u + 2-, sion relation has the form f,r = 2fi
(3.30) it follows that (3.37)
ecl and Y,_tboth go to zero, and that the disper-
YC,I+ 0(‘9,](2 - u)).
(3.39)
The meaning of this result is clear; it is just the Taylor expansion of the dispersion law e,r = ~(‘3’~~) of our family of kink solutions. This dispersion law is not simple because of the lack of Galilean invariance of RFT. In A@4 the action, being invariant under Lorentz transformations, does not depend on the propagation speed u of the
Fig. 6. The action per unit rapidity for the various families of classical kink solutions, as a function of (A) solutions inside the triangle: full curve; (B) solutions outside the triangle: dashed curve; (C) single-field solutions: dot-and-dash curve.
V. Alessandrini et al. /Kink
states in supercritical RFT
441
moving kinks. The dispersion law for a moving kink is then a direct consequence of Lorentz invariance. For our family of kinks solutions with -m < IJ < 2, the action does depend on u, the u dependence of the solution is non-trivial and the general dispersion law cannot be obtained in analytic form. However, as u + 2-, or, what is the same, for small classical momenta, the dispersion law of the classical solution becomes a phonon-type dispersion law, with a zero energy gap. Besides the family of classical solutions we have just discussed, we must recall the existence of kink solutions outside the triangle, which have negative action. Moreover, as discussed in appendix A, there are also classical kink solutions of the uncoupled field equations - with one of the fields identically zero - which connect one of the (0, 1) or (1,O) fixed points to the perturbative vacuum. The action, energy and momentum of these kink solutions is obviously zero, and their propagation speed is bigger than 2 *. The critical propagation speed is then reached from above when one considers the uncoupled field equations, and from below, as we have discussed at length, when one starts from the coupled case. We summarize in fig. 6 the action per unit rapidity interval of the various families of classical solutions discussed in this paper. As we have done in fig. 3, we exhibit the perfect continuity in u by choosing the negative u branch when convenient.
4. Quantization
of kink states
The purpose of this section is to set up a quantization scheme in RFT by expanding around one of the classical solutions discussed in the previous section. We shall therefore adapt to RFT the methods introduced to quantize X$4 theory in 1 + 1 space-time dimensions around a moving kink. The different methods used - canonical quantization or path-integral formulation - are of course all equivalent, and their main task is to isolate the collective coordinates associated with translational invariance and, to integrate over the translational mode. We shall adopt in this paper the path-integral formulation of Gervais and Sakita [6]. In this formalism the global translational invariance is transformed into a local gauge invariance by introducing an extra degree of freedom and a constraint. The collective coordinate degree of freedom can then be easily integrated away when one considers matrix elements of the evolution operator between eigenstates of total momentum. In our case, after projecting total momentum in the initial and final states of eq. (2.17) we get the following representation for the matrix element
(4.1) l
The ones with 0 < v < 2 have an oscillating
field.
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V. Alessandrini et al. /Kink
states in supercritical RFT
In this equation +r,k and qr,k are the reduced wave functionals for the initial and final state, the plane wave factors of the collective coordinate part of the wave functional have already been used in the collective coordinate integration. The reggeon fields have already been transformed to the body-fixed system, namely, p = p@,y) and 4 = q@,y). The first delta functional represents the constraint, and fixes the quantum momentum operator (4.2) equal to the total momentum eigenvalue k. Moreover, Q@, 4) = 0 is the gauge condition, to be chosen so as to eliminate the zero mode, and J is the corresponding Jacobian. Taking advantage of the presence of the constraint in the functional integration, we can rewrite the S-matrix as:
X exp
fdy[-i&k
+6j(qp
1%
eff> dpl
,
(4.3)
0
where the effective Hamiltonian
density 91,rr is given by: (4.4)
Since p is the transverse coordinate in the body-fixed system, the classical static fields that make stationary the effective action of eq. (4.3) satisfy the equations of motion
691eff --0,
MP)
_
&%ff
_
&P(P) -OF
(4.5)
which, given the form (4.4) of BI,n, are precisely the classical equations of motion (3.la,b) discussed in the previous section. Their solutions are then the moving kinks propagating with velocity u. Notice, however, that these moving kinks are local minima of the unconstrained action. One could still try to fix u, which plays the role of a Lagrange multiplier, in such a way that the constraint k = P is satisfied, but this will not be possible in general given the imaginary nature of the classical momentum of our families of kinks. We shall come back to this problem in the next section, where we shall discuss at length the choice of the classical solution used to shift the reggeon fields. The classical kink solutions contain both fields qCi and pcl, and it will prove convenient to introduce a spinorial formalism in order to simplify the subsequent expressions. Moreover, we want to treat creation and destruction operators in a sym-
V. Alessandrini et al. /Kink states in supercritical RFT
metrical way. We then introduce
f=
449
a spinor field E by
0 ;
(4.6)
and a metric tensor 0
u=
-1
( ) 1
(4.7)
0
in such a way that
This is typical of “first order” formalisms even for Hermitian theories [S]. Notice that c[ = qp - p4 is zero at the classical level, but is of course the commutator of p and 4 at the quantum level. If the scalar product is defined as (r, 8 = /-dp (Q’P - ~‘4) = -6
F’) 2
(4.9)
then the transformations that leave this scalar product invariant at the Lagrangian level are equivalent to transformations that leave the commutation relations invariant at the Hamiltonian level, which are of course Bogoliubov transformations. Moreover, from eq. (4.9) it follows that for any c-number spinor [, (4.10) and therefore, in order to define a non-vanishing norm we shall be led to introduce a spinor dual to .& which is not gn. Let us now perform a shift of the reggeon fields, by defining the shifted fields according to (4.11) where (4.12)
(do) and a) being the solutions action density SQ0) becomes:
of eqs. (3.4a,b). In terms of the shifted fields the
(4.13)
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V. Alessandrimi et al. /Kink states in supercritical RFT
where Syz is a differential
Bc2
operator of the form
(4.14)
=lJ
where
B = 24~) >
B=2&p),
O(u)=u$d2
+2(cpt$)-
-dp2
1,
DT (u) = D(-u) .
(4.15)
In order to properly incorporate in the shifted action density (4.13) the gauge condition and the constraint, let us investigate separately the degrees of freedom associated with the translational invariance of the RFT action. It follows, as usual, from translational invariance that
lo@) =No %
(4.16)
is a zero-energy eigenvector of 5X2, i.e. g,to
&I~2
= 0,
(4.17)
=o,
No being a normalization factor to be discussed later on. Moreover, differentiating with respect to u the classical equations of motion (3.4a,b) we get
j&l
8
2
,/pEcl --
a~
-0
ap
(4.18)
3
which implies that the spinor x0 defined as
Lo being a new normalization 9f,Xo(P)
= -P
5 No
(o(P)
factor, satisfies ’
(4.20)
From eqs. (3.17) and (3.20) it follows at once that both to and x0 are orthogonal to any eigenvector of 91, with non-zero eigenvalue. Notice that the vector x0 just defined is not a zero-frequency eigenvector. It will play however an important role because it is the conjugate vector to the zero mode eigenvector to, and it will have to be kept in order to have a complete set of eigenfunctions to expand the shifted field. The arbitrary normalization constant Lo will be fixed by the condition (X0> Eo) = 1.
(4.2 1)
V. Alessandrini et al. /Kink states in supercritical RFT
451
We want to emphasize that the formulation we are describing here is by no means specific to RFT. The same pattern emerges when one performs a kink quantization of Lp4 theory, for example, using a first-order formalism. We can now discuss the choice of the gauge condition Q = 0. As is usually done, we impose a gauge condition that ensures that the spinor field [ has no component on the zero mode. Our gauge condition is then, as suggested by eq. (4.2 1)
Q=(f?o,t)=O.
(4.22)
The classical part of ,!jin eq. (4.11) will satisfy by itself the gauge condition provided that the over-all translational parameter a left undetermined by the equations of motion in ,&r(p t a) is adequately chosen. One can easily check that for the class of kink solutions discussed in the previous section our gauge condition fixes a = 0. Therefore, eq. (4.22) reduces to a condition for the shifted field l’, and the most general solution is 114
E =;
EC1 +
(a 1 !-
Pox0
+
al 3
(4.23)
where v is the genuine quantum fluctuation, namely, the part of the shifted field having components on the remaining non-zero modes:
(xo,17)=(~oo,77)=o. Once the gauge condition
(4.24)
is chosen, the Jacobian J of eq. (4.3) is given by (4.25)
and, by using the expansion
J=
(4.23) we obtain
v[;$ +($y”(a~~,xo)f(~)1'4~o(a,w0.XO)].
We can next use the constraint in eq. (4.3) to fix the x0 component field. The constraint equation reads, in our spinor formalism,
(4.26)
of the shifted (4.27)
i(P-k)=~(~,a,U-ik=O, which in terms of the shifted fields become of the form R-F[A,]
=o,
where we have introduced
(4.28) the notation: (4.29)
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V. Alessandrini et al. /Kink states in supercriticalRFT
Wol =-A,
(”y4[:j$-) + y4 @,ii, x0) 2
(4.30) If the constraint delta functional 6 [R - F(Ao)] is used to perform the A0 integration, one gets a new Jacobian factor of the form [lI,F’(Ao)]-’ , evaluated at the value of A0 that satisfies the constraint, namely 2&7&Y) II2 a,(v)=-fW[f2W+ @o,apxo)] 1
g 1’4 X +
fW=
0
p
t
(4.3 1)
(a,q, x0)
~~0,~,x0~
(4.32) *
We have chosen the root of eq. (4.28) that guarantees that A0 vanishes when R = 0, that is to say, when the constraint is saturated by the classical momentum and the quantum fluctuation 1) is zero. By computing F’[A,] from eq. (4.30) and using (4.3 1,32) we then find that this new Jacobian exactly cancels - up to a constant the first one given by eq. (4.26), i.e. CX’114 (4.33) ~ FfrAol = y 1 = constant . l--Y 1 This result confirms the internal consistency of our procedure. In the general formulation of the collective coordinate method the over-all Jacobian one gets in the path integral can be written as the commutator of the gauge condition with the constraint, Calling A0 the to component of g, we have written the constraint and the gauge condition as:
+l,(v)+f
- (f2
+;;$Qg’]
~LJOWI
(4.34)
and the fact that the over-all Jacobian is a constant means that A0 and Ao, considered as operators, satisfy canonical commutation relations. Therefore the identification of x0 as the conjugate vector to to is fully justified. We then conclude that the constraint and the gauge condition can be integrated, without adding any extra terms to the action density in eq. (4.13). All what one has to do is to shift the fields according to eq. (4.23), with A0 fured by eqs. (4.3 1,32). The n field is to be integrated in the subspace orthogonal to the zero mode. Let us now turn our attention to the normal modes which diagonalize the quadratic part of the action, and which will provide us with a basis to expand the n field. They are solutions of the eigenvahre equation %r;,
= %En,
(n # 0) .
(4.35)
V. Alessandrini et al. /Kink
We next define the components
states in supercritical RFT
453
of &, by
(4.36) and, using the explicit form of 91z given by eqs. (4.14, 15) the eigenvalue equation reads (4.37a)
1
-2qF,,+ u$+-$+l-2(qtlp)
Fn=T
F,,.
(4.3713)
By inspection of this system of equations, and using also the fact that the classical solutions we are considering have the property that cp(p) = a-p), it follows that if .$, is an eigenfunction of 91 2 with eigenvalue o,, then ES-P) E-“@)=(F is an eigenfunction RZt-n@)
n
(4.38)
(_J of g2 with eigenvalue (--0,)
= -%t-n(P)
(n # 0)
.
(4.39)
This doubling of eigenfrequencies is typical of first-order formalisms, which deal with creation and destruction operators (or coordinate and momenta) as independent variables. We still have to find the left eigenfunctions of $X2. It also follows by inspection that t*n 9!2
= Q+ztkn
9
(4.40)
which means that the spinor dual to .$, is g_n. It is obvious then that the correct normalization conditions to be imposed on our eigenfunctions is (L,~,)=-(~n,~~)=6n,-mun,
(4.4 1)
where on is the sign of n, and we adopt the convention that positive n corresponds to positive 0,. The .diagonalization of the quadratic part of the RFT effective Hamiltonian proceeds then as usual with Bogoliubov transformations. From the completeness relation (cf. appendix B)
55
n>O
=
[tn@) E-n@‘) - t-n@) inCP’>l + EO@)XO(P’)- XO@)tO@')
1 %J -P’) ,
(4.42)
4.54
V. Alessandrini et al. /Kink
it follows from eqs. (4.20,35 tor $X2 is 812
=
c n>O
States in supercritical RFT
and 38) that the spectral decomposition
un[tn(P)E-nCP’)
+ E-n(P)&@‘)1
+ dOSO@)
tO:o@‘)
of the opera-
.
(4.43)
If we now define, for n > 0 ‘4, = C_, 3771, then the expansion
n(P,.Y)=n$i
An
= (En
977)t
(4.44)
(n >o> 3
for the spinor field 77becomes
(4.45)
[A,01)~,(p)-A,01)5-n(p)l
and therefore the eigenfunction
HPJ) =f Ed(P)+
expansion
for the full reggeon field [ reads
0$ 1’4
(4.46)
The coefficients A, and 2, must be considered as the integration variables in the path-integral formulation, while, as we emphasized before,do is given by the constraint equation (4.32) in terms of the integration variable n, and the gauge con. dition has fured Ae = 0 in an extra term of the form Aoto that could have been added in eq. (4.46). In a canonical formalism x, and A, become, as we shall soon confirm, creation and destruction operators of a quantum of excitation energy w, corresponding to the small oscillations around the classical solution. Indeed, they satisfy the commutation relations [An9 Aml
= 6n.m
(4.47)
7
which are a consequence of the original commutation relations of the reggeon fields and of the completeness of our set of eigenfunctions (cf. appendix B). One has to be careful, however, with the ordering of operators, which is unambiguously fixed by the ordering of operators in the original Hamiltonian. In terms of the shifted fields, the action density (4.13) becomes (4.48) where the energy E is given, in the quadratic approximation E = dcl t ivfi
k -
$sdp
by
[A020 + fj] 5& [AOXO+ VI+ MT,
where 6E are ordering terms that we shall discuss later. By using eqs. (4.20,2
(4.49) 1 and
states in supercritical RFT
V. Alessandrini et al. /Kid
455
24) one obtains E=sQcItivflpkt~A$--and from eqs. (4.43,44
Lo No
-i
s
dpijG&qtfiE
and 47)
E=s&+ivfik+$Aa$+
c
o,(~,A,+~)+cSE.
(4.5 1)
n=l
The ordering term 6E is fixed by remembering that the unshifted Hamiltonian is ordered in such a way that the perturbative vacuum is an exact eigenstate. If one shifts the reggeon fields by keeping the ordering of the operators, one gets the quadratic form $‘D(v) $’ •t&j/l* + B$‘* ,
(4.52)
where +’ and 5’ are the components of the shifted spinor [‘, defined in (4.11). There fore, by comparing this quadratic form with the one we have actually diagonalized, one can see immediately that the ordering terms are what is needed in order to replace $‘D(v)$’ by a symmetrized form. Then, (4.53)
the w:(v) being the eigenvalues of the same Schriidinger problem as before, but without off-diagonal coupling. The final result for the energy is, then OD
E=dpQd tivfik+$.d$$t
c
n=l
u,(v).&An
+; 5
n=l
[tin(v) -w:(v)] (4.54)
5. Discussion of the spectrum 5.1. Choice of the parameter v In order to calculate the energy spectrum from eq. (4.54) we must calculate the eigenfrequencies w, by solving the eigenvalue equations (4.37a, b) which depend explicitly on the classical solution {q”(p), &&I)} used to shift the reggeon fields. Also PQ,_toccurring in (4.54) depends on the classical solution in a rather non-trivial way, as we discussed at length in sect. 3. We must therefore face the problem of the actual choice of the classical solution. In principle any one is equally good: quantum fluctuations will span the whole path integral. Our purpose is however to construct eigenstates of the Hamiltonian by treating all nonperturbative effects in the semiclassical limit. We have therefore to
.
456
V. Aiessandrini et al. /Kink
states in supercritical RFT
require after shifting around the solutions chosen, that the Hamiltonian operator contains the quantum fluctuations in a positive definite quadratic form. This restricts our choice, because for some of them the quadratic form is not positive, and for others linear terms in the quantum fluctuations (tadpoles) can be present due to the momentum conservation constraint. A problem which is similar in some aspects to ours, arises in Xp4 theory, related to the arbitrary choice of the kink propagation velocity. Due to Lorentz invariance the classical action in Xv4 theory is independent of u (u can be changed by a Lorentz transformation). However, the classical energy and momentum depend on u, the different terms in the expansion of the Hamiltonian are then u-dependent and to check the independence of u of the sum is highly non-trivial. Since u is introduced essentially as a Lagrange multiplier (see eq. (4.31)), the quantization is usually done by choosing u in such a way that the constraint is saturated at the classical level, i.e. 9et(u) = k. Only in this case the moving kink is a minimum of the constrained action and the expansion of the Hamiltonian does not contain linear terms in the shifted fields. If u is not chosen in this way the constraint equation introduces tadpole terms which are perturbative in the sense that they are proportional to the small coupling constant h, and which must be explicitly summed. The result of this summation has a u dependence that compensates the u dependence of the classical energy. The previous remarks may be intuitively understood as follows: in X(p4 theory, the kink energy is given by
Ekink(k) = Jk* + (MO + AM)*+ O(h)
,
where Me = 0( l/X) and &l4 is a renormalization the constraint, with the result
(5.1) mass * . If u is chosen to saturate
then the result for the Hamiltonian that is obtained is equivalent to the expansion in AM of eq. (5.1). For other values of u, the form of the Hamiltonian is equivalent to an expansion of Ekink(k) in powers of a combination of AM and the fraction of k not saturated by PC,. In our case, the situation is complicated by two facts. In the first place the theory is not Galilean invariant, and therefore the classical action depends on u. As a consequence, we do not expect a simple particle-like dispersion law to hold for an arbitrary value of u, as it is the case in eq. (5.1) for hp4 theory. On the other hand, the classical momentum is imaginary, and consequently we cannot hope to saturate the constraint k = iTc;p,lwith our classical solutions for physical (real) values of k. This implies that in general we are bound to have tadpoles coming from the momentum conservation constraint. Fortunately, it turns out that we can dispose of both difficulties at once by choosl
See, e.g., eqs. (4.15) and (4.24)
of Baacke
and Rothe,
ref. 171.
V. Alessandrini et al. /Kink
states in supercritical RFT
457
ing to quantize the theory around solutions of class (A) exhibited in fig. 3 - namely, the ones inside the triangle - having a propagation velocity u such that u + 2-. That is to say, we pick the classical solutions of class (A) having a vanishing classical momentum 5, = 0((2 - u)~), according to our estimates of sect. 3. We shall then show that no tadpole occurs in this case, for any value of the quantization momentum k, and that the phonon-like dispersion formula e,r - V&
PC, = SQ,, = 0 [9e,(2 - u)] + 0
-
(5.3)
is not destroyed by the quantum fluctuation, in the sense that they do not introduce, for example, an energy gap. A first intuitive justification for the absence of tadpoles can be given by the following argument, Going back again to the general form of the energy given by eq. (4.54), we can see that tadpoles will in general show up through the Ai term, which is fixed by the momentum conservation constraint and given by eq. (4.3 1). Indeed, it is easy to see that A; contains linear terms in the quantum fluctuation n. Let us first discuss the role played by the Ai term at the purely classical level, by neglecting the quantum fluctuation n. Let us then fix for simplicity k = 0 and compare the energy gap J!?~~@ = 0) predicted by eq. (4.54) for two different values of the velocity u, i.e. u = 2 and u < 2. We expect them to be the same, at the expense of the fluctuation in the direction of x0, that is to say, A o, which corresponds to changing 3’er in the classical action. This is indeed what happens, because for u = 2 both 34,r and Ai are identically zero (if n = 0) and E,,,i,,(k = 0) = 0. If, instead, u < 2, then A0 as given by (4.31) turns out to be of order dYCr, and the contribution of the A; is of the form (5.4) which corresponds to rewriting the previous result for u = 2, Emin (k = 0) N &(P = 0), by expanding 9QCl(3 = 0) around ;P= PCt # 0. This shows then that the A0 contribution is minimized at the classical level if sQ,t is stationary with respect to Y.-t, which happens only for u + 2-. We shall also show later in this section that also the quantum corrections to A o vanish in this limit. The aim of this rather lengthy discussion was to motivate and partially justify our choice of the u + 2- kink solutions inside the triangle to shift the reggeon fields. A full justification of our choice will emerge as we proceed along this section. We shall not consider the class outside the triangle. They correspond to classical fields which are not monotonous functions’of p, implying that the zero-mode wave function has nodes. At the quantum level this means that the zero mode is not the ground state, and therefore they are not stable. Moreover, these solutions do not yield for Ai the result (5.4), not even for u --f 2 - , implying that one has also tadpoles in this case. As for the family of solutions of the single-field equations, the action, as well as the energy and the momentum, vanish for any u. For u < 2, the zero mode wave function again has nodes, and we expect them to be unstable. The case u > 2 and
458
V. Alessandrini
et al. /Kink
states in supercritical
RFT
especially u -+ 2+ is the one closest to our choice (u + 2-). The main drawback of this possibility is that it does not satisfy explicitly the p * q+ symmetry of RFT. Therefore, the quantum states dual to the ones found by shifting around, say, (JJ, 4) = (C&A, 0) are to be obtained by performing a different shift around (0, &~/l/h).This poses technical problems and we prefer to use an explicitly symmetric procedure. We would like to mention at this point the entirely different choice of classical solution made by White [ 111, to discuss the same problem we are concerned with. In our language, he chooses to quantize around the limiting solution u + -m of the class (a), which is of the form
cp@)-- ;<1- t&y), aP)--
;<1+t&Y),
(5.5)
i.e. reduces to the “instanton” along the hypothenusa of the triangle in fig. 3. As we have already discussed in sect. 3, such a solution has the opposite time evolution to the ones we consider, that is to say, it satisfies &=-oo)=O,
cpo,=+=)=O,
which are to be compared with the perturbative our kink solutions cp(j=-w)=O,
&V=+-)=O.
(5.6) boundary
conditions
satisfied by
(5.7)
White claims that, in order to have analyticity in the parameter /J for an infinite volume V in impact parameter space the boundary conditions (5.7) have to be changed, and that (5.6) ends up to be the only possible choice. Our approach in this and in the preceding papers [ 1,2] has been, on the contrary, to keep the boundary conditions fixed and demand analyticity in p only at a finite volume I’ of impact parameter space, and fixed rapidity Y. Analyticity in w can be proved in this case [8,9], and the matching with ordinary perturbation theory can be argued [9]. The S-matrix (2.17) is in this case stable against small, p-independent variations of the boundary condition (5.7) because the vacuum l&J is the only ground state at finite V. When I/ + 00 a phase transition has been found [l] due to the ground state degeneracy, and a non-analytic behavior in /J shows up at some critical value /.Icrit. We therefore consider White’s approach as basically different from ours, and from our point of view is open to criticism. Notice for example that the IJ-+ -00 kink has an infinite action per unit time that White subtracts away. This infinite action per unit time corresponds, in the language of sect. 2, to the exponentially vanishing normalization factor of the new ground state l$t), given by eq. (2.15). In subtracting it away, White keeps a finite norm for states which for us are of zero norm. It is then easily understandable that, if his shifting procedure is convergent, he will get a spectrum completely different from ours.
V. Alessandrini et al. /Kink states in supercritical RFT
459
5.2. Eigenfrequencies and eigenfunctions for v --,2We now discuss the calculation of the eigenfrequencies and eigenfunctions corresponding to quantum fluctuations around a classical kink solution with velocity v -+ 2-. In order to deal with the eigenvalue problem defined by eqs. (4.37a,b) we shall rewrite them in a more suitable form by eliminating the first derivative from the differential operators involved. Let us then define the functions f,(p) andfn(p)
(5.8) where C, is a normalization factor to be defined later on. Then, the relation (4.38) that gives the components of &n(p) in terms of those of&(p) implies the obvious relation f-,(p)
= U-p)
(5.9)
between the (+c+) and the (-w,) solutions. Introducing (5.8) in (4.37a,b), tem reduces to a coupled Schrijdinger problem of the form
1-$ + z 1 &XP) cup
f,(p)
W(p) -
f,(p)
where the potential D,--
2
+ - $ [
, 7
the sys-
+ Q(p) emuPfn(p) = 0
(S.lOa)
1
(5.10b)
+ W(p) + F
s,(p) = 0
W(p), defined in terms of the operator D(v) of eq. (4.15) by
d2 w(p) +
=
emupI D(v) evp12 ,
(5.11)
dp is given explicitly W(p)=+?-
in terms of the classical solution 1 t2[&)+&)].
{cp,(p} by (5.12)
For small values of (2 - v), the shape of the potential is sketched in fig. 7. It looks roughly speaking like a well centered at the origin p = 0 of height 2 and range 2L. Due to the fact that 40) = p(O) 2: 0(2 - v) and that cp(p) behaves as ~(0) exp(ivp) in the perturbative region cp(p) << 1, we can estimate the range L to be L = log(2 - v) .
(5.13)
Moreover, at the bottom of the well, near p = 0, IV@) takes a very small value of the order of (2 - v). We notice next that the coupled system of eqs. (5.10a,b) can be decoupled as v N 2- because in this case one of the non-diagonal potentials is always small. Let us consider for example the region p > 0. From our previous discussions in sect. 3,
V. Alessandrini et al. /Kink states in supercriticalRFT
460
Fig. 7. Shape of the potential W of eq. (5 .l 1) and of the classical fields. G is represented by a dashed line, and q by a dotted-dashed one.
we know that in this region cp@) grows first like ~(0) exp(iup) and then saturates at the value cp= 1. Since the exponential growth of cp@) is overcompensated by the factor exp(-up), it follows that when p > 0 the nondiagonal potential cp(p) exp(--up) is of order cp(O),that is to say, (2 - u). Always in the region p > 0, we can easily see that the other nondiagonal potential, -a) exp(up), can be of order 1. Indeed, from the known approximate solution for-&) given by eq. (3.32), it follows that m)
cup N $@)
(5.14)
and in the region p - L (b(p) is of order 1. The opposite results about the magnitude of the nondiagonal potentials hold in the region p < 0. Let us start by considering positive energy solutions, on > 0. Because of the previous discussion, in the region p > 0 eq. (5.10a) reduces to &fXJ)
(5.15)
= %LMP)
whereDs is the Schrodinger operator of eq. (5.11). Eq. (5.10b) becomes an inhomogeneous equation that determines.& in terms off,, . Indeed, DS being a positive semidefinite operator, there is no non-trivial solution of the homogeneous equation (Ds t w,)f = 0 when o, > 0. Also because of this fact, when we consider the region p < 0 where the inhomogeneous term - proportional to q exp(up) - can be neglected, we conclude that the only solution of (5.10b) is &l(P) = 0,
p
n>O.
(5.16)
Plugging this result back into eq. (5.10a), we obtain again eq. (5.15) for p < 0. There fore, eq. (5.15) is valid everywhere, i.e. in the whole region - < p < 00. We are therefore left with an ordinary Schriidinger equation involving a potential IV(p) which is approximately a well of height 2 and range L. When (2 - u) is small, L is very large [cf. eq. (5.13)] and the low-lying eigenvalues will be approximately given,forn >O,by n=2m-1 (5.17) n=2m. 1)n/L >
\(2m+
461
On the other hand the ground state w. should be of order 0(2 -- u), because we know that the exact coupled equations have an exact zero mode arising from translational invariance, and they differ from the present approximate ones by 0(2 - u). In deriving the zero-mode eigenfunctions from translational invariance one realizes that the shape of the potential is extremely important in order to have the translational mode correctly placed at o = 0, so the estimate (5.17) cannot be valid in this case. The fact that the ground state o. of our Schrijdinger problem is related to the zero mode arises of course from the observation that the zero-mode eigenfunctions, given by (4.16), have no nodes. The stability of our classical solutions - and the fact that Ds is positive semidefinite is then guaranteed. Since the Schrijdinger potential IV(p) is an even function of p, the Schrijdinger eigenfunctions f,(p) will satisfy the parity condition fn(-P)
=
(-l)"fn@)
(n = 0, l> ..*) .
,
They represent a complete orthonormal
5
n=O
fn(P)fn(P’)
= WP -P’)
(5.18)
set of functions
satisfying
(5.19)
3
(f.,fm)=SdPf.@)f,@')=Sn,m _ca
(5.20)
.
In what concerns f,&), we have already established that it is determined unique solution of eq. (5. lob)
by the
fn@)= [Ds + ~nl-12ti>e”Pfn@) =
5
fm (P)(fm
m=O
,2@ euP’fn)
where we have used the completeness
(fm
,V
e”“’f,)
= J
d/J’ fm b’)
(5.21)
>
%+~rn
relation (5.20) and the obvious notation
V(P’)
_m
euP’ fn b’)
(5.22)
.
We notice that, as expected, eq. (5.22) continues smoothly into eq. (5.16) for p < 0. Indeed, the support of&r’) exp(up’) is p’ large and positive. Having discussed the solutions of eqs. (5.10a,b) for on > 0, we can use eqs. (4.38,39) to derive at once the eigenfunctions for the case w, < 0:
C;_n@) = C_“( i
YEi)
= C-n( ::I::,
n
?I
= C-n(iI:
r:II,2)
.
(5.23)
462
Using the reflection properties of f,(p), = (-l)“C,, we can rewrite l_,Cp) as
~-,co)
given by eq. (4.18)
and fixing C_,
(5.24)
(n > 0) ,
= G(
where g,@) is now given by g
(p)
=
n
5 fm(PI Cr,,2~ cup’fn> m=O
=
%+Wm
(-1Y L-P)
(5.25)
and, of course, &@) = 0 for p > 0. We can now use the normalization (4.41) to determine C,, :
=c;s,.,
‘C,
= 1)
condition
(5.26)
where use has been made of the orthogonality relation (5.20) and of the fact that the functions&@) and &@) have everywhere a negligible overlap, namely, Xk%Z@) N 0. We now turn to a discussion of the zero-mode. We have to identify the zero-mode wave functions to and its conjugate vector x0, defined in terms of the classical sohrtions by eqs. (4.16) and (4.19), with the eigenfunctions we obtained for n = 0. Notice that we have two eigenfunctions available, by setting n = 0 in .$ or ,$_, . We discuss this problem at length in appendix B, where the approximate forms of lo and x0 are given for (2 - u) small. We are more concerned now with an estimate of the normalization factors No and Lo as IJ+ 2-, given the role they play as coefficients of the ,4; term in the general form of the energy given by eq. (4.54). In order to compute No we recall the fact (see appendix B) that, for (2 - IJ) small, the correct identification of E. is f. (-p)
evp12
fo(-p)
emuP”
Q-P>
Therefore, by identifying (4.16) we obtain foc0)
= NOW)
for
eMupI
p >
,
0 the upper component
(P>O)
.
(5.27)
of to with the exact result
(5.28)
V. Alessandrini et al. /Kink states in supercriticalRFT
463
and, since fe@) is an even function
l= [ dp fi@) -00
= 2Nz s
dp (b2 eevp .
(5.29)
0
This integral has already been discussed in sect. 3 [cf. eq. (3.34)] and proved to be of order (pg up to irrelevant logarithms. Therefore, No = O(&)
.
(5.30) constant, Lo, is determined
The remaining normalization condition (4.2 1)
_m
1 =(~o,~o)=LoNo
fdp
g$
-;
1
m-a$
s
ap - dp. _m au ap
= 2LoNo
(5.31)
We have made use of the fact that &I) = -$(-+I), acp(p) av=--’
$
from the normalization
and that
a$(+)
(5.32)
au
This last statement is just a consequence of the reflection back to eq. (5.31), we observe next that, for p > 0
aim am
_vpl2
aU=ave
_ 4 u&O)
e-vP/2 .
property (3.7). Going
(5.33)
The second term of the right-hand side is of order cpo, but the first is of order cpo/(2 -II). Since we know also that &J) is of 0(2 - V) for p > 0, we conclude that 1
_=o &No and therefore, Lo ,-0(2-u). 1”o
(
$u
(5.34) 1
from eq. (5.29) we obtain (5.35)
We have now all the information we need in order to evaluate the energy spectrum (4.53) in the limit IJ+ 2-. We note that: (i) The classical action density Act = 0 [ PCl(2 - u)] vanishes as a consequence of the phonon-like dispersion law. (ii) The constraint term (pi~Lo/No) also vanishes, because of (5.35). Notice that, away from u = 2, A: contains a tadpole which by eqs. (4.3 1,32) is proportional to
464
V. Alessandrini et al. 1 Kink states in supercritical RFT
N;‘($, aPxo). It is only for IJ + 2- that, thanks to (5.30),,4; is quadratic in n, and moreover gives a vanishing contribution to the energy because of the behavior of (Lo/No) given by (5.35). (iii) The zero-point energy
(5.36) vanishes as well, because Ok + o,(u) for u + 2-. We have in fact previously shown that u -+ 2- the eigenvalue problem for w, reduces to (5.15) whose eigenfrequencies, by (5.1 l), are 0:. The correction (0, - WE) can be computed to first order of perturbation theory, and the result for AEo is mo
= c’
un >2v e-uofk>(fky
Gfn)
wn + ak
n,k
= o(2
_ u>
9
(5.37)
where C’ excludes the term n = k = 0 in the sum. By the results above, the final formula for the energy becomes E=2ifik+p
c
w,A,A,+O(h).
(5.38)
n=1
Note that the quantum fluctuations have a small gap wr cx 7r/2L and for u + 2(L 51 iln(2 - u)l + 00) give rise to a continuum headed by the kink state. The above formula for the spectrum of RFT represents the main result of this section. Let us analyze the procedure that allowed us to establish it. We have seen the importance of keeping the triple pomeron coupling h and (2 - u) as independent parameters. As we discussed at the beginning, the quantization of RFT should be independent of u and therefore independent of the classical solution chosen to shift the reggeon fields. However, it is only for u + 2- that we succeeded in having, after shifting, a bona fide quadratic form, and the reason for this to happen is easily understood. For every u < 2 there is a classical kink solution that represents an unconstrained minimum of the action corresponding to some classical momentum ;p,l. If the theory could be quantized in such a way that k = P = i f&, then of course the expansion around the classical solution would involve only quadratic terms. This is not possible in our case, but the results of this section show that at the optimal value u = 2 the term in the Hamiltonian arising from the constraint not only is quadratic in the quantum fluctuation but also gives a vanishing contribution. Truly enough, this is achieved at the expense of having a continuum spectrum for the quantum fluctuations. The physical meaning of u = 2 is that the classical action density has an extremum at YCt= 0, where the phonon-like dispersion formula eC1= 2fl@ 3,t + O(@‘) holds. This dispersion law is preserved in the quantum case because there is no mass renormalization of the kink arising from zero-point fluctuations. We therefore find
V. Alessandrini et al. /Kink states in supercriticalRFT
465
that kink states with u = 2 give rise to a zero energy gap for k + 0. It must be remarked that, if our shifting method is properly convergent, the kink states and its quantum fluctuations are bona fide eigenstates of the non-hermitian RFT Hamiltonian. The same conclusion was reached in the spin model [2] where, only in the infinite volume limit, the kink states were found to have the dispersion law E(k) 0: [ 1 - exp(-ika)] = ika, a being the interlattice distance. However the continuous spectrum arises in that case in a discontinuous way, the finite volume spectrum being highly degenerate, but with a finite k = 0 gap. The same situation seems to occur in ref. [4]. Therefore, the occurrence of the phonon-like dispersion law appears more neatly in the present method than in the spin model.
6. Quantum
states and the expanding disk
Let us summarize the structure of the lower-lying spectrum. For y > pC we have the two degenerate ground states discussed in sect. 1 (the vacuum IGo> and its companion 1I)~> of eq. (2.8)) and, furthermore, two bands of states whose energy gap E = 52ikfivanishes for k + 0. Both ( I)~) and the kink states can be thought of as the lowest members of families of states obtained by a quadratic expansion of the Hamiltonian around the corresponding classical solutions. Let us stress again the difference between our situation and the similar one occurring, e.g., in hp4 theory. In an hermitian field theory the expansions around the space independent fixed points and the kinks define superselected Hilbert spaces. The existence of the topological charge VP(-) - cp(--00) is just a sign of that. In RFT we shall find instead that also the kink states, as the ground state /Go), are connected to the perturbative vacuum I@,,)by field operators. This is due to the peculiar role of the metric operator % of eq. (2.1 l), and implies that the vacuum I&,), l$e) and the kink states belong to the same (metric) space. Two logical consequences follow from the previous remark. On the one hand all states found so far are to be considered together in the time evolution operator and in the completeness relation. This is what we shall do in the following, and we shall make sure that we do not double-count by checking that energy eigenstates are properly orthogonalized. On the other hand it is not possible, in our case, that the semiclassical expansion we have used be equivalent to a “bona-fide shift”. By this we mean that some quantum fluctuations must be large when we compute matrix elements of ageneral operator. This also means that, in principle, one should be able to find, say, the kink state, starting from vacuum excitations if large fluctuations are allowed (in a sense, this has been done in the spin model). Nevertheless, the basis of our approach is that some classical solutions exhaust all the non-perturbative content of the quantum states, and that therefore a quadratic expansion is acceptable - at least locally - for the Hamiltonian operator. This is enough in order to identify the lowest lying states mentioned before, which in this respect are analogous to the breather modes found in the vacuum sector of Xv4 theory. A different question is how to take into account excited states around the
466
V. Alessandrini et al. / Kink states in supercriticalRFT
classical solutions. We shall discuss this point briefly later on, and we shall first concentrate on finding the contribution of kink states in the completeness relation. 6.1. The quantum kink state The quantum energy operator
kink states correspond to the ground state of momentum of eq. (5.38). They are defined by
k of the
l
Elk) = 2ikfilk)
,.
The second condition
Plk)=klk).
is automatically
implemented
(6.1) by writing the kink states as
_m
Using eq. (5.38) the kink state I&r>is then determined
&l&,)=0,
up to O(h) corrections
(n+O) *
by (6.3)
We remind that the gauge condition 2 o = (j&, , l) = 0 has fixed the arbitrary translation parameter a that occurs in the classical solution to, say, a = 0. Notice that eq. (6.3) assumes that Ec~,A,A, commutes with P, which is true approximately to the order XJp to which we are working. Let us recall that, from eqs. (4.44) and (4.36) &I =(C,,~-ed)=(~-,,p-~~)=(F_,,q-~~),
(6.4)
where F_, and F_, are given explicitly in eqs. (5.23) and (5.24,25) for the u + 2solutions we are using. By representing the operator p(p) by -a/aq(p), we can look for solutions of eq. (6.3) of the form /J I&I)=
ew[--(a,q)+
1 2@7,
h
p4)11@0),
(6.5)
where (Y@)is an unknown function and /3 an operator with symmetric kernel /3(p, p’). Then, by using eq. (6.4) we find that eq. (6.3) reduces to (F-n,o-@+(F-n,@)=O,
(6.6) nf0
(%,
p4) - (L,,
q) = 0 .
(6.7)
By using the explicit expressions (5.24,25) for F_, and F_n, as well as the fact that &I) exp(ivp) is of order 0(2 - u), it follows that the term (F_, , @) is of order 0(2 - u). We then find o(p) = cp(p) t constant &I) t 0(2 - u) , where the (b@) term comes from the fact that eq. (6.6) does not determine l
We use different
the zero-
symbols for the (same) Hamiltonian operator in order to distinguish its form in terms of the unshifted fields 4 and p from its form E of eq. (4.50) in terms of the shifted ones II and fi.
H of eq. (2.5) written written
(6.8)
V. Alessandrini et al. /Kink
states in supercritical RFT
467
mode components of both Q@) and p(p, p’). These components correspond to an infinitesimal translation in (6.5). Since we have to perform this translation anyway in order to obtain momentum eigenstates, we can assume that CYand /3have no zeromode components. With this specification al(P) = cp(P) + O(2 - u) ’
(6.9)
In the same way we obtain from eq. (6.7)
Ph P’>=n
mfi, P,, euP/*fn(p) euP/*fm(p’) ,
pnm = pm, = _
‘-,‘yyfn n
.
(6.10)
(6.11)
m
By putting eqs. (6.2), (6.5) and (6.9) together, we finally get
Ik)=JdaeCika exP (Pa =
[
- t (Vap 4) + i(4, Pa4)
cP(P - a>, ba =NP - a, P’-
The field cpis approximately, forp>O: $=cp-cp*
t@,
)3 1I@0
a>.
for LJ-, 2-, a solution of the single-field equation
= -2pco)
(6.13)
(b(O) = V(O)
and the kernel fl(p, p’) is a solution of the differential [ua, - a; + 2V@) -
(6.12)
equation
11&J,P’)+ [$I - $3 + MP’) - 11P(PP’>
6(p - P’) + (2V)oo
fo@uafJ') .
(6.14)
It is interesting to notice that - aside from the elimination of the zero mode in eq. (6.14) - both eqs. (6.13) and (6.14) can be derived from the generalized coherent state approach developed in appendix A. Indeed, they are essentially the first two correlation functions, in the particular case in which these correlation functions are kink-like, i.e., they depend on y only through the body-fixed variable p. As discussed in appendix A, these equations are the semiclassical approximations to the exact equations for the correlation functions, and therefore they do not include renormalization effects *. This means that, provided we limit ourselves to the null gap states with u = 2, the approximation of expanding the equations of motion in powers of 4 is essentially correct for the calculation of right eigenstates (cf. sect. 2). * This indicates a method for including renormalization effects in the construction of the states. For instance, the addition of the term p(p, p’) in (A.4) to (6.12) would give a mass renormalias already found in the spin zation. The ordered phase would occur for p > ccc = 0(,X*/m), model.
468
V. Alessandrini et al. /Kink states in supercriticalRFT
6.2. Approximate
evolution operator and S-matrix
We want now to investigate the normalization properties of kink states. We first notice that for each kink state with speed v = 2 and energy 2ikG, which we shall denote by Ik), , there is an antikink state Ik)_ with speed -2 and energy 2ikd&, due to the reflection symmetry (3.6) and (3.7). This antikink state Ik)_ is simply obtained by the replacement p + -p in cp@) and p. The states Ik>+and Ik)_ propagate in space in opposite directions: left for Ik)+ and right for IkL, corresponding to a single direction in time, namely, the one for which p -+ p/h asy + 00. This is exhibited in fig. 8.
a)
b)
Fig. 8. Time evolution of (a) the kink and (b) the antikink states.
For the purpose of this section we shall neglect the correlation (PC(/~,and we shall write Ik)+=Jdue-‘kaexp Ik)_ = rda
-; [
emika exp
c
-:
1 1
dpq(P-a)&) dp 4-p
(6.15a)
I&o>,
+ a> 4(p)
Ido)
0 with respect to
.
(6.15b)
They satisfy: Elk),
= +iv,kl k>+_,
where we have introduced
Plk), = klk),
,
(6.16)
the critical speed (6.17)
ve=24&.
We want now to find the left eigenstates corresponding to (6.16). By introducing the parity operator I, which obviously commutes with E and anticommutes with P, we note that from (6.16) and the definition of the metric operator %! (see eq. (2.11)) it follows that _(-klZE = _GkIEI= _(-klZP=k
_(-kll.
ivok _(-kll,
(6.18) (6.19)
469
V. Alessandrini et al. f Kink states in supercritical RFT
Moreover, one can check that: _Ckll=
+(-klI=
+&I ,
(6.20)
_(kl ,
which means that in the left eigenstates t and - exchange their roles. This fact is confirmed by the evaluation of the scalar products according to the metric operator (2.11). While I k)+ and Ik)_ are themselves null states in the thermodynamic limit +Uclk)+ = _Uclk)_ = exp[-
~(p/A>2Vl s
0
(6.2 1)
the scalar product _(kl k>+ is not trivial, because it contains only a partial overlap of cp@) and cp(--p). Indeed, one easily obtains _&‘)+
(6.22)
= 2n6(k - k’) N(k) ,
where
(e --f o+)
(6.23)
and we have roughly approximated the integral in the exponent by @(-a) for A<, with the ground states i$e> and I I),,) = I #,,) - I c#q) discussed in sect. 2. Of course the kink states are orthogonal to the ground states for k # 0, but in order to keep this property for k = 0 we have to project away the vacuum component by subtracting a 6 function part. Then, we define Ik): = Ik>+ - 2d(k)
lG1> , (6.24)
Ikf_ = Ik)_ - 2n8(k)l@t), in such a way that the new states are automatically
orthogonal
to both I& > and
($0). By taking into account the normalization factors (6.21) and (6.22) we are now able to write the contribution of ground states and kinks to the completeness relation. We get:
t
s-=dk -
0
{-ik e
ivOkY
Ikf_ +(kl’ t ik emivokY Ik): _(kl’}
(6.25)
2n
+ quantum
excitations
.
* This is true if q(p) is approximated by a step function centered at p = 0. Strictly speaking q(p) should be centered at the transition region p = L. This would give rise to an extra factor exp(2ilk) iniV(k), which, however, cancels against similar factors in the matrix elements.
V. Alessandrini et al. /Kink
470
states in supercritical RFT
From this completeness relation we can calculate the pomeron Green functions. for /AY-+ =. We shall define the propagator as D(j3, Y) = -(@cl p(O,O) eiPB eCEYq(O, O)l&> .
(6.26)
This definition is stable against addition of time-dependent g@e) spread over a volume V, provided we put
@‘“‘(B, Y)p,.
= j_im_ ,lir,
[analytically
continued
external sourcesf(-vo),
DgLyoj,
gtyoj(B, Y)]
(6.27) where the analytic continuation from 1_1<0 to P > 0 is unambiguously made at finite space volume. In fact, I q?~,, ) is the true ground state at finite V and is automatically picked up by the y. + m limit l . With this proviso, the ground state contribution to (6.26) is, by (6.25) and (2.16) and always neglecting relative corrections of O(h/p) 2 DC4
-~~~lpI~~~~-~~~~,l4l~o~
r> z B
=
$
(6.28)
.
0
fixed
The limit (6.28) is asymptotically reached at finite B. However, one should also have that D(B, Y) + 0 for I B I + m at finite Y, by the cluster decomposition of the correlation function. Therefore, we need additional states of small momentum which cancel the contribution (6.28) at large impact parameters. We shall now show that in fact the kink states provide their cancellation l * . By neglecting for the time being quantum excitations, we get ,from (6.25) and (6.26): 2
dk
0 J
D(B, r> = z
-
2n ecikB [x+(k) x_(k)
kuokY + c.c.]
(6.28)
,
where xi (7~)= Q. Ip(0) Ih’>*. Using (6.15) and (6.24) we find x+(k) =x(k)
=f
x_(k)=X,(k)=x
[Jda evikaO(a)[~dae-ikaO(-a)-
2n6(k)] 2n6(k)]
= -$ =-!
,
& &
.
(6.30)
Therefore, by using the inverse transforms: O(z) =
&j&
eikz
hk =&jdk ewikz &
,
(6.3 1)
* Ambiguities both in the analytic continuation and in the ground state may arise only if the V-t - limit is taken first. l * The same cancellation mechanism was independently exhibited in the particular case of RFT with “magic” quartic couplings studied in ref. [4].
V. Alessandrini et al. /Kink states in supercritical RFT
471
we finally get: [i -O(B-LJ,Y)-C~(-B-U,Y)]
D@, Y)=
020(voY-
= f
,B,),
(/Jr>>
1,
h<
(6.32)
This is the final result of our calculation. It clearly shows the role of the kink state in cancelling the ground state contribution at large 1B I. The pomeron propagator of eq. (6.32) is a grey disc of opacity (P/A)~ and radius R(Y)=voY=2fiY.
(6.33)
The J-plane structure of this result, in which kink excitations are neglected, can be analyzed as follows. The contribution of ]rJo ) is a fixed pole at J = 1, (J = 1 - E), while the kink and antikink are complex conjugate poles with residues which are singular for k + 0. They combine together to give
A2 0;
W k) =
J+l27r6(k) +
1 ik(J-
1
1 + ikuo)
+ C.C.
2
= (J - l)(J - 1 t iv&) which is the J-plane structure kink excitations, as discussed kink poles are really speaking The S-matrix corresponding evaluated by the same method
+ Cc. = (J-
1)2 t u;k2
’
(6.34)
of the expanding disk for D = 1. The introduction of later on, gives rise to J-plane cuts and the kink and antithe edge of those cuts. to eikonal-type couplings, given by eq. (2.17), can be and, for point-like sources is asymptotically given by *
Scf, g; B, Y) - 1 - [ 1 - eVgW/‘] [I - e-fP/h]
O(ue Y - iBI) ,
(6.35)
as in the quantum spin model. This provides inside the disk a factorized opacity bounded by one. Notice also that this result for the S-matrix is really valid in the weak coupling limit, namely X << p. For larger values of h/p, renormalization effects neglected here would eventually replace p/X by up/h, where u is an order parameter which vanishes at p = po. Moreover, the precise form of the asymptotic S-matrix changes due to correlation effects in IrJo> (see Cardy, ref. [3]).
* This simply follows from the approximate representation exp(-b/h)
gdd)
h~cP
1 - [ 1 - exp(-gcl/Ql O(P).
412
V. Alessandrini et al. /Kink states in supercriticalRFT
6.3. Excited states Taking into account excited states is not easy to for several reasons. The first question of principle that arises is whether or not to count both the excitations around I Go>(shifted pomerons) and those around the kink states. This difficulty is shared with other theories which have in the same superselection sector several minima of the action which are not perturbatively connected. We can advance some arguments which indicate that the expansion around the kinks could provide, at least approximately, all the elementary excitations of the theory. Let us in fact try to identify the excited modes around the kink. We have in the first place the lower-lying states inside the well (see fig. 7) with dispersion law w, = /Ai, (ki << 1). These are a band of states with a small gap or , which collapse into a continuum in the limit u + 2- (L + 00). The intuitive picture that we have of these states is that they correspond to unshifted pomerons excitations (around the perturbative vacuum I&)) in the presence of the kink, and viewed from a reference frame moving at the critical speed uo. First of all, notice that the bottom of the well corresponds to the perturbative vacuum and that, as IJ approaches 2, our kink solutions spend a very long “time” 2L in this state. Secondly, let us see how an unshifted pomeron looks like in the kink rest-frame. The transformation b + b + uoy is a Galilean transformatinn with imaginary velocity iv0 = 2ifi, because rapidity y corresponds to imaginary time. Correspondingly, the pomeron dispersion law can be rewritten in terms of momentum k’ in the moving frame as follows 2
= iuok + cx’k” , where iuo/2cw’is the center of mass momentum corresponding to a pomeron “mass” 1/2cX’. Formally, this is exactly the spectrum that we get by quantizing around the kink if k’ is identified with &? k,. However, since k’ = k - iuo/2cu’ - as it must be for free pomerons - eq. (6.36) is just a trivial restatement of the original dispersion law. What happens in our case is that unshifted pomerons are in interaction with the kink, which eats up their momentum in such a way that k’ is not fixed by the momentum conservation constraint, and can be interpreted in fact as &? k,. Momentum is taken away by the zero mode A0 which, according to the results of sect. 5, does not contribute to w(7c) for u + 2-. This is the mechanism by means of which the kink stabilizes the unshifted pomerons. It is also easy to understand the higher modes o, > 2~1,which correspond to a true continuum above the kink well. They are essentially fluctuations around the value cp= p/X of the fields, and can therefore be interpreted as shifted pomerons in interaction with the kink, viewed also from the kink rest frame. Their characteristic frequency is-shifted to 2~ by the same mechanism that pushes the unshifted pomerons up to zero frequencies, namely, the transformation of energies under a Galilean transformation. To sum up, since we have around the kink both shifted and unshifted
V. Alessandrini et al. /Kink states in supercriticalRFT
473
pomerons [with dispersion law (6.36)] we feel that these excitations can provide all relevant corrections. Another question of principle that we have to face is the evaluation of corrections introduced by the excited states to the expanding disk just found. The most important are clearly those arising from small w values (distorted pomerons). From the point of view of the J plane they give rise, in the limit u + 2-, to cut singularities at J(w) = 1 + 2ikue - w,
(6.37)
whose branch points (w = 0) coincide with the kink singularities. These contributions will not spoil the disk result provided the cut discontinuity (due to excited states) is less singular than o -iI2 for any k, including k = 0. We have performed a rough estimate of the contribution of the nth mode to the evolution operator by approximatingf,@) of (5.15) by eigenfunctions of a square well of range 2L. The result indicates that the cut discontinuities vanish at the branch points for any real k. Therefore we expect our results to be valid both for Y >> max(uu IB I, 1/p) and for lB I >> Y/ue. The excitations contribute mostly in the transition region IBI - ue Y. More precisely, in our rough calculations, they are most important outside the disk where they quantitatively reproduce - for IB I > u. Y >> uo/p - the unshifted pomeron tail, thus correcting the disk result utot a Y (forD= 1) by an inverse power of Y = log s. A proof of this indication would show the matching with ordinary perturbation theory, which is expected to be correct outside the disk. We are still unable to understand whether these distorted pomerons which in our approach correspond to bona fide eigenstates of the Hamiltonian, are relevant to the fulfillment of t-channel unitarity for t > 0 (k2 < 0). A detailed analysis of the role of the excited states is left to future investigation.
7. Conclusions In order to find the lower-lying spectrum of RFT, which determines the asymptotic properties of the amplitudes, we have introduced a quantization procedure around classical solutions, suitable for the supercritical phase (o(O) > ar,). We have been able to identify the relevant solutions in the case of a one-dimensional impact parameter space. We have shown that kink-like classical solutions exist for a range of values of the speed u up to a critical value 2, for which the dispersion law is phonon like. This particular value is the best suited for our quantization procedure, even if track must be kept of (2 - u), which acts as an infrared regulator parameter and can be sent to zero only at the end of the calculation. We have then obtained the kink-like quantum states and the quantum excitations around them. The spectrum of the kink states is E = *2ifi k, while the excitations give rise to a (real) continuum starting from there.
V. Alessandrini et al. /Kink states in supercriticalRFT
414
We have finally computed the contribution to the Green functions of the lower lying states, i.e., the vacuum, its degenerate companion and the kink states. The contribution of the latter was found to cancel the one of the degenerate ground states for lB1 > 2fi Y, while for 2fi Y > IBl, the Green function was still asymptotically constant. This is just the disk behavior typical of RFT, with the expected expansion velocity. We argued that the lower lying excited states can be interpreted as perburbative pomeron excitations viewed in the moving kink frame, and that their contributions do not substantially modify the expanding disk behavior. On the light of our results we think that a basic characteristic of thistheory is the simultaneous interplay of different states which cannot be obtained perturbatively from each other. The quantization around classical solutions is well suited to isolate those nonperturbative effects. In this way we were able to spot the contribution to the evolution operator of various quantum states corresponding to different classical solutions. This implies that the theory cannot be interpreted as a perturbative expansion around a single, suitably chosen ground state (“shifted” vacuum). We wish to thank E. Edberg and K. Kijlbig for their help in numerical
calculations,
Appendix A We shall first discuss in this appendix a generalized coherent state approach to the solution of the Schrodinger equation (2.18) of RFT. After having shown the relevance of the single field equation of motion in the semiclassical limit, we shall give a detailed discussion of its solutions. A. Generalized coherent states The most general state vector leti)) HI \k(Y)) = -
$
solution of the RFT Schrijdinger
equation
(A.1)
I\k@))
can, without any loss of generality, be written in the form I@@)) = epAcy) !@) , where l&,> is the perturbative A(y) =Nfj
+
Jdbi
(A-2) vacuum, and A@) an operator given by .. . dbN s(bi)
.. . &N)
G&i
.. . bN; u)
(A.3)
in terms of an infinite number of correlation functions G,. The Schrijdinger equation (A.l) can now be transformed into an infinite set of coupled differential equa-
V. Alessandrini et al. /Kink states in supercriticalRFT
475
tions of first order in “time” y for these correlation functions. These equations are obtained by computing both sides of eq. (A.l), using the RFT Hamiltonian given by eq. (2.5) and the commutation relations (2.4) and by identifying powers of creation operators. Although the general equation for a+~ is not difficult to obtain, we write below only the first two equations for simplicity
- aG,(b,y) =-~tol'vf>G,(b,y)tXG:(b,y)+~Gz(b,b,~),
(A.4)
aY
-
aGz(b,
b’;y) =-6~ t a’v;)G2
ay
+ 2X[G1 (b, y) + Cl
- (j,.i t cth$)Gz
+ 2M(b
(b’,y)lGz + 2%@, b, b’;y)
-
b’)G1@A (A-5)
and so on. In general, a,,G, depends on G,+, and GN_t . For an initial coherent state of the form considered in sect. 2, these equations are to be solved with the initial condition
G,@,0) =g@), GN(bl,
.. . . bN; 0) = 0
forN>
2,
(A-6)
g(b) being the initial source distribution. We are still at the starting point in the sense that we have not yet made any approximation that allows us to handle the problem under consideration. We now proceed to compute a solution in the semiclassical approximation which, as discussed in the text, should be good in the weak coupling limit for o(O) > 1. The semi-classical limit for the coupled equations for the correlation functions is easily obtained, because in this case A(V) is just the classical action and, considered as a functional of the dynamical variable q(b) it satisfies the Hamilton-Jacobi equation
(A-7) Replacing the expansion (A.3) for A(y) and identifying again powers of q(b) one can easily obtain the semi-classical limit for a,G,. The result is neatly phrased by saying that the semi-classical limit corresponds to dropping the “feedback” terms in the exact equations, that is to say, terms proportional to GN+~ in the equation for a,G,. What is left is then a set of uncoupled equations for the correlation functions of the form
a(%_(p + dV;)G1 --
ay
- XG: ,
(A.@
V. Alessandrini et al. /Kink states in supercritica~RFT
416 =2 -=
[@ + dv;,
+ 01 •t ~‘v3 + 2VG,(b,y)
+GI(~‘,YNIG~
au
+-2AS(b-b’)Gl(b,y).
(A.91
The only non-linear equation is that of G r ; G, satisfying a linear differential equation that has as an input the lower correlation functions. By means of an adequate resealing of the correlation functions one can check that the semi-classical equations are independent of X, and that the feedback term (radiative corrections) are perturbative in A. We are therefore naturally led to study the equation for Gr , which is exactly the single-field equation for p,t(b, y). Higher correlations take into account the fact that the classical field qCt@, y) is non-zero, and they can in principle be obtained by solving linear equations once Gr is known. A.2. The single-field equation We shall henceforth restrict to a one-dimensional impact parameter space, and study the single-field equation (A.8). We want to illustrate how a wave generated by a source approaches asymptotically solitons with a well-defined (critical) speed. Let us consider the well-defined boundary value problem for the single-field equation
(A.lO)
By introducing t = py, x = @b and up= (p/A) pCl we eliminate all dimensional constants, and the problem reduces to + = cp- cpa t cp” )
dO,x) =p
fhmix)= %@c).
(A. 11)
Non-linear diffusion equations of this kind have been widely studied in the mathematical literature [ 121, and the qualitative features of (A.1 1) have already been given in the context of RFT [13]. They are: (i) The only fixed points, i.e. static solutions of (A.1 l), are cp= 0 and cp= 1. (ii) If the source f(b) is everywhere smaller than p/h, namely, 0
V. Alessandrini et al. /Kink states in supercriticalRFT
417
the negative (p2 term in the differential equation, provided that the source ue (x) is everywhere smaller than the saturation value of cp. We want to clarify how cp= 1 is approached for large times. We shall see that this approach occurs with a well-defined speed, which picks up one among many soliton solutions of (A.1 1). Let us therefore look for homogeneous waves that interpolate between the two fixed points, of the form * (A.13)
cp(P) = cp(ut *x) , such that d-=)=0,
cp(+-)=l.
They satisfy the differential
(A.14) equation
Ip=-cptcpatu+
(A.15)
and solutions to this equation with boundary conditions (A.14) exist for any positive speed 0 < u < 00 l’ . In fact, d-p) can be interpreted as the coordinate of a particle of unit mass moving in a potential (A.16)
V(p) = &a - $3
and subject also to a friction force with viscosity coefficient u. Indeed, by reversing the sign of “time” p the term u$ in (A.15) changes sign and becomes then a dissipative force. The shape of the potential V(q) is exhibited in fig. 9. Since for the timereversed motion cphas to start at cp= 1 and end up at ‘p = 0, it is obvious from the mechanical analogy that this will happen for any value of u. The time-reversed motion will correspond to the particle moving downhill and performing damped oscillations around the bottom of the well for small values of the viscosity coefficient u, or reaching very slowly cp= 0 without oscillating for large values of u. The borderline between these two regimes corresponds to a critical velocity u, that we discuss next. But it is quite clear that solutions of (A. 15) exist for any positive u. When cpbecomes small, the potential V(p) approaches a harmonic one, and the problem becomes that of a damped oscillator. Linearizing the equation and looking for solutions of the form cp(p) = exp(k,p) we get k; -uko
+ 1 =0,
:.
j+$u+@T.
(A.17)
We have therefore damped oscillations for d-p) around cp= 0 for u < 2, and at most one oscillation for u > 2. The phase-space trajectory corresponding to the solutions of eq. (A. 15) can be
*Cf. footnote after eq. (3.2). ** We are indebted to G. Parisi for the argument that follows. More rigorous derivations can be found in the mathematical literature [ 121.
V. Alessandrini et al. /Kink
478
states in supercritical RFT
Fig. 9. The potential V(q) of the single field equation (A.15).
found by introducing
the velocity field W(q) = Cp,which satisfies the equation (A.18)
and in order to define W(p)uniquely we shall give a boundary condition at cp= 1. From the linearized form of eq. (A.15) near cp= 1, it follows that q is of the form (A.19) but k_ has to be discarded because from (A.14) Cpco)must reach 1 as p +m. Therefore, it follows that
W((P) -=k+=&@t1, 1-q
asp-+1
(A.20)
Eqs. (A.18) and (A.20) define a unique trajectory in the (W,cp) plane that is sketched in figs. 10a and b for u < 2 and u > 2 respectively. For u < 2, fig. 10a shows the existence of an infinite number of damped oscillations for the time-reversed motion. The origin is then a spiral point for W(q). On the contrary, for u > 2 we observe that no damped oscillation can exist. In fact, in such a case the phase space trajectory would cross the cp= 0 axis in fig. lob at some point W,,.But then the corresponding unique trajectory would lie necessarily above the line W = k&o and could not possibly end up in q = 1. a)
b)
Fig. 10. Phase-space (IV, 9) trajectories for the kink solutions of the single-field equation. (a) u<2,(b)u>2.
V. Alessandrini et al. /Kink
states in supercritical RFT
479
This last statement can also be easily understood by means of the mechanical analogy. Let us think of cp(p) as the coordinate of a particle. After the particle crosses cp= 0 with a finite speed to the right, it will be subject to a restoring force (-cp), an anti-dissipative force u$ that feeds energy and repels it to the right, and the nonlinear force cp2 that further repels it to the right. Therefore, the velocity corresponding to the linear problem where cp2 is neglected gives a lower bound to the actual velocity of the particle. Going back to fig. lob, we remark that for u > 2 the phase space trajectory is asymptotically tangent to W = k&, kg being the smaller exponent of eq. (A. 17), which is asymptotically dominant for p -+ -00, Therefore, we conclude that the soliton for IJ > 2 lies always in 0 < cp< 1. It is also clear that its steepness increases when u decreases from large values to the critical value LJ= 2. Indeed, by identifying the work performed by the non-conservative force u$ between cp= 0 and cp= 1 with the variation of the mechanical energy of the particle, one gets
U
s
‘b2dp=V(1)-V(O)=;,
(A.2 1)
-m
which shows that, as u decreases the steepness of p(p) increases if cpis a monotonous function of p. The different shapes of solitons of u > 2 are exhibited in fig. 11. We are now in a position to discuss the asymptotic propagation speed in the original initial value problem (A. 11). The basic idea is to rewrite the differential equation (A.1 1) in a frame moving with speed u in the positive x-direction, for example. Then we have cp(x, t) = u(ut -x,
t) = U(P, t) ,
(A.22)
au=u-uutu-u2, .. .
at
u(-x,
0) = 240(x).
(A.23)
The solitary waves we discussed before are now static solutions of (A.23), and it is clear that they are possible asymptotic limits for u@, t) as t + 00 at fixed p. The mechanism that chooses precisely the u = 2 soliton for the asymptotic limit of u@, t) is given by the following results which are proved in ref. [ 121. (a) For any u > 2, if the initial condition ue (xx>vanishes more rapidly than any
Fig. 11. Different shapes of kink solutions of the smgie-ileid equation, for v > 2.
V. Alessandrini et al. /Kink
480
states in supercritical RFT
Fig. 12. The way in which the solution of the diffusion equation (A.ll)
gets asymptotically
trapped between kinks with u > 2 and v < 2.
exponential for x + 00, it is possible to find a soliton solution $>(p) such that u(p, t) < @,@). Moreover, a&_~, t) < 0. (b) For any u < 2, there is an (oscillating) soliton solution with @<@e) = 0 such that U(JJ, t) > u<(p) -@ > pe) - and sufficiently large time. Moreover, atub, t) >O. The situation for cp(x, t) is therefore the one depicted in fig. 12. What points (a) and (b) are telling us is that, in a frame moving with velocity u > 2 one would see z&, t) - bounded from above by &,@) - decreasing with time t, while in a frame moving with velocity u < 2 one would see u(p, t) - bounded from below by #c(p) for p >pe - increasing with time. From (a) and (b) and known theorems, it follows that
cp(ut- p, t) = u(p, t) --f 0
if
u > 2,
cp(ut-p, t) = u@, t) --f 1
if
u<2.
(A.24)
In other words, rp(x, t) gets trapped between solitons with u > 2 and u < 2, and has to propagate with the asymptotic speed u = 2. Without attempting a proof of these statements here, let us notice that the properties of u@, t) discussed in the preceeding paragraph are indeed shared by the perturbative solution (A.12). For the special case uo@) Q:S(x) we have in fact lpo(ut-P7
t)=& W-J t[
=&
exp[t(l
(ut 4t
- $3)
PI2 1 t iu2p - p2/4t] .
(A.29
Completely similar arguments hold for x < 0. For simplicity we shall then summarize the results described above by the statement (not strictly proved in the mathematical literature) that cp(x, t) approaches asymptotically the soliton with u=2. In other words ~x,t)~[@2(2t-X-~)t$2(2t+X+b)-1]
(A.26)
for some values of the translational parameters a and b which depend on the specific form of the initial condition u. (xx>.This is essentially equivalent to the equation (2.20) of the text.
V. Alessandrini et al. /Kink states in supercritical RFT
Appendix
481
B
Zero modes and completeness The purpose of this appendix is to relate the zero-mode wave function .&@) and its conjugate vector x,,(p) to the lowest eigenfunctionfe(P) of the SchrSdinger operator DS of equation (5.11). We are particularly interested in understanding their role in the completeness relation, in order to prove the spectral decomposition of the identity given by eq. (4.42) as well as the spectral decomposition of the Hamiltonian % s given by eq. (4.43). Our strategy will be to prove these results from the known completeness of the eigenfunctions of DS. Let us recall that [&I) and X,-&I) are, by eqs. (4.16) and (4.19)
(B.1)
03.2) From the known properties of our classical solutions as functions of p and u (see eq. (5.32) as well as the comment preceding it) we deduce that the components of [e and x0 have the symmetry property
FobI = -G-P)
(B-3)
3
.
Go(P) = Go(-P)
(B-4)
Therefore, in order to satisfy the symmetry property (B.3) between the upper and lower components of go, we identify it with the eigenfunctions introduced in sect. 5 as follows: fob) to(P)
evp12
=(
fo@) emupI
1
d(P)
-
(
J;b(--p)
eup12
fo(--p)
emupI
1
0(-P)
*
(B-5)
Likewise, in order to satisfy (B.4) we write x0@) as go (-p)
go@) evpt2 xocp) =
( go (J.I) e-vp/2 1
B(P) +
where go@), go(p) are functions read Dsfo +2~e-VP~o 2g evPf.
+
( go(-p)
evp12 emvp12
yet to be determined.
1
0(-P)
(B.6)
Eqs. (4.17) and (4.20) now
=0,
Dsfo = 0 ,
(B-7)
V. Alessnndrini et al. / Kink states in supercriticalRFT
482
Dsg,-, + 29 ewvp go = Szf,, , 2@ evPgo t Dsgo =
(a=-&O/NO)>
-CZfo,
w3)
Let us now try to express, for (2 - u) << 1, all the components terms of
fo(p) = N0 cb(p) e-vp/2 @>O) .
of go and x0 in
(B.9)
The function so defined, which is the exact expression for the upper component of the zero mode for p > 0, is also approximately equal to the ground-state wave functions of Ds introduced in sect. 5, corresponding to the eigenvalue w. = 0(2 - u), and for which we have used the same notation. In fact, both can be identified as u + 2-. Since the ground-state wave function of DS is an even function, its derivative at the origin vanishes. We check that this is approximately true for fob). By using (3.29) and the equation of motion at p = 0, we obtain fO(0)=fvO[~O)-$0e]
= N&u-
We therefore extend the definition function
fo(-P) = fo(Ph
(P> 0)
of
l)%
t&j]
=0(2-u).
(B.10)
fo@) to p < 0 by demanding it to be an even
3
(B.11)
having then the property that Dsfo = 0(2 - u),
fo(0) = 0(2 - u) .
(B.12)
We next go back to eqs. (B.7) which reduce, to leading order in (2 - u), and for p >o,to
Dsfo= 0, (P >Oh
2$fc, + Dsfo = 0 ,
where use has been made of the fact that, for p > 0, cpexp(-up) that (p = (b exp(-up). Due to the fact that, for p > 0
O=$[Dsfol the two independent
=Dsfo
is of 0(2 - u), and
(B.14)
solutions of (B.13) are
0
fo
(B.15)
a
4
+2;fo,
(B.13)
fo
IT
(b)
( .I’ fo
V. Alessandrini et al. /Kink
states in supercritical RFT
483
and the zero mode &-, must be a linear combination of the two, which is determined by using eq. (B.3). This equation implies that, at the origin,fe(O) = -fe(O). Therefore in order to satisfy this relation at p = 0 between the upper and lower components of to, we must have the antisymmetric combination, i.e.: f. evp12 &)(p)=No
fg
_i cf, _ fe) e-VP/2
For p < 0, .$e is automatically
1
(p >o;
u-2_).
(B.16)
’
given by (BS), i.e.
=(
[o(P)
(p CO;
u+2_)
(B.17)
Notice that the upper component of .$,@I) in p < 0 contains also f,-,, and therefore it is neither even nor odd under p * -p as expected from the known shape of p(p). A similar situation occurs for x0(p), in the limit u -+ 2-. In fact, by eq. (5 35) (B.18) and therefore the inhomogeneous term in the right-hand side of (B.8) vanishes to leading order in (2 - u). The equations determining the components of x0(p) reduce then in this limit to eq. (B.13). In other words, x0(p) becomes, for u + 2-, a second zero-mode solution. This is not so surprising if one remembers that the two components of the classical solution, &) and $@), are less and less correlated as u + 2-, and that in the limit u = 2 they are independent solutions of single field equations. However, this time the boundary condition at p = 0 is given by eq. (B.4), which implies g(O) = g(O), and therefore x0 is proportional to the symmetric combination of the two independent solutions (B.15):
%I
X0(P)= Lo z
f. evpi2 = :
(jJ >o; Q. t fo) e-vp/2
where the factor 1 is determined
(x07 to) = 2 s
by the normalization
[Fo(n)Go(~)
-
u+2_),
(B.19)
’ condition
~oW~oW1 c-b
0
=2s 0
f;(p) dp = 1.
(B.20)
484
V. Alessandrini et al. / Kink states in supercritical RFT
Finally, the form of x0 for p < 0 is obtained
from eq. (B.6)
v. U-+2_).
(PC%
X0(P) 21 ;
(B.2 1)
Let us remark that the expression for x0 differs from the one for to essentially because of the replacement f. + -f.in the lower component. It must also be noticed that, since fo(0)= 0 and fo(p) is of order 1 for p z L, the term proportional to f. in the lower component of (B.19) is negligible compared to the other term for moderate values of p. On the other side, in this region the lower component of x0 can be estimated as follows (for 0 < p
e
-up/*
-L -
z Oau’
Lo
g
-2f-Lo-
v-2
cbe
--VP
.
(B.22)
2-v
Therefore, by comparing it with f. givenby (B.9) we get --
Lo --=;(2-u) _a No I-1
(B.23)
in agreement with the argument in the text. The explicit expressions we have obtained for the vectors E. and x0, together with the similar ones for .&, (n # 0) discussed in the text, allows us to prove the completeness relation:
njjl[MP)L(P')
- L(P)~n(P')l
(B.24)
+ 5o(P)jio(P') - xo(P)5_dP')= IS(P -PI).
In fact, by using the eqs. (5.8) and (5.9) of the text, we get for the left-hand side of (B.24) a matrixM@, p')which, for the case p, p' > 0 is given by M,, =Mz2 = 2 e"P~2fn(p)e-"P~2fn(p'), n=O Ml*=09
M2r = {C n=l
e--vD/2 g,(P) e-"P"*fn(P')
t e-"P/2fo(p)e-vP"2fo(p')- (p * P’)} It is clear that, by the completeness
.
relation for the hermitian
(B.25) operator Ds given by
V. Alessandrini et al. /Kink states in supercritical RFT
485
eq. (5.19) in the text, we get Ml1
‘M22
= SC0 -P’)
(B.26)
9
because the similarity transformation exp(iup) the other hand, from eq. (B.14) we get fo@) =-
c f,(p)(2G),a nf0
+ constant
leaves the h-function
.fa(p) .
unchanged.
On
(B.27)
an
However, only the first term contributes to M2 r , the second one being canceled when the expression is antisymmetrized in p and p’. Then, using (B.27) and the expression (5.25) for g,(p), we get
M21
= fIl
Sk
f&3)&@‘)
e-*+p’)/2
-(p
*P’)
s 0.
(B.28)
because the first term is symmetric in (p, p’). This completes the proof of eq. (B.24). From the completeness relation we also get the spectral decomposition of the operator SK2 on both sides of (B.24), it follows that 81’2x0 = ato
3
912to
= 0
(B.29)
and acting with $X2 on both sides of (B.24), it follows that
References [l] D. Amati, M. Le Bellac, G. Marchesini, M. Ciafaloni, Nuclear Phys. 8112 (1976) 107. [2] D. Amati, G. Marchesini, M. Ciafaloni, G. Parisi, Nucl. Phys. B114 (1976) 483. [3] J.L. Cardy, Nucl. Phys. B115 (1976) 141; R.C. Brower, M.A. Furman and K. Subbarao, UCSC preprint, 1976. [4] J.B. Bronzan and R.L. Sugar, U.C. Santa Barbara preprint TH 77-l. [5] N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606. [6] R. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DlO (1974) 4114,4130, Dll (1975) 3424; J.L. Gervais and B. Sakita, Phys. Rev. Dll (1975) 2943. [7] J. Baacke and H.J. Rothe, Nucl. Phys. B125 (1977) 108. [8] V. Alessandrini, D. Amati and R. Jengo, Nucl. Phys. B108 (1976) 425; R. Jengo, Nucl. Phys. B108 (1976) 447; J. Bronzan, J. Shapiro and R. Sugar, Phys. Rev. D14 (1976) 618. [9] M. Ciafaloni, M. Le Bellac and G.C. Rossi, Nucl. Phys. B130 (1977) 388. [lo] P.W. Anderson, Rev. Mod. Phys. 38 (1966) 298. [ll] A.R. White, CERN preprint TH. 2259 (1977), Nucl. Phys. B, submitted. [ 121 D.G. Aronson and H.F. Weinberger, Non-linear diffusion etc., in Lecture Notes in Math., 446 (Springer Verlag, 1975) and references therein. [13] M. Ciafaloni and G. Marchesini, Nucl. Phys. B109 (1976) 261.