PHYSICAL REVIEW D VOLUME 48, NUMBER 9 1 NOVEMBER 1993

Two-dimensional field theory description of a disoriented chiral condensate

Ian I. Kogan* Physics Department, Princeton University, Princeton, New Jersey 08544

(Received 2 June 1993)

We consider the effective ( 1 + 1 )-dimensional chiral theory describing fluctuations of the order param- eter of the disoriented chiral condensate (DCC) which can be formed in the central rapidity region in rel- ativistic nucleus-nucleus or nucleon-nucleon collisions at high energy. Using ( 1 + 1 )-dimensional reduc- tion of QCD at high energies and assuming spin polarization of the DDC one can find the Wess- Zumino-Novikov-Witten model at the level k = 3 as the effective chiral theory for the one-dimensional DDC. Some possible phenomenological consequences are briefly discussed.

PACS number(s): 24.85. +p, 11.30.Rd, 12.38.Mh, 25.75. +r

There exists some connection between high-energy scattering in quantum chromodynamics (QCD) and effective (1 + 1)-dimensional field theory [I]. I t was shown recently in a very elegant way [2] how to derive the effective (It-1)-dimensional theory describing the high energy interaction between two quarks in the limit s >>t >>AQcD. To get the two-dimensional picture one can split four coordinates into two longitudinal coordi- nates x a and two transverse coordinates x ',

with x ' = t f z, and then perform the rescaling of the lon- gitudinal coordinates,

with h- 1 / d i -0. In this limit the QCD Lagrangian will reduce to some effective (1 + 1 )-dimensional Lagrang- ian (for more details see paper [2]).

The aim of this paper is to apply similar ideas to anoth- er high-energy process-the formation of the disoriented chiral condensate (DCC) [3] in a relativistic nuclear col- lision. It is well known that the QCD Lagrangian is in- variant (approximately, if nonzero masses for the light Nf quarks are taken into account) under global chiral S U N f ), X SU( Nf I R , where Nf is the number of the light flavors. This symmetry is spontaneously broken down to vector SU(Nf I V which leads to ~ f 2 - 1 (quasi) Goldstone bosons, pions (if Nf =2), or pions, kaons, and an 7 meson (if Nf '3). The order parameter for this breaking is the vacuum expectation value of the quark condensate ( $I) ). However one can imagine that under some special condi- tions in a finite volume V during the time interval T the vacuum condensate may be disoriented in isotopical space. I t is convenient to describe chiral dynamics by a u model with isoscalar 5 and isovector a fields (in the case of N f = 2 ) and the constraint 02+a2=f2, . In vacuum one has ( a ) = f, and since u is an isoscalar there is an isoscalar condensate ($I)) only. However one can consider another configuration - ( u ) = f ,cos0 and

.rr= f,nsin0, where n is some unit vector in isospace, which describes the DCC, i.e., some classical pion field configuration, which is metastable and decays after some time into pions-the signature for this event will be the large number of either neutral (TO) or charged (T') pions [3]. I t will be interesting to formulate an effective low- energy theory describing this condensate and the small fluctuations around it. One obvious candidate is the usu- al four-dimensional chiral model; however we must remember that we are considering now the small fluctua- tions of the order parameter not around the 0(3,1)- invariant vacuum, but around some new metastable ground state arising immediately after the collision where neither 3 + 1 Lorentz invariance nor rotational invariance are valid. Thus the effective chiral model may be aniso- tropic and one can think about some new universality classes.

I t is important to remember that in the Bjorken model of hydrodynamical expansion in the central rapidity region [4] (see also [5]) one has approximate 1 + 1 Lorentz invariance with respect to the longitudinal boosts x'-exp(_fO)x' which is based on the fact that initial conditions for the hydrodynamical expansion are invariant with respect to longitudinal Lorentz boosts.' I t was estimated in [4,5] that until the time R , /us, where R , is the nucleus radius and v, is the speed of the sound waves in nuclear matter (for a uranium nucleus this time is approximately 10 fm) the expansion will be largely 1+ 1 dimensional. It is precisely during this time the DCC may be formed and we shall try to find an effective (1 + lbdimensional theory describing the DCC in the cen- tral rapidity region, where the rapidity is defined as y = + l n ( x + / x - ) .

To derive this effective theory we shall use the same approach as in a derivation of a 3+ 1 chiral Lagrangian from QCD (see, for example, [7] and references therein), i.e., consider pions 7ia as the elementary external fields in- teracting with quarks. The effective low-energy action is

*on leave of absence from ITEP, B. Cheremyshkinskaya 25, Moscow, 117259, Russia.

' ~ h e s e initial conditions are different from the initial condi- tions in the Landau model [6] which lead to an explosion at t,z = 1 fm.

0556-282 1/93/48(9)/3971(5)/$06.00 - 48 R3971 @ 1993 The American Physical Society

R3972 IAN I. KOGAN

where n = n a t a / f , and t a are the generators of the axial SU(Nf), f,=95 MeV is the pion coupling constant and s [ A ] =$J d 4 x t r( F,,,F~") is the gluon action, F,,,=d,AV--a,,A,+e[A,, A,,] is the non-Abelian field strength, A,= A i r a and ra are the generators of the color SU(3),. We also include here the quark mass term ~ M V , where M =diag(rnl . . . rnh7 ) is the quark mass

f matrix.

We can rewrite the fermion part of the action as

where the left and right gauge fields are

and U = e x p ( i n ) . The first factor in the direct products refers to flavor SU(Nf), while the second refers to color SU(3), .

We are looking now for some new (quasi-)one- dimensional chiral models and will try to derive some effective (1 + 1)-dimensional action W , + , [ n ] describing the fluctuations of the chiral order parameter in the cen- tral rapidity region arising after nucleus-nucleus collision.

To extract that part of the action that is relevant to the high-energy collision we shall use a rescaling of the light- cone coordinates suggested in [2] x'-hx'. The com- ponents of the gauge potential are transformed under re- scaling as A,- A, and A,-~-:A,, while the quark fields are transformed as $-$//A. Hence the rescaled gluon action can be written in the form

where E,, is an auxiliary field. The rescaling action de- scribes the same physics as the original one if one also re- scales s -h2s; thus the high energy limit s - co corre- sponds to h-0 in the rescaled theory [2] and we get the truncated action

and now the auxiliary field E becomes a Lagrange mul- tiplier imposing the zero-curvature constraint

Now let us consider the fermion action after rescaling:

Taking the limit A-0 we see that the mass term disappears and quarks propagate in the longitudinal directions only and couple only to Li and R* components which become pure gauge (let us remember that color and flavor commute with each other):

To get the effective two-dimensional action we must rewrite four-dimensional fermions l in terms of two-dimensional ones. Using the chiral basis for the y matrices,

it is easy to get

48 - TWO-DIMENSIONAL FIELD THEORY DESCRIPTION OF A . . . R3973

where the two-component left and right spinors YL and YR were defined as YL=($- ,$+) and WR = ( $ + , $ - I and there are two left (in a two-dimensional sense) $+ and 4, fermions and two right $- and 4- fermions.

When we neglected the transverse part of the fermion action we assumed that the quark transverse momenta are sufficiently small hlpll << 1 and it looks reasonable to consider fermion fields as independent of xi-one has decoupled one-dimensional systems, and to calculate the effective action W1+l ( n - ) we must calculate the two- dimensional determinants of the Dirac operators:

the z axis: $* and 4+. For $ fermions (which are com- ponents of the left spinor W L ) the direction of spin is the same as the direction of momentum; for 4 , it is the oppo- site. Thus removing, let us say, $- and 4, we have one left mover $+ and one right mover 4-, but both of them have the same direction of spin and this additional reduc- tion leads to the spin-polarized vacuum. One can esti- mate the spin in this state as a volume in a phase space occupied by these fermions S = R LA^, where R , is the transverse scale, L is the longitudinal one, and A is the ultraviolet cutoff for the effective chiral model which is defined by the inverse "size of the pion," i.e., the pion

Wl+ 1 (n - )= ln det[aa(aa+La )I +In det [aa(aa+Ra ) I . coupling constant f, = 95 M ~ V . one can estimate the The determinant of the Dirac operator in a non-Abelian gauge field A , has been calculated by Polyakov and Wiegmann [8]:

where Wk(G) is the action [9] of the Wess-Zumino- Novikov-Witten (WZNW) model at level k,

and M is a three-dimensional disk the boundary of which is our two-dimensional space. The chiral field G ( x ) takes the value in the direct product SU(3), X SU(Nf).

Using (10) one can see that for both L * and R * fields GLo,=HLol and thus W,+l(n-)= W ( G ~ . H C ' ) + W ( G R . H i l ) = O ; i.e., we do not obtain any nontrivial two-dimensional chiral action-the chiral dynamics of the DCC will be described by the usual four-dimensional chiral Lagrangian. The reason for this follows: after the reduction to a two-dimensional problem the chiral (in a four-dimensional sense) spinors YL and YR were transformed into left-right symmetric (in a two- dimensional sense) pairs $* and 4t interacting with two vector fields L * and R *; thus, in the resulting theory the original chiral rotations look like vector gauge transfor- mations and have no anomalv. which means that there is * . no two-dimensional chiral action. To obtain a nontrivial W1+,(n-) one must have a two-dimensional anomaly for the chiral transformation and it is possible to obtain it by removing the left- (right-) moving fermion from one pair and the right- (left-) moving fermion from another one.* What does this further reduction mean? Let us remember that chiral fermions have a definite spin pro- jection on the momentum direction n (neglecting the fer- mion mass) (u.n)\IIL,,! = +\IIL,R. Then after reduction one has four ( I + 1)-dimensional fermions moving along

2 ~ e cannot remove both $+,4 ,, or $-,4- because in this case there will be an anomaly for color SU(3), too, which is ab- solutely forbidden.

transverse radius as R l W l / f , comparing the two- dimensional reduction of the four-dimensional action f:Sd4xa,n-a,n- and the two-dimensional one J d 2~ a,n-aan-. The fact that R, f, - 1 means that we really have a one-dimensional phase space (as it should be) and transverse fermion excitations are irrelevant. For spin one gets the estimate S--Lf,; for example, for L = 10 fm, one gets S = 10.

We can suggest that such a spin-polarized state can ap- pear in ultrarelativistic collisions with a polarized nu- cleus. To support this idea let us use the Walker argu- ments [lo] about the heavy ion as a color vacuum cleaner-due to the strong interaction of each colored parton in the target with soft gluons from the projectile all colored degrees of freedom will be swept out of the target. The "nothing" that is left behind the leading par- ticles in the central rapidity region carries no memory of the valence degrees of freedom, but carries four- momentum P, and it is in this region that the creation of the DCC is possible. However the same region can carry angular momentum too-and thus one can imagine that after the collision of the polarized nucleus there will be a large spin polarization in the central rapidity region. Then one can think that a reduced (1 + 1)-dimensional model with the action (again we are assuming that there is no dependence on the transverse coordinate x')

will describe a spin-polarized DCC. The effective action W,+,(n-)=ln det [oa(aa+ An ) ] is nontrivial now, be- cause the gauge fields are

SO G = U @ g , H = U - ' O g , and G . H - ' = u ~ o I . The gluon degrees of freedom are decoupled from the effective action and the trace over the color group gives us the fac- tor Nc = 3 in front of the WZNW action. Thus the effective one-dimensional chiral dynamics .of the spin- polarized disoriented chiral condensate is governed by the SU(Nf) WZNW model at level k =3:

R3974 IAN I. KOGAN 48

where we changed the definition and denote u2 as U, which is now defined as U =exp(2irata/f,). Maybe it is also possible to get WZNW models at levels 2 and 1-in the case when the spin-polarization is not complete and in addition to the $+ and 4 fields there is a (though smaller) number of $-,+, pairs; however, this is not completely clear now.

Thus long-range fluctuations of the order parameter U are described by the two-dimensional conformal field theory; this is important and means that one can have a quasi-one-dimensional chiral condensate which will not be destroyed by the infrared effects. The WZNW model is exactly solvable and the spectrum of anomalous dimen- sions and correlation functions are known [ l l ] , so we can use this information to study the correlation between the order parameter values in the different space-time re- gions. The two-point correlation function (up to some normalization factor O is

i, j, k, 1 are SU(2Vf) indices, and the anomalous dimension of the chiral field U equals

where the constant cg is defined as tuta=c,I. For flavor SU(2) and k = 3 one gets cg =$ and A = 6. It is more convenient to rewrite the correlation function (18) in proper time rapidity coordinates [4] r2=t - z 2,

y = f ln ( t + z ) / ( t -z) and to study the correlation at equal proper time 7. Then

( U ~ ( ~ , ~ ) U / ~ ( T , ~ ' ) ) = c s ~ s ~ ( ~ ~ ~ ) - ~ ~

and in the central rapidity region when y and y ' are small we shall get a simple scaling law (y -y1)-3'5. Because the direction of the order parameter gives us the ratio of neutral to charge pions arising after the decay of the DCC, the two-point correlation function (20) gives us a distribution of the events with a given ratio of neutral to charged pions in a given rapidity interval. We also can consider more complicated correlation functions; for ex-

ample, four-point correlation functions which were calcu- lated in [ l l ] . These functions will give us the multiparti- cle correlations. It will be extremely interesting to look for such a correlation either in nucleus-nucleus collisions or even at high-energy nucleon-nucleon collisions at the Superconducting Super Collider (SSC). It is also interest- ing to understand what will be the phenomenological manifestations of the spin polarization.

In conclusion we would like to briefly discuss the possi- bility of taking into account the effect of leading particles (the regions in the rapidity space with the nonvanishing baryon number) in this picture. What we have is two- dimensional conformal field theory defined on a finite in- terval. The baryon charge is distributed at the boun- daries of this interval and one can try to take it into ac- count by adding some boundary vertex operators; if this is possible one can apply the well elaborated methods of conformal field theory to study the dynamics of the con- densate at large rapidities. The effect of the baryon charge in this region could be described as the soliton charge of the boundary vertex operators. Using this ap- proach one can study correlations between leading parti- cles moving in opposite directions. It is also interesting to understand how to take into account the interaction between fluctuations of the DCC order parameter U and the hydrodynamical fluctuations during the ( I f 1)- dimensional expansion. The hydrodynamical fluctuations are connected with the group of the one-dimensional diffeomorphisms DiffR ' and it may be that the large scale fluctuations are governed by two-dimensional quantum gravity. These and related questions deserve more de- tailed investigations.

Noted added. While this paper was being prepared for publication I became aware of a recent paper of Khlebni- kov [12] in which the two-dimensional description of the DCC was considered and the WZNW model was suggest- ed as an infrared fixed point of the general asymmetric chiral model. It was also suggested that one can get the WZNW model from the reduction of the 3+ 1 model with the Wess-Zumino term provided there is topologi- cally nontrivial configurations of kaons in transverse directions. However the level of the WZWN model in [12] was suggested to be 1, not 3, which leads to nonin- teger topological charges of the kaon configuration Q =4/3. Let us note that if one takes k = 3 instead of k = 1 the charge will be integer Q = 1. The relation be- tween this reduction and the spin-polarization picture suggested in this paper is unclear to us now.

I am grateful to A. A. Anselm for introducing me to this subject and to A. Bilal, I. Klebanov, A. Polyakov, and H. Verlinde for interesting and stimulating discus- sions. This work was supported by the National Science Foundation Grant No. NSF PHY90-21984.

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