Fortschritte der Physik / Progress of Physics: Band 29, Heft 7 [Reprint 2022 ed.] 9783112655962

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 7 • 1981 . B A N D 29

A K A D E M I E - V E R L A G 31728

EVP 10,- M

.

B E R L I N ISSN 0015 - 8208

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Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. D r . Frank Kaschluhn, Prof. Dr. Arthur Lösche, Prof. D r . Rudolf Ritsch], Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr.: 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 7400 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 120,— M). Preis je H e f t 15,— M (Preis f ü r die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/29/7. © 1981 b y Akademie-Verlag Berlin. Printed in the German Democratio Republic. A N (EDV) 57 618

ISSN 0 0 1 5 - 8 2 0 8 Portschritte der Physik 29, 303 - 3 4 4 , (1981)

Two-Dimensional Field Theories A. V. B e r k o v , Ytr. P.

Nikitin

Engineering Physical Institute, Moscow, USSR I. L .

Rosental

Space Research Institute, Moscow, USSR

Abstract Quantum field theories in space with the dimensionality 1 + 1 are considered. Quantum electrodynamics, quantum chromodynamics, sufficiently nonlinear models (with soliton solutions) and gravitational theory are discussed from the common viewpoint. The possible correspondence of two-dimensional field models to physical reality is analyzed.

Contents 1. Introduction

304

2. The 2.1. 2.2. 2.3. 2.4. 2.5.

306 306 308 310 311 314

string model r String as a model of QCD Kinematics of the string Several solutions of the string equations Relativistic string quantisation Spin degree of freedom of the string

3. Two-dimensional quantum electrodynamics (QED-2) 3.1. Main properties of QED-2 3.2. Electrons and photons in two-dimensional world. Lagrangian of the model 3.3. Vacuum polarisation and "photon" mass 3.4. Interaction with the external scalar field

315 315 315 318 320

4. Two-dimensional quantum chromodynamics (QCD-2) 4.1. Introduction 4.2. Structure of QCD-2-model 4.3. Charge confinement in QCD-2 4.4. Meson states spectrum in QCD-2 4.5. Numerical results and extrapolation to the four-dimensional space

322 322 323 325 325 327

5. Two-dimensional models with soliton solutions 5.1. The simplest model 5.2. Sine-Gordon model (model sin R=WReel2,

W = W8'2.

(3.10)

The current components in cone coordinates are j+ = tpR*ipR,

(3.11)

j~ = fL*yL

and they are transformed according the law (3.9). If the two-component field is introduced

W the law of its transformation will take the form corresponding to the spin 1/2 field transformation : y-^e^lty where a 3

=

(3.12)

^j • Presenting the current in the form ji = yyiy,

(i = 0, 1) where the matrices y° = ^

, y1 = ^

1 o)

(3.13) anC^

^

=

^

easy

see that its components = l/2(j° ^ j1) coincide with (3.11). It should be mentioned that in cone coordinates the algebra of y-matrices in the twodimensional world is as follows : y± =

y2

(y° ± y 1 ),

y± 2 = 0,

y + y - + y - y + = 0,

Spy+y-=2.

(3.14)

All calculations are sharply reduced in such variables. So in the two-dimensional world the free Dirac field can be introduced with the free Lagrangian J? = iyiêy = i(yR+d+y)R + yL+d_ipL), (3.15) 8 8 ±—. d ± = d o ± d l = — The Dirac equation for free particles split into two independent equations i8+yR = 0 ,

id_ipL = 0

(3.16)

whose solutions describe right-moving and left-moving particles. It should be noted that antiparticles of fields ipR and ipL will be also right-moving and left-moving respectively.

318

A. V. Berkov, Yu. P. Nikitin, I. L. Rosental

The minimal (gauge invariant) Lagrangian with the electromagnetic interaction can be easily written. I t takes the usual form *==

- pi) y -

1

(3.17)

where g is the fermion field charge. As the dimension of the Lagrangian density $ is L~2 (L is the length) or m 2 , the field tp dimension is [y] = L~112 = m 1 ' 2 , potential A{ is dimensionless then the dimension of g2 is equal to L~2 = m 2 . According to the formal criterion the quantum field theory with such Lagrangian i does not involve irremovable divergencies. In Coulomb gauge for the potential (A1 = 0) the Dirac equation with the external electric field takes the form =

(3.18)

I f the left side of the equation is multiplied by we obtain id+y)R

=

id-y>L

=

g A

0

after the transition to ipn and ipL f

R

,

(3.19) g^oWL-

Thus, the right and left fermions do not mix in the presence of the external electromagnetic field. 3.3. Vacuum polarisation and "photon" mass Simplicity of QED-2 with massless fermion is formally related to the following property •of the two-dimensional gamma matrices y° and y1: yiykyl

. . . ymyi

0.

=

(3.20)

This relationship and gauge invariance of the theory sufficiently decreases the number of possible Feynman diagrams that makes it possible to obtain a series of exact solutions for the physical values in QED-2 [28]. T h e free Green function of two-component fermion field in the 1 + 1 - s p a c e takes the form G(p) = j

(3.21)

or in the coordinate representation

«W-B-GL

(0)]j

(3.33)

where D(x) is the Green function of boson with mass ¡x and D0(x) is the Green function of boson with zero mass. Using the form of 0(x) (3.22) and DJx) = l/4:wln {—x2), we obtain that for x 0 S(x)

=

2 f2 - L

exp [In (-a; 2 ) -

4iD{x)].

(3.34)

Thus, the first addend in the exponent cancels the factor l/x 2 before the exponent caused by the fermion Green function. From here it follows that the Schwinger model's spectrum consists only of massive neutral excitations and fermions are not generated at all. This situation (charge confinement) is associated with the fact that strong polarisation of vacuum at small distances does not allow charged particles to move at large distances where only the bound states of fermions can exist. However, the expression (3.34) has a defect, i.e. S(x) 4= 0 with x ->• oo [30]. This suggests that the vacuum in the diagrammatic solution (absence of massless fermions and "massive photons") is not a physical vacuum for the Schwinger model. This is possible if phase transition occurs in the model. In fact, physical vacuum of the model is the superposition of the states with different chirality 0, ¿ 2 , ¿ 4 , ... To observe the transition from chiral symmetric vacuum of the diagrammatic solution to chiral noninvariant vacuum of the QED-2 model in [25] is studying the imaginary part of the Fourier-component S(x) written as a sum over all states of the model *(«) = E ¿' = e i > r S f j j acquiring the factor e2li. Therefore, spontaneous breaking of chiral invariance takes place in this case. Thus, the studies of the basic properties of the QED-2 model show that in terms of this model one can observe the action of two essential mechanisms which assist in explaining the properties of the real world of hadrons. First, it is the charge confinement mechanism caused by strong polarisation of vacuum at small distances. Second, it is the mechanism of spontaneous breaking of vacuum symmetry. Though the QED-2 model does not correspond to physical reality, the study of its mathematical properties can promote the understanding of the abovementioned mechanisms in the more realistic four-dimensional constructions (discussion on QED-2 as a hypothetical structure of hadron dynamics and detailed list of literature see [32]). 4. Two-Dimensional Quantum Chromodynamics (QCD-2) 4.1. Introduction As it has been mentioned above QCD is the most probable candidate for the theory of strong interactions. The hypothesis that the physical states of hadrons are the color singlets while quarks and gluons have the quantum numbers related to the color oo) with the fixed value of g2N (g is the QCD coupling constant). The mentioned properties of the two-dimensional model arise just in the main order by 1 ¡N. In this case only planar diagrams with quarks in the end remain in the theory and the fermion loops do not arise. The topological structure of the diagrams in the main order by 1 ¡N of the two-dimensional model is similar to that of the dual resonance model of the string. So, one can expect that the

Two-Dimensional Field Theories

323

limit of large N in four-dimensional gauge theories will provide the dynamical substantiation of the string model. I t is shown in [35] that the model proposed b y ' T HOOFT satisfies the most essential physical requirements of unitarity, analyticity, current conservation etc. Besides, it is shown that the quarks which disappear asymptotically from the spectrum of physical states appear at small distances, as one could expect in asymptotically free theories, and the confinement phenomenon observed in such a model is independent of the method of introduction of infrared cutoff. Finally, it is shown that all qualitative features of the model also are conserved in higher orders of expansion by the parameter 1 ¡N. The approximate Bjorken scaling in the deep-inelastic scattering of leptons [36] was found in the model, the properties of "charmonium", the behaviour of meson formfactors etc. were studied in a qualitative agreement with the effects observed. We guess that the most essential point is t h a t ' t Hooft's model, if one may say so, is the most "realistic" among the other two-dimensional models. It is mostly close to the four-dimensional model of QCD and has such attractive properties as superrenormalizability, confinement of free quarks, possibility to predict a hadron spectrum. In accordance with the character of our review, we shall describe below following mainly the original paper [34] only the most essential features of the model not going into technical detailes. 4.2. Structure of the QCD-2 model The model considered is described by the following Lagrangian I = qa°-(iyiDi - ma) q* + j GJaGik;

(4.1)

where GiA = Mti - Mil + g[Ai, Ak]{ A?«" = 8iq» + gAfof A,i(x) = -Aftx).

(4.2) (4.3) (4.4)

In these formulae a = 1, 2, ..., F is the number of various types of quarks; tx = 1,2,...» N is the number of various colors; Lorentz indeces i, j = 0 , 1 ; qaa are the wave functions of quarks; A£ is the gauge colored vector gluon field. The gauge group relative to which the Lagrangian (4.1) is invariant is assumed to be the group U(N) (in many papers the group SU(N) is considered; to the first order of expansion by 1 /N this difference is insignificant); g is the constant with dimensionality of mass. Such a theory is superrenormalizable since the renormalisations of mass and coupling constant are finite. The property of asymptotic freedom is also easily checked. Essential simplification due to only two dimensions consists in the possibility to choose such a gauge of fields A£(x) that the commutator [Ait Ak] approaches zero, i.e. there is no self-action of gluons any longer. Theory becomes still more simple if we choose the infinite momentum frame, or, respectively, introduce the cone coordinates (as in ch. 3). In this case (^o ± Ai),

p± = -i- (p0 ± Pi),

PI = P+1- + P-9+

(4-5 ) (4.6)

324

A . V . BERKOV, Y U . P . NIKITIN, I . L . ROSENTAL

and the algebra of /-matrices is given by the formulae (3.14). Then the gauge on the light cone means that we choose A_ = 0. (4.7) As is known [37, such a gauge facilitates discussion of the bound states problem in relativistic theories since there are no topologically complicated diagrams in this gauge that makes it difficult to analyze integral equations for bound states. Besides, in QCD-2 this gauge excludes the "ghost" states related to the "unnecessary" components of gauge field. In the cone gauge the Lagrangian (4.1) takes the form X =

- m0 +

qa(iy%

igy_A+)

"coordinate", then from (4.8) it is seen that A+(r, z) is not an independent dynamical variable since its time derivative does not enter into the Lagrangian. However, this quantity determines the Coulomb force of interaction between quarks. Indeed, the equation of motion obtained in standard manner from (4.8) takes the form 8jA

+

( r , z) =

~ g j . ( r , z)

(4.9)

where = qay_qa is the quark current. The general solution of (4.9) for the point source j_ = (5 oo, g2N = const, only the planar diagrams without the quark loops should be taken into account. All the lines of gluons must be between the quark lines and cannot cross each other (self-action is absent). All the diagrams of higher orders are the ladder diagrams. «

a

-i

Fig. 5

-/'

325

Two-Dimensional Field Theories

4.3. Charge confinement in the QCD-2 Let us consider a total sum of quark self-energy diagrams in the limit IV oo, g2N = const. (Fig. 6). The quark propagator S(p) in this approximation satisfies the integral equation d2k

—

/

1

0(|fc-| - A) y-S(p + k)y_—.

(4.11)

Here, because of bad behaviour of the gluon propagator in zero (on the light cone) we introduce the infrared cutoff Si A proposed by 't Hooft. All the Green functions of the gauge invariant operators turn out to be independent of A with A 0, since they are free of the infrared divergence.

Fig. 6

The solution of the eq. (4.11), obtained b y ' t Hooft, has the form ) =

/ „2 A7\ / I^T

V

le

•

( 4 -!2)

\

Now, approaching A -s- 0, we obtain, that the pole of the "dressed" quark propagator is shifted to infinity that corresponds to the infinite self-energy of a quark. Thus, quarks are eliminated from the spectrum of physical states of the model. It is convenient to rewrite eq. (4.12) as

2p+p-

-

where =

M*

2

-

92N I p.

71

A

q2N 71

(4.14)

4.4. Meson states spectrum in QCD-2 To determine the structure of bound states spectrum for quark and antiquark in the QCD-2 model 't Hooft solved the homogeneous Bethe-Salpeter equation for quarkantiquark scattering. This equation in 't Hooft notation is graphically shown in Fig. 7. Here the arbitrary vertex (shaded "circle) from where exit quark a with mass m l and momentum p and antiquark b with mass m2 and momentum r — p, is given by some function y>„{p, r) which satisfies the following homogeneous equation determining the

326

A . V . BEBKOV, Y U . P . NIKITIN, I . L . ROSENXAL

two-particle states spectrum: , Wn(p,

x

r)

=

éig2N — J ^ f

2p+p_

, (P-

-

-

r

-)

V-

2(p+

q2N M S - 2 — HA

-

r+)

(p_

\p_\ +

-

r.)

-

M

V>n(p

ie "

I

*

-

+

h

^

r)

k j

f

-

IJJL -

dk+dk_.

r_| +

ie

(4.15)

Index n denotes that y>n{p, r) is the vertex function corresponding to the observation of quark and antiquark in some bound state n on the mass shell.

p+k-r Fig. 7

If we determine the function 9>n(s)

=

/

Vn(P,

r)

(4.16)

dp+

where x =

(4.17)

P-jr_

varies from 0 to 1 and has the meaning of the portion of momentum carried away by the quark in the left-moving frame on the light cone, then one can show that = 0 .

(5.10)

Though the coupling constant enters the Lagrangian £ so that the classical equation of motion (5.10) does not contain it, this constant appears in final results after quanti-

Two-Dimensional Field Theories

331

zation since it is contained in the canonical Poisson brackets {