On the dynamics of tumbling gauge theories

Volume 123B, number 6 PHYSICS LETTERS 14 April 1983

ON THE DYNAMICS OF TUMBLING GAUGE THEORIES

V.P. GUSYNIN, V.A. MIRANSKY and Yu. A. SITENKO Institute for Theoretical Physics, 252130 Kiev-130, USSR

Received 25 June 1982

The dynamics of symmetry breaking in gauge theories without fundamental scalars is investigated. The method of determining the way of the vacuum rearrangement in these theories is elaborated.

At present much interest is inspired by tumbling gauge theories [1 ]. The problem central in these theories is connected with the spontaneous breaking of the initial symmetry due to the binding of fermions and the formation of vacuum condensates. Since the dynamics o f the condensate formation appears to be rather complicated, the development of a qualitative picture o f this phenomenon is desirable. The first step in this direction has been undertaken in ref. [2] where the criterion o f the most attractive channel (MAC) is suggested. The MAC criterion leads to some interesting predictions and promises a way of solving the gauge hierarchy problem. At the same time a sequence o f crucial questions concerning the dynamics o f condensa- tion is left unanswered. Among them: what is the con- nection of this dynamics with confinement [3] ;why, in some cases, is the MAC criterion unable to determine unambiguously the way of symmetry breaking [4] ?

The purpose of the present letter is to make a further step to get insight into the dynamics of tumbling. We consider the dynamical mechanism of the condensate formation which is a generalization of the previ- ously proposed mechanism of the dynamical chiral symmetry breakdown in QCD [5,6]. This mechanism starts from the hypothesis [7,8] about the dominating role of the supercritical Coulomb-like forces for the generation of fermion masses in gauge theories. Actual- ly this hypothesis appears to be a special case of the MAC criterion for vector-like gauge theories (as QCD is) and the model [5,6] provides a dynamical realization o f it. Therefore it seems reasonable to extend the model to the general case of asymptotically free tumbling gauge theories.

The resolution of the problem of dynamical symmetry breaking can be achieved in two stages. In the first place, one must exhibit the instability o f the symmetric phase, i.e. the existence o f tachyons in it. Sec- ondly, one must find the stable phase in which all tachyons disappear, in particular, those with quantum numbers o f the spontaneously broken symmetry generators have to be transformed into Goldstone bosons. Tachyons and goldstonions are bound states of fermions with large squared mass defect D = 4m 2 - M 2 1> 4m 2 (where m is the fermion mass, M is the bound state mass; M 2 < 0 for tachyons and M 2 = 0 for goldstonions; m = 0 and m > 0 in the symmetric and symmetry broken phases, respectively). At the same time the forces increasing with distance which are usually associated with the confinement mechanism can only decrease D, i.e, they act against the condensate formation. Therefore the size of tachyon and Goldstone bound states should be less than the distance at which the confining forces domi- nate .1 . Thus we are led to the assumption that the dynamics forming such states is caused by the supercritical Coulomb-like forces acting at the middle distances.

A soluble model realizing our assumptions consists of the following. The dynamics o f the fermion binding is described by the Bethe-Salpeter (BS) equations. Ac- cording to the hypothesis we introduce cut-off parameters in momentum integrals of BS equations in order to pick out the ranges o f momenta which are responsi- ble for binding. The kernels of BS equations are chosen

,1 Note that a goldstonion has a size of order m -1 [6]. In particular, a pion (an approximate goldstonion of the chiral symmetry) has a size of order (350 MeV) -1, while the QCD confining forces are essential at distances r > (150 MeV) -1.

407 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland


to be in the ladder approximation with the values of coupling constantg equal to the mean values of run- ning coupling ~(q2) in the appropriate momentum ranges. The ultraviolet cut off A{R ) is determined for each channel of binding of two fermions {N} × {N'}

(R} ({N}, {N'} and (R} are the gauge group representations of constituents and a composite, respectively) through the following relation

c{R~(N, ) g2(A~R})/4~" ~ l, (1)

where the value C{(NR}){N, } [2] is 1 (C(N}+ C(N, ) - C{R}), C{N ~ is ~he quadratic Casimir invariant of the {N} representation. The consistency of our approximation scheme is ascertained by the fact that BS equations in the ladder approximation have the solu- tions for bound states with D >~ 4m 2 if C({NR}}(N,- ~ g2/ 47r > rr/3 only (see refs. [5,6] and below):Th~ irifrared cut o f f g which is common for all binding channels is identified with the confinement scale. Special attention has to be paid to the gauge choice since the approximation used should not have to break explicitly the symmetries investigated (i.e. the validity of the Ward identities must be sustained). Our approximation is self-consistent in this sense if we choose a Landau (transverse) gauge: the corresponding Ward identities are valid there (see below).

It can be shown that in the symmetric phase the BS equation of our dynamical model coincides with the BS equation for a composite meson in the chiral symmetric phase of QCD (with massless quarks assigned to (3} o f the colour SU(3) group) [6] if one substitutes

£R} 2 4 2 -- ~l) 2 C~ ' g R / 4 r r f o r ~ g ]4~r=C . g /47r. , ,v){~, ). { .T {3:~{3 } 1his equatton has a tachyonic solution at c{~)x{~,~ g}R~/4~ > a c = ~r/3 [ 6 ] L e t us a s s u m e n o w

rife uriive'rsal n'ature of the d.yn.amics of binding, i.e. assume that the quantity C(tNR))(N, ~ g2R~/41r (which is the mean value of C ({NR)){N,~g~qf)[4~ ~t t~ 2 < q2 < A~R1) is equal to a comn~on constant tx > a e for all channe'~s of binding with A~R } > /a 2 (ct is a free param- eter of our model). Then the channels are different by the scale parameters A {R} only. In particular, the quo- tient of squared masses o f tachyons from different

2 2 _ 2 2 channels is equal to M{R}/M(R,, } - A{R}/A{R,). Each tachyon determines its own way of rearrangment o t the symmetric phase. The probability o f rearrangement increases with increase of IM2R}h i.e. with the increase of C{{NR}}{N, } (this is, in factt, reflected by the

MAC criterion). However, the knowledge of tachyon spectrum does not permit to determine the residual symmetry of the stable vacuum. For this purpose one must consider the dynamical equations for bound states in the phases with different spontaneous symmetry breaking condensates. In this way we obtain not only the dynamical realization o f the MAC criterion but also a method of determining the stable vacuum and its symmetry in cases when this criterion is ambig- uous.

The bound state equations in a phase with spontaneous symmetry breaking can be obtained using the effective action method [9]. In the present case this method is reduced to the following: the mass terms which break symmetry in any possible way are introduced in the initial lagrangian. The mass parameters which are interpreted as the mean values of the fermion mass functions at ~t 2 < q 2 <A~R } are determined from the condition of the exister~ce" of goldstonions corresponding to the spontaneously broken symmetry generators. The stabilizing role of a fermion mass is connected with the general fact that, for a given dynamics o f interaction, the squared mass M 2 of a composite increases with the increasing masses m (i) of its constituents. At some critical value m~ i) a tachyon bound state (M 2 < 0) is transformed into a Goldstone one (342 = 0). At m (i) > m~ i) it would become a massive bound state with M 2 > 0 *2. In the stable phase some tachyons become goldstonions while the other (playing here the role of Higgs par- ticles) acquire a real mass. Evidently, an arbitrary way of spontaneous symmetry breaking does not necessar- ily lead to the stable phase, i.e. in some cases the mag- nitude o f a fermion mass can turn out to be insufficient for the elimination of all tachyons. Thus the problem consists in finding a phase in which the squared masses of all composites are nonnegat!ve (M 2 ~> 0) ,3. How-

,2 This can be illustrated by the mass formula for pseudo- scalar mesons in QCD [6]: Mb = 2(m(i)m (f) - m2), where M~/. is the mass of a meson consisting of ith quark and/th antNuark with the masses rn0) and mq)whi le m is the dynamical (critical) mass of a quark.

,3 The absence of tachyons guarantees that a vacuum corre- sponds to a local minimum of the effective potential. In principle, several local minima could possibly exist. In this case the tunneling processes have to be taken into account and the true vacuum is determined by the absolute minimum. This issue requires a special discussion and will not be touched here.

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ever a considerable difficulty is encountered along this way: the BS equations for massive bound states composed of massive fermions are so complicated that at present it seems impossible to solve them in the general case. To bypass this obstacle, we shall perform the following trick. For a given way of spontaneous symmetry breaking (i.e. for given values of the fermion dynamical masses m(i)) let us define the critical masses m (i) for a bound state In) as the values of the constitu-

c n

ent masses at which this state would be massless. There upon, if these critical masses are not larger than the fermion dynamical masses m (i), then (due to the fact that M 2 increases with increasing of the constituent masses) the squared mass of this bound state is non- negative, otherwise this state is tachyonic. Thus we re- duce the problem to the determination of critical masses for different bound states, i.e. to the analysis of BS equations for massless composites which are con- siderably simpler than BS equations for massive composites.

Let us apply now our method to some examples. The known example [4] o f the MAC criterion ambigu- ity is the SU(3) gauge theory with left fermions ~bab = q~ba (a, b = 1,2, 3) in the sextet representation of SU(3) *4. The only attractive channel compatible with Fermi statistics and Lorentz invariance is {6} X {6} -+ {6"} ~ 6ab ( x , y ) = eacee bdg ~cd(X) ~T (y) C (here C is the charge conjugation matrix, T is the sign of the

. . . . . . - . ~6" ' s of Dlrac Indices) with C(6}'}a },~ =-$. In the transposition symmetric vacuum there exist twelve taclhyons {6} {6}. Possible symmetry breaking patterns determined by the nonequivalent orientations of o ab condensate are: SU(3) ~ SU(2), SU(3) -+ SO(3), SU(3) ~ 0(2). The MAC criterion cannot find which of the possibili- ties is realized. We shall show that the stable vacuum is SO(3) symmetric.

Consider first the phase with the SU(2) residual symmetry. The effective lagrangian is:

/geff= ~ab (~c~d .a + a;{c.d + ".c ~d. t a b 10 g aOb gg~ b a)~C d

1 , 3ac 3bd-T e ~abc~Tcd~" -- '~mte e ~abC~cd+e3ac 3bd J"

(2) ,4 In this model the Adler-BeU-Jackiw anomalies are present,

which are not, however, revealed in our approximation. The anomaly free modification (which consists in addition of seven left fermion antitriplets) has been also considered by us (for results see below). We start with this toy model to demonstrate our approach in the most simple way.

Since the mass term is of the Majorana type, then the Majorana bispinor field has to be introduced

~bct # = I~a/3 + e3ao, e3# 8 ct~T~/8 , ct,fl,')',t~ = 1,2. (3)

The wave functions of bound states corresponding to the twelve tachyons of the initial symmetric phase are

~p = F [(01Tq~u# ~t~lP)] (q), (4)

6~ ' r /~ = F [(0l T 4 ~ t~ 313IP, c~')] (q), (5)

6 ~ ' p~ = F [(0l Te3at3 ~03~ q~tle; a ' )] (q), (6)

tg# = .q(1) ,o.(2) (7) "a~ -- va# '

where

6~;~ '~ ' 0 (1)~ = F [(0[ T~t~(~33 + ~:3 C) IP; ~'fl')] (q),

1 c (~T37 t~3ti +~T36 1~33')) +~" e3a3, e3# 6

x le; a'Y)] (q);

here P and q are the total and relative momenta of constituents, respectively;F is the sign of the Fourier transform. The general structure of a wave function of a massless bound state (Pu = 0) is (for euclidean momenta)

f ( q , P = 0) = fS(q2) + ~.5fP(q2) + i•fV(q2)

+ i(l 'YsfA(q2). (8)

In the framework of our dynamical model the BS equations following from Z? eff (2) are (men are the critical masses)

(q2 + m21) S _r/P = 9 S _r/P) ,

7..s +,,'>)t +.'>)<,j I ~ = o ,

( ~ v _ ~ A ) : _- _ (mcllq 2) (~S _ ~ P ) ~ ,

(pV _ p A ) a = _ (mc l /q2 ) (pS + pP)e, (9)

(q2 + m22)2 ~ oS = 9{~oS}, ~ oV = 2mc2 ~o s , (10) - q2 m22 m z ,,--~2

c2 - "~

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(q2 +_~ 2 S+ P + , m23(~S + ~P)~# mc3)(O _O )~# 5

= 9 ((0 S + OP)o~},

(o v ± oA)~I~) = - (mcdq2)(O s + OP)~,

_ 0 ,av =0, (11)

where

~ . 1 f d4q ' 9{f(q2)} = ac (2zr)2J (q - q')2 f(q,2),

~6.} ~2(A2) ... l; i.t 2 < q ' Z < A 2 , C {6} z[6 }

aa# = o(al; + 2 0(2).

To obtain eqs. (9) - (11) we have used the identity

f d4 q ' 7U ~l'Tv ffq'2)c'D £ (q - q ' ) = O,

tr t~v(k) = k -2 (6uo kt~kv/k2); (12)

note also that (rl S + r/P)a = (~/V + r/g)a = (pS _ PP)a = (0V+ oA)a = 0 due to the definition (5) and (6).

An equation of type (9) has been analyzed in ref. [5]. At a > % it has a nontrivial solution with m21 = A2Z(ct), where Z(a) is the positive definite function (Z(ct) = 16 exp[-zr(a/a c - 1) -1/2 ] at (a - % ) / % ,~ 1). Thus eqs. (9) describe five Goldstone bound states corresponding to the spontaneous symmetry breaking SU(3) ~ SU(2) and the fermion triplet q~# acquires the dynamical mass m = mcl as a result of this breaking *s. Now we have to compare the values of other critical masses mc2 and mc3 with that of re. An equa-

*s The importance o f the use o f Landau gauge in our approximation is due to relation (12) which ensures the validity of the Ward identities. In other covariant gauges with

long 4 c1) = dekt.tkv/k 4= 0 this approximat ion violates the /.to . . . . . . . . . Ward ~dentmes and ~t ~s Impossible to obtain the consistent system of equat ions for goldstonions. We note also that in this approximat ion the interaction between colour gold- s tonions and gluons which generates the mass tZg for five gluons is no t taken into account. However, since the order o f magni tude of this mass does not exceed the inverse size o f goldstonions (gg < m), one may expect that the disre- gard of~tg will no t affect essentially the dynamics of bind- lng.

tion of type (10)has been analyzed in ref. [10] in con- nection with the investigation of the spectrum of scalar mesons in QCD. There we show that 3.6 m22 < m21 . Thus eq. (10) defines a massive bound state. Eqs. (1 1) couple the wave function Oa# with the wave function

- = O (1) + 2 0 (2) corresponding to the repulsive chan- c~lJ t~/J /J t r J c , l

nel {6} X {6} ~ (15 } with C(~1~} = - 4/3. Therefore, the additional equations appear

(_2.q -r~l m23) (~S +oP)¢~# + 32 '"c3~2 i,~,r'qS _+ oP)a#

' 9 ( 0 s + ~P)~} . (13)

The analysis of (11) and (13) (details will be published 2 elsewhere) yields me3 > m 2. Thus eqs. (11) define six

tachyon bound states and, consequently, the phase with the SU(2) residual symmetry appears to be un- stable.

Let us consider now the spontaneous symmetry breaking SU(3) ~ SO(3). The SU(3) s e x t e t t~ab is de- composed into the SO(3) quintet and singlet

1

where

, - s g ~ed,

s c d _ I c d c d 1 ~cd. ab -'~ (8afb ÷ 668a) --~ 8ab

The effective lagrangian takes the form

..~ - ab c d .^ ^ c d + ~ . . ~ c s d x = ~ ( t S a ~ b l ~ + g . 9 ~ a 5 b , 5y t b a)~bed

x m(1) txac 8 bd a,T (5}t-,i, {5} -- -i ~" Wab '-, wed

- ab - Tcd + ~)ac~bd ~{5)C~(5} )

' m (2) ~ c ~ T ) . (14)

We introduce the Majorana bispinor fields corresponding to the mass terms in (14)

5}+ -Ted + C~ T. (15) ,v,, b = q,~ ~ S o d C q ' (s } , "~ = q,

The wave functions of the bound states in terms of these bispinor fields are

X = F [(0l TxI'aO Cpablp)] (q),

K = F [(01T~ff [e>] (q),

8 ab'a'b' ~Oa3 = F [(01T SaC~ 8dg~ec ~eg Ie; a'b'>] (q),

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8ab;a'b'ta b = F [<01T~ab fly Ie; a'b')] (q),

8 ab;a'b' "~ = F [<01Tfac 8bd ~ c d IP; a'b')] (q). (16)

The analysis of the BS equations ~ r these functions shows that m (1) = m (2) and tab -- tab = 12Wab + 7tab + 7~ab = 6X + 5t¢ = 0. The equations for the nontrivial combinations of the wave functions take the form

3X P - 2t¢ P

(q2 + m21)( (6mp ~p "~p ) - ) , , b

{( 3 x P - 2xP )}

= 9 (6¢0P-- ~P-- ~P)ab

3X A - 2t¢ A = 0, (6mA _~A "~A)a b = 0,

(q2 + rn22)2 [ 3X S _ 2r S ~"

q - mc2

(17)

9 !~ ( 6 ~ s _ ~s _~s)a b

2mc 2 3× v _ 2K v - (3xs - 2Ks),

m 2 _q2 c2

(6mV_~V_'~V)a b _ 2mc2 (6mS_~S_'~S)ab. m 2 _q2

c2 (18)

As is clear from the foregoing argument, eqs. (17) define six Goldstone bound states ,6 and eqs. (18) define six massive bound states. No tachyons appear, hence the phase with the residual SO(3) symmetry is stable. Since the O(2) group is the SO(3) subgroup, then the further symmetry lowering [SO(3) + 0(2)] does not take place.

#6 Since the UL(1) anomaly is not taken into account, there exists the global UL(1 ) symmetry in the SU(3) symmetric phase. Under the symmetry breaking SU(3) ~ SU(2) the UL(1) symmetry is substituted by the exact U~(1) symmetry with generator B - 2/x/~ ha and the number of goldstonions (five) is equal to the number of generators SU(3)/ SU(2). Under the breaking SU(3)~ SO(3) the UL(I) symmetry is also broken and an additional (sixth) goldstonion appears.

This example demonstrates clearly that a phase with broken symmetry gets more stable not only with increase of fermion dynamical mass (that is reflected by the MAC criterion) but also with increase of the number of massive fermions permitted by the residual symmetry: in the phase with the SO(3) symmetry all fermions acquire a mass, while in the phase with the SU(2) symmetry a mass is acquired by fermions from the triplet only. This conclusion seems to be rather general and could prove to be efficient for more realis- tic tumbling gauge models.

We have also considered some other examples: (a) The anomaly free SU(3) gauge theory with left

fermions in

7 A

{6) + {3") +.. .+ {3");

there are channels of binding (6) X (6} --* (6*), (6) X (3") --> {3} with equal A. The results are that the fermion sextet is combined with some of the fermion antitriplets to acquire the Dirac-type mass and the other fermions remain massless; the global U(1) X SU(7) symmetry is lowered to U(1) X SU(4) and the gauge SU(3) symmetry is substituted by the SU'(3) symmetry which is the diagonal subgroup of SU(3) X SUgl(3 ) [SUgl(3 ) is a subgroup of the global SU(7)],

(b) The SU(5) gauge theory with left fermions in (10) + (5*); there are two channels of binding {10} X {10} -+ {5") and (10} X (5") -+ {5} with different A. The results agree with those of the MAC criterion [2].

(c) QCD with the chiral SUL(K ) X SUR(K ) symmetry. In this case the formation of the chiral condensate ({3} X (3 '} -+ (1)) suppresses the formation of the colour condensates ((3) X (3) -+ (3"}) (see ref. [101).

For theories without fundamental scalars the approximation used by us plays apparently the role of the tree approximation in theories with fundamental Higgs fields. To what extent can the improvement of this approximation affect the final results? In chiral gauge theories the tachyons of the symmetric phase are in nonsinglet representations of the gauge group and therefore the confinement forces could play some role for the determination of their fate. In this connec- tion, note that (providing that some conditions are satisfied) "complementarity" between the confinement

411


and Higgs phases has been elaborated in ref. [ 11 ]. Also, the tumbling dynamics has been found to be realised in some two-dimensional models [12]. All this indi- cates that the qualitative aspect of the tumbling scheme is rather reasonable. This, in turn, allows us to hope that the present approach will prove to be useful for further investigation of theories without fundamental scalars.

We should like to thank A.I. Akhiezer, A.A. Anselm, B.A. Arbuzov, D.I. Dyakonov, A.T. Filippov, P.I. Fomin, I.V. Krive, B.V. Struminsky and D.V. Volkov for fruitful discussions.

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On the dynamics of tumbling gauge theories