Renormalization group and dynamical symmetry breakdown in abelian gauge theories

Nuclear Physics B109 (1976) 526-546 © North-ttolland Publishing Company

RENORMALIZATION GROUP AND DYNAMICAL SYMMETRY

BREAKDOWN IN ABELIAN GAUGE THEORIES V.P. G U S Y N I N and V.A. M I R A N S K Y Institute for Theoretical Physics, Ukrainian Academy of Sciences, USSR Received 17 September 1975

Generalized renormalization group equations are used to analyze the dynamical mechanism of particle mass generation in terms of the Cornwall-Norton model both with and without cut-off. We look for solutions which contain non-zero physical masses of the two fermions (m 1, m2) and of one of the vector bosons (~) when the bare masses m 1A, m2A, ~A approach zero. For a theory without cut-off we obtain results which are similar to those of Cornwall. For a theory with cut-off the mass generation mechanism may only oc cur when a bare coupling constant c~A of the A~z vector boson, which remains massless, exceeds some critical value c~c. In this case the fermion masses turn out to be of the superconductivity type. The model's "memory" of the nature of spontaneous symmetry breaking limml A, m2A-,0 mlA/m2A~l is an indispensible factor for the vector boson to acquire a mass.

1. I n t r o d u c t i o n The dynamical nature o f particle mass generation which is analogous to the mechanism o f the energy gap generation in a s u p e r c o n d u c t o r was suggested by N a m b u and Jona-Lasinio [1 ]. Since then these ideas have been widely spread. Lately, the problem o f mass generation for vector mesons has been of great i m p o r t a n c e in the light o f recent progress in gauge theories o f weak, electromagnetic and strong interactions. In these theories a vacuum polarization tensor Iluv(p ) acquires a pole at p2 = 0, which constitutes the basis for the appearance o f massive vector mesons. This possibility was first shown by Schwinger [2]. A pole at p2 = 0 also appears in the polarization tensor via the Higgs m e c h a n i s m [3]. However, the t h e o r y would then involve a certain n u m b e r o f scalar fields which is, generally speaking, undesirable f r o m the physical point of view. For a series o f renormalizable models in the field t h e o r y , the Schwinger m e c h a n i s m has been recently investigated in a pure f o r m w i t h o u t introducing scalar fields in refs. [ 4 - 7 ] . When studying problems o f the dynamical mass generation, two trends should be distinguished. The first trend assumes that the renormalizable field theories are self526

V.P. Gusynin, V.A. Miranskv /Dynamical symmetry breakdown

527

consistent, i.e. they lead to finite expressions for the S-matrix elements as well as for the particle mass spectrum when cut-off (A ~ oo) is removed. The Johnson-BakerWiley-Adler (JBWA) program for the finite quantum electrodynamics refers Io this trend [8]. Accodring to the second trend such theories are not self-consistenl. This requires some cutoff A to be introduced. In this case particle masses can depend on A; for example, for a superconductivity-type solution we have m ~ A exp[--,:/g 2] [1,9,10] where g is a renormalized coupling constant, c is a number. The papers [6,7], in which the Schwinger mechanism is studied in terms of Abelian gauge field theories, refer to the first trend. Our treatment is an anal ~sis (initiated in ref. [10]) of the problem of the dynamical mass generation b y means of modified renormalization group differential equations generalizing the BogoliubovShirkov equations [11 ]. We investigate the Abelian gauge theories [6] both with and without cut-off. When a cut-off is absent we obtain very easily results which are similar to those of ref. [7], but which are more exact. In the theories with cut-off one of the vector bosons also acquires a mass owing to a pole in the polarization :ensor. A new result is that such a mechanism can hold only for the case when one of the bare couplings constants exceeds some critical value (a n ~ c% > 0). This agrees with the original Schwinger hypothesis, the difference being that aA is not a coupling constant of the boson Bu acquiring mass, but that of the boson A u which remains massless. In addition, unlike the case without cut-off (when the physical mass is an arbitrary parameter), in the theory with cut-off, masses turn out to be of the suFerconductivity type. We finally note that the model must possess a " m e m o r y " of the nature of spontaneous s y m m e t r y breaking for the Schwinger mechanism to exist. In other words, for two fermion bare masses, m l A and m2A , we have limmlA, m2A~ 0 mlA/t~!2A 4: 1. In addition, if in the model with cut-off liram ~A, m 2A~ 0 m 1A/m 2A > 0, ther a less physical condition limml~,m2oo_~ 0 r n l ~ / m 2 o ° = - 1 takes place in the model[ without cut -off. The plan of our paper is as follows: in sect. 2 we briefly summarize the main properties of the models [6], in sect. 3 mass equations are analyzed, in sects. 4 and 5 these equations are solved in a certain approximation for theories with and ~ i t h o u t cut-off. In conclusion, the results are discussed and the present approach is compared with the conventional generalized Hartee-Fock method [1,6,9] and wil h the Callan-Symanzik equations [7].

2. Cornwall-Norton model We shall now consider the Cornwall-Norton model [6]. The s a ~ ; :,'-Johnson model [6] without 3'5 anomalies is equivalent to the former. The model kagrangian is of the form .m : ~(,,,,,,'a,., - too)C, - ~G~ m-,,' - ¼ c,,,c,-'"

528

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown +gO~ru$A"

(1)

+gO~7,27.$B",

Fur = 3~A v - O.A u ,

Guy = OuB v - 3 . B u • 1

Here ~, is the eight-component spinor field written as ~ = ( ; z ) ; A . and B u are vector fields; 72 is the Pauli matrix 7-2 = -

0

The Lagrangian is invariant under the local gauge transformations -+ eic4x)~ ,

A u -+ A u - l o

-~ e ir2°(x) ~ ,

~uOz(x) ,

B u -+ B u - ~

~uO(x) .

(2a)

(2b)

gB

This invariance corresponds to the conservation of currents i~"(~vu~) = 0 ,

~"(~7,2%~ ) = 0 ,

which leads to the Ward-Takahashi identities k ~ ' p A ( p , p + k) = G - l ( p + k) - G - l ( p ) ,

(3)

k " P ~ u ( p , p + k) = 7 , 2 G - l ( p + k) - G - l ( p ) 7 , 2 ,

(4)

where G ( p ) is a two-dimensional (in the 7,-space) fermion propagator, PuA and PuB are vertex (two-dimensional) functions. Owing to the invariance under transformations (2) with constant phases and under discrete transformations -~ 7-1 t) ,

B u -+ - B u ,

qJ-+ ir,3~ ,

Bu~-B

u ,

Au ~ Au , Au--" A u ,

the functions G, pA, IF,B u and ~ take the form G(p) = G o ( P ) l ,

I"uA ( p , p + k ) = I ' .ao ( p , p + k ) I ,

lpB(p, p + k) = PB2(p, p + k) 7-2,

2;(p) = Y~o(P)I,

(5)

where G-l(p)=p

m0-2;(p).

The Green function structure (5) can be obtained if the vacuum is assumed to be invariant under transformations (2b) with a constant phase. If the vacuum is not invariant (spontaneous symmetry breaking), then the terms violating formal Lagran-

KP. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

529

gian symmetry can emerge inthe Green functions. In particular, N(p) = N o ( P ) l + 523(P)r 3 .

The Ward identity (4) yields kupB(p,P

+k)[k= 0 = It2, r3]Z3(P) = 2 i r l £ 3 ( P ) ,

i.e. FB(p, p + k) has a pole F u ( p , p + k) -__£~ZrlFl k" " ~

+k) + ... , r ~ ( p , p ) = 22;3(p) .

Then the Schwinger-Dyson equation

HBv(k)

-

gOB2fd4p T r [ p B ( p , p

(2rr)4i

+ k ) G ( p + k)'),vr2G(P)]

implies that the polarization tensor II~ has a pole as well. As a result the ve :tot boson Bu acquires a mass [2,6]. Our aim is to find a fermion mass splitting and the vector particle mass. F'or obtaining mass equations the method of modified renormalization group equalions will be applied [10].

3. Mass equations We express Lagrangian (1) in terms of fields ~(I'2),A~,B~ for which the, Green functions are normalized at the point X [10] ./2= ~il)(iTuOu - mx)ff(1) + ~i2)(iTuOu - mx)t~(h2) 1F

p~t~

lg'~,

/2, btv q..oh,7"~,

+,XTX ,T,(2)~ ~(2)A~ ~SA'-'lAV'X ~u

w(1)~

d,(1)AU

. x - a . 7(1) ~(a2)B~ /gBZ1B~. 7u

+ •;6B~lBWh , a T x 7(2)~t~wh ,t,(1)~u ~X + counter terms

(6)

The Green functions will be determined as follows Ok(p) -

s j(p2)

p - mxs~/(p2 ) '

D(A, B)t ~ ( Xuv r e , = - g~v

j = 1,2,

PuPV]D(A, PuPv ~ ] ~ B ) ( p 2 ) _ d l P--~--'

(7)

(8)

530

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown D(xA,B)(p2 ) _ d~,, B(p2) _ 1 p2 p2(1 _ rr~,B(p2 ) '

(9)

with the normalization conditions

@.(p2 = _)~2)= 1,

i,j = 1 , 2 , (10)

d~,B(p2 =--X 2) = 1 .

where )t2 > 0, d l is a gauge parameter. x These conditions enable us to describe the theory by different setsgA, B and m x when the parameter X is varied. Such an extension of the renormalization group is necessary for the determination of the physical mass m. When sufficiently large values for X are chosen (X = A), it can be given the same meaning as a cutoff para;t=A , gBk=A play the role of bare masses and meter A [1 0]. The parameters rex=a, gA charges, respectively. We put in the following X = A. To find the spontaneous breakdown of the gauge symmetry (2b), we introduce the different bare fermion masses mlA , m2a and the boson mass/JA 4 : 0 in (6). Then Lagrangian 22 transforms into an asymmetric Lagrangian, .t2N = 22 + A.67. Further, after solving equations for physical masses m 1, m2,/J, we shall assume m l i , m2A,/l A to be zero, thus reconstructing the symmetry (2b). The existence of the solution m 1 4= m 2, tt 4= 0 in this limit manifests the spontaneous s y m m e t r y breaking. The absence of gauge invariance at the first stage, which is due to Ward identity breakdown, makes the Lagrangian 22N, generally speaking, unrenormalizable. To avoid this difficulty, we shall use the fact that not only Lagrangian 22N but many other Lagrangians, ~Oeff = 22+ A22eff, with 6m A =- g1 ( m l a -- m2n ) , / l a 4= 0 can be constructed, which coincide with 22 when 6rn A -- bta = 0. It follows from the discussion in sect. 2 that it is sufficient for our purposes that the term A22eff violates invariance under transformation (2b) only with a phase 0 = const. Such a Lagrangian was constructed in refs. [7]. The}e A22eff violates invariance under (2b) with a phase O(x) satisfying the equation DO(x) = 0 (for instance, with 0 = const), but it conserves invariance for [~O(x) ~ O. This turns out to be sufficient to provide renormalizability of L e ft . Moreover, in the transverse gauge (d l = 0) this 22eft coincides with 22N = ~O+ A22. In the rest of this paper we use for convenience only this gauge. The introduction of/~a and m l A 4= m2A into the Lagrangian changes the structure of Green functions G~, D(BA). Instead of (7) and (9) we have

G]A(p) -

s~/'(P 2) A

(11)

2 '

p - mjAs2/(p ) =

D(AB)(p2)

1

p2 _ ~2 _ i i ~ ( p 2 )

_

I

p2 _ ~2 _ p2~r~(p2 ) _ ~r~(p2)

,

(12)

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

531

with the normalization condifi-ons

sOl.(p2 = - A 2) = 1 ,

rr~/(p 2 = - A 2) = 0 .

(1 3)

Renormalization group equations for the Lagrangian ~ N are derived analogously to that of ref. [10]. The derivation of renormalization group equations is given in the appendix. Here we only make some remarks. The matrix structure G q ( p ) enables equations to be written for the funclions S~X/.(p2) and s~/(p2), each separately

4 = z2;(t) 4.

A,2

t= A2'

,

mia4" For function D A(B) we have DA(B)

=

Z3A(B)(t)DAA(, B) .

Let us rewrite DBA in the form d ~ ( p 2)

DB(p2) -

p2 _ 0 2 h a ( p 2 ) '

(12')

where by definition

d E = Z3B(t)d~'. Hence,

o hA -- A, hA As was proved in ref. [7], divergences in the asymmetric theory can be removed by constants Z from the symmetric one. In what follows we use for convenience the functions d E and Z3B from the symmetric theory, i.e. (see eqs. (9), (12)) dE -

1 1-7r~'

ha - 1 02+7r~ 021 -Tr~

A dA(B), A h A depend on the following arguments The functions s2i,

s ~ - s 2 / ( x , y , z, u,

%, &),

d~ ~ dA(X, y, Z, U, aA, flA)' dE - d B ( X , y ' a A, 13A), h A ~ h(x, y, z, u, a A, ~A) ,

(14)

532

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

where we introduce the notations p2 ----=x, A2

m2A A2

6mA =u ,

ratA

-y,

-z,

m2A

gA tXA - 47r '

13A - 4zr "

These functions satisfy the renormalization group equations (A.14)-(A. 16) (see appendix). Physical masses m], tl are defined by the poles of the Green functions G~(p), DBAuv(p), respectively. Therefore,

m2 m] =mjAS2j(--~,y,z,u,O~A,13A)

/a2_- taAn~--2 . [ ~/-12 , y , z , u ,

~A,CJA

]

,

I

As shown in the appendix, these equations may be considerably simplified. For this task we introduce functions sa/, dah,B , h a independent of argument y. These functions by definition, satisfy the renormalization group equations (A.20)-(A.22) with boundary conditions (A.26). A definition of this kind provides the coincidence of a dA,B, a h a and si], A d AA,B, hA, respecthe asymptotical expansions for functions si/, tively, in the range A 2, [p2[ > > m 2,/a2. If the renormalized charges a and 13are assumed to be small, then the following mass equations take place (see appendix)

m/

=m/A sa2/'~-A--~ , z, U, a A, 13A) 2 a /-t2

/-t2 =/IAh

(U,

,

Z, b/, 0~A,/~A) •

(15)

These equations will be solved in the next sections.

4. Theories without cut-off In this section we shall study the Cornwall-Norton model in the framework of finite theory [6,7]. This theory assumes the existence of zeros of the Gell-Mann-Low functions ~PA and ~>B(see eq. (A.23) at points a ~ and/300. Therefore, when the cutoff (A -+ oo) is removed, the charges a A- and 13A approach the values a ~ and/3~, re-

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

533

spectively. Moreover, for the-problem in which we are interested, we may recognize with a sufficient accuracy thatdA(B) -= 1 [8]. In the lowest order of perturbation theory we obtain for ~ j and H a (see eqs. (A.24), (a.2 5)) 3

~ 2 ( z , u, o~,/3~) = - ~

Ha(z, u, a~, ~ )

3

4~

= -

(a~ +/3~z) ,

(17)

u.

(18)

7f

The calculation of the functions qfl2] is similar to that in symmetric theory. In order to find the H a function, a polarization operator of vector particle B u was ccmputed in the lowest approximation

1

I I ~ ( p 2) = 213--~Af dv 7r 0

In~ p2v(1 - v) - m 2AU - rn2A(l - v) 1 LA2v(1

u)

- - ml2av - m2A(1

U-)

X [2p2v(1 - v) +mlAm2A - m 2 a v - m2A(1 -- V)]. Then we singled out the function h A (see eq. (14) and assumed d A -= 1. The result is 4~

ha ~-- l - - - u

lnx.

7r

Using this equation and (A.25) we obtain expression (18) for H a. Eqs. (A.21), (A.22) take the form

1 3 Io~ +/3~z l s ~ 2 1

~lns~l_ OX

X 4zr

a lns~2 _ 3x

1 3 +/3 z-~-21 x 37r a= S22A ,

Oha Ox -

~

(19)

s~ 1 _] '

[

1 4~u 1 x 7r ( z - l )

sa 1

(20)

2 [ z s ~ l - s~212'

(21)

In order to find S~l and s~2, it is convenient to proceed to the equation for the funct i o n f = S~l/S~2, ~lnf_ Ox

13/3~ 1 x 47r z f

--2f

,

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

534

which is solved as 1 - z - (1 + z)x -3~12~r

(22)

1 - z + (1 + z)x -3¢3~/2rr Substituting now (22) into (19) and (20), we obtain for functions S~l and s~2 s~l = {(1 + z-1)x -~°12 + ½(1 - z - 1 ) x -6'/2 ,

(23)

s~2 =½(1 +z)x -6°/2 +½(1 - z)x -~'12 ,

(24)

where we introduce 80 = 3(a= + t3~)/2rr, 81 = 3(a= - ~=)/27r. Let us now integrate eq. (21) directly. Then for h a we have

(25)

h a = 1 + 4n~lU (x-6X - 1). Mass equations (15) and eqs. (23)-(25) imply (A2'~ 6°/2 [A2~ 61/2 ml --~(1 +z-1)\~212 ] +½(1 - z - 1 ) ~ m 2 ] , mlA

[ASIa°/2 +_~(1

m2 -½(1 +Z)~m2 ] m2A

IA2\ 6'/2

-z)t~2)

(26)

(27)

,

- 1-1 413~ 8m2A r( A2',~X_/~'--l J-1 .

(28)

We now express bare masses m l a , m2A, •a given by eqs. (26.)-(28) through physical masses m 1, rn 2, #

mlB 2 + m2B 1 1 ;~mA=_~(mlA_m2A)-I mA==-g(mlA +mZA)-A1B2 +A2B 1

-

41~ (rnlA2 - rn2Al'l 2

#a2 =~2 --~-~1 \ A-----1B~2+AzB1 ] (C- 1), where

=[A2~ 5'/: Ai

tmZ!

,

n; tmTS

,

C-

\u 2 ]

°

-

mlA2 - m2A 1 AIB2 +A2B1

KP. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

535

If A -+ ~, then =[A2,~-ao/2 ml

mA \ m 2]

+rn2(ml/rn2) -6'

1 +(ml/m2) 6°

(A2]. - 6 ' / 2 ml 6mA

km2]

8'

/a2~

=raG - [m 2)

m2(rnl/m2) -6°

1~2 = , 2 _ ~ 1

\ml"

2 61 " l [m;~

(29)

'

/A2'~-6,/2

1 +(ml/m2) 61-6°

4/3=[m~2m2(ml/m2)

8o/2

,

(30)

-60 2

+(ml/m2)61-6o ] \ ~2 ]

(31)

Here E, G are cut-off-independent functions of renormalized mass ratios and coupling constants. Eqs. (30), (31) imply that for the spontaneous symmetry breaking to oc,'ur (6m A = 0,/a A = 0) two conditions should be satisfied: (i) 81 > 0 (am >/3=) from which it follows that it is necessary to introduce an interaction with the field Au; and (ii) the physical masses/~, m l , m 2 should obey the relation -6o 2

2 61

4/3~[ "ml- - m2(ml/m2) " l [ mi ] .2 =-~1 1 +(ml/m2) 61-8° j \ 02 ] which at

(32)

'

8m/m < < 1 takes the form

U2 m

4471

(6rn) 2 (1 + 260 - 261

6m).

Relation (32) is a generalization of the corresponding relation obtained in ref. [7]. They coincide in the lowest order in a=,/3=, 6m/m. Note that the ordinary differential equations considered here admit less arbitrariness than the CS equation applied in ref. [7], which is a partial differential equation. This enables functions E. G, in particular, to be determined. We also emphasize that, accroding to eqs. (29) and (30), 6mA

Z- 1

mA

z + 1 A--+

i.e. the bare mass ratioz

:

mlA/m2A

A~

> --1.

5. Theories with cut-off

In theories with cut-off the A parameter is kept finite. To obtain mass equations, one must solve the system (A.20)-(A.22). In contrast with the theories without cut-

V.P. Gusynin, V.A.Miransky/ Dynamicalsymmetrybreakdown

536

off, this system uses invariant charges ~ and ~ instead of a~ and/3~. For the func. tions ~ and s0~ we have in tile lowest order 2132 ~,(z,

u, %,/3A)

-

37r '

~f~(z, u ,

%, &)

-

3rr '

which coincide with the expressions in the symmetric theory. Evaluation of gives in the lowest order

~a2j,H a

3 ~ 1 (Z, U, ~A'/3A) = - - ~ (&A +/3AZ-1) ' 3 ~2(Z, U, aA'/3A) = --~-~g(0~A+/3AZ) ' 2/3A

4/3A - Ir u.

Ha(z, u, a A,/3A) - 3~

(33)

The system (A.20)-(A.22) is solved in the limit mlA, m2A,/~A ~ 0 with z 0 = limmlA, m2A~ 0 mlA/m2A = const, u 0 = limmlA, m2A,UA__,0 (SmA//aA)2 = const. We have the equations O~a Ox

12 [~a]2 , x 37r

i = A, B,

(34)

~31ns~l _ 1 3 [~t + ~ z-1 ~sa] 0~ x 4rr S~l_J ' ~31n s~2 0X

1 3 [~+ x 4rr

Oha _ 1 2

[h

ilx - x 3 ~ ~

a

(35)

~;~3zS~ll,

(35)

s~2_j

6u

--(S~l (z - 1)2

-s~2)2

]

•

(37)

/3A

(38)

From (34) we obtain

~

_

% 1

--

(2aA/3rr) In x '

~ = 1 -- (2/3A/37r) In x

Now eqs. (35), (36) can be easily solved S~l= (1 - ~2°tA - - n in x) 9/8 (1 - -~-~--n 2/36 in x}\-9/8

X 2Z-1

[z

-- 1 +(1 +Z)

(1

2 / 3 A ,9/4-] --~7-n l n x ) J ,

(39)

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

537

Sg2 = (1 ---~--2°~AIn x)9/8 ~1 -- ~2/3A In x)'-9/8

E,

X½

- z + ( 1 +z) 1 -~-~--lnx

.

(40)

Turning to the equation for function h, we note that it is a linear unhomogeneous differential equation which may be solved conventionally. Finally we have ha = 1

J(x) = J dv 0 In a given approximation the system (34)-(37) is thus explicitly integrable. The mass equations (15) and eqs. (39)-(41) imply me-(1 mlA X l z -1

In x

1 +~J(x)[

2aA ,9/8 (1 +2/3AL1)-9/8 + ~-~--~LI~ 37r Ia

,9/4-]J ,

( 2]3A - 1 +(1 +z) 1 +~--n L1)

(42)

m2 - [1 + ~---L2) 2a A \9/8 (1 + -2/3A ~ L2)\-9/8 m2A

i

X½ 1 - - z + ( l + z )

( l + - ~2/3A ,9/4-]J, n L2)

~2_ ( 1 2/3A , - 1 i1

+

(43)

4/3AUj~/221q

(44)

where

AA, L 1 =Into2

L 2 = I n ,aA ,2,

A2 L3 =ln~-~

I

It follows from eqs. (42)-(44) that a purely dynamical solution for masses of fermions and a boson is possible only at a A = o~ when mlA = 0, m2a = 0,/1A = 0. This condition is analogous to the condition c~ >/3= which occurs in the theory without cut-off. In the limit a A -->0% z -+ z0, u -+ u 0, eqs. (42)-(44) imply m 1 _[L1~9/8 (1 + 2•AL1/3rr] -9/8 z 0 --1 +(1 +z0)(1 + 21~AL1/3rr) 9/4 m2

\L2]

1 +213ALz/37r]

1 --z 0 +(1 +Zo)(l+21~ALz/3rr)9/4 '

(45)

538

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

//2 - 16

m2

131r/3A(Z0-- 1) 2

L1314(1 +2/3AL ~-1 (1 +2/3AL ~9/4 ~

(

2/3A

37

~

27

\9/4-] - 2

X 1-z0+(l+z0)I+-~-L2)

J

.

(46)

Let us show that (i) I f z 0 = 1, then m 1 = m 2 , / / = 0 irrespective of whether interaction with field By occurs (3A 4: 0) or not (/~A = 0). (ii) If z 0 > l(z 0 < 1), 3A = 0, then m 1 > m2(m I < m 2 ) , / / = 0. (iii) I f z 0 > l(z 0 < 1), 3A 4: 0, then m 1 > m2(m 1 < m 2 ) , / / 4 : 0. The results are given in table 1. Let us start with z 0 = 1. Then eq. (45) takes the form ml

/Ll19/8

m2=\~]

+23ALi137rt918 ( ~ T ~ ] "

(47)

It is evident that the solution is m 1 = m 2. Other solutions do not exist, since, if m 1 > m2(m 1 < m 2 ) , then L 1 < L 2 ( L 1 > L2), and when the right-hand side of (47) is less (more) than a unit, the left-hand side is more (less) than a unit. In addition, from (46) it follows that/2 = 0 at z 0 = 1 both when 3A 4:0 and when 3A = 0. Now let z 0 4: 1. For definiteness, we choose z 0 > 1. Then m 1 > m 2. Indeed, when m 1 < m 2, the first term on the right-hand side of (45) would exceed a unit, and the remaining two terms would give

T 2 z o _ l +(l +zo)T 2 T1 1 - z 0 + ( 1 + z 0 ) T 2 > 1 '

(

213A )9/8

Ti = 1 + ~ L

i

,

because z 0 > 1 and T 1 > T 2. For a vector particle mass eq. (46) g i v e s / / > 0 only at 3A > 0, z 0 4: 1. In other cases//= 0.

Table 1 flA

~A = 0

[3A 4=0

Zo = 1

ml = m2, tt = 0

ml = m 2 , t t = 0

Zo:~l

ml ~ m 2 , N = 0 m l > m 2 ( m l A > m2A) ml < m 2 ( m l A K m 2 A )

m l ¢ m2,/~.a 0 ml > m 2 ( m l A > m2A) ml < m 2 ( m l A < m 2 A )

Zo

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

539

Thus (see table 1), the Bu vector particle acquires a mass when a " m e m o r y " about the initial splitting o f fermion masses is present (z 0 4: 1). If z 0 = 1, the fermion mass splitting does not occur. When z 0 4- 1, it takes place regardless of the presence o f the Bu field. In particular, at z 0 4: 1, t~ splitting occurs in quantum electrodynamics. The condition of the dynamical origin o | fermion masses a A = o~ allows us to express both fermion masses in terms of cut-off parameter. Indeed, from (38) we have aA

37r

~A x=m ~/Az = -- %ni = 1 + (2aA/3n") ln(A2/mi2) e a-- ~ 2 ln(A2/m 2) ,

(48)

whence m2 =A 2 exp'-37r

1.

(49)

L ZOLrni ] The physical requirement m 2 < < A 2 gives am i < < 1. Since

°~ml =%n2

1+-~

|nm~- ] ,

then in the limit C~m2~ 0 we have C~ml -- C~m2 -~ a --' 0, where c~ is a renormalized coupling constant. Thus, (49) is a superconductivity-type solution singular at the point c~ = 0. The fact that the dynamical solutions for masses in the approximation considered are available only for c~a = ~ raises the problem of making this statement cc,mpatible with the well-known fact that the renormalization group is applicable ordy when c~A < < 1 [l 1 ]. This problem is discussed in detail referring to the electrodynamics in ref. [10]. The main conclusion derived there implies that the superconductivitytype solution for rn A = 0 is possible only if the bare coupling constant a A e~:ceeds some critical value (c~a >i c% > 0). Only qualitative arguments will be given t-ere as evidence o f this. Suppose the contrary, i.e. that ~c = 0. Then, if sufficiently small values of c~a are chosen the approximation considered here describes the Green functions with accuracy. However, as already known, it gives c% = ~ . We thus come to a contradiction. Thus, the presence o f the critical coupling constant in a given model turns out to be a condition necessary for the fermion and vector particle masses to appear. This agrees with the initial Schwinger hypothesis differing from it in that c~A is a coupling constant o f the Au boson which remains massless. As for the coupling constant t3A of the field Bu, it may be chosen sufficiently small. We note that in the range o f its applicability (0~a < < 1, m a 4 = 0) the renormalization group equations also yield superconductivity-type solutions. Let us consider, for example, the case z = 1, t3A = 0. Then from (42) and (43) we have

m A i - (1

20tA

~-9/8

-amt ,9/8

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

540 or

mA t"

/

2Cbni

- ~1 -

mi

A2 \9/8 ln~D

~

.

mi /

(51)

By virtue of olA << 1, eq. (50) implies that the parameters mAi/m i satisfy the inequality 1 > rn~a'mi> > (O~mt)9/8.

.

(52)

Therefore, from (51), it is convenient to express m i through A and mAi/mi,

mi

= A e x p [ - ~ - - ~ i (1

(mai~9/8~l -[---~i ] I 3 "

(53)

The existence of the superconductivity-type solution in the range where the renormalization group is applicable allows us to hope that it is also true for a more interesting case, rna/= 0 and c~n ~> a c.

6. Conclusions The method applied here to analyze the dynamical mechanism of particle mass generation seems to possess a number of advantages over those commonly used (the generalized Hartree-Fock method [1,6,9] and the CS equation method [71). Its main benefit is due to the fact that the solutions of renormalization group equations of the (A.20)-(A.22) type ensure the equivalence of renormalized and unrenormalized representations for the I.agrangian, whereas when the Hartree-Fock method is used the equivalence is absent. This equivalence makes it possible to define whether the spontaneous symmetry breaking takes place if higher orders of perturbation theory are taken into consideration. It proves also to be helpful in setting up restrictions for the parameters of the theory at which such a symmetry breakdown exists. The indicated equivalence is provided by the CS equation method as well. However, the CS equation as a partial differential equation admits higher arbitrariness for solutions than the ordinary differential equations (A.20)-(A.22). The conditions derived here for the spontaneous breakdown of gauge symmetry within the Abelian theories prove to be comprehensible from the physical viewpoint. Both theories with and without cut-off demand that the Au vector field should be available. This field accounts for the emergence of the "Cooper pairs" in the model, i.e. rearrangement of the ground state of the system. It is noteworthy that in the theory with cut-off the constant a A must exceed some critical value of ac" This constitutes a contrast as compared to the non-relativistic theory of superconductivity,

541

V.P. Gusynin, V.A. Miransky / D y n a m i c a l s y m m e t r y breakdown

where the rearrangement of the ground state takes place at an infinitely small value of the coupling constant. The essential difference between the theory with cut-off and without cut-off is determined by the fact that masses of the former theory depend on the cut-off parameter A and have a superconductivity form, while masses of the latter theory turn out to be arbitrary parameters. The other important distinction results from the fact that in the absence of cutoff the spontaneous breakdown of symmetry occurs only if the ratio of bare fermion masses is as follows: limml~,mz~-~O rnl~/m2~o = - 1 . Within the theory with cut-off, a more physical condition (limmlA, m2A-, 0 m l A / m 2 A > 0) takes place. We would like to express our gratitude to Dr. P.I. Fomin for his stimulating interest in our work and Yu.A. Sitenko for fruitful discussions. We are also indebted to Mrs. Shirley Hargreaves for reading the manuscript.

Appendix Derivation o f renormalization group equati~ons A

A

The equations are derived analogously to those in ref. [10]. The functions sij, d A, d E and h A introduced in sect. 3, are normalized in the following way Si/(x,y,z,u,

dA(X,y,z,U,O~A,~A)[x= 1 = 1 ,

OIA,~A)[x= 1 = 1 ,

dB(X,Y, aA,[3A)Ix= 1 = 1 ,

h ( x , y , z , U , aA, f3A)IX= 1 = 1 .

(A.1)

B B B We shall use the vertex function ['A2(I~Au = ')'Ul'2['A2 + ...), which is assuraed to be normalized as follows:

pB(xl, x2, x 3,y, z, u, aA, 13A)-= I'BAz(P2, k2, (P + k)2) = 1 at x 1 = x 2 = x 3 = 1 ,

(A.I')

where x 1-

p2 A2,

x 2-

k2 A2,

x3-

(p + k)2 A2

Passing now from the residue point A to A' in the usual way, and using the relations of renormalizations, we get

dA(X'Y'Z'U'aA'[3A) =Z3A(t)dA

' t m2A ' m 2 N ' 112, 'O~A"J3A' ' (A.2)

542

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

Slj(X'Y'Z'U'aA'flA)=Z2j(t)Sl]

X y m 2 A ' mla' 6m2' ) 7 ' t m2A 'm2A,' /22, 'aA"/3A' ' (A.3)

-m s Ix ym2A ' mlA' 8m2' mjAS2j(x,Y,Z,U, aA,~A)- jA' 2 / ~ T , 7 ~ 1 2 A , m2A,, /j2, , aA', t3A' ] ,

dB(X, Y, aA,

~A) =

1.1.4)

2 .... [X V rtllA' Z3Btt)ClB[7, t ~ ' aA'' t3A'] ' \ N1A

(A.5)

t22h(x'Y'Z'U'~A'[SA )=t22'h/x17 Vm2A' mlA, 6m2A,~ fA' , , m2 A ,m2A, ,t/2A, ) A' "

~

-

-

•

,

Ol

,

(A.6)

P~(x 1, X2, X3 ,y, Z, U, (ta,/3A) rn{Xl x2 x3 ym21A, mla, 6m 2, ) 2 ~ 7 ' t ' t ' t m2A 'm2A,' p2, '0~A"fA' "

=Ill(t) For charges

aA

(A.7)

and/3 A the relations

aA' = °IAZ3A(t) ,

fin = fA Z21 (t)Z22(t)Z3B(t)Z12(t)

(A.8)

hold [71 . Using the conditions of normalizations (A. 1) and (A. 1 '), we obtain

Z2l(t ) = Slj(t, y, z, u, 0¢A, f A )

'

Z3A(t ) = dA(t, y, z, u, a A, fA), Z3B(t) = dB(t,Y, aA, flA),

z l(t)

= r (t, t, t,y, z , . ,

fa),

mjA, = mjAS2j(t, y, Z, U, 0~A,/3A) , I12, = #2 h(t, y, z, u, aA, ~A) . Let us introduce the invariant charges ~A and ~B ~A(t, y, Z, u, a A, ~A) ~- aAdA(t, Y' Z, U, a A, flA) ,

~B(t,y, z,

u , 0tA,'t/~A) --

~ASll(t,y, Z, U, aA, ~A)Sl2(t,Y , z, U, aA,/3A)

(A.9)

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

543

X d B(t, y, a A,/3A)I~ 2 (t, t, t, y, z, u, c~A, 13A).

(A. 10)

Then (A.3)-(A.10) imply ~A(B)(X, y, 2, u, ~A, 13A) = ~A(B)

' 7(0, P (t), o(t), ~A(t), ~B(t

Sij(x,y,z,U,O~A,[3A)=Sij(t,y,z,U,OLA,[3A)Sij

,

(A.1 1)

,3"(t),p(t),o(t),~A(tL~B(t) (A.12)

h(X,y,Z,U,aA,13A) = h ( t , y , z , U , ~ A , [ j a ) h )~-, X "f(t),

p(t), o(t), ~A(t), ~B(t)] , (A.13)

where ~A(B)(t), 7(t), p(t), o(t) are the values of the functions ~A(B), 7 - (Y/X)S~l, p =- z sZl/S22, o -- u [(z s21 - Sz2)/(z - 1)] 2 h -1 at x = t, respectively. It is canvenient to rewrite the functional relations (A.10)-(A.13) in a differential fo:m a~A0~)(x, y , z, u, c~a, flA) aX In sij (x, y, z, u, a A,/~A) ax

-

1 X ~0A(B)['/' p' 0", ~A, ~B ] '

(A. 14)

1 x ¢ i j D , o, a, ~A, ~B],

(A.15)

3 In h(x,y, z, u, a A, 13A) = --Jill u,3,, aX X p' O, ~A' ~B] '

(A.16)

where the functions ~PA(B), ~if, H are defined as

~0A(B)(y, z, u, a A, ~ A ) ~ii(Y, Z, U, aA,/3A) = H(y, z, u, aa, t3A) -

-

0~A(~)(x' y ' z, u, a A,/31) x : 1 OX

lnsij(x'Y'Z'U'aA'~A) 13X

x=l

In h(x, y, z, u, aA, 13A) x= l ~}X

'

'

(A.17) (A.18) (A.19)

Eqs. (A.14)-(A.16) differ from conventional equations for the renormali~ation group [11 ] by the fact that they contain not only invariant charges ~A, ~B but invariant masses mjAs2/, t.tAh 1/2 as well. According to perturbation theory [11 ] we assume that the Y(7) dependerce of the functions si., ~A(B), h(~ij, tPA(B), H) becomes inessential in the range A 2 . Ip21 > > m 2, p~. Therefore, we can p u t y = 3' = 0 in this range. As will be shown this allows mass equations to be obtained in a convenient form. Let us now define new functions ~A(B), ~a sij, a i,l - a satisfying the following equations

,

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

544

at all values of the momentum p, (),~k(B)(X, Z, U, 0~A, f/a)

0 in s~. (x, z, u, OtA,'~A) 0X 0 In

l

a

(A.20)

=X 79A(B) [pa, oa, ~aA, ~f3],

0X

ha(x, z, u, o~A, f/A) ~X

-

X

'

(A.21)

'

1

=_X H a [pa, oa, ~ , ~31

(A.22)

'

where the functions 0~,(B)(X, z, u, a A,/3A) x=l

~0~0~)(z, u, % , ~A) -

ax

(A.23)

'

0 in s~(x, z, u, ~a, f/A) ] q,~(~, u, % , f/A) -

Ox

Ha(z,'u, ~a, f/A) -

3 In ha(x,z,u,otA,[JA) 0x

(A.24)

,~,=, '

x=l

(A.25)

'

are equal to the functions CA(B)Iy=0, t)illy=O, Hlv=0 from (A.17)-(A.19), respectively. The normalization conditions for ~(B), s~l'~ha should be chosen as 1

O~A x=l

=1,

s~/-Ix=1 = 1 ,

~

f/A

x=l = 1

(A.26)

ha[x=l = 1 .

These conditions guarantee the coincidence of the asymptotical expansions of ~A(B), a

s a.,.,~h a ~with the ,asy pm totical expansiOnSa of ~Afm,sii,a . . . . . . .h' respectively,, in the region A z, [pZ[ > > m~, Oz . The functions ~A(B), sq, h a defined in the above way satisfy the following functional relations at all values of the momentum p, ~(B)(X, z, u, C~A,f/A) = ~A(B)

,Pa, oa, ~aA, ~3 ,

(A.27)

Sf/(X, Z, U, OtA, f/A) = Z2/(t)S~j( t , Pa, oa, ~aA, ~ ) , m,A < ( X , Z, U,

f/a> -n2,ha[ x

. 2 a h a ( x , z , u, % , & ) = ,'A "" ~, t '

p°, o°, , oa,

(A.28) ,

!

(A.29 ) (h.30)

V.P. Gusynin, V.A. Miransky /Dynamical symmetry breakdown

545

Since the relations (A.2)-(A.6) and (A.27)-(A.30) are the same, when momentum p is asymptotically large, the p-independent parameters (m/A, I~A, etA, ~a, Z2j) coincide in (A.2)-(A.6) and (A.27)-(A.30). L e t us now introduce the renormalized Green functions s~/(P 2) of(p)

-

fi - mjs~j(p 2) '

(A)c ~, (g Duv ( F ' j = - uv

(g

(B) c . ,

Duu ( " J = -

•u

PuPv~ dA(p2) p2 ]

p2

'

P,Pu_~ dB(P 2) p2 ] p2-t.t2hC-~(p2) '

with the conventional normalization conditions [11 ]. For s~j, h c the equations

mj s~j (p 2) = mj A s~l.(p2 ) ,

(A.31)

p2hC(p2) =/12hA(p2) ,

(A.32)

hold. We can define the functions s~j c and h a in a similar manner to that which was used above for s~/, h a, so that the equations

mjS~jc(p2) = mjAS~j (p2) ,

(A.33)

p2ha(p2) = 112hAa(p2),

(A.34)

bold. The normalization conditions which guarantee the coincidence of the as~mptotical expansions of S~jc, h a with the asymptotical expansions of s~j, h c, respectively, have the form [10]

S~jc(p2)[ pZ=_rna 7 " = 1 + k,l=O ~ c(kl)°ck~l 2] k +l>~l hca(p2)l

t

p2=_/t2

= 1+ ~

H(kl)ak~ l

(A.35)

k,l=O k+l~l

where ~, t3 are the renormalized charges, and the coefficients t~2j~(kl),H (kl) can be calculated in perturbation theory [10]. Then.eqs. (A.33)-(A.34) yield the mass equa-

V.P. Gusynin, V.A. Miransky / Dynamical symmetry breakdown

546 tions

m/

m =

s a [m2

)

jA 2 j ~ ' Z ' U ' ( ~ A ' / 3 A

122 = Id2A ha[P 2 , 2, u, ~---~

~A,/3A)

/$~jc Ip 2- - m j 2'

/ ha p2 =_la2 "

(A.36)

(A.37)

Assuming the normafized charges c~ and/3 to be small, from (A.35) we have "~ 1 and hal p2=-~,2 "~ 1 which leads to

S~jclp2=_m~ mj

=m

s a {m2

jA 2 j ~ - ~ - , z, u, ~A,/3A

p2

z u oA A)

)

,

(A.38)

(A.39)

References [1] Y. Nambu, Phys. Rev. Letters 4 (1960) 380; Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345. [2] J. Schwinge~, Phys. Rev. 125 (1962) 397. [3] P.W. Higgs, Phys. Rev. 145 (1966) 1156; F. Englert and R. Brout, Phys. Rev. Letters 13 (1964) 321; G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Letters 13 (1964) 585. [4] E. Eichten and F. Feinberg, Phys. Rev. D10 (1974) 3235. [5] F. Englert, 1. Fr~re and P. Nicoletopoulos, Preprint CERN, TH-1933 (1974). [6] R. Jackiw and K. Johnson, Phys. Rev. D8 (1973) 2386; J.M. Cornwall and R.E. Norton, Phys. Rev. D8 (1973) 3338. [7] J.M. Cornwall, Phys. Rev. D10 (1774) 500. [8] K. Johnson and M. Baker, Phys. Rev. D8 (1973) 1110; S. Adler, Phys. Rev. D5 (1972) 3021. [9] P.I. Fomin, JETP Letters (Soy. Phys.) 6 (1967) 972; P.I. Fomin and V.I. Truten, Yad. Fiz. 9 (1969)838; V.I. Truten and P.I. Fomin, Teor. Mat. Fiz. 5 (1970) 219. [101 P.I. Fomin, V.P. Gusynin, V.A. Miransky and Yu.A. Sitenko, Nucl. Phys. B (1976) in print. [11 ] N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Nauka, 1973).