IL NUOV0 CIMENTO VoL 42 A, N. 1 1 Novembre 1977

Covariant Formulation of Spontaneous Symmetry Breaking in Chiral Quantum Field Theory.

G. I:~USCHE]~PLATT

D E S Y - Hambu~'g

(ricevuto il 25 Maggio 1977)

S u m m a r y . - - For the (SU2 (~ SU2)-invariant model of Giirsey a formula- tion is given in terms of a direct integral of Hilbert spaces. The structure of matrix elements, due to the spontaneous breakdown of the ehiral sym- metry is discussed.

l . - I n t r o d u c t i o n .

In order to clarify the meaning of spontaneous s y m m e t r y breaking (SSB) in quan tum field theory , the algebraic approach (1) is appropr ia te . Bu t it

needs a highly sophist icated ma themat i ca l machinery . So in pract ical cases, of which chir~l dyllamics (~) is a p rominen t example , ~nd to which we shall

restr ict ourselves in the following, more naive methods like current algebra (~) are used to ex t rac t low-energy theorems as consequences of the spontaneous

breakdown of SU2 ×SU~. However , now we shall give a fo rmu la t i on - -based

on covariant f ie lds-- for a field theory with spontaneously broken 2U2 × SU2

which, first, incorporates the knowledge obta ined f rom algebraic studies and,

second, is applicable to field-theoretic models.

A modification of the usual field-theoretic f rame when SSB is involved is

(1) See, for example, R. HAAG: NUOVO Cimento, 25, 287 (1962); R. F. STREATE:F~: Proc. Roy. Soc., 287A, 510 (1965). (2) M. W]~INSTEIN: Chiral symmetry, in Spri,n,ger Tracts in Moder.t~ Physics, 60 (1971). (3) A. R. DASHEN: Curre,ttt Algebras (New York, N. ¥., 1968).

15

1 6 G. ~ U S C H E N P L A T T

necessary because of two linked phenomenona: the divergence of the axial- charge integral (4), which means tha t the axial t ransformations cannot be unitari ly implemented in the usual Hilbert space carrying an irreducible repre- sentation of the field algebra, and the degeneracy of the ground state (5). For this we propose the Hitbert space H for ~n (SU~xSU~)-invariant theory to be a vector bundle over the hypersphere S ~ =-- SU 2 x SUJSU~ with SU~, the isospin group, being the diagonal subgroup of SU, xSU2:

(1.1) H ~ {F; c-~g(c) , c e S ~, g(c) ~ H~ Hilbert space with irreducible re-

presentat ion of the field algebra, fd~(c)(g(c), g(c))< ~ } .

Then the whole (SU~X SU~)-group is unitari ly implemented ill H (s) and in this direct integral representat ion the degeneracy of the ground state occurs in a natural way.

Such a description is justified in tha t the model of three free scalar fields with spontaneously broken ISOa symmet ry (7) exhibits exact ly this s t ructure with S 4 replaced by IS03/SOa. Moreover, we expect tha t in the LSZ-asymp- tot ic limit (s) the SU~ × SU~ symmet ry contracts (9) to azl IS03 symmetry .

In order to be specific, we consider in the following the model of Giirsey (lo)

with the chiral (SU2xSU~)-symmetric Lagrangian

(1.2) i i

1 + ~ Tr ~, exp [i21~q~] ~ exp [-- i2]~q~],

± u l -

~ three scalar fields, yJ~ two spinor fields. Arguments for the spontaneous breakdown of the axial symmet ry in this

model are given in (ix).

(~) See, for example, T. W. KIBBLE: The Goldstone Theorem, Bochester Con]erence, 1967 (New York, N.Y., 1967). (5) Y. NAMBV and G. JOZ~A-LASINIO: Phys. Rev., 122, 345 (1961). (6) G. W. MACK]~Y: Induced Representations o/ Groups and Quantum Mechanics (New York, N.Y., ~nd Torino, 1968). (7) H. Joos and E. WEIMAR: NUOVO Cimento, 32A, 283 (1976). (s) H. L]~I-I-~IA~T~T, K. SYMANZlK and W. ZIMM]~RMAN~T: NUOVO Cimento, 1, 425 (1955). (9) E. WEIMAr: Nuovo Cimento, 15 B, 2i5 (1973). (lo) F. Gi3zcs]~Y: Nuovo Cimento, 16, 230 (1960). (11) G. RUSCm~.Z,;~'LATT: Dissertation Hamburg (1976).

COVARIANT FORMULATION OF SPONTANEOUS SY~VIMETI~,Y B R E A K I N G ETC. 1 7

2 . - (SU~×SU~)-eovariant fields and Hilbert space.

I n our t r e a t m e n t of the Gfirsey model (1.2) the (SU~×SU~)-group is uni tar i ly implemented in the Hi lber t space H (1.1) and nonlinearly realized (~) on the fields. I f the elements of SU~×SU~ are denoted b y (R, R'), R, R'6

SU~, t hen the i sosp in- -v ~ (R~ R), R -+ v ~ v~ @ i ( v ~ ) - - a n d a x i a l - - a =~ (R', R'-x), R'--~a ~ a~ @ i ( a~ ) - - cova r i an t t ransformat ions of the fields are

{ U(v) -~ exp [i2Jw] U(v) = vi exp [i2]vq~]v-~,

(2.1) U(~) -~ exp [ i 2 J ~ ] U(a) = a~ exp [i2]v~o]a~,

U ( v ) - i ~ U ( v ) = v~y , ,

U ( a ) - ~ o g ( a ) " ~

b:" - - 1 (b~ @ 1 @ i(b~)) ~ _ 1 V2(b~ + 1~ ' %/2(b~ -t- 1) (b~ @ 1 +iTs(b'~)) ,

respectively. I n writ ing this we used the spinor calculus (n) : to point a e S 4 with Eucl idean co-ordinates a = (a~, a) is associated a 2 ×2 ma t r ix a - - a~-~-

i(a~) ; to reach a f rom the nor th pole co ~ (1, 0) e N 4, one has to app ly the

axial t r ans format ion a = a ~ Co a ~. SSB of ~ U2 × S U~ implies reducibil i ty of the representa t ion of the field algebra.

Hence H has the following s t ructure (~a). The field algebra which is genera ted

b y the fields ~, ~o (1.2) has different irreducible representat ions in different

factors H~ of H. I n each Ho there is a vector lc}- - the (( v a c u u m ,)--which is

cyclic under the field algebra. Bu~ all He are physical ly equivalent in the sense tha t they car ry the same representa t ion of the Lorentz group (~). Since the H~ are supposed to be separable, so is H in contras t to a formulat ion of SSB

given in (14). Fo r fur ther e laborat ion of the s t ructure of H, we consider the opera tor

(2.2) - - l im d V : e x p [i2Jw]" , F--o-co

V

which is an e lement of the centre of the field algebra. I t is a ma t r ix in isospin

space. Under S U2 × S U~ it behaves like

(~o.a) U(v)-I~U(v) = v ~ v - ~ , U(a)-l~U(a) = a '~a ~ .

AS central e lement there is a c-number in every He and its (~ eigenvalue ma t r ix )~ is a 2 ×2 ma t r ix of the fo rm d ( c ) = d~(c)~ i (d(c)x).

(12) j . WESS: Spri~ger Tracts in Moder,~ Physics, 50, 132 (1969). (la) H. A~AxI: Prog. Theor. Phys., 32, 844 (1964). (14) G. KnA~EI~ m~d W. F. PAL~IER: Phys. Rev., 182, 1492 (1969).

2 - I I N u o v o Cirae~do A .

18 G. RUSCHE~PLATT

Since the isospin subgroup SU~ is not spontaneously broken, there is an H~_~ o invariant under SU~ with a SU~-invariant vacuum Ic ~ 0>. This SU~ invarianee of H~___ o implies d(c =--O)= Co. So with the redefinition H~___o~

H~., Ic ~ 0> ~- ]Co>, we get

(2.4) @H% = coil%, especially ~]co> = Co[Co>.

F rom the definition Ho =~ U(a ~-- c)H% and (2.3), it follows tha t

(2.5) ~H~ = cH~, especially ~]c> = c[c>.

With (2.3) and (2.5) the ~SU~×SU2 t ransformations of ~he factors H~ of H are explicit ly determined: ~he uni ta ry representat ion of SU~×SU~ in H is a n - - b y SU~--induced one in the sense of Maekey (~).

The isospin-invariant He. can be identified with the usual Hilber t space

of field theory. Let us come to the description of the physical vectors of H. The connected

par t of the Poincar6 group commutes with SU~×SU~. Hence the four-mo- men tum p and the spin s do not depend on c, and H can be decomposed into

simultaneous eigenveetors of 19, s and G. They t ransform under the translat ion U(a) and the Lorentz t ransformat ion

U(A) as

U(a)]p, s, s3; c> = exp [inS'p#] IP, s, s3; c>,

(2.6) u(A)[p, s, s~; c> = ~ D(A, p)~,olAp, s, s'~; c>.

Since the pion is pseudoscalar, the space reflection U(z) transforms ~:

U(~r) - l~U(Jr) = G-1. Therefore

(2.7) Y(~)l(po, p), s, s3; c> = I (Po , - P), s, s3; c-1>

with ¢-1[¢; c-1> = c-1]¢; c-1>. Also the t ime reversal T and the charge conjugation C simultaneously trans-

form p, s and c; but , because of the complexi ty of the t ransformat ion prop-

erties of many-part ic le states, we restrict our discussion to one-particle states

in a later section. SU2 × SU2 simultaneously transforms the isospin I and c. I a order to get

well-defined properties, we first consider the eigenstates of I, I in He°:

(2.8) P[j ,m; co>=j(jq-1)lj , m; Co>, I, lj, m; co>=mlj, m; co>.

We define the corresponding vector in Hc by

(2.9) ]j, m; c> ~ U(a ~'-c)]j, m; co>,

COVARIANT F O R M U L A T I O N OF S P O N T A N E O U S SYMMETRY B R E A K I N G ETC. 1 9

which leads to the relatively complicated isospin t ransformat ion for a rb i t ra ry c

(2.~.o) U(v)lj, m; c> : ~ D(m, m'; v; c)lj, m' ; c'> in"

with v c = c ' k , k~SU~. This definition of I, I~ in H~ has its counterpar t in the definition of spin

in relativity.

3. - Matrix e l e m e n t s and discrete t rans format ions .

An immediate consequence of the formulat ion given above is t ha t one gets the s tructure of mat r ix elements which is due to S U2 × S U2 covariance. We show how this works in case of the tensor operator :exp [i2]~v]:; other ex- a m p l e s - t h e Fermi field~ the SU~ x SU~ cur ren t s - -a re t rea ted in the appendix. In the matr ix elements between H,. and H~, we split off the d-function which expresses the orthogouali ty of the two spaces: <H~IH~,} = (~(c-- c')<H~IH~,}~ (r stands for reduced). Throughout we use the nota t ion of (is).

The 1-point function is explicitly known:

(3.1) <el :exp [i2]~s] "lc>~ : (~l-o --~ c .

I f the theory contains only the pions described by the ~-fields, the 1-pion matr ix elements take the form

(3.2)

with

<cl : e x p [ i 2 i ~ ( x ) ] : ]L,,,:~>r =

_ / a ~: ~ =p,I/( Z

V . ZT~

(27r)32oJv

In an expansion in ], the renormalizat ion constant is

(3.3) v/z~ = 2i] q- a(]~).

With the 2-point function we shall show the different implications which

the symmetries SU~, ~qU~><SU~ and spontaneously broken SU2xSU2 have

(15) j . ]~JORKEN ~nd S. DRELL: Relativistic quantum ]ields, International Science in Pure and Applied Physics.

9.0 G. R U S C t t E N P L A T T

on its structure. In all three cases it is of the form

(3.4)

(c I :exp [i2fl~(x)] : :exp [i2]vq~(O) :{c), =

= 9~(x)c ® c ÷ 2~(x)(x ® ~ - - 1 ® 1) ,

co

~ ( x ) ~ K , + fdm~qi(m ~) A+~(~) (e, not necessarily positive!) 0

ZI+ = ~ 1 '~ fd 1 - - ~p ~ exp [-- ipx] (~o;)"- = p" ÷ m ~ , \2:~] 3 20)~

where the constants K~ originate f rom the vacuum as intermediate state in the spectral representation. Isospin invariance alone gives Ks = 0, Kx, ~, a rb i t ra ry ; here c = Co ~ 1 necessarily. Ordinary S U s x S U s invariance would lead to Ks = 0, ~ = 0, & arb i t ra ry ; in this ease too c ~ c0 = 1. Spontaneously broken S U ~ x S U s symmet ry , however, demands quite different properties. The s t ructure due to isospin invariance in H~o and a chiral boost f rom co -+ c yield ~ arbi t rary , except for I£~ = o. Now in our model taking apart the 1-

pion contr ibut ion according to (3.2)--we get

(3.5) <c[ :exp [i2h(p(x)] • :exp [i2h~(O)]:lc> ~ = [1 ÷ zr¢A+(x)] c ® c ÷

÷ :~:ZIo4(x)(T (D 'v -- 1 @ 1) ÷ ~l(x) c ® c ÷ -~z('v ® 'v -- 1 ® 1),

~ , being wi thout 1-i)ion contribution. A perturbati( ,r t-theoretic calculation, with ~ the free field, gives the same

s t ructure (11):

(3.6) <el :exl [ i2 /~(x) ] : :exp [i21~(o)] "It>, =

4 2 + = {(1-- ~] 3 0 (x)) cosh (4]~A+(x)) ÷

+ (4/%+(x)- ~) sinh (4/%+(x))} c ® c - {~ sinh (4/-%+(x)) ÷

÷'" -d+,x , eosh (41%+(x))}[~ ® ¢ - ~ ® 1] = { 1 - 41~4+(x)} c ® c -

-- 4]2A+(x)[x ® x - - 1 ® 1] ÷ a(f 3) (~ is the free field, z = 1).

So up to the 1-pion contr ibution or up to order a(/s) in per turbat ion theory, respectively, the result in our model for spontaneously broken S U s × S U ~ is

(3.7) ~ = 1 - 41~ a + o ÷ ~ ( p ) ,

n

~,~ = _ 4/~ z a + o + ~ ,

~ = - 4l~ a+o + (r (p) .

C O V A R I A N T F O R S I U L A T I O N OF S P O N T A N E O U S S ¥ ] ~ I M E T R ¥ B R E A K I N G E T C . 21

I f t he L a g r a n g i a n (1.2) is s y m m e t r i z e d b y m e a n s of c o m m u t a t o r s or b y

n o r m a l order ing, t h e t h e o r y is i n v a r i a n t u n d e r space ref lec t ion P, t i m e reversa l

T, charge con juga t i on C a n d G-par i ty . The b e h a v i o u r of the vec to r s u n d e r these opera t ions is d e t e r m i n e d b y the t r a n s f o r m a t i o n s of ~ :

(3.s)

hence

V A C U a :

_PIe> = ]c-1>, TIc > = ]v-l~'> , CIc > = leT>, Gle> = ]e-l> ;

1 - fe rmion s ta tes :

(3.9)

~o,p) ,s ,J=~,m;c / = (Vo,--p ,s,;=},'rn; c -z f l

T ~¢;) \ - - exp [ - - i~ + (p, s~l I~(~) , \ ~o,p) , (so ,s)d=½.m;cf = JA I(VO,--p),(So,--S),i=u,m;c-IT~ (--)

C ~(q) \ :1)m+~- s~] I~(q~ \ .... ~=~,,~;°/: - - (-- e x p [ - - i¢(p, ,~,~.~,j=~,_~;o~/, (+ ) (+)

G ~(~) \ - - e x p [ - - i¢(p, s~l 1~(~) \ .p,s,j=.~,m;e / = ] J ~ms,j=~,m;c-~ / (+) (+)

q is t he fe rmion , q t he an t i f e rmion . The phases are f ixed wi th t he he lp of t he m a t r i x e l emen t s (A.2).

4. - T h e l i m i t f---> 0.

I ~ t he a d i a b a t i c l imi t ] -~ 0, t he LagTAngian (:1.2) equals t he /JagrangiAn

of t he f ree field; t he (SU2 x SU2)-group con t r ac t s to the ISO~-group:

(4.1) [L, ~] = i~_~,~, [L, ~ ] = i~A_~, [~ , A~] = i # 2 ~ L -~ O

I , A are t he isospin a n d chiral gene ra to r s ; A = 2]A_.

These pu re ly g roup - theo re t i c a l re la t ions are also t rue if one t a k e s t he I a n d A to be t he f o r m a l charges (11)

(4.2) I =fd'xJ°(~, =o/, A=fa~xJ?,(~, Xo), ~a ~a

jo , $o are g iven in (A.5).

2 2 G. RUS CH:EIqPLATT

Whether this is valid in the LSZ-asymptotic limit too is not known.

The author is indebted to Prof. Dr. H. Joos for many helpful discus- sions.

A P P E I ~ D I X

In this appendix we collect briefly the results for the structure of several matr ix elements. The 2-point function of the Fermi field is

(A.~) <cfiv(x) vT(0)[c>~ = i ~ ( x ) - c-~ d:(X) mo

with mo the mass of the free Fermi field. Here, as in the free Dirac equation (iV~ ~- moS-x)iv = 0, the c-dependence

is connected with the mass term. The same behaviour exhibits the propagator S~ (p)~ = c-~S~ ' (p) ,c -~, but the pole of i t - - a n d hence the mass of the iv-field-- is independent of c, showing the unphysical nature of the e-dependence of the mass in the Dirac equation. This solves the paradox in (~).

The 1-fermion matr ix elements are

(A.2)

<clive(x) I~,~,~=~..~o>~ =

> < c [ ~ ( x ) l~,~,j={,~;o ~ =

-~ l~ Zq OIbq

- V z~m~

i zame exp [-- ipx](-- 1) ~+~ F¢(p, s) [~-'~](~)~

, ~ _ ~ . ~ exp [ ipx]~(p , s ) tc J~ ,

- - e x p [ipx](--1)~+m[~-~]~_,.)v~(p, s) ,

where E~ : p ~ + m~, q is the fermion, ~ the antifermion, ~,.. . the spin index and a, ... the isospin index.

For the fermion-antifermion Bethe-Salpeter amplitudes, one gets

(A.3)

)]^1 <C[Tiv(X)iv(y)l~,j=O;c>r = ~-~ F~XZ~(Z , y c - : ,

<clYiv(x)(v(y)l~,~=~:o>r = c-~ / '~X~,(x, y , LC~

COVARIANT FORMULATION OF SPONTANEOUS SXMMETRY B R E A K I N G :ETC. 2 ~

M is the mass of the bound s ta te ;

for F ~ = 7", 7~yu,

for 1"~ = i~ y~, au ~ ,

~ ~ - - 1 ~_ a~¢ 2ic~ y~ __ ~--~- for / '~ = 7", 7~7~ , ~/2(c~ + 1)

for F ~ = 1, y~, a . ~ .

The un t i symmet r i c S04 tensor

(A.4) J ' : J ~ = ~ J ~ ' ~ ,

which is buil t f rom the isospin and chiral currents

(A.5)

Jf '~ = - - Cf),. ~ yJ -J- sin 2 ( 2 ] ~ / ~ ) ( ~ , ~ × ~ ) ~ ,

~u~ cos (2J ~ - ~ ) sin ( 2 / ~ ) @

cos ( I )sin

hns[2-point funct ions of the fo rm

(A.6) @]J~, (x )J ,~(y ) lc )~ = [~,~dT~ - - ~,~(}k,~]A'~(x - - y) @

@ [CiCm~n - - CiCn~km @ CkCn~im - - CkCm~ik ] z~V(X - - y) •

The kernel K ~ of the fe rmion-ant i fe rmion Bethe-Sa lpe te r equat ion

(A.7) <elTv4xl)~(x~)l~; c>, =

= f d 4 y l . . , d 'y~ SF(x~ - - y~) K c(yl, Ys; Ys, Y~) (elTy~(Y2) ~(Ys) ln; c)r S , (y4 - - x2)

has as an a m p u t a t e d funct ion the t r ans fo rma t ion p r o p e r t y

(A.S) K o = ~ ® F~K oo ~ ® ~ .

A discussion of the Bethe-Salpe ter kernel in the ladder app rox ima t ion is given in (n).

24 G. RUSCHENPLATT

Q R I A S S U N T 0 (*)

Si presen¢a una formulazione per il modello di Gtirsey invariante per SU~® S U 2 in termini di un integrale diretto degli spazi di Hilbert. Si diseute la s t rut tura dcgli elementi di matrice, dovuta alla rot tura spontanea della simmctria chirale.

(*) Traduzione a eura della Redazione.

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TOBOfi TeopllH HOYlSI.

PeamMe (*). - - IIpe~naraeTcn ~opMyn~IpoBra S U 2 Q S U~-nHBapHaHTnOfi Mo~earl Fiop- ce~ B TepMI4HaX IIpnMOFO i4HTerpaJia rlaJIb6epToBbIX IIpOCTpaltCTB. O6cy~Kj~aeTca cTpyKTypa MaTp~IqHt,IX 9YieMeltTOB, 06ycJIOBJIe/~na~ CIIOHTaR~II, IM HapymenHeM rHpaab- HO~ CI4MMeTpHH.

(°) IlepeaeOetto pe3amtue~t.