Renormalization of gauge theories

> L(P2) = -W/P2) Qp(P)*

305 (109)

On the other hand the @17 equation of motion (106) yields: PQP”)

- P2Lc(P2)

= ~cTc(P”).

(110)

It follows that:

which shows that this determinant

has a double zero at p2 = rnz, .

C. The Physical S Operator : Gauge Invariance

According to the LSZ asymptotic theory, the physical S operator is given in the perturbative sense in terms of the Green’s functional 2 = exp(i/fi)Zc, by: Spllys

=

sPhYs(GQGl

(112)

where (113) In the above formula, uin, KU; aTin , IPUB* are the canonically quantized asymptotic fields and differential operi%rs involved in their equations of motion, derived from the asymptotic Lagrangian determined by r, which in the present case, can be read off from the tree approximation. We want to prove: asphys -= am:,

-asphys = 0. 3K

(114)

Owing to Lowenstein’s action principle [3], one has

where h is one of the parameters K, m$, and d,, is a dimension

four insertion

306

BECCHI,

ROUET,

AND

STORA

obtained by differentiating sdx -&f(x) with respect to A. Using the Slavnov identity, we are going to show that A,, can be written as: ,.A, = i

C;*‘Aio + i

i=l

Ch”*iA,”

(116)

i = l,..., 5,

(117)

i=l

where the Aio’s are such that &I&WA

7) = 0,

and therefore leave unaltered the physical normalization conditions (lOla, b), (102a, b), (103). In the following these insertions will be called nonphysical. On the other hand the Ais’s are symmetric insertions, namely ~“oi”z($,

q> = 0,

i = l,..., 5.

(118)

Thus applying Eqs. (116), (117) to the physical normalization conditions (IOla, b), (102a, b), (103), yields a linear homogeneous system of equations of the form k C;~~d,“*j = 0

j = l,..., 5,

t=l

(119

for which we are going to see that [I 1, l] det [I A,S*jII # 0 since this will prove to be true in the tree approximation. c,s*” = 0

i=l

,***, 5,

(120)

Hence it follows that (121)

and the gauge invariance of the S operator follows from Eqs. (112), (113), (115), (116), (117). The decomposition of A, given in Eq. (116) follows first from the Slavnov identity: o = a, YZ = (aA9) z + (i/h) YAnZ. (122) Hence (i&Y) z = - 8 j- P(x) SJacaJdx Z (123) = f [A,, 9’1 Z.

RENORMALIZATION

OF

GAUGE

THEORIES

307

we define AI0 by

4OZ = f j dx [J”(x) SJa(,)+ y”(x) &,,I Z which is a dimension [3]. Hence

(125)

four insertion as follows from Lowenstein’s action principle [A, - (ap/ax)dlo, 91 = 0

(126)

A, = (ap/ah) A,0 + AAS

(127)

thus

where AAsis a dimension four-symmetric insertion. We are thus left with finding a basis of symmetrical insertions. Since we know that, given the Slavnov identity 5$rr depends on nine parameters, namely four to specify the couplings with the external fields and five to specify the remaining part of the Lagrangian, there are nine independent symmetrical insertions. We shall first construct the four missing unphysical insertions. The method consists in constructing insertions which are relized by differential operators as a consequence of the action principle, and study their commutators with Y. First consider [l I], [I]

and define die, i = (.rr, aL), by &Z

= :YQ,? Z

(129)

where as usual the dots indicate the substraction of the disconnected part. One has:

[A;‘, yp] Z = - .T%‘~Q~‘Z = - [,P, pi’] Z

These commutators

have obviously

finite limits

as E ---f0. Consequently

the

308

BECCHI,

ROUET,

AND

infinite part of die is symmetric. Substracting limit E + 0 some Ai’s (i = T, aL) such that

STORA

the infinite

parts, we get in the

Furthermore

Now let A? = Ai’ - poi(hf2) A,’ - P&,‘) where the symmetrical

A,”

(134)

insertions

respectively coincide with J dx P(x) CjJql) and J dx J;$SJ;y under application of z phys. and restriction at $ = 0. After what was seen before the finite part Ai0

RENORMALIZATION

of dy is nonphysical one can check that

OF

GAUGE

THEORIES

and satisfies Eq. (131). From these commutation

309 relations

(136)

are symmetrical,

(and obviously nonphysical). In the same way one can check that

4” = I dx [P(x) $a,,) + P(x) &J is symmetrical

(obviously nonphysical). O;‘Z

(137)

Finally dt” defined by

= 1 dx Jo(x) Z

(138)

which can be realized by performing a u field tranlation, is certainly symmetrical and nonphysical. These four symmetrical nonphysical insertions are independent as can be checked in the tree approximation (di” has a 03 term which the others do not have; 0:’ has a 7~~term, not contained in the others; OT*’ has a CC%, term which is not in O$“). To complete the basis of symmetrical insertions there remains to find five others which are independent on the physical normalization points. According to the general argument, i.e., the implicit function theorem for formal power series [l], we know that there are four of them whose tree approximations are the four terms of g&v. (98). A fifth one is das. Their independence on the normalization points, i.e., Eq. (120) can be decided at the tree level and can indeed be verified. This completes the proof. D. Radiative Corrections

: Unitarity

of the Physical S Operator

Usually, the unitarity of the S operator follows from the relationship between time ordered and antitime ordered products, together with the hermiticity of the Lagrangian. Here we are investigating the unitarity of the physical S operator so that we have to show the cancellation of ghosts in intermediate states. Furthermore the Lagrangian is not hermitian. We shall however show that the unitarity of the S operator can be derived from the usual relation between time ordered and antitime ordered products, thanks to an additional symmetry property of the Lagrangian, and the Slavnov identity. Together with (139) we introduce

310

BECCHI,

ROUET,

AND

STORA

where T and T respectively denote time ordered and antitime ordered products. We first define the S operator in the full Fock space (including the ghosts) through the LSZ formula: s = ~wqhl where Z = exp s dx dy ‘Pi:) Ki:S& Sphys . One has:

: which is the straightforward

m(y) and if the Lagrangian

(141)

extension of

* #zz(Jq = 0

(142)

is hermitian,

Let now E,, be the projector on the bare physical subspace. By application Wick’s theorem one obtains: zphys

z($)

exp

d&he

z(y)

=

4,

of

(144)

where d = ifi

the index g indicating

s

dx [&-.h+g

that the summation

* K~6~g](x)

(145)

is restricted to the ghost fields, and ghost field commutators.

ifiS+g being the positive frequency part of the asymptotic

We are now first going to see that

ELm+ = @xA

(146)

where $t+ is the coefficient of ly,+ in $ . Y, G?t = .f, @f = -{, %?y = $+ otherwise. This is a consequence of the corresponding Yifr

=

property at the Lagrangian

%%ff

(147)

level: (148)

where V?C= UC = %‘v = %?I) =

C, -c, q+ (all classical fields), *+ otherwise.

(149)

RENORMALIZATION

OF

GAUGE

311

THEORIES

This property is due to the fact that the V operation and the hermitian conjugation transform the Slavnov identity in the same way, leave the normalization conditions unchanged (the V and t operations are defined in a natural way on r), and that the theory is uniquely defined by the normalization conditions and the Slavnov identity (lOl), (102), (103), (104), (105) (96). Thus .Zphys is hermitian since the asymptotic Lagrangian is hermitian, which follows from (146) hence (142) can be rewritten z Phys

z(f)

[exp

&‘I

f&hys

z($>l’

=

(150)

&

where .dC = ifi

s

dx [iJgigS+g

* k.!~Q,](x)

(151)

with #!I = @?g

(152)

Let us consider U(h) = exp hYQZV

(153)

Then &%h,s($)

u(x)

$hys(f)j$=o

=

&h&f)

d”CU@)

(154)

$hys($>iy,o

Introducing (155)

whose variation under a Slavnov transformation reduces on mass shell to Cm, and using the restricted ‘t Hooft gauge [22] defined by

in which the ghost propagators d*

= i?i j dx [&(iiS.+. * i)&j + &(iS+

have only simple poles [23], allows to rewrite & + 8,(iS+

* i) @fjj i& + i&s+

* i&e

&

* i>@ if + 6c,(iS+ * i,,,

t+](x).

(157)

Now making use of the Slavnov identity on the ghost mass shell and of the vanishing source restriction allows [l] to reduce (154), (157) to a&h,*(cf?

u@> %m(~)l/=o

312

BECCHI, ROUET, AND STORA

which upon integration

with respect X leads to:

Since the expectation value between physical states of the time ordered product of an arbitrary number of gauge operators is disconnected [21] &hys$hys

= &,

WV

i.e.,

Similarly,

EJE, SE, = E,,

(161)

E,SE, SE, = E,

(162)

one can prove that:

which shows that the physical S operator

EJE,

obeys perturbative

unitarity.

CONCLUSION

Gauge theories can be characterized by the fulfillment of Slavnov identities when the underlying Lie algebra is semisimple, i.e., is sufficiently rigid against perturbations. Then, simple power counting arguments are sufficient to prove that indeed Slavnov identities can be fulfilled, in the absence of Adler Bardeen anomalies. In particular, we have shown, on the SU2 Higgs Kibble model whose particle interpretation can be completely analyzed that the gauge independence and unitarity of the physical S-operator follow. It is believed that both the lack of rigidity of the underlying Lie algebra and the possible occurrence of Adler Bardeen anomalies can only be mastered by more sophisticated tools based on a closer analysis of the consistency conditions involving the behaviour of the theory under dilatations [7, 201. The analysis of gauge independent local operators, although not touched upon here [l] should also be tractable in terms of the methods used here. APPENDIX

A: COHOMOLOGY

OF LIE ALGEBRAS

This appendix is a brief summary of definitions are not easily found in classical text books [24].

and results needed here which

RENORMALIZATION

313

OF GAUGE THEORIES

Let h be a Lie algebra with structure constants ftB, which is the DEFINITION. sum of a semisimplealgebra 9’ and an abelian algebra rd. A cochain of order n with value in a representation space V on which h acts through a completely reduced representation: lj 3 ha--+ f” is a totally antisymmetric tensor built on lj whose components

are elements of V. The set of such cochains is called C”(V). coboundary operator d”:

We define the

C”(V) dn_ P’-l( V)

(Al)

by: n+1 (,,,),,'-=n

L, =

&

nr1 (-)"-1

pkp-+.%+l

+

&

(-)h‘fZ

f~r"lrA"l...~r...61...a,~il

CA3 in which capped indices are to be omitted. The fundamental property is: &+l o d” = 0

(43)

a consequence of the commutation relations [P, P] = f.y

(A4)

and the Jacobi identities. An element r of C”(V) is called a cocycle if d”r = 0.

(AS)

The set of cocycles is denoted Z”(V). An element r of C”(V) is called a coboundary if r = d”-ll’

W)

for some f E C”-‘(V). Obviously every coboundary is a cocycle (cf. Eq. (A3)). The converse is not always true. However in the present casewhere the representation is fully reduced, the parametrization of all coboundaries can be found as follows: We first split I’ into an invariant and a non invariant part:

r = rtl + r,

(A7)

314

BECCHI,

ROUET,

AND

STORA

such that t2ri, = 0,

t2rb # 0.

Here we have defined t2 = tv,

(A@

where indices are raised and lowered by means of a non degenerate invariant symmetrical tensor. The restriction of Eq. (A5) to P, yields through multiplication by tal and summation over 01~: I’, = d”-ll$

(A9

where
pyl”%l*

(AlO)

(Use has been made of the commutation rules between the t”‘s.) Next we look at r, . The cocycle condition reduces to: Aol..+~.dt,.w

n+l

=

0.

(All)

We define the two operations

and

(60 transforms F according to the sum of n adjoint representations One can check that n+lIo

o &a

+

&-1

o nIo

=

n@

of b and P). (A14)

and furthermore d” 0 qp = me0 o dn.

6415)

(A161

RENORMALIZATION

315

OF GAUGE THEORIES

Hence (“Qzr,

= &l-l o +l~o o nIor, .

Reducing the antisymmetrized

(Al7)

product of n adjoint representations

according to

rq = rq’l+ l-hb

(‘418)

with

(qw

(fqzrab# 0

= 0,

we find that

and Fe 6 is arbitrary. To sum up, in the case of a completely reduced representation and a Lie algebra which is the sum of a semisimple and an abelian algebra, every cocycle is a coboundary up to totally invariant cochains. When lj is semisimple however there is no such cochain for II = 1,2. (In this last case, the invariance which implies the cocycle condition and the non degeneracy of the Killing form yield the result.) APPENDIX

B

We want to show here that the equation: s4(yi(x) P,(x) f P(x) &,(x1> + s(f(x) Ai -- YitX)[tdjtY) -

tpLU)

A- P(x) A,(x))

sb,(s) - dmtY) sC,(v)) pit-y> sQj(~)

+

-t PW[-4v)

pdY)

sC&4))

Oi(x>l

&scv,~d4 + &8(Y)&yY,4~41 = 0

031)

implies the structure:

=

--b”(f(X)

Pi(X)

+

i”(x)

&a(x))

+

St&c)

fl,(x)

-’

E”W

mx))l

C,(Y))pitx) + tpjtY> sdj(v)+ pfxt.Y)sCa(~)) nfCx>l - 5’(.NuY) ~gcY,fvx) + e3o(Y> SC,du~)). 0-W

Y’ts)[tnj(Y)

S6,(y)

+

17,(Y)

s

From this result it follows that Eq. (59) which is a perturbation of order hd of Eq. (82) is a consequence of the consistency condition Eq. (58) which is a perturbation of the same order of Eq. (Bl).

316

BECCHI,

ROUET,

AND

STORA

Let us recall the notations Pi(X) = ti”j[@jcJ(X)

+ qfTJx)

(B3)

where : (B4) Since qBais an invariant tensor it can always be chosen of the form: (B5) Also, R(x) = mGQ(x)

036)

A,, A,, I!, ,17, are given by -d,(x) = $[O$P&C,

+ C,ZgT&g](x),

d,(x) = - QIy(C+qC$,)(x), (B7) I&(x) = [&DjC,

+ 2;c,](x),

I&(x) = ~Pfyc&,)(x), with

Following these definitions, we shall reduce (Bl) and (B2) to c numbers. First, substituting Eqs. (B3), (B6), and Eq. (B7) into Eq. (Bl) yields:

RENORMALIZATION

OF

GAUGE

317

THEORIES

where, for convenience, we have replaced (Y,p, y, 6 of Eq. (B9) by aI , J?, !y3, 1”. Finally, g(T”p);

- (pp);

- f;“.Jy

- 2@,“37,”

-L

2Iy-$,“]

=

0

(B13)

where -ra is defined by

(P&P); = t;,,zy + f’l;‘Ty

(Bl4)

In Eqs. (BIO), (B12) we have followed the conventions introduced in Appendix A, Thus, taking into account the definition of the cocycle condition given in Appendix A-and putting (tQre6y)A = f py,

(Pwqij

= [P, Wlij

(t’i?s)l,

= &!$,6,

(P/P);

= (Tom;

,

CB15)

)

we see that Eq. (BlO), (Bl l), (B12) can be written as: (&yy4 (dw);;-

.595/9W-3

= 0

+ r;‘““““t;j

0316)

= 0

(B17)

318

Comparing

BECCHI,

ROUET,

AND

STORA

the coboundary operators which operate on @$ and L’zB we see that:

Hence, taking into account Eq. (B17), Eq. (B18) assumes the form: [&y

-L @F)]y3

= 0.

0-W

Finally Eq. (B13) writes: (&q,“4~’ - 2q(@,*“*’ - p/y

= 0.

WI

In much the same way as for Eq. (Bl) expressing Eq. (B2) in terms of its coefficients yields : p%% = -(&yz”3,

0322)

oy’

=_ -[(cl’@y

Fl”i a

= -[(d’{,J?

-

&F])y”

+ (d’@)“b,“” Fb - yt:,F,],

= -[(#{,g

-

&))y

-

Jy

= -2[(&),B

- fa”V,rj],

+ q@p

(~23)

@TFJ)

(~24)

- ly)].

W5)

We are now going to show that Eqs. (B22), (B23), (B24), (B25) are consequences of the results of Appendix A, and that consequently 1?, 6, 2 can be chosen linear in r, 0, 2. First, Eq. (B22) is indeed a consequence of Eq. (B16) since the cochain r takes values in the adjoint representation of b which is semisimple so that ri = 0. Then, owing to the relation

Equation

(B17) becomes [d”(O - f+AtA)];;-

= 0.

(~27)

Since no antisymmetric invariant tensor of rank two on 1~exists if h is semisimple we see that Eq. (B27) and Eq. (B20) imply Eq. (B23) and Eq. (B24) respectively. Finally substituting Eq. (B23) into Eq. (B21) we get (d’,z),“flJ, + 2q(d’@),“fl,Y + 2q(ly’

- pyy)

= 0.

0328)

RENORMALIZATION

OF GAUGE THEORIES

319

Now considering e as a cochain of order one with values in the tensor product of the adjoint representation with itself and applying the coboundary operator d we get: (dlf);o*Y Comparing -p$

= f,““fs,Y_ f yy

r f-ply -_.f yp _ f;y,

(B29)

with Eq. (B22), =.=fp+p’

+ f,aqy

f f;lYAp3 _ ,,qy

__ fpp’Y _ fjwp,

(B30)

we see that Eq. (B28) has the form [d’(2 + 2q{O - f})p

= 0.

Since no invariant tensor of rank one on 1) exists if Ij is semisimple, can be solved according to

(B31

Eq. (B31

The possibility of choosing I?, 2, 6 linear in r, 0, ~7 stems from the explicit construction given in Appendix A. ACKNOWLEDGMENTS Two of us C. B., A. R., wish to thank CNRS for the kind hospitality extended to them at the Centre de Physique Theorique, Marseille, where this work was started. A. R. wishes to thank C. E. A. for financial support and Prof. W. Zimmermann for the kind hospitality extended to him at the Max Planck Institut ftir Physik und Astrophysik, in Munich. Two of us, A. R., R. S., wish to acknowledge the collaboration of E. Tirapegui, L. Quaranta at a very early stage of this work. We wish to thank the Service de Physique theorique, CEN Saclay where some results quoted here were obtained. One of us, R. S. wishes to thank G. ‘t Hooft, B. W. Lee, C. H. Llewellynn Smith, M. Veltman, J. Zinn Justin for informative discussions on gauge theories. We wish to thank J. H. Lowenstein and W. Zimmermamr, M. Berg&e and Y. M. P. Lam, F. Jegerlehner, and B. Schroer, B. Zuber and H. Stern for keeping us currently informed on their work, prior to publication, as well as H. Epstein, V. Glaser, K. Symanzik, and J. Iliopoulos for their kind interest in this work.

REFERENCES A. ROUET, AND R. STORA, Phys. Letr. 52B (1974), 344; Renormalization of the Abelian Higgs Kibble Model, Commun. Moth. Phys., to be published. 2. W. ZIMMERMANN, Ann. Phys. 77 (1973), 77 (1973), 570. 3. J. H. LOWENSTEIN, Commun. Math. Phys. 24 (1971), 1. 4. Y. M. P. LAM, Phys. Rec. D6 (1972), 2145; D7 (1973), 2943. M. BERGERE AND Y. M. P. LAM, to be published. 1. C. BECCHI,

320

BECCHI,

ROUET,

AND

STORA

5. H. EPSTEIN AND V. GLASER, Ann. Inst. Henri Poincarb, XIX, no 3, p. 211 (1973); in “Statistical Mechanics and Quantum Field Theory,” (C. de Witt, R. Stora, Ed.) Les Houches Summer School of Theoretical Physics, 1970, Gordon and Breach New York 1971; Colloquium on Renormalization Theory, CNRS, Centre de Physique ThBorique, Marseille June 1971, CERN TH 1344, June 1971. 6. M. GOMES AND J. H. L~WENSTEIN, Phys. Rev. D7 (1973), 550. 7. J. H. LOWENS~EIN AND B. SCHROER,Phys. Rev. D7 (1973), 1929; C. BECCHI, Commun. Math. Phys. (1973), 33-97. 8. A complete bibliography about the theory of gauge fields can be found for instance in: E. S. ABERS AND B. V. LEE, Phys. Reports 9C (Nov. 1973), no. 1; M. VELTMAN, Invited talk presented at the International Symposium on Electron and Photon Interactions at High Energies, Bonn, 27-31 August 1973 (The algebraic structure referred to here is however along the lines of [l].); The results of the present paper have been announced by C. Becchi, A. Rouet, R. Stora, CNRS Centre de Physique Thkorique, Marseille, Colloquium on Recent Progress in Lagrangian Field Theory, June 1974 and summarized in J. ILIOPOULOS, Progress in gauge theories, Rapporteur’s talk, 17th International Conference on High Energy Physics, London 1974. The symmetry property described in [l] is applied to traditional renormalization procedures in J. ZINN JUSTIN, Lectures given at the International Summer Institute for Theoretical Physics, Bonn 1974. 9. L. C. BIEDENHARN, in Colloquium on Group Theoretical Methods in Physics, CNRS Marseille June 5-9 (1972) (where however the implications of renormalizability thanks to which all algebraic problems reduce to essentially finite dimensional ones, have not been investigated); C. H. LLEWELLYNN SMITH, Phys. Lett. 46B (1973), 233, and private discussions; J. WESSAND B. ZUMINO, Phys. Lett. 37B (1971), 95. 10. S. L. ADLER, Phys. Rev. 177, 2426 (1969); “Lectures on Elementary Particles and Quantum Field Theory” (S. Deser, M. Grisaru, H. Pendleton Ed.), 1970 Brandeis University Summer Institute of Theoretical Physics, M.I.T. Press, Cambridge, Mass., 1970; W. BARDEEN, Phys. Rev. 184 (1969), 1848. 11. J. H. LOWENSTEIN AND B. SCHROER, Phys. Rev. D6 (1972), 1553. 12. J. H. LQWENSTEIN, to be published; C. BECCHI, to be published; T. CLARK AND A. ROUE~, to be published. 13. J. H. LOWENS~EIN AND W. ZIMMERMANN, to be published; PH. BLANCHARD AND R. SENEOR, CERN/TH 1420 preprint, Nov. 1971; CNRS Centre de Physique Thdorique Marseille, Colloquium on Renormalization Theory, June 1971, and to be published in Ann. Inst. Henri Poincar& 14. A. A. SLAVNOV, T M 0 10 (1972), 153. 15. L. D. FADDEEV AND V. N. POPOV, Phys. Lett. 25B (1967), 29. 16. G. ‘T HOOFT, Nucleur Phys. B35 (1971), 167; G. ‘T HOOFT AND M. VELTMAN, CNRS Marseille, Colloquium on Yang Mills Fields, June 1972, CERN/TH 1571. 17. SCminaire Sophus Lie, l?cole Normale Supbrieure Paris 1954: Tht?orie des Algebres de Lie. In [9], the relevance of cohomology theory in the present context can be detected. 18. G. ‘T HOOFT AND M. VELTMAN, CERN Yellow Report TH/73/9 “Diagrammar” (1973); Nuclear Phys. BSO (1972), 318. 19. The treatment of non linear gauges would require the introduction of two more multiplets of external fields, one coupled to the renormalized gauge function, one coupled to the renormalized Slavnov variation of the gauge function, with appropriate dimensions and quantum numbers. Cf. Ref [l]. 20. C. BECCHI, A. ROUET, AND R. STORA, Lectures given at the International School of Elementary Particle Physics, Basko Polje, Yugoslavia 14-29 Sept. 1974, and to be published.

RENORMALIZATION

OF GAUGE

THEORIES

321

21. Using the LSZ formula, it is enough to compute aZ,($)/ah, (X = K, m&J, and, from the Legendre transform structure of Z&Q to show that the matrix elements of the gauge function between physical states vanish, as a consequence of the Slavnov identity. More generally, one can check that physical matrix elements of time ordered products of gauge functions are disconnected. 22. G. 'T Hoorr, Nuclear Phys. FE35(1971), 167. 23. The treatment of double poles given in [l] can easily be adapted modulo the modifications due to the non hermiticity of the Lagrangian described here. 24. The main reference is: “Seminaire Sophus Lie 1. 1954/1955, “Theorie des Algebres de Lie,” &ole Normale Sup&ieure, Secretariat Mathematique, 11, rue Pierre Curie, Paris 5e. A summary can be found in N. Bourbaki, Groupes et Algebres de Lie, Chap. I, Sect. 3, Problem no. 12, Hermann, Paris, 1960.