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Transformation properties of proper vertices in gauge theories
Volume 46B, n u m b e r 2
PHYSICS LETTERS
17 September 1973
TRANSFORMATION PROPERTIES OF PROPER VERTICES IN G A U G E T H E O R I E S B.W. LEE* CERN Geneva, Switzerland and Institute for Theoretwal Physics , State University o f New York, Stony Brook, USA •
Received 2 July 1973 The Ward-Takahashi ldentltms for (single-particle irreducible) proper vertices are derived for non-Abelian gauge theories; t h e renormalizatlon o f such theories, previously discussed m terms o f the Ward-Takahashi identities for Green f u n c t m n s , can n o w be considerably simplified.
In discussing the renormalizability and unitafity of the S-matrix in (spontaneously broken) gauge theories, the Ward-Takahashi (WT) identities satisfied by Green functions play a central r6le. The renormalization procedure is usually stated in terms of proper (i.e., single-particle irreducible) vertices, so that the WT identities written in terms of proper vertices would enormously simplify the discussion of the renormalization procedure. It is the purpose of ttus note to provide a derivation of such identmes. In previous discussmns [ 1, 2] of the renormalizabilffy of these theories, one has to contend witht i the WT identities for Green functions, and a considerable amount of work was necessary to extract information on proper vertices therefrom. We shall adopt the convention previously used [3] that all fields [gauge fields b~(x) and scalar fields Ca(X)] are represented by ¢i where i stands for all attributes of the fields. Thus for the gauge field b Otu (x) i stands for the group index tx, the Lorentz index p and the spacetime variable x, collectively. Summation and integration over repeated inchces shall be understood. Thus, for example,
=fd4x( -
~b~(x)b:(x)g uv + ~t~a(x)~a(x)) ot
g I
a
where coa = ~Oa(Xa) is the space-time dependent parameter of a compact Lie group G. A~'as of the form As ["1"]~ 3 4 t = [ g J au (X - x ) , =0
ifOi = b3u(x)' otherwise,
where ga~ =gafaO Is the gauge coupling matrixt 2. We have
k + A?) where f ~ is the structure constant of the local gauge group. The invariance of the Lagrangian under this gauge transformation may be formulated as (A~ + ti/¢]) ~
= 0.
(1)
We shall always use a real basis for ¢~, so that the representation matrix of the Lie algebra [t~]i]= Wil.is real antisymmetric. In quantizing such a Lagrangian, we must choose a gauge. We consider a gauge condition linear in ¢. F [ O ] =F
Oi = O.
a
The infinitesimal local gauge transformation of ¢i may be written as * Supported in part by the NSF grant GP-32998X. Permanent address.
214
¢i -* ¢; = ¢i ÷ (AT +
t 1 Tiffs was especially true m t h e first-cxted reference m [ 1 ]. t 2 If t h e group G is a dtrect product o f n simple groups, GI (~ Ga ODD " ® Gn, .there are in general n gauge couphngs gl, g 2 , . . . ,gn" Within the same factor group Gi, £or course gtv = g3"
Volume 46B, number 2 Usually, one chooses
PHYSICS LETTERS
Fo,i to be
i.e., Xq [A ] are the propagators when the fields $ are constrained to have the vacuum expectation values A. Now we are in a position to write down the WT Identity for P[A]. First observe that
_!l g~A.~ if ¢i is a gauge field,
M-1III 6~-]exp(1Z[J])M-31[A+ Ii ~--J
~,
(2)
if ~bi is a scalar field, where c~, ~ are gauge fixing parameters, and ~ a r e some constants. In this gauge, the Feynman rules for constructing Green functions are deduced from the effective action Serf [4] Serf[Q ] = L[q~]- ½{F~[q~]}z + ~-M ~[¢] ca
(3)
where ~- and c are fictitious anticommuting fields which generate the so-called de Witt-Faddeev-Popov ghost loops [5, 6] and M ~ is given by
A +1 ~ji=A +iXi/[Al 6 1 8_~_exp (iZ [j])" = exp(tZ[J]) ~l
Recalling that Fa is linear in ¢, we write eq. (4) as - F [A ] - 8A--~-.
The operator M is in general not Herminan so that the ghost line is orlentable. In the following we shall be dealing with unrenormalized fields and coupling constants, and dimensionally regularized [7] quantities. As has been shown elsewhere [3], the generating functional W[J] of Green functions satisfies the WT identityt 3 ~LT~J
:o.
i
(7)
where
XM
{-
17 September 1973
;lf
+iX~
~*
,]
+
[AI 5
(8) =0.
Now define Fo[A ] by
rtA] = totAl - ½(F[AI }2.
(9)
Then from eq. (8) follows
(10) We must now examine
(4) The quantity M -1 [5/i5J] W[d] is the Green function for the fictitious field c in the presence of the external source J. The functional Z[J] defined by
•
6
=it~X/k[A]8~kM#2[A+iX-ff-~]
(11)
W[J] = exp(iZ [J]) is the generating functional of the connected Green functions. The Legendre transform I'[A] ofZ[d]:
ViAl =ZIJ] - Sdli, A, = 8 Z I J I / S J v
-Ji = ~r'tAl/SAi,
(5) is the generating functional of proper vertices [8]. For later use we define )(//.[.4] by
XiitA ] = -8Ai/8~
Upon definingt 4
G~[A] =M -1
+ iX
1
so that Gas [A] is the ghost propagator when the fields ¢ are constrained to have the vacuum expectation values A, we can solve eq. ( 1 I) for ~/~ [A ] :
(6)
Xt* [A i ~2 Pt A I/6A ~c6~ = *i/ ~s An eqmvalent diagrammatic derivation of eq. (4) is given in ref. [2].
t"4 Here one must dmtmguish between an operator equation such as (l+x a) 2 = 1+ ~+ x 2a2 and the equation resulting from its action on a certain function of x, such as (l+x~) 21 =1.
215
Volume 46B, number 2
77[A]
PHYSICS LETTERS
=-i~X/k[A]G#~[A]~G~ I
tAI.
02)
k Now eq. (10) may be written as 6 LaiIA ] ~ i
Fo[A] = 0
(13)
[ A I = A ~ ' +t~IA/+77[A ]
(14)
and -1 [.4] = F~trai[A ] . G~.
(15)
Note that G -1 [A] is the generating functional o f proper vertices with two ghost lines, so that
-~ - G ~ [ 0 ] GOa Is the reverse ghost propagator: =.
62G-I #.a
7#a, i-- 6A i
[0]
is the proper vertex of two ghosts at a and ~ and the field at i, etc. The system, ¢qs. ( 1 2 ) - ( 1 5 ) , is our main result. Eq. (13) defines a non-linear gauge transformation which leaves Fo [A] invariant. It is generalization o f eq. (1) and reduces to it in the tree approximation. On the basis o f eqs. (12)--(15), the considerations o f ref. [1] can be generalized readily to any compact Lie groups and arbitrary representations o f scalar
216
fieldst s. Furthermore it can be shown that the divergent parts o f the renormalization constants are insensitive to the parameter ~ in eq. (2) and therefore the renormalized S-matrix is independent of the gauges~'6'7. The details have been worked out and will be reported elsewhere. I thank S. Coleman, G. 't Hooft, M. Veltman and J. Zmn-Justin for much help they have given me.
where
L
17 September 1973
t s The discussmns m refs. [1] were tailored to the group SU(2) and other completely reducible compact Lie groups 0.e, groups in which the product of two irreducible representaUons R and R' contams a third R" at most once). ~-6 The renormahzablhty m the most general case, and the gauge independence of the renormahzed S-matrix have also been worked out, prior to me, by G. 't Hooft and M. Veltman by means of thetr diagrammatic analysis (private communication) and by J. Zlnn-Justin (unpubhshed manuscript) on the basis of eq. (4). t7 I have not full investigated the case in which Fa ts of dimension two and not hnear in fields.
References [1] B.W. Lee and J, Zmn-Justin, Phys. Rev. D5 (1972) 3121, 3137, 3155. [21 G, 't Hooft and M, Veltman, Nuclear Phys. B50 (1972) 318. [3] B.W. Lee and J. Zmn-Justin, Phys. Rev D7 (1973) 1049. [4] G. 't Hooft, Nuclear Phys. B33 (1971) 173. [5] B.S. de Wltt, P~ys. Rev. 162 (1967) 1195, 1239. [6] L.D Faddeev and V. Popov, Phys. Lett. 25B (1967) 29. [7] G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189. [8] G. Jona-Lasimo, Nuovo Clm. 34 (1964) 1790.
PHYSICS LETTERS
17 September 1973
TRANSFORMATION PROPERTIES OF PROPER VERTICES IN G A U G E T H E O R I E S B.W. LEE* CERN Geneva, Switzerland and Institute for Theoretwal Physics , State University o f New York, Stony Brook, USA •
Received 2 July 1973 The Ward-Takahashi ldentltms for (single-particle irreducible) proper vertices are derived for non-Abelian gauge theories; t h e renormalizatlon o f such theories, previously discussed m terms o f the Ward-Takahashi identities for Green f u n c t m n s , can n o w be considerably simplified.
In discussing the renormalizability and unitafity of the S-matrix in (spontaneously broken) gauge theories, the Ward-Takahashi (WT) identities satisfied by Green functions play a central r6le. The renormalization procedure is usually stated in terms of proper (i.e., single-particle irreducible) vertices, so that the WT identities written in terms of proper vertices would enormously simplify the discussion of the renormalization procedure. It is the purpose of ttus note to provide a derivation of such identmes. In previous discussmns [ 1, 2] of the renormalizabilffy of these theories, one has to contend witht i the WT identities for Green functions, and a considerable amount of work was necessary to extract information on proper vertices therefrom. We shall adopt the convention previously used [3] that all fields [gauge fields b~(x) and scalar fields Ca(X)] are represented by ¢i where i stands for all attributes of the fields. Thus for the gauge field b Otu (x) i stands for the group index tx, the Lorentz index p and the spacetime variable x, collectively. Summation and integration over repeated inchces shall be understood. Thus, for example,
=fd4x( -
~b~(x)b:(x)g uv + ~t~a(x)~a(x)) ot
g I
a
where coa = ~Oa(Xa) is the space-time dependent parameter of a compact Lie group G. A~'as of the form As ["1"]~ 3 4 t = [ g J au (X - x ) , =0
ifOi = b3u(x)' otherwise,
where ga~ =gafaO Is the gauge coupling matrixt 2. We have
k + A?) where f ~ is the structure constant of the local gauge group. The invariance of the Lagrangian under this gauge transformation may be formulated as (A~ + ti/¢]) ~
= 0.
(1)
We shall always use a real basis for ¢~, so that the representation matrix of the Lie algebra [t~]i]= Wil.is real antisymmetric. In quantizing such a Lagrangian, we must choose a gauge. We consider a gauge condition linear in ¢. F [ O ] =F
Oi = O.
a
The infinitesimal local gauge transformation of ¢i may be written as * Supported in part by the NSF grant GP-32998X. Permanent address.
214
¢i -* ¢; = ¢i ÷ (AT +
t 1 Tiffs was especially true m t h e first-cxted reference m [ 1 ]. t 2 If t h e group G is a dtrect product o f n simple groups, GI (~ Ga ODD " ® Gn, .there are in general n gauge couphngs gl, g 2 , . . . ,gn" Within the same factor group Gi, £or course gtv = g3"
Volume 46B, number 2 Usually, one chooses
PHYSICS LETTERS
Fo,i to be
i.e., Xq [A ] are the propagators when the fields $ are constrained to have the vacuum expectation values A. Now we are in a position to write down the WT Identity for P[A]. First observe that
_!l g~A.~ if ¢i is a gauge field,
M-1III 6~-]exp(1Z[J])M-31[A+ Ii ~--J
~,
(2)
if ~bi is a scalar field, where c~, ~ are gauge fixing parameters, and ~ a r e some constants. In this gauge, the Feynman rules for constructing Green functions are deduced from the effective action Serf [4] Serf[Q ] = L[q~]- ½{F~[q~]}z + ~-M ~[¢] ca
(3)
where ~- and c are fictitious anticommuting fields which generate the so-called de Witt-Faddeev-Popov ghost loops [5, 6] and M ~ is given by
A +1 ~ji=A +iXi/[Al 6 1 8_~_exp (iZ [j])" = exp(tZ[J]) ~l
Recalling that Fa is linear in ¢, we write eq. (4) as - F [A ] - 8A--~-.
The operator M is in general not Herminan so that the ghost line is orlentable. In the following we shall be dealing with unrenormalized fields and coupling constants, and dimensionally regularized [7] quantities. As has been shown elsewhere [3], the generating functional W[J] of Green functions satisfies the WT identityt 3 ~LT~J
:o.
i
(7)
where
XM
{-
17 September 1973
;lf
+iX~
~*
,]
+
[AI 5
(8) =0.
Now define Fo[A ] by
rtA] = totAl - ½(F[AI }2.
(9)
Then from eq. (8) follows
(10) We must now examine
(4) The quantity M -1 [5/i5J] W[d] is the Green function for the fictitious field c in the presence of the external source J. The functional Z[J] defined by
•
6
=it~X/k[A]8~kM#2[A+iX-ff-~]
(11)
W[J] = exp(iZ [J]) is the generating functional of the connected Green functions. The Legendre transform I'[A] ofZ[d]:
ViAl =ZIJ] - Sdli, A, = 8 Z I J I / S J v
-Ji = ~r'tAl/SAi,
(5) is the generating functional of proper vertices [8]. For later use we define )(//.[.4] by
XiitA ] = -8Ai/8~
Upon definingt 4
G~[A] =M -1
+ iX
1
so that Gas [A] is the ghost propagator when the fields ¢ are constrained to have the vacuum expectation values A, we can solve eq. ( 1 I) for ~/~ [A ] :
(6)
Xt* [A i ~2 Pt A I/6A ~c6~ = *i/ ~s An eqmvalent diagrammatic derivation of eq. (4) is given in ref. [2].
t"4 Here one must dmtmguish between an operator equation such as (l+x a) 2 = 1+ ~+ x 2a2 and the equation resulting from its action on a certain function of x, such as (l+x~) 21 =1.
215
Volume 46B, number 2
77[A]
PHYSICS LETTERS
=-i~X/k[A]G#~[A]~G~ I
tAI.
02)
k Now eq. (10) may be written as 6 LaiIA ] ~ i
Fo[A] = 0
(13)
[ A I = A ~ ' +t~IA/+77[A ]
(14)
and -1 [.4] = F~trai[A ] . G~.
(15)
Note that G -1 [A] is the generating functional o f proper vertices with two ghost lines, so that
-~ - G ~ [ 0 ] GOa Is the reverse ghost propagator: =.
62G-I #.a
7#a, i-- 6A i
[0]
is the proper vertex of two ghosts at a and ~ and the field at i, etc. The system, ¢qs. ( 1 2 ) - ( 1 5 ) , is our main result. Eq. (13) defines a non-linear gauge transformation which leaves Fo [A] invariant. It is generalization o f eq. (1) and reduces to it in the tree approximation. On the basis o f eqs. (12)--(15), the considerations o f ref. [1] can be generalized readily to any compact Lie groups and arbitrary representations o f scalar
216
fieldst s. Furthermore it can be shown that the divergent parts o f the renormalization constants are insensitive to the parameter ~ in eq. (2) and therefore the renormalized S-matrix is independent of the gauges~'6'7. The details have been worked out and will be reported elsewhere. I thank S. Coleman, G. 't Hooft, M. Veltman and J. Zmn-Justin for much help they have given me.
where
L
17 September 1973
t s The discussmns m refs. [1] were tailored to the group SU(2) and other completely reducible compact Lie groups 0.e, groups in which the product of two irreducible representaUons R and R' contams a third R" at most once). ~-6 The renormahzablhty m the most general case, and the gauge independence of the renormahzed S-matrix have also been worked out, prior to me, by G. 't Hooft and M. Veltman by means of thetr diagrammatic analysis (private communication) and by J. Zlnn-Justin (unpubhshed manuscript) on the basis of eq. (4). t7 I have not full investigated the case in which Fa ts of dimension two and not hnear in fields.
References [1] B.W. Lee and J, Zmn-Justin, Phys. Rev. D5 (1972) 3121, 3137, 3155. [21 G, 't Hooft and M, Veltman, Nuclear Phys. B50 (1972) 318. [3] B.W. Lee and J. Zmn-Justin, Phys. Rev D7 (1973) 1049. [4] G. 't Hooft, Nuclear Phys. B33 (1971) 173. [5] B.S. de Wltt, P~ys. Rev. 162 (1967) 1195, 1239. [6] L.D Faddeev and V. Popov, Phys. Lett. 25B (1967) 29. [7] G. 't Hooft and M. Veltman, Nuclear Phys. B44 (1972) 189. [8] G. Jona-Lasimo, Nuovo Clm. 34 (1964) 1790.