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Consequences of gauge invariance for the interacting vertices in non-abelian gauge theories
Nuclear Physics 0 North-Holland
B164 (1980) 152-170 Publishing Company
CONSEQUENCES OF GAUGE INVARIANCE FOR THE INTERACTING VERTICES IN NON-ABELIAN GAUGE THEORIES
S.K. KIM and M. BAKER
l
Utzirersity of Washir~gtor~,Seattle, WashingtorI 98195. USA Received
13 September
1979
The most general form of the triple-gluon vertex consistent with the requirements of gauge invariance is determined in both axial and covariant gauges.
1. Introduction
Gauge invariance imposes powerful constraints on the basic vertex functions of non-abelian gauge theories. These constraints, the Slavnov-Taylor identities [I], which play an essential role in demonstrating the renormalizability of Yang-Mills theory, are also important for non-perturbative studies of the theory. In this paper we find the most general solution of these constraints for the triple-gluon vertex I’3 in Yang-Mills theory. In sect. 2 we show that the triple-gluon vertex in the axial gauge is the sum of a kinematic singularity free longitudinal part determined in terms of the gluon propagator and an undetermined transverse part which vanishes when any one of the external momenta approaches zero. A special case of this solution has been used in the Schwinger-Dyson equations to obtain a closed set of equations for the gluon propagator [I] which is being used to study the infrared behavior of the theory. In covariant gauges the Slavnov-Taylor identities also involve the ghost propagator and the ghost-ghost-gluon vertex. In sect. 3 we show that the triple-gluon vertex in covariant gauges is the sum of a kinematic singularity free longitudinal part constructed from the gluon and ghost propagators and the ghost-ghost&on vertex, and an undetermined transverse part which vanishes when any one of the external momenta vanishes. In sect. 4 we calculate the longitudinal part in lowestorder perturbation theory in the Feynman gauge. Comparing this result with a perturbation theory calculation of the full triple-gluon vertex in the Feynman gauge now in progress [3] will yield the transverse part of the vertex. This would then give an understanding of the importance of the transverse part of r3 to the infrared behavior of the theory. l
John Simon Guggenheim
Memorial
Foundation
Fellow.
152
S.K. Kim, ill. Baker / Gauge irwariance
153
We conclude this introduction by writing the Feynman rules (see fig. 1) and the Slavnov-Taylor (ST) identities. In the covariant gauge, the first ST identity is f
GE”(l),
- +G) G&AK P : 41 GP2)
4, r) =J(~2)~“,~2
+ J~~2x&2
- 4*4/J cgJ4?
The second ST identity
P : y) G(P21 .
(1.1)
[I] is
6 s) = J(q2)k%72 - 4*qV) ~~~%5(q~ P : c s> G(P2) + JCr2Xg$-2 -
r$,,)W$&dcP
: 4,
+J(s2)(g;s2 - sAssa) l-$$&,(s,p : q,
s) Gb2) r)
G(p2)
- r;f&j,(q, ST-4 - s) rg&
+ s, P : r) G(P2)
- %&d4,
+ r. P : s) G(P2)
r9 -4 - r) E$fl(q
- rl;dc$(r, s, -r - s) r$&(r ghost
propagator
- - - -> - -ba P gluan
(1.2)
t s, p : q) G(p2) ,
r&(p)
= Sap-L P2
propagator PA pp
DX,“b(p)=-~[gA$~-a)T
I
ghost-ghost-gluanvertex
0.P triple
gluan
vertex
r.cc
A PAa
g,r$);JP.w) q
-g,fObC q,/O
[g +(p-d,+ g Jr
g&g-r)A+
-p),]
four gluan vertex S3.d
r.y,c
g~C$~,~(p~w,s) =-ig, 2 [ f obefcde(gXvg/18-gXSg/J
x PAa
g.@
+f accfbde(gXpg,8-g*ggJ +fadefcbe(gX”g~g-gX~g~~)]
Fig. 1. Feynman
rules in covariant
gauges.
where J(q*) is the gluon self-energy which is defined in terms of the inverse gluon propagator as &%g)
= (q2$U -
(1.3)
G(p*) is the full ghost propagator,
g a Pb” aghuv(p, q, Y) is the proper triple-gluon
vertex,
g~f’$$(‘, q, r, s) is the proper four-gluon vertex: p”P$&, q : r) is the proper vertex for the coupling of a gluon of momentum Y, polarization V, and color index c, with two ghosts, of momentum p and q and color indices a and b, $“I-a,f$&@, q : r; s) is the proper vertex for the coupling of two gluons of momentum r and s, polarization v and 6, and color indices c and d, with two ghosts, of momentum p and cl and color indices a and b, respectively. In the axial gauge, we use the notation employed by Kummer [4]. The full gluon propagator is written as a;;(q)
-1
= q2(g2
~
74 [ tt2(1
n2f2)
n*q, ’t?
(
-“/s)
1 +
k2
-
((cl
~)*/q*)f*l
( gpv
+
_-
n*q/
q. II ,I 2
1
2 4
I(lJ
tzP -__q.n
Yqtlptl”
l-y,
[ IIZ
qpq”t~
+m-
$.lqu
+ t1,q/17 q’tl
ii
(1.4) and g2 in general depend on where y4 = tz*q*/(q tl)‘. The invariant functionsj; and (q . tz)*. The inverse propagator, connected to A,, by the relation
q*
(1.5) can be written
up, = k2&.I”g2
+ (4.
nj*f2
The ST identities
[
-
gpu
4/A”) -
2 4 tip4 ___ (q n)*
+ Slcer;:;ffi
+ ttuq@ (4.n)
1
(1.6)
’
are
p”Gb$V (P, q, r) = -Pbc P”P:b;:s(P.
qjltt, +
[q&7)
- rr,,(r)l
ecd ( P + q, q, r? s) = i]f” be rs& (P
+ r,
4,
s)
In the axial gauge, the Feynman covariant gauge.
+ fUdTg”
(p
r,
(1.7)
, s)
+ s,
q, r)]
(1.8)
rules for the gluon vertices are the same as in the
2. The triple-gluon vertex in the axial gauge In the axial gauge we define the arguments
of a function
with color indices as
F abc E FabC(p, q, r) . We use number indices to represent the function F, = F[p2,
(P.
F, -F[q2
,(q.
without color indices, i.e.,
n121, /z)~] ,
F3 = F [r2 , (r. H)~] . In the axial gauge the most general tensor structure which satisfies Bose symmetry is the following: r:“Gy (p, q, r) = -i[-gu(Anbcpv
of the triple-gluon
vertex
t Abacq,)
+ (A@p”
t fvbac&J yApp t i(P abc tPncb) (Gphqurp t Sphqg-“) t i(pabC - Pacb) S(rhp,q, - qhpd,) + $I Y/K/~h~~p\~r, + Rybc gxc,~z, t (Tybq,
t Tfbap,) nAn,, t S fbUrl-lr,,flht SfMqPqV,lh t (Sp” - SF)
x r,y,rJ,,
+ (Sf”” + .sp
- Sqbca) qvr,nA]
+ cyclic symmetric
terms , (2.1)
where P abc = pbca = pcab ,
abc _
-R;“,
RI
(S, - s3ybc = (S, - Qac,
SqabC = sp
.
(2.2)
and Xabc IS completely symmetric under the interchange of any pair of variables and color indices. By substituting eqs. (1.6) and (2.1) into the first ST identity, we obtain the following scalar equations: ~ (AabC _ Abac) _ (Aacb ~ ACab) + p
r (pacb t Ncba _ lVnb t NaCb)
t p . q(Pbc t Nbca - NbaC t Nnbc) t p . n (Sp
-
Sp)
=0 ,
(2.3)
- Aabc - Aacb t p rNacb t p . qNabC t p . ,,(Sy”” t Sic’ - S,b’“) = 0 , (2.4) - ACbap . r ~ AbCap . q t Rfbap
tJ = fabc [q2g2
- (q
n)2.f2
- r2g3
t (r. n)2f3. (2.5)
~ (,@bc
- AbE)
- Aacb + p
q(NabC
- Nb’c)
+
p . y(pacb
+ Nacb)
t p . nsfCa = -jabcg2 ,
(2.6) _ (A“Cb - Acab) - Aabc t p . r(Nacb - NCub) t p . q(pabc t ,Vabc) t p . nSfba =fnbcs3
,
(2.7)
156
SK. Kim, M. Baker / Gauge invariance
_ R;bC +p.
r(Sy
_R;bct p . rSfbc
- S;“) +p
+ Sibcp.
(I - Tpp
+p.
q (SYb -Syb)+p.
_ Rfb
tp.
qS’fcb +p.
rSTb -p.
rSFb +p.
IIf3 ,
p . 11= -fabcr
qSgbc + ( Tpm - Tf”)
_Rrb
rzf2 ,
. H = ybcq.
nfi,
- T~ba)=~bcq.
p. rTybc + p. qTfcb + p. nXabc = fabc(-q2f2 t r2f3) .
(2.11j
(2.12)
the sum ofeqs. (2.3) and (2.4) from that of eqs. (2.6) and (2.7) yields
Subtracting p
(2.9)
nf3, (2.10)
n Tfba=-jQbcr.
n(Tfab
(2.8)
rNcba + p . qNbCa - p . nSp
= fabc(g2
- g3).
(2.13)
If we write N cba = ;(Ncba ~ Nb‘=) + $(NCba + Nbca) , eq. (2.13) allows us to determine ;(NCba _ Nbca) = &
the antisymmetric [fabc(g2 -8s)
combination
+p.
(Ncbn - Nbca) as
nS,bca t ;p2(Nbc=
+Ncb”)]
.
Thus we can determine Ncba as 1 N cba _- z(N lzba + Nbca) + &
[jab”(g2
- g3) + p. r&y
+ ip2 (Nbca + Ncba)] (2.14)
By subtracting
eq. (2.4) from eq. (2.6), we obtain
~~~ _ p . q~baC + p rpacb _ p . n ($ba _ s;ba) = -g2ybc
.
(2.15)
From eqs. (2.14) and (2.1.5), we can then determine Aabc as A abc = fabCgl + q$f?$
[fnbc(gl - g2) + r . nSibc t
$r2(Nabc
+ $p . q(NabC t Nbac) - q. rPabc + q. ,I (Sy” - Sy”) . Substitute
1
-p.
[r . I@~
&
,z(S~
- Sp)
~ Ss(“‘) + q . rz(Sy” - Syb)
- 4. r(Pbc
+ Pacb)] .
eqs. (2.13) and (2.15) into eq. (2.5). We then obtain
RbC‘, = 1
(2.16)
eq. (2.15) into eq. (2.4). We can then determine SFba as
Scba _-
Substitute
+ Nbac)]
+ y ’ rSica
(2.17)
SK. Kim, hf. Baker 1 Gauge invariance
+
-L [p. qr.
P.n
-p.
t$qbc - St”‘) + p
rq
157
n(Syb - S,“““)
rp. q(PabCtPmb)].
(2.18)
Then, from eqs. (2.8) and (2.18), we obtain TbcQ =
1
-~
rq
[(p . n) fl f
1 p.
n r. II
(q
. n) fil
[p.rq.n(Ly-
- E*
(Sb;lc - SF,
SYb) t 4 r/J. n (sf””
- S,‘““)
- q . t-p . r(Pabc t Pncb)] . Finally, if we substitute
(2.19)
eq. (2.19) into eq. (2.12)
Xabc=f=bCf,(E
tps)t(;:
+ cyclic symmetric
Xabc is determined
as
:*;I ; (Sf”” 4,““‘)
terms .
(2.20)
Those functions N, A, St, R,, T, and X given in eqs. (2.14) through (2.20) satisfy eqs. (2.3) through (2.12) for the arbitrary values of S,, Sq, P and the symmetric part of N. However, all the functions we have determined are not manifestly free of kinematic singularities. As an example, we see that the part of Rp in eq. (2.18) determined by the gluon inverse propagator has a kinematic singularity at p. n = 0. As a first step toward removing these kinematic singularities of l/p. n type, we define a function Ry as p~qr.nlabCtp.rq~nInCb-p.rp.q(PabctPacb) ~f”bcq.t~r.n(fj-f~)tR~,
(2.21)
where I abc G (,qbc _ sQbc),
RbCa 2
= Rcba 2
.
We then obtain from eqs. (2.18) through (2.2 1) R~=~bc[q.nf2-rInf~]
+q.rSp+:4,
(2.22)
(2.23)
158
S.K. Kim, M. Baker / Gauge invariance
X abc = +q.
From terms ever, other
1 p. nq.
nr. n
rq. r(Pabc tPacb)
[p. 4~.
rRtM +p. rRyb +p. qR;bc] .
(2.24)
the above we see that the parts of the functions R, and T, determined in of the function fare free of kinematic singularities of the 1/p . n type. Howthey have other types of kinematic singularities. I and R2 are related to each by eq. (2.21). Thus we can determine Iabc/q. n - facb/r. n as
pyf3
t ~ q.
1 nr.
n
-f2)+&?*
+p . rp . q (Pabc t Pacb)]
[Rp
Then the function Icba which determines can be determined immediately as
=
p$$
+Ar2 2
.
(2.25)
I
the function
[R;”
fabc(f2 - fl)t p%
[cab I cba __ t-
(
q.tz
p.n. )I
I t+q.n
cba
-+-.
( ,q.
n
Tf”” partly by eq. (2.23)
t p. rq . r(Pabc + Pacb)]
Icab
p. n.)
(2.26)
Note that a kinematic singularity at p* = q* is introduced into Tyb by the function Icba given in eq. (2.26). From eqs. (2.14), (2.16) and (2.26) we see that the only remaining kinematic singularities are of the types q+
&[q*,(q.
)I)*1-g[r*,(r. n)*lI,
--!---Cf[q2,C7~n)21 -f[r*,(r~n)*lI.
(2.27)
(2.28)
q* - r*
To remove the kinematic singularities of the types given in eqs. (2.27) and (2.28), we define the following functions g2,3 and f2,3 as g2,3 = +k[q2,
(4. n>*l
-dq*,
Cr.
n)*l +g[r*, (4. n)*l -dr*, Cr.n)*l it
(2.29j
f2,3 'k(f(q*,(q.fl)*l
-f[q*?(r'n)*l
tf[r2T(q'n)21
-f[r2p(r'n)21).
(2.30)
SK.
Note that the functions g2,3 andf,,, are antisymmetric q and r, and vanish when p . II = 0, i.e., g2.3
p
0 3
+ n-0
g2.3
+ 92,,2
under the interchange
+ 0. p n+O
fi,3
Furthermore g2,3 andf2,3 limit of q* = r2, i.e.,
159
Kittt, hf. Baker / Gauge inrariancc
(2.31)
approach (g2 - g3) and (f2 - f3) respectively
f2,3 q2
’
g2 -g3
of
f2 -
in the
(2.32)
f3 ’
It then follows that &&2
-g3
-g2,3)9
g-y2
are regular at q2 = r2. We next redefine the undetermined p . ns~
+ ip2 (~bc= + @a)
(f2
functions
-f3
-f2,3)
given in eqs. (2.14) and (2.25) as
E _ fCbcg2,3 + Fbc= ,
(2.33)
(2.34) where FbCa and Gab’ are arbitrary FbC‘,
= FCba
G
symmetric
functions,
i.e.,
abc = G bat
(2.35)
By eliminating write the functions (Icba/q ’ n t Icab/p triple-gluon vertex
S, and R2 from eqs. (2.14) (2.16) (2.17), (2.22) (2.24) we can A, N, R ,,X and S, in terms of the functions g, f, (Nabc t Nb”), ’ n), Pabe, G and F.From these functions we can express the r3 in the form,
rfl;i,(~
4, r)
4, r) = r,L”,b,“(p, 4, r) + r$cb,‘(p,
where the longitudinal
part rp)
1
+adb
4 - qAPg,)+ 1
[
P.
nfi
-
4.
(2.36)
is
r$$Yp~ 4* r) = -ifabc -gw +(23&P.
,
1
[
pa
- 4d2 + (P, - qJ “,:‘“h:
z2 (gr -g2
P.Y
nf2 -Gnl,2
-
g1,2)+nhnpnv~fl,2
p.
+- r.
nq.
n
n fi,2]
(f,
- f2 -f,,2)]
160
S.K. Kim, M. Baker / Gauge invariance
+ rzhn,
[
pufi -
(fi -
cr,f2 - (P, - qv)p*
tq%r.nfl,3-p%
fz- fu)
r.nf2,3 1
‘P - p/J
g1.2
+ppqhnv ~+2 h - qh -22
.
rhw
n (fi-f3
-fi,3)
--r
n(f2 -f3
rgbq.
--r
4
P
+ cyclic symmetric
-f2,3)
I
terms,
(2.37)
and the transverse part lYiT) is defined as @$F(P,
4, r) = Fabc(p.
t $(Nabc
gw - pcrqA)
t Nbu )@.m,
(
-Ppqh)(ru-hr2)
+ Gabc p.nq.n(g~P.r2q.ntp.qn^n~-p.nqhnl, -4.v,nd pv-qv’ 42
( +
x
4
r. n
1 IZ
)
+ Icba
-)khPp. q.n
i2p.n
nq. n+p.
qw,
nqhn,
-p.
- 4. v,d
X r, - r2 n, r. n ) (
t ;(Pabc
t ; (Pabc t Pmb)
-
( - 4.
v,m
-
P.
2~. rq . r p. nq.
t
vvtP'
cyclic symmetric
nr.
n
w.
n&up.
n + P. ww,
wkq.A
+ 2~. 4p. rq. r nhn,n, 3~. n q. nr. n +gA&(q.
a) + ~kdwh - wwJ1
rp, - P.
- Pacb)[ghJq.
%-PAqvrp
terms .
2q. r - wPpv P.
n t$Aqprv I
(2.38)
161
SK. Kitn, M. Baker / Gauge invariance
The amplitude rF) defined in eq. (2.37) is manifestly free of kinematic singularities because of eqs. (2.31) and (2.32). rp) satisfies the homogeneous ST identity pAI?@
(P. 49 4 = 0
(2.39)
1
while f’iL) satisfies the inhomogeneous phr$L$“c (P, 4, y) = -Pbc
eq. (1.7), i.e.,
ST identity,
[aPA)
-
npvWl .
(2.40)
Since the complete r3 which is the sum of I’iL) and I’?) must be free of kinematic singularities, r3CT) itself must be free of kinematic singularities. We now differentiate eq. (2.39) with respect to p”I and obtain 4, Y) + P” $
r$;r(p,
rc,T,l”,b”(p,
4, Y) = 0
(2.41)
.
Since rr) is free of kinematic singularities, p”@/ap”) F’gp vanishes in the limit of p + 0. We then conclude from eq. (2.41) that the transverse part I’scT)vanishes in the kinematic regime where any one of three momenta approaches zero. Thus in this regime the total triple-gluon vertex f’s is uniquely determined in terms of gluon propagator. Note that if 7~~~is independent of n, then from eq. (1 S), f= 0 and g is a function only of q*. In this case the general expression for l?iL), eq. (2.37), reduces to the simple form - &.l(P”g,
r~~?n,bc(P, 49 4 = -ifabe
-q&T*)
[ +(gPipP.
4 -4aPJp2
+ cyclic symmetric
E!Lz!qg, _42
-g*)
1 (2.42)
terms .
The result, eq. (2.42), was used in ref. [2] to analyze the infrared behavior of YangMills theory.
3. The triple-gluon
vertex in the covariant gauge
In the covariant gauge, the most general tensor structure which satisfies Bose symmetry is rf&(p,
4. I) = -gU(AabcpV
+ (NabCpv + Nbacq,) +
of the triple-&on
vertex
+ Abmq,)
qhp,,
@abc+Pacb) (-Phcld+,+ $w7Mrv)
+ $ (pabc - Pacb) $hp,q, - w-w,) + cyclic symmetric
terms,
(3.1)
162
S.K. Kim, hf. Baker / Gauge invariance
where Pk is completely color indices. The ghost-ghost-gluon
symmetric
under the cyclic permutation
(p, 4 : r) (fig. 1) has the general form
vertex p”r$$,
(p, q : r) = AibcgA, + BibCpflv.+
f$v
of its argument and
CibCrhrv
+ (Dibc t Cibc) rhpu + Eibcrvph = Aibcgh,
t B,“bcpflp, - CibCrhqv
t Dzbcr,,pg, t Eibcpkrv
.
(3 4
Substituting eqs. (3.1) and (3.3) into the first ST identity obtain the following equations:
_p .rAcba _ p . qAbca = r2Arb + q2Ap
given in eq. (1.1), we
,
(3.3)
A abc f AaCb _ ACab _ p. r(Na=b _ Ncab) _ p. q(pabc f Nabc) + q. r(DP”
=“f”
- q2CF
t Cf”)
,
(3.4)
Aabc + AaCb _ Abac _ p . q(lvabc _ Nbac) _ p . r(pmb + Nacb) =A,b’lc+q.
r(~~aC+C~)-r2C~b,
AabC+Aacb_p.qNabC_p.rNacb=q.r(Cpbac A
abc
_
AbaC
+ AaCb
_
ACab
_
_ p. q(pabc + NbCQ+ pbc
p
(3.5) tcp),
(3.6)
. r (pacb + @‘a + NaCf’ _ pb) _ Nbac) = _r’cDP”
+ cr”)
- q2(Dim
t CF) (3.7)
where we defined
r-cab 3g -p2G(p2)J(r2)
rgb
,
,
i.e., 1:”
=p2G(p2)J(r2)
Afb
?r”
fpzG(p2)J(r2)
C,“” ,
-p” D
=p2G(p2)J(r2)
Dfb
Subtracting yields
.
(3.8)
the sum of eqs. (3.4) and (3.5) from the sum of eqs. (3.6) and (3.7)
p.rNcb”tp.qNb”=A~b+Agb~_~.rD~b_~.qD$C.
(3.9)
The form of eq. (3.9) is similar to that of eq. (2.13). Thus by writing N cba = ;(Ncba + NbCa) + $(Ncba _ NbcQ) ,
(3.10)
S.K. Kim, hf. Baker / Gauge invariance
163
we can determine Ncba as
+
&
Subtracting
;(Ncba
t NbcQ)t :(NcbQt NbcQ).
(3.11)
eq. (3.5) from eq. (3.6) yields
AbaC _p.qNbQC+p.rPQcb=_~~_-_.r~Cgc(lb
- q.rDF.
(3.12)
Next consider the sum: (3.3)fp.r
-P.
(3.12)+q.
r(3.9)
y[(3.12)-(=)(3.6)].
(3.13)
Eq. (3.13) yields A~b+~~bQ-p.qC~bQ+q.rD~b+~.rD~b~=~.
Eq. (3.14) gives us a constraint gator and the ghost-ghost-gluon obtain (P
abc + pub)
= _ (C;bc
Finally, if we substitute A
abc -_
-nbc 4
-q.
(3.14)
among the invariant functions of the ghost propavertex. Next from eqs. (3.6), (3.12) and (3.14) we + 2;,b”
+ “fb
+ Czcb + C,“”
+ ,;bQ)
eqs. (3.11) and (3.15) into eq. (3.12) ,.ClbQ
_p.
(3.15)
we find
rjjibc
Qbc + Nbm)
_
!_g
r (pQbc _ pQcb)
.
(3.16)
Now it is straightforward to show that the invariant functions N, P and A given in eqs. (3.1 l), (3.15) and (3.16) satisfy eqs. (3.3) through (3.7) for arbitrary values of (NQbc + Nbac) and (Pabc - PQcb). We can now express the triple-gluon vertex r3 as the sum of two parts, a longitudinal part rgL) and a transverse part r$r) rf&(p,
q, r) = r$L$Vbc(~, 4, r) f r$T$vbyp, 4, r) ,
(3.17)
164
S. K. Kim, M Baker / Gauge invariance
where, from eqs. (3.1 I), (3.15) and (3.16), 4, r) = gh, [p,(ZibC + 4. r Ciba + p . r Dibc)
rSL$jc(p, +q,(Ap
,Pv
tp.
r
CP”
tq.
rD,b”)]
- 4v
q2-_p2(qhPp-g~P.q)(Apb”tA~b-q.rb~ca-p.rD~b)
x
(Cagbc
+
“g””
+
CgCab+ "p"" t
Cgbac + Fib”>
terms ,
t cyclic symmetric
(3.18)
1 t : (PQbC - Pacb)
+ cyclic The amplitude <
P
hJq.
riLzF (p,
- P.
t $(rhpu
-
(gir’
while L’iT) satisfies the homogeneous (P, 4, r) = 0
4)
wwJ1
defined in eq. (3.18) satisfies the first ST identity, q, r) = J(r2)
gA&p.
(3.19) eq. (l.l),
i.e.,
- rhrv) K’gbu(r, p : q) G(p’)
+J(q2)
rq,)
terms .
symmetric
@)
rp,
(4h4P-
: r) G(p2)
1
(3.20)
ST identity (3.21)
.
It is easy to see that I’cL) given in eq. (3.18) is manifestly free of kinematic singularities. Thus, in a similar way as we did in sect . 2, the undetermined transverse part vanishes in the kinematic regime where any one of three momenta vanishes.
4. Perturbative
calculation
of the triple-gluon
vertex
We will construct the longitudinal part of r3 perturbatively. We first calculate the ghost propagator, the gluon self-energy and the ghost-ghost-gluon vertex to order g:. We can then construct riL) by using eq. ( 3.18). The ghost-ghost-gluon vertex rjg is represented by the graphs of fig. 2. In the Feynman gauge the integral expres-
S.K. Kim, hf. Baker / Gauge ittvariatzce
The ghost -ghost -gluon
P.0
165
vertex
k-P
Fig. 2. The second-order
ghost-ghost-gluon
vertex.
sion for r3g is l?$Yti(-p,
g : r) = gti,fabc-
i i(g&>
-PArv + ITrAp, + khrv + 2pp^k, - 3r&,
- k^P, + gh,(k + r) (k - P) -.
k*(k ~ P)*(k - r)*
(4.1)
with -p + q t r = 0, where we take t sign for outgoing momentum
and C2 is defined
by
facdfbcd = C2hab. We see that there are three types of intergrals in eq. (4.1). They are defined as
‘(P*?q*?r2)E[d4k
k2(k
Fh =
s
d4k
_
:j2
k^ k2(k - p)* (k - r)*
(k
_
r)2
“F,(p2,
(4.2)
’
q*,
r*)pA
+F2(p2,q2,
r*)rh,
(4.3) J(P*,
q*, r2) =Jd4k
The integration J=rli(log$
(k _ p):
(k _ r)~ .
(4.4)
for J is easily carried out. The result is + I) ,
(4.5)
S.K. Kim. M. Baker / Gauge imariance
166
where AZ is an ultraviolet cut-off. The other two integrals I and F, cannot be expressed in terms of elementary functions. However, the three invariant functions I, FI and F2 are related to each other. One trivial relation is F,(P*,
4*, r*)
q2,
P*)
=MP*,
r*) .
q*,
(4.6)
both sides of eq. (4.3) by ph yields
Multiplying F,p*
=F~(Y*,
r=
+F2p.
j
P.k k* (k - p)* (k - r)*
d4k
k* +p* -(k-p)*
=_; jd4k
(4.7)
k*(k - p)*(k - r)* r*
1. *It$r*ilog~. =*P
eq. (4.3) by rh yields
Multiplying
2
ir*I t in*i log< = Flp. 4
r + F2r2 .
(4.8)
From eqs. (4.7) and (4.8), we obtain I(P*,
q*, r*) =
=I_~(P*,
-2~.
qF, - 2q. rF, t n*i log(r*/p*) r2 -p*
q*, r*) + IA@*,
q*, P*> +
q*
[IA(r*,q*,
P*)
-IA(P*,
q*, r*N
+ n*i los(r*/p*)
r* -p* (4.9)
Now recall the definition IV&-p,
of the invariant
functions
of I’ss given in eq. (3.2)
q, r) = AibCgb + Bibcphp, t CibC thrv - (Dzbc + Clbc) r^p, - EibC pkrv . (4.10)
Comparing eq. (4.10) with eq. (4. l), we find A,(P*,q*J*) =
- 1 = jd”k
(k+r).(k-p) k2(k
_
d4kk2-p.rt~[(k-p)2-p2-k2] j
+ c&7*
p)2
Ck
_
r)2
-i[(k-r)*-r*-k*]
k* (k - p)2 (k - r)*
P*)Q*, 4*.r*> ,
(4.11)
S.K. Kim.
&(P2.q*, r2)=F, C,(p’,
r2) = -?-I+
167
/ Gauge irwariar~ce
(4.12)
>
42, r2) = -2F,
D&2,$,
hf. Baker
)
(4.13)
3(F, + F,) )
(4.14)
E&p*, q*, r2) = I - F, - 2F,
(4.15)
where we have omitted an overall factor -&(C2/32n4) from all above functions. We now change the momentum-conservation law from -p t q t r = 0 top + q t r = 0 in the following calculations, since all the invariant functions of the ghost-ghostgluon vertex are expressed in terms of p2, y2 and r* in eqs. (4.11) through (4.15). The ghost propagator G(p*) and the gluon self-energy are, to order&,
C(p’)=S
(,I+%
log$).
(4.16) We first note that the perturbation expressions eqs. (4.1 l), (4.13), (4.14) and (4.16) satisfy the general consistency condition, eq. (3.14) as required by gauge invariance. From eqs. (4.11) through (4.16) we find A;b= = J($)
$G
(q2) AEbC
=/.bc[l+g$$
(-~log$+?log$
2
+flog$+l
.2 ) -rg, &4
CbC =J(P2) _
i&2
=J(p2)
q*G(q*)
From these three functions, late -$C A
+q.
1,
(4.17)
CibC
2F2fabc ,
32n4 -;bc D
q2G(q2)
(iq” - p*)l
rczba
=P”‘[l+$$
(4.18) k82 D;bC = _ __
32n4
we can construct
abc L-21
f
log;
3(F, + F,)l
(4.19)
rk given in eq. (3.18). We first calcu-
+p. rB:bc
(- y
+
+21,,$
SK. Kim, hf. Baker / Gauge invariance
168
+=logg+ 1)I r2 - p2
2 IA(r2, ~
-i%pbc[-p.rq
+(p.
r-q’
q2, p2> - I,4(p2, r2 _p2
q2. r2)
1
- r2)I_4(p2,q2,r2)
[email protected]+q2
(4.20)
-p2)I_4(r2,q2,p2),
where we used eqs. (4.6) and (4.9). Next we calculate
+_J;ba _ q. rD;ba A-;bc_ p, q jj;bc
&
-
2
3g:c2 --Tf-32n
q2 [IA@‘,
q2, p2> -I_4b2,
q2, r2)1
1.
(4.21)
We can then express riL) as
( ( &2
rFzvbc (P, 4, r) =fabcgu,pv 1+
‘?;cz
--1 s
-p
. rq
A2
- p log3
3
q2, P’) - r~(p’,
2 I,(r2,
+2log7
q2, r2)
r2 - p2
t(p.r-q2
- r2) rA(p2,
+(p.rtq2
-p2)IA(r2,q2,p2)
q2, r2)
1)
+
fabc
-kh,(q + q,w,
L-g*@. . ‘P,
rq,
+ 4.
+ p . 4r,
+ r,p,qh
-
I+
ipy) phw,
Ph4gvl
-
g,,(P.
vi
+ P.
rqh)
A2
S.K. Kim, M. Baker / Gauge invariance
169
s;c*I_4(P*,4*,r*)
X - i32 [
1
+ (3! - 1) remaining
permutations
Comparing eq. (4.22) to the perturbation the transverse part K’$$$“(p, 4, Y).
. result for r$&(p,
(4.22) 4, r) [3] then gives
In the limit when any one of the three momenta vanishes, I$-) vanishes and the full vertex is determined by r3(L) . In this limit, the explicit log p*, log q2 and log r* terms in eq. (4.22) are cancelled by corresponding logarithms in I,, and I’cL) is finite in that limit. More explicitly we find FI (p*, 4*, rZ) = IA($,
4*, r2)
n*i ---2
P
r+O
p*=q2
---$$
(1 +log$)
p*=,* -$(l+log$).
(4.23)
;2:y.2
From eqs. (4.22) and (4.23), we obtain
+
(4~
-
cd
(r,q,
-
g,,q.
4
~
-
(4.24)
5. Summary We have found the general solution of the Slavnov-Taylor identities for the triple-&on vertex r3 in both covariant and axial gauges. The solutions for the quadruple-gluon vertex will be presented in a future publication. These results combined with the Schwinger-Dyson equations then provide a non-perturbative approach to the study of the infrared behavior of Yang-Mills theory.
We would like to thank R. Anishetty and J.S. Ball for valuable conversations. One of us (MB) would like to thank the John Simon Guggenheim Memorial Foundation for support during the course of this work.
170
SK. Kim, 151.Baker / Gauge invariance
References [l]
A.A. Slavnov, Theor. Mat. Fiz. 10 (1972) (Theor. Math. Phys. J.C. Taylor, Nucl. Phys. 10 (1971) 99. [2] R. Anishetty, M. Baker, J.S. Ball, S.K. Kim and F. Zachariasen, [3] J.S. Ball and T.W. Chiu, to be published. [4] W. Kummer, Acta Phys. Austriaca 41 (1975) 315.
10 (1972)
99;
Phys. Lett. 86B (1979)
52.
B164 (1980) 152-170 Publishing Company
CONSEQUENCES OF GAUGE INVARIANCE FOR THE INTERACTING VERTICES IN NON-ABELIAN GAUGE THEORIES
S.K. KIM and M. BAKER
l
Utzirersity of Washir~gtor~,Seattle, WashingtorI 98195. USA Received
13 September
1979
The most general form of the triple-gluon vertex consistent with the requirements of gauge invariance is determined in both axial and covariant gauges.
1. Introduction
Gauge invariance imposes powerful constraints on the basic vertex functions of non-abelian gauge theories. These constraints, the Slavnov-Taylor identities [I], which play an essential role in demonstrating the renormalizability of Yang-Mills theory, are also important for non-perturbative studies of the theory. In this paper we find the most general solution of these constraints for the triple-gluon vertex I’3 in Yang-Mills theory. In sect. 2 we show that the triple-gluon vertex in the axial gauge is the sum of a kinematic singularity free longitudinal part determined in terms of the gluon propagator and an undetermined transverse part which vanishes when any one of the external momenta approaches zero. A special case of this solution has been used in the Schwinger-Dyson equations to obtain a closed set of equations for the gluon propagator [I] which is being used to study the infrared behavior of the theory. In covariant gauges the Slavnov-Taylor identities also involve the ghost propagator and the ghost-ghost-gluon vertex. In sect. 3 we show that the triple-gluon vertex in covariant gauges is the sum of a kinematic singularity free longitudinal part constructed from the gluon and ghost propagators and the ghost-ghost&on vertex, and an undetermined transverse part which vanishes when any one of the external momenta vanishes. In sect. 4 we calculate the longitudinal part in lowestorder perturbation theory in the Feynman gauge. Comparing this result with a perturbation theory calculation of the full triple-gluon vertex in the Feynman gauge now in progress [3] will yield the transverse part of the vertex. This would then give an understanding of the importance of the transverse part of r3 to the infrared behavior of the theory. l
John Simon Guggenheim
Memorial
Foundation
Fellow.
152
S.K. Kim, ill. Baker / Gauge irwariance
153
We conclude this introduction by writing the Feynman rules (see fig. 1) and the Slavnov-Taylor (ST) identities. In the covariant gauge, the first ST identity is f
GE”(l),
- +G) G&AK P : 41 GP2)
4, r) =J(~2)~“,~2
+ J~~2x&2
- 4*4/J cgJ4?
The second ST identity
P : y) G(P21 .
(1.1)
[I] is
6 s) = J(q2)k%72 - 4*qV) ~~~%5(q~ P : c s> G(P2) + JCr2Xg$-2 -
r$,,)W$&dcP
: 4,
+J(s2)(g;s2 - sAssa) l-$$&,(s,p : q,
s) Gb2) r)
G(p2)
- r;f&j,(q, ST-4 - s) rg&
+ s, P : r) G(P2)
- %&d4,
+ r. P : s) G(P2)
r9 -4 - r) E$fl(q
- rl;dc$(r, s, -r - s) r$&(r ghost
propagator
- - - -> - -ba P gluan
(1.2)
t s, p : q) G(p2) ,
r&(p)
= Sap-L P2
propagator PA pp
DX,“b(p)=-~[gA$~-a)T
I
ghost-ghost-gluanvertex
0.P triple
gluan
vertex
r.cc
A PAa
g,r$);JP.w) q
-g,fObC q,/O
[g +(p-d,+ g Jr
g&g-r)A+
-p),]
four gluan vertex S3.d
r.y,c
g~C$~,~(p~w,s) =-ig, 2 [ f obefcde(gXvg/18-gXSg/J
x PAa
g.@
+f accfbde(gXpg,8-g*ggJ +fadefcbe(gX”g~g-gX~g~~)]
Fig. 1. Feynman
rules in covariant
gauges.
where J(q*) is the gluon self-energy which is defined in terms of the inverse gluon propagator as &%g)
= (q2$U -
(1.3)
G(p*) is the full ghost propagator,
g a Pb” aghuv(p, q, Y) is the proper triple-gluon
vertex,
g~f’$$(‘, q, r, s) is the proper four-gluon vertex: p”P$&, q : r) is the proper vertex for the coupling of a gluon of momentum Y, polarization V, and color index c, with two ghosts, of momentum p and q and color indices a and b, $“I-a,f$&@, q : r; s) is the proper vertex for the coupling of two gluons of momentum r and s, polarization v and 6, and color indices c and d, with two ghosts, of momentum p and cl and color indices a and b, respectively. In the axial gauge, we use the notation employed by Kummer [4]. The full gluon propagator is written as a;;(q)
-1
= q2(g2
~
74 [ tt2(1
n2f2)
n*q, ’t?
(
-“/s)
1 +
k2
-
((cl
~)*/q*)f*l
( gpv
+
_-
n*q/
q. II ,I 2
1
2 4
I(lJ
tzP -__q.n
Yqtlptl”
l-y,
[ IIZ
qpq”t~
+m-
$.lqu
+ t1,q/17 q’tl
ii
(1.4) and g2 in general depend on where y4 = tz*q*/(q tl)‘. The invariant functionsj; and (q . tz)*. The inverse propagator, connected to A,, by the relation
q*
(1.5) can be written
up, = k2&.I”g2
+ (4.
nj*f2
The ST identities
[
-
gpu
4/A”) -
2 4 tip4 ___ (q n)*
+ Slcer;:;ffi
+ ttuq@ (4.n)
1
(1.6)
’
are
p”Gb$V (P, q, r) = -Pbc P”P:b;:s(P.
qjltt, +
[q&7)
- rr,,(r)l
ecd ( P + q, q, r? s) = i]f” be rs& (P
+ r,
4,
s)
In the axial gauge, the Feynman covariant gauge.
+ fUdTg”
(p
r,
(1.7)
, s)
+ s,
q, r)]
(1.8)
rules for the gluon vertices are the same as in the
2. The triple-gluon vertex in the axial gauge In the axial gauge we define the arguments
of a function
with color indices as
F abc E FabC(p, q, r) . We use number indices to represent the function F, = F[p2,
(P.
F, -F[q2
,(q.
without color indices, i.e.,
n121, /z)~] ,
F3 = F [r2 , (r. H)~] . In the axial gauge the most general tensor structure which satisfies Bose symmetry is the following: r:“Gy (p, q, r) = -i[-gu(Anbcpv
of the triple-gluon
vertex
t Abacq,)
+ (A@p”
t fvbac&J yApp t i(P abc tPncb) (Gphqurp t Sphqg-“) t i(pabC - Pacb) S(rhp,q, - qhpd,) + $I Y/K/~h~~p\~r, + Rybc gxc,~z, t (Tybq,
t Tfbap,) nAn,, t S fbUrl-lr,,flht SfMqPqV,lh t (Sp” - SF)
x r,y,rJ,,
+ (Sf”” + .sp
- Sqbca) qvr,nA]
+ cyclic symmetric
terms , (2.1)
where P abc = pbca = pcab ,
abc _
-R;“,
RI
(S, - s3ybc = (S, - Qac,
SqabC = sp
.
(2.2)
and Xabc IS completely symmetric under the interchange of any pair of variables and color indices. By substituting eqs. (1.6) and (2.1) into the first ST identity, we obtain the following scalar equations: ~ (AabC _ Abac) _ (Aacb ~ ACab) + p
r (pacb t Ncba _ lVnb t NaCb)
t p . q(Pbc t Nbca - NbaC t Nnbc) t p . n (Sp
-
Sp)
=0 ,
(2.3)
- Aabc - Aacb t p rNacb t p . qNabC t p . ,,(Sy”” t Sic’ - S,b’“) = 0 , (2.4) - ACbap . r ~ AbCap . q t Rfbap
tJ = fabc [q2g2
- (q
n)2.f2
- r2g3
t (r. n)2f3. (2.5)
~ (,@bc
- AbE)
- Aacb + p
q(NabC
- Nb’c)
+
p . y(pacb
+ Nacb)
t p . nsfCa = -jabcg2 ,
(2.6) _ (A“Cb - Acab) - Aabc t p . r(Nacb - NCub) t p . q(pabc t ,Vabc) t p . nSfba =fnbcs3
,
(2.7)
156
SK. Kim, M. Baker / Gauge invariance
_ R;bC +p.
r(Sy
_R;bct p . rSfbc
- S;“) +p
+ Sibcp.
(I - Tpp
+p.
q (SYb -Syb)+p.
_ Rfb
tp.
qS’fcb +p.
rSTb -p.
rSFb +p.
IIf3 ,
p . 11= -fabcr
qSgbc + ( Tpm - Tf”)
_Rrb
rzf2 ,
. H = ybcq.
nfi,
- T~ba)=~bcq.
p. rTybc + p. qTfcb + p. nXabc = fabc(-q2f2 t r2f3) .
(2.11j
(2.12)
the sum ofeqs. (2.3) and (2.4) from that of eqs. (2.6) and (2.7) yields
Subtracting p
(2.9)
nf3, (2.10)
n Tfba=-jQbcr.
n(Tfab
(2.8)
rNcba + p . qNbCa - p . nSp
= fabc(g2
- g3).
(2.13)
If we write N cba = ;(Ncba ~ Nb‘=) + $(NCba + Nbca) , eq. (2.13) allows us to determine ;(NCba _ Nbca) = &
the antisymmetric [fabc(g2 -8s)
combination
+p.
(Ncbn - Nbca) as
nS,bca t ;p2(Nbc=
+Ncb”)]
.
Thus we can determine Ncba as 1 N cba _- z(N lzba + Nbca) + &
[jab”(g2
- g3) + p. r&y
+ ip2 (Nbca + Ncba)] (2.14)
By subtracting
eq. (2.4) from eq. (2.6), we obtain
~~~ _ p . q~baC + p rpacb _ p . n ($ba _ s;ba) = -g2ybc
.
(2.15)
From eqs. (2.14) and (2.1.5), we can then determine Aabc as A abc = fabCgl + q$f?$
[fnbc(gl - g2) + r . nSibc t
$r2(Nabc
+ $p . q(NabC t Nbac) - q. rPabc + q. ,I (Sy” - Sy”) . Substitute
1
-p.
[r . I@~
&
,z(S~
- Sp)
~ Ss(“‘) + q . rz(Sy” - Syb)
- 4. r(Pbc
+ Pacb)] .
eqs. (2.13) and (2.15) into eq. (2.5). We then obtain
RbC‘, = 1
(2.16)
eq. (2.15) into eq. (2.4). We can then determine SFba as
Scba _-
Substitute
+ Nbac)]
+ y ’ rSica
(2.17)
SK. Kim, hf. Baker 1 Gauge invariance
+
-L [p. qr.
P.n
-p.
t$qbc - St”‘) + p
rq
157
n(Syb - S,“““)
rp. q(PabCtPmb)].
(2.18)
Then, from eqs. (2.8) and (2.18), we obtain TbcQ =
1
-~
rq
[(p . n) fl f
1 p.
n r. II
(q
. n) fil
[p.rq.n(Ly-
- E*
(Sb;lc - SF,
SYb) t 4 r/J. n (sf””
- S,‘““)
- q . t-p . r(Pabc t Pncb)] . Finally, if we substitute
(2.19)
eq. (2.19) into eq. (2.12)
Xabc=f=bCf,(E
tps)t(;:
+ cyclic symmetric
Xabc is determined
as
:*;I ; (Sf”” 4,““‘)
terms .
(2.20)
Those functions N, A, St, R,, T, and X given in eqs. (2.14) through (2.20) satisfy eqs. (2.3) through (2.12) for the arbitrary values of S,, Sq, P and the symmetric part of N. However, all the functions we have determined are not manifestly free of kinematic singularities. As an example, we see that the part of Rp in eq. (2.18) determined by the gluon inverse propagator has a kinematic singularity at p. n = 0. As a first step toward removing these kinematic singularities of l/p. n type, we define a function Ry as p~qr.nlabCtp.rq~nInCb-p.rp.q(PabctPacb) ~f”bcq.t~r.n(fj-f~)tR~,
(2.21)
where I abc G (,qbc _ sQbc),
RbCa 2
= Rcba 2
.
We then obtain from eqs. (2.18) through (2.2 1) R~=~bc[q.nf2-rInf~]
+q.rSp+:4,
(2.22)
(2.23)
158
S.K. Kim, M. Baker / Gauge invariance
X abc = +q.
From terms ever, other
1 p. nq.
nr. n
rq. r(Pabc tPacb)
[p. 4~.
rRtM +p. rRyb +p. qR;bc] .
(2.24)
the above we see that the parts of the functions R, and T, determined in of the function fare free of kinematic singularities of the 1/p . n type. Howthey have other types of kinematic singularities. I and R2 are related to each by eq. (2.21). Thus we can determine Iabc/q. n - facb/r. n as
pyf3
t ~ q.
1 nr.
n
-f2)+&?*
+p . rp . q (Pabc t Pacb)]
[Rp
Then the function Icba which determines can be determined immediately as
=
p$$
+Ar2 2
.
(2.25)
I
the function
[R;”
fabc(f2 - fl)t p%
[cab I cba __ t-
(
q.tz
p.n. )I
I t+q.n
cba
-+-.
( ,q.
n
Tf”” partly by eq. (2.23)
t p. rq . r(Pabc + Pacb)]
Icab
p. n.)
(2.26)
Note that a kinematic singularity at p* = q* is introduced into Tyb by the function Icba given in eq. (2.26). From eqs. (2.14), (2.16) and (2.26) we see that the only remaining kinematic singularities are of the types q+
&[q*,(q.
)I)*1-g[r*,(r. n)*lI,
--!---Cf[q2,C7~n)21 -f[r*,(r~n)*lI.
(2.27)
(2.28)
q* - r*
To remove the kinematic singularities of the types given in eqs. (2.27) and (2.28), we define the following functions g2,3 and f2,3 as g2,3 = +k[q2,
(4. n>*l
-dq*,
Cr.
n)*l +g[r*, (4. n)*l -dr*, Cr.n)*l it
(2.29j
f2,3 'k(f(q*,(q.fl)*l
-f[q*?(r'n)*l
tf[r2T(q'n)21
-f[r2p(r'n)21).
(2.30)
SK.
Note that the functions g2,3 andf,,, are antisymmetric q and r, and vanish when p . II = 0, i.e., g2.3
p
0 3
+ n-0
g2.3
+ 92,,2
under the interchange
+ 0. p n+O
fi,3
Furthermore g2,3 andf2,3 limit of q* = r2, i.e.,
159
Kittt, hf. Baker / Gauge inrariancc
(2.31)
approach (g2 - g3) and (f2 - f3) respectively
f2,3 q2
’
g2 -g3
of
f2 -
in the
(2.32)
f3 ’
It then follows that &&2
-g3
-g2,3)9
g-y2
are regular at q2 = r2. We next redefine the undetermined p . ns~
+ ip2 (~bc= + @a)
(f2
functions
-f3
-f2,3)
given in eqs. (2.14) and (2.25) as
E _ fCbcg2,3 + Fbc= ,
(2.33)
(2.34) where FbCa and Gab’ are arbitrary FbC‘,
= FCba
G
symmetric
functions,
i.e.,
abc = G bat
(2.35)
By eliminating write the functions (Icba/q ’ n t Icab/p triple-gluon vertex
S, and R2 from eqs. (2.14) (2.16) (2.17), (2.22) (2.24) we can A, N, R ,,X and S, in terms of the functions g, f, (Nabc t Nb”), ’ n), Pabe, G and F.From these functions we can express the r3 in the form,
rfl;i,(~
4, r)
4, r) = r,L”,b,“(p, 4, r) + r$cb,‘(p,
where the longitudinal
part rp)
1
+adb
4 - qAPg,)+ 1
[
P.
nfi
-
4.
(2.36)
is
r$$Yp~ 4* r) = -ifabc -gw +(23&P.
,
1
[
pa
- 4d2 + (P, - qJ “,:‘“h:
z2 (gr -g2
P.Y
nf2 -Gnl,2
-
g1,2)+nhnpnv~fl,2
p.
+- r.
nq.
n
n fi,2]
(f,
- f2 -f,,2)]
160
S.K. Kim, M. Baker / Gauge invariance
+ rzhn,
[
pufi -
(fi -
cr,f2 - (P, - qv)p*
tq%r.nfl,3-p%
fz- fu)
r.nf2,3 1
‘P - p/J
g1.2
+ppqhnv ~+2 h - qh -22
.
rhw
n (fi-f3
-fi,3)
--r
n(f2 -f3
rgbq.
--r
4
P
+ cyclic symmetric
-f2,3)
I
terms,
(2.37)
and the transverse part lYiT) is defined as @$F(P,
4, r) = Fabc(p.
t $(Nabc
gw - pcrqA)
t Nbu )@.m,
(
-Ppqh)(ru-hr2)
+ Gabc p.nq.n(g~P.r2q.ntp.qn^n~-p.nqhnl, -4.v,nd pv-qv’ 42
( +
x
4
r. n
1 IZ
)
+ Icba
-)khPp. q.n
i2p.n
nq. n+p.
qw,
nqhn,
-p.
- 4. v,d
X r, - r2 n, r. n ) (
t ;(Pabc
t ; (Pabc t Pmb)
-
( - 4.
v,m
-
P.
2~. rq . r p. nq.
t
vvtP'
cyclic symmetric
nr.
n
w.
n&up.
n + P. ww,
wkq.A
+ 2~. 4p. rq. r nhn,n, 3~. n q. nr. n +gA&(q.
a) + ~kdwh - wwJ1
rp, - P.
- Pacb)[ghJq.
%-PAqvrp
terms .
2q. r - wPpv P.
n t$Aqprv I
(2.38)
161
SK. Kitn, M. Baker / Gauge invariance
The amplitude rF) defined in eq. (2.37) is manifestly free of kinematic singularities because of eqs. (2.31) and (2.32). rp) satisfies the homogeneous ST identity pAI?@
(P. 49 4 = 0
(2.39)
1
while f’iL) satisfies the inhomogeneous phr$L$“c (P, 4, y) = -Pbc
eq. (1.7), i.e.,
ST identity,
[aPA)
-
npvWl .
(2.40)
Since the complete r3 which is the sum of I’iL) and I’?) must be free of kinematic singularities, r3CT) itself must be free of kinematic singularities. We now differentiate eq. (2.39) with respect to p”I and obtain 4, Y) + P” $
r$;r(p,
rc,T,l”,b”(p,
4, Y) = 0
(2.41)
.
Since rr) is free of kinematic singularities, p”@/ap”) F’gp vanishes in the limit of p + 0. We then conclude from eq. (2.41) that the transverse part I’scT)vanishes in the kinematic regime where any one of three momenta approaches zero. Thus in this regime the total triple-gluon vertex f’s is uniquely determined in terms of gluon propagator. Note that if 7~~~is independent of n, then from eq. (1 S), f= 0 and g is a function only of q*. In this case the general expression for l?iL), eq. (2.37), reduces to the simple form - &.l(P”g,
r~~?n,bc(P, 49 4 = -ifabe
-q&T*)
[ +(gPipP.
4 -4aPJp2
+ cyclic symmetric
E!Lz!qg, _42
-g*)
1 (2.42)
terms .
The result, eq. (2.42), was used in ref. [2] to analyze the infrared behavior of YangMills theory.
3. The triple-gluon
vertex in the covariant gauge
In the covariant gauge, the most general tensor structure which satisfies Bose symmetry is rf&(p,
4. I) = -gU(AabcpV
+ (NabCpv + Nbacq,) +
of the triple-&on
vertex
+ Abmq,)
qhp,,
@abc+Pacb) (-Phcld+,+ $w7Mrv)
+ $ (pabc - Pacb) $hp,q, - w-w,) + cyclic symmetric
terms,
(3.1)
162
S.K. Kim, hf. Baker / Gauge invariance
where Pk is completely color indices. The ghost-ghost-gluon
symmetric
under the cyclic permutation
(p, 4 : r) (fig. 1) has the general form
vertex p”r$$,
(p, q : r) = AibcgA, + BibCpflv.+
f$v
of its argument and
CibCrhrv
+ (Dibc t Cibc) rhpu + Eibcrvph = Aibcgh,
t B,“bcpflp, - CibCrhqv
t Dzbcr,,pg, t Eibcpkrv
.
(3 4
Substituting eqs. (3.1) and (3.3) into the first ST identity obtain the following equations:
_p .rAcba _ p . qAbca = r2Arb + q2Ap
given in eq. (1.1), we
,
(3.3)
A abc f AaCb _ ACab _ p. r(Na=b _ Ncab) _ p. q(pabc f Nabc) + q. r(DP”
=“f”
- q2CF
t Cf”)
,
(3.4)
Aabc + AaCb _ Abac _ p . q(lvabc _ Nbac) _ p . r(pmb + Nacb) =A,b’lc+q.
r(~~aC+C~)-r2C~b,
AabC+Aacb_p.qNabC_p.rNacb=q.r(Cpbac A
abc
_
AbaC
+ AaCb
_
ACab
_
_ p. q(pabc + NbCQ+ pbc
p
(3.5) tcp),
(3.6)
. r (pacb + @‘a + NaCf’ _ pb) _ Nbac) = _r’cDP”
+ cr”)
- q2(Dim
t CF) (3.7)
where we defined
r-cab 3g -p2G(p2)J(r2)
rgb
,
,
i.e., 1:”
=p2G(p2)J(r2)
Afb
?r”
fpzG(p2)J(r2)
C,“” ,
-p” D
=p2G(p2)J(r2)
Dfb
Subtracting yields
.
(3.8)
the sum of eqs. (3.4) and (3.5) from the sum of eqs. (3.6) and (3.7)
p.rNcb”tp.qNb”=A~b+Agb~_~.rD~b_~.qD$C.
(3.9)
The form of eq. (3.9) is similar to that of eq. (2.13). Thus by writing N cba = ;(Ncba + NbCa) + $(Ncba _ NbcQ) ,
(3.10)
S.K. Kim, hf. Baker / Gauge invariance
163
we can determine Ncba as
+
&
Subtracting
;(Ncba
t NbcQ)t :(NcbQt NbcQ).
(3.11)
eq. (3.5) from eq. (3.6) yields
AbaC _p.qNbQC+p.rPQcb=_~~_-_.r~Cgc(lb
- q.rDF.
(3.12)
Next consider the sum: (3.3)fp.r
-P.
(3.12)+q.
r(3.9)
y[(3.12)-(=)(3.6)].
(3.13)
Eq. (3.13) yields A~b+~~bQ-p.qC~bQ+q.rD~b+~.rD~b~=~.
Eq. (3.14) gives us a constraint gator and the ghost-ghost-gluon obtain (P
abc + pub)
= _ (C;bc
Finally, if we substitute A
abc -_
-nbc 4
-q.
(3.14)
among the invariant functions of the ghost propavertex. Next from eqs. (3.6), (3.12) and (3.14) we + 2;,b”
+ “fb
+ Czcb + C,“”
+ ,;bQ)
eqs. (3.11) and (3.15) into eq. (3.12) ,.ClbQ
_p.
(3.15)
we find
rjjibc
Qbc + Nbm)
_
!_g
r (pQbc _ pQcb)
.
(3.16)
Now it is straightforward to show that the invariant functions N, P and A given in eqs. (3.1 l), (3.15) and (3.16) satisfy eqs. (3.3) through (3.7) for arbitrary values of (NQbc + Nbac) and (Pabc - PQcb). We can now express the triple-gluon vertex r3 as the sum of two parts, a longitudinal part rgL) and a transverse part r$r) rf&(p,
q, r) = r$L$Vbc(~, 4, r) f r$T$vbyp, 4, r) ,
(3.17)
164
S. K. Kim, M Baker / Gauge invariance
where, from eqs. (3.1 I), (3.15) and (3.16), 4, r) = gh, [p,(ZibC + 4. r Ciba + p . r Dibc)
rSL$jc(p, +q,(Ap
,Pv
tp.
r
CP”
tq.
rD,b”)]
- 4v
q2-_p2(qhPp-g~P.q)(Apb”tA~b-q.rb~ca-p.rD~b)
x
(Cagbc
+
“g””
+
CgCab+ "p"" t
Cgbac + Fib”>
terms ,
t cyclic symmetric
(3.18)
1 t : (PQbC - Pacb)
+ cyclic The amplitude <
P
hJq.
riLzF (p,
- P.
t $(rhpu
-
(gir’
while L’iT) satisfies the homogeneous (P, 4, r) = 0
4)
wwJ1
defined in eq. (3.18) satisfies the first ST identity, q, r) = J(r2)
gA&p.
(3.19) eq. (l.l),
i.e.,
- rhrv) K’gbu(r, p : q) G(p’)
+J(q2)
rq,)
terms .
symmetric
@)
rp,
(4h4P-
: r) G(p2)
1
(3.20)
ST identity (3.21)
.
It is easy to see that I’cL) given in eq. (3.18) is manifestly free of kinematic singularities. Thus, in a similar way as we did in sect . 2, the undetermined transverse part vanishes in the kinematic regime where any one of three momenta vanishes.
4. Perturbative
calculation
of the triple-gluon
vertex
We will construct the longitudinal part of r3 perturbatively. We first calculate the ghost propagator, the gluon self-energy and the ghost-ghost-gluon vertex to order g:. We can then construct riL) by using eq. ( 3.18). The ghost-ghost-gluon vertex rjg is represented by the graphs of fig. 2. In the Feynman gauge the integral expres-
S.K. Kim, hf. Baker / Gauge ittvariatzce
The ghost -ghost -gluon
P.0
165
vertex
k-P
Fig. 2. The second-order
ghost-ghost-gluon
vertex.
sion for r3g is l?$Yti(-p,
g : r) = gti,fabc-
i i(g&>
-PArv + ITrAp, + khrv + 2pp^k, - 3r&,
- k^P, + gh,(k + r) (k - P) -.
k*(k ~ P)*(k - r)*
(4.1)
with -p + q t r = 0, where we take t sign for outgoing momentum
and C2 is defined
by
facdfbcd = C2hab. We see that there are three types of intergrals in eq. (4.1). They are defined as
‘(P*?q*?r2)E[d4k
k2(k
Fh =
s
d4k
_
:j2
k^ k2(k - p)* (k - r)*
(k
_
r)2
“F,(p2,
(4.2)
’
q*,
r*)pA
+F2(p2,q2,
r*)rh,
(4.3) J(P*,
q*, r2) =Jd4k
The integration J=rli(log$
(k _ p):
(k _ r)~ .
(4.4)
for J is easily carried out. The result is + I) ,
(4.5)
S.K. Kim. M. Baker / Gauge imariance
166
where AZ is an ultraviolet cut-off. The other two integrals I and F, cannot be expressed in terms of elementary functions. However, the three invariant functions I, FI and F2 are related to each other. One trivial relation is F,(P*,
4*, r*)
q2,
P*)
=MP*,
r*) .
q*,
(4.6)
both sides of eq. (4.3) by ph yields
Multiplying F,p*
=F~(Y*,
r=
+F2p.
j
P.k k* (k - p)* (k - r)*
d4k
k* +p* -(k-p)*
=_; jd4k
(4.7)
k*(k - p)*(k - r)* r*
1. *It$r*ilog~. =*P
eq. (4.3) by rh yields
Multiplying
2
ir*I t in*i log< = Flp. 4
r + F2r2 .
(4.8)
From eqs. (4.7) and (4.8), we obtain I(P*,
q*, r*) =
=I_~(P*,
-2~.
qF, - 2q. rF, t n*i log(r*/p*) r2 -p*
q*, r*) + IA@*,
q*, P*> +
q*
[IA(r*,q*,
P*)
-IA(P*,
q*, r*N
+ n*i los(r*/p*)
r* -p* (4.9)
Now recall the definition IV&-p,
of the invariant
functions
of I’ss given in eq. (3.2)
q, r) = AibCgb + Bibcphp, t CibC thrv - (Dzbc + Clbc) r^p, - EibC pkrv . (4.10)
Comparing eq. (4.10) with eq. (4. l), we find A,(P*,q*J*) =
- 1 = jd”k
(k+r).(k-p) k2(k
_
d4kk2-p.rt~[(k-p)2-p2-k2] j
+ c&7*
p)2
Ck
_
r)2
-i[(k-r)*-r*-k*]
k* (k - p)2 (k - r)*
P*)Q*, 4*.r*> ,
(4.11)
S.K. Kim.
&(P2.q*, r2)=F, C,(p’,
r2) = -?-I+
167
/ Gauge irwariar~ce
(4.12)
>
42, r2) = -2F,
D&2,$,
hf. Baker
)
(4.13)
3(F, + F,) )
(4.14)
E&p*, q*, r2) = I - F, - 2F,
(4.15)
where we have omitted an overall factor -&(C2/32n4) from all above functions. We now change the momentum-conservation law from -p t q t r = 0 top + q t r = 0 in the following calculations, since all the invariant functions of the ghost-ghostgluon vertex are expressed in terms of p2, y2 and r* in eqs. (4.11) through (4.15). The ghost propagator G(p*) and the gluon self-energy are, to order&,
C(p’)=S
(,I+%
log$).
(4.16) We first note that the perturbation expressions eqs. (4.1 l), (4.13), (4.14) and (4.16) satisfy the general consistency condition, eq. (3.14) as required by gauge invariance. From eqs. (4.11) through (4.16) we find A;b= = J($)
$G
(q2) AEbC
=/.bc[l+g$$
(-~log$+?log$
2
+flog$+l
.2 ) -rg, &4
CbC =J(P2) _
i&2
=J(p2)
q*G(q*)
From these three functions, late -$C A
+q.
1,
(4.17)
CibC
2F2fabc ,
32n4 -;bc D
q2G(q2)
(iq” - p*)l
rczba
=P”‘[l+$$
(4.18) k82 D;bC = _ __
32n4
we can construct
abc L-21
f
log;
3(F, + F,)l
(4.19)
rk given in eq. (3.18). We first calcu-
+p. rB:bc
(- y
+
+21,,$
SK. Kim, hf. Baker / Gauge invariance
168
+=logg+ 1)I r2 - p2
2 IA(r2, ~
-i%pbc[-p.rq
+(p.
r-q’
q2, p2> - I,4(p2, r2 _p2
q2. r2)
1
- r2)I_4(p2,q2,r2)
[email protected]+q2
(4.20)
-p2)I_4(r2,q2,p2),
where we used eqs. (4.6) and (4.9). Next we calculate
+_J;ba _ q. rD;ba A-;bc_ p, q jj;bc
&
-
2
3g:c2 --Tf-32n
q2 [IA@‘,
q2, p2> -I_4b2,
q2, r2)1
1.
(4.21)
We can then express riL) as
( ( &2
rFzvbc (P, 4, r) =fabcgu,pv 1+
‘?;cz
--1 s
-p
. rq
A2
- p log3
3
q2, P’) - r~(p’,
2 I,(r2,
+2log7
q2, r2)
r2 - p2
t(p.r-q2
- r2) rA(p2,
+(p.rtq2
-p2)IA(r2,q2,p2)
q2, r2)
1)
+
fabc
-kh,(q + q,w,
L-g*@. . ‘P,
rq,
+ 4.
+ p . 4r,
+ r,p,qh
-
I+
ipy) phw,
Ph4gvl
-
g,,(P.
vi
+ P.
rqh)
A2
S.K. Kim, M. Baker / Gauge invariance
169
s;c*I_4(P*,4*,r*)
X - i32 [
1
+ (3! - 1) remaining
permutations
Comparing eq. (4.22) to the perturbation the transverse part K’$$$“(p, 4, Y).
. result for r$&(p,
(4.22) 4, r) [3] then gives
In the limit when any one of the three momenta vanishes, I$-) vanishes and the full vertex is determined by r3(L) . In this limit, the explicit log p*, log q2 and log r* terms in eq. (4.22) are cancelled by corresponding logarithms in I,, and I’cL) is finite in that limit. More explicitly we find FI (p*, 4*, rZ) = IA($,
4*, r2)
n*i ---2
P
r+O
p*=q2
---$$
(1 +log$)
p*=,* -$(l+log$).
(4.23)
;2:y.2
From eqs. (4.22) and (4.23), we obtain
+
(4~
-
cd
(r,q,
-
g,,q.
4
~
-
(4.24)
5. Summary We have found the general solution of the Slavnov-Taylor identities for the triple-&on vertex r3 in both covariant and axial gauges. The solutions for the quadruple-gluon vertex will be presented in a future publication. These results combined with the Schwinger-Dyson equations then provide a non-perturbative approach to the study of the infrared behavior of Yang-Mills theory.
We would like to thank R. Anishetty and J.S. Ball for valuable conversations. One of us (MB) would like to thank the John Simon Guggenheim Memorial Foundation for support during the course of this work.
170
SK. Kim, 151.Baker / Gauge invariance
References [l]
A.A. Slavnov, Theor. Mat. Fiz. 10 (1972) (Theor. Math. Phys. J.C. Taylor, Nucl. Phys. 10 (1971) 99. [2] R. Anishetty, M. Baker, J.S. Ball, S.K. Kim and F. Zachariasen, [3] J.S. Ball and T.W. Chiu, to be published. [4] W. Kummer, Acta Phys. Austriaca 41 (1975) 315.
10 (1972)
99;
Phys. Lett. 86B (1979)
52.