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Non-analytic mass behaviour and the renormalization group
Nuclear Physics B124 (1977) 109-120 © North-Holland Publishing Company
NON-ANALYTIC MASS BEHAVIOUR AND THE RENORMALIZATION
GROUP N.K. NIELSEN Nordita, Copenhagen, Denmark Received 16 February 1977 In 't Hooft-Weinberg renormalized spinor quantum electrodynamics, the leading mass singularity for m ~ 0 is found to be m3(-ln m) 10 for gauge-invariant vertex functions at non-exceptional momentum configurations, and rn2(-ln m) 7/2 for gauge-dependent vertex functions, where rn is the renormalized mass parameter. The corresponding result inside the Gell-Mann-Low renormalization scheme, where the renormalized mass parameter is the physical electron mass mphys, is mphys(-ln mphys) -9/4 in both cases. Analogous results can be found in an asymptotically free non-Abelian gauge theory, provided an infrared-stable fLxed-point in coupling-constant space is present. The results are obtained by means of a renormalization group method developed by Symanzik.
1. Introduction Renormalization group methods have by now become prominent tools in many branches o f physics and have consequently been formulated in several rather different fashions. In high energy physics, the interest has been concentrated on the original Gell-Mann-Low renormalization group equation [ 1 ] , as well as on the CallanSymanzik equation [2], and also on the equations of 't Hooft [3,4] and Weinberg [5] (cf. also ref. [6]) which are closely related to each other. Whereas the fields in the Callan-Symanzik approach are normalized on the mass shell, this is not the case for the Gell-Mann-Low or 't Hooft-Weinberg equations, which therefore are convenient for the description of theories containing massless particles, such as quantum electrodynamics or asymptotically free non-Abelian gauge theories [7], since infrared divergences are not introduced via the field renormalizations. Off-shell renormalization of the fields is also responsible for another feature o f the Gell-Mann-Low and 't Hooft-Weinberg renormalization groups, with which we shall be concerned in the present paper. The leading mass singularity, when the renormalized mass parameter goes to zero, o f all vertex functions at non-exceptional momenta can be determined by renormalization group methods and is universal in the sense that it does not depend on the number of external legs. The analysis has already been carried out b y Symanzik [8,9] for the scalar ~b4 theory, and what is done below is es109
110
N.K. Nielsen / Non-analytic mass behaviour
sentially only an adaption of Symanzik's methods to spinor quantum electrodynamics and the asymptotically free quark-gluon field theory, with the 't Hooft-Weinberg equation as the starting point instead of the Callan-Symanzik equation. The method can also be used to obtain non-trivial mass corrections to asymptotic forms [10]. It has been a general belief that the unrenormalized vertex functions at non-exceptional momenta in spinor electrodynamics and its non-Abelian counterpart are twice differentiable in the bare mass at the origin, with the consequence that the correspond ing vertex functions in the 't Hooft-Weinberg renormalized theory are twice differentiable in the renormalized mass parameter at the origin [5,11 ]. However, it has recently been shown by Papatzacos [12] that the unrenormalized fourth-order electron propagator in QED is only differentiable once with respect to the bare mass at the origin because of a vacuum polarization insertion in a photon line. In agreement with this result, we find different leading mass singularities forgauge-invariant and gaugedependent vertex functions at non-exceptional momenta, namely, ma(-ln m) l°
(1.1)
in the gauge-invariant case, and m2(-ln m) 712
(1.2)
otherwise. These estimates are possible because in QED the origin in coupling-constant space is an infrared-stable fix point. For the non-Abelian theory one has a similar distinction between gauge-invariant and gauge-dependent vertex functions, but here further assumptions are in general necessary for the determination of the leading mass singularities, since infrared-stable fix points only are known in some special cases [13]. If a spacelike normalization point is used [ 1] and the physical fermion mass mphys is chosen as the renormalized mass parameter of the theory, one gets for QED instead of (1.1) and (1.2) mphys(--ln mphys)-9/4
(1.3)
for gauge-invariant as well as gauge-dependent vertex functions. It is also possible here to make a statement concerning the non-Abelian case, provided an infrared-stable fix point is assumed present. The layout of the paper is the following: in' sect. 2 't Hooft-Weinberg renormalized gauge invariant vertex functions are analyzed, in sect. 3 the gauge-dependent ones are considered, and finally in sect. 4 the case where a spacelike normalization point is used is dealt with.
2. Leading mass singularities of gauge-invariant vertex functions We shall now give a detailed treatment of the mass-singularity analysis for gaugeinvariant vertex functions. The starting point is the renormalization group equation,
N.K. Nielsen / Non-analytic mass behaviour
2n
~
111
x
Fig. 1. A term in the Wilson expansion of a 2n point photon vertex function with four insertions Z2Z4fd4x ~tp(x) (indicated by crosses). The Wilson coefficient C is a connected Green function with 2n amputated photon legs as well as two amputated fermion legs with zero momentum flow and where fermion indices are averaged over, and the quantity K is a five-point function of Z2Z 4 f d4x ~qJ(x), apart from unimportant factors. of
which in the 't Hooft or Weinberg formulations can be written [ 3 - 6 ]
(X ~ - [3(g)-~g+ Dr(g) + (1-~(g)) m ~m)P(X; la,g, m) = O .
(2.1)
Here r is a gauge-independent vertex function, g and m are the renormalized coupling constant and mass,/a is an auxiliary mass unit, and/3(g), 6(g) and Dr(g) are functions of the coupling constant to be determined in perturbation theory. D r ( g ) becomes a matrix if I" contains operators which are not multiplicatively renormalizable, and for g -+ 0 it reduces to minus the mass dimension of r . The momentum dependence of the vertex function is suppressed, but the quantity X is a common scale factor of all the momenta upon which P depends, and we always assume that the momenta are non-exceptional. By differentiating eq. (2.1) three times with respect to m, one obtains
(~-~-~(g)~g+Dr(g)+3(134 = - ( 1 - (5(g))m ~
~i(g)))
a3 r ( x ; p, g, m)
r(x; u, g, m ) .
(2.2)
The right-hand side of (2.2) would be negligible for large values of X by Weinberg's theorem [ 14] if the number o f mass derivatives had been lower than four. Its behaviour must in the present instance instead be estimated by a Wilson expansion (see ref. [8] and refs. given there) as shown in fig. 1 (note that mass differentiation generates insertions of the finite composite operator Z2Z 4 f d4x ~ 0 ( x ) , where ~ is the fermion field and Z2Z 4 a renormalization constant) 34 (1 - 8(g))m ~ P(X; p, g, rn) = C(~,;/a, g) K(,u, g, m) + .... (2.3) In eq. (2.3), the Wilson coefficient CQt;/l, g) is mass independent and fulfils the equation
(X--~ - lS(g) ~ + Dr(g) + a + 6(g)) C(X; la, g) = O
(2.4)
N.K. Nielsen~Non-analyticmassbehaviour
112 with the solution
{/<, dX
C(;k;p,g)=exp-
~
}
Dr(g(X))+3+8(g(X'))] C(1;p,g(X)),
(2.5)
where the effective coupling constant g(X) as usually is defined by X -dg(X) - ~ =/3(g(X)). ~
(2.6)
Returning to (B3/bm3)P(~.; ~t, g, m) we find that its asymptotic part, which in conformity with [8, 9] is denoted [(a3/bma)F] as (~;/~, g, m) (the underbar indicates an asymptotic form at exceptional momenta) fulfils an inhomogeneous partial differential equation with the solution [ 15] (X;~,g, m) = exp -
[Dr(g(X')) + 3(1 - 6(g(X')))] dX'
t
X [ ~ F]as (1;/~,g(~), m ) - if ~7 e x p / - l f
XlX'd?," -~
[Dr(g(;k"))
+ 3(1 - 6(g(X")))l} K~, g(3./~,'), m) C(~,';/a, g(k/X'))
= exp -
~
[Dr(g(X')) + 3(1 -6(g(;k')))]
-C(1;l~,g(X))fX7
(1;/l,g(X), m)
6 (g(X")) K(la,g(X/X'),m
4
1
as
,(2.7)
'
where in the last step eq. (2.6) was used. A statement on the leading mass singularity now follows, if eq. (2.7) is compared with what one gets by treating eq. (2.2) as a homogeneous partial differential equation. Introducing the effective mass m
( XdX'~
m(X)=T expd
}
8(g(X'))s'
(2.8)
one has /)3 [ x dX' bm 3 F(~.; p, g, m) = exp ~ - f -~r [Dr(g(Jk')) + 3(1 - 6(g(;k')))l } ~3 X
am(;~)3 r(l; u, g(X),m(X)).
(2.9)
N.K. Nielsen / Non-analytic mass behaviour
113
Comparing this result with (2.7) and substituting g(?~- l) for g one finally gets
f
0fiT(k) 303
7 exp / - 4 f ' "-~ d?~" 6 (gCA"))}r ~ , g(k'-'),m), I'(1 ;/a, g, ~(k)) = Xd?~' 1 ~ X'-I (2.10)
where m { rh(X) =--~ exp -
dX' -~- 6(g(X'))
(2.11)
is the effective mass evaluated with the coupling constant g replaced by g(X-1). The further analysis of eq. (2.10) presupposes' the presence of an attractive infrared fixed point in coupling constant space. For quantum electrodynamics the origin is an infrared-stable fixed point, so the only required information is the values of/3 and 6 in lowest order of perturbation theory: _ g3 /3(g) - 1 - ~ 2 + O(gS) ,
3g2 8(g) = ---~2 + O(g4),
(2.12)
(it should not lead to confusion that the electric charge here is called g), as well as the fact that the quantity K has already an O(g °) term. For large k we then have g(~-
67r2 lnk-1 ,
1)2 ~_ _ _ _
(2.13)
and thus
f1
exp - 4
~
6(g(~,")) ~0~, g(~,,-1 ), m)
h'-I h
c~ f
dk' (In ;k')9 = (In ;k)l o ,
(2.14)
and m
r~(3,) ~-~- (ln k) -9/4 .
(2.15)
The two last results enable us to conclude from eq. (2.10) that the leading mass singularity of any gauge-invariant vertex function at non-exceptional momenta renorrealized according to the prescriptions of 't Hooft [3, 4] or Weinberg [5] is given by P(1; U, g, m) • m3(-ln m) l0 .
(2.16)
In the asymptotically free non-Abelian case we must in general rely on assumptions. If/3 has an infrared-stable zero with 6 = 6o,
~ = Ko ,
(2.17)
114
N.K. Nielsen / Non-analytic mass behaviour
then we have f -~7 exp - 4 1
-~-.6(g(X"))) K ~ , g ( X ' - l ) , m ) c x X -4~° ,
(2.18)
-1
fit(h) cc mX - l + a ° ,
(2.19)
so the leading mass singularities of the perturbation series here sum to a power P(1 ; ~, g, m) ~x m 3 × m 46 o/( 1-8 o),
(2.20)
provided 60 :/: 1. 3. Gauge-dependent vertex functions The analysis of sect. 2 does not apply to gauge-dependent vertex functions. The reason is that a Wilson expansion is required for the estimation of the asymptotic behaviour already after three mass differentiations. The expansion is sketched in fig. 2 for the case of the electron propagator. Via the expansion, gauge-dependent composite operators of mass dimension 2 arise. In QED, the only operator of this kind appearing is Au2 , where A u is the photon field, but in the non-Abelian case one also gets a composite operator consisting of Faddeev-Popov ghost fields. But since these operators are gauge-dependent they do not occur in the Wilson expansion of gaugeindependent vertex functions, so the neglect of them in sect. 2 is justified. With the modifications mentioned above, the analysis now runs as in sect. 2. We concentrate on QED, where the relevant composite operator is Au2 . Instead of (2.2) we now have the following equation for 02P/am2: O
0+
\
02
03 = -(1 - 6(g)) m ~
r(x; u, g, m, ~),
(3.1)
where ~ is a real parameter fixing the gauge within the usual class of Fermi gauges. A Wilson expansion is now performed on the right-hand side of (3.1) as indicated for the electron propagator in fig. 2, Oa
m-~m3 I'(X;tl, g, m, ~) = C(X; ta, g, ~) k'O; g, m) + ....
(3.2)
That the quantity g is independent of ~ is already plausible from fig. 2 and can be verified by the Ward-Takahashi identity. ~ has in contrast to the corresponding function K encountered in sect. 2 no O(g °) term in perturbation theory, but only an O(g2) term.
N.K.Nielsen/Non-analyticmassbehaviour
115
Fig. 2. A term in the Wilson expansion o f the electron propagator with three insertions o f
Z2Z 4 f d4x ~0 (x). The Wilson coefficient Chere has two amputated photon lines with zero
momentum flow and with vector indices averagedover. Correspondinglyan insertion of f d4x A2(x)occurs in the quantity ~'. The new Wilson coefficient C(X; p, g, ~) fulfils
(X ~-~(g) ~---~+2~(g)~~--~+Dr(g)+2+2/3(g)]c(x;/l,g,~)=0,g,
(3.3)
with the solution { /dX'( 2fl(g(X'))] / C(X;/.i, g, ~) = exp -ff Dr(g(X')) + 2 + g(X') ]J (3.4)
X CO;U, gqA), ~CA)), where ~(X) = (g-~X))2 ~,
(3.5)
so corresponding to eq. (2.7) we have in the present case ?" dX'
/
[~B-~ZmZ P]as(X;l~'g'm'O=expl-/-~- [Dr(g(X'))+ X
(
['
)
1
+
(1; la, S(X), m, ~(X)) - C(1 ;/a, g(X), ~(X)) as
X
d 1
+dX' exp
dX"
2
m)+,,.
. fl(g(~."))~ 6 (g(X")) -v~ j ~ /a, g
,
(3.6)
h.lJk'
Therefore ~2
a~h(?,)2 I"(1; u, g, r~(X))
,~ f--~expa dX' 1
f -2 f ~
t dX" X" --~ \(5(g(X"))+/~(g(..X, i l ) ) ))/ ~l), g ( X ' - t ) , g t ^
h '-1
m),
(3.7)
116
N.K. Nielsen/Non-analyt& mass behaviour
where we have for large k, using (2.12) and..(2.13) and the fact that the perturbation series for ~ starts out with a second-order term,
1
fl
XdX'exp(_2 f dX" / i " ~ I, x'-' ~ 7 ; [ (g(X))
o:
+ fl(g( ))~I ~-k" ~]/'
"
.~.,-1,, ~,gl, ) m)
x dX' f -~- (In X') s/2 = (In X)7/2 ,
(3.8)
so the leading mass singularity of gauge-dependent vertex functions at non-exceptional momenta and renormalized according to the 't Hooft-Weinberg methods is in QED given by P(1;/2,g, m, ~) ~ m2(-ln m) 7/2 .
(3.9)
It is remarkable that the leading masstsingularity of gauge-dependent vertex functions is independent of the gauge-fixing parameter ~. As seen from the derivation, this is due to the fact that the quantity ~ as well as the anomalous dimension of the composite field Au2 , which is -2~/g, are independent of ~. Gauge-dependent vertex functions of a non-Abelian theory can also be dealt with under the assumption already made in sect. 2. There is here a small mixing problem due to the presence of a composite operator made out of ghost fields, but apart from that the calculations run as before. 4. Gell-Mann-Low renormalized vertex f u n c t i o n s In the original version of the renormalization group [1, 16], a spacelike normalization point is used for the field and coupling constant renormalizations. It is shown in the appendix how the vertex functions renormalized according to such a prescription are related to 't Hooft-Weinberg renormalized vertex functions. The mass singularity analysis is carried out by means of eq. (A.3), with m first chosen as mphys defined by (A.7). When m is replaced by fit(X) given by (2.11), the following relation is obtained: P(2P'B)(1 ; M, gMCA), mphys(X), ~M(X)) = cb(2F'B)(M;/2,g, rh(X), ~) x r (2F,B) (1;/2, g, rh(X), ~),
(4.1)
where gM(X) and ~M(X) are found from gM and ~M defined by eqs. (A.5) and (A.6), respectively, by substituting ffz(k) for m, and mphys(~k ) is obtained by an analogous modification of (A.7) mphys(~k) = ff/(~k) b(--mphys(~k) 2 ;/2, g, rnGk)) .
(4.2)
The leading mass singularity of F (2F'B) (1 ;M, gM, mohys, ~M) is found by taking
N.K. Nielsen ~Non-analytic mass behaviour
117
the derivative of (4.1) with respect to m. On the right-hand side, m only enters through rh(~.), so a .p(2F,B) (1 ;M, gM(~), mphys(~k), ~M(~.)) am _
rh(h) a ¢(2F'B)(M; p, g, rh(~.), ~) F (2F'B) (1 ;p, g, rh(?,), ~). m arh(X)
(4.3)
It is seen from sects. 2 and 3 and the construction in the appendix that the mass derivative on the right-hand side of (4.3) goes to a finite value for X -->0% so we have in this limit a p(2F, B)( 1 ; M, gM(X), mphys(~k), ~M(X)) cc /~(~k) am m
(4.4)
By the same argument, the terms arising through the dependence ofgM(X ) and }M(X) on m are also proportional to rhOO/m, so eq. (4.4) can be written amphys(~k) a ~(2F, B)(1 ;M, gMOk)' mphys(~k), ~M()k) ) cc /7/0~) am amphys(~k ) m
(4.5)
The term mphys(~k) is found from (4.2) via the transformation law for b(p 2 ;/a, g, m) under the renormalization group, which can be found by comparing (A.2) with (2.1) and solving the resulting partial differential equation for b(p 2;/~, g, m). The outcome is
mb(X2p 2;/a, g, m) = Mn(X) b(p 2 ;/a, g(~), m(X)).
(4.6)
Here we substitute g ~, g(X - 1 )
and apply the result to (4.2) ~"nphys(~k) = mb(-3, 2 mphys(~k)2 ,/.t, g(X- 1), m ) .
(4.7)
The behaviour of the solution of (4.7) for X -->oo depends on the infrared stable fixed points of the coupling constant. The solution in QED, where the origin is infrared stable, is mphys(~k) "- m/~k.
(4.8)
In QED the leading mass singularity is now found by means of (4.5) and (2.15),
F(2F'B)(1; M, gM, mphys, ~M) OCmphys(_ln mphys)-9/4 .
(4.9)
If we assume for the asymptotically free non-Abelian gauge theory that mphy s makes sense and that a non-trivial infrared stable fixed point exists, we also have eq. (4.8)
N.K. Nielsen ~Non-analytic mass behaviour
118
fulfilled, apart from a proportionality factor, so leading mass singularities here, according to (2.19), sum to a power 1--5 0
P(2F'B)(1; M, gM, mphys, ~M) (x mphy s
(4.10)
If the new mass parameter ~ instead is fixed by the definition (A.8), then we hav~ on the other hand a situation where the vertex functions in the formulation of the theory using a spacelike normalization point, as well as all parameters characterizing them, are connected with the corresponding quantities of the 't Hooft-Weinberg renormalized version of the theory by relationships which are regular also in the massless limit. Consequently, the estimates (2.17) and (3.7) as well as (2.20) are valid her~ too. I am most grateful to Professor K. Symanzik for instructing me on his methods, for making unpublished material available to me, and for pointing out the relevance of ref. [12] and its interpretation in terms of the Wilson expansion. Also I wish to thank Dr. R.J. Crewther for a stimulating correspondence which led to the comple- . tion of this work, most of which was carried out several years ago. Appendix In this appendix we briefly record the connection between the vertex functions renormalized by means of a spacelike normalization point [ 1,16] with those renorrealized according to the prescriptions of 't Hooft [3,4] or Weinberg [5]. The two-point vertex functions with two bosons (photons or gluons) and two fermions, respectively, are in the 't Hooft-Weinberg schemes denoted: r (t~L; ° 2)rk ~ ; 11, g, m ) = i ( 6 ~ v k 2 - k ~ k v ) d - l ( k i ' 1 1 ,, g , m ) + gauge terms, p(2,O)(p; 11,g, m) = i( i'yp + mb(p 2 ; 11, g, m) ) a(p 2 ;11, g, m, ~) .
(A.1) (A.2)
The functions d and b will in QED be independent of the gauge-fixing parameter ~ but this is not so in the non-Abelian theory. A general vertex function F(2F'B)(X; M, gM, m, ~M) with 2F fermion legs and B boson legs, renormalized by the prescription using a spacelike normalization point, is then found from the corresponding 't Hooft-Weinberg renormalized vertex function F(2F'B) (,k; 11,g, m, ~) by F(2F'B)(~k; M, gM, ~Z, ~M) = dp(2F'B)(M; 11, g, m, ~) r(2ic'n)(X; 11, g, m, ~) , (A.3)
where • (2F'B)(llt;11, g, m, ~) = [a(M2; 11,g, m, ~)]-F[d(M2;11,g, m)] B/2.
Here M is a real parameter.
(A.4)
N.K. Nielsen ~Non-analytic mass behaviour
119
In eq. (A.3) the coupling constant gM, the mass ~ and the gauge fixing parameter ~M still have to be specified. Consider first the coupling constant. In QED the equation fixing gM is g ~ = g2 d(M2; la, g, m) .
(A.5)
In the non-Abelian case also the value of p ~ , 3 ) (p, q, r;/~, g, m, ~) for p2 = q2 = r 2 = M 2 is necessary for the definition o f g M [ 17]. The gauge-fixing parameter ~M is now given by ~M = ~ d - l ( M 2 ; p , g , m) ,
(A.6)
both in QED and in the non-Abelian theory (cf. (3.5)). This leaves the choice o f the new mass parameter ~ . One possibility, realized in the original formulation o f the renormalization group [1 ] , is to use the physical mass defined from (A.2) by mphys = mb(--m~hys; la, g, m) ,
(A.7)
as the mass parameter of the theory. In perturbation theory, this definition certainly makes sense in quantum electrodynamics. Whether this is also the case in the nonAbelian theory is open to conjecture. The new mass parameter r~ can also be fixed by [16] = m M = m b ( M 2 ; la, g, m)
(A.8)
in analogy to (A.5) and (A.6).
References [1] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300; N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959); K.E. Wilson, Phys. Rev. D3 (1971) 1818. [2] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227. [3] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [4] J.C. Collins and A.J. Macfarlane, Phys. Rev. D10 (1974) 1201; M.J. Holwerda, W.L. van Neerwen and R.P. van Royen, Nucl. Phys. B75 (1974) 302. [5] S. Weinberg, Phys. Rev. D8 (1973) 3497. [61 M. Gomes and B. Schroer, Phys. Rev. DI0 (1974) 3525. [7] G. 't Hooft, Nucl. Phys. B62 (1973) 444; H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346; D.J. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343. [8] K. Symanzik, Comm. Math. Phys. 23 (1971) 49. [9] K. Symanzik, Comm. Math. Phys. 34 (1973) 7. [10] G. Mack and K. Symanzik, Comm. Math. Phys. 27 (1972) 247. [11] B. Lautrup, Nucl. Phys. B105 (1976) 23.
120
N.K. Nielsen ~Non-analytic mass behaviour
[12] P. Papatzacos, Nuovo Cimento Letters 17 (1976) 347. [13] W.E. CasweU, Phys. Rev. Letters 33 (1974) 244; D.R.T. Jones, Nucl. Phys. B75 (1974) 531. [14] S. Weinberg, Phys. Rev. 118 (1960) 838. [15] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 2 (Interscience, New York, 1962). [16] K.E. Eriksson, Nuovo Cimento 30 (1963) 1423; M. Astaud and B. Jouvet, Nuovo Cimento 63A (1969) 5. [17] T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856.
NON-ANALYTIC MASS BEHAVIOUR AND THE RENORMALIZATION
GROUP N.K. NIELSEN Nordita, Copenhagen, Denmark Received 16 February 1977 In 't Hooft-Weinberg renormalized spinor quantum electrodynamics, the leading mass singularity for m ~ 0 is found to be m3(-ln m) 10 for gauge-invariant vertex functions at non-exceptional momentum configurations, and rn2(-ln m) 7/2 for gauge-dependent vertex functions, where rn is the renormalized mass parameter. The corresponding result inside the Gell-Mann-Low renormalization scheme, where the renormalized mass parameter is the physical electron mass mphys, is mphys(-ln mphys) -9/4 in both cases. Analogous results can be found in an asymptotically free non-Abelian gauge theory, provided an infrared-stable fLxed-point in coupling-constant space is present. The results are obtained by means of a renormalization group method developed by Symanzik.
1. Introduction Renormalization group methods have by now become prominent tools in many branches o f physics and have consequently been formulated in several rather different fashions. In high energy physics, the interest has been concentrated on the original Gell-Mann-Low renormalization group equation [ 1 ] , as well as on the CallanSymanzik equation [2], and also on the equations of 't Hooft [3,4] and Weinberg [5] (cf. also ref. [6]) which are closely related to each other. Whereas the fields in the Callan-Symanzik approach are normalized on the mass shell, this is not the case for the Gell-Mann-Low or 't Hooft-Weinberg equations, which therefore are convenient for the description of theories containing massless particles, such as quantum electrodynamics or asymptotically free non-Abelian gauge theories [7], since infrared divergences are not introduced via the field renormalizations. Off-shell renormalization of the fields is also responsible for another feature o f the Gell-Mann-Low and 't Hooft-Weinberg renormalization groups, with which we shall be concerned in the present paper. The leading mass singularity, when the renormalized mass parameter goes to zero, o f all vertex functions at non-exceptional momenta can be determined by renormalization group methods and is universal in the sense that it does not depend on the number of external legs. The analysis has already been carried out b y Symanzik [8,9] for the scalar ~b4 theory, and what is done below is es109
110
N.K. Nielsen / Non-analytic mass behaviour
sentially only an adaption of Symanzik's methods to spinor quantum electrodynamics and the asymptotically free quark-gluon field theory, with the 't Hooft-Weinberg equation as the starting point instead of the Callan-Symanzik equation. The method can also be used to obtain non-trivial mass corrections to asymptotic forms [10]. It has been a general belief that the unrenormalized vertex functions at non-exceptional momenta in spinor electrodynamics and its non-Abelian counterpart are twice differentiable in the bare mass at the origin, with the consequence that the correspond ing vertex functions in the 't Hooft-Weinberg renormalized theory are twice differentiable in the renormalized mass parameter at the origin [5,11 ]. However, it has recently been shown by Papatzacos [12] that the unrenormalized fourth-order electron propagator in QED is only differentiable once with respect to the bare mass at the origin because of a vacuum polarization insertion in a photon line. In agreement with this result, we find different leading mass singularities forgauge-invariant and gaugedependent vertex functions at non-exceptional momenta, namely, ma(-ln m) l°
(1.1)
in the gauge-invariant case, and m2(-ln m) 712
(1.2)
otherwise. These estimates are possible because in QED the origin in coupling-constant space is an infrared-stable fix point. For the non-Abelian theory one has a similar distinction between gauge-invariant and gauge-dependent vertex functions, but here further assumptions are in general necessary for the determination of the leading mass singularities, since infrared-stable fix points only are known in some special cases [13]. If a spacelike normalization point is used [ 1] and the physical fermion mass mphys is chosen as the renormalized mass parameter of the theory, one gets for QED instead of (1.1) and (1.2) mphys(--ln mphys)-9/4
(1.3)
for gauge-invariant as well as gauge-dependent vertex functions. It is also possible here to make a statement concerning the non-Abelian case, provided an infrared-stable fix point is assumed present. The layout of the paper is the following: in' sect. 2 't Hooft-Weinberg renormalized gauge invariant vertex functions are analyzed, in sect. 3 the gauge-dependent ones are considered, and finally in sect. 4 the case where a spacelike normalization point is used is dealt with.
2. Leading mass singularities of gauge-invariant vertex functions We shall now give a detailed treatment of the mass-singularity analysis for gaugeinvariant vertex functions. The starting point is the renormalization group equation,
N.K. Nielsen / Non-analytic mass behaviour
2n
~
111
x
Fig. 1. A term in the Wilson expansion of a 2n point photon vertex function with four insertions Z2Z4fd4x ~tp(x) (indicated by crosses). The Wilson coefficient C is a connected Green function with 2n amputated photon legs as well as two amputated fermion legs with zero momentum flow and where fermion indices are averaged over, and the quantity K is a five-point function of Z2Z 4 f d4x ~qJ(x), apart from unimportant factors. of
which in the 't Hooft or Weinberg formulations can be written [ 3 - 6 ]
(X ~ - [3(g)-~g+ Dr(g) + (1-~(g)) m ~m)P(X; la,g, m) = O .
(2.1)
Here r is a gauge-independent vertex function, g and m are the renormalized coupling constant and mass,/a is an auxiliary mass unit, and/3(g), 6(g) and Dr(g) are functions of the coupling constant to be determined in perturbation theory. D r ( g ) becomes a matrix if I" contains operators which are not multiplicatively renormalizable, and for g -+ 0 it reduces to minus the mass dimension of r . The momentum dependence of the vertex function is suppressed, but the quantity X is a common scale factor of all the momenta upon which P depends, and we always assume that the momenta are non-exceptional. By differentiating eq. (2.1) three times with respect to m, one obtains
(~-~-~(g)~g+Dr(g)+3(134 = - ( 1 - (5(g))m ~
~i(g)))
a3 r ( x ; p, g, m)
r(x; u, g, m ) .
(2.2)
The right-hand side of (2.2) would be negligible for large values of X by Weinberg's theorem [ 14] if the number o f mass derivatives had been lower than four. Its behaviour must in the present instance instead be estimated by a Wilson expansion (see ref. [8] and refs. given there) as shown in fig. 1 (note that mass differentiation generates insertions of the finite composite operator Z2Z 4 f d4x ~ 0 ( x ) , where ~ is the fermion field and Z2Z 4 a renormalization constant) 34 (1 - 8(g))m ~ P(X; p, g, rn) = C(~,;/a, g) K(,u, g, m) + .... (2.3) In eq. (2.3), the Wilson coefficient CQt;/l, g) is mass independent and fulfils the equation
(X--~ - lS(g) ~ + Dr(g) + a + 6(g)) C(X; la, g) = O
(2.4)
N.K. Nielsen~Non-analyticmassbehaviour
112 with the solution
{/<, dX
C(;k;p,g)=exp-
~
}
Dr(g(X))+3+8(g(X'))] C(1;p,g(X)),
(2.5)
where the effective coupling constant g(X) as usually is defined by X -dg(X) - ~ =/3(g(X)). ~
(2.6)
Returning to (B3/bm3)P(~.; ~t, g, m) we find that its asymptotic part, which in conformity with [8, 9] is denoted [(a3/bma)F] as (~;/~, g, m) (the underbar indicates an asymptotic form at exceptional momenta) fulfils an inhomogeneous partial differential equation with the solution [ 15] (X;~,g, m) = exp -
[Dr(g(X')) + 3(1 - 6(g(X')))] dX'
t
X [ ~ F]as (1;/~,g(~), m ) - if ~7 e x p / - l f
XlX'd?," -~
[Dr(g(;k"))
+ 3(1 - 6(g(X")))l} K~, g(3./~,'), m) C(~,';/a, g(k/X'))
= exp -
~
[Dr(g(X')) + 3(1 -6(g(;k')))]
-C(1;l~,g(X))fX7
(1;/l,g(X), m)
6 (g(X")) K(la,g(X/X'),m
4
1
as
,(2.7)
'
where in the last step eq. (2.6) was used. A statement on the leading mass singularity now follows, if eq. (2.7) is compared with what one gets by treating eq. (2.2) as a homogeneous partial differential equation. Introducing the effective mass m
( XdX'~
m(X)=T expd
}
8(g(X'))s'
(2.8)
one has /)3 [ x dX' bm 3 F(~.; p, g, m) = exp ~ - f -~r [Dr(g(Jk')) + 3(1 - 6(g(;k')))l } ~3 X
am(;~)3 r(l; u, g(X),m(X)).
(2.9)
N.K. Nielsen / Non-analytic mass behaviour
113
Comparing this result with (2.7) and substituting g(?~- l) for g one finally gets
f
0fiT(k) 303
7 exp / - 4 f ' "-~ d?~" 6 (gCA"))}r ~ , g(k'-'),m), I'(1 ;/a, g, ~(k)) = Xd?~' 1 ~ X'-I (2.10)
where m { rh(X) =--~ exp -
dX' -~- 6(g(X'))
(2.11)
is the effective mass evaluated with the coupling constant g replaced by g(X-1). The further analysis of eq. (2.10) presupposes' the presence of an attractive infrared fixed point in coupling constant space. For quantum electrodynamics the origin is an infrared-stable fixed point, so the only required information is the values of/3 and 6 in lowest order of perturbation theory: _ g3 /3(g) - 1 - ~ 2 + O(gS) ,
3g2 8(g) = ---~2 + O(g4),
(2.12)
(it should not lead to confusion that the electric charge here is called g), as well as the fact that the quantity K has already an O(g °) term. For large k we then have g(~-
67r2 lnk-1 ,
1)2 ~_ _ _ _
(2.13)
and thus
f1
exp - 4
~
6(g(~,")) ~0~, g(~,,-1 ), m)
h'-I h
c~ f
dk' (In ;k')9 = (In ;k)l o ,
(2.14)
and m
r~(3,) ~-~- (ln k) -9/4 .
(2.15)
The two last results enable us to conclude from eq. (2.10) that the leading mass singularity of any gauge-invariant vertex function at non-exceptional momenta renorrealized according to the prescriptions of 't Hooft [3, 4] or Weinberg [5] is given by P(1; U, g, m) • m3(-ln m) l0 .
(2.16)
In the asymptotically free non-Abelian case we must in general rely on assumptions. If/3 has an infrared-stable zero with 6 = 6o,
~ = Ko ,
(2.17)
114
N.K. Nielsen / Non-analytic mass behaviour
then we have f -~7 exp - 4 1
-~-.6(g(X"))) K ~ , g ( X ' - l ) , m ) c x X -4~° ,
(2.18)
-1
fit(h) cc mX - l + a ° ,
(2.19)
so the leading mass singularities of the perturbation series here sum to a power P(1 ; ~, g, m) ~x m 3 × m 46 o/( 1-8 o),
(2.20)
provided 60 :/: 1. 3. Gauge-dependent vertex functions The analysis of sect. 2 does not apply to gauge-dependent vertex functions. The reason is that a Wilson expansion is required for the estimation of the asymptotic behaviour already after three mass differentiations. The expansion is sketched in fig. 2 for the case of the electron propagator. Via the expansion, gauge-dependent composite operators of mass dimension 2 arise. In QED, the only operator of this kind appearing is Au2 , where A u is the photon field, but in the non-Abelian case one also gets a composite operator consisting of Faddeev-Popov ghost fields. But since these operators are gauge-dependent they do not occur in the Wilson expansion of gaugeindependent vertex functions, so the neglect of them in sect. 2 is justified. With the modifications mentioned above, the analysis now runs as in sect. 2. We concentrate on QED, where the relevant composite operator is Au2 . Instead of (2.2) we now have the following equation for 02P/am2: O
0+
\
02
03 = -(1 - 6(g)) m ~
r(x; u, g, m, ~),
(3.1)
where ~ is a real parameter fixing the gauge within the usual class of Fermi gauges. A Wilson expansion is now performed on the right-hand side of (3.1) as indicated for the electron propagator in fig. 2, Oa
m-~m3 I'(X;tl, g, m, ~) = C(X; ta, g, ~) k'O; g, m) + ....
(3.2)
That the quantity g is independent of ~ is already plausible from fig. 2 and can be verified by the Ward-Takahashi identity. ~ has in contrast to the corresponding function K encountered in sect. 2 no O(g °) term in perturbation theory, but only an O(g2) term.
N.K.Nielsen/Non-analyticmassbehaviour
115
Fig. 2. A term in the Wilson expansion o f the electron propagator with three insertions o f
Z2Z 4 f d4x ~0 (x). The Wilson coefficient Chere has two amputated photon lines with zero
momentum flow and with vector indices averagedover. Correspondinglyan insertion of f d4x A2(x)occurs in the quantity ~'. The new Wilson coefficient C(X; p, g, ~) fulfils
(X ~-~(g) ~---~+2~(g)~~--~+Dr(g)+2+2/3(g)]c(x;/l,g,~)=0,g,
(3.3)
with the solution { /dX'( 2fl(g(X'))] / C(X;/.i, g, ~) = exp -ff Dr(g(X')) + 2 + g(X') ]J (3.4)
X CO;U, gqA), ~CA)), where ~(X) = (g-~X))2 ~,
(3.5)
so corresponding to eq. (2.7) we have in the present case ?" dX'
/
[~B-~ZmZ P]as(X;l~'g'm'O=expl-/-~- [Dr(g(X'))+ X
(
['
)
1
+
(1; la, S(X), m, ~(X)) - C(1 ;/a, g(X), ~(X)) as
X
d 1
+dX' exp
dX"
2
m)+,,.
. fl(g(~."))~ 6 (g(X")) -v~ j ~ /a, g
,
(3.6)
h.lJk'
Therefore ~2
a~h(?,)2 I"(1; u, g, r~(X))
,~ f--~expa dX' 1
f -2 f ~
t dX" X" --~ \(5(g(X"))+/~(g(..X, i l ) ) ))/ ~l), g ( X ' - t ) , g t ^
h '-1
m),
(3.7)
116
N.K. Nielsen/Non-analyt& mass behaviour
where we have for large k, using (2.12) and..(2.13) and the fact that the perturbation series for ~ starts out with a second-order term,
1
fl
XdX'exp(_2 f dX" / i " ~ I, x'-' ~ 7 ; [ (g(X))
o:
+ fl(g( ))~I ~-k" ~]/'
"
.~.,-1,, ~,gl, ) m)
x dX' f -~- (In X') s/2 = (In X)7/2 ,
(3.8)
so the leading mass singularity of gauge-dependent vertex functions at non-exceptional momenta and renormalized according to the 't Hooft-Weinberg methods is in QED given by P(1;/2,g, m, ~) ~ m2(-ln m) 7/2 .
(3.9)
It is remarkable that the leading masstsingularity of gauge-dependent vertex functions is independent of the gauge-fixing parameter ~. As seen from the derivation, this is due to the fact that the quantity ~ as well as the anomalous dimension of the composite field Au2 , which is -2~/g, are independent of ~. Gauge-dependent vertex functions of a non-Abelian theory can also be dealt with under the assumption already made in sect. 2. There is here a small mixing problem due to the presence of a composite operator made out of ghost fields, but apart from that the calculations run as before. 4. Gell-Mann-Low renormalized vertex f u n c t i o n s In the original version of the renormalization group [1, 16], a spacelike normalization point is used for the field and coupling constant renormalizations. It is shown in the appendix how the vertex functions renormalized according to such a prescription are related to 't Hooft-Weinberg renormalized vertex functions. The mass singularity analysis is carried out by means of eq. (A.3), with m first chosen as mphys defined by (A.7). When m is replaced by fit(X) given by (2.11), the following relation is obtained: P(2P'B)(1 ; M, gMCA), mphys(X), ~M(X)) = cb(2F'B)(M;/2,g, rh(X), ~) x r (2F,B) (1;/2, g, rh(X), ~),
(4.1)
where gM(X) and ~M(X) are found from gM and ~M defined by eqs. (A.5) and (A.6), respectively, by substituting ffz(k) for m, and mphys(~k ) is obtained by an analogous modification of (A.7) mphys(~k) = ff/(~k) b(--mphys(~k) 2 ;/2, g, rnGk)) .
(4.2)
The leading mass singularity of F (2F'B) (1 ;M, gM, mohys, ~M) is found by taking
N.K. Nielsen ~Non-analytic mass behaviour
117
the derivative of (4.1) with respect to m. On the right-hand side, m only enters through rh(~.), so a .p(2F,B) (1 ;M, gM(~), mphys(~k), ~M(~.)) am _
rh(h) a ¢(2F'B)(M; p, g, rh(~.), ~) F (2F'B) (1 ;p, g, rh(?,), ~). m arh(X)
(4.3)
It is seen from sects. 2 and 3 and the construction in the appendix that the mass derivative on the right-hand side of (4.3) goes to a finite value for X -->0% so we have in this limit a p(2F, B)( 1 ; M, gM(X), mphys(~k), ~M(X)) cc /~(~k) am m
(4.4)
By the same argument, the terms arising through the dependence ofgM(X ) and }M(X) on m are also proportional to rhOO/m, so eq. (4.4) can be written amphys(~k) a ~(2F, B)(1 ;M, gMOk)' mphys(~k), ~M()k) ) cc /7/0~) am amphys(~k ) m
(4.5)
The term mphys(~k) is found from (4.2) via the transformation law for b(p 2 ;/a, g, m) under the renormalization group, which can be found by comparing (A.2) with (2.1) and solving the resulting partial differential equation for b(p 2;/~, g, m). The outcome is
mb(X2p 2;/a, g, m) = Mn(X) b(p 2 ;/a, g(~), m(X)).
(4.6)
Here we substitute g ~, g(X - 1 )
and apply the result to (4.2) ~"nphys(~k) = mb(-3, 2 mphys(~k)2 ,/.t, g(X- 1), m ) .
(4.7)
The behaviour of the solution of (4.7) for X -->oo depends on the infrared stable fixed points of the coupling constant. The solution in QED, where the origin is infrared stable, is mphys(~k) "- m/~k.
(4.8)
In QED the leading mass singularity is now found by means of (4.5) and (2.15),
F(2F'B)(1; M, gM, mphys, ~M) OCmphys(_ln mphys)-9/4 .
(4.9)
If we assume for the asymptotically free non-Abelian gauge theory that mphy s makes sense and that a non-trivial infrared stable fixed point exists, we also have eq. (4.8)
N.K. Nielsen ~Non-analytic mass behaviour
118
fulfilled, apart from a proportionality factor, so leading mass singularities here, according to (2.19), sum to a power 1--5 0
P(2F'B)(1; M, gM, mphys, ~M) (x mphy s
(4.10)
If the new mass parameter ~ instead is fixed by the definition (A.8), then we hav~ on the other hand a situation where the vertex functions in the formulation of the theory using a spacelike normalization point, as well as all parameters characterizing them, are connected with the corresponding quantities of the 't Hooft-Weinberg renormalized version of the theory by relationships which are regular also in the massless limit. Consequently, the estimates (2.17) and (3.7) as well as (2.20) are valid her~ too. I am most grateful to Professor K. Symanzik for instructing me on his methods, for making unpublished material available to me, and for pointing out the relevance of ref. [12] and its interpretation in terms of the Wilson expansion. Also I wish to thank Dr. R.J. Crewther for a stimulating correspondence which led to the comple- . tion of this work, most of which was carried out several years ago. Appendix In this appendix we briefly record the connection between the vertex functions renormalized by means of a spacelike normalization point [ 1,16] with those renorrealized according to the prescriptions of 't Hooft [3,4] or Weinberg [5]. The two-point vertex functions with two bosons (photons or gluons) and two fermions, respectively, are in the 't Hooft-Weinberg schemes denoted: r (t~L; ° 2)rk ~ ; 11, g, m ) = i ( 6 ~ v k 2 - k ~ k v ) d - l ( k i ' 1 1 ,, g , m ) + gauge terms, p(2,O)(p; 11,g, m) = i( i'yp + mb(p 2 ; 11, g, m) ) a(p 2 ;11, g, m, ~) .
(A.1) (A.2)
The functions d and b will in QED be independent of the gauge-fixing parameter ~ but this is not so in the non-Abelian theory. A general vertex function F(2F'B)(X; M, gM, m, ~M) with 2F fermion legs and B boson legs, renormalized by the prescription using a spacelike normalization point, is then found from the corresponding 't Hooft-Weinberg renormalized vertex function F(2F'B) (,k; 11,g, m, ~) by F(2F'B)(~k; M, gM, ~Z, ~M) = dp(2F'B)(M; 11, g, m, ~) r(2ic'n)(X; 11, g, m, ~) , (A.3)
where • (2F'B)(llt;11, g, m, ~) = [a(M2; 11,g, m, ~)]-F[d(M2;11,g, m)] B/2.
Here M is a real parameter.
(A.4)
N.K. Nielsen ~Non-analytic mass behaviour
119
In eq. (A.3) the coupling constant gM, the mass ~ and the gauge fixing parameter ~M still have to be specified. Consider first the coupling constant. In QED the equation fixing gM is g ~ = g2 d(M2; la, g, m) .
(A.5)
In the non-Abelian case also the value of p ~ , 3 ) (p, q, r;/~, g, m, ~) for p2 = q2 = r 2 = M 2 is necessary for the definition o f g M [ 17]. The gauge-fixing parameter ~M is now given by ~M = ~ d - l ( M 2 ; p , g , m) ,
(A.6)
both in QED and in the non-Abelian theory (cf. (3.5)). This leaves the choice o f the new mass parameter ~ . One possibility, realized in the original formulation o f the renormalization group [1 ] , is to use the physical mass defined from (A.2) by mphys = mb(--m~hys; la, g, m) ,
(A.7)
as the mass parameter of the theory. In perturbation theory, this definition certainly makes sense in quantum electrodynamics. Whether this is also the case in the nonAbelian theory is open to conjecture. The new mass parameter r~ can also be fixed by [16] = m M = m b ( M 2 ; la, g, m)
(A.8)
in analogy to (A.5) and (A.6).
References [1] M. Gell-Mann and F.E. Low, Phys. Rev. 95 (1954) 1300; N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959); K.E. Wilson, Phys. Rev. D3 (1971) 1818. [2] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227. [3] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [4] J.C. Collins and A.J. Macfarlane, Phys. Rev. D10 (1974) 1201; M.J. Holwerda, W.L. van Neerwen and R.P. van Royen, Nucl. Phys. B75 (1974) 302. [5] S. Weinberg, Phys. Rev. D8 (1973) 3497. [61 M. Gomes and B. Schroer, Phys. Rev. DI0 (1974) 3525. [7] G. 't Hooft, Nucl. Phys. B62 (1973) 444; H.D. Politzer, Phys. Rev. Letters 30 (1973) 1346; D.J. Gross and F. Wilczek, Phys. Rev. Letters 30 (1973) 1343. [8] K. Symanzik, Comm. Math. Phys. 23 (1971) 49. [9] K. Symanzik, Comm. Math. Phys. 34 (1973) 7. [10] G. Mack and K. Symanzik, Comm. Math. Phys. 27 (1972) 247. [11] B. Lautrup, Nucl. Phys. B105 (1976) 23.
120
N.K. Nielsen ~Non-analytic mass behaviour
[12] P. Papatzacos, Nuovo Cimento Letters 17 (1976) 347. [13] W.E. CasweU, Phys. Rev. Letters 33 (1974) 244; D.R.T. Jones, Nucl. Phys. B75 (1974) 531. [14] S. Weinberg, Phys. Rev. 118 (1960) 838. [15] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 2 (Interscience, New York, 1962). [16] K.E. Eriksson, Nuovo Cimento 30 (1963) 1423; M. Astaud and B. Jouvet, Nuovo Cimento 63A (1969) 5. [17] T. Appelquist and J. Carazzone, Phys. Rev. D l l (1975) 2856.