PHYSICAL REVIK% D. VOLUME 19, NUMBER 6. 15 MARCH 1979. Renormalization group and infrared behavior of quantum chromodynamics.
PHYSICAL REVIK%
VOLUME
D
Renormalization
19,
15 MARCH 1979
NUMBER 6
group and infrared behavior
of quantum chromodynamics
David Shalloway* Laboratory
of Nuclear
Studies, Cornell University, Ithaca, New York 14853 (Received 6 December 1978)
A diagrammatic implementation of the generalized renormalization group is developed and used to study the infrared divergences of the photon-quark-quark amplitude in quantum chromodynamics. The leading-log result is simply obtained, We show that nonleading terms cause no qualitative change and that the leadinglog result is accurate down to the nonperturbative region. The infrared behavior of QED is discussed as a pedagogical example.
I.
INTRODUCTION
The problem of quark confinement in quantum chromodynamics (QCD) has motivated a number of studies of the infrared (IR) divergences of the theory. (See Ref. 1 for a pedagogical review and references. ) These divergences appear as powers of irk, where X is an effective IR momentum cutoff (e.g. , a gluon mass or term keeping external lines off the mass shell), or as powers of x-=I/(d —4) = dimensionality of space-time used to provide (d — a gauge-invariant IR cutoff). Kinoshita and Ukawa2 have studied (in weak-coupling perturbation theory) the divergence structure of diagrams with an arbitrary number of massive quarks and color-singlet external sources and have derived. a differential equation governing the leading-log divergences. For example, for the simplest nontrivial case, the production of a mass =m quark-antiquark pair by a color-singlet current carrying finite momentum q, the leading-log behavior of the renormalized amplitude T is described by
dT(P, P', t)
g(t)
dt
(4~)'
O'=P
C
P, . t) '~ m'e''I~T~ j
-P
turn-space cutoff and gauge-invariant results such as Eq. (1.1) is discussed below. ] The boundary condition is provided by T(P, P';0), the amplitude calculated with coupling constant g and IR cutoff perturbation theory is still Ao (where low-order valid). The'behavior of the solution is similar to that found in QED: exponential suppression of the soft-photon-free production amplitude with e2 replaced by g, the effective coupling constant determined by the pure gauge field sector of the theory. Whether or not this behavior is altered by the nonleading terms and the relevance to the problem of confinement have been open questions. To illustrate this point and clarify the objective of this work, consider an example where the O(g2) amplitude is
T„, —1+g I CO
(1.2)
q
[t =In(AO/X)
and Co is a numerical coefficient of O(1)). This result is valid for g t «1, but we would like to study the region g «1, t+&1, g t~ 1 where the behavior could be qualitatively different. In this domain it seems sensible to group the terms of the formal expansion in g in the form T =CO(g't) + g2C, (g't) + g4C2(g2t) + ~ ~ ~
2
g' ( ) =-1
22C 3
4),
= eigenvalue of the quadratic Casimir C~(C„) — operator of the gauge group for the
fermion (adjoint) representation, du
E(s) =—(2e — In their work ently use
t=x=
+ 4.
I-d),
/(
%but we can equival-
t = -ln(y/Ao) and Ao X is an effective IR cutoff momentum is an IR reference momentum [A20«m, q'; but (g ln(AO/m) «1].[The compatibility of a momen-
where
~
where Co, C&, .. are functions calculated by nextsumming the leading-log, next-leading-log, next-leading-log, . terms. [We have assumed, as in QED and QCD, that a maximum of n powers of t can appear in an O(g ") diagram ) However, while we may hoPe that Eq. (1.3) can be treated as a perturbation expansion in g with coefficients C„(g2t), there is no reason to suppose that the C„ are bounded since each function represents the sum of an infinite number of O(1) terms. Thus, singularities in the C„could overpower the additional powers of g', i.e. , the leading-log term could be dominated by the next-leading-log or some higher term. For a specific example, summing only the leading logs could give
C„.
zt,
..
&
762
Qc
1979 The American Physical Society
RENORMALIZATION TLL
GROUP AND INFRARED BEHA VIOR OF. . .
Cp(g f)
while including the next-leading logs could give the qualitatively different result
~ ~~ =~ p(g'f) +g'Cg(g'f) g' =e' "p I
1+ I-a,g't) '
Such a possibility of practical concern in QCD since the combination of two competing effects, infrared gluon emission and gluon coupling-conmakes the prediction of stant renormalization, nonleading terms difficult. An independent problem is the effect of the multiplicity of higher-order diagrams. We roughly expect that there will be about n! O(gP"} diagrams and that, as a result, the usual perturbation series is, at best, an asymptotic series for small t. This will be reflected in Eq. (1.3) as a growth in the C„ for large n since most diagrams give nonleading contributions. In this paper we apply the renormalization group (RG) (in the generalized sense popularized by Wilson ) to this problem and show that the effect of any finite number of C„ terms in Eq. (1.3) can be bounded and is small in the region g2t ~ 1. We do not consider the problem of the proliferation of high-order diagrams for large n and the convergence of the entire series. The RG replaces the infinite number of diagrams contributing powers of 1 by the iteration of RG transformations involving an infinite number of effective vertices but no powers of t. That is, the diagrams which define the renormalization group are independent of t even for RG transformations are constructed in Minkowski, rather than Euclidean, space by a diagrammatic procedure in which propagators are split into high-and low-frequency parts. Only the high-frequency part, which is defined to have both IR (X„) and ultraviolet (A„) cutoffs, is integrated in one stage of the transformation generating, at most, factors of ln(A„/X„} O(1). The ratio b =A JX„will be held fixed independent of n, and A„, —X„. The cutoff masses are chosen to be complex, not real. The cutoff (mass)' are -i Xp and -i A„' so that the spurious poles and thresholds due to the cutoff masses are off the real axis. This makes it- easier to obtain upper bounds for complicated diagrams in Minkowski space. The powers of t arise as the result of many iterations of the transformation due to the summation of many factors of lnb. The
t»
t-~.
&
effective vertices are described by a system of coupled differential equations representing a generalization of Eq. (1.1). The system is manageable because the IR scaling properties of the effec-
1763
tive vertices can be used to classify them as relevant, marginal, ox irrelevant. The relevant and marginal vertices correspond to the primitively divergent diagrams of the usual perturbation theory; all others are irrelevant. The irrelevant vertices are controlled by simple power-counting (scaling) arguments which do not require the detailed solution of their respective differential equations. The behavior of the remaining (relevant or marginal) variables is described by equations such as Eq. (1.1) but where we can demonstrate that all errors are bounded. The bounded errors cannot significantly alter the qualitative behavior of the solutions. Since we construct the RG transformations perthe effective coupling constant, turbatively in our results are not applicable to the nonperturbative region where g is large. This still represents a significant extension of perturbation theory into the IR domain compared to perturbation theory in a fixed coupling constant or leading-log sums without controls on nonleading terms. The method clearly shows why there is no justification for extending the results obtained by perturbative methods into the strong-coupling region. As a byproduct we see how the RG provides a transparent but rigorous derivation of the leadinglog results. For simplicity we discuss only the case of the color-singlet current-quark-quark amplitude. The QED amplitude is studied first as a pedagogical example. In Sec. II we review the generalized RG procedure' (for the case of a single scalar field) and discuss its domain of validity. In Sec. III we present a functional representation of the propagator splitting procedure and define the construction of the RG transformation in detail. In Sec. IV we study the IR behavior of the QED three-point vertex and see that there are only three marginal vertices requiring detailed use of the BQ recursion relation: the primitively divergent electron self-energy, the electron-electronphoton vertex and the electron-electron-external current vertex. Ward identities (maintained in leading order) remove two of these leaving a single recursion relation equivalent to the QED form of Eq. (1.1) plus bounded terms coming from approximation errors. We do not study the detailed cancellations of these nonleading terms, but show that they can be of no physical significance by evaluating their maximal effect after the iteration of many HG transformations. In Sec. V we study the color-singlet currentquark-quark vertex in QCD. The gluon and ghost masses are relevant terms and require special discussion. Gauge invariance is maintained but is nonmanifest. There are eight marginal ver-
g,
DA V ID SH ALLOWAY
1764
tices:
divergent diagrams and the vertex. The color-singlet current-quark-quark Slavnov-Taylor identities reduce the problem to two coupled recursion relations corresponding to Eq. (1.1) plus nonleading approximation error the primitively
terms as in QED. These error terms are similarly shown to be of no physical significance. In Appendix A we show that a single RG transformation gives at most factors of (g' ln b)". The special treatment of self-energy corrections is presented in Appendix B. II. REVIEW A. The generalized renormalization
group
As emphasized byWilson, the essential feature ef the RG procedure is that a calculation involving a large number of degrees of freedom is replaced by the iteration of many intermediate calculations each involving only a small (RG tranformations), number of degrees of freedom. Thus, large coefficients such as ln(X/m) (coming from the simultaneous inclusion of a logarithmically large momentum range) never occur in an individual RG transformation (RGT) but result from iterating the transformation many times. Individual RGT's are approximated in the same way as conventional perturbative calculations, but the absense of any uncontrolled (arbitrarily large) terms means that the approximations are always sensible. The price paid is that errors may accumulate during the iterative procedure. This requires a new, RG specific, analysis which introduces the classification into relevant, marginal, and irrelevant variables depending on whether or not accumulation can take place. Only the relevant and marginal effective vertices corresponding to the primitively divergent vertices need be studied in detail. The RG method is practical because the usual dimensional arguments show that there are only a small number of these. We now discuss the RGT. Given an action S„defining a Green's functional W(j) by g/-(j)
[dy]
i
s„(eiJ
(2. 1)
&
an RGT R defines a new action S
such that Iii'(
j) —
[d y]& i 8 „& (4
4
&
(2.2)
S„ is
expanded in terms of the coefficients y of, in general, arbitrarily complicated effective vertices. R is defined by a set of coupled equations expressing the y„„ in terms of the y~. If S„ is defined with a physical cutoff momentum A„, is constructed by integrating out the high-frequency
S„,
19
degrees of freedom for X„&p"&A„.' A. „(=A„„) is then the physical cutoff momentum for all remaining integrations. [The (t&(p") are either completely or partially integrated depending on whether a sharp or smooth cutoff is employed. ] Integrating the original vertices over Q(f&") generates a new set of effective vertices. By rescaling the remaining dummy variables, Q(P) f, (t&„(i&AQ&„), we restore the original formal domain of functional integration. 7 The rescaled set of effective vertices defines The method of integration of high-frequency components is not unique. In this paper we follow a suggestion of Wilson' and use a diagrammatic procedure to construct the effective vertices: An initial gauge-invariant renormalization stage (see Sec. IIIB) gives an ultraviolet (UV) cutoff propagator of the form &f&Q~)
-
S„„.
i 0
A'p
p2(p2+iA 2)
(2.3)
'
This propagator is then split into high- (h) and low(I) frequency parts Do
D~r
+Dp
.
For example,
(P'+i&0')(P'+iA(&')
'
(2.4)
The first term on the right-hand side is D,' and the second term is D,". Imaginary (mass)' is used to preclude denominator zeros coming from the Minkowski space metric. (We are interested in onshell Green's functions and cannot rotate to Euclidean space. ) New effective vertices are calculated using the old vertices and Dp~ propagators. These will generate the same Green's functions as the original theory when connected together with D
The detailed reconstruction of an individual Feynman graph of conventional perturbation theory can be quite complicated. For example, the completely integrated graph of Fig. 1a is reconstructed using the effective vertices of Fig. 1b (which are partially expanded to show the terms needed for this particular graph). For instance, when the second term in the expansion of the four-point function is contracted diagonally with one D,' we get a contribution corresponding to Fig. 1c. In Sec. III we develop a functional representation of this procedure to simplify the derivation of the rules for generating effective vertices and to prove the equivalence of this two-step calculation to the
one- step calculation.
RENORNIAI, IZATION
19
GROUP AND INFRARED BEHAVIOR OF. . .
p=b
h
~hh h
p',
e(P) =b' e.(P'),
h
h
/ +'"
(2.6)
h ~
~
h
Ph
n[
we get
(c) 4
Xhl
h
Qh) hl
f h
+~~~
/
d'P'I
n+i( Ps)
(2)))42 ~B(P )
a/2
+(scaled effective vertices).
FIG. 1. Heconstruction of a Feynman graph in terms of effective vertices. (a) A typical Feynman graph. P) Effective vertices involved in the reconstruction of (a). (c) A typical term in the reconstruction. h and l on lines denote high and low propagators; circled h's denote effective vertices after the high-frequency integration. Dots denote amputated external legs in all figures.
This procedure is iterated many times. That is, instead of integrating with D, in one step, we make the change of notation D,'-D, and apply the above procedure again. At each stage we choose A/X„ =b so that after N stages of iteration we have integrated out all high-frequency components above a physical cutoff A„=b "A, . We study the small A„behavior by increasing N while holding b constant. The transformations described are not yet RGT's since the explicit momentum scale fixed by the propagator masses changes at each stage. This is avoided by including a momentum rescaling in the RGT such that the formal propagator masses are invariant. We denote these masses by &(=&,) and &(=&0) and use the unsealed subscripted variables A„(=b "A,) specifically to refer to the physical mass scale. A field amplitude renormalization is also included so that the renormalized propagators are invariant. ' Thus, we can drop the propagator indexing subscripts. Specifically, following the D" integration we have
'
P"(P" +iA')
+
(b)
+ (effective vertices).
(2. 5)
Changing to rescaled variables
~(
1765
(2.7)
The rescaling allows the direct comparison of interaction terms in S„with those in S„„since the same propagators are used in both cases. (We are compensating for the decreased phase space but increased mean square fluctuations of the lower-frequency field components. ) It also removes the unbounded parameter A„ from the coupled equations defining R. The "RG philosophy" is that R should be a bounded transformation containing no large parameters or uncontrolled regions of integration. This is now the case since the integrals are UV cutoff at A and IR cutoff at ~ and can generate only powers of ln b -1 instead of powers of t.' Once the RGT is calculated we study the behavior of S„=R"(S,). Representing each S„as a point in the multiply infinite-dimensional space of all cutoff actions 8,' all the physics is summarized in the discrete, but in principle continuous, trajectory S(t); S(t) =S„ for t =n lnb. A sequence of approximations to W(j) improving as t-~ can be extracted by expressing S(t) in terms of the unscaled source term. (See Sec. IIIE for details. ) The intermediate S„provide interpolating functionals between the totally unintegrated action S, = Jd xL((t), j) (Ref. 11) and the totally integrated
j„
2'(j).
Each RGT generates a small segment of the trajectory so RGT approximation errors are introduced at each step. For accurate overall approximation we must determine what type of errors are amplified and what type are deamplified. As this has been extensively discussed by Wilson, ' we only provide a cursory review: 3 can be parametrized by. making a Taylor's series expansion of the possible n-point functions:"
'J', „'.(())~- ())'(())+Z ', J'2';. " ",',;~ (I:()((),)" ((),
()=)-,
)
—,
x +
ya'i''' n
Q 2
2,
' 'ap'i 1
. . .p'a+
.j.(P)It.(P)i.(P) .
d'p
(2, )4 4 (p)j.(p) (2.8)
DA VID SHAI I OWAY
1766
The expansion coefficients y„provide a basis. The ROT equations will be of the form
~:„="(~„+N[&r„H) (n= u, , f,).
(2.9)
-f„.
The N represent terms that are known in an iterative calculation of the entire ROT trajectory. The a are rescaling factors. An iterative treatment of the entire RG trajectory (and not just of the individual RGT's) is required to calculate the relevant variables since these variables are calculated by a reversed recursion relation &p, „'=f(5p~, . .); see Ref. 3b, Sec. V. We do not actually implement the iterative procedure in this paper since the relevant variables do not influence the leading IR terms in QED or QGD. The significance of the procedure is that it provides, in grinciple, a well-defined method for calculating the relevant variable contributions and shows that these contributions are also bounded. In general, Eq. (2.9) would be expressed in terms of the eigenvectors F of the matrix of linear terms of the transformation S„,=R(S„). These are linear combinations of the y 's. In that case, the N vnuld be at least quadratic in the F were known [which begin in O(g)] so that if the to O(g ), then the N would be known to O(g "). The a 's would be the eigenvectors of the linear matrix. However, for simplicity we work in terms of the y™'sthemselves and include off-diagonal linear terms in the N 's. This is possible in this case because the matrix of linear terms in the Each k- point y representation is triangular: vertex will have linear contributions obtained only by contracting external terminals of m ~ k-point effective vertices with bare propagators. Thus, the diagonal elements in the y representation, the a 's are the correct eigenvalues of the complete linear matrix. Also, since only a finite number of n-point functions are nonzero to any given order in g, Eqs. (2.9) can still be solved iteratively by first solving for the k-point function is deand then working down. [k where k =k termined by the skeleton graphs and, for our work, is given by k =m+2 in O(g ).] Depending on whether a &1, =1, or &1, we call or relevant. irrelevant, marginal, —sv„ Consider an irrelevant vertex variable y„= with a' =b '&1. The solution of Eq. (2.9) is'+'
', .
I"
"
n
w„=b "mo+gb~
"N
[(ylj].
(2.10)
The small weighting factors b "and b~ make I)„ insensitive to ~i, , andto the y~ for k «n. There can be no accumulation of individually small effects in the sum. Thus, the sum is roughly of the same size as the b 'N term and zv„. is sensibly approxi-
mated in terms of the N [1yJ], k = n. For b»1 we can replace the recursion relation Eq. (2.10) by the simple equation
.„=~'N H~B.
(2.11)
This is not the case for a marginal vertex variable y~ =-g„with a~=1. Now the solution to Eq. (2.9} is n-1
g. =g. + EN'[O', H.
(2.12)
Equation (2.12) cannot be used to get an accurate approximation for g„since the undamped sum (and, as importantly, the undamped sum of errors coming from the approximation of each NB) is unbounded as n ~. This is the well-known origin, in RG terms, of the logarithmic divergences in conventional perturbative calculations. ' Instead, we will study the effect of errors on marginal variables by using approximations to Eqs. (2.9) directly. That is, we study the sensitivity to z~ and g~ of the equations
-
yB
—(yl+ c8) +NB ~
[&
.". H
gy
+c
H
(2.13)
for y~, a marginal vertex variable. pa and q@ represent bounded error terms: g„8 represents vertex variable errors while g8 represents errors in the RG equations (e. g. , due to perturbative approximation). We emphasize that irrelevant y 's may appear in the leading terms of Eqs. (2. 10)-(2.13) [e.g. , see Figs. 3(c) and 8, and the associated discussions]. The appellation irrelevant only means that Eq. (2. 11) can be used to replace Eq. (2.9) for these variables. Similarly, irrelevant (as well as marginal) variable approximations introduce accumulating errors in the marginal variables. The point is that their effect can accumulate and become large only by way of induced marginal variable
errors. The only relevant variables appearing in this study are the effective gluon and ghost masses. Relevant variables generally become large and important for f ~. However, in QGD, gauge invariance forces these variables to go to zero as f ~ (Ref. 15) and so they are unimportant for
-
-
our calculation (see Sec. V). A word on gauge invariance: The original UV cutoff propagator is introduced by a gauge-invariant renormalization procedure. While manifest gauge invariance is broken, all Ao dependence is canceled in calculations of W(j) by the appropriate Ao-dependent counterterms and renormalization factors appearing in the effective action S, (see Sec. IIIB for details). Similarly, while the RGT partial momentum integrations break mani-
19
GROUP AND INF RARED BEHA VIOR OF. . .
RENORMALIZATION
"
fest gauge invariance, the invariance is still only Because hidden since all S„yield the same W(j). of this and the fact, as we will show, that we can place bounds on all errors and their possible accumulation, there is no danger that our results are distorted by gauge-breaking terms. The lack of manifest gauge invariance does result in one restriction on the domain of validity of our result, i.e., g/m' o0 (where q' is the finite momentum of the external photon or color-singlet current). This is because the Ward and Slavnov-Taylor identities are respected by the leading approximation to the RG transformation equations and imply cancellations at q' =0 which may not hold for the nonleading terms. Thus, the relative error at q'/m'= 0 can be large. (See Secs. IV and V for details. ) B. Perturbative renormalization
group
The right-hand side of the RGT Eqs. (2.9) can be approximated by expanding the N in powers of the y„(in particular, in terms of g„, the effective coupling constant in discrete notation) assuming This differs from a that the latter are small. conventional expansion in that high-order terms in this equation can never become significant due to unbounded logarithmic coefficients. Therefore strict limits can be placed on the error terms in Eq. (2.13). Error limits exist even if b is fairly large (say 10'). In this case we get contributions of O((g„'lnb)') (see Appendix A) which are still small. Thus we can take b large but bounded and make the additional approximation of keeping only the leading terms in ln b. For example, O(g„') terms are neglected relative to O(g„' ln b) terms. Errors
"
"
are bounded by O(g„').
The approximation does fail if g„[or, in a more general case, any of the marginal y~'s that appear nonlinearly in Eq. (2.9)] becomes of O(l). This does not occur in QED so the perturbative RG is applicable for the entire trajectory S(t = ln (A, /A„)). Therefore, the conclusion of our pedagogical QED study (Sec. IV), that nonleading IR divergences have negligible effect, is valid for any value of IR cutoff. . (The significance of this result is that it is obtained without any study of the detailed cancellations due to gauge invariance. ) The domain of the perturbative RG in QCD is bounded by the IR growth of g„, but is significantly larger than the usual perturbatiye domain. We review conventional lore to clarify this point. Assuming initial renormalization at a mass m, one-step perturbative calculations are accurate for g, ' « I, g, ' ln (m/X) & «& l. b is a small parameter that we leave arbitrary since we know nothing about the radius of convergence of the
1767
series. The leading-log behavior gives an indication of the extended behavior but can be invalidated by nonleading contributions since the latter terms are uncontrolled for &/m-e ' '0 . However, in an RGcalculation all terms are controlled as long as For g'(t) -g, '/(1 —Cg, 't), we have a g (t) & bound A.„Jm -e '~c~o . C, in contrast to 5, is known. This allows us to determine the structure of the theory right up to the point where it can undergo qualitative nonperturbative charges. Calculations within the perturbative RG region are approximated by including only the dominant RGT contributions in g„and ln b to each vertex variable. The included contributions may be of different orders for different variables. For example, the included three-gluon vertex correction is O(g„' lnb) while the four-gluon vertex correction is O(g„' ln b). This independence of approximation reflects the independence of the manner in which vertex variables enter the RGT equati. ons. In algebraic terms, the components of dS(t)/dt along each axis of 8 are independent and are independently approximated. The perturbative expansion is implemented in terms of effective vertices. To determine the order of a given graph, each effective vertex is counted according to the order in g„of its leading term. In general, the leading order is determined (where k is the by skeleton graphs and is number of external lines) in QED and QCD. This means that most irrelevant vertices are not involved in evaluating the leading corrections to marginal vertices. Of course, the effective vertices implicitly contain higher-order corrections coming from previous RGT's. As a result, the approximation procedure, while simple in terms of effective vertices, is complex when expanded in terms of go. To clarify the nature of the approximations, we enumerate the O(g, ') corrections to the threepoint vertex in a two-step QED calculation [Figs. 2(a), 2(b), and 2(c)] and show which diagrams are included and excluded after all approximations. For simplicity in these figures we omit self-energy subgraphs and only split photon propagators. The graphs of Fig. 2(a) are included since the regions enclosed by dashed lines are incorporated as parts of effective vertices beginning in O(g, ) giving an overall graph of O(g, ' ln b). The graphs of Fig. 2(b) are excluded since they are of fifth order in effective vertices. The graphs of Fig. 2(c) involve four-point functions; they come from the effective vertex graphs shown in Fig. 2(d}. The four-point effective vertices begin in O(g, ') so they are counted as O(g, '}. Thus, the overall graphs of Fig. 2(c) are of fourth order in effective vertices and of fifth order in g, and are excluded.
"
l.
g„""
DAVID SHALLOWAY
1768
(a)
(a)
~
X
(b)
r
~
(b)
(c)
~
I
h
I/
I I
~ 130Nele
X ISED+
I
(c)
~
~
~ 4%HI
~
I
~l
I
s
I
L
l I I
I I
FIG. 3. Examples of effective vertices which (a), and (c) do contribute to lnb terms. Solid lines represent fermions, wavy lines photons, dashed lines ghosts, and curly lines gauge bosons. (b) do not contribute,
III. THE RENORMALIZATION-GROUP FIG. 2. (a), (b), (c) Fifth-order corrections to the three-point vertex in a two-step @ED calculation. Dashed lines encircle effective vertices. Self-energy corrections and fermion propagator splitting are omitted. (d) Effective vertex graphs corresponding to (c). Circles denote effective vertices from previous integrations.
The second approximation, to keep only the leading order in ln b, further reduces the graphs that For excontribute to the leading approximation. ample, the graphs of Fig. 3(a) are omitted from the QCD calculation since the disparate cutoffs for the high and low propagators in the loop prevent any logarithmic integrals from developing. More generally, even the use of effective fourpoint vertices (constructed with high propagators) in graphs generalizing Fig. 3(a) cannot build logarithmic divergences. The graph of Fig. 3(b) is
"
similarly omitted. [Fermion propagators are split relative to their on-mass-shell momenta rather than to zero momentum (see Sec. III D for details) but the analysis is similar. ] The graphs of Fig. 3(c) are unique exceptions and are included (cf. Fig. 8 and the associated discussion). These effective vertices do contribute to the ln b terms: Because of the original quadratic divergence of the graphs of Fig. 3(c) (in the absence of cutoffs), a logarithmic integral remains even with the mismatched cutoffs. This gives an ln-b term.
TRANSFORMATION: FORMAL DISCUSSION A. Single-field RGT
Consider the nth-stage action S„ for a single
scalar field:
(3.1)
yh(Dh)-lych
(We use an obvious abbreviation for momentumspace integrals. ) If we define a new action for two fields (suppressing indexing subscripts)
$(yh yl g) —
&
where D" +D' =D, then" gr(~) —
[dltlh][dpi] 8 l g fS '&4~5
)
S (h "e4
d
)
(3.3)
Equation (3.3) is proved by rewriting it in terms of lt' = lt h+ p' and evaluating the quadratic p in-
tegral. This corresponds to the diagrammatic propagator splitting procedures if we perform the lt h and lt ' integrations sequentially. For example, if
GROUP AND INFRARED BEHAVIOR OF. . .
RENORMALIZATION D =(P2+ib)
',
' —(p'+iA')
P=&
1
Dl —(p2+26)-1
(3.4)
(p2+2y2)1
yh —
(3.9)
Powers of b and Z, are absorbed into the vertex variables by rescaling 4
3
nhg
~n
From Eqs. (2.9) and (3.10) and we see that &0. —y4-k-m
1yh+-1
l
e"V(i" fe'
l -1
(D ) y
jD"]-)
1
'j
(3.10)
where
g+j Dh
~
0, (P'),
(P2 + 2A2)-1
Integrating (t)h gives S((t)„j), the unscaled effective action. Clearly S (0, j) is diagrammatically constructed by the usual Feynman rules using D" instead of D. The rules with Q' x0 are obtained by expressing Eq. (3.2) in terms of the shifted variable
yh(Dh)
'P'
4(p) = b'Zh'"
we choose D h —(P2 + 2y2)
1769
(3.11)
The source terms are rescaled by -(D-" Ii)~
(3.12)
222(p') =b 'Zh'
(3.5) Since
j only
couples to ((I))'-j D"), we see that
s ((
j) =-'*
(t)h
We also have
in the combination
' '~(f ) jD'+& jy („'(D')
'(+
(
)'
(&)')-
D'(P) so that all factors of
(-j D") =Z'(j)+-'
j D"j .
labeling
Zh(j) =-i ln W "(j) is the connected graph generating functional constructed conventionally but using D" instead of D. V((t)) is the generator of the bareconnected Green's funcpropagator-amputated
tions. S could now be expressed in terms of a rescaled this form is in(t) to define the RGT. However, complicated contains (t), jconvenient because V mixing vertices and j-only partial Green's func-. tions that would have to be carried along as we iterated RGT's. Instead we rewrite S in terms of
'
D(p) D "(p)
(3.6)
where V
-,
, -' D'(P) -'Z D{
(t)~-g,
renormalized S as S as desired.
Re-
b and
Z, are absorbed.
„we
and defining the get Eq. (3.1) with
j~-j „etc.,
n-n+1
(3.13)
B. Initial stage
The above procedure assumes a cutoff input propagator. Thus, the initial stage, where S equals S„~, the canonical action with conventional propagators, required special treatment. For simplicity assume SUv
2
0 (P +26)+ VUv(4)
(3.14)
We define
y
—.
yl
j Dh
(3.7)
Dl —(p +26)
so that
S=2 0(D') ' 4'+ (3.8)
j
The explicit coupling to is now trivial. The final step is to rescale p and renormalize to bring S to the form of Eq. (3.1) with a Q and new effective vertex functional. The quadratic terms in V modify the propagators so an additional field renormalization by Z, is required. (The definition of Z, and the treatment of quadratic interaction terms is discussed in Appendix B.) The free field rescaling [Eqs. (2.6)] is replaced by
j
Dh = (p2+2A 2} 1 1
(3.15) (p +2A 2)
(3.16)
where A, is in the IR region. The D"„~calculations are performed in the presence of a conventional gauge-invariant ultraviolet cutoff, for example, by dimensional regularization in dimension 4-z. The presence of (mass)' -iA, 'does not affect the usual renormalization arguments so a renormalization at p' =0 removes all UV-cutoff dependence (i.e. , all s dependence). We use the same field renormalization conventions as in the RGT's (see Appendix B) but do not rescale by b The initial. renormalization constant is called Z3 UQ The output action S, has the cutoff propagator D =—Dlv and has the correct form to be an RGT input. It is expressed in terms of the renormalized coupling
DA VID SHALLOWAY
1770
(p = @'=m), and where p represents an I& momentum (p' «m'). The initial cutoff fermion propagators for the IR calculations have the form
constant g, and contains a finite mass-shift term 6m, ' =O(g, 'A, ') from the breaking of manifest gauge invariance. This is the initiating counterterm for the discussion of the running massshift term Sn„' in Appendix B. Am, ' and the other manifest gauge-breaking terms generated by the act as counterterms for the followDU~y diagrams ing calculation to ultimately restore the hidden gauge invariance. It is clear that the gauge invariance is only hidden since S, generates the same W(j) as the manifestly gauge invarian't Soy. The proof of this ia identical to the proof of Eq. (3.3) but is carried out in 4-s dimensions,
P is
C. Double subtraction
«m,
D~ p(p) = [(p+p}+m]([(P+p)'- m'+i&]
—[(I +p)'
symbolic for either P or P' and the cutoff is in the IR region (A, «m). For p and A, both A, D~, ~ simplifies to
=~
— + m
p
P
D jj= 2(p2+fy2)
&
(p2+ f2/2)
—2(p'+iA') '+ (p'+i2A')
', ',
(3.17)
&
'.
The previous discussion is easily generalized this case.
to
D. Multiple field RG'f's and fermion propagator splitting
When S contains more than one field we define a complete ROT by propagator splitting each field in sequence followed by momentum rescaling and renormalization. The scalar field discussion applies directly to the ghost field and is trivially generalized to the photon and gluon fields. [Only the 1/p' part of the spin-1 propagators is split. D, D~, and D' all include the numerator factor (a'. —& p.p.&p') ] The massive fermion propagator splitting requires special attention since fermion momenta are near the mass shell rather than near zero. Since we can omit fermion loops, all internal fermion momenta are either P+p or P'+P where P and P' are the external fermion particle momenta of T, the overall amplitude under study
.
"
-fA, )']"). (3.18)
The single subtraction of Eq. (3.4) is not sufficient to cut off the boson self-energy graphs in @CD. Instead we use a double subtraction to split the propagator. That is, D =(p2+i5) ' —2(p'+iA') '+ (p'+i2A') D'=(p'+i 0) ' —2(p'+i&') "+(p'+f2&')
(m
'
—+
) il&
(3.19)
This propagator is obtained after an initial highlow propagator split and integration of the initial high-momentum propagator (which covers all momenta above A, including both momenta -m and momenta»m). The procedure is similar to that discussed in Sec. III B. Equations (3.18) and (3.19) are the first places where the IR specific nature of the problem appears since there is no distinction between the IR and UV formalism in the absence of the fermions. P and P' appear only in normalized form as component selecting vectors and cancel in vertex momentum conservation relations. Thus, they can be treated as fixed quantum numbers distinguishing two types of fermions which can be separately functionally integrated. We call p, the relevant momentum variable, the "reduced fermion momentum" and distinguish between P and P' fermion propagators, D~, p and D~, ~,. We omit the P and P' subscripts when no confusion will
arise. The linear dependence on P of the individual propagator terms requires special attention when studying the bounds on higher-order diagrams (see Appendix A), but the propagator splitting and rescaling analysis parallels that already presented. The action now contains fermion propagators and source terms. For one boson and one fermion field, Eq. (3.1) is replaced by
(3.20)
GROUP AND INFRARED BEHAVIOR OF. . .
RENORMAI. IZATION
19
where g„and g„are rescaled fermion sources and R~ „ is the term explicitly bilinear in fermion sources. The fermion terms are symbolic for boih the P and P' type fermions. For example, )DEp represents ( gEDE EgE+ g p DE E gE ). Fermion propagators are split into low and high parts D~ and D~k, respectively, where
f
f
Equation (3.23) fixes the rescaling factors a of Eq. (2.9). The a are given by &e
b4-k»-(8/2) kf -nf
f
jA p P/m(p P/m+i~)' i-(A
E. Relation to the complete theory
Given S„[Eq. (3.1)] we define the approximate generating functional Z„by a steepest-descents Taking" approximation.
(3.21)
A
X)
(p P/m +i X)(p P/m +i IA)
(3.26)
=-D j
a nd using
(3.2'7)
Eq. (3.12) to get
I
The proof of equivalence of the two-step to onestep calculation goes through as before. The generalization of Eqs. (3.5)-(3.8) to include fermions is obvious. The requirement that D„' plus the relevant quadratic interaction terms be transformed into D~ by the momentum rescaling P =b P' implies that Eq. (3.9) must be augmented by
g(p)=b' 'Z, '
'p
(p'),
vertex variables yk&2k' eh&'
' '
n
which are the coefficients of terms with —m
zg
powers of momentum in the Taylor series expansion of effective vertices with k~ boson terminals and k/ fermion terminals (k -=k, +k/). Equations (3.9) and (3.22) imply that all factors of b, Z„and Z, are absorbed if rescaled z„~ are defined by ~kg kf fl, ' ' ' ck —b4-ky- (3 / 2) kf - m 2
&
~
et+
xZ
kk/2Z
k//2 Zkkkk/111
''
&k
(3 23)
if source terms are rescaled by R
b-3/2Z 1/2 D E(p) /)(p) 2 Dl (p)
(3.24)
and if we define
EekR
k'j='o
)
I+p2/iA 21 &&
1+p2/;A
2
(3.28)
ls. (p),
p„=b" p, A„=b "Ao, we have
(3.22)
where Z, is the fermion wave-function renormalization for the RGT under consideration (see Appendix B). [Note that it is fortuitous that the IR scaling of g given by Eq. (3.22) is the same as the UV scaling since Eq. (3.22) is derived from a different form of the propagator which could, in principle, be multiplied by mass constants. ] Equation (2.8) is trivially generalized to include
e-1
"I',2, , -
1„(PJ=&
j. j.
Z.[j.(p) ]=s.(-D(P.) (P.), (P.)) .
(3.29)
j,(p) is the
zeroth-stage source term expressed as a function of the unscaled (physical) momentum It has been UV renormalized by Z, „~ and is equivalent to the renormalized source term of a conventional calculation (renormalized with IR cutoff A, ). The scaling factors in Eq. (3.28) combine with powers of b from the effective vertices (appearing when the latter are reexpressed in terms of p instead of p„) to convert the scaled external propagators D(p„) to unscaled propagators D(p) Z„ includes all momentum integrations over the region p&A„. If g„were not growing, we would expect Z„[j] to be a good approximation to Z[j], the complete connected generating functional, for (p)&p&A„. This is because the relatively large momenta [of O(p)] carried by the terms provide an effective IR cutoff at O(p) so that further integration with a UV cutoff A„& O(p) would give negligible additional contributions. That is, propagators with A„&O(p) are suppressed because of the flow of the relatively large external momenta. However, because of the unknown growth of g„ in QCD, we can make no rigorous statements about the approach to Z[j]. We only note that Z„[j] includes all contributions from internal momenta above A„.
'4.
j
"
j
IV. QED
(p&) —b-1 Z 2
~
-1
( )
ERE' D (p)
DE(p) DE(p)
D'(P)
(3.25)
To illustrate the application of the RG method we consider the on-mass-shell vertex in QED (with external fermion lines of momenta P and P'). As shown in Appendix A, all graphs are rendered at most of O((g„21n b)") by the use of singly subtracted fermion and photon propagators. Only
DA VID SHALLOWAY
1772
of the photon propagators
the denominators split. That is,
1
1 p2
p2
ZA2
f
v
1
1
p2
@2+
+ ~+
2
are
y2
2
~2
~p
4
D -Dl+Dh
where" Z„, =-g„+pJ„/p'. Graphs
"
with fermion loops need not be considered. As discussed in Sec. II, we can reliably approximate the RGT by calculating only the leading contributions to the equations for each relevant and marginal vertex variable. The relevant and marginal variables are those-for which the IR rescaling factors a' given by Eq. (3.26) are )1 or =1, respectively. We have a =b . D is equal to the UV degree of divergence in this case (and in QCD) because of the fortuitous identity of the IR and UV scaling behaviors. The only relevant variable is the electron mass shift. However, this is IR finite so all shifts after the initializing renormalization are of O(e„A„) and are negligible compared to m. The only marginal variables are the electron wave-function renormalization constant Z, „, the photon- electron- electron vertex renormalization constant Zy „, and the external source-electron-
electron on-mass-shell vertex T„(q'), q' -=(P- P')'. These are displayed in Fig. 4. T„ is distinguished from F„because q' is large. This distinction can be formalized by regarding P+P and P'+p fermion lines as distinct species [P'= P" =m' and P» O(A, )]. T„ is the P fermion =P' fermion-photon vertex while 1"„is the P(P') fermion-P(P') fermion-photon vertex. We take Z„, F„, T„, etc. as input vertex functions and define E„, 1 „, T„, etc. as the unP+p
Z„(P)= (Z, '„-I) y
n
19
scaled output functions including the nth stage corrections. (only the marginal terms in these vertices are calculated but it is convenient to use this familiar nomenclature. ) Following Eq. (3.6) and the associated discussion, these vertices are calculated by amputating the bare external propagators from the respective connected graphs of a Feynman diagram calculation using D". The p' term in Z„ is actually zero (i.e. Z, „=1) since , we renormalize fermion propagators at each stage. However, Z„ is unrenormalized:
„—1)P+O(P P /A
Z„=(Z
These variables are calculated to O(e„' ln 5) or O(e„' ln b) [where e„=l „(0,0) =Z, from the diagrams shown in Fig. 5. (For convenience, we include the coupling constant in the definition of the vertex. ) No irrelevant variables contribute to the marginal variable equations in leading order; we need only calculate the "zero moment,um" values of Z„, f'„, and T„[i.e. , 2, „", Z, and T„((P- P')')] since only these contribute to the ln b terms in Fig. 5. Thus, the mathemstical calculation is simple since the marginal terms which enter the equations correspond to rescaled "bare"' vertices and propagators. External leg corrections [e.g. , Fig. 5(b)] which, in principle, are included in the effective vertices, do not contribute to the O(ln b) corrections (see Appendix B), and graphs such as that of Fig. 3(b) do not contribute to the ln b terms because of the differing propagator cutoffs. The logarithmically divergent terms of a one- step calculation become the ln b terms of the RGT and after a straightforward calculation we get
„']
„',
(0)
~n= Zn
+"..
In ~+&
~l.„
P+p n
).
+
~
+
~
~ ~ ~
~ ~ ~
(p p')= z, '„Xp+
tn
=
Tn
~
0~ ~
(b) ~
P+ p
P'+p' j 'fn
~
Tn (P+ p,
Pap
FIG. 4. @ED marginal variables. are in the IR region.
) * Tn
(f P-P
j ) Yp, +
I' =p' =m2; p, p'
FIG. 5. (a) Lowest-order corrections to @ED marginal variables. Only leading corrections are shown. (b) Example of a nonleading correction to I'„.
RENORMALIZATION
19
GROUP AND INFRARED
(4.2)
Z,
",
„=e„1+6e„' ln
T.(q') = T.(q')
O'I
~+O(e„', e„' ln' b),
(t e ' » 4 2
(P(q'&
') -4)
(4.3) )
+O(e„', e„' ln' b),
(4.4)
where
We complete the ROT by rescaling P by a factor of b as in Eq. (3.9) and renormalizing the electron field according to Eqs. (3.22)-(3.25). The renormalized equations are simplified by the Ward identities which are maintained in the leading approximation. We do not expect the Ward identities to hold in higher orders of approximation since the manifest gauge invariance is broken, so the error terms in Eqs. (4.2) and (4.3) are independent and we have
e„Z, „=Z, „+O(e„',e„' ln' b). Thus, we get for the renormalized
(4.5) variables
(4.6)
2g n+1
e„„=e„+O(e„', e„' ln' b),
T, (q )= (1+ ",»b e„'
(4.7)
P(q /
)) T„('q')'
+O(e„', e„'ln' b).
(4.8)
Equations (4.7) and (4.8) are converted into a differential equation in f = ln(A, /A„):
BT
f)
e
-)4, P(q't)e )+t)
T(q, t)+eq,
(4.9)
where c and q are error terms of O(e'/ln b, e' lnb) at A, . and e'=—e,', the charge renormalized Equation (4.9) is the usual QED IR equation for the renormalized amplitude with bounded error
terms. We can explore the effects of the g and g error terms by solving Eq. (4.9) for constant z and )I. 'The solution
is tee
T(q', t)=T(q*, q)exp
.p(q*i
*)+4 tI
(exp(I(e'/16m')F(q'/m') + z]t}-1) (e'/16m')F(q'/m') + z
(4.10) The significance'of
Eq. (4.10) is that it shows
BEHAVIOR OF. . .
(without detailed calculation)
that the nonleading
terms can have no effect on the qualitative IR behavior of T except in two specific ways: (1) The g term is obviously small relative to the leading term for E&0. However, for E&0, it can mask the exponential suppression at f -~ in O(e' lnb) or O(e(ln b) '). But in this case, it cannot grow with f (as is a Priori possible given only the leading-log result) and remains small. (2) The c term becomes relatively important in the vicinity of q'/m'=0, the only zero of F(q2/m2). This is the only significant effect of the lack of manifest gauge invariance. The zero of F at q'/m2 =0 is a, direct result of the fact that the Ward identity Z, „=e„Z, „ is maintained by the leading approximation so that there is no IR renormalization of the electric charge by the leading terms. However, the Ward identity is not well defined as the IR cutoff goes to zero for the soft-photon-free pnmass-shell fermions because of the IR divergences of the renormalization constants. Thus, a priori (ignoring the known cancellations of the nonleading terms in QED), the nonleading terms of the softphoton-free vertex could give an apparent IR charge renormalization resulting in a large relative error at q'/m'=0. (This presumably would vertex be canceled when soft-photon-containing amplitudes were included so that there would be no physical IR charge renormalization. ) Note that we have no real numerical control of the errors. That is, when exp[(P/16))')F (q'/m')f) is large, errors due to g and c will be numerically large. The important point is that no new poles or radically different analytic behavior can be introduced by the nonleading terms. This result essentially depends on the absence of explicitly f-dependent error terms in Eq. . (4.9).
V. QCD
The QCD calculation is similar to the QED calculation but is complicated by the ghost field and additional couplings. Both the gluon and ghost field denominators are split by Eqs. (3.17) to ensure convergence of the self-energy diagrams. We use Landau gauge so that the gauge parameter. is invariant;. Prom Eq. (3.26) we see that the relevant variables with a &1 are )he fermion, gluon, and ghost mass shifts. The fermion mass shift is always of O(g 'A) after the initial mass renormalization and is negligible relative to m. The gluon and ghost mass shifts 6I(,„' and 6v„' are of O(q„'A') at each state (see Appendix B). They act as counterterms for the gluon (II„) and ghost (II„) self-energy corrections and combine to give
DA VID SHALLOWAY
„'= bp„'+II„(0) =O(g„'X'), bv„' = bv„2+ Iio(0) =O(g„»X') . bp,
The &ti„' and
' insertions
&v
'
(5.1)
are of O(g
'/p')
2I(.
(with I(. «p' «A') when combined with the gluon and ghost high propagators and do not contribute to the leading (ln b) terms
The marginal variables are the seven fundamental multiplicative renormalization constants: Z3, Z3 ZQ Z~, Z4, Zy and Z, and the on- mass- shell color-singlet current-quark-quark vertex T((f'). Notation is fixed by Fig. 6 (indexing subscripts are suppressed). (Z„Z»o, and Z, equal one because of the field rescalings. The intermediate renormalization factors before rescaling are denoted by Z„Zg, and Z„respectively. ) These variables are calculated to O(g„' ln b) where The diagrammatic expansion g„=T'„(0, 0) =Z, is the same as that of conventional perturbation theory except that (1) the fermion, gluon, and
„'.
SELF -ENERGY TERMS
~vu
~
(p) bo(("((qq
ghost propagators D~, D, and D~ are replaced by the high propagators D~'», D», and D~'», and (2)
bare vertices are replaced by effective vertices as in Fig. 5. Unlike QED, we need three irrelevant variables in addition to the marginal variables to calculate the )eading corrections to the marginal variables. These are the O(p'/A') terms in the expansions of r "', Z'" ', and Z'"" ', the four-gluon, fourghost, and two-gluon- two-ghost vertices. They are needed for the calculation of II and II~. [Cf. Fig. 3(c).] They are approximated by Eq. (2.11). The O(g„' ln b) marginal variable corrections come from the same diagrams as those calculated in the one-loop approximation to the P function, except that high propagators are used. Since only the "bare" (but renormalized) parts of the propagators and vertices contribute, the actual mathematical calculations are relatively simple. As in the case of QED, the logarithmically divergent terms of a one-step calculation become the ln b terms of the RGT. Denoting the unscaled output variables by tildes we get
qq(4*!6(("pv
Z p
e
vr
~
=
(p )
-
3g
X (p) =$ (Zi-I
'+p'((Z ~Q)+
=1+g
"
", — 2
ln b 13
g„2ln
b
)+5
VERTICES
INTERN+
Op, q)=iZ, 'f'
C„+O(g„', g'ln'b),
0
'
+ 2 per, +
(p-q)„g)
g„
"
C, +O(g„', g„' ln'b), (5.2)
ln 5 17
C
f' p'"(p qr, )= Z„' 4 2perm + f', ry
Pqa
ln 5b
+O(g„', g„' ln'b),
I
I
~Pqp'
I'
b,
g„'
f"'(~Xp~pf -94gpp)
(p, q)
i(Z~ )-
f
Z(' —ZG+ O(g
p&+. .
q(q'i (p, p')= (2. , )
'
X
y+
+
= q(q*)
3
g»
((+
ln2b)
", l
b
('
(q'I(q)-m61)
+O(g„', g„' ln'b), EXTERNAL
T
COLOR SINGLET
-
CURRENT
' T(P+p, P+p')
=
VERTEX
T((P-Pl
)you.
+".
FIG. 6. @CD relevant and marginal variables. Curly lines are gluons, dashed lines are ghosts, solid lines are fermions: P =P' =m;p, p', q, x, and s are in the IR region. is the structure function. ~ is the ——g» +pQ„/p . fermion representation matrix: J» =
f
where C~(C„) =eigenvalue of the quadratic Casimir operator of the gauge group for the fermion (adjoint) representation. We rescale p and renormalize the electron, gluon, and ghost fields according to Eqs. (3.9), (3.12), (3.13), and (3.22)-(3.25). The equations are greatly simplified by the Slavnov-Taylor identities which are maintained in the leading approximation. We get
RENORNIALIZATION
19
GROUP AND INFRARED BEHAVIOR OF. . .
nontrivial equations of Eqs. (5.3) are adequate for our purposes. We convert these into differential equations in
3 0+1
~3
y
1775
n+j.
2 yn+1,
—
sg(t) g~(t) 11O C~+g ( t ) 6, 2
n 3
C„lnb g„,
"1-~ml 2 ~4s n+1 (Z g, tte), )
(Z, )
T,
gtte1
()e
" =)eb
C
O
2 a E(q'/m')) T„(q')
(error terms are suppressed). As in QED, we do not expect the nonleading terms to be manifestly gauge invariant so the error terms in Eqs. (5.3) are independent. To handle the nonleading terms would therefore require fiVe instead of two nontrivial equations. (The equations for Z~, Z~~, and Zo would become nontrivial. ) However, the two T(q', t) = T(q',
d(t)0(t t
tt'(t')
(5.5)
(",
q q'(t) )0 b) . lnb '
Equations (5.4) and (5.5) without error terms give the leading-log result Eq. (1.1). The RG derivation bounds the nonleading terms thereby showing that they can not alter the qualitative structure of the solutions down to the nonperturbative region. To make this clear, consider the solution to Eq. (5.5) with error terms present:
Cv 0)exp, E(q'/m')t(l)+E(t)I
t
t (t)
T(q', t)
+g(t}g, where &, g, and g are error terms of
0
'=g„„,
(q )
, CtE(q'/m')+e)
(5.3)
(5.4)
dt ,
)exPI)0'. E(q''/m')(t(t) =-Z
(t)
f
t
e'll')
t(t ))e(E(t)
(5.6)
dt',
where g(t) is the solution to Eq. (5.4). 6, s, and g will be functions of the vertex variables and have implicit t dependence. We see that the nonleading terms, in the sense described in Sec. I, cannot affect the qualitative IR behavior of T until g'(f) becomes of O(1} but for the same two exceptions found in the QED study; that is, (1) the exponential for F (0 can be masked in suppression at t O(g'(t) ln b) or O(g(f) ln f) ') by the q term, and (2) the E term can become relatively important in the vicinity of q'/m' =0 since the manifest gauge invariance of the leading terms results in E(0) =0. The discussion presented in Sec. IV is directly applicable to this case as well. We reiterate that the results do not provide numerical control but provide control over the general analytic behavior of the amplitude. The RG derivations make it clear that the results cannot be extended beyond the point g (t) ~ 1 since the e(equations they are derived from change beyond this point. This is independent of any perturbative rear rangements. Because these results are limited to the perturbative region, they do not directly pertain to the
-~
-E(t''))I«'-
question of quark confinement. They do show that the perturbative IR behavior is governed only by the behavior of g(t) and that little additional information is to be gained by further perturbative study. ACKNOWLEDGMENTS
I thank Kenneth Wilson for suggesting this problem, for many helpful conversations, and much advice, and thank him and Toichiro Kinoshita for a critical reading of the manuscript. I also thank Akira Ukawa for many helpful conversations. APPENDIX A: BOUNDS ON HIGH-ORDER DIAGRAMS
The neglect of higher-order diagrams in an RG transformation requires that all coefficients from momentum-space integrals be bounded. We now show that the bounded IR (&) and UV (A) cutoffs incorporated in D~ restrict the integrals to powers of ln (A/&) =ln b and also argue (without rigorous proof) that there will be only one factor of ln 5 per loop integration. Actually, detailed knowledge of the bounds on high-order diagrams is unimport-
l9
DA VID SHALLOWAY
&776
ant as long as we can show that they are given in terms of b (which is controlled) and not f (which grows without bound). We proceed inductively and assume that all effective vertices are finite for boson legs at zero momentum and fermion legs on-mass-shell (we call this the zero-momentum vertex). We need not include diagrams with massive fermion loops. Now consider the nth RG transformation with effec& and UV cutoff 4-i A. tive masses The complex masses prevent any zeros from appearing in the denominators of the integrands so all IR divergences are regulated by ~. Because of the form of the IR fermion propagator [Eq. (3.19)], it is p, the reduced fermion momentum, rather than P+p (P'+p) that'is relevant for the discussion. By the usual power-counting arguments all unscaled effective vertices will have the form
"
~i
(A1)
where D =4 —k&- 3/2 k& for an effective vertex with (We assume k~ boson and kz fermion terminals. that the integrals are UV finite; this will be proved below. } The RGT momentum and field rescaling [see Eqs. (3.9), (3.22), and (3.23)] will convert the factors of X~ to factors of A~ and the p'/X~ dependence to p"/A' dependence. Since the maximum mass scale for the following RGT is also A, the P'/&' dependence and XD terms for D ~ 0 can only give terms of O(1). Thus, apart from the &~, D&0, terms (these self-energy terms receive special discussion below), only the A'/&' dependence of y is important in bounding the perturbative RGT's. We now show that the A'/X' dependence of y is bounded by O((ln A'/&')"}. It is immediately obvious from the'IR structure of the effective graphs that the behavior in ~ can be no worse than ~" where x is finite and depends on the number of internal propagators. This somewhat restricts the A/X dependence (assuming that the integrals are UV finite), but it is difficult to tighten this bound by studying the IR behavior. Instead we consider the UV behavior; limiting the maximum A dependence will automatically fix the maximum A/X dependence. We assume that an analytic continuation has been performed so that the real parts of all the boson propagator denominators are negative. This continuation is described below. Were it not for the linear form of the fermion propagator [Eq. (3.19)], the UV behavior" of the diagrams could be bounded by power counting and Weinberg's theorem. In the absence of fermions, these familiar arguments show that all UV divergences are controlled by the double propagator subtraction Eq. (3.17). (The
double subtraction is required only for the gluon and ghost self-energy graphs; a single subtraction suffices for QED. ) In this case all graphs give at most powers of ln A except for the gluon and ghost self-energy graphs which give powers of A'. In Appendix B we show that the O(A') mass-shift terms cancel stage by stage (due to gauge invariance) so that the output mass shifts are actually of O(V). These act as counterterms in the next RG transformation where they are canceled by new self-energy diagrams. Thus they do not con-
tribute powers of b. Subgraphs containing fermion propagators require special attention. The fermion propagators only depend on a single timelike component of the reduced fermion momenta (parallel either to P or P'). The other (spatial) components of the reduced fermion momenta can be arbitrarily large without affecting the propagator so we cannot include fermions in the power-counting argument. But all loops containing a fermion propagator and at least two boson propagators can have no worse than logarithmic behavior even when. we count only boson denominator factors, since all reduced fermion momenta are completely linearly dependent on the boson and external momenta. (This argument presupposes a rotation to Euclide space; see below. ) Thus we just need to specif'cally consider subgraphs with only one boson propagator. (All subgraphs have at least one boson propagator since fermion loops can be omitted. ) These can be enumerated (Fig. 7) and shown to have only»A behavior by explicit calculation: The mass-shift part of the self-energy graph [Fig. 7(a)] is IR finite and therefore is essentially removed by the initial stage renormalization. The remaining portion of the
a)
b)
c)
]i
~
FIG. 7. Subgraphs with only one internal boson propagator. The ultraviolet behavior of these graphs requires special attention.
RENORMALIZATION
19
GROUP AND INF RARED BEHAVIOR OF. . . I
self-energy graphs and the vertex graph [Fig. 7(b)] give only factors of ln A by explicit calculation. Graphs with additional boson legs [Fig. 7(c)] can only be more UV convergent than the vertex graph. The previous analysis assumes that the "boson propagators"' are in Euclidean space. However, because of the linear form of the fermion propagators, the integration contours will cross fermion poles if we attempt a Wick rotation. It is impossible to shift all the fermion poles out of the way of the rotation. To avoid these complications we replace the Wick rotation with an analytic continuation in P which we now describe. This gives real negative p' and allows the use of the power-counting arguments. We can ignore external reduced momenta in the discussion and set them equal to zero since they are of O(X) and do not affect the UV behavior. We take P=m(+1, 0, 0, 0)andP'=ym(1, 0, 0, -P) where 0&P&1 and y=(1 —P') 'I'. First we transform to a new momentum representation and introduce coordinates" Pg
m yP
(A2) I
—Pi
myP {AS)
The high propagators
are (omitting i subscripts)
',
D, (P) =(~+in)-'- (~+&A}-~, D„" ~ (P) = (~'+f &)-'- (w'+f A) ', D"(P) = [-(n'+~" +P ',)+2 ye~' —(X-A)
(A4)
=
',
(- '[(1 —y)(w + m')'+ —,
"
',
'"',
where 8= arg[y(pe'+)]. m/4& e(pe'~)& 0 so, referring toEqs. (A4) we see thatnopropagator poles are crossed. [Note that e(P) = e(iP) =0 so the integration contours finally return to the real axis with no overall rotation. ) The power-counting arguments previously presented can now be used to bound the UV behavior for 0&y&1. .Positive powers of A cannot be generated by the continuation back to y&1 since this would require terms (such as A'" ') with essential singularities in y. But we see from the analytic structure of the integrals that the only singularities as a function of (P P')', -and hence, as a function of y, will be poles and branch cuts. Thus the logarithmic bounds continue to hold. The above arguments show that we get at most powers of ln A (implying powers of ln b} from loop integrations. In fact, we expect only one power of ln A per loop since the four-momentum components are not independent but are tied together by the quadratic propagator terms. Multiple logarithms per loop integration typically occur only in exceptional cases when the integrals factorize, e.g. , due to mass singularities associated with the kinematics of massless particles. Since all effective masses are nonzero we do not expect such factorization, However, we have not proved this (for our argument, inessential) fact.
8:
QUADRATIC INTERACTION TERMS
Equations (S.4) or (3.17) are clearly the simplest types of splits of the zeroth-order (in g„) propagator, but the situation is more complicated when S„contains a higher-order quadratic interaction term, ' Z„(P)—P'(P), as part of V'„[with respect to Eq. (3.8)]. In particular, we must specify whether this term is to be included as part of the subsequent propagator D (and split in the propagator splitting procedure)" or treated as a counterterm, i.e. , included in . Different specifications will result in different HOT's. The specific definition of the ROT is fairly arbitrary: We only insist that D contain no low-frequency singularities and that D' drop off sufficiently rapidly above
,f
(1+ y)(w —v')'] (A5)
we see that all real terms are negative for jy &1. We cannot continue from y&1 to y &1 along the real axis because of the singularity from the Jacobean at y = 1 (P =0) so we continue along the path P-e'~P, 0~ @ & m/2. This implies that y follows a curved path in the upper half plane around y=1. We make a simultaneous rotation of the integration contours to avoid crossing the poles in the boson propagators. As y is continued along the path ~
8"),
P, -P, e
APPENDIX
—p, 2+i~2P —fX-A]
~(P) -~(P
e-ge/2
i&+']-'
and the integral gets a Jacobean factor (4m'yP) where N is the number of loops. If we rewrite D~ in the form
D'(p)
we rotate the contours to
V,
"
momentum ~. Our prescription is to include the marginal p' term of Z„ in the propagator (and then to absorb it by a wave-function renormaiization) and to treat
the remainder of Z„as a counterterm. the relevant (&p„') and marginal (Z,
„'-To1) isolate terms
DA VID
1778
in
terms) and letting pa- g, P'-P we get the renormalized rescaled output action
Z, we write z„(P') -=-6) '+ (g, '-1)P'+ z„.(P'),
. .
z,
„=0.
, z,
„(0)=0,
SHALL0%AY
We define a modified low propagator D„'
by"
'+(t, , . ' —&)(t'+, ~,
(B2)
reexpress Eq. (3.8) in terms of D„' rather than D', and supplement the free field rescaling [Eqs. (2.6)] by a wave-function renormalization which restores to its original form:
the propagator
P'=&P
~
-'+&
=/-///
)t/'. +//
„M/l.
.
j terms are rescaled according to Eqs. (3.12) (3.13) of the main text. ] &p, „' requires special attention because it is a relevant variable. Consider the integration of
P' at stage n We get additional self-energy contributions i Z„'(P') so the full high-frequency propagator is (we omit indexing subscripts)
]+g DA i(A'-
X')
' (P2+ $2)(P2 P A2)
Defining
(B4a)
z, „,(p") =z, „b'z, „(p'), U, (4, ) = U.(4) (where U, represents
,
j terms,
+
[The
' D'(P') =D(P")
(P") =b '2
D'
'+V,
and
„'"j(P)
y, (P')=f 'Z,
'
*f(
(t)
4
(t/!) '=(//')
D
where
+=0
I
' S„„=-,
(Bl)
(B4b)
(B4c) interaction
the nonquadratic
3
(P
(B5)
where"
z, (p') =z(p')+z'(p') -=-6) '+z.(p')+ z'(p'),
'-
V = )I. z(P
t'
Z(-t' X'),
i(A' —)I.') )
(
2+'
t(P
A2)
z, (p ) =z(p')-z(-& ~')
INPUT
+&g)„=
i(4pL
—(P'+i
+I/2=
I/2
n-I
m~0,
1+„P.z
(Z, -I) g+
IVI" -UV
"1
~(,) ;-tI(+&c
i+
—
NEW DIAGRAMS WITH
=
iD
ioh
——
+I/2 ;Oh st
I
+
-i
[(p+I) ) -(p+ih
-
+
)
]
iO
it,'=I// Ut
—
'
1+Z,D~
D
4
OUT PUT
(unrenormalized)
iZ„=i(++ X~+Qq
+ /2 I
i+ &I):~Z, +Z'go" n-I '
-iZ
=
-I)p'+
(Z. 3lll OsOt ills
~
~
n
-
~
~
Un
+
Un
=
UV
+ ~p
X
-~+
FIG. 8. Schematic reconstruction of the 0 (g„) quadratic interaction terms. Lines with X or & represent propagators i(p +X ) or i(p +i A) 1 , respectively. Factors of i associated with the u„ are sup-
pressed.
-&
~']
z.(P') t,
(P'+i &')
The new unrenormalized quadratic interaction term appearing as the ROT output is Z
'
2' &'), dp
'
(B6)
By dimensional argument, the only place where A' terms can appear in the unscaled output action S is in &p.„', the mass-shift part of Z. The important point is that & p, „' will be of O(g„'A') and will cancel the O(g„'A') term in the mass-shift part of Z„' to give 5)t„' of O(g„')I.'). (The cancellation occurs within Z, .) Thus the rescaling of &p„' by b' [Eq. (B4a)] gives an output 6)t„„'which again is of O(g„'A'). This self-consistent cancellation i, s dictated by the manifest gauge invariance of the leading terms which implies that the physical mass shift tends to zero with more ROT's. We show how the cancellations appear diagrammatically to O(g ') in Fig. 8. The inputs at stage n are twoand four-point functions which have mass scale A. Irrelevant skeleton graph corrections to the
GROUP AND INFRARED BEHAVIOR OF
RENORMALIZATION
19
four-point vertex play an important term
role.
~
l
~
779
" The iD
comes from the previous BGT renormalizations [to O(g„')] and cancels the p' term of the Selfenergy diagrams. The initial renormalization term (Z, ~,» —1) is included in the summation. The new diagrams calculated with D~ combine to cancel all A dependence giving an unrenormalized output of the same form but with mass scale ~. The ~'s will be converted to A's and the self-enerterm will become gy renormalization Z3
~
I
U I
+XjD
ig
I+XqQ
'-1 p'
are renormalized. Quadratic insertions also appear in the external legs of the effective vertices since U™, the vertex functional after the P integration, is constructed by amputating D" rather than D~'. That is,
when the output variables
I'J is the completely amputated vertex functional. These insertions plus the quadratic interaction insertions in the propagators of the next stage of calculation combine to reconstruct the single- step calculation result. We show how this occurs for the simple case of an unscaled twostep calculation in Fig. 9. The low propagator s will include Z corrections and give terms
iph I
=
— 0 it
+X ) D"
0" +O
I+Et(D
)—
+(0
FIG. 9. Diagrammatic combination of quadratic interaction terms in a two-step calculation. Dashed circles denote effective vertices; cross-hatched circles denote the completely amputated effective vertices a; lines with solid blobs in the middle denote complete propagators; lines with solid blobs at the edge of an effective vertex denote partially amputated propagators D" /D" . The corrections for the uncontracted legs are
where
not shown.
.
corrected propagators (Dh+Dt)
1+Z, (D +D') (We omit additional
low-frequency
self-energy
'
terms. ) The terms formed by joining effective vertices z7(p, ' ' ' p') ([1+Z,(P, )D"(p, )] ' ' [1+Z,(p~)D" (p')] '] with D" (upper part of Fig. 9) combine with larger effective vertices with in-
The external leg insertions do not contribute to the leading contributions because they always connect high- and low-frequency propagators in series. The resulting momentum nonoverlap gives
a factor 1/b.
ternal D~' propagators joining the same totally amputated vertices z7(p, p~) (lower part of Fig. 9) to give an overall. result with the properly
The previous discussion can be trivially generalized to the cases of fermions and spin- 1 bosons required for the QED and QCD studies.
+Present address: Sidney Farber Cancer Institute, 44 Binney St. , Boston, Mass. 02115. 'Vil. Marciano and H. Pagels, Phys. Rep. 36C, 139 (1978). 2T. Kinoshita and A. Ukawa, Phys. Rev. D 15, 1596 (1 977); 1 6, 332 (1977) . 3(a) K. Wilson and J. Kogut, Phys. Rep. 12C, 75 (1974). (b) K. Wilson, Rev. Mod. Phys. 47, 773 (1975) . K. Wilson, &ew Pathways in High Enn gy Physics,
edited by A. Perlmutter (Plenum, New York, 1976), Vol. II, and personal communication. The high-low
propagator spl. itting technique has been used by J. Lowenstein and P. Mitter, Ann. Phys. (N. Y.) 105, 138 (1977), to define a renormalized field theory. P. Mitter and G. Valent, Phys. Lett. 70B, 65 (1 977), have developed a manifestly (non-Abel ian) gauge-invariant form of the propagator split by introducing an additional ghost field.
DAVID SHALL0%AY
l 780
emphasize that the generalized RG approach is more powerful than conventional "Callan-Symanzik" RG techniques. [See, e.g. , E. Poggio, H. Quinn, and J. Zuber, Phys. Rev. D 15, 1630 (1977), for a discussion of the QCD IR problem by that method. ] Gon= comitantly, it is less well known. We suggest familiarity with the contents of Ref. 3(a), Secs. 3, 4, and 5 and Ref. 3(b), Secs. 4 and 5. [In the terminology used there, we are studying the second-order perturbations of marginal operators about a Gaussian (free field) fixed point. ] In practice, S«, is only useful for determining Green's functions for P» A«yf where &«+f is the neer effective
5We
&
momentum scale, A„+& &A«. use a diagrammatic procedure in which the cutoff is introduced in the propagator rather than in the domain of functional integration. In this case the rescaling serves to restore the propagator in S«-, to a form with the same cutoff as the propagator in S„. We denote the invariant scaled propagator cutoffs by A and X in accord with field-theoretical notation although we could take 6= 1. The physical cutoff h„ appears explicitly only in the source terms of 'the rescaled action. P and P will represent either unscaled or scaled variables depending on the context. The following discussion is simplified by ignoring selfenergy corrections to D' and sealing of j(p) source terms. In particular, Eq. (2. 6) is modified in O(g ) by self-energy corrections. These matters will be .discussed in Sec. III. PA sharper cutoff than that of Eq. (2.4) may be necessary to ensure finite integrals. The required generalization is discussed in Sec. III. Bounds on high-or-. der diagrams are derived im Appendix A.
~We
&
' See Bef. 3(a), Sec. 12.
'As we only study the IR segment of the trajectory we will actually define Sp to incorporate integration down to an original IB cutoff &p. See Sec IIIB. ~ The IB cutoff permits the expansion aboutP;= 0. The in terms of and the method of resdefinition of to the last two terms is presented in tricting all ~
j„
Sec. III.
j
j„'s
'3The variable is also marginal if a~=1. The analysis of Eq. (2.12) also applies in this case. If Nr ({yz ) j contains an O(g& l term (as it will in our study) the expansion of g„ in terms of gp is of the form
g„=gp+cgp n+d gp
n
+' '
'.
The factors of n = fin(X p/&)](lnb) correspond to the logarithmic divergences in Feynman diagrams as t While all lnt terms appear essentially equivalent in a Feynman diagram calculation, they are naturally divided into two classes in an RG calculation. All lnt terms are generated by marginal variables, but those coming from variables that only appear linearly in the RGT equations (such as the photon-quark-quark vertex in QCD) cause no difficulty while those coming from variables that appear nonlinearly (such as the effective coupling constant in QCD) eventually limit the domain of the applicability of the perturbative renormalization group by their gr'owth. '~In a theory without such symmetry we could choose the original mass shift counterterm 6 pp2 so that lim„~ corresponding to fixing 4 pp at its critical value (Ref. 3). In practice, 6 p«would be calculated by inverting 6 p«((5 p«&, ) and calculating 5 p„g p«+f ) [see Ref. 3(b) Sec. V]. Fortunately, we do not require an accurate determination
p„0,
.
y
of 6 p„ for our calculations. ~cour calculation is nor more or less well defined than a conventional calculation in regard to the usual ambiguities of gauge invariance coming from the lack of a well-defined on-mass-shell coupling constant in QCD. We ignore this complication. ' Heuristically, we can see how the leading RGT terms dominate to all orders after many iterations by considerating a simple example where the only RGT equatiog. is T«+&= (1+cg2+dg +eg ' ) T«. The coefficients will be bounded. After N iterations we c, d, e, have
7'„= (1+cg'+ dg4+
~ ~ ~
) "7.'p
' —1) c+Nd 1+N cg~+ N(N
2
g4+
The c term in the BGT equation dominates all higherorder BGT equation terms by powers of N in every order in the final result. [Higher orders in g are significant in the final result because of the large factors of N(corresponding to factors of t). ] There will be a relative error of the form e/(fg« lnb) = c (lnb) '+dg„ lnb where e is the error term and c, and d are coefficients «O(1). We can minimize this
f,
by choosing lnb = (c/dg„)' =O(1/g„). We do not consider possible instanton effects. Cf. the discussion of Fig. 5.2. in Ref. 3(a). If either of the propagators is of negative norm, e.g. , D"= —(p —nz an analytic continuation in the associated p is needed to obtain a well-defined integral. We ignore this inessential formality. ~2K. Symanzik, DESY Report, No. T-73/3, 1973 (unpublished); T. Appelquist and J. Carazzone, Phys. Rev. D 11 2856 (1975) ~Equation (3. 27) is only the O(1) approximation to &S/&Ql z = 0. However, the tree diagram corrections neglected by using Eq. (3.27) are small for sources in the momentum region where we expect Z„ to be accurate,
+is)",
J.
e' j
i (p)ap If we define p= —D„'j„where D„'" ~(p«)=D" &(p„) ~ Z«(p«) (Z«= quadratic interaction part of V) then D„', the j„rescaling, and the effective vertex external leg quadratic insertions (discussed in Appendix B) combine to give effective propagators DI; that correctly include the self-energy corrections down to A„. Cf. Ref. 3(a), Sec. 11. This is only a formal result for the exact Z„. Z„, as calculated in practice, will be a poor approximation for most processes with p » ~«since the approximation procedure at every stage assumes low-momentum (or reduced fermion momentum) sources except for particular sources (such as the color-singlet current in our stady) which are explicitly recognized as having large momenta. Z„ is accurate for the other sources in the region
pRA„.
6The choice of gauge is arbitrary
in QED. All gauge specifications are invariant under an RGT since there are no IR vacuum polarization corrections. 2~Note that by "UV behavior" we mean the behavior in the region p~-O(A2) for which the approximate form of the fermion propagator, Eq. (3.19), is still valid. The singularity of the transformation at P= 0 corresponds to the qualitatively different behavior (linear IR divergences) found in perturbation theory at this point. We do not study this case. 2PWe restrict the use of "self-energy term" (as opposed to quadratic interaction term ) at stage n to the quadratic terms generated by integrating D" at stage n.
19
RENORMALIZATION
C
ROUP AND IN F RARED BEHA VIOR OF. . .
3
If the quadratic interaction terms are to be included in the subsequent propagator, the propagator split must be modified in higher O(g„) so that the total rescaled propagator D is restored to its simple original form. This adjustment cpn be iteratively performed in perturbation theory. We do not pursue this approach. We use a singly subtracted propagator for simplicity. The discussion easily generalizes to the doubly subtracted case. 3~In a conventional calculation we may equivalently treat the additional quadratic term as part of the propagator [i.e. , D = (D~) + Z] or as a counterterm t.i.e. , D = (D )" '], the only effect being a regrouping of pertur'bation terms. However, these rearrangements do not commute with propagator splitting in a two-step calculagion: Different RGT's of 8 are defined (for the same choice of &, the splitting scale) depending on the definition of D" Since the final is invariant, we obtain Ward identities for the two-step calculation. For example, the Ward identity functionals for propagator resplitting and counterterm adjustment are
.
~(j)
=0~
6Z Dh
1781
[dy'] ldll']
DT
Xe's
r
I
2
Dl J (See also P. Mitter and G. Valent, Ref. 4.) We use a singly subtracted propagator for simplicity. The discussion easily generalizes to the doubly sub—1) P'/A tracted case; The irrelevant term {Z3 is included to avoid an explicit shift in the position of the upper pole in D. The exact upper mass depends on. Z, „(p ) but is not needed for our calculations. 3 To OP, /A= 1/5) we can replace Z(i& ) and (d/ dP )Z ) h2 by &t l;h& and (&/dP ) Z&. l;h2 resPeetlvely. The latter'replacement depends on the fact that Z& g)h
„~
&
6m2+Z~+Z'=OP
)+O(p2)+'
+When the four-point vertex is the external momenta p, only relevant. The skeleton graph 8 is irrelevant since it begins
expanded in powers of the constant term is correction to u„ in Fig. in O(p /X ).
