Exploring arbitrarily high orders of optimized perturbation theory in QCD with nf -> 16.5
EExploring arbitrarily high ordersof optimized perturbation theoryin QCD with n f → P. M. Stevenson
T.W. Bonner Laboratory, Department of Physics and Astronomy,Rice University, Houston, TX 77251, USA
Abstract:
Perturbative QCD with n f flavours of massless quarks becomes simple in the hy-pothetical limit n f → , where the leading β -function coefficient vanishes. TheBanks-Zaks (BZ) expansion in a ≡ (16 − n f ) is straightforward to obtain from per-turbative results in MS or any renormalization scheme (RS) whose n f dependence is‘regular.’ However, ‘irregular’ RS’s are perfectly permissible and should ultimatelylead to the same BZ results. We show here that the ‘optimal’ RS determined bythe Principle of Minimal Sensitivity does yield the same BZ-expansion results whenall orders of perturbation theory are taken into account. The BZ limit provides anarena for exploring optimized perturbation theory at arbitrarily high orders. Theseexplorations are facilitated by a ‘master equation’ expressing the optimization con-ditions in the fixed-point limit. We find an intriguing strong/weak coupling duality a → a ∗ /a about the fixed point a ∗ . a r X i v : . [ h e p - ph ] J un Introduction
The initial impulse for these investigations was a concern with the compatibility of the Banks-Zaks (BZ) expansion [1]-[4] with renormalization-scheme (RS) invariance [5]. In dimensionalregularization the β function naturally has a term − (cid:15)a which strongly affects any zero near theorigin. Can one safely take (cid:15) → then take n f → , or do these limits somehowclash? Our results here basically resolve those concerns; the BZ expansion appears to be fullycompatible with RS invariance in the sense that “optimized perturbation theory” (OPT) [6],which enforces local RS invariance in each order, ultimately yields the same BZ results.The BZ expansion is normally discussed only within a restricted class of ‘regular’ schemes.However, infinitely many schemes – and in some sense most schemes – are not ‘regular.’ Inparticular, the “optimal” scheme is not. In ‘regular’ schemes one needs only k terms of theperturbation series to obtain k terms of the BZ expansion, but in other schemes the informationneeded is distributed among higher-order terms [7]. In general all orders are required. Turningthat observation around, the BZ expansion can be viewed as a “playground” in which onecan analytically investigate arbitrarily high orders of OPT in QCD. Admittedly, this adoptsthe “drunk-under-the-lamppost” principle of looking, not where we really want to, but wherethere is enough light to make a search. The deep and difficult issues that we would like tostudy – “renormalons” and factorially growing coefficients – are simply absent in the BZ limit.Nevertheless, we believe our search provides some interesting insights and employs some methodsthat may have wider applicability.Infrared fixed points and divergent perturbation series were no part of the motivation forOPT [6], but OPT has important consequences for both these topics. Fixed points in OPT arediscussed in Refs. [8]-[14]. Such infrared behaviour was found for R e + e − at third order for all n f [9, 10], though error estimates at low n f are large. The role of OPT in taming high-order perturbation theory was investigated in Ref. [15]. Atoy example, involving an alternating factorial series, showed that even when the perturbationseries is badly divergent in any fixed RS, the sequence of optimized approximants can converge.This “induced convergence” mechanism (related to the idea of “order-dependent mappings”[16]) has been shown to operate [17] in the anharmonic oscillator and φ field theories in thevariational perturbation theory of Refs. [18]-[20]. In QCD “induced convergence” of OPT hasbeen investigated in the large- b approximation [21]. It has also been shown [22] that adjustingthe renormalization scale with increasing order — which happens naturally in OPT [15] — can Also, other physical quantities behave rather differently [11]. The idea [4, 7] that the BZ expansion can beextrapolated, crudely, to low n f no longer seems tenable [14]. The “freezing” behaviour at small n f , confirmedat fourth order [12]-[14], seems instead to stem from somewhat different physics. b approximation (the BZ limit), where the issues are rather different. In particular,the role of optimizing other aspects of the RS, besides the renormalization scale, come to thefore.The plan of the paper is as follows. Following some preliminaries in Sect. 2, the BZ expansion,as obtained from ‘regular’ schemes, is summarized in Sect. 3, and we note that it is suffices toconsider two infrared quantities, R ∗ and γ ∗ . Sect. 4 briefly reviews OPT. Sect. 5 presentsOPT results in the BZ limit, up to 19 th order. Sect. 6 describes analytic methods for studyingOPT at arbitrarily high orders. It also introduces a crude approximation, “NLS,” and a betterapproximation, “PWMR.” These approximations, applied to the BZ limit, are explored in detailin Sects. 7 and 8. From these results we see that OPT, taken to all orders, does reproduce theexpected BZ-limit results, and we gain some insight into how OPT’s subtle features conspire toproduce accurate results and a rather well-behaved series for R ∗ . In Sect. 9 we show that all-orders OPT reproduces higher terms in the BZ expansion correctly, and in Sect. 10 we point outan intriguing a → a ∗ /a duality. Our conclusions are summarized in Sect. 11. (Two appendicesdiscuss (a) some subtleties associated with the critical exponent γ ∗ [23]-[25] and (b) the pinchmechanism [14], which is a way that a finite infrared limit can occur in OPT without a fixedpoint. This mechanism is probably not directly relevant in the BZ limit, though it nearly is.) Consider a suitably normalized, perturbatively calculable, physical quantity R with a perturba-tion series R = a (1 + r a + r a + r a + . . . ) , (2.1)where a ≡ α s /π is the couplant of some particular renormalization scheme (RS). (More generally R can start a P (1 + . . . ) but in this paper we will consider only P = 1.) The physical quantity R could be a function of several experimentally defined parameters. One may always single outone parameter, “ Q ,” with dimensions of energy and let all other parameters be dimensionless.(The precise definition of Q in any specific case may be left to the reader; it is needed onlyto explain which quantities are, or are not, Q dependent.) For dimensional reasons the r i candepend on Q and the renormalization scale µ only through the ratio µ/Q .The physical quantity R is independent of RS [5], but both the couplant a and the coefficients r i depend on the arbitrary choice of RS. In particular, a depends on the arbitrary renormalizationscale µ : µ dadµ = β ( a ) = − ba B ( a ) , (2.2)3here B ( a ) = 1 + ca + c a + c a + . . . . (2.3)The first two coefficients of the β function are RS invariant and are given by b = (33 − n f )6 , c = 153 − n f − n f ) . (2.4)The higher β -function coefficients c , c , . . . are RS dependent: they, together with µ/ ˜Λ, can beused to parametrize the RS choice [6]. Certain combinations of R and β -function coefficientsare RS invariants [6]. (Their definition, and that of ˜Λ, will be discussed in Sec. 4.) The first feware: ˜ ρ = c, and ρ ( Q ) = b ln( µ/ ˜Λ) − r , ˜ ρ = c + r − cr − r , (2.5)˜ ρ = c + 2 r − c r − r r + cr + 4 r . The ˜ ρ i are Q independent, since the µ/Q dependence from the r i ’s cancels out. The specialinvariant ρ ( Q ) depends on Q and can be written as ρ ( Q ) = b ln( Q/ ˜Λ R ) , (2.6)where ˜Λ R is a scale specific to the particular physical quantity R . At n f = = 16 the leading β -function coefficient b vanishes. For n f just below 16 the β function has a zero at a very small a ∗ , proportional to (16 − n f ). Its limiting form, a ≡ b = 8321 (cid:0) − n f (cid:1) , (3.1)serves as the expansion parameter for the Banks-Zaks (BZ) expansion [1]-[4]. To proceed, onefirst re-writes all perturbative coefficients, eliminating n f in favour of a . The first two β -functioncoefficients, which are RS invariant, become: b = 1078 a , (3.2) c = − a + 194 . (3.3)Note that c is large and negative in the BZ context.4e will consider a class of physical quantities (dubbed ‘primary’ quantities) for which the˜ ρ i invariants have the form ˜ ρ i = 1 a (cid:0) ρ i, − + ρ i, a + ρ i, a + . . . (cid:1) . (3.4)Within the class of so-called ‘regular’ schemes [3, 4], the β -function coefficients ( bc i ) are analyticin a so that c i = 1 a (cid:0) c i, − + c i, a + c i, a + . . . (cid:1) . (3.5)Note that this equation is a property of the scheme, irrespective of the physical quantity, whereasEq. (3.4) is a property of the physical quantity, irrespective of the scheme. For ‘primary’ quan-tities in ‘regular’ schemes we have r i = r i, + r i, a + r i, a + . . . . (3.6)[In fact, for certain quantities the numerator of Eq. (3.4) is a polynomial whose highest termis ρ i,i a i +1 , and in certain ‘rigid’ schemes, such as MS, the a series for c i and r i truncate afterthe c i,i − and r i,i terms. These properties are unimportant here, but are crucial in the oppositelimit, the large- b approximation.]Expanding in powers of a the zero of the β function is found to be a ∗ = a (1 + ( c , − + c , ) a + . . . ) , (3.7)and hence the infrared limit of R is R ∗ = a (1 + ( r , + c , − + c , ) a + . . . ) . (3.8)Since the BZ expansion parameter a is RS invariant the coefficients in the R ∗ series are RSinvariant and can be written in terms of the ρ i,j : R ∗ = a (1 + ( ρ , − + ρ , ) a + . . . ) . (3.9)Note, though, that a ∗ is not a physical quantity and its a expansion has RS-dependent coeffi-cients.At a finite energy Q the result for R to n th order of the BZ expansion can be expressed asthe solution an equation of the form [4] ρ ( Q ) = 1 R + 1ˆ γ ∗ ( n ) ln (cid:18) − RR ∗ ( n ) (cid:19) + c ln ( | c | R ) (3.10)for n = 1 , ,
3. (For n ≥ R ∗ ( n ) andˆ γ ∗ ( n ) are the n th-order approximations to R ∗ and ˆ γ ∗ ≡ γ ∗ b . The critical exponent γ ∗ governsthe manner in which R approaches R ∗ in the Q → R ∗ − R ) ∝ Q γ ∗ . (3.11)5ormally γ ∗ is identified with the slope of the β function at the fixed point [23], and that is truein the present context. (Some subtleties with γ ∗ [24, 25] are discussed in Appendix A.) The BZexpansion of γ ∗ is ˆ γ ∗ ≡ γ ∗ b = a (cid:0) g a + g a + O ( a ) (cid:1) , (3.12)where the g i ’s are the universal invariants of Grunberg [3]: g = c , = ρ , ,g = c , − c , − − c , − = ρ , − ρ , − − ρ , − . (3.13)They are universal in that they do not depend on the specific physical quantity R being con-sidered, and invariant because they can be expressed as combinations of the invariants ρ i,j (combinations in which all the r i,j terms cancel).Close to the BZ limit R remains almost constant over a huge range of Q about ˜Λ R . Thisconstant value is not R ∗ but 0 . R ∗ [4]. More precisely, it is R ∗ / (1 + χ ) where ln χ + χ + 1 = 0,a result that follows from Eq. (3.10) to leading order in a with ρ ( Q ) = 0, corresponding to Q = ˜Λ R . Only when Q/ ˜Λ R becomes extremely small does R abruptly rise up to R ∗ , andonly when Q/ ˜Λ R becomes extremely large does R very slowly decrease to zero, as required byasymptotic freedom. (See Fig. 1.)Fig. 1. Schematic picture of R as a function of Q close to the BZ limit showing thethree regions (i) the “spike” at very low energies, (ii) the huge flat region where thetheory is “nearly scale invariant,” and (iii) the slow approach to asymptotic freedomat very high energies. (Region (iii) is shown on a log scale.)Since Eq. (3.10) completely characterizes the Q dependence of R in low-orders of the BZexpansion, it suffices to consider R ∗ and ˆ γ ∗ , both of which are quantities defined in the Q → Optimized perturbation theory
Since it is a physical quantity, R satisfies a set of RG equations [6] ∂ R ∂τ = (cid:18) ∂∂τ (cid:12)(cid:12)(cid:12)(cid:12) a + β ( a ) b ∂∂a (cid:19) R = 0 , “ j = 1” , (4.1) ∂ R ∂c j = (cid:18) ∂∂c j (cid:12)(cid:12)(cid:12)(cid:12) a + β j ( a ) ∂∂a (cid:19) R = 0 , j = 2 , , . . . . The first of these, with τ ≡ b ln( µ/ ˜Λ), is the familiar RG equation expressing the invariance of R under changes of renormalization scale µ . The other equations express the invariance of R under other changes in the choice of RS. The β j ( a ) functions, defined as ∂a/∂c j , are given by[6, 12] β j ( a ) ≡ a j +1 ( j − B j ( a ) , (4.2)with B j ( a ) = ( j − a j − B ( a ) I j ( a ) , (4.3)where I j ( a ) ≡ (cid:90) a dx x j − B ( x ) . (4.4)The B j ( a ) functions have expansions that start 1 + O ( a ). (Note that for j → + one naturallyfinds B ( a ) = B ( a ).)As mentioned earlier, certain combinations of r i and c j coefficients form the RS invariants˜ ρ i . (See Eq. (2.5)). Dependence on Q enters only through ρ ( Q ) = b ln( Q/ ˜Λ R ). The scale ˜Λ R is related by ˜Λ R = ˜Λ exp( r µ = Q ) /b ) to a universal but RS-dependent ˜Λ parameter that arisesas the constant of integration in the integrated β -function equation: b ln( µ/ ˜Λ) ≡ τ = K ( a ) , (4.5)where K ( a ) ≡ a + c ln( | c | a ) − (cid:90) a dxx (cid:18) B ( x ) − cx (cid:19) . (4.6)(This form of K ( a ), completely equivalent to our previous definition [6, 12], is more convenientwhen c is negative [14].) The ˜Λ parameter thus defined is RS dependent, but it can be convertedbetween different schemes exactly by the Celmaster-Gonsalves relation [26].The β function is RS dependent. The conversion between two schemes (primed and un-primed) is given by β (cid:48) ( a (cid:48) ) ≡ µ da (cid:48) dµ = da (cid:48) da µ dadµ = da (cid:48) da β ( a ) . (4.7)7or any specific physical quantity R one can always define the “fastest apparent convergence”(FAC) or “effective charge” (EC) scheme [27] in which all the series coefficients r i vanish, sothat R = a EC (1 + 0 + 0 + . . . ). As a special case of the previous equation we have β EC ( R ) = d R da β ( a ) . (4.8)The ˜ ρ n invariants can conveniently be defined to coincide with the coefficients of the EC β function. Thus, defining β EC ( R ) = − b R B EC ( R ), with B EC ( R ) ≡ ∞ (cid:88) n =0 ˜ ρ n R n , (4.9)the invariants ˜ ρ n can be obtained by equating coefficients in B EC ( R ) = a R d R da B ( a ) , (4.10)which we shall refer to as the “invariants master equation.”The ( k + 1) th -order approximation, R ( k +1) , in some general RS, is defined by truncating the R and β series after the r k and c k terms, respectively: R ( k +1) ≡ a k (cid:88) m =0 r m a m , B ( k +1) ≡ k (cid:88) j =0 c j a j , (4.11)with r ≡ c ≡
1, and c ≡ c . Because of these truncations, the resulting approximant dependson RS. “Optimization” [6] corresponds to finding the stationary point where the approximant islocally insensitive to small RS changes, i.e., finding the “optimal” RS in which the RG equations(4.1) are satisfied by R ( k +1) with no remainder. The resulting optimization equations [6] havebeen solved for the optimized ¯ r m coefficients in terms of the optimized couplant ¯ a and theoptimized ¯ c j coefficients [12]. (The overbars denote quantities in the optimal scheme, but wewill generally omit these henceforth, except to distinguish ¯ a from a generic a .) To present thatsolution it is convenient to define S ≡ d R da = 1 + s a + s a + . . . , (4.12)with coefficients s m ≡ ( m + 1) r m . The optimized s m coefficients are given by [12]: s m ¯ a m = 1 B k (¯ a ) ( H k − m (¯ a ) − H k − m +1 (¯ a )) , m = 0 , , . . . , k, (4.13)where H i ( a ) ≡ k − i (cid:88) j =0 c j a j (cid:18) i − j − i + j − (cid:19) B i + j ( a ) , i = (1) , , . . . , k, (4.14)8ith c ≡ c ≡ c . H is to be understood as the limit i → H k = B k and we define H ≡ H k +1 ≡ Q →
0. Afinite infrared limit in optimized perturbation theory (OPT) can occur in at least two ways: (i)through a fixed point (a zero of the optimized β function) [8, 9, 10] or (ii) through an “unfixedpoint” and the pinch mechanism [14]. The latter case is discussed in Appendix B, but seems tobe only tangentially relevant in the BZ limit.In the fixed-point case the infrared limit of the optimized couplant is a ∗ , which is the firstzero of the optimized β function: B ( k +1) ( a ∗ ) = 0. The above solution for the optimized s m ≡ ( m + 1) r m coefficients in terms of the optimized c j coefficients simplifies greatly to [12]ˆ s m = 1( k − ( k − m )ˆ c m − m (cid:88) j =0 ˆ c j , (4.15)where ˆ s m ≡ s m a ∗ m , and ˆ c m ≡ c m a ∗ m . Explicit results for the infrared-fixed-point limit of OPT were obtained in Ref. [8] for k = 2and 3. Extending the calculation to higher orders is made easier by the formula (4.15), whichcan be used to substitute for the optimal-scheme r m ’s in the ˜ ρ invariants. From the resulting ˜ ρ expression one can solve for the optimal-scheme c in terms of a ∗ , c, ˜ ρ . Then, making use of thatresult, one may solve for c in terms of a ∗ , c, ˜ ρ , ˜ ρ , and so on up to c k − . The last coefficient, c k can then be found from the fixed-point condition B ( a ∗ ) = 0. Substituting in the expression for˜ ρ k then produces an equation for a ∗ that involves only the invariants c, ˜ ρ , . . . , ˜ ρ k . One can thenfind a ∗ numerically as the smallest positive root of that equation. Finally, the expressions forthe c j ’s in terms of a ∗ and the invariants can be substituted in the formula (4.15) to determinethe r m ’s. Hence, one can find R ∗ .The preceding discussion pre-supposes that the perturbative calculations have been done to( k + 1) th order, so that the numerical values of the invariants up to ˜ ρ k are known. The greatsimplification in the BZ limit is that we can effectively set almost all the invariants to zero: thiscan be seen as follows. As a → ρ i is of order 1 /a , buteach ˜ ρ i enters the analysis along with a factor of a ∗ i that is of order a i . Thus, to find the leadingterm in the BZ limit, we can effectively set to zero all the invariants except c . (Furthermore,only the − /a piece of c will contribute.) To obtain the next-to-leading correction in a wewould also need the piece of c along with the ρ , − /a piece of ˜ ρ (whose value depends onthe specific R quantity under consideration). 9or k = 2, following the procedure in the first paragraph of this section, we find r = − (cid:18) ca ∗ a ∗ (cid:19) , r = − c , (5.1)from the optimization condition, Eq. (4.15). Then c can be found from B ( a ∗ ) = 0 as c = − ca ∗ a ∗ . (5.2)Substituting in the expression for ˜ ρ in Eq. (2.5) yields the equation for a ∗ : − ca ∗ − c a ∗ a ∗ = ˜ ρ . (5.3)(When comparing with Refs. [8, 10] note that the “ ρ ” used there is ˜ ρ − c ). In the BZ limitwe can set ˜ ρ = 0 so that the a ∗ equation becomes( ca ∗ + 1)( ca ∗ − ) = 0 . (5.4)Hence, we find a ∗ = − /c → a . The coefficients c , r , r all vanish, so, in an a posteriori sense, the k = 2 OPT scheme is ‘regular’ in the infrared (fixed-point) limit. The final result for R ∗ is R ∗ = − c → a . (5.5)Thus, exactly as in any ‘regular’ scheme, we find that a ∗ and R ∗ tend to a in the BZ limit.The same is true for ˆ γ ∗ , obtained from the slope of the β function at the fixed point.At higher orders, though, the OPT scheme is not ‘regular’ — the optimized r i coefficients,for instance, have 1 /a i pieces — and the story is more complicated. For k = 3 the optimizationcondition gives r = − a ∗ , r = − (1 + ca ∗ + 2 c a ∗ )6 a ∗ , r = − c . (5.6)Proceeding immediately to the BZ limit, we set ˜ ρ = ˜ ρ = 0. Substituting into ˜ ρ = 0 gives c = (11 − ca ∗ )32 a ∗ , (5.7)and then the last coefficient, c can be found from B ( a ∗ ) = 0; after using the previous equation,this gives c = − (43 + 28 ca ∗ )32 a ∗ (5.8)The equation for a ∗ in the BZ limit then follows by substituting in ˜ ρ = 0. We could haveexpected a cubic equation, but in fact we find83 + 52 ca ∗ = 0 . (5.9)10hus, we do not get a ∗ = − c → a , but a ∗ → a = 1 . a . The final result for R ∗ is not a but is a = 1 . a , which is remarkably close.Results for higher orders are shown in Tables 1 and 2. The even- k results are significantlybetter than those for odd k . Note that a ∗ /a increases, apparently towards 4. It is perfectlyacceptable for a ∗ to differ from a , since a ∗ is inherently scheme dependent. However, R ∗ is aphysical quantity so it is reassuring that R ∗ /a is always close to 1. In Sect. 7 we will find asimple explanation for a ∗ /a → R ∗ /a → k → ∞ . k a ∗ a R ∗ a ˆ γ ∗ a · · · · · · · · · · · · · · · · · · · · · · · · OPT results in the BZ limit for k = even. k a ∗ a R ∗ a ˆ γ ∗ a · · · · · · · · · · · · · · · · · · · · · · · · · · · OPT results in the BZ limit for k = odd. The situation with ˆ γ ∗ is less clear. This is also a physical quantity (with the caveats ofAppendix A) so we should have ˆ γ ∗ /a → k → ∞ . The numerical results in the tablescannot be said to support that contention, but neither are they inconsistent with it; one can11ake good fits to the data with functions of k that very slowly approach 1 as k = ∞ for botheven and odd k .It is hard to go to much larger k with the method described in this section, so we turn to ananalytic approach in the next sections. Our results – albeit in approximations to OPT ratherthan true OPT – support the claim that a ∗ /a → R ∗ /a and ˆ γ ∗ /a tend to 1 as k → ∞ : they also provide valuable insight into the workings of OPT at arbitrarily high orders. To make progress analytically with OPT in ( k + 1) th order it helps greatly to deal with functionsand differential equations rather than with 2 k individual r i and c i coefficients. The set of ˜ ρ i invariants naturally follow from a single “master equation,” Eq. (4.10), and what we need is toalso formulate the k optimization conditions as a “master equation.” For general Q this wouldbe a daunting task. In the infrared fixed-point limit, however, it is relatively simple — and,happily, that suffices in the present context since, as noted in Sect. 3, in the BZ limit and forthe first three terms of the BZ expansion, the entire Q dependence of R is characterized by thetwo infrared quantities R ∗ and ˆ γ ∗ .We now show that the optimization conditions in the fixed-point limit, Eq. (4.15), followfrom equating coefficients in the following “fixed-point OPT master equation:” d R da = B ( a ) − a ( k − (cid:18) dB ( a ) da + B ( a )( a ∗ − a ) (cid:19) . (6.1)(Superscripts “ ( k +1) ” on R and B ( a ) are omitted for brevity.) Note that a here is merely adummy variable, while a ∗ is the optimized couplant in the infrared limit.The first step of the proof is to note that, by the definition of a ∗ , the polynomial B ( a ) hasa factor of a ∗ − a and can be written as B ( a ) = ( a ∗ − a ) a ∗ k − (cid:88) n =0 (cid:16) aa ∗ (cid:17) n ˆ t n , (6.2)where ˆ t n is a partial sum of β -function terms:ˆ t n = n (cid:88) j =0 ˆ c j (6.3)with ˆ c j ≡ c j a ∗ j . Note that ˆ t n − ˆ t n − = ˆ c n and that ˆ t k = 0 by virtue of the fixed-point condition.To show Eq. (6.2), expand the right-hand side, then use ˆ t k = 0 and define ˆ t − ≡ k (cid:88) n =0 (cid:16) aa ∗ (cid:17) n ˆ t n − k − (cid:88) n = − (cid:16) aa ∗ (cid:17) n +1 ˆ t n . (6.4)12ow put n = n (cid:48) − k (cid:88) n =0 (cid:0) ˆ t n − ˆ t n − (cid:1) (cid:16) aa ∗ (cid:17) n = k (cid:88) n =0 ˆ c n (cid:16) aa ∗ (cid:17) n = k (cid:88) n =0 c n a n , (6.5)which is B ( a ), as claimed.To prove Eq. (6.1), equate powers of ( a/a ∗ ) m , using Eq. (6.2) to write B ( a ) / ( a ∗ − a ) as apolynomial. This leads to ˆ s m = ˆ c m − k − (cid:0) m ˆ c m + ˆ t m − (cid:1) . (6.6)Using ˆ t m − ˆ t m − = ˆ c m again and simplifying leads to the fixed-point optimization conditions,Eq. (4.15), completing the proof.Unfortunately, Eq. (6.1) proves difficult to deal with. To make progress we have resortedto two approximations, designated PWMR and NLS, that we now explain. Ref. [12] has shownthat the series expansion of H i ( a ) − H i ( a ) − k − i + 2 k c k − i +1 a k − i +1 (1 + O ( a )) , (6.7)which quickly leads to s m = k − mk c m + O (¯ a ) , (6.8)a result first obtained (in a quite different manner) by Pennington, Wrigley, and Minaco andRoditi (PWMR) [28]. Dropping the O (¯ a ) term leads to the PWMR approximation which iseasily formulated as a “master equation”: d R da = B ( a ) − k a dB ( a ) da (PWMR) . (6.9)Looking at the above equation, or the original equation (6.1), it is tempting to suppose that,as k → ∞ , they reduce to d R da = B ( a ) (NLS) . (6.10)We shall refer to this as the “na¨ıve limiting scheme” (NLS). It corresponds to a well-defined RSin which s m = c m , so that the coefficients r m = s m / ( m + 1) of the R series decrease by a factor1 / ( m + 1) relative to the coefficients of the B series.Clearly, this idea is very na¨ıve. In the PWMR case the actual relation is s m = k − mk c m ,which only reduces to s m ≈ c m for m (cid:28) k ; that is, for the early part of the series only.Nevertheless, there may be a kernel of truth here, for if the series are “well behaved” the earlyterms should dominate. In any case, adopting this na¨ıve idea leads us in a fruitful direction.Our investigations below will lead us to conclude that, at least in the BZ context, the NLS does13ield the all-orders limit of OPT, although it is a poor guide to how fast results converge to thatlimit.Using the NLS equation above to eliminate B ( a ) in the invariants master equation (4.10)leads directly to B EC ( R ) = a R (cid:18) d R da (cid:19) . (6.11)Taking the square root leads to d R da = R a (cid:112) B EC ( R ) , (6.12)which is immediately integrable.The BZ limit provides us with a nice “playground” for exploring further, since it effectivelycorresponds to the case B EC ( R ) = 1 + c R . We continue this analysis in the next section. In the BZ limit the only one of the ˜ ρ n invariants that contributes is c , which is negative: c = − /a + O (1) as a →
0. We may set B EC ( R ) = 1 + c R in this limit. (The terms neglectedcan only contribute to O ( a ) corrections, as argued in Sect. 5.) It is convenient to define u ≡ − ca , v ≡ − c R . (7.1)In these variables, the NLS condition is B = dvdu and Eq. (6.12) becomes dvdu = vu √ − v, (7.2)which leads to (cid:90) dvv √ − v = (cid:90) duu . (7.3)Performing the integral and then exponentiating both sides gives1 − √ − v √ − v = u, (7.4)where the constant of integration has been fixed by requiring v → u as u →
0, correspondingto the R series beginning R = a (1 + . . . ). Inverting this equation (assuming u ≤
1) gives v = 4 u (1 + u ) . (7.5)Hence, B = dvdu is given by B = 1 − u (1 + u ) . (7.6)14The two formulas above are key results. They show an interesting u → /u duality that wewill discuss in Sect. 10.)The fixed point, where B = 0, is at u ∗ = 1. Recalling Eq. (7.1), we see that a ∗ is − /c → a .Nevertheless, because u ∗ = 1 in Eq. (7.5) leads to v ∗ = 1, we find R ∗ = − /c → a , in agreementwith the regular-scheme result.Evaluating the slope of the β function at the fixed point gives − b (cid:18) − c (cid:19) u ddu (cid:18) − u (1 + u ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) u =1 = − b c → ba , (7.7)which seemingly gives ˆ γ ∗ ≡ γ ∗ /b = a . Here the subtlety discussed in Appendix A comes intoplay. The critical exponent γ ∗ is really the infrared limit of an effective power-law exponentgiven at finite Q by [25] γ ( Q ) = dβda + β ( a ) d R da (cid:46) d R da . (7.8)Normally the second term drops out in the infrared limit because β ( a ) vanishes at the fixedpoint. However, in the NLS the denominator d R da also vanishes because it is B ( a ) = β ( a ) / ( − ba ).Therefore, in the NLS case the second term contributes − ba d R da = − ba dBda which contributesan equally with the first term, thus rescaling the previous result by a factor of 2. Hence, we findˆ γ ∗ = a , in accord with the regular-scheme result.The preceding discussion corresponds to the NLS result re-summed to infinite order. Onemust now ask: Do the finite-order NLS results converge to their infinite-order form – and, if so,how fast? At ( k + 1) th order the B and v series are truncated, and v ∗ is found by evaluatingat u ∗ , the zero of the truncated B . Luckily, as with a simple geometric series, the sum of finitenumber of terms can be expressed fairly simply. The truncated B series is B ( k +1) = k (cid:88) j =0 ( j + 1) ( − u ) j = 1 − u (1 + u ) + ( − k k u k +1 (1 + u ) (cid:18) O ( 1 k ) (cid:19) . (7.9)Only for odd k do we get a zero. (We will discuss even k near the end of this Section.) The zeroof the truncated B is just before u reaches 1. If we put u = u ∗ ≡ − η ( k ) k (7.10)with η ( k ) (cid:28) k , we find (noting that u k +1 → e − η ( k ) ) that η ( k ) = 3 ln k − ln(ln k ) − ln(3 /
4) + O (cid:18) ln ln k ln k (cid:19) . (7.11)The truncated v series is v ( k +1) = 4 u k (cid:88) j =0 ( j + 1)( − u ) j = 4 u (cid:18) u ) + ( − k k u k +1 (1 + u ) (cid:18) O (cid:18) k (cid:19)(cid:19)(cid:19) . (7.12)15hen we substitute u = u ∗ we find a cancellation of the η ( k ) /k terms which leaves v ∗ ≈ −
94 ln kk . (7.13)This is in good accord with the numerical results in Table 3. k u ∗ = a ∗ a v ∗ = R ∗ a ˆ γ ∗ a · · · · · · · · · · · · · · · · · · NLS results in the BZ limit.
A similar analysis for ˆ γ ∗ (including the factor of 2 discussed above) leads toˆ γ ∗ = a (cid:16) − k +1 ln k + . . . (cid:17) , (7.14)which indicates that the NLS results for ˆ γ ∗ do not converge – the nominal limit of a is “cor-rected” by a ln k term arising from the series-truncation effects. We indeed see this in thenumerical results in Table 3.Returning to Eq. (7.9) we see that the truncated B ( u ) function closely approximates itslimiting form − u (1+ u ) until u gets close to 1. For odd k the ( − k “truncation effect” term causes B to suddenly dive down, producing a zero. For even k this term causes B to suddenly shootupwards and there is no zero. This means that there is no finite infrared limit in these orders;the “spike” in R goes all the way up to infinity. However, since B has a minimum very close tozero the running of the couplant “almost stops” here and if we were to evaluate v at this valueof u we would find a result close to the R ∗ /a obtained in the previous odd- k order. A relatedobservation is that, with only a slight change of RS, we would find an infrared limit arising froma pinch mechanism (see Appendix B).We conclude that the NLS provides a lot of insight into OPT as k → ∞ , but is only arather crude approximation to true OPT. We move on to the PWMR approximation in thenext section. 16 All-orders PWMR in the BZ limit
As before we have B EC ( R ) = 1 + c R in the BZ limit and we use u ≡ − ca and v ≡ − c R . In thesevariables the invariants master equation (4.10) becomes B = v u (1 − v ) dvdu , (8.1)and the PWMR master equation (6.9) becomes14 dvdu = B − k u dBdu . (8.2)We will proceed to solve these two coupled differential equations, treating k as an ordinaryparameter: only later will we consider the other k dependence coming from the truncations ofthe resulting series at ( k + 1) th order. (We have explicitly checked that at low k this two-stepapproach does produce the same results as a PWMR version of the OPT procedure describedin Sect. 5.)We begin by making an ansatz : B = 14 dvdu ξ , (8.3)where ξ depends on u . (We will actually want to view it as a function of a new variable X ,introduced below, that itself is a function of u .) Substituting in Eq. (8.1) leads, in the same wayas in the NLS case, to (cid:90) dvv √ − v = (cid:90) duu ξ, (8.4)which leads to v = 4 X (1 + X ) , (8.5)with the new variable X defined by X ≡ exp (cid:90) duu ξ, (8.6)or more specifically, enforcing X → u as u → X ≡ u exp (cid:90) u d ¯ u ¯ u ( ξ − . (8.7)Note that dXdu = Xu ξ, (8.8)so that the inverse relationship is u = X exp (cid:90) X d ¯ X ¯ X (cid:18) ξ ( ¯ X ) − (cid:19) . (8.9)17e will now want to consider ξ as a function of the new variable X .We can now find dvdu as dvdX dXdu and substitute back in the ansatz (8.3) to get B = (1 − X )(1 + X ) Xu ξ . (8.10)From this we can calculate dBdu , which, after some algebra, reduces to dBdu = Bu (cid:18) (1 − X + X )(1 − X ) ξ − − X dξdX (cid:19) . (8.11)Substituting this, and dvdu = ξ B from the ansatz (8.3), into Eq. (8.2), leads, after cancelling afactor of B , to an equation for ξ ( X ):1 − ξ = 2 k (cid:18) (1 − X + X )(1 − X ) ξ − − X dξdX (cid:19) . (8.12)Remarkably, this nonlinear, first-order differential equation is soluble. The trick is to write ξ inthe form ξ = 1 − k X F d F dX . (8.13)This substitution, because of a cancellation of ( F (cid:48) / F ) terms, leads to a linear second-orderequation for F . A further substitution, F = (1 − X ) F, (8.14)leads to a Gauss hypergeometric equation, revealing that F = F ( − n, , − n −
12 ; X ) , (8.15)where n ≡ k/ −
1. We will focus on the case of even k . (Curiously, the roles of odd and even k are reversed relative to the NLS case.) For even k the F function is a polynomial of degree n in X : F = n !(2 n + 1)!! n (cid:88) i =0 (2 i + 1)!! i ! (2( n − i ) + 1)!!( n − i )! ( X ) i . (8.16)The first few F ’s are shown in Table 4. Note the ‘reflexive’ symmetry i → n − i , meaning thatthe coefficients are symmetric about the middle. In the n → ∞ limit F approaches (1 − X ) − / ,except near X = 1, where its behaviour involves a modified Bessel function I (see Table 4).To find u in terms of X it is helpful to use another representation of ξ , namely1 ξ = 1 − n + 2 X P d P dX , (8.17)so that Eq. (8.9) will immediately lead to u = X P − n +2) . (8.18)18 n F X X + X X + X + X
10 4 1 + X + X + X + X ∞ ∞ (1 − X ) − / ( X (cid:54) = 1) √ n √ π e − x I ( x ) x ( X = 1 − xn )Table 4: The first few F polynomials and their form for large k = 2 n + 2 . Substituting the above form for ξ into the ξ equation (8.12) leads again to a linear equation.One can verify that this equation is satisfied by setting P = (1 + X ) P (8.19)with P = 1( n + 1) 1(1 + X ) (cid:18) [ n + 1 − ( n − X ] F − − X ) X dFd ( X ) (cid:19) . (8.20)The numerator turns out to have a (1 + X ) factor, so that P is a polynomial of degree 2 n in X .The first few P ’s are shown in table 5. These polynomials also have a ‘reflexive’ property. k n P − X + X − X + X − X + X − X + X − X − X − X + X
10 4 1 − X + X − X + X − X + X − X + X ∞ ∞ (1 − X ) − / (1 + X ) − / ( X (cid:54) = 1) √ n √ π e − x I ( x ) ( X = 1 − xn )Table 5: The first few P polynomials and their form for large k = 2 n + 2 . Yet another expression for ξ is ξ = (1 + X )(1 − X ) PF , (8.21)which can be proved by substituting for P and simplifying to reach Eq. (8.13). Using this formof ξ in Eq. (8.10) gives B = (1 − X ) F P − ( n +1 n +2 ) . (8.22)19s noted in the tables, both F and P polynomials have simple limits as k → ∞ , providedthat X (cid:54) = 1. It is easy to see that X → u and that all formulas revert to their NLS forms in thislimit. Thus, it is clear that v ∗ must ultimately tend to 1, so that R ∗ = a in accord with theBZ limit.However, to go further analytically and determine how fast the finite-order PWMR resultsapproach their infinite-order form is beset with difficulties; the subtleties when X ∼ B and v as series, not in X but in u ; then find u ∗ from the zero of the truncated B series; and then evaluate the truncated v series at u = u ∗ . Nevertheless, we can explore these issues numerically with Mathematica. Wehave been able to explore up to k ≈
100 and the numerical results are presented in Table 6. Itappears that v ∗ approaches 1 significantly faster than in the NLS case: v ∗ ∼ − A ln k/k k , (8.23)with A ≈ .
08 and k ≈ .
5, roughly. k u ∗ = a ∗ a v ∗ = R ∗ a ˆ γ ∗ a · · · · · · · · · · · · · · · PWMR results in the BZ limit.
The ratio of v to its NLS form v NLS ≡ u (1+ u ) stays very close to 1 in the entire relevantrange 0 < u < u ∗ , although it strongly deviates thereafter. See Fig. 2.The v series is also much better behaved than in NLS, where the magnitude of the coeffi-cients increased in arithmetic progression: v NLS = 4 u (cid:80) j ( j + 1)( − u ) j . In the PWMR case, thecoefficients v j in v = 4 u k (cid:88) j =0 v j ( − u ) j (8.24)are plotted in Fig. 3 for k = 100. The initial ( j + 1) growth is suppressed by a more-than-exponential decay (a crude fit is ( j + 1) exp( − . j / )). The middle coefficient j = k isexactly zero because of the k − j factor in the PWMR relation between s j and c j coefficients,20ig. 2. Plot of v divided by v NLS ≡ u (1+ u ) as a function of u for PWMR at k = 100.The curve is shown dashed beyond u = u ∗ = 0 . u j factor, even at u = u ∗ , the largestrelevant u , and it actually plays a beneficial role. This can be seen in Fig. 4 which plots thepartial sums of n max terms of the v series, Eq. (8.24), at u = u ∗ in the case k = 100. The serieshas pretty well converged after 50 terms, but including 25 more terms significantly reduces theerror. The very last term makes an unexpectedly large correction, but this further reduces theerror and means that the last term provides quite a realistic error estimate.Fig. 3. Coefficients v j in the series expansion of v ( u ) = 4 u (cid:80) kj =0 v j ( − u ) j , for PWMRwith k = 100. The inset shows the higher-order coefficients on a finer scale.21ig. 4. The partial sums 4 u (cid:80) n max j =0 v j ( − u ∗ ) j versus n max for the v ∗ series in the case k = 100. The plots use three different scales, so as to show that (a) the series hascrudely converged after 50 terms but (b) a slight adjustment from 50 to 75 termsreduces the error quite significantly, and (c) the last term makes an unexpectedlylarge change, given the trend of the preceding terms, but this further improves theresult and means that the last term is, within a factor of 2, a good measure of theactual error. 22he series for ˆ γ ∗ , which is just dβ/da | ∗ , is much worse behaved. Also the sequence of resultsfor ˆ γ ∗ in Table 6 appear to diverge, though at a much slower rate than in NLS. It is reasonableto hope that the extra subtleties in full OPT would lead to ˆ γ ∗ converging to a , albeit very, veryslowly, in view of the low-orderOPT results in Tables 1 and 2.We have not been able to extend the analysis to the full fixed-point master equation, (6.1).One can get to an equation similar to Eq. (8.12), but with an extra term involving u/ ( u − u ∗ )that seems intractable. Moreover, the parameter u ∗ can only be fixed after the B ( u ) functionis found, and expressed as a truncated series, so the interaction between analytic subtleties andtruncations effects is even more complicated and delicate. Setting aside the difficult issue of how fast results converge as k → ∞ , the results of the lastsection confirm that the simple NLS formulas from Sect. 7, v = 4 u (1 + u ) , (9.1) B = 1 − u (1 + u ) , (9.2)represent the all-orders limit of PWMR – and presumably of true OPT too – in the BZ limit. Aspreviously noted, these formulas give the same BZ limit for R ∗ and ˆ γ ∗ as ‘regular’ schemes. Wenow show that higher terms in the BZ expansion are reproduced correctly by all-orders NLS.Before discussing the general proof it is instructive to look at next-to-leading order in theBZ expansion. At this level we now need two of the invariants, c and ˜ ρ so we take B EC = 1 + c R + ˜ ρ R . (9.3)(In fact, only the ρ , − piece of ˜ ρ would contribute when we re-expand the results in powersof a . However, it will not be necessary to carry out that step explicitly, since once we showequivalence to the EC scheme, a ‘regular’ scheme, we are bound to get the same BZ expansionto the corresponding order in a .) Recall that the NLS condition and the invariants masterequation together lead to Eq. (6.12), d R da = R a (cid:112) B EC ( R ) , (9.4)which now gives (cid:90) d RR (cid:112) c R + ˜ ρ R = (cid:90) daa . (9.5)23ntegration yields ln (cid:32) R c R + 2 (cid:112) c R + ˜ ρ R (cid:33) = ln a, (9.6)where the constant of integration has been fixed so that R = a (1 + . . . ) as a →
0. One can nowexponentiate and solve for R , and then B ( a ) can be found from d R /da . As before we define u = − ca/ v = − c R . The zero of B is at u ∗ = 1 (cid:113) − ˜ ρ c , (9.7)and in terms of these variables we find v = 4 u (cid:16) u + u u ∗ (cid:17) , (9.8) B = 1 − u u ∗ (cid:16) u + u u ∗ (cid:17) . (9.9)It is now straightforward to check that v evaluated at u = u ∗ gives R ∗ = − v ∗ c = − c ρ (cid:32) − (cid:114) − ρ c (cid:33) , (9.10)which is the root of B EC ( R ) = 0. Thus, the R ∗ of all-orders NLS agrees with the R ∗ of the ECscheme. Also, ˆ γ ∗ , defined as the infrared limit of Eq. (7.8), which leads toˆ γ ∗ = − a dBda (cid:12)(cid:12)(cid:12)(cid:12) ∗ , (9.11)with the factor-of-2 subtlety as in Sect. 7, can be shown to reduce toˆ γ ∗ = −R dB EC d R (cid:12)(cid:12)(cid:12)(cid:12) ∗ , (9.12)which is the ˆ γ ∗ of the EC scheme.The general proof is really just a special case of the general formal arguments that R ∗ andˆ γ ∗ (properly defined) are invariant under RS transformations [25]. From Eq. (9.4) we can seeimmediately that B ( a ), equal to d R /da in NLS, must vanish when B EC vanishes; thus the R evaluated at a = a ∗ in NLS must agree with the R ∗ defined as the zero of the EC β function.Furthermore, the equivalence of the two equations for ˆ γ ∗ above can be proved just from the NLScondition B = d R /da and Eq. (9.4), without assuming any specific form for B EC .24 a → a ∗ /a duality It is easily verified that under u → u ∗ /u the v of Eq. (9.8) remains invariant, while the B ofEq. (9.9) transforms to − ( u /u ∗ ) B . These properties are even easier to spot in Eqs. (9.1, 9.2),in the BZ-limit case, where u ∗ = 1.Let us try to trace the origin of these properties. Consider a transformation a −→ λ a , (10.1)with some positive constant λ . We postulate that R and all the ˜ ρ i invariants remain invariantand that the β -function equation, µ dadµ = β ( a ) maintains its form. The latter condition meansthat dadτ = − a B ( a ) , (10.2)where τ = b ln( µ/ ˜Λ), must transform to ddτ (cid:18) λ a (cid:19) = − (cid:18) λ a (cid:19) B T ( a ) , (10.3)where B T ( a ) ≡ B ( λ a ). This requires B T ( a ) = − a λ B ( a ) . (10.4)If B ( a ) vanishes at a = a ∗ then B T ( a ) must too. Thus λ /a ∗ must be a zero of B ( a ). If weassume that there is only one zero, then we must take λ = a ∗ .The transformation of d R da would be d R da −→ d R d (cid:16) λ a (cid:17) = − a λ d R da . (10.5)Note that this is the same transformation rule as for B above. Thus, the NLS scheme-fixingcondition, d R da = B ( a ), transforms into itself. It is straightforward to check that the same is trueof the invariants master equation Eq. (4.10). It thus seems that an a → a ∗ /a duality is notspecial to the BZ limit, but is a general property of all-orders NLS and hence of all-orders OPT.
11 Conclusions
The BZ expansion and RS invariance appear compatible. While BZ results are most simplyobtained in a restrictive class of ‘regular’ schemes, the same results emerge from ‘irregular’schemes, though they then require consideration of all orders of perturbation theory. Results in25PT for the fixed-point value R ∗ are never far from the BZ result and converge quite nicely toit. The error at ( k + 1) th order shrinks as ln k/k in NLS, as ln k/k in PWMR, and probablyslightly faster in true OPT. Our explorations provide some insight into how the subtle featuresof OPT conspire to improve finite-order results.It might be claimed that the EC scheme, or any ‘regular’ scheme is clearly better than OPTin the BZ limit, since their results converge immediately to the right result. This is true, butone should keep in mind that the BZ limit, where n f is infinitesimally less than 16 , is not aremotely physical theory, even in principle. It is an open question whether or not OPT givesbetter results than the EC scheme for n f = 16, the closest physical case.The situation with the critical exponent γ ∗ is much less satisfactory. While the all-orders NLSformulas produce the correct result, the finite-order NLS and PWMR results do not actuallyconverge. In true OPT the results might converge but, if so, the convergence is extremelyslow. The problem may stem from trying to obtain γ ∗ as a by-product of the optimization of R ∗ . If one is principally interested in γ ∗ itself, then one should construct its own perturbationseries and optimize that. However, our reason here for studying γ ∗ was not for its own sake,but as a shortcut to obtaining R ( Q ) at non-zero Q , relying on Eq. (3.10), which holds for thefirst three orders of the BZ expansion. That was very convenient because we only needed theoptimization conditions at the fixed point, and these are analytically much simpler than forgeneral Q . However, the natural procedure is to optimize R ( Q ) itself. There is no reason tosuppose that the convergence of OPT for R ( Q ) at non-zero Q is significantly worse than for R ∗ ;indeed, as Q gets larger we expect convergence to become much better. Thus, our difficultieswith γ ∗ are probably a technical, mathematical issue, rather than a problem of physical concern.The investigations in this paper have gone off in a number of different directions and revealnew territories worthy of further exploration. A key result is the “fixed-point OPT masterequation” (6.1) which opens a route to an analytical treatment of arbitrarily high orders ofOPT, given knowledge of the ˜ ρ i invariants — although here we have only been able to makeprogress in two simplifying approximations, NLS and PWMR. It appears that the simple NLSapproximation does yield the all-orders limit of OPT, although it is a poor guide to the rate ofapproach to that limit. The NLS formulas, (7.5, 7.6) at leading order in the BZ expansion, and(9.8, 9.9) at next-to-leading order, are remarkably simple. They illustrate a general a → a ∗ /a duality property of all-orders OPT that is intriguing and deserves further study.We close by mentioning some important developments [30, 31] which combine RS optimiza-tion with the optimization of a variational mass parameter, as in the φ anharmonic oscillatorproblem [18]-[20]. Perhaps the methods discussed here can be extended to investigate theseapproaches at high orders. 26 ppendix A: The critical exponent γ ∗ The critical exponent γ ∗ governing the approach of R its infrared limit R ∗ :( R ∗ − R ) ∝ Q γ ∗ . (A.1)is normally thought to be the slope of the β function at the fixed point [23]. That is not quitetrue [24]. The puzzle is resolved in Ref. [25], whose main points we briefly summarize.Since R is a physical quantity and Q is a physical parameter, the successive logarithmicderivatives of R : R [ n +1] ≡ Q d R [ n ] dQ (A.2)for n = 1 , , , . . . , with R [1] ≡ R , must be RS-invariant quantities (at any Q ). In particular,the combination γ ( Q ) ≡ R [3] R [2] = 1 + Q d R dQ (cid:46) d R dQ (A.3)is RS invariant. It is the exponent of the local-power-law form of R ( Q ) around a specific Q .Standard RG arguments, relating Q and µ dependence, lead to γ ( Q ) = dβda + β ( a ) d R da (cid:46) d R da , (A.4)and one can verify explicitly that this quantity is invariant under RS transformations [25].The critical exponent γ ∗ is the infrared-fixed-point limit of γ ( Q ). Since β ( a ) vanishes in thislimit one might think that the second term in Eq. (A.4) always drops out. While this is often thecase, it is not always true, and the NLS, where d R /da also vanishes at the fixed point, is a casewhere the second term contributes (see Sect. 7). Quite generally, it is important to recognizethat dβ/da | ∗ is not RS invariant; the second term in Eq. (A.4), even though it may vanish in alarge class of schemes, is crucial to the RS invariance of γ ∗ .Another issue arises with finite-order approximations, because then the equivalence betweenEqs. (A.3) and (A.4) is not necessarily preserved. In OPT the two are generally not the sameat finite Q , but, remarkably, they do coincide at Q = 0 [14]. We have not investigated whetherthis is also true for NLS and PWMR, which would entail explicitly considering R at finite Q and then investigating its Q → Appendix B: Pinch mechanism infrared limit
As discussed in Ref. [14], a finite infrared limit in OPT can occur through a pinch mechanismwhereby the evolving B ( a ) function of the optimized scheme develops a minimum that “pinches”the horizontal axis at a “pinch point” a p , which ultimately becomes a double zero of B ( a ). The27nfrared limit of the couplant, however, is at an “unfixed point” a (cid:63) > a p that is not a zero ofthe β function. The approach to the infrared limit is not a power law, but rather [14] R (cid:63) − R = 1 b | ln Q/ ˜Λ R | as Q → , (B.1)which corresponds to γ (cid:63) = 0 since Q d R dQ ∼ − b ir ( R (cid:63) − R ) / (B.2)for R close to R (cid:63) . In the k = 3 case, the coefficient b ir was found to be b ( k =3)ir = (cid:113) a p (3 + ca p ) (cid:16) a p a (cid:63) (cid:17) bπ , (B.3)and in the e + e − case the pinch mechanism was operative for 6 . < n f < . n f → , the pinch mechanism does not seem to occur in true OPT, atleast as far as we have been able to explore it in Sect. 5. However, the mechanism is probablyclose to being relevant because in the BZ limit the critical exponent γ ∗ ∼ ba tends to zero. Asmall or zero γ ∗ gives rise to a sharp infrared “spike” in R plotted versus Q , as in Fig. 1.The NLS and PWMR approximations to OPT seem to have fixed points only in every otherorder (for odd k in NLS, and even k in PWMR). In these orders, as discussed in Sect. 7, the B ( u ) function closely approximates its limiting form (1 − u ) / (1 + u ) until u gets close to 1,when it suddenly dives down, producing a zero. In the alternating orders B ( u ) suddenly shootsupwards and there is no zero. However, B ( u ) then has a minimum very close to the horizontalaxis, so only a slight modification of the scheme would produce a “pinch point.”We first show that that, in circumstances where the pinch mechanism does govern the infraredlimit of OPT, the master equation that replaces Eq. (6.1) is d R da = (cid:18) − a/a (cid:63) − a/a p (cid:19) (cid:20) B ( a ) − a ( k − (cid:18) dB ( a ) da + B ( a )( a p − a ) (cid:19)(cid:21) . (B.4)(Superscripts “ ( k +1) ” on R and B ( a ) are omitted for brevity.) Except for the pre-factor, andthe fact that a p (not a (cid:63) ) replaces a ∗ in the last term, this equation is identical to (6.1).The derivation is as follows. As Q → B ( a ) function nearly vanishes at the pinch point a p and close to a p can be approximated by the form [14] B ( a ) ≈ η (cid:0) ( a − a p ) + δ (cid:1) , (B.5)where δ vanishes ∝ / | ln Q | as Q → η is some positive constant. The integrals I j ( a ) ofEq. (4.4) are dominated by a huge peak in their integrands around a p : I j ( a ) ≈ (cid:90) dx x j − ( η (( a − a p ) + δ )) ≈ a j − η π δ . (B.6) Note the slightly different notation ( (cid:63) instead of ∗ ) for infrared-limiting quantities according to whether theycorrespond to an unfixed or a fixed point. δ → B j ( a ) and hence the H j functions [14]. (Notethat the B ( a ) /a j − factor in Eq. (4.3) will involve the limiting value of a , which is a (cid:63) and not a p .)While the B j ’s and H j ’s diverge, the 1 /δ factors cancel out, as does η , in Eq. (4.13), leavingfinite limiting values for the optimized r m coefficients. Instead of Eq. (4.15) of the fixed-pointcase, we find s m a (cid:63)m = 1( k − (cid:18) a (cid:63) a p (cid:19) m m (cid:88) j =0 ( k − m − j − c j a j p − (cid:18) a (cid:63) a p (cid:19) m − m − (cid:88) j =0 ( k − m − j ) c j a j p , (B.7)where s m ≡ ( m + 1) r m . Using a dummy variable a we can then form the function S ( a ) = d R da = k (cid:88) m =0 s m a m . (B.8)Reorganizing the resulting double summation over m and j so that the latter becomes the outersummation, the inner summations become finite geometric series or derivatives thereof. Theouter j summation then produces terms that are B ( a ) or dB/da or B ( a p ) or dB/da | a = a p . Thelast two vanish in the infrared limit since a p is then a double zero of the B ( a ) function. Aftersome further algebraic tidying up the result reduces to Eq. (B.4) above.Note that the na¨ıve large- k limit of Eq. (B.4) is not the NLS condition (6.10) but d R da = (cid:18) − a/a (cid:63) − a/a p (cid:19) B ( a ) (NLS (cid:48) ) . (B.9)If we proceed in parallel with the analysis in Sect. 7 we find, instead of Eq. (7.3), (cid:90) dvv √ − v = (cid:90) duu (cid:115) − u/u (cid:63) − u/u p . (B.10)Note that the above equations correspond to the ansatz form used in the PWMR analysis ofSect. 8 with ξ replaced by ξ → (cid:115) − u/u (cid:63) − u/u p . (B.11)Doing the integrations, exponentiating both sides, and solving for v leads to v = 4 U (1 + U ) , (B.12)where U = (cid:18) u (cid:63) u p u (cid:63) − u p (cid:19) (cid:113) − u/u (cid:63) − u/u p − (cid:113) − u/u (cid:63) − u/u p + 1 (cid:32) √ u (cid:63) − u + √ u p − u √ u (cid:63) + √ u p (cid:33) (cid:113) u p u(cid:63) . (B.13)29ote that when u > u p (which is relevant since u ranges from 0 to u (cid:63) , which must exceed u p )this formula for U develops an imaginary part. However, recall that both v and B , B = (1 − U )(1 + U ) Uu (cid:115) − u/u p − u/u (cid:63) (B.14)(Cf. Eq. (8.10)), have to be expanded as series in u and then truncated after k terms, makingthem inevitably real.These formulas are hard to deal with, even at low orders, especially since u p and u (cid:63) have tobe determined by the requirements that the truncated B and its derivative vanish at the pinchpoint u p . For k = 2 , k it appears there is. Anticipating that both u p and u (cid:63) will tend to 1 as k → ∞ , we define δ ≡ u p − u (cid:63) (B.15)and proceed to expand to lowest non-trivial order in δ . This gives U ≈ u (cid:18) − δ − u ) (cid:19) , (B.16) v ≈ u (1 + u ) − δu (1 − u )(1 + u ) ln(1 − u ) , (B.17)and B ≈ − u (1 + u ) − δ (cid:18) u (1 + u ) + (1 − u + u )(1 + u ) ln(1 − u ) (cid:19) . (B.18)Remarkably, one can find analytic expressions for the truncated-series versions of v and B andthereby explore numerical results up to very high k values. These results (see table 7) show thatindeed there a valid solution (with u (cid:63) > u p ) exists with δ tending to zero as δ ∼ (2 / ln 2)(1 /k )and R (cid:63) /a tending to 1. k u p u (cid:63) δ v (cid:63) = R (cid:63) a
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