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Fixed and Unfixed Points: Infrared limits in optimized QCD perturbation theory

Perturbative QCD, when optimized by the principle of minimal sensitivity at fourth order, yields finite results for R(e+e-)(Q) down to Q=0. For two massless flavours (n_f=2) this occurs because the couplant"freezes"at a fixed point of the optimized beta function. However, for larger n_f's, between 6.7 and 15.2, the infrared limit arises by a novel mechanism in which the evolution of the optimized beta function with energy Q is crucial. The evolving beta function develops a minimum that, as Q ->0, just touches the axis at a_p (the"pinch point"), while the infrared limit of the optimized couplant is at a larger value, a^star (the"unfixed point"). This phenomenon results in R approaching its infrared limit not as a power law, but as R ->R^star-const./|ln Q|^2. Implications for the phase structure of QCD as a function of n_f are briefly considered.


Introduction
Countless textbooks explain how key properties of a renormalizable field theory follow from a graph of its β function. Fig. 1, for instance, supposedly represents an asymptotically free theory with an infrared fixed point at a = a * . The problem, though, is that there is no such thing as "the β function." It is a myth that there is a unique β function characterizing a given theory. In fact, away from the origin, β(a) depends strongly on the arbitrary choice of renormalization scheme (RS); that is, it depends on the definition adopted for the renormalized coupling constant (couplant) a ≡ α s /π. While the first two terms of β(a) are unique, all the higher coefficients are RS dependent [1]. Whether or not the β function has a fixed point is an entirely RS-dependent question [1,2,3].
Renormalization-group invariance [4] means that any physical quantity, R, is, in principle, independent of the RS choice. However, finite-order perturbative approximants to R are RS dependent. The idea of "optimized perturbation theory" (OPT) [5] is to find -for a given R at a given energy Q and at a given order of perturbation theory -the "optimal" RS in which the perturbative approximant is locally invariant; i.e., stationary under small changes of RS. At second (next-to-leading) order this optimization is simply a precise formulation of the familiar and powerful idea that the renormalization scale µ should not be kept fixed but should "run" with the experimental energy scale Q. At higher orders, though, the optimization procedure also determines optimal values for the higher-order β-function coefficients, and these evolve as the energy Q is changed. Thus, the optimized β function itself evolves.
Hitherto this last point had seemed -even to this author -a technicality, unlikely to overthrow the basic picture that a finite infrared limit in QCD only occurs if "the β function" has a fixed point. Such a fixed-point limit of OPT was analyzed in Ref. [2] and was later found to occur in QCD in the third-order R e + e − case [6,7]. The recent calculation of the fourth-order correction to R e + e − [8] has allowed us in Ref. [9] to investigate OPT at fourth order. For the phenomenologically relevant case of two massless flavours (n f = 2) we again found fixed-point behaviour with the couplant freezing to a modest value, with the third-order result 0.3 ± 0.3 [7] now refined to 0.2 ± 0.1 [9]. See Fig. 2. Continuing our investigation to higher n f values, however, produced a surprise: a finite infrared limit in OPT can also occur by a quite different mechanism in which the evolution of the β function plays an essential role. This "pinch mechanism" produces an extreme "spiking," rather than a "freezing," of the couplant as Q → 0; see Fig. 3. The main purpose of this paper is to describe the pinch mechanism and to present numerical results for the infrared limit as a function of n f .
In discussing the infrared behaviour of perturbation theory in QCD, one must of course recognize that the results are not directly physical. There exist large nonperturbative, highertwist terms that perturbation theory is completely blind to. However, it is a longstanding idea [10,11] that perturbation theory corresponds to some kind of average over hadronic resonances.
As shown in Ref. [7], the low-energy e + e − data agrees very nicely, in this sense, with the prediction of OPT that the couplant freezes to a modest value. Moreover, there is a wealth of phenomenological evidence for freezing. 1 We therefore believe that there is some real-world relevance to studying the infrared behaviour of perturbation theory.
It is also interesting theoretically to consider QCD with n f flavours of massless quarks for various n f . One reason is to compare with extrapolations from n f = 16 1 2 (the Banks-Zaks (BZ) or "small-b" expansion [13]), or from n f = −∞ (the "large-b" approximation [14,15]). The other reason is the whole issue of the phase structure of QCD, and other gauge theories, as a function of n f [16] - [20].
We begin by briefly reviewing the key ingredients of OPT [5] in Sect. 2. (See Ref. [9] for a fuller account.) Sections 3 and 4 respectively describe the fixed-point and pinch mechanisms whereby fourth-order OPT produces a finite Q = 0 limit. Numerical results are presented in Sect. 5. Sect. 6 describes the approach to the Q = 0 limit and Sect. 7 briefly discusses the possible implications of the results.

Optimized perturbation theory
The β function, of some general RS, is written as Recently it was pointed out that a fixed point in the n f = 3 theory provides a simple and appealing explanation of the ∆I = 1 2 rule for Kaon decays [12].
TheΛ parameter in τ arises as the constant of integration in the integrated β-function (int-β) This form of K(a), completely equivalent to our previous definition [5,9], is more convenient when c may be negative. TheΛ parameter thus defined is RS dependent, but it can be converted between different schemes exactly by the Celmaster-Gonsalves relation [23].
The physical quantity considered here is the e + e − hadronic cross-section ratio R e + e − (Q) at a total c.m. energy Q. Neglecting quark masses, this has the form R e + e − = 3 q 2 i (1 + R) with R = a(1 + r 1 a + r 2 a 2 + r 3 a 3 + . . .). (2.7) (Actually, in this paper we will consider only the "non-singlet" part of R; that is, we drop the terms proportional to ( q i ) 2 . This makes very little difference for 0 ≤ n f ≤ 6 and allows us to discuss higher n f 's without needing to specify the electric charges of the fictitious, additional quarks.) Since it is a physical quantity, R satisfies a set of RG equations [5] ∂R where The B j (a) functions have expansions that start 1 + O(a). (Note that for j → 1 + one naturally finds B 1 (a) = B(a).) The RG equations (2.8) imply that certain combinations of R and β-function coefficients are RS invariants. Up to fourth order these are: The numerical values of theρ 2 ,ρ 3 invariants (see Table 1 below) can be obtained from the MS calculations of c 2 , c 3 [24,25] and r 1 , r 2 , r 3 [26,27,8], with all dependence on the arbitrary MS choice dropping out.
Q dependence enters only through ρ 1 (Q), which can be conveniently expressed as whereΛ R is a characteristic scale specific to the particular physical quantity R. It can be related back to the traditionally defined Λ MS parameter by the exact relation (2.14) Note that the infrared limit corresponds to ρ 1 (Q) → −∞.
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. Because of these truncations, the resulting approximant depends on RS. "Optimization" [5] corresponds to finding the stationary point where the approximant is locally insensitive to small RS changes, i.e., finding the "optimal" RS in which the RG equations (2.8) are satisfied by R (k+1) with no remainder. The resulting optimization equations [5] were recently solved for the optimizedr m coefficients [9].
[A note about notation: An overbar can be used to explicitly distinguish optimized from generic quantities, but we shall generally omit these below, leaving it understood that all quantities are the optimized ones at (k + 1) th order. The one exception is the symbol "a," which we employ merely as a dummy argument. Thus, we can discuss "the β(a) function" in basically the traditional sense as a function of a single variable, a, with definite coefficients, c j =c j ; the key difference, though, is that thec j coefficients will themselves evolve as the energy Q changes.] The optimized r m coefficients are given in terms of the optimized couplantā and the optimized c j coefficients by [9]: H 1 is to be understood as the limit i → 1 of the above formula, and H 0 ≡ 1 and H k+1 ≡ 0. At fourth order (k = 3) the H's are explicitly given by 17) and the optimized r m coefficients are given by The optimized r m and c j coefficients must also be constrained to yield theρ n invariants of Eq. (2.12). The iterative algorithm outlined in Ref. [9] can be used to solve numerically for the optimized coefficients, and thereby obtain the optimized result, at any given Q value. In the Q → 0 limit these steps can be carried out analytically, as discussed in the next two sections.

Fixed-point mechanism
A finite Q → 0 limit for R(Q) can occur by essentially the familiar fixed-point mechanism, with the optimized B(a) function manifesting a simple zero at a = a * (see Fig. 4). The limiting behaviour can be analyzed as follows [2]. For a close to a * one can linearize B(a) as where σ is some positive constant (directly related to γ * , the slope of the β function at its fixed point; γ * = ba * 2 σ). The integrals I j (a) of Eq. (2.11) will then diverge in the infrared limit, a → a * : . This result corresponds to ∂a * /∂c j → a * j /σ, which indeed follows directly [2] by taking ∂/∂c j (with the other c i 's held constant) of the (k + 1) th -order fixed-point condition The slope parameter σ is given by At fourth order (k = 3) one obtains By substituting in the definitions ofρ 2 ,ρ 3 , Eq. (2.12), one can then find c * 2 , c * 3 in terms of a * and those invariants. The fixed-point condition above can then be expressed entirely in terms of invariants as [2] 83 64 The relevant a * is the smallest positive root of this equation. The final result for the limiting value of R at fourth order can then be simplified to [2] R * = a * 249 256 Eq. (3.7) turns out to have no acceptable root when n f is 7, . . . , 15.
(For n f = 15 there is a positive root but it gives a negative slope σ, which is unphysical.) Nevertheless, going to ever lower Q values with the optimization procedure, one does find that the optimized result remains bounded as Q → 0. How this happens is the topic of the next section.

Pinch mechanism
The essence of the pinch mechanism is illustrated in Fig. 5, which shows the evolution of the optimized B(a) function in the n f = 8 case. As Q is lowered the optimized c 2 , c 3 coefficients change so that B(a) develops a minimum -which, in the limit Q → 0, just touches the horizontal axis at a "pinch point," a p . Although this point is then a double root of B(a) = 0, it does not represent a fixed point. The infrared-limit of the optimized couplant is not a p but a larger value, a ⋆ , dubbed the "unfixed point" to stress that it is not a zero of the β function. One can understand this infrared behaviour analytically as follows. B(a) can be approximated around its minimum (at, or nearly at, the pinch point a p ) by where δ → 0 as Q → 0 and η is some positive constant. Thus the integral for the K(a) function in Eq. (2.6) becomes dominated by a "resonant peak": Therefore, in the Q → 0 limit (where ρ 1 (Q) = K(a)−r 1 tends to −∞), the δ parameter vanishes The integrals I j (a) of Eq. (2.11) are also dominated by a huge peak in their integrands around a p : At fourth order (k = 3) one finds The infrared limit of the fourth-order B(a) function is The pinch point a p is where this function touches the a axis (see Fig. 5) and hence satisfies the two equations B ⋆ (a) = 0 and dB ⋆ /da = 0 at a = a p . These two equations yield These two equations determine a p and a ⋆ in terms of the invariants c,ρ 2 ,ρ 3 . One can manipulate these equations to find a p in terms of a ⋆ as with a ⋆ given by a 6th-order polynomial equation: The final result for the infrared limit of R at fourth order can be expressed as Note that a ⋆ ≥ a p is needed for this solution to be relevant. One can check that the special case a ⋆ = a p is indeed the boundary between the pinch mechanism and the fixed-point mechanism, and corresponds to where γ * = 0. From such an analysis one can determine the precise n f values where the switchover from one mechanism to the other takes place.
It is possible, in principle, for the pinch mechanism to occur at third order; see Appendix A.

Numerical results
The inputs to our numerical calculations are collected in Table 1, which lists the RS-invariant quantities c,ρ 2 ,ρ 3 for integer n f from 0 to 16. These values are obtained from the Feynmandiagram calculations of Ref. [8] and earlier authors [22], [24]- [27]. (The singlet terms, proportional to ( q i ) 2 have been dropped.) Table 2 gives our results for the infrared limit of R for n f = 0, . . . , 16. The quoted error estimate on R corresponds to the last term, r 3 a 4 , of the truncated perturbation series, evaluated in the optimized RS [7,9]. Also listed are values of the fixed-point, or the unfixed-point and pinch-point. (The critical exponent γ * will be discussed later.) The fixed-point mechanism operates for n f < 6.727, then the pinch mechanism takes over until n f = 15.191, when the fixed-point mechanism returns and operates until n f = 16 1 2 when a * → 0. Table 3 gives the optimized coefficients, weighted by the appropriate power ofā, in both the β-function and R series. This information is important for anyone wishing to check our results and also displays the behaviour of the truncated series for both R and B(a). This behaviour is, at best, only marginally satisfactory: Clearly, by going to the Q → 0 limit we are pushing low-order perturbation theory well beyond its comfort zone. Nevertheless, all things considered, we believe that the results are credible within the large uncertainties quoted in Table 2 and illustrated in Figs. 2 and 3. In particular, we believe that the dramatic Q → 0 spike produced by the pinch mechanism is real; the very large error estimate just cautions that the height of the spike is very uncertain; it might be somewhat smaller, or it might well be considerably bigger. 0.013 ± 0.001 0.001 Table 2: Infrared-limit results for R e + e − (non-singlet) in OPT at fourth order for different n f values. For n f = 0, . . . , 6 and n f = 16 the limit is governed by a fixed point at a * : For n f = 7, . . . , 15 it arises from the pinch mechanism, with an "unfixed point" at a ⋆ and a "pinch point" at a p . (The a ⋆ equation has solutions outside this range, giving the values in parentheses, but these violate the a ⋆ > a p requirement. Also, the fixed-point equation has a solution for n f = 15, but one that violates the γ * ≥ 0 requirement.) The last column gives values for the critical exponents γ * which characterize the power-law approach of R to its fixed-point limit; Table 3: Terms in the optimized β-function and R series in the infrared limit (a = a * or a ⋆ , as appropriate).  where the pinch mechanism produces really dramatic spiking of R as Q → 0, as seen in Fig. 3 for n f = 8. If, instead of R * , we had plotted R(Q) for some low, but finite Q -say around 1 2Λ R -the bump would not have appeared and the points would have been close to the smaller, fainter points.
Those smaller points are the infrared-limiting results in the FAC (fastest apparent convergence) or "effective charge" [28] scheme. That scheme is defined such that all the r m coefficients vanish, giving R = a FAC (1 + 0 + 0 + . . .). The FAC β function's coefficients then coincide with theρ n invariants (and so can be read off from Table 1). Since those coefficients do not evolve with Q, the infrared limit in FAC is simply obtained by finding the fixed point of the FAC β function. Many authors (e.g. [29]) have observed that, at low orders, FAC seems to yield very similar results to OPT. That observation holds true here, certainly at low n f and close to n f = 16 1 2 . It also holds in the range 7 n f 13 at energies Q 1 2Λ R , as noted above. However, the FAC scheme does not see the extreme spiking at Q = 0. While it is still true, because the error estimates (see Table 2) are so large in this region, that OPT and FAC infrared results agree within the error estimate, it is fair to say that the presence or absence of the spike is a qualitative difference in the predictions of the two schemes. (We expect other distinct differences between OPT and FAC to emerge at higher orders since the FAC β function is almost certainly factorially divergent, whereas an "induced convergence" mechanism is conjectured to operate in OPT [30].) 6 Approach to the Q = 0 limit Proper analysis of the subleading terms governing the approach to the Q = 0 limit, in both the fixed-and unfixed-point cases, is surprisingly subtle and intricate. We postpone details to a future publication and report here only the main results.
The usual lore is that the approach to a fixed point is described by a power law with a critical exponent given by the slope of the β function at the fixed point: The derivation of this result, in a fixed RS, and the proof that β ′ (a * ) is invariant under RS changes [31] is subject to some caveats -which, as Chýla [3] has rightly pointed out, are not necessarily to be viewed as very rare exceptions. 2 In OPT it is far from obvious that the above result will hold because the optimized couplant and optimized r m and c j coefficients have ǫ ln ǫ corrections as they approach their fixed-point limits, where ǫ ≡ B(a). Remarkably, though, the ǫ ln ǫ terms cancel in R, leaving in (k + 1) th order. From the int-β equation, (2.5), together with the ρ 1 (Q) definition in Eqs. (2.12), (2.13), one sees that ln ǫ = γ * ln Q + const., (6.3) so that ǫ, and hence R * − R, is proportional to Q γ * where γ * is the slope of the optimized β function at its fixed point: that is, γ * = ba * 2 σ with σ given by Eq. (3.5), evaluated in the optimized scheme. At fourth order this corresponds to The numerical γ * values are reported in Table 2. Note that γ * is around 2 or 3 for 0 ≤ n f ≤ 6, so the resulting low-Q behaviour (Fig. 2) is appropriately described as "freezing" of the couplant. 2 Regarding some other comments in Chýla's paper, note that it was written before the correct result for r MS 2 [27] was published. An earlier, incorrect result had made it seem thatρ2 was large and positive, so that third-order OPT apparently failed to yield a finite infrared limit, unlike the actual situation [6,7].
However, when γ * is very small, as in the n f = 16 case, one sees instead "spiking" at Q → 0, though not quite as extreme as the logarithmic spiking produced by the pinch mechanism.
In the unfixed-point case we find, after a lengthy calculation, the simple result: From the int-β equation and ρ 1 (Q) definition we obtain (as in Eq. (4.2)) Therefore One way to look at the result is to note that for R close to R ⋆ . Thus, the "physically defined β function" associated with R is predicted by OPT to have neither a simple nor a double zero, but something in between. An even more intriguing interpretation is to see the low-energy prediction as with λ ∼ 1/(b ir ln Q) viewed as the running coupling constant of some infrared effective theory whose β function starts b ir λ 2 (1 + . . .).

Discussion
We now briefly discuss the implications of our results. The abrupt change between n f = 6 and n f = 7 seems indicative of a phase transition. For n f ≤ 6 the phase is presumably the one we are familiar with in the real world; colour is confined and chiral symmetry is broken, with the associated goldstone bosons (pions) being massless. Vector mesons (ρ's, etc.) have masses of orderΛ and their resonant contribution dominates e + e − → hadrons at low energies. Although the actual R e + e − ∝ 1 + R is very different from the smooth perturbative prediction (Fig. 2), the two agree well after Poggio-Quinn-Weinberg (PQW) smearing [11] is applied to both [7].
For n f > 7 the effective low-energy theory seems to be a renormalizable theory with a mass scale appearing only in logarithms. The extreme spiking of R as Q → 0 (Fig. 3), if viewed as a resonant peak in the vector channel, hints that massless vector bosons are now present. These might be the gluons of an unconfined phase, or they might be massless, colourless vector mesons of a confined phase, perhaps with unbroken chiral symmetry. It is beyond the scope of this paper to attempt a detailed comparison with the literature, but we do see some points of resemblance with other approaches [16,17,18] and with some firmly established results in supersymmetric QCD [19,17]. There is also a large literature on lattice Monte-Carlo studies of QCD at large n f values (for recent work, see [20]).
A quick look back at third-order OPT results is in order. There the results for R * decreased roughly linearly from 0.4 to 0 as n f increased from 0 to 16 1 2 (see Fig. 1 of Ref. [32]). Since the uncertainties were large (∼ 100% at low n f ), sizeable changes at fourth order were not unexpected. Nevertheless, it is an interesting surprise to find qualitatively different features -particularly the spiking phenomenon produced by the pinch mechanism, responsible for the prominent bump around n f ≈ 9 in Fig. 6. Previously, the good agreement of the third-order results with the leading 16 1 2 − n f (BZ) expansion led us to suggest [32] that that expansion might remain good down to very low n f . That suggestion no longer seems tenable. We would now expect the BZ expansion to break down around n f ∼ 9, if not sooner.
The fact that the fourth-order results show a rise of R * with n f at low n f is interesting. At third order the OPT fixed-point equations would give R * ∼ 2.19/b in the n f → −∞ (b → ∞) In closing we would like to stress that the results in this paper are directly the result of applying the method of Ref. [5] to the Feynman-diagram results for R e + e − . We have invented nothing new, nor tweaked the method in any way. The freezing or spiking, depending on n f , is just what happens when one solves the optimization equations [5] at ever lower Q values.
Achieving a finite infrared limit was no part of the motivation for OPT (and was never considered in Ref. [5]), so the fact that it happens is a genuine prediction -and a non-trivial one, as history shows. 3 The "pinch mechanism" (Fig. 5) is another remarkable consequence of OPT. It has serious implications beyond perturbation theory, because it suggests that the phase structure of QCD may not be understandable in the traditional language of fixed points of "the β function." 3 see the previous footnote