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    Serie de dyson Serie de dyson Document Transcript

    • RESEARCH REVIEW PROJECTConvergence of Perturbation Series in Quantum Field TheorySupervisor: Prof. James Stirling; Candidate Number: 6941V Abstract. Perturbation expansions are ubiquitous in quantum field theory. They are a standard tool for determining field theory pre- dictions to high accuracy. The purpose of this paper is to review considerations of perturbation series as mathematical entities: their radius of convergence; their large-order behaviour; what physical in- terpretation can be assigned to divergent perturbation series; and recent research on the renormalon divergence.1. IntroductionWe consider the perturbative expansion of a function f (g), ∞ f P T (g) ≡ f (0) + cn g n . (1) n=1Here g is the strength of an interaction, and f P T (g) − f (0) measures deviations to the stateof the noninteracting system within the radius of convergence of f P T . In quantum fieldtheory it is sometimes assumed that, for g small, f (g) = f P T (g). The justification is experi-ence: in the framework of the standard model, perturbative predictions have had enormoussuccess in predicting experimental results. However, agreement between experiment and theperturbation series truncated at the Nth power in g, where N is a small number, does notimply f (g) = f P T (g). Indeed, it is well worth knowing about the divergence of perturbationseries in quantum field theories because the divergence may eventually limit the accuracy ofpredictions, and because the limit of the perturbation series becomes ill defined. In Section2, Dyson’s argument for the divergence of perturbation series in Quantum Electrodynamics(QED) will be reviewed. In Section 3, large-order estimates of coefficients cn will be discussedfor the anharmonic oscillator in quantum mechanics and for various quantum field theories.Section 4 focuses on attempts at resumming divergent perturbation series in the frameworkof Borel summation. Recent research on the renormalon divergence is reviewed in Section 5.Conclusions are drawn in Section 6.2. The Dyson InstabilityIn 1952, Dyson published a paper [1] on the convergence of perturbation series in QED. He e2suggested that in a fictitious world with a negative QED coupling constant, α = 4π < 0;i.e., imaginary charge e, the vacuum is a metastable state. This gedanken experiment ismotivated by the fact that any function, expanded perturbatively around the point at whichthe coupling constant α is zero, is influenced by the negative coupling region. 1
    • In such a fictitious world, the Coulomb force is reversed, such that like-charged particlesattract one another, and oppositely charged particles repel. Dyson argues that one canthen construct a state constituted of N electrons in one region of space and N positronsin a separate region of space such that each particle’s rest- plus kinetic-energy is smallerthan the absolute value of its potential energy. This potential energy owes to the attractiveCoulomb force of the other N-1 like-charged particles in the vicinity. Dyson claims thatone can construct such a state without having to use very small distances or large chargedensities, so the classical Coulomb force is a valid approximation. In this world, the vacuumis no longer the state of lowest energy. Furthermore, in a quantum mechanical system, sucha state is always accessible through quantum tunneling. Hence there would be a non-zerochance for the vacuum to decay into such a state in any finite amount of time. Once thevacuum has decayed, the electron and positron regions would build a potential between themthat would facilitate further decay, resulting in a rapid disintegration of the vacuum.Following this reasoning it appears that a microscopic negative coupling introduced intothe system can have a macroscopic effect, which is an event that cannot be described per-turbatively. Since there is a finite probability of quantum tunneling from any state into adecayed-vacuum state in a finite amount of time, no function f (α) will be analytic when thecoupling constant is negative.It is a general property of the Taylor expansion for complex-valued functions that: 1) theregion of convergence is a circle centered on the expansion point; and 2) the function isanalytic within the entire region of convergence. Suppose f (α) has non-zero radius of con-vergence. Then for some range of imaginary α, f (α) is analytic, which contradicts the aboveargument. This shows that any perturbation series in QED has zero radius of convergence.Dyson claimed that the decay of the vacuum involves the interaction of many particles, andhence does not influence very small orders of the perturbation expansion. His approachof examining the properties of a field theory at small negative coupling is the foundationon which most large-order estimates of perturbation theory coefficients in field theories arebased. The Dyson instability is not exclusive to quantum electrodynamics. It is in fact ageneral feature of most quantum field theories (see Fischer [2] for details).3. Large-Order Estimates of cnIn the early 1970s, Bender and Wu [3, 4] derived the exact asymptotic behavior of theperturbation series coefficients for the Kth energy level of the one-dimensional anharmonicoscillator in quantum mechanics (in units h = 1) ¯ d2 x2 x4 + + λ − E K (λ) ψ(x) = 0. (2) dx2 4 4The coupling here is the strength λ of the quartic power perturbation to the harmonicoscillator Hamiltonian.Using carefully established analyticity properties of the energy of the oscillator, Bender andWu were able to link the nth -power perturbation coefficient cn to the imaginary part of the 2
    • Figure 1: Comparison of anharmonic oscillator potential at small positive and negativecouplingenergy at small negative coupling. This imaginary part of the energy arises because thereis a possibility of quantum tunneling out of the oscillator altogether; i.e., probability is notconserved even in the ground state of the oscillator (see Figure 1).Bender and Wu obtained the imaginary part of the energy for an arbitrary energy level via aWKB analysis of transmission through the potential barrier. Using a perturbative expansionaround the energy for λ = 0 ∞ 1 E K (λ) = K + + cK λn , (3) 2 n=1 nthe result is √ 12K 6 1 lim cK = (−1)n+1 3n Γ(n + K + ), (4) n→∞ n K!π 3/2 2where the Γ-function is a generalization of the factorial function g(n) = n! to non-integervalues. With increasing order of expansion, the coefficients of the perturbation series forthe energy asymptotically approach a factorially divergent series with alternating sign. Ben-der and Wu found agreement with results from a computer simulation to 150th order inperturbation theory and with the results of an alternative approach for the lowest energylevels.The extension of this approach to quantum field theories is non-trivial. Lipatov [5] showedthat the problem could be simplified by approximating the paths of the particles at smallnegative coupling by their corresponding classical paths with small quantum fluctuations.He calculated the degree of divergence of massless renormalizable scalar field theories. Hisapproach was extended by many others: Brezin, Le Guillou and Zinn-Justin [6] were ableto rederive and extend the result obtained by Bender and Wu for the anharmonic oscillator;Parisi [7] extended the method to fermion fields; Itzykson, Zuber, Parisi and Balian [8,9] 3
    • as well as Bogomolny and Fateyev [10] extended the result to QED. A discussion of large-order estimates in field theories, including Quantum Chromodynamics (QCD), is providedin Fischer [2] and references therein.The general feature is a factorial growth of the expansion coefficients at large order. Giventhis fast divergence of the perturbation series, we will next describe the manner by whichone might formally extract the exact physical quantity f (g) from the divergent perturbationseries f P T (g) via the method of Borel summation.4. Borel summation of the divergent perturbation seriesDivergence of a perturbation series signals that the quantity being studied is not analytic atthe point about which the expansion is being performed. The possibility that there exists abijective correspondence between f P T and f is not excluded by the divergence of the formerseries. In other words, as long as there is a unique mapping φ : f P T (g) → f (g), physicalresults to arbitrary precision can be obtained from f P T .The standard process of Borel summation is an attempt to obtain this information: ∞ φ : f P T (g) → h(g) = dt e−t B(gt), (5) 0where we have defined ∞ ∞ cn B(gt) = (gt)n using the cn of f P T (g) = cn g n . (6) n=0 n! n=0The motivation lies in the integral representation of the gamma function at integer points: 1 ∞ 1= dte−t tn . (7) n! 0Inserting this index-n dependent identity at each term in the sum of f P T (g) and moving thedivergent sum inside the integral (thereby changing the function, which is what we aim toachieve), we obtain the above mapping. Notice that the sum B(t) has improved convergenceby scaling cn down by a factor of n!. Under certain analyticity conditions on the functionf (g), this procedure yields a unique function h(g) whose perturbative expansion matchesf P T (g). We would then claim h(g) = f (g) and use non-perturbative results and experimentto verify our claim.Despite arguments that renormalizable theories are not in general Borel-summable, whichwe will come to later in this section, in particular cases Borel-summation has been usedsuccessfully. For example, Ogievetsky [11] in 1956 calculated the contribution of vacuumpolarization by a constant external magnetic field to the QED Lagrangian using Borel re-summation of the perturbation series. His resummed series agrees with the nonperturbativeresult found by Schwinger in 1951. Brezin, Le Guillou and Zinn-Justin [12, 13] used Borelsummation in condensed matter field theory to resum the divergent Wilson-Fisher expan-sion for critical phenomena. At the time they obtained the most precise theoretical valuesof the critical exponents for phase transitions in Ising-like systems. 4
    • Figure 2: In order to ensure uniqueness of the Borel transform h(g), there are differentrequirements on the minimum opening angle of the region of analyticity of f (g), dependingon the steepness of the divergence of f P T (g). Note that only the opening angle at the originmatters, not the extent of the domain. (a) A minimum opening angle of π for factorialdivergence; (b) a minimum opening angle 0 < θ(ρ) < π for divergence as (n!)ρ for ρ < 1;and (c) a minimum opening angle of 0 for divergence as (ln(n))nWhile the Borel transform works well for some problems, in QED and QCD one oftenfinds that the Borel sum B(gt) has singularities, which have to be integrated over to geth(g). Since the integration is along the positive real axis, these singularities introduce anambiguity through different possible choices of contour around these poles. In order to avoidthe poles and unambiguously define the Borel transform h(g), the function f (g) needs tobe analytic within a certain opening angle around the origin in the complex g plane. Inensuring convergence, there is a competition between the magnitude of the opening angleand the steepness of the divergence of the perturbation series. Nevanlinna[14] shows that fora factorially divergent series, the analyticity region of f (g) must have an opening angle of πat the origin. Weaker statements of Nevanlinna’s Theorem require for example a divergenceas slow as cn ∝ (ln(n))n for a wedge of zero opening angle at the origin (See Figure 2 fordetails). For an overview of Borel summation, see the review by Fischer [2] and referencestherein.’t Hooft [15] showed that the region of analyticity of any Green function in QCD is limited toa horn-shaped wedge of zero opening angle. He used the fact that analyticity in the couplingconstant is related to analyticity in momentum and, using a coupling parameter such thatthe Gell-Mann Low function had only two terms, proved that the renormalization groupequations impose the above constraint on the analyticity region. His result was verifiedby Khuri [16], who showed explicitly that the region of analyticity is independent of theparticular choice of coupling parameter. The horn-shaped analyticity region correspondsto choice (c) in Figure 2 and is related to a maximum divergence cn ∝ (ln(n))n for Borelsummability, which is not satisfied by QCD (see references in Fischer [2]).The zero opening angle is believed to be a general property of renormalizable theories. Inother words, the divergence of f (g) at the origin is too strong to allow for a reconstructionof the series via Borel summation. There are two alternative interpretations. AlternativeA: the problem lies with the Borel resummation technique, and there exists a different map 5
    • that formally recovers f (g) from f P T (g); viz., φ : f P T (g) → f (g) for g < gmax . (8)In this case, the problem is merely of a technical nature. The function f (g) could be uniquelydefined for small values of the coupling g and there would be no ambiguity in the theory.Alternative B: the perturbation series in QCD (or QED) could suffer from inherent ambigu-ities and therefore cannot be resummed uniquely even at small coupling. There is no way offormally defining the limit of the perturbation series without some non-perturbative input.The Borel representation h(g) and its poles provide the formalism in which many contem-porary authors (see for example Beneke [20] and Fischer [2]) discuss asymptotic estimates.We will now turn to the renormalon divergence arising from the contribution of a particulartype of diagram in renormalizable theories.5. Renormalon divergenceIn perturbation theory, the number of Feynman diagrams usually grows with the order ofexpansion, oftentimes as n!, where n is the order of expansion (see for example Jaffe [17]or references in Lautrup [19]). Giving each Feynman diagram an amplitude of order 1 andsumming over all diagrams at every order with no cancellation, we can naively imagine away in which the large-order divergence of the perturbation series predicted in Section 3,cn ∝ n!, could appear in the terms of the series.This picture may not work in renormalizable theories: Gross and Neveu [18], Lautrup [19]and ’t Hooft [15] found diagrams of a particular type whose contribution to the perturbationseries f P T at large order grows factorially with the order of expansion. Since this divergenceoccurs only in renormalizable theories, it has been termed the renormalon divergence. Insome cases, for example QCD, the divergence due to these diagrams is claimed to be stronger(see Beneke [20] and references therein) than that calculated by Lipatov’s method of usingthe saddle-point technique around classical solutions at small negative coupling.In diagrams contributing to the renormalon divergence in QED, photon lines are modifiedby insertions of a large number of lepton loops (pair creation and annihilation of leptons),see Figure 3. One may wonder how a small collection of all diagrams can produce a strongerdivergence than all diagrams taken together. There are two possible answers to this question: 1. The divergence of the perturbation series has been underestimated: Lipatov’s method of finding the divergence of f P T , due to all Feynman diagrams at each order of the perturbation series, relies on the saddle-point technique around the classical particle path and hence does not take into account contributions to the integral from the tail, far away from the classical path. It is possible that the integral over the tail yields a result which is comparable or bigger than the saddle-point approximation (see Figure 4). In this case, Lipatov’s method would not be applicable and a stronger divergence than that calculated by the saddle- point technique may be possible, allowing for a renormalon-type divergence of the total series. 6
    • (a) Figure 3: Renormalon diagrams in QED (b) Figure 4: An example of a function for whose contribution to f P T at large order n which the saddle-point technique around x0 is proportional to n! is inapplicable Dyson’s fundamental argument would still hold: the perturbation series is expanded around a non-analytic point owing to the instability of the vacuum at negative coupling. In this case one cannot, however, evaluate Green functions at quantum fluctuations near the classical path for the purpose of calculating large-order terms of the pertur- bation series. For an overview of papers whose authors advocate the significance of renormalon di- vergences, see Beneke’s review [20] and references therein. Beneke associates the renor- malon divergence with non-perturbative power corrections to the perturbations series, and claims that the magnitude of these non-perturbative corrections is comparable to the magnitude of resummed renormalon terms. 2. Suslov [21] argues that cancellation among diagrams prevents the renormalon diver- gence when all Feynman diagrams are taken into account and that Lipatov’s method for calculating large-order terms in the perturbation series is valid despite the faster divergence of a subclass of diagrams.In either case, the general claim put forth by Dyson is not questioned: the perturbationseries is divergent in most quantum field theories. We will now summarize what we havelearned.6. ConclusionsFischer [2] has posed three questions to quantify the relationship between f (g) and f P T (g): 1. Convergence: in what domain of g is f (g) = f P T (g)? Dyson’s argument [1], that the vacuum instability at negative coupling leads to non-analyticity of f (g) at the origin, leads us to answer: f (g) = f P T (g) only at the origin. 2. Truncation: what are the properties of the remainder functions, RN (g), defined in Eq. (9)? N RN (g) ≡ |f (g) − f (0) − cn g n |. (9) n=1 7
    • N Since we want to use the truncated perturbation series f (0) + cn g n to approximate n=1 f (g) at low coupling, RN (g) must be of order g N +1 or higher: RN (g) = O(g N +1 ) as g → 0. If this is true for every N, then f P T (g) is an asymptotic expansion of f (g). By definition, estimates of the remainder function have to be obtained from information outside of the perturbation series. We did not have space to review estimates of RN (g) here. See Fischer [2] for more details. 3. Uniqueness: how much physical information is contained in the set {cn } defined in Eq. (10)? ∞ f P T (g) ≡ cn g n (10) n=0 Can the function f (g) be unambiguously reconstructed from this set for small values of g, or is there a fundamental limit to the precision of the perturbation series? In other words, do we lose information by perturbing around the non-analytic origin? We have reviewed Borel resummation, and some of its successes. In general for renor- malizable theories, ’t Hooft [15] has argued that the method fails because of insufficient analyticity properties of the function f (g) in the vicinity of the origin.We have also learnt about Lipatov’s method of approximating large-order terms in the per-turbation series by evaluating Green functions, at negative coupling, only at the classicalpath and quantum fluctuations nearby. Applying this approach, we find that most quantumfield theories show a Γ(bn + c) divergence of the large-order coefficients with the order ofexpansion n, where b,c are constants.Finally, we have learned about a class of diagrams contributing to the particularly strongrenormalon divergence and we have encountered different interpretations of this divergence.It may or may not be true that Lipatov’s method of approximating the divergence of theperturbation series gives only a lower bound to the divergence of the series, yet this does notcontradict Dyson’s original argument for the divergence of the perturbation series.References[1] F. J. Dyson, Phys. Rev. 85 (1952) 631[2] J. Fischer, PRA-HEP 97/06 hep-ph/9704351 (1997)[3] C.M. Bender and T.T. Wu, Phys. Rev. Lett. 27 (1971) 461[4] C.M. Bender and T.T. Wu, Phys. Rev. D. 7 (1972) 1620[5] L.N. Lipatov, Pis’ma Zh. Eksp. Teor. Fiz. 25 No.2 (1977) 116-119[6] E. Brezin, C.J. Le Guillou and J. Zinn-Justin, Phys. Rev. D 15 (1977) 1544[7] G. Parisi, Phys. Lett. B 66 (1977) 382[8] C. Itzykson, G. Parisi, J-B. Zuber, Phys. Rev. D 16 (1977) 996 8
    • [9] R. Balian, C. Itzykson, G. Parisi, J-B. Zuber, Phys. Rev. D 17 (1978) 1041[10] E.B. Bogomolny and V.A. Fateyev, Phys. Lett. B 76 (1978) 210[11] V.I. Ogievetsky, Doklady Akademii Nauk 109 (1956) 919[12] E. Brezin, C.J. Le Guillou and J. Zinn-Justin, Phys. Rev. D 15 (1977) 1544[13] C.J. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95[14] F. Nevanlinna, Jahrbuch Fort.Math. 46 (1916−1918) 1463[15] G. t Hooft in: The Whys in Subnuclear Physics, Proc. Erice Summer School, 1977, ed. A. Zichichi (Plenum Press, New York, 1979), p. 943[16] N.N. Khuri, Phys.Rev. D 23 (1981) 2285[17] A. Jaffe, Commun Math. Phys 1 (1965) 127[18] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235[19] B. Lautrup, Phys. Lett. B 69 (1977) 109[20] M. Beneke, CERN-TH/98-233 hep-ph/9807443 (1998)[21] I.M. Suslov, J. Exper. and Theor. Phys. 100 (2005) 1188 9