The Quark-Hadron Duality

QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003

M.A.Shifman

William I. Fine Theoretical Physics Institute, University of Minnesota 116 Church Str. S.E. Minneapolis, MN 55455, USA

I review the notion of the quark-hadron duality from the modern perspective. The proper theoretical framework in which the problem can be formulated and treated is Wilson’s operator product expansion (OPE). Two models developed for the description of duality violations are considered in some detail: one is instanton-based, another resonance-based. The mechanisms they represent are complementary.

Prelude

Curiously, this is the tenth talk I give in Italy, since I came here for the ﬁrst time in the late 1980’s. The Soviet Empire was still in existence, albeit its ideo- logical foundation — “scientiﬁc communism”1 —was already tumbling. It was not just Italy, but a charmed corner — Capri, the land of fairy tales — where, by a perverted twist of fortune, I found myself in 1988 with Vladimir Naumovich Gribov.

This magic word, Capri, was familiar to every So- viet schoolboy. Our school lessons of Russian literature were ideologically charged. We spent much time studying the heritage of the “greatest proletarian writer of all times,” Maksim Gorky. In 1906 Gorky “... The sun melts in the blue midday sky, pouring settled in Capri where other proletarian leaders, such its hot rainbow-hued rays on to see and earth. The as Lenin, used to come for a vacation. He spent many drowsy see exhales an opalescent mist, the blue water years in Capri, with intermissions, until in 1931 he was gleams like steel, and a strong scent of brine is wafted finally lured by Stalin back to the USSR, “worker’s ashore. The waves plash lazily against the grey boul- paradise.” Apparently, human feelings were not alien ders, spill over their backs and on to the whispering to the proletarian writer, despite his preoccupation pebbles; they are small waves, as transparent as glass with the socialist idea. He fell in love with Capri and and untouched by foam. A purple haze enwraps the Italy at large; as a result of this love affair twenty seven mountain, the grey olive leaves are like old silver in the Tales of Italy appeared in 1906 – 1917 presenting a sunlight, the dark velvety green of the gardens terrac- strange mixture of socialist propaganda with stories of ing the hills is lit up by the golden glow of lemons and larger-than-life human passions and colorful sketches oranges, the scarlet pomegranate blossoms smile their of Capri, the Bay of Naples, Sorrento and other roman- vivid smile and there are flowers everywhere...” tic places, drenched in golden sunlight, with craggy “... The rocky coast descends in jagged ledges down cliffs and sweet scent of flowers wafting from the hills. to the see; it is covered with a luxurious tangle of He cherished the breath of antiquity which blurred the dark vines, lemons and figs... Flowers, gold, red and distinction between the past and the present. white, smile gently through the dense foliage dropping steeply down to the sea, and the yellow and orange I do not know whether it was in the curriculum, or our fruits remind one of stars on a warm moonlight night literature teacher was just overzealous, but we had to when the sky is dark and the air is moist...” learn a couple of Gorky’s tales by heart, and some pas- “... The cicadas are strumming. It is as if thousands of sages produced such a strong impression on me that I metal strings were stretched taut among the thick fo- remembered them once and for all. liage of the olive trees, the wind stirs the brittle leaves, 1This combination of words is, of course, oxymoronic, but that’s they touch the strings and this light, ceaseless contact the way I learned it at school. fills the air with intoxicating sound. It is not exactly 2 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 music, yet it seems as if invisible hands were tuning hundreds of invisible harps, and one waits in tense ex- pectancy for the tuning to cease, and for a grand string orchestra to strike up a triumphant hymn to the sun, sky and sea...” “... A bluish mist heavy with the sweet scent of the golden furze floats skyward from the rocky island. Standing in the midst of the dark mass of sleepy water under the pale cupola of the sky, the island is like an Figure 1. An illustration of duality: a fragment of Es- altar to the Sun God...” cher’s “Sky and Water.”

I kept thinking: “was it on my planet? and if Gorky, [1]. As I have mentioned, this aspect of QCD is ex- the proletarian writer, could see this, why couldn’t tremely important in a wide range of practical applica- I?” The surrounding reality and my everyday experi- tions — e+e− annihilation, deep inelastic scattering, ences taught me that that was a poetical exaggeration. τ And even if there was some truth in this description, I jet physics, decays, inclusive heavy flavor decays, to name just a few. At the same time it is rarely dis- would never ever become a part of it. So, it was better cussed by theorists; bits and pieces that you can find to forget ... but I couldn’t. in textbooks reflect the understanding of this subject My journey to Italy took thirty years. When I fi- of twenty years ago, at best. A considerable progress nally descended on Capri for the first time in 1988 I was made in the understanding of the quark-hadron instantly realized there was no exaggeration, it was duality in the 1990’s [2–10]. But then it again fell in all real — sweetly intoxicating scent of flowers, melt- neglect. The organizers of this Workshop invited me ing sun, melodic Italian speech (I did not understand to review the subject. At first I hesitated whether to a word, but I heard in them the voice of a vio- accept it, due to the lack of progress in the last three lin, and, sometimes, a passionate cello), washed-out years. But then I decided that QCD@Work 2003, jagged rocks, dark-eyed curly-haired Italian women, (with the emphasis on “Work,” right?) provides an with mystery in their eyes, undoubtedly the most excellent forum for resurrecting this topic. My task beautiful in the world... is to get you involved, so that you could develop the subject further. I came to Capri for the second time in 1994; in a sense this encounter brought even more excitement in my life. This is a separate story, however. 2 Early inception and a long latent period

1 Duality Quantum chromodynamics (QCD) is a very strange theory. All theoretical calculations are done in terms It is time to interrupt my recollections and turn to of quarks and gluons. At the same time, quarks and physics which can be no less captivating. Dictionaries gluons are never detected experimentally. What is define duality as something with seemingly contradic- actually produced and detected in experimental de- tory qualities which are in fact manifestations of one vices are hadrons: pions, kaons, protons, etc. The and the same phenomenon seen from complementary quark-hadron duality allows one, under certain cir- angles. With Seiberg’s duality, Seiberg-Witten’s dual- cumstances, to bridge the gap between the theoretical ity, string dualities, and so on, it is obvious that our predictions and experimentally observable quantities. age is the age of duality. I will discuss today a partic- The idea was first formulated at the dawn of the QCD ular topic going under the name of quark-hadron du- era by Poggio, Quinn and Weinberg [10], who sug- ality — the one whose understanding is most needed gested that certain inclusive hadronic cross sections by our experimental colleagues. The notion of the at high energies, being appropriately averaged over an quark-hadron duality exists in high-energy physics for energy range, had to (approximately) coincide with 30 years or so. Apparently, artistic imagination dis- the cross sections one could calculate in the quark- covered it much earlier. The above statement is il- gluon perturbation theory. In spite of the dramatic lustrated by Fig. 1 exhibiting a fragment of Escher’s developments in QCD in the subsequent two decades “Sky and Water.” Here the gaps between the birds are the notion of the quark-hadron duality remained at the fishes, while the gaps between the fishes are birds. 1976 level, with the very basic questions unanswered. These questions are: The last time I delved in this problem in earnest was in 2000 when I was working on Handbook of QCD • What energy is considered to be high enough for the QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 3 quark-hadron duality to set in, and what accuracy is q q to be expected? • What weight function is appropriate for the averaging of the experimental cross sections? Figure 2. The one-loop graph determining the polariza- • If the theoretical prediction includes only perturbation operator and the spectral density in the leading (par- tion theory, should one limit oneself to some particular ton) approximation. The “photon” momentum is denoted order in the αs series? by q. • Do we have to include known nonperturbative effects (e.g. condensates) in the theoretical prediction? (In fact, this is the isovector part of the actual electro- • Given a definition of the quark-hadron duality, can magnetic current). We define the two-point function one estimate deviations from duality and how? Πµν Of course, not all of these questions are independent. iqx 4 † For instance, answering the last question, one will si- Πµν = i e d x0|T {Jµ(x)Jν (0)}|0 . (2) multaneously learn the boundary energy. Systematic explorations of these and related issues Here q is the total momentum of the quark-antiquark started in earnest [2] in 1994. As in many other in- pair. Due to the current conservation Πµν is transver- stances, this was dictated by practical needs. Pre- sal, − 2 2 viously, the accuracy of the experimental data on Πµν =(qµqν q gµν )Π(q ) . (3) hard inclusive processes was rather modest, so that The experimentally observable quantity is the imagi- the Poggio-Quinn-Weinberg prescription was good nary part of Π(q2) at positive values of q2 (i.e. above enough. By 1994 the data, mostly associated with the physical threshold of the hadron production), the the b quark physics, became so precise and the ques- spectral density, tions raised so pressing, that a much better theoretical 12π understanding became imperative. ρ(s)= Im Π(s) ,s≡ q2 . (4) Nc In the problem of quark-hadron duality one cannot expect help from lattices. The duality violation is a Up to a normalization, the expression above coin- phenomenon inseparable from the Minkowskian kine- cides with the cross section of e+e− annihilation into matics; numerical Euclidean approaches, such as lat- hadrons measured in the units σ(e+e− → µ+µ−), the tice QCD, have nothing to say on this issue. Analytic famous ratio R. methods are needed. Theoretically one can calculate Π(q2) in the deep Eu- I will discuss the modern formulation of the problem clidean domain, at negative q2. For instance, from the based on Wilson’s operator product expansion (OPE). free quark loop of Fig. 2 one gets Then I will review models which were designed to give Nc us a certain idea of (and a degree of control over) Π(Q2) →− ln Q2 ,Q2 ≡−q2 . (5) the deviations from duality. There are two classes 12π2 of such models: instanton-based and resonance-based. Performing the analytic continuation to the They are complementary to each other. Finally, I will Minkowski domain and taking the imaginary part present sample applications in the processes of the cur- one arrives at rent interest, such as the hadronic τ decays. ρ(s)theor → 1 ,s→∞. (6) 3 The quark-hadron duality: what does that This is the spectral density in the theory with free mean? quarks, i.e. αs = 0. There are various corrections to the free quark result (5). The perturbative gluon 2 exchanges give rise to the αs(Q ) series. Nonpertur- Let us consider an idealized theory — QCD, with two bative (power) corrections come from the quark and massless quarks, u and d. We are interested in the + − gluon condensates and from other sources, e.g. the total hadronic cross section of the e e annihilation. small-size instantons. A systematic method of han- The “photon” in our theory is idealized too, its cou- dling the theoretical calculations of Π(Q2)inthedeep pling to the quark current has the form Euclidean domain is provided by Wilson’s operator uγ¯ µu − dγ¯ µd product expansion (OPE) [11]. Essentially, this is a Jµ = √ . (1) 2 bookkeeping procedure: one consistently separates the 4 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 short-distance contributions (i.e. those coming from have to be truncated at some finite order. A few distances ≤ µ−1) from the large distance contribu- lowest-dimension condensates that can be captured, tions (i.e. those coming from distances ≥ µ−1). Here are known approximately. The best we can do is ana- µ is a theoretical parameter (usually referred to as the lytically continue the truncated theoretical expression, normalization point) separating the two domains. The term by term, from positive to negative Q2.Foreach choice of µ is a matter of convenience — observable term in the expansion the imaginary part at positive quantities do not depend on it. q2 (negative Q2) is well-defined. We assemble them together and declare the corresponding ρ(s)theor to be The short-distance contributions determine the coeffi- dual to the hadronic cross section ρ(s)exp. In the given cients Cn(q) of OPE, context “dual” means equal. 2 D(q )= Cn(q; µ)On(µ) , Let me elucidate this point in more detail. Assume 2 2 4 n that Π(Q ) is calculated through αs and 1/Q , while 2 2 2 2 3 6 D(Q ) ≡−(4π )Q (dΠ/dQ ) . (7) the terms αs and 1/Q (with possible logarithms) are dropped. Then the theoretical quark-gluon spectral The normalization point µ is indicated explicitly. The density, obtained as described above, is expected to sum in Eq. (7) runs over all possible Lorentz and coincide with ρ(s)exp, with the uncertainty of order 3 3 gauge invariant local operators built from the gluon O[(αs(s)) ]andO(1/s ). The uncertainty in the the- and quark fields. The operator of the lowest (zero) di- oretical prediction of this order of magnitude is natural mension is the unit operator I, followed by the gluon since terms of this order are neglected in the theoret- 2 2 condensate Gµν , of dimension four. The four-quark ical calculation of Π(Q ). If the coincidence in this condensate gives an example of dimension-six opera- corridor does take place, we say that the quark-gluon tors. prediction is dual to the hadronic spectral density. If there are deviations going beyond the natural uncer- At short distances QCD is well described by the quark- tainty, we call them violations of duality. Needless to gluon perturbation theory. Therefore, as a first ap- 2 say that, once our calculation of Π(Q ) becomes more proximation, it is reasonable to calculate Cn pertur- −1 precise, the definition of the “natural uncertainty” in batively, in the form of expansion in αs(Q) ∼ (ln Q) . ρ(s)theor changes accordingly. This certainly does not mean that the coefficients Cn are free from nonperturbative (nonlogarithmic) terms. This is the most clear-cut definition I can suggest. The latter may and do appear in Cn’s; they are of the From the formal standpoint, it connects the duality type ∼ Q−γ where γ is a positive number, not neces- violation issue with that of analytic continuation from sarily an integer. Such terms in Cn’s are generated, for the Euclidean to Minkowski domain. Negligibly small instance, by the small-size instantons. Another source corrections (legitimately) omitted in the Euclidean of the power terms in Cn, of a technical rather than calculations may and do get enhanced in Minkowski. dynamical nature, is the normalization point µ: in cal- Before I proceed to explain, from the physical stand- culating Cn(µ) one must remove all soft contributions point, where the violations of duality come from, and with off-shellness less than µ. their pattern, I would like to make a remark impor- The condensate terms in Eq. (7) give rise to correc- tant in the conceptual aspect. The necessity of trun- n tions of the type (ΛQCD/Q) where n is an integer ≥ 4, cation of the αs and condensate series is not just due the (normal) dimension of the operator On, modulo to our practical inability to calculate high-order terms. logarithms associated with the anomalous dimensions. (For the vast majority of the theorists, myself includ- ΛQCD is the scale parameter of QCD (sometimes, I ing, “high-order” begins at the next-to-next-to-leading will drop the subscript “QCD” for brevity) entering level.) Assume we have a “sorcerer’s stone” which through the matrix elements of On’s. would allow us to exactly calculate any term in the 2 expansion we want. Still, we would not be able to find If one could calculate Π(Q ) in the Euclidean domain 2 Π(Q )exact because the both series are factorially di- exactly, one could analytically continue the result to vergent. The αs series has at least two known sources the Minkowski domain, and then take the imaginary of the factorial behavior: first, the number of the Feyn- ρ s theor part. The spectral density ( ) obtained in this man graphs grows factorially at high orders [12]; sec- exact way would present the theoretical prediction for ond, there are graphs with renormalons [13] (of which There would the measurable hadronic cross section. we will not bother in what follows since the infrared be no need in duality. renormalons — the only ones which are potentially In practice, our calculation of Π(Q2) is approximate, dangerous — are totally eliminated by the introduc- for many reasons. First, nobody is able to calcu- tion of the normalization point µ). The condensate 2 late the infinite αs(Q ) series for the coefficient func- series is divergent too. The factorial divergence of the tions, let alone the infinite condensate series. Both condensate series is studied even to a lesser extent than QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 5

that of the αs series. The only fact considered to be space; therefore, it has no impact on the OPE expan- firmly established [2,3] is the divergence per se.Itis sion at large Q2. This contribution is clearly miss- not Borel-summable, which sets a limit on the theoret- ing. The Fourier transform of Eq. (8) at large Eu- ical accuracy. Including more and more terms in the clidean Q2 falls off as exp(−Qρ). Thus, this term is series would not help, and even an optimal truncation smaller than any of the terms in the condensate ex- 2 2 would leave a gap between Π(Q )theor and Π(Q )exact. pansion. We should not forget, however, that our final goal is a prediction in the Minkowski domain. Upon analytic continuation, the exponentially small term 4 Where the duality violations come exp(−Qρ) looses its suppression and becomes oscil- from? lating, sin(−Eρ), where E stands for the total energy, E = q2. If it were not for the power suppression in Theoretical calculations of inclusive processes in QCD the pre-exponent, such exponential/oscillating terms are performed through Wilson’s OPE in the Euclidean would be a total disaster. Since they are not seen in domain. Therefore, to understand what is included in OPE, any prediction for the inclusive cross sections such calculations and what is left out, one should have made through OPE (which is equivalent, in practice, a clear picture of limitations of OPE. to perturbation theory plus a few condensates) would Let us return to the consideration of the correlation be grossly wrong. There would be no quark-hadron function of two currents, duality at all. All calculations of the hard processes — from the total hadronic cross sections in e+e− an- † 0|T {Jµ(x)Jν (0)}|0 , nihilation, to jet physics, to the heavy quark decays — would be valid roughly up to a factor of two, no which determines the polarization operator in Eq. (2), matter how large the energy release is. at negative (Euclidean) x2. Wilson’s OPE is nothing † Fortunately, one avoids the disaster due to the but an expansion of 0|T {Jµ(x)Jν (0)}|0 in singular- 2 fact that the duality-violating exponential/oscillating ities at x = 0. It properly captures all terms of the −κ 6 2 2 2 terms are, in fact, suppressed as E sin(−Eρ)where type 1/x times logarithms, or ln x ,orx ln x ,and κ is a positive index, which depends on the process un- so on. Every term of this type is represented in the der consideration and is typically rather large. This condensate series provided that the calculation is car- justifies theoretical predictions based on a few first ried out to a sufficient order. terms in the condensate expansion. Needless to say Let me note in passing that limiting oneself to sin- that determining κ is one of the most important tasks gular terms (and, in particular, to the leading singu- in the issue of the quark-hadron duality. lar term) is a reasonable approximation in the theo- 2 Singularities at x →∞also lead to the exponen- ries in which the asymptotic conformal regime sets in tial/oscillating terms of a somewhat different form, of at short distances in the power-like manner. In fact, 2 2 the type exp(−Q ρ ) in Euclidean. In the Minkowski OPE in its original formulation was designed by Wil- domain one gets an oscillating function of the type son for applications in such theories (see the first work −η 2 2 E sin(−E ρ ), where η is another positive index. in Ref. [11]). In QCD the approach to the asymptot- In one and the same process, one can expect both ically free regime is very slow — logarithmic. That’s duality-violating components to show up. Generally why the precision of the leading (parton) approxima- speaking, the indices κ and η are unrelated; at least, tion is sufficient only for very rough estimates, and at the moment we do not see any obvious relation be- that’s why the high-order terms, both logarithmic and tween them. (The coincidence of κ and η in the prac- power, must be kept. The time when the QCD prac- tically important problem of the inclusive hadronic τ titioners would be satisfied by the leading, or even the decays, κ = η = 6, seems to be accidental, see Secs. 7 first subleading approximation, is long gone. and 9). † It is clear that the function 0|T {Jµ(x)Jν (0)}|0 is not 2 2 How can one isolate the singularities at x =0from fully determined by its singularities at x = 0. Gen- others diagrammatically? To answer this question it erally speaking, one may have, additionally, isolated 2 is convenient to pass to the momentum space. In singularities at finite x , or a singularity at infinity, the leading approximation the polarization operator which are not reflected in the truncated Wilson’s OPE. 2 2 Π(Q ) is presented by the graph of Fig. 2. The large Consider, say, a singularity at finite x of the form Euclidean momentum q (remember, Q2 →∞)flows 1 in the photon line on the left, propagates through the . (8) x2 + ρ2 fermion lines, and leaves the graph through the photon line on the right. The virtual momenta of the fermion The expansion of this function (truncated at any or- lines typically scale as q. The first αs correction is pre- der) generates derivatives of δ(q2) in the momentum sented in Fig. 3. In this graph the virtual momenta 6 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003

q q q k q

Figure 3. The one-gluon correction in the polarization operator. The gluon momentum is denoted by k. soft field of all lines in the loops, the fermion and the gluon, scale as q. Thus, the diagram of Fig. 3 determines the leading logarithmic correction to the parton result (5) or (6). Figure 4. Transmitting a large external momentum through a soft field. Although the leading contribution to the integral comes from the domain where all virtual momenta are proportional to q, there are nonvanishing contribu- condensate series [3]. This is the usual story. The ex- tions from other kinematical domains. For instance, ponentially small terms in Euclidean convert into an one can consider a “corner” where the gluon virtual oscillating function in Minkowski. | | momenta k in Fig. 3 is small, k <µ, and does not I will use the fixed-size instantons for the purpose of scale with q. Certainly, at small k the gluon propa- modeling this mechanism of duality violations. The gator is not given by perturbation theory. The gluon observation that “soft” instantons generate an oscillat- line must be cut. This corresponds to the gluon con- ing component ascends to Refs. [15,16]. By no means I densate term [14] in the condensate expansion. imply that the instantons are the dominant soft fields In general, any term in the condensate expansion can in the QCD vacuum. True, there are models in which be interpreted in this way, in terms of factorization in they are assumed to be dominant, the so-called in- the momentum space. The large external momentum stanton liquid models [17]. Their status in the range q is transmitted through one or several hard lines — of questions I am interested in (the duality violations) their momenta scale with q. This is the definition of has yet to be clarified. I will use instantons for the “hardness.” The remainder of the graph is a soft part, purpose of orientation, in hope that at least some fea- which factors out and gives rise to gluon, quark and tures of the results obtained in this way will be more mixed condensates. The virtual momenta in this part general than the model itself. of the graph do not scale with q, they are limited by a fixed parameter µ (this is the definition of “softness”). 5 Model or theory? The hard part of the graph is responsible for the coefficient functions. The fact that not all lines in the given The topic I address — the quark-hadron duality vi- graph are hard results in the power suppression of the olations — has a unique status. By definition, one corresponding coefficient function. Letting more and cannot build an exhaustive theory of the duality vio- more lines to be soft, one obtains consecutive terms in lations based on the first principles. Indeed, assum- the condensate expansion. What is missing? ing there is a certain dynamical mechanism (which goes beyond perturbation theory and condensates) for It is conceivable that the number of lines through which such a theory exists, one will immediately in- which q is transmitted becomes so large, that though clude the corresponding component in the theoretical Q →∞, neither of the lines is hard. Of course, in this 2 calculation. The reference quantity, Π(Q )theor, will case one cannot speak of the separate lines in the Feyn- be redefined accordingly. After the analytic continu- man graphs. It would be more relevant to say that the ation to Minkowski, this will lead, in turn, to a new external momentum q is transmitted from the incom- theoretical spectral density to be used as a reference ing to the outgoing photon through a coherent soft ρ(s)theor in the duality relation. field fluctuation, see Fig. 4. An example is provided, for instance, by a fixed-size instanton. It is clear that Thus, by the very nature of the problem, it is bound this mechanism is conceptually related to the trun- to be treated in models of various degrees of funda- cated tail of the condensate series. Indeed, as one pro- mentality and reliability. This is because the dual- ceeds to higher condensates, more lines become soft. ity violation parametrizes our ignorance. Ideally, the Eventually we arrive at the situation when all lines are models one should aim at must have a clear physical soft. Mathematically, the exponential/oscillating con- interpretation, and must be tested, in their key fea- tribution is related to the factorial divergence of the tures, against experimental data. This will guarantee QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 7 a certain degree of confidence when these models are component to be a combination of (i) and (ii), or (i) applied to the estimates of the duality violations in and (iii). From the theoretical standpoint it is quite the processes and kinematical conditions where they difficult to consistently define the duality violating had not been tested. component of the type (iii). An operational definition I might suggest is as follows: Start from the limit ∞ 6 The physical picture behind the du- Nc = and identify the component of the type (ii). Follow its evolution as Nc becomes large but finite. ality

Before delving into technical details I will describe the 7 An instanton-based model phenomenon from a slightly different perspective. The quark-hadron duality takes place in those processes The basic features of the formalism [4] to be used where one can isolate two stages in the process under below to model exponential/oscillating terms (which consideration, occurring at two distinct scales. A ba- present deviations from duality) are as follows: sic transition involving quarks (gluons) must typically (i) One considers quarks propagating in the instanton occur at a short scale regulated by external parame- background field. The instanton size ρ is assumed to + − ters such as Q, mQ, etc. For instance, in the e e be fixed. Alternatively, one can say that an effective annihilation the basic transition is the conversion of instanton measure has a δ-function peak in ρ. the virtual γ intouu ¯ or dd¯ . Then, at the second stage, (ii) For the light quarks, most instanton amplitudes the quarks (gluons) materialize in the form of hadrons, are suppressed by powers of the light quark masses. at a much larger scale. In the appropriate frame, the This suppression is due to the fermion zero modes first time scale is of order 1/Q while the second of or- 2 of the Dirac operator, occurring in the instanton-like der Q/Λ . By that time, the original quarks are far backgrounds. We ignore these factors, as well as all away from each other — a residual interaction cannot other pre-exponential overall factors coming from the significantly alter the transition cross section which instanton measure. We will only trace dependences on was “decided” at the first (quark-gluon) stage. large momenta relevant to the problems under consid- The duality violations are due to (i) rare atypical eration (Q in e+e− annihilation, the heavy quark mass events, when the basic quark transition occurs at large mQ in the case of the heavy quark decays, and so on). rather than short distances; (ii) residual interactions (iii) We ignore all singularities of the correlation func- occurring at large distances between the quarks pro- tions at x = 0. The corresponding contributions are duced at short distances. In the first case appropriate associated with the power terms in the condensate ex- (Euclidean) correlation functions develop singularities 2 pansions. The instantons-based models are presum- at finite x , while the second mechanism is correlated 2 ably not precise enough to properly capture the con- with the x →∞behavior. densates. We will isolate and calculate only those con- In both cases the duality violating component follows tributions that come from the finite distance singu- the pattern I have discussed above — exponential in larities at x2 = −ρ2. In the momentum space they Euclidean and oscillating in Minkowski. Three dis- produce the exponential/oscillating terms sought for. tinct regimes were identified and considered in the literature so far: It is seen that our evaluation of deviations from duality is based on the most general aspects of the instanton (i) Finite-distance singularities formalism, and, in essence, does not depend on details. √ s−κ/2 sin( s) ; (9) The advantage of the model is its simplicity. The only formula we will need is 2 ν−2 (ii) Infinite-distance singularities (Nc = ∞) 4 1 iqx 2π Qρ K2−ν (Qρ) d x 2 2 ν e = 2ν−4 −η/2 (x + ρ ) Γ(ν) 2 ρ s sin(s) ; (10) (12) in the Euclidean domain, which implies that in the 2 (iii) Infinite-distance singularities (Nc large but finite, Minkowski domain, at large q , →∞ s ) √ 4 1 iqx ν − 5 ∝ 2 4 − 1 Im d x 2 2 ν e s sin( sρ δ) , exp (−αs)sin(s),α= O 1 . (11) (x + ρ ) Nc s = q2 . (13)

These regimes are not mutually exclusive — in con- Here K is the McDonald function, and δ is a constant crete processes one may expect the duality violating phase which is of no concern to us here. 8 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003

To explain how it works it seems best to consider q concrete examples. Let us start from e+e− annihilation. The polarization operator defined in Eq. (2) τ ν τ is given by the graph of Fig. 2. In Sec. 3 we evaluated this graph for free quarks and found that ρ(s) tends to a constant at asymptotically large ener- q gies. Now, we evaluate the very same graph using the quark Green functions in the background instanton Figure 5. The transition operator Tˆ determining the −1 field, rather than the free quark Green functions. For hadronic τ width, Γhadr(τ)=Mτ Im τ|Tˆ|τ. massless quarks the Green functions Ginst are known exactly [18], but we will not need the full expression. We will only need to know that Ginst is a sum of terms Equation (16) implies that the exponential component of the following structure of Π(q2), defined in Eq. (3), is

1 1 Ginst(x, y)= ∆Π ∝ exp(−2Qρ) , (17) [(x − y)2]2 Q3 √ × 1 ˜ 2 2 2 2 G, which implies, in turn, that at large E = s the os- [(x − z) + ρ ] 1 [(y − z) + ρ ] 2 cillating component of the spectral density is (14) 1 ∆ρ(s) ∝ sin(2Eρ) . (18) where z is the instanton center, and E3 1 3 1,2 = or , 1 + 2 =2, (15) Equation (18) reproduces the high-energy asymptotics 2 2 of the exact result [19] for the polarization operator in and the numerator G˜ is a polynomial of x, y.As the one-instanton approximation. was explained above, the singularity of Ginst(x, y)at Does the s−3/2 fall off of the oscillating (duality violat- x − y = 0 is irrelevant for the exponential/oscillating ing) component make sense? I will confront theoret- terms. Of importance are the singularities in the com- ical expectations with experiment in Sec. 11. Here I plex plane coming from the second factor in Eq. (14). just refer the reader to Fig. 10 (see the dashed curve), The integrals which one has to take can (and must) postponing a more detailed discussion till after both, be evaluated at a saddle point; a simple analysis [4] the instanton-based and the resonance-based models, shows that at the saddle point the instanton center z is are considered. exactly in the middle between x and y. Then the relevant singularities of the quark correlation functions The next example to be analyzed is the total hadronic 2 2 are at (x − y) = −4ρ (see Eq. (14)). At the sin- τ width. This exercise is quite similar to previous ex- 4 gularity the first factor 1/(x − y) (which is singular ercises. The corresponding transition operator is de- 4 2 2 at the origin) can be replaced by 1/(16ρ ) and then picted in Fig. 5 (its imaginary part at p = Mτ is safely omitted together with all other prefactors. proportional to the width of the hadronic τ decay). We can proceed now to the calculation of the expo- The neutrino Green function clearly does not “feel” nential/oscillating component of the polarization op- the background gluon field, so that the neutrino prop- agator is that of a free fermion. The same is valid for erator Πµν . The polarization operator is the product of two Green functions (14); therefore, at large Eu- the τ lines. Therefore, we immediately conclude that clidean momenta 1 ∆Γ(τ → ν +hadr.) ∝ sin(2Mτ ρ) , (19) 4 iq(x−y) 4 Mτ Πµν ∝ d xe d z cf. Eq. (16). The asymptotic (parton-model) predic- 1 5 × tion in this case is Γ(τ → ν +hadr.) ∝ Mτ ,sothat 2 2 2 2 3 [(x − z) + ρ ][(y − z) + ρ ] the oscillating component in Rτ scales as 1 1 ∝ 4 iqx × 4 iqz 1 d xe 2 2 d ze 2 2 3 ∝ x + ρ (z + ρ ) ∆Rτ 6 sin(2Mτ ρ) , (20) Mτ ∝ ∝ 1 − K1(Qρ)K−1(Qρ) exp( 2Qρ) . (16) where Q − → ≡ Γ(τ ντ +hadrons) Rτ − − . (21) Note that once the integration over the instanton cen- Γ(τ → ντ e ν¯e) ter is carried out, the integral factorizes. QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 9

q Then the total width is proportional to the imaginary part of the transition operator, b ν b 1 Γ= HQ|Tˆ|HQ , (22) MHQ

l where Figure 6. The transition operator Tˆ determining the total ˆ ˜¯ ˜ − semileptonic width of B mesons. T = i Q(x)G(x, y)Q(y)D(x y) eimQ(x0−y0)d4(x − y)d4z, (23) Note that the pre-exponential suppression factor in + − ∆Rτ is significantly stronger than in ∆R(e e ), − −6 −3 G(x, y) is the light quark Green function, D(x y) de- namely, Mτ vs. E . Of course, to make quan- scribes the propagation of colorless objects (the lepton titative statements it is not sufficient to establish the pair in the case at hand), while z is the instanton cen- scaling laws; one needs absolute normalizations. Here, ter. The subscript 0 marks the time component. we are basically in uncharted waters. At best, we have [4] some educated guesses, which, if true, im- Both, the light quark Green function G(x, y)andthe ˜ ply that ∆Rτ /Rτ ∼< 5%. This estimate is rather close heavy quark fields Q are to be considered in the back- to what one obtains for duality violations in τ in the ground gluon field. Taken separately, they are not resonance-based model, see Sec. 9. gauge invariant; the product in Eq. (23) is. In the leading order in the heavy quark expansion, the heavy In Sec. 2 I mentioned that the current interest to the quarks propagate only in time; therefore, problem of the quark-hadron duality was driven, to a x0 i A0(τ,x)dτ large extent, by a significant progress in the experi- Q˜(x0, x)=Te 0 Q˜(0, x) ≡ U(x)Q˜(0, x) . ments on the inclusive heavy flavor decays. Theoreti- + − (24) cally, they can be treated along the same lines as e e It is convenient (although not necessary) to impose annihilation or τ decays. The distinctions are techni- the condition that at large distances from the instan- cal. Let us start from the total semi-leptonic width of ton center the quark propagation becomes free. This the b flavored hadrons. At the quark level the process condition implies that the singular gauge is used. An is described by the transition explicit expression for U(x) can be found in this gauge; b → q −ν, it is rather cumbersome, and I will not quote it here, referring the reader to the original publication [4]. At where q is either u or c quark, and stands for the the saddle point (which again corresponds to the in- electron, muon or τ lepton. We will first neglect the stanton situated exactly in the middle between the masses of the final fermions; this is an excellent ap- points, x and y, i.e. z =(1/2)(x + y)) the product proximation for q = u and = e, µ. The impact U −1(x)...U(y) in Eq. (23) reduces to unity, at least of the final quark (lepton) masses will be considered in the part which is singular at (x − y)2 = −4ρ2. later. Note, that the matrices converting the nonsingular- The relevant transition operator is depicted in Fig. 6. gauge Green function G(x, y) to the singular gauge The differences compared to the case of the τ decay can be ignored too. This implies that one may con- are as follows: tinue to use Eq. (14). (i) There is one (rather than two) light-quark line car- Thus, the heavy quarks decouple from the instan- rying a large external momentum; ton background in the calculation of the exponential/oscillating terms. This is the consequence of the (ii) Unlike the τ lines, the b quark lines do experience fact that we exploit the heavy quark expansion and interactions with the background gluon field. limit ourselves to the leading in 1/mQ terms. In the The analysis of the exponential/oscillating component subleading terms this decoupling does not necessarily in the heavy flavor decays combines standard elements take place. Generally speaking, the replacement of the of the heavy quark expansion (for a review see e.g. z integral by the value of the integrand at the saddle Ref. [20]), and the instanton calculus (a pedagogical point is not warranted in the next-to-leading orders. review can be found in Ref. [21]). It is convenient to This effect will not be further pursued, however. choose the rest frame of the heavy hadron at hand, Summarizing, the heavy quark expansion makes the and single out the large “mechanical” part in the x heavy quarks sterile with respect to the duality vio- dependence of the heavy quark field, lating component. This is certainly not counterintu- Q(x)=e−imQtQ˜(x) . itive. The duality violating component in the heavy 10 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 quark inclusive decays originates from the light-quark + − → → → propagators. Its calculation reduces to Eq. (14) and e e X τ νX HQ νX the routine outlined after this equation (plus the basic 1 1 1 E3 M 6 M 8 formula (12)). Starting from Eq. (23) we arrive at τ HQ

1 −2m ρ ˆ Q → → → T ∝ 3 e (25) HQ X HQ Xγ HQ HQ X mQ 1 1 1 M 4 M 6 ∆M 6 in the Euclidean domain, which results in HQ HQ ( HQ )

1 Table 1. The index of the power fall-oﬀ of the oscillat- ∆Γsl ∝ 3 sin(2mQρ) . (26) mQ ing (duality violating) component in various inclusive processes, normalized to the corresponding asymptotic (par- 5 Since the total width scales as Γ ∝ mQ, the oscillat- ton model) formulae, in the instanton-based model. The ing component of the semileptonic branching ratio is capital X denotes the light-quark hadronic states. 8 obviously suppressed by mQ,

1 ∆Br(HQ → ν +hadrons)∝ sin(2mQρ) . (27) The exact dependence on the masses of the final m8 Q quarks or leptons is rather sophisticated. Although it is calculable in principle, the corresponding calcu- It is easy to generalize this formula to include an ar- lations are much harder to perform than the simple bitrary number of the light quarks in the final state. estimates presented above. Moreover, this is hardly Each extra light quark adds two powers of mQ in the necessary. Given a crude nature of the model, which numerator. Thus, in the total nonleptonic branching is intended for orientation rather than for precision es- ratio, with three light quarks in the final states, timates, it seems reasonable to treat u, d and s quarks 1 as massless while c as heavy, and the c lines as free ∆Br(HQ → light hadrons) ∝ sin(2mQρ) . (28) propagation. Then the only impact of the c quark m4 Q mass is kinematical, it results in the replacement of For the sake of completeness I will also mention the the total energy mQ by a relevant energy release in radiative decays of the type b → s + γ. Assuming that the light quarks in the process at hand. 2 these decays are induced by local operators These are just a few of important applications which are under discussion in the current literature. The re- ¯ bσµν (1 + γ5)sFµν , sults are collected in Table 1. Our instanton-based model of the duality violation is user-friendly — it is where Fµν is the photon field strength tensor, we im- very easy to evaluate the exponential/oscillating com- mediately conclude that ponent in other inclusive processes not included in Ta- 1 ble 1, would such a necessity arise. ∆Γ(b → s + γ) ∝ 3 sin(2mQρ) , (29) mQ vs. precisely in the same way as in Eq. (26). The parton 8 Global local duality expression for Γ(b → s + γ) scales as Usually by local duality people mean point-by-point 3 Γ0(b → s + γ) ∝ mQ ,mQ →∞. (30) comparison of ρ(s)theor and ρ(s)exp, while global duality compares the spectral densities ρ(s) averaged over As a result, our instanton-based model predicts that some ad hoc interval of s, with an ad hoc weight func- → the duality violating component in b s + γ is sup- tion w(s), pressed as s2 s2 ≈ → → ∝ 1 ds w(s) ρ(s)theor ds w(s) ρ(s)exp . ∆Γ(b s+γ)/Γ0(b s+γ) 6 sin(2mQρ) . (31) s1 s1 mQ

Here I would like to comment on a very common mis- So far it was assumed that the quarks and leptons conception which travels from one paper to another. produced were massless. What changes if we decide Many authors believe that global duality defined in to take into account finite (nonvanishing) masses of this way has a more solid status than local duality. the quarks and leptons? Some authors go so far as to say that while global 2In actuality this is true only for a part of the amplitude. duality is certainly valid at high energies, this is not QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 11 necessarily the case for local duality. This became a This would lead us nowhere, however. First of all, routine statement in the literature. Well, routine does the integral in Eq. (32) runs all the way down to not mean correct. zero, while Eq. (18) is valid at asymptotically large E. Even if the lower limit of integration were cho- In fact, both procedures have exactly the same theo- sen to be high enough, this would not help to find retical status. The point-by-point comparison, as well ∆Rτ from Eq. (32). Indeed, ∆ρV,A is an oscillating as the comparison of ρ(s)’s (with an ad hoc weight function of s, any smearing inevitably entails cancel- function), must be considered as distinct versions of lo- lations, and the result would depend on subtle details cal duality. The distinction between the “local” quan- + − of ρ(s), which are certainly beyond theoretical con- tities, such as R(e e ) at a certain value of s and the trol. The cancellations are seen, in particular, from integrals of the type involved, say, in Rτ is quantita- the fact that asymptotically ∆Rτ (Mτ ) falls off faster tive rather than qualitative. Comparison of Eqs. (18) −6 −3 than ∆ρV,A(s), namely, M versus E . There is no and (20) makes this assertion absolutely transparent: τ way one could predict the change of the index from 6 in both quantities there is a duality violating compo- to 3 based solely on the integral (32). nent, the only distinction is a concrete index of the power fall-off (3 vs. 6). At the same time, our instanton-based model does allow one to predict the index in Rτ . To this end one The genuine global duality applies only to special in- must analyze the appropriate transition amplitude in tegrals which can be directly expressed through the Eu- the Euclidean domain performing the analytic contin- clidean quantities. For instance, if the integration in- uation to the Minkowski domain at the very end. The terval extends from zero to infinity, and the weight model captures enough intricate features of QCD to function is exponential, the integral “know” the result of the smearing as a whole. Note ∞ that ∆Rτ (Mτ ) depends on the highest scale in the ds exp{−s/M 2}ρ(s) , 0 problem at hand, Mτ , rather than on any intermedi- ate or low scales one encounters in process of integra- reduces [14] to the Borel transform of the polarization in Eq. (32). This is a general feature which will 2 tion operator Π(Q ) in the Euclidean domain (i.e. at be valid in any process and will persist in any model 2 positive Q ). For such quantities, duality cannot be based on evaluating singularities off the origin. violated, by definition. There is one more aspect of averaging (smearing) 9 A resonance-based model which is not fully understood in the literature and requires comment. Let us consider again Rτ .Itcanbe Now we will acquaint ourselves with another approach expressed in terms of spectral densities ρV and ρA in based on the resonance saturation of the colorless n- the vector and axial-vector channels, respectively, point functions. That this is a distinct dynamical 2 2 source of duality violations follows from the discussion Mτ ds − s in Secs. 4 and 6. Shortly we will confirm this by ob- Rτ = 2 1 2 0 Mτ Mτ serving that the functional form of oscillations comes s out different compared to that in the instanton-based 1+2 2 [ρV (s)+ρA(s)] . (32) Mτ model. Theoretical analysis is most transparent in the limit Nc = ∞. Later I will allow Nc to become finite 2 (More exactly, the integration over s runs from Mπ, albeit large (Sec. 10). but in the chiral limit we stick to, the pion mass van- Let us first summarize what we expect to get for the ishes.) The spectral densities ρV and ρA are normal- polarization operator defined in Eq. (2) in the limit ized in such a way that their asymptotic (free-quark) of large Nc. In multicolor QCD, Nc →∞, the res- values are onance widths vanish. The spectrum of excitations in the given channel is expected be (asymptotically) ρV (s) ,ρA(s) → Nc at s →∞. equidistant. A string-like picture of the color confine- 0 Correspondingly, the asymptotic value of Rτ is Rτ = ment naturally leads to (approximately) linear Regge Nc. trajectories. For each primary trajectory there are infinitely many daughter ones. The daughter trajec- In the instanton-based model, the duality violating −3 tories are parallel to the primary trajectory and are component in ρV,A scales as E . It is tempting to shifted by integers (for a review see e.g. Ref. [22]). As say then, that the duality violating component in Rτ a result, the excitation spectrum in the given channel can be obtained by integrating the duality violating takes the form components of ρV,A with the weight function specified 2 2 2 2 in Eq. (32). Mn = M0 + σ n, σ ≡ 2/α , (33) 12 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 where α is the slope of the trajectories. Note that the not affect the estimate of the duality violations. (For neighboring resonance states in the given channel are an additional remark on the residues see Sec. 11.) separated by the interval 2/α rather than 1/α.This The infinite sum in Eq. (34) reduces to a well-known is due to the alternating signatures of the daughter Euler’s ψ function, the logarithmic derivative of Γ, trajectories. 2 2 Nc Q + M All these properties can be extracted from the Π(Q2)=− [ψ(z) + Const] ,z≡ 0 , Veneziano amplitude found in 1968 (see Sec. 7.4 in 12π2 σ2 Collins [22]), which gave rise to the modern string the- (35) ory. (see e.g. Ref. [24]). The irrelevant (subtraction) constant on the right-hand side is infinite, strictly speak- 2 Equation (33) implies, in turn, that Π(q ) can be pre- ing. The occurrence of the Γ function reminds us of sented as an infinite sum Veneziano’s amplitude. 2 2 ∞ At positive values of Q an asymptotic representation 2 −Ncσ 1 Π(q )= 2 2 − 2 exists for the ψ function, 12π n=0 q Mn ∞ 2 ∞ Ncσ 1 1 B2k −2k = − . (34) ψ(z)=lnz − − z , (36) 2 2 − 2 − 2 2z 2k 12π n=0 q σ n M0 k=1

where B2k stand for the Bernoulli numbers, In the problem at hand there are two large quantities: k−1 2(2k)! the number of colors and the energy. If one fixes s B2k =(−1) ζ(2k) ; (37) (2π)2k and lets Nc grow, one eventually arrives at the comb of infinitely narrow δ functions, as in Fig. 7. On the here ζ is the Riemann function. (In some textbooks other hand, if Nc is fixed (no matter how large it is), k+1 (−1) B2k is called the k-th Bernoulli number and is with increasing s one eventually finds oneself in the denoted by Bk.) Equation (36) defines the asymptotic energy region where the resonance widths cannot be expansion of the polarization operator, neglected; in fact, the resonances start overlapping, and ρ(s) gets smoothed. The approximation of the ∞ 2 −→ − Nc 2 Ck infinitely narrow resonances badly fails here, and must Π(Q ) Q2→∞ 2 ln Q + 2 k , (38) 12π k=1 (Q ) be amended. These two limits (Nc →∞with s fixed, and Nc fixed with s →∞) are not interchangeable. where the coefficients Ck can be expressed through Later I will include the nonvanishing widths (Sec. 10). B2k (see Eq. (36)) in a relatively straightforward man- For the time being I put Nc = ∞, so that all resonance ner. The leading logarithmic term exactly coincides widths vanish. with the free quark loop of Fig. 2 presented in Eq. The equidistant spectrum only holds if both, the pri- (5). This explains our choice of the resonance residues. mary and all daughter trajectories, are exactly linear The next-to-leading term is 1/Q2, followed by higher and parallel. This is not fully realistic. Even putting power corrections. Equations (36) and (37) highlight Nc = ∞ does not help. The low-energy parts of the the factorial divergence of the condensate series which Regge trajectories in QCD are not exactly linear, in I have already mentioned several times. particular, due to the spontaneous breaking of the chi- One might want to eliminate the 1/Q2 term from ral symmetry and the emergence of the massless pions. Π(Q2)tomakethe1/Q2 expansion realistic — it This shortcoming of the model affects the condensate is known [14] that, in QCD with massless quarks, expansion at low orders but has no impact on devi- the terms of the first order in 1/Q2 do not appear ations from duality at high energies. Since the phe- in Π(Q2). The power series starts from G2/Q4 nomenon under discussion is related to the factorial where G2 is the gluon condensate. One could elim- 2 2 divergence of the high-order terms, letting the low- inate 1/Q , say, by fine-tuning the parameter M0 .If 2 2 2 lying excitations “breathe” does not change the fac- M0 = σ /2=1/α ,the1/Q term cancels. Although torial behavior [23], which is in one-to-one correspon- this might seem desirable, in fact this is hardly worth dence with the spectral formula at large n, where the the bother because the model with exactly linear tra- 2 2 4 spacings Mn+1 − Mn must be constant. Therefore, it jectories is too rigid to be realistic anyway. The 1/Q is okay to use the linear pattern (34). For the very correction will come out way too large. To accommo- same reason the residues of all resonances in Eq. (34) date the gluon and mixed condensates properly one are taken to be equal. Fluctuations of the residues would need a more flexible model, with more than one would show up in the condensate expansion; they do adjustable parameter. Therefore, I will feel free to QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 13

ρ(s) where the moments In are deﬁned as

M 2 15 n In(M)= dss [ρV (s)+ρA(s)] . (41) 0 12.5 To estimate the oscillating contribution to Rτ which 10 constitutes duality violation that cannot be seen in a 7.5 truncated OPE we treat Mτ as a free (large) parameter. It will be seen momentarily that for Nc = ∞ and 5 Mτ large, yet ﬁnite, the duality violation in Rτ scales 6 2.5 as 1/Mτ . In other words, the vanishing power index 3 s in ρ translates in η =6inRτ . Certainly, this distinc- 2.5 5 7.5 10 12.5 15 tion is quantitative rather than qualitative, but, sure enough, it is quite important from the practical side. Figure 7. The spectral density in the resonance model. The sum over resonances in Rτ is easily calculated For clarity I gave a tiny width to the δ functions. analytically: for the spectral density of Eq. (39) it is

OPE osc Rτ = Rτ +∆Rτ , 2 simplify the model further by putting M0 =0and 4 discarding the term with n = 0 in Eq. (34). OPE 2 2 Rτ − σ 1 σ =1 2 + 2 , Nc Mτ 30 Mτ The spectral density corresponding to the inﬁnite sum of the equidistant resonances is shown in Fig. 7, osc 2 3 ∆Rτ − − − σ = x(1 x)(1 2x) 2 Nc M ∞ τ s 2 2 ρ V (s)=ρA(s)=Nc · δ − n ; σ = . 2 4 2 2 2 1 σ n=1 σ α − − + x (1 x) 2 , (42) (39) 30 Mτ

Let us truncate the power expansion (38) at some fi- where nite order, and examine the theoretical prediction for M 2 ρ(s) obtained from the Euclidean side. It does not x = fractional part of τ ,x∈ [0, 1) . matter in which order we truncate. Any power term σ2 (1/Q2)n is invisible in ρ(s) at positive s: analytically continuing to positive q2 and taking the imaginary I presented the result as a sum of two functions of part one ends up with the δ function and its deriva- 2 OPE Mτ . The first one, Rτ , is a smooth function ex- tives. The only imaginary part at positive s comes 2 osc 2 pandable in 1/M . The second one, ∆R , oscillates from the analytic continuation of ln Q . Thus, in the τ τ with the period σ2; its average vanishes, see the plot model at hand ρ(s)theor = 1. Comparing this with the of ∆Rosc/R0 in Fig. 8. Although ∆Rosc is not a pure comb of the δ functions in Fig. 7 we conclude that the τ τ τ sine — it contains higher harmonics — the coefficients point-by-point duality is maximally violated: between of the higher harmonics are numerically suppressed. the resonances the spectral density is grossly overesti- With the accuracy of a few percent one can write mated (the “experimental” curve runs below the “theoretical” expectation) while at the peaks it is grossly osc 2 3 2 ∆Rτ − √1 σ Mτ underestimated (the “experimental” curve runs above = 2 sin 2π 2 . (43) Rτ M σ the “theoretical” expectation). What is most crucial, 3 12 τ the deviations from duality do not die off with energy. The power index vanishes! I pause here to make a comment. The contribution of any particular resonance of mass Mk to Rτ , accord- All this is pretty trivial. A somewhat less trivial ques- ing to Eq. (32), is given by a simple polynomial in 2 2 − 2 tion worth examining is the impact of the ad hoc 1/Mτ (times the step function θ(Mτ Mk ) ). Vari- smearing. For instance, let us average the spectral ations of parameters of a given resonance (or reso- 2 density in Fig. 7 with the weight function appropriate nances) change only the regular terms of the 1/Mτ to Rτ , expansion, but have no impact on the oscillatory component. From Eq. (32) it is clear that such variations

2 2 2 3 I0(Mτ ) − I2(Mτ ) I3(Mτ ) I remind that the power index η was deﬁned in Sec. 4; in the Rτ = 2 3 6 +2 8 , (40) R /R ∼ M −η M 2/σ2 Mτ Mτ Mτ case at hand ∆ τ τ τ sin τ . 14 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003

OPE expansion for Rτ is immediately recovered provided

0.004 that one substitutes the appropriate value for the appropriate Bernoulli number, B2 = −1/30.

0.002 A few words on the numerical aspect. Our consideration is admittedly illustrative. One should not take too literally the numbers which ensue for many rea- 2 0 sons: in particular, Mτ is not much larger than the spacing between the resonances, Nc = 3 is probably not large enough to warrant the zero width approx- -0.002 imation, and so on. I would not put too much con- 3 4 5 6 7 8 9 10 fidence on particular numbers. I would settle on the statement that the above estimate of the oscillation osc 0 Figure 8. Oscillations in Rτ . The plot of ∆Rτ /Rτ is component is valid, say, up to a factor of two or so. 2 2 osc presented as a function of Mτ /σ . Taking our formula for ∆Rτ at its face value and us- 2 2 ing the actual value of the τ mass (Mτ /σ ∼ 1.5) we osc ∼ 2 6 8 obtain that ∆Rτ /Rτ 3%. It is rather rewarding change only coefficients of the 1/Mτ ,1/Mτ and 1/Mτ to see that this estimate is in the same ball park as terms. OPE for Rτ must exactly reproduce these three that obtained in the instanton-based model. It seems expansion coefficients, and so it does. safe to conclude that duality violations in the total OPE In fact, it is not difficult to demonstrate that Rτ co- hadronic τ width are expected at a level of a few per- incides with the OPE prediction in the model at hand. cent. We do not know whether two mechanisms add The power corrections can be presented as follows: constructively or destructively. Given this additional uncertainty, it will be no exaggeration to assert that I˜0 I˜2 I˜3 OPE − the overall theoretical uncertainty Rτ = Nc + 2 3 6 +2 8 , (44) Mτ Mτ Mτ ∆Rτ /Rτ =3%. where the “condensates” Iñ are ∞ The 3% uncertainty in the hadronic τ width translates n ∼ Iñ = dss [ρV (s)+ρA(s) − 2Nc] . (45) into 20% uncertainty in αs(Mτ ), which entails the 0 uncertainty of about 6% in αs(MZ ). These integral representations for the “condensates” In summary: Iñ follow from Eqs. (32), (41) if one assumes that the spectral densities approach their asymptotic lim- • The resonance structure of the hadronic spectrum its faster than any power of 1/s. In the model at hand, associated with the confining properties of QCD leads with the comb-like spectral density, the integral rep- to a distinct exponential/oscillating component invis- resentation (45) requires regularization. As a regular- ible in the truncated OPE. − ization one can introduce the weight factor exp( s), • This component is related to singularities of the ap- taking the limit → 0 at the end. With this regular- OPE propriate n-point functions at infinite separations (in ization, Rτ from Eq. (42) is reproduced. the coordinate space). The same result for the coefficients of the power terms • OPE The corresponding duality violations are maximal in Rτ could be obtained directly from the expansion at Nc = ∞, when the resonance widths vanish. The of Π(Q2) in Eq. (35). We agreed to put M 2 =0for 0 power index which was introduced in Sec. 4 is calcula- simplicity. Then the expansion coefficients of Π(Q2) ble in many instances. Generally speaking, the power are those of the ψ function in Eq. (36). To find the indices obtained in the instanton-based and resonance- power terms in Rτ one may analytically continue the based models do not coincide with each other. Rτ power terms in Eq. (36) to Minkowski, take the imag- seems to be an exceptional case. We do not under- inary part and convolute with the weight function pre- stand the reasons explaining the coincidence of the sented in Eq. (32). It is easy to see that the only rel- power indices in Rτ ; probably, it is accidental. evant terms in Eq. (36) are 1/z and 1/z4 giving rise to δ(s)andδ(s) in the imaginary part. The terms of • Inclusion of the nonvanishing resonance widths will the higher order in 1/z drop out because the weight replace the power suppression in the pre-exponent by function is a polynomial of the third order; it contains a weak exponential suppression, see Sec. 10. The no s4 or higher terms. The term 1/z2 in Eq. (36) change of the regimes will probably have little numeri- 2 (it would generate δ (s)) drops out because the weight cal impact in the τ decays since Mτ is not large enough function does not contain terms linear in s. Then, the for the exponential regime to develop in earnest. The QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 15

2 2 3 factor (σ /Mτ ) in Eq. (43) will be replaced by Now comes a crucial assertion. At large n the resonance width must scale as Γn ∼ Mn/Nc, i.e. the 2 −2πBMτ n dependence of Γn is the same as that of Mn,the exp 2 . Ncσ square root of n. This behavior was predicted long ago [25] on the basis of a simple qualitative picture; Both factors are rather close numerically. much later it was confirmed [5] by numerical studies in the two-dimensional ’t Hooft model. This formula can be explained as follows. For highly excited states 10 The impact of the resonance widths a quasiclassical treatment applies. When a meson is created by a local source, it can be considered, qua- As was already mentioned, no matter how large Nc siclassically, as a pair of (almost free) ultrarelativistic is, at sufficiently high energies one cannot neglect the quarks; each of them with energy Mn/2. These quarks resonance widths. One should also remember that in are produced at the origin, and then fly back-to-back, the real world the number of colors is three — not creating behind them a flux tube of the chromoelec- 2 very large by any count. One can expect that the tric field. The length of the tube L ∼ Mn/Λ where nonvanishing resonance widths, once the resonances Λ2 represents the string tension. The meson decay start to overlap, provide an additional suppression of width is determined, to order 1/Nc, by the probabil- the duality violating component. ity of producing an extra quark-antiquark pair. Since Below I will show that, inside the domain of overlap- the pair creation can happen anywhere inside the flux 4 ∼ 2 ping, the power regime (10) is replaced by an exponen- tube, one expects that Γn LΛ /Nc. Taking into ∼ 2 tial one, see Eq. (11), even if the power index vanishes account that L Mn/Λ one arrives at (as is the case for the spectral density (4) presented B Γn = Mn , (48) in Fig. 7). The basic idea is that the nonvanishing Nc resonance widths shift the poles away from the physical cut, to unphysical sheets, which automatically re- where B is a dimensionless coefficient of order one. sults in a smoother imaginary part on the physical cut, Naively extrapolating Eq. (48) to n =1andNc =3 much closer to that obtained from OPE. (well beyond the limits of its applicability) and nor- malizing by the ρ meson for which Γ/M ∼ 0.2, one I return to the equidistant resonances considered in can make an educated guess, Sec. 9 in the leading order in 1/Nc, with the intention to include the next-to-leading 1/Nc effects. For any B ∼ 0.5 . (49) −1 given excitation n, the widths scale as Γn/Mn ∼ Nc . Below we will analyze an infinite sequence of reso- The Breit-Wigner resonance formula now takes the nances for which we will need to know the depen- form (in the Euclidean notation) dence of this ratio on the excitation number n.For 2 −1 the time being, however, let us focus on an “isolated” 2 2 2 BMn −gn Q + Mn − i . (50) resonance. Nc In the leading order its contribution to the polarization One must exercise certain care in using the Breit- operator Π(q2)is Wigner expression for the resonance contribution in 2 gn the polarization operator far away from the pole. It 2 2 ; (46) q − Mn must be adjusted in such a way that the analytic properties of Π(q2) are not spoiled. Namely, Π(q2)must 2 2 here gn is the residue. If Γn = 0, in the vicinity of remain analytic everywhere in the complex q plane, the pole one can use the Breit-Wigner formula which with a cut along the positive real semi-axis. No sin- replaces Eq. (46) by gularities are allowed on the physical sheet, all poles 2 must be shifted to unphysical sheets. This is relatively gn easy to achieve. To this end let us replace Eq. (50) by 2 2 . (47) q − Mn + iΓnMn −1 B Q2 2 2 − 2 2 − ˜ 2 Strictly speaking, gn and Mn in Eqs. (46) and (47) are gñ Q 1 ln 2 + Mn . (51) πNc σ different: in the former the residue and the mass must be taken in the leading order in 1/Nc while in the latter On the upper side of the physical cut near the pole 2 2 they include O(1/Nc) corrections. This effect is less (i.e. at Q in the vicinity of −Mn − i) the imaginary important than that due to Γn’s. In many instances 4 2 2 Let me note in passing that the 1/Nc corrections due to cre- (but not always, see below) the 1/Nc shifts of gn and ation of two quark pairs are of order L2/N 2 within this picture. 2 √ c √ Mn can be neglected. Since L ∼ Mn ∼ n, the expansion parameter is n/Nc. 16 QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003

parts of both expressions (50) and (51) coincide, pro- 2 ρ 2 2 2 (s) B=0.5 vided thatg ˜n and M˜ n are appropriately adjusted (˜gn 2 2 2 Nc =3 and M˜ n diﬀer from gn and Mn by a 1/Nc correction; in what follows we will omit the tilde, making the ad- 1.5 justment at the very end). Thus, both expressions are equally legitimate for the Breit-Wigner description of the resonances. Moreover, Eq. (51), in turn, can be 1 replaced by

−1 − B 0.5 2 πNc − 2 2 Q 2 gn Q 2 + Mn . (52) σ s σ2 0 / 1 2 3 4 5 6 7 The distinction between Eqs. (51) and (52) is of the 2 order O(1/Nc ). We do not track such terms anyway. Figure 9. The spectral density for the equidistant reso- This latter formula has the required analytic proper- nances with finite widths specified in Eq. (48). ties — it is nonsingular everywhere in the complex Q2 plane, except the cut at negative real Q2.Onthe physical sheet of Q2, the variable be 1− B 2 2k Q2 πNc B2k σ z ≡ (53) 2 (zero-width approx.), σ2 2k Q never becomes real and negative, so that the pole sin- now becomes gularities in Eq. (52) are indeed shifted to unphysical 2 2k 2 B2k σ 2kB Q B 1 sheets. 2 1+ ln 2 + + O 2 . 2k Q πNc σ πNc Nc Summing over the infinite chain of the equidistant res- (55) onances, as in Eq. (34), we arrive at The impact of the next-to-leading 1/Nc terms on the power expansion is quite insignificant. In particular, 2 2 the (B/Nc)lnQ correction mimics the logarithmic 2 Ncσ 1 Π(Q )= 2 anomalous dimension typical of OPE in QCD. 12π 1 − B/(πNc) −1 ∞ − B c 2 πNc At the same time, the 1/N effects radically affect × 2 Q 2 Q 2 + Mn ImΠ on the physical cut (the spectral density). In- n=1 σ stead of the comb of the δ functions of Fig. 7, we now get at high energies a smooth function, with − Nc 1 1 mild oscillations. The plot of the spectral density = 2 ψ(z)+ , (54) 12π 1 − B/(πNc) z ρ(s) = (12π/Nc) ImΠ (with Π determined by Eqs. (54) and (53)) is presented in Fig. 9. − where z is defined in Eq. (53), and 1 B/(πNc)inthe At asymptotically large s the spectral density tends to denominator reflects the adjustment of the residues unity, plus power corrections corresponding to OPE discussed above (this factor can be established by de- plus an oscillating component which cannot be ob- 2 →∞ manding the correct asymptotic behavior at Q , tained in the power expansion. One can readily isolate cf. Eq. (5)). it by exploiting the well-known reflection property of How does this compare with the zero-width approxi- Euler’s function, mation of Sec. 9? Formally both results for the polar- 1 ization operator look similar; the only difference5 is in ψ(z)+ = ψ(−z) − πcotπz . (56) z the definition of the variable z. In the deep Euclidean 2 domain the power expansion of Π(Q ) ensues from the On the physical cut (in units of σ2) asymptotic representation (36). The k-th term of the expansion, which in the leading order in 1/Nc used to 1− B B 1 − πNc z = s 1+i + O 2 , (57) Nc Nc 5The 1/z term which was absent in Sec. 9 is due to the fact that 2 I put M0 = 0 and start the summation from n = 1 rather than n and the imaginary part of the left-hand side of Eq. from = 0. These simplifications resulting in the occurrence of − the 1/z term, are irrelevant in the studies of the duality violating (56) is essentially given by that of πcotπz.The component at high energies. imaginary part of ψ(−z) gives rise to non-oscillating QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 17 power corrections s−2k in the spectral density corre- ρ 5 Aleph data sponding to OPE. Alternatively, these OPE correc- Pert. theory tions can be obtained by a direct analytic continuation Instanton model 4 to Minkowski of the power terms (55). Resonance model

As a result, the finite-width resonance model leads us 3 to 2 → −2πsB 2πs ρ(s) 1+power corr.+2 exp 2 cos 2 , σ Nc σ (58) 1 where it is assumed that s, GeV2 2πsB ln s 0 1 2 3 4 5 6 7 2 1 but 1 . (59) σ Nc Nc Figure 10. Experimental data (ALEPH) on the spec- ∞ tral density in the vector channel and the duality violation If s is fixed while Nc is set to we recover unsup- models. pressed oscillations, as in Sec. 7. In the opposite limit Nc fixed and s large we observe an exponential suppression, with a weak exponent proportional to 1/Nc. Fig. 10. I will first explain what is depicted, and then offer several comments. A key question one can ask in connection with the above analysis is as follows. For the given value of Nc, The ALEPH experimental data [26] I show correspond to the spectral density in the vector isovector channel. what is the boundary energy marking the onset of the 2 + − exponential suppression? The best I can say at the There exist some data above 3 GeV from e e anni- moment is that this boundary energy scales as s0 ∝ hilation, but the error bars are so large that plotting Nc. A reliable determination of the proportionality these points would just obscure the picture. Directly coefficient is a task for the future. A naive estimate measurable in the hadronic τ decays is the sum of the following from Eq. (59), vector and axial spectral densities. To obtain the spectral densities separately one has to sort out all decays 2 σ Nc 2 by assigning specific quantum numbers to each given s0 ∼ ∼ 2GeV at Nc =3, 2πB final hadronic state. In the majority of cases such an assignment is unambiguous. For instance, two pions if correct, would mean that the resonances essentially (whose contribution is the largest) can be produced overlap (and, thus, smear the spectral density), start- only by the vector current. Some processes, however, ing from the first or second excitation. can occur in both channels, for instance, the KKπ production. Using certain theoretical arguments it 11 How does all this match experi- was decided [26] that around 3/4 of all KKπ yield must be ascribed to the vector channel. Other theo- ment? retical arguments [27], which seem more convincing, tell that virtually all KKπ production takes place in In spite of the current extremely high demand on the- the axial channel. Therefore, in fitting the data, I sub- oretical estimates of duality violations, there is sur- tract the KKπ yield from the data points presented in prisingly little effort to elucidate the issue by direct Fig. 10. I hasten to add that the subtraction affects experimental studies. A high-precision measurement only the energy range s>2GeV2, i.e. the second + − of the ratio R in e e in a broad range of energies, peak in Fig. 10, and even in this energy range the 2 from threshold up to, say, s =10GeV in a dedi- effect is small, ∼< 5%. The subtraction is not essential cated experiment with the proper absolute normaliza- for a general picture I draw here. tion would give an enormous boost to this issue. Alas, The solid curve at s<2.7GeV2 is the best fit of the such measurements have never been undertaken... The data points thus obtained by the sum of two Breit- most accurate data on the spectral densities were ob- Wigner peaks (the first one, the ρ meson, is a modified tained in τ decays [26], where they (naturally) extend 2 Breit-Wigner taking into account threshold effects im- only up to s =3GeV . This energy range is way too portant for the ρ meson). For my purposes the two- narrow to be helpful in perfecting the models which resonance fit presents an excellent approximation to are currently in use or in the design of new models. the experimental spectral density at s<2.7GeV2. Still, confronting the data with current theoretic ideas might give a feeling of whether or not we are moving The tails of the curves at 2.7

(the solid and dashed curves, respectively) which I dis- become instantly obsolete. Then the exact asymp- cussed in Secs. 10 and 6. The solid curve is a modu- totic behavior of inclusive cross sections would be lation of ρ(s)pert by the factor known, and the very concept of duality violations would become irrelevant. 1+1.22 E−3 sin(2ρE − δ) ,

−1 ρ =3GeV ,δ=1.32 ,Ein GeV , Acknowledgments cf. Eq. (18). The dashed curve is a modulation of 6 ρ(s)pert by I am deeply grateful to Pietro Colangelo, Fulvia De Fazio, and Giuseppe Nardulli for the kind hospitality extended to me in Conversano. − −2πsB 2πs − 1 1.24 exp 2 sin 2 3.08 , σ Nc σ

2 2 σ =2GeV ,B=0.5 ,Nc =3, References 1. M. Shifman (Ed.), Handbook of QCD “‘At The cf. Eq. (58). Finally, the thick solid curve in Fig. 3 Frontier of Particle Physics/Boris Ioffe 10 displays ρ(s)pert calculated through order αs, with (3) Festschrift”, Vols. 1–3, (World Scientific, Λ = 200 MeV. MS Singapore, 2001, 2151 pp). It is clearly seen that, qualitatively, both models 2. M.A. Shifman, in Continuous Advances in QCD, match the data well. In fact, in actuality I would ex- Ed. A. Smilga (World Scientific, Singapore 1994), pect an oscillating component which is a combination p. 249 [hep-ph/9405246]. of the solid and dashed curves, since the mechanisms 3. M. Shifman, in Particles, Strings and Cosmology, of the duality violations they represent are comple- Eds. J. Bagger et al., (World Scientific, mentary. The instanton mechanism leads to a visibly Singapore 1996), p. 69 [hep-ph/9505289]. slower (power) fall off, with rearer oscillations. It is 4. B. Chibisov, R.D. Dikeman, M. Shifman and A12 also clear that experimental measurements at the per- N. Uraltsev, Int. J. Mod. Phys. (1997) cent level of accuracy will most certainly provide us 2075 [hep-ph/9605465]. with the material needed for construction of a reliable 5. B. Blok, M. Shifman and D. Zhang, Phys. Rev. D57 D59 and well calibrated model of the duality violations. (1998) 2691; (E) (1999) 019901 [hep-ph/9709333]. 6. M. Shifman, Prog. Theor. Phys. Suppl. 131, 12 Conclusions (1998) 1 [hep-ph/9802214]. 7. I. Bigi, M. Shifman, N. Uraltsev and I identified general mechanisms causing deviations A. Vainshtein, Phys. Rev. D59(1999) 054011 from duality and derived scaling laws governing the [hep-ph/9805241]. damping of the duality violating component in a vari- 8. I. Bigi and N. Uraltsev, Phys. Lett. B457 (1999) ety of inclusive processes as a function of energy (mo- 163 [hep-ph/9903258]; Phys. Rev. D60, (1999) mentum transfer). The estimates of the absolute nor- 114034 [hep-ph/9902315]. malization one can perform at the moment are less 9. R. F. Lebed and N. G. Uraltsev, Phys. Rev. D62 certain. This explains why I was so cautious in my (2000) 094011 [hep-ph/0006346]. discussion of the numerical situation in Rτ , and com- 10. E.C. Poggio, H.R. Quinn and S. Weinberg, Phys. pletely avoided this issue in other inclusive processes. Rev. D13(1976) 1958. The Getting a better idea of the absolute normalization Poggio-Quinn-Weinberg work ascends, in turn, to is absolutely necessary for all practical applications of A. Bramon, E. Etim and M. Greco, Phys. Lett. the theoretical constructions presented here. Precision B41(1972) 609; J.J. Sakurai, Phys. Lett. B46 measurements of ρ(s) in a wide energy range would be (1973) 207. of enormous help in this question and would probably 11. K.G. Wilson, Phys. Rev. 179 (1969) 1499; Phys. lead to a speedy solution. Rev. D3 (1971) 1818; K.G. Wilson and J. Kogut, 12 As I have already noted, duality violations Phys. Rept. (1974) 75. parametrize our ignorance. Were an analytic solu- 12. This phenomenon is explained and illustrated in tion of QCD found, the contents of this review would great detail in the collection Large Order 6 Behavior of Perturbation Theory, Eds. J.C. Le As we know from Sec. 10, the equidistant resonances with Guillou and J. Zinn-Justin (North-Holland, equal residues result in oscillations of this type superimposed on ρ = 1. By a trivial adjustment of the residues one can readily Amsterdam 1990). One can find there all achieve that the oscillations (58) are superimposed on ρ(s)pert. classical works and an extended commentary. QCD@Work 2003 - International Workshop on QCD, Conversano, Italy,14–18 June 2003 19

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