∗ 7 1/NC Expansion of QCD Amplitudes 0 0 2 J.J. Sanz-Cillero n a Department of Physics, Peking University, Beijing 100871,P.R. of China J e-mail: [email protected] 9 1 ThistalkcommentsthemainfeaturesofahadronicdescriptionofQCD inthelimitoflargenumberofcolours. Wederiveaquantumfieldtheoryfor 1 mesons based on chiral symmetry and a perturbative expansion in 1/NC. v 3 Some large–NC and next-to-leading order results are reviewed. 5 1 PACS numbers: 12.39.Fe, 11.15.Pg 1 0 1. Introduction 7 0 The success of large–N determinations based on ’t Hooft’s limit of / C h QCD [1] has led to an extraordinary development of the field and has nat- p urally raised the question about subleading effects [2]–[7]. The analysis - p beyond leading order in 1/NC (LO) is essential to validate the large–NC e limit. A formally well defined 1/N expansion can be easily obtained C h by implementing the proper N scalings of the hadronic couplings and : C v masses [1, 3, 4, 5]. On the other hand, naively, loops with heavy reso- i X M2 nance are expected to produce corrections of the form R , order 1/N 16π2F2 C r but numerically large. However, previous phenomenological analyses have a shown that the next-to-leading order contributions (NLO) remain under control [3, 5]. Rewriting the resonance parameters in terms of widths (Γ ) R an masses (M ), one observes that the 1/N expansion is actually an ex- R C pansion around the narrow-width limit. The corrections to the large–N C amplitudes are suppressed by Γ /M 1/N . However, it is not yet clear R R C ∼ how broad-width states like the σ meson fit in this pattern [8, 9]. Since the QCD action is chirally invariant, one needs to construct a chi- ral theory for resonances (RχT) that preserves the symmetry. This feature, common to several phenomenological lagrangians [7, 10, 11, 12, 13], ensures the recovery of Chiral Perturbation Theory (χPT) [14] at low energies even ∗ TalkpresentedatEffective theories ofcolours andflavours: from EURODAPHNEto EURIDICE,24-27 August2006, Kazimierz(Poland). Thiswork hasbeensupported byEURTNContract CT2002-0311 andChinaNationalNaturalScienceFoundation undergrant number10575002 and 10421503. (1) 2 proc-JJSC printed on February 2, 2008 at the loop level. Likewise, the validity of the 1/N expansion at all ener- C gies allows to match QCD at short distances, where it is described by the operator product expansion (OPE) [15]. 2. Large–NC, next-to-leading order and resummations Inthelarge–N limit,QCDcontainsaninfinitetowerofhadronicstates, C the resonances (R) and the Goldstones from the spontaneous chiral sym- metry breaking (φ). The Green-functions are provided by the tree-level exchanges [1]. Other observables like the form-factors are derived from them and they are also given by the tree-level diagrams. Their absorptive contribution is a series of delta functions , so theamplitudes are determined by the masses (position of the real poles) and the corresponding couplings (residues). At NLO in 1/N , the perturbative amplitudes contain two-meson ab- C sorptivecutstogetherwithsingleanddoublerealpolescoming,respectively, from diagrams with one and two tree–level propagators [3, 5]. Pure pertur- bation theory, i.e., without resummation, is valid when no intermediate particle is near its mass-shell. However, the perturbativeexpansion breaks down in theneighbourhood of the resonance poles at any finite order in 1/N and a Dyson-Schwinger C summationisrequired[16]. Thisshiftstherealresonancepolesintounphys- ical Riemann sheets. The particles gain a finite width and the amplitude becomes finite. In the past, the attention has been focused either on the large-N limit C or on resummed descriptions. However, the previous step to any resumma- tion is the perturbative calculation and only a few examples of it exist by themoment[2]–[7]. By-passingthisintermediate pointmayleadtostrongly model dependent resummations and, therefore, incorrect determinations. 3. Resonance Chiral Theory 3.1. Leading order in 1/N C In general, one is forced to work within a minimal hadronical approx- imation with a finite number of states (MHA) [17]. This is an acceptable approximation in the case of amplitudes that are chiral-order parameter, provided that we include a minimal number of light states, enough to re- produce the short-distance QCD power behaviour. Since we work within a large–N framework, the particles are classified C in U(n ) multiplets [18]. The Goldstones from the spontaneous chiral sym- f metrybreakingφ= (π,K,η ,η )areincorporatedthroughcovarianttensors 8 0 (φ) [10, 11, 14]. The lightest 1−−, 1++, 0++, 0−− resonance fields are in- G cluded, beingthe spin–1mesons represented through antisymmetric tensors proc-JJSC printed on February 2, 2008 3 Rµν [10, 12]. The last ingredient of RχT relies on the assumption that operators with a large number of derivatives are forbidden and only (p2) O chiral tensors are to be considered. The addition of higher power operators is expected to lead to a wrong growing behaviour of the Green-functions at large momenta. These ingredients yield the general lagrangian, ′ = (φ) + (R,φ) + (R,R,φ) +... (1) RχT G L L L L R R,R′ X X The operators with just Goldstones are given by χPT at (p2) [14]: O F2 (φ) = u uµ + χ . (2) G µ + L 4 h i The operators linear in the resonance fields were derived in Ref. [10]: F iG (R,φ) = V V fµν + V V uµuν + c Sχ +... (3) L 2√2h µν + i √2 h µν i dh +i R X The analysis of three-point functions and form-factors have requires the introduction of operator with two and three resonance fields [5, 11, 19]. In order to make the theory dual to QCD, it must be matched at the regions whereitis calculable, this is, atlow andhigh energies. Therecovery of the low energy QCD structure, described by χPT, is trivial once chiral symmetry has been properly incorporated. On the other hand, RχT must reproducetheOPEatshortdistances[15]. Forinstance,thematchingofthe V A correlator yields the well known Weinberg sum-rules and establishes − a relation between resonance parameters at LO in 1/N [20]. C 3.2. Next-to-leading order The one-loop diagrams give place to NLO contributions . They produce ultraviolet(UV)divergencesthatrequirenewoperatorswithNLOcouplings in order to be renormalised. Many of these operators can be actually re- moved throughtheuseoftheequationsofmotion[3,4]. Furthermore,ithas been proved that some matrix elements do not need local χPT operators to fulfill the renormalisation [6]. We will refer here to the example of the correlator Π(t) Π (t) Π (t) [5]. ≡ SS − PP The first step is to examine the absorptive part of the amplitude by means of the optical theorem. The contribution from a two-particle inter- mediate state M M to the spectral function, shown in Fig.(1), is in general 1 2 proportional to some squared form factors [5]: 2 ImΠ(t) (t) . (4) M1M2 ∝ FM1M2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 proc-JJSC printed on February 2, 2008 Fig.1. One-loopabsorptivecontributions to Π(t). All the lines stand for tree-level meson propagators in our perturbative calculation. The vanishing of (t) at infinite momentum [21] makes ImΠ(t) 0 ππ ππ F → when t . Demanding a similar vanishing behaviour for each separate → ∞ absorptive contribution ImΠ(t) leads to a series of constraints for the M1M2 form-factors at LO in 1/N [5]. C After the preliminary large–N analysis of the absorptive subdiagrams, C one is ready to aboard the renormalisation of the one-particle-irreducible vertex-functions entering in our amplitude [3, 4]. Imposing the proper UV asymptotic conditions in the absorptive part of Π(t) leads to the absence of new UV divergent structure [6]. No new operators are needed for the renormalisation, just a 1/N shift of the parameters existing at LO. C An alternative way to calculate NLO amplitudes [2, 3, 7] is the use of dispersion relations [5]. In our case, it is possible to write down an unsubtracted dispersive integral for Π(t). The two-meson cuts contribute to the correlator with a finite part, ∆Π(t) , and Π(t) depends now on M1M2 the renormalised resonance masses and couplings: 8cr2 8dr2 F2 Π(t) = 2B2 m m + + ∆Π(t) . (5) 0Mr2 t − Mr2 t t M1M2 S − P − MX1M2   The NLO contribution ∆Π(t) only depends on ImΠ(t) [5] and, at M1M2 M1M2 high energies, behaves like F2 F2M2 t 1 ∆Π(t) = δ(1) + S δ(2) +δ(2) ln − + . (6) t NLO t2 NLO NLO MS2! O(cid:18)t3(cid:19) e The matching of the one-loop RχT correlator to the OPE yields a NLO generalisation of the first and second Weinberg sum-rules [5]: 8cr2 8dr2 F2(1+δ(1) ) = 0, m − m − NLO 8cr2Mr2+8dr2Mr2+F2M2δ(2) = 8δ, (7) − m S m P S NLO − whereδ 3πα F4/4turnsouttobenumericallynegligible[e20]. Thematch- s ing is co≡mpleted by demanding that the 1 ln( t/M2) term also vanishes, t2 − S e (2) this is, δ = 0. These relations fix the renormalised resonance couplings NLO cr , dr in terms of the renormalised resonance masses Mr [5]. m m R e proc-JJSC printed on February 2, 2008 5 One-loop RχT reproduces at low energies the one–loop χPT structure, generating the proper running for the chiral couplings. Thanks to this, it is possible to provide predictions for the renormalised Lr(µ) at any µ in terms i of the renormalised RχT parameters. In our example, the short-distance matching of the form-factors at large–N and the correlator at NLO fixes C the chiral coupling Lr(µ) in terms of the renormalised masses Mr, yielding 8 R the prediction [5]: Lr(µ ) = (0.6 0.4) 10−3, for µ = 770 MeV. (8) 8 0 ± · 0 Themain error, also present at LO, comes from the scalar and pseudoscalar masses. The uncertainty on the saturation scale is completely removed. One must keep in mind that any large–N estimate of the LECs contains C an inherent theoretical error dueto theNLO runningfrom theloops. There is no particular saturation scale for all the χPT couplings. This uncertainty can be only removed by taking the calculation up to the one-loop level. 4. Open questions Although it is possible to extract some information about the couplings of highly excited mesons (M Λ ) [22]–[25], one still needs to specify R QCD ≫ the structure of the spectrum at high energies. It can be solved in some models [25]–[27] but, in general, the QCD spectrum is badly known in the ultraviolet. This forces to work under a MHA [17],introducing uncertain- ties [22,24]thatarereflected insomeproblemsintheshort-distancematch- ing of three-point Green functions [28]. An improved way to perform the matching would be desirable. In addition to making MHA a complete and self-consistentdescription,itwouldallow theexplorationofGreen-functions that are not order parameter and are actually dominated by the high part of the infinite series of resonances [22, 25]. Alaststandingproblemistheequivalencebetweenlarge–N lagrangians. C The spin–1 mesons can be described through different formulations [7, 12, 13, 29]. 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