Strings in AdS_4 x CP^3: finite size spectrum vs. Bethe Ansatz
Davide Astolfi, Gianluca Grignani, Enrico Ser-Giacomi, A. V. Zayakin
PPrepared for submission to JHEP
ITEP-TH-51/11
Strings in AdS × C P : finite size spectrum vs.Bethe Ansatz Davide Astolfi, a Gianluca Grignani, a Enrico Ser-Giacomi, a A.V. Zayakin a,b a Dipartimento di Fisica, Università di Perugia,I.N.F.N. Sezione di Perugia,Via Pascoli, I-06123 Perugia, Italy b Institute of Theoretical and Experimental Physics,B. Cheremushkinskaya ul. 25, 117259 Moscow, Russia
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We compute the first curvature corrections to the spectrum of light-cone gaugetype IIA string theory that arise in the expansion of AdS × C P about a plane-wave limit.The resulting spectrum is shown to match precisely, both in magnitude and degenerationthat of the corresponding solutions of the all-loop Gromov–Vieira Bethe Ansatz. The one-loop dispersion relation correction is calculated for all the single oscillator states of thetheory, with the level matching condition lifted. It is shown to have all logarithmic diver-gences cancelled and to leave only a finite exponentially suppressed contribution, as shownearlier for light bosons. We argue that there is no ambiguity in the choice of the regular-ization for the self-energy sum, since the regularization applied is the only one preservingunitarity. Interaction matrices in the full degenerate two-oscillator sector are calculatedand the spectrum of all two light magnon oscillators is completely determined. The samefinite-size corrections, at the order J , where J is the length of the chain, in the two-magnonsector are calculated from the all loop Bethe Ansatz. The corrections obtained by the twocompletely different methods coincide up to the fourth order in λ (cid:48) ≡ λJ . We conjecturethat the equivalence extends to all orders in λ (cid:48) and to higher orders in J . Keywords:
AdS-CFT correspondence, Penrose Limit and pp-wave background a r X i v : . [ h e p - t h ] A p r ontents × C P : the procedure 7 SU (2) × SU (2) subsector 184.2 How to deal with auxiliary roots 194.3 Claim to exactness 21 References 26
The appearance of integrable structures both at strong and weak coupling has given hopefor a complete solution to the spectral problem of the AdS / CFT duality in the planarlimit. This is the best-understood example of a duality between gauge theory and stringtheory, it states the equivalence between the IIB superstring theory on AdS × S and N = 4 super Yang-Mills (SYM) theory in dimensions. In [1, 2] an all-loop asymptotic Betheansatz has been proposed for the AdS / CFT duality and starting from the mirror versionof the Beisert-Staudacher’s equations, further the Y-system was formulated that alllowscomputation of anomalous dimensions for opertaors of any length [3–6]. This Y-systempasses some very important tests: it incorporates the full asymptotic Beisert-Staudacher’sBethe ansatz at large length J and it reproduces all known wrapping corrections.Even if the AdS/CFT correspondence is at present best understood for AdS , alsoin the more recent AdS / CFT duality the solution to the spectral problem, at least forthe sector described by a coset space, seems to be at reach in the planar limit thanks tointegrability [7]. The AdS / CFT correspondence is an exact duality between type IIAsuperstring on AdS × C P and a certain regime of the ABJM-theory [8]. The ABJMtheory is an Chern-Simons N = 6 gauge theory with matter dual to M-theory compactifiedonto AdS × S / Z k . It possesses a U( N ) × U( N ) gauge symmetry with Chern-Simons like– 1 –inetic terms at level k and − k ; if the ’t Hooft coupling λ = Nk is (cid:28) λ (cid:28) k the gravityside being effectively rendered as a IIA superstring on AdS × C P .The integrability of the AdS / CFT duality due to the reduced number of supersymme-tries, offers interesting new challenges. The sector of the theory described by a coset spacewas proven to be classically integrable, but classical integrability for the whole theory hasstill to be demonstrated [9–13]. Nevertheless, the semiclassical and quantum integrabilityof some sectors of the theory have received plenty of attention both at weak [14–23] and atstrong coupling [24–30]. In particular an all-loop asymptotic Bethe Ansatz has been pro-posed [7] and a Y-system has been suggested also for the AdS / CFT duality [3, 4, 6, 31, 73].More recently nontrivial evidence for the scattering amplitudes/Wilson loop duality for thistheory has been given [32–35].According to the AdS/CFT correspondence energies of excited states of superstrings inspecific curved backgrounds should coincide with the anomalous dimensions of appropriateoperators of the corresponding gauge field theory. For the planar limit the coupling iszero, however the string is still in a curved space and thus its two-dimensional world-sheettheory is not interaction-free. Calculating the superstring spectrum in such backgrounds istherefore still a complicated problem. Nevertheless the nontrivial interactions become smallwhen one takes the Penrose limit of the metric [36, 37]. Corrections to the free spectrumcan then be computed perturbatively as an expansion in inverse powers of the backgroundcurvature radius R .This idea was suggested by Callan et al. in [38, 39] for AdS / CFT correspondence.The outcome of [38, 39] and the respective studies on the field theory side [40], have beenimportant for understanding the integrability of the AdS/CFT correspondence. In [38, 39]it was shown for the first time that there is a disagreement between field theory operatorsanomalous dimensions and the respective string energies at three loops. The disagreementwas afterwards interpreted as a breakdown of a double scaling limit and resolved by includ-ing the dressing factor that interpolates nontrivially from weak to strong coupling in theBethe equations describing the spectra of the gauge and the string theory [2, 41–43].In [44] a complete calculation of the curvature corrections to the pp-wave energy ofthe two oscillator non-degenerate bosonic states in the decoupled SU(2) × SU(2) sector oftype IIA superstring on AdS × C P was performed. This study was initiated in [26] andreexamined in [28, 45]. In [46] the interacting Hamiltonian for oscillations in the near planewave limit of AdS × C P was calculated. This is a crucial tool for the computations ofthis Paper and it is given in[44] by a perturbative expansion in terms of /R powers H = H ,B + H ,F + 1 R ( H ,B + H ,BF ) + 1 R ( H ,B + H ,F + H ,BF ) + . . . . (1.1)where R is the C P radius. For brevity we shall further refer to “third-order Hamiltonian” H ≡ H ,B + H ,BF and “fourth-order Hamiltonian” H = H ,B + H ,F + H ,BF .The quadratic Hamiltonian term, H B + H F , is the plane-wave free Hamiltonianfrom[44] where fermionic and bosonic fields are fully decoupled [15, 24, 47, 48]. A pe-culiarity of this theory is that in the pp-wave limit the eight massive bosons and eightmassive fermions have different worldsheet masses. Four fermions and four bosons are– 2 –heavy”, while the remaining four fermions and four bosons are “light” having a world sheetmass which is / of that of the heavy ones.The unique feature of the AdS case is the presence of cubic terms H in the Hamil-tonian [26]. This yields extra terms for the matrix element of some arbitrary | f (cid:105) state inaddition to the expectation value of the quartic Hamiltonian H , (cid:104) f | H | f (cid:105) : now the energycorrection δE (2) f looks like δE (2) f = 1 R (cid:88) | i (cid:105) |(cid:104) i | H | f (cid:105)| E f − E | i (cid:105) + (cid:104) f | H | f (cid:105) (1.2)where | i (cid:105) is an intermediate state and summation is done in all admissible channels. Thefirst term in (1.2) gives rise to extra logarithmic divergences. However the total answer mustbe finite. This result can be achieved by imposing a unique normal ordering prescriptionas in [44], the ordering prescription being the Weyl prescription.For readers’ convenience, let us briefly describe here the main characters of this work,referring to [44] for exact definitions. There are four light bosonic oscillators a , a , ˜ a , ˜ a ;four heavy bosonic oscillators ˆ a i , i = 1 . . . ; light fermions d α , heavy fermions b α where α is the Dirac ten-dimensional index. The dispersion laws are summarized in the Table 1. Table 1 . Dispersion laws
State Energy a , a ω n − c/ a , ˜ a ω n + c/ a i Ω n d ω n b, even Ω n − c/ b, odd Ω n + c/ where the frequencies are ω n = (cid:113) n + c , Ω n = √ n + c . (1.3)Referring to the “ SU (2) × SU (2) sector” we mean states solely consisting of either a , a or ˜ a , ˜ a . Parameters of the theory are c , which is meant to be large c = 4 JR , (1.4)the curvature radius R R = 4 π √ λ = 4 πJ √ λ (cid:48) , (1.5)and the Frolov-Tseytlin coupling constant λ (cid:48) = λJ . (1.6)– 3 –he main result of this Paper is in fact the extension of the results of [44] to thewhole set of degenerate two-oscillator light bosonic states, those states whose energies canbe also compared directly to the corresponding solutions of the Bethe equations. Theseare: 8 states built up by two bosonic oscillators and 16 made of two fermionic excitations.They have degenerate plane-wave energy, thus the procedure for computing the spectrumis straightforward, yet technically much more complicated than that in [44]: one must,as in standard quantum mechanical perturbation theory, diagonalize the mixing matrix ofthe perturbation, solve the secular equation and find eigenvectors and eigenvalues, whichare the finite size corrections. The spectrum of such excitations, which we do not displayhere for brevity (see the Tables on page 15), can be computed exactly in λ (cid:48) and thencompared with the corresponding solutions of the asymptotic Bethe equations: this involvesthe analysis of configurations carrying auxiliary roots and thus provides a test of the Betheprogram even more stringent than the one carried out in [26], where the only activatedroots where the fundamental ones, carrying the physical momentum. The Bethe equationsmust be solved perturbatively, by a judicious Ansatz for the expansion of the momentum inpowers of λ (cid:48) = λJ and J around the asymptotic free solution, and employing a consistentregularization technique for the configurations with auxiliary rapidities 0 or ∞ [49]. Havingsolved the Bethe equations for the momenta, one plugs the solution in the dispersion relationand gets the spectrum: we obtained it up to O (cid:0) λ (cid:48) (cid:1) but it can be improved with somemore computational effort. Actually we consider the O (cid:0) λ (cid:48) (cid:1) sufficient, since the dressingphase factor interpolating from weak to strong coupling starts at order λ (cid:48) and there is nophysical mechanism entering at higher orders other than those already encountered. Thuswe consider such a matching a very satisfying test to consider it an all order result.We defer to the main body of the Paper the detailed discussion of the basis that diag-onalizes the string theory perturbation Hamiltonian and the corresponding Bethe Ansatzconfigurations. To summarize we display here the Tables 2,3 of spectrum identifications.They refer to the Bethe configurations in the language of [7]. The integers K i are multi-plicities of the i -th Bethe root. The energies of the identified submultiplets are identical atthe order /J ; this is our main result, which is derived in the main body of the paper. Table 2 . Boson-boson state identification
Multiplicity Corresponding BA states Corresponding ST states K K ¯4 K K K State nr. branch 1 ,
74 2 0 1 1 1 branch 2 , , ,
81 1 1 1 1 branch 1 branch 2 , (1.7)Yet this is not the end of the story about the spectrum of the two-oscillator bosonicstates: Each of the 8 bi-bosonic and 16 bi-fermionic state has a further contribution to theenergy which is given by the same infinite sum appearing in the eqs. 1.2 and 1.3 of [44].– 4 – able 3 . Fermion-fermion spectrum comparison Multiplicity Corresponding BA states Corresponding ST states K K ¯4 K K K State nr. ,
248 1 1 2 2 0 9 , , , , , , ,
222 0 2 1 0 branch 1 branch 2 branch 1 , , , , ,
161 1 2 1 0 branch 2 (1.8)The computation of such term was one of the results of [44] and it is therefore appropriateto recapitulate its interpretation and inquire whether the further developments carried outin this Paper might shed more light about it.There are no divergences in the eqs. 1.2 and 1.3 of [44] due to a nontrivial, yet natural,ordering prescription for the quantum operator associated to the classical quartic Hamilto-nian, which gives infinite sums in the spectrum cancelling those divergencies arising fromthe cubic Hamiltonian evaluated at second order in perturbation theory. This scheme ap-plies unchanged for the 24 bosonic states considered in this Paper, thus providing furtherevidence of the naturalness of such ordering prescription.In [44] it was shown that the infinite sum of Eqs. 1.2 and 1.3 appears diagonally in themode numbers for the states having an arbitrary number of light bosonic oscillators. Fur-thermore, if one considers a single-impurity light bosonic state, without the level matchingcondition which would otherwise have forced its mode number to be vanishing, the energyof this state displays the same kind of contribution. Its natural interpretation is there-fore as a correction to the dispersion law of a single magnon. This exponential one-loopeffect must be similar to the Lüscher terms coming form a field theory or Bethe Ansatzcalculation. One should be able to directly compute it from the Lüscher formula (see thereview [50] and references in it). This effect is yet another example of the exponentiallysmall finite size corrections to the magnon dispersion relation that, for type IIA superstringon AdS × C P , were first computed in the giant magnon limit in [25] (see also [51–54])and derived from Lüscher’s corrections in [55–60]. Finite-size effects were also calculatedfor spiky strings in AdS × C P and for giant magnons in the presence of an arbitrarytwo-form B field. Alternative methods for dealing with giant magnons on AdS × C P byemploying the so-called dressing method were suggested in [61–64].On these grounds it is quite crucial to inquire whether a state built by a light non levelmatched fermionic oscillator indeed displays the same kind of contribution to the spectrum:such a computation for a fermionic mode actually also involves the issue of quantum orderingof classical terms quartic in the fermions, which was not addressed in [44] since there, they– 5 –ere not relevant. The generalization to such terms of the Weyl ordering is remarkably theunique choice leading to a finite spectrum. This confirms the naturalness of our orderingprescription. Even more remarkably, for each light fermionic oscillator of a state havingan arbitrary (including just one) number of them, one obtains a contribution which is thesame infinite sum of Eqs. 1.2 and 1.3 of [44]. This clearly reinforces its interpretation as afinite size correction to the magnon dispersion relation.The light-magnon dispersion relation is fixed by symmetries of the theory E = (cid:114)
14 + 4 h ( λ ) sin p (1.9)but the scaling function h ( λ ) [15, 24, 47] that interpolates from the strong to the weakcoupling is not. The magnon dispersion relation [65, 66] in the AdS / CFT duality is E = (cid:114) f ( λ ) sin p , (1.10)where f ( λ ) happens to be equal to λπ at both strong and weak coupling. For the AdS / CFT duality the function h ( λ ) looks like λ + O ( λ ) at weak coupling [14, 15, 47] and like (cid:113) λ + O ( λ ) at strong coupling [15, 25, 47]. It has been computed up to 4 loops onthe field theory side in [21, 22, 67]. Quasiclassical calculations for the spinning and foldedstrings have yielded [27, 58, 68–70] h ( λ ) = (cid:114) λ a WS1 + O (cid:18) √ λ (cid:19) where a WS1 = − log 22 π , λ (cid:29) (1.11)the superscript WS standing for the world-sheet.Gromov and Vieira, on the other hand by means of the semiclassical Bethe Ansatz[7, 71], extrapolating to the strong coupling of the all loop Ansatz of [67], obtained h ( λ ) = (cid:114) λ a AC1 + O (cid:18) √ λ (cid:19) where a AC1 = 0 , λ (cid:29) (1.12)where the superscript AC means the algebraic curve.The different values for h ( λ ) come from different regularizations used. If one treatsall modes in a uniform way, one gets a (cid:54) = 0 . If one takes care of heavy and light modesdifferently, and remembers that heavy modes are kind-of bound states [29] of the lightmodes and therefore must be cut off at a twice higher value of the momentum as the lightones, one gets the zero a . In this work we argue that there is a definitive evidence from theunitarity preservation requirement to choose a unique regularization, the one with differentcutoffs.Namely, since a “heavy-light-light” vertex is present in the S-matrix, the same cutoff onmode numbers of light and heavy states will render the regularized S-matrix non-unitary.Only cutting the self-energy summation off in such way that preserves unitarity at eachlarge but finite value of the cutoff is acceptable. For conventional global symmetries weknow that a regularization breaking a symmetry of a theory results in an anomaly. Unitarity– 6 –s a different kind of symmetry, realized on quantum level solely, and not at the level ofthe classical Lagrangian. However, there is a great degree of resemblance between the log 2 pieces in the self-energy sums due to broken unitarity by regularization and the presence ofthe anomalous non-zero parts in otherwise classically zero divergences of Noether currentsat quantum level.Actually the curvature corrections to the string state energies that have been computedin [44] and in this Paper would notice the presence of an a term in h ( λ ) . In the BMN limitthe momentum is p = πnJ and expanding for large J at nonzero a in h ( λ ) yields E = (cid:118)(cid:117)(cid:117)(cid:116)
14 + 4 (cid:32)(cid:114) λ a (cid:33) sin p (cid:39) (cid:114)
14 + 2 λ (cid:48) n π + √ a J (cid:104) π n √ λ (cid:48) − π n λ (cid:48) / + 96 π n λ (cid:48) / + O (cid:16) λ (cid:48) / (cid:17)(cid:105) (1.13)namely there will be J = cR term with semi-integer powers of λ (cid:48) . Such a term could arisein the finite size energies of two light magnons (see eqs. 1.2 and 1.3 of [44]) however, withthe regularization we use, it does not. The λ (cid:48) power expansion of the first terms in the eqs.1.2 and 1.3 of [44] yields integer powers of λ (cid:48) , which are basically due to the interactionsbetween magnons, while the Bessel function sum with produces non analytic terms whichare exponentially suppressed like ∼ e − J √ λ , thus they are compatible with a = 0 solely.The Paper is organized as follows. In Section (2) we discuss the dispersion relationsof single oscillator states with lifted level matching condition, the regularization procedureand the unitarity argument that proves its consistency. In Section 3 we explicitly computethe energies of the string states and construct the mixing matrix for two-oscillator lightbosonic states. In Section 4 we derive the solutions of the Bethe equations correspondingto the string states discussed in Section 3. In Section 5 we draw our conclusions. × C P : the procedure The type IIA superstring of interest to us lives in the AdS × C P background where atwo-form and four-form Ramond-Ramond fluxes are present. The corresponding geometryis described in the appendix (A) of [44]. We compute the corrections at the order /R =1 / (4 πJ √ λ (cid:48) ) to the energies of the one- and two particle sectors. We keep J large, λ largeand λ (cid:48) = λJ = fixed . Thus we can say we are in the near-BMN limit. Our corrections areperturbative quantum mechanical corrections. Unlike the pure BMN case, we already havesome interaction in the system. Unlike the folded or rotating string case, semiclassics arenot applicable here, so exact quantum-mechanical analysis is due. Unlike the giant magnoncase, finite size effects do not decouple from the quantum effects. Thus our limit is in someway unique since it resides at strong coupling, yet is perturbatively treatable.Our basic string configuration is a point-like IIA string moving in the SU(2) × SU(2) subsector of C P and along the time direction R t on AdS [15, 24, 47]. The specific plane-wave background, has been obtained in [24] and discussed extensively in [26, 46], therefore– 7 –e describe it only in the appendix (B) of [44]. The quantization procedure for the freeplane-wave Hamiltonian that has been done in the Section 2.1. of [44]. The derivation ofthe interaction Hamiltonian is found in the Appendix (D) of [44]. All geometry, Gammamatrices, and quantization notations are similar to those of [44]; with the exception of the H F , all other Hamiltonian pieces are taken directly from there. u impurity We start with heavy bosonic states. Consider the single impurity state, non level matched | u (cid:105) = ( a u n ) † | (cid:105) (2.1)Cancellation of divergencies for bosons is a crucial test for the validity of our theory. Toillustrate how divergencies cancel in the dispersion relation for the fourth heavy boson, weshow the partial contributions of different sectors in the Table (4) below. The boson energyis E u n = Ω n + 1 R Ω n (cid:88) q (cid:15) u q . (2.2) Table 4 . Cancellation of divergencies for bosons
Hamiltonian piece (cid:15) u q H − q cω q + n ω q ω n + q + nqω q ω n + q H light ˆ u ˆ u BB − n q c ω q − nq Ω n c − n cω q − q cω q H heavy ˆ u ˆ u BB nq Ω n c − q Ω n c Ω q H u − n q c Ω q + nq Ω n c − n c Ω q − q c Ω q H light B F q Ω n c ω q − nq Ω n c H heavy B F q Ω n c Ω q − nq Ω n c + n c Ω q Total n c (cid:16) q − ω q (cid:17) The “total” line of the table refers to the sum both over partial contributions and thesummation mode index. This lets the expressions be additionally simplified, since the dumbvariable allows constant shifts, leading to extra cancellations. Also note exact cancellationof quadratic divergencies. By summing over partial channels shown above the dispersionrelation up to finite size becomes E u = (cid:114) n c + 2 n c Ω n R N (cid:88) q = − N q − N (cid:88) q = − N ω q + – 8 – cR Ω n N (cid:88) q = − N (cid:26) q ω q − (2 ω q + ω q + n + ω q − n ) + c (cid:18) ω q + 1 ω q + n + 1 ω q − n (cid:19) − c (cid:18) ω n + q ω q − ω q ω n + q + ω − n + q ω q − ω q ω − n + q (cid:19) (cid:27) (2.3)We already know very well how to treat the sum in the first line in eq. (2.3) since it isexactly the one appearing in [44], giving rise to the Bessel functions series. For this state,this sum appears uniquely from contributions due to the quartic Hamiltonian. The otherterms in the equation are organized as follows: their cutoff is N because the sum is over alight mode.Since the sum in the second and third lines of (2.3) are convergent and have the samecutoff N , we can safely send it to infinity and performing some shifts we can see that allthe terms sum up to zero. Therefore the dispersion relation is rather the following simplerone: E u = (cid:114) n c + 2 n c Ω n R N (cid:88) q = − N q − N (cid:88) q = − N ω q (2.4) Single u impurity The interaction Hamiltonian is not explicitly invariant with regard to u → u i replacement,since the fourth direction is special it belongs to C P while u i ∈ AdS for i = 1 , , . There-fore we must consider now the single impurity heavy state with a u oscillator separately | u (cid:105) = ( a u n ) † | (cid:105) (2.5)We see by an explicit calculation that its dispersion relation up to finite size is the same asfor the u single oscillator state: E u = (cid:114) n c + 2 n c Ω n R N (cid:88) q = − N q − N (cid:88) q = − N ω q (2.6)By virtue of the same argument as above we can see that all divergencies cancel, whereasthe remaining term contains only an exponentially small correction in J . Fermions
Consider now the fermionic states: the light one | d (cid:105) = d † αn | (cid:105) , (2.7)and the heavy one | b (cid:105) = b † αn | (cid:105) . (2.8)We check the fermion dispersion relation perturbatively and demonstrate the results in theTable (5). – 9 – able 5 . Cancellation of divergencies for fermions. Separate sectors give divergent results, theremnant is finite. Hamiltonian piece Light state energy correction (cid:15) d Heavy state state energy correction (cid:15) b H light loop − n cω q − q cω q + q c Ω q − n cω q H heavy loop − n c Ω q + q cω q − c ω q − q c Ω q − c q H B F light loop − q n c ω q − n cω q − q cω q + c ω q − q n c ω q − n cω q − q cω q − c ω q H B F heavy loop − q n c Ω q − n c Ω q − q c Ω q − q n c Ω q − n c Ω q − q c Ω q H F light loop q n c ω q + n cω q + q cω q q n c ω q + n cω q + q cω q + c ω q H F heavy loop − n cω q + q n c Ω q + n c Ω q − q cω q + q c Ω q + c q q n c Ω q + n c Ω q + q c Ω q Total n c Ω q − n cω q n c Ω q − n cω q The superficial divergences present in the loop contributions to fermions do cancel indeed,only a finite exponentially suppressed part (as e − constJ ) remaining. Here we demonstratehow various contributions cancel in order to leave a finite piece only. For light states E d = ω n + 1 R ω n (cid:88) q (cid:15) dq , (2.9)for heavy states E b = Ω n + 1 R Ω n (cid:88) q (cid:15) bq . (2.10)The partial (taken over separate channels) (cid:15) dq and (cid:15) bq are given in the Table (5). Unitarity-preserving Regularization
The sums of the light and heavy self-energies (cid:80) q (cid:15) dq , (cid:80) q (cid:15) bq are convergent, but have to beregularized to be ascribed a numerical value. We use here the most natural “algebraic-curve”regularization prescription suggested by the form of the cubic Hamiltonian [44]. That is, wecut the heavy modes at a cutoff N , and the light modes at a cutoff N , where N is afterwardssent to infinity. The arguments that have been used in literature for this regularization havebeen reiterated below and will be discussed yet once more in the Conclusion; here we wishto bring in a very generic argument, which leaves this “unequal-frequency” regularizationas the only permissible one. Zarembo has shown [29] that the heavy-to-light vertex isorganized in our theory in such a way that the light two-particle cut starts exactly in thepoint where the heavy particle pole is. Thus an on-shell decay heavy-into-two-light modesis seen to be possible. Consider now the most general requirement of validity of a quantumfield theory, the equation on the unitarity of the S-matrix SS † = 1 (2.11)– 10 –et us write it down more explicitly in the 1-particle heavy sector (indices , i ) and the2-particle sector (indices , jk ) S , im S , † mi (cid:48) + S , i,jk S , † jk,i (cid:48) = 1 , i,i (cid:48) (2.12)where summation is meant over the repeating Hilbert space indices. The second part ofthe left-hand side must be taken into account due to the mentioned result by Zarembo.The relation is valid at any mode number. Suppose we regularize the theory now. Thiseffectively means that both for the unit operator in the Hilbert space and for the S-matricesall elements above some N cutoff are filled in with zeros. Suppose that N cutoff is special foreach of the sectors. Thus we have below the cutoffs S , n, n S , † n, n (cid:12)(cid:12)(cid:12) n
F H B F + H F F B H BBF H F + H F F B (3.8)where F and B represent the two-fermion and two-boson oscillator states. This Hamiltonianis symbolically depicted in Fig. (1).The corresponding pieces of the mixing matrix are shown below.– 13 – ++ F B + B F Figure 1 . Schematic diagram of the contributing sectors of string Hamiltonian
The four-boson contribution to the BB sector is the following × matrix given in termsof the basis states u . . . u : H B = 2 n cω n Diagonal (cid:16) − n c − , − n c − , − n c − , − n c − , − n c , − n c − , − n c , − n c − (cid:17) . (3.9)The H B contributes via three types of intermediate states H B = (cid:88) j =1 H B j , (3.10)where the intermediate states s j are: | s (cid:105) = | ˆ a † | (cid:105) , | s (cid:105) = | ˆ a †− a − b a † a ˜ a † b | (cid:105) , | s (cid:105) = | ˆ a †− a − b − c − d a † a ˜ a † b a † c ˜ a † d | (cid:105) . (3.11)The following matrix (in terms of the same bosonic basis as above) is the contribution ofthe first channel H B = 4 n cω n Diagonal (cid:16) , , ω n + c , , , , , (cid:17) . (3.12)The second channel gives H B = 2 n cω n Diagonal (cid:16) , , , , , − , , − , (cid:17) . (3.13)Finally the third channel gives H B = − n cω n Diagonal (cid:16) , , ω n − c , , , , , (cid:17) . (3.14)Switch now to the two fermionic oscillators. The quartic purely fermionic Hamiltonianis H ,F = − i (cid:16) ¯ θ Γ Γ + M θ (cid:48) + ¯ θ Γ + M Γ θ (cid:48) (cid:17) − c ( A ,σ − ˜ A ,σ ) − A + ,σ ( ˜ C + − + ˜ B +56 + ˜ B +78 ) + 14 ˜ A + ,σ ( C + − − C ++ + B +56 + B +78 ) − c (cid:88) i =1 C i − c (cid:88) i =5 (cid:104) C + i − s i B +4 i + 12 (cid:88) j =5 (cid:15) ij B + − j (cid:105) . (3.15)– 14 –nlike the other sectors where we have referred the reader to [44] for the Hamiltonianexpression, we show H ,F explicitly, since we correct a misprint of the earlier version.In the basis u . . . u the × matrix of the mixing in the fermionic sector is H F = − n cω n Diagonal (cid:16) n c + , n c − , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c + , n c − , n c − (cid:17) . (3.16)The mixing term in the FF sector coming from the H F F B is the following: H F F B = 2 n cω n Diagonal (cid:16) , − , , , , − , − , − , , , , , , , , (cid:17) . (3.17)This term comes from the long intermediate channel; the short channel yields identically0. Of the 16 fermionic states, u . . . u only states u , u have Lorenzian AdS spin s = 0 and thus could have in principle mixed with bibosonic states, all of which have spin 0. Thismixing indeed is vanishing. There could be in principle a mixed term H BBB H BF F , with abosonic zero mode intermediate state as shown in Fig. (1), but explicit calculation shows itis zero. Summing the contributions we get the full mixing matrix, which our judicious choiceof the basis has automatically brought to a diagonal form. We have therefore obtained theset of eigenstates and eigenvalues exact in λ (cid:48) . The following is the finite-size spectrum oftwo bosonic oscillations, where, on the rightmost column, in order for further comparisonwith the Bethe Ansatz, we have expressed the spectrum in terms of λ (cid:48) and J , expanding upto fourth order in λ (cid:48) : see table 6. The finite-size spectrum of the two fermionic oscillations Table 6 . Finite-size spectrum of two bosonic oscillations state spectrum expansion of the spectrum u − n Ω n c R ω n J (cid:0) − n π λ (cid:48) + 96 n π λ (cid:48) − n π λ (cid:48) + 6144 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n Ω n c R ω n J (cid:0) − n π λ (cid:48) + 96 n π λ (cid:48) − n π λ (cid:48) + 6144 n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) is given in the Table 7 below. The Tables above show that each energy has an even,at least double, multiplicity, as one expects from the symmetry of the Bethe frameworkconfigurations, as we shall discuss in the Section below. Thus the above string spectrum,one of the main results of this Paper, can be consistently compared with the solutions of– 15 – able 7 . Finite-size spectrum of two fermionic oscillations state spectrum expansion of the spectrum u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n ( c +2 n ) c R ω n J (cid:0) − n π λ (cid:48) + 32 n π λ (cid:48) − n π λ (cid:48) + 2048 n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u − n c R ω n J (cid:0) − n π λ (cid:48) + 256 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u
23 4 n ( c − n ) c R ω n J (cid:0) n π λ (cid:48) − n π λ (cid:48) + 768 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) u
24 4 n ( c − n ) c R ω n J (cid:0) n π λ (cid:48) − n π λ (cid:48) + 768 n π λ (cid:48) − n π λ (cid:48) + . . . (cid:1) the Bethe equations, providing a significant test of them and of the integrability of thestring sigma model in the near-BMN limit. Notice also that the results above, togetherwith the eight bosonic states addressed in [44], sheds a complete light on the spectrum ofall the 32 bosonic two-impurity light states of the theory. Finally, for comparison with theBethe Ansatz framework, the string spectrum above, exact in λ (cid:48) , can be expanded in powerseries, as shown in the Table 8 below. The c i are the coefficients of the λ (cid:48) expansion of thespectrum defined as (cid:15) = (cid:15) + 1 J (cid:88) i =1 c i λ (cid:48) i (cid:0) π n (cid:1) i + O (cid:18) J (cid:19) , (3.18)where (cid:15) is the term of order /J , n is the mode number of the two-oscillator state.– 16 – able 8 . Coefficients c i and state multiplicities. State type (BB,FF) and Nr. i c c c c c Multiplicity
F F : 23 ,
24 1 −
32 32 −
32 32 F F BB : 5 , F F : 9 , , , , , , ,
22 0 −
12 12 −
12 12
10 = 8
F F + 2 BB BB : 2 , , , F F : 11 , , , , , − −
12 12 −
10 = 6
F F + 4 BB BB : 1 , − −
32 32 − BB (3.19) The Bethe roots are quantized through the algebraic equations [7]: K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 − /x ,k x +4 ,j − /x ,k x − ,j K ¯4 (cid:89) j =1 − /x ,k x +¯4 ,j − /x ,k x − ¯4 ,j , K (cid:89) j (cid:54) = k u ,k − u ,j − iu ,k − u ,j + i K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i , K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 x ,k − x +4 ,j x ,k − x − ,j K ¯4 (cid:89) j =1 x ,k − x +¯4 ,j x ,k − x − ¯4 ,j (cid:32) x +4 ,k x − ,k (cid:33) L = K (cid:89) j (cid:54) = k u ,k − u ,j + iu ,k − u ,j − i K (cid:89) j =1 − /x − ,k x ,j − /x +4 ,k x ,j K (cid:89) j =1 x − ,k − x ,j x +4 ,k − x ,j × (4.1) × K (cid:89) j =1 σ BES ( u ,k , u ,j ) K ¯4 (cid:89) j =1 σ BES ( u ,k , u ¯4 ,j ) , (cid:32) x +¯4 ,k x − ¯4 ,k (cid:33) L = K ¯4 (cid:89) j =1 u ¯4 ,k − u ¯4 ,j + iu ¯4 ,k − u ¯4 ,j − i K (cid:89) j =1 − /x − ¯4 ,k x ,j − /x +¯4 ,k x ,j K (cid:89) j =1 x − ¯4 ,k − x ,j x +¯4 ,k − x ,j ×× K ¯4 (cid:89) j (cid:54) = k σ BES ( u ¯4 ,k , u ¯4 ,j ) K (cid:89) j =1 σ BES ( u ¯4 ,k , u ,j ) , where the spectrum of string energies is expressed in terms of the roots u and u ¯4 , whichcarry momentum, as follows: E = h ( λ ) Q , (4.2)the conserved charges being expressed in terms of the roots as Q n = K (cid:88) j =1 q n ( u ,j ) + K (cid:88) j =1 q n ( u ¯4 ,j ) , q n = in − (cid:18) x + ) n − − x − ) n − (cid:19) . (4.3)The Zhukovsky variables are defined in terms of the roots as x + 1 x = uh ( λ ) , x ± + 1 x ± = 1 h ( λ ) (cid:18) u ± i (cid:19) . (4.4)– 17 –ecalling that p j = i log x +4 ,j x − ,j and ¯ p j = i log x +¯4 ,j x − ¯4 ,j , we have E = K (cid:88) j =1 (cid:18)(cid:114) h ( λ ) sin p j − (cid:19) + K ¯4 (cid:88) j =1 (cid:32)(cid:114) h ( λ ) sin ¯ p j − (cid:33) . (4.5)At large ’t Hooft coupling we have h ( λ ) (cid:39) (cid:112) λ/ . (4.6)The rapidity variable expressed in terms of the momentum of the roots is given by u ,j = 12 cot (cid:16) p j (cid:17)(cid:114) h ( λ ) sin (cid:16) p j (cid:17) . (4.7)In the near plane wave limit, the BES kernel [2] reduces to the AFS phase factor [42]: σ AFS ( u j , u k ) = e iθ jk , (4.8)where θ jk = ∞ (cid:88) r =2 h ( λ ) (cid:2) q r ( x j ) q r +1 ( x k ) − q r ( x k ) q r +1 ( x j ) (cid:3) . (4.9)The Bethe equations can be solved for the momenta p j in the near plane wave limit onlywith a judicious perturbative Ansatz as p j = 2 πn j J + AJ + Bλ (cid:48) J + Cλ (cid:48) J + Dλ (cid:48) J . . . , (4.10)where we solve order by order determining the expansion coefficients. Eventually we plugthe solution for the momenta in the dispersion relation (4.5) to get the spectrum. SU (2) × SU (2) subsector Consider the cases ( K u , K u ¯4 , K u , K u , K u ) = (2 , , , , and ( K u , K u ¯4 , K u , K u , K u ) =(0 , , , , , which are clearly identical . Due to the level matching condition, we have onlyone independent momentum, p . Plugging the expansion (4.10) in the Bethe equations, onegets, up to order λ (cid:48) and J : J (cid:2) A − πn + λ (cid:48) (cid:0) B + 8 n π (cid:1) + λ (cid:48) (cid:0) C − n π (cid:1) + λ (cid:48) (cid:0) D + 192 n π (cid:1)(cid:3) = 0 , (4.11)which completely determines the momentum up to the desired perturbative order. We have A = 2 nπ, B = − n π , C = 32 n π , D = − n π , (4.12)which plugged in the dispersion relation (4.5) gives the spectrum: E = 4 n π λ (cid:48) − n π λ (cid:48) +32 n π λ (cid:48) + 1 J (cid:0) n π λ (cid:48) − n π λ (cid:48) + 448 n π λ (cid:48) − n π λ (cid:48) (cid:1) . . . (4.13)– 18 –hich is the spectrum of the string states | s , (cid:105) = (cid:16) a , n (cid:17) † (cid:16) a , − n (cid:17) † | (cid:105) , addressed in [26].Consider now the case ( K u , K u ¯4 , K u , K u , K u ) = (1 , , , , . Similarly, due to thelevel matching condition, there is only one independent momentum, p . Yet we can buildtwo different configurations, which shall be degenerate: the u root carrying momentum p and the u ¯4 carrying − p , or viceversa. The perturbative expansion of the Bethe equationsreads: J (cid:2) A + λ (cid:48) B + λ (cid:48) (cid:0) C + 16 n π (cid:1) + λ (cid:48) (cid:0) D + 128 n π (cid:1)(cid:3) = 0 , (4.14)which gives A = 0 , B = 0 , C = − n π , D = − n π , (4.15)and therefore E = 4 n π λ (cid:48) − n π λ (cid:48) + 32 n π λ (cid:48) − J (cid:0) n π λ (cid:48) − n π λ (cid:48) (cid:1) + . . . , (4.16)which is the spectrum of the string states | t , (cid:105) = (cid:16) a , n (cid:17) † (cid:16) a , − n (cid:17) † | (cid:105) , addressed in [26].Actually the matching, which can be perturbatively checked at arbitrary high ordersin λ (cid:48) , between these solutions to the Bethe equations and the near plane wave spectrumof string states in the SU (2) × SU (2) subsector, discussed in [26], has provided one of theearliest tests of the all loop asymptotic Bethe Ansatz proposed in [7]. The one-magnon bosonic states correspond to the configurations: ( K u , K u ¯4 , K u , K u , K u ) =(1 , , , , , (0 , , , , , (1 , , , , and (1 , , , , . The one-magnon fermionic config-urations are instead ( K u , K u ¯4 , K u , K u , K u ) = (1 , , , , , (0 , , , , , (1 , , , , , (0 , , , , . Out of these, 32 two-magnon states may be formed. We are interested inthose having degenerate energies in the plane-wave limit, since they correspond to thestring configurations we have studied in Section 3. In the boson-boson sector these are ( K u , K u ¯4 , K u , K u , K u ) = (1 , , , , , (2 , , , , and (0 , , , , ; in the fermion-sector these are ( K u , K u ¯4 , K u , K u , K u ) = (1 , , , , , (1 , , , , , (1 , , , , , (2 , , , , , (2 , , , , , (2 , , , , , (0 , , , , , (0 , , , , and (0 , , , , .Taking into account the exact Z degeneracy due to p → − p symmetry and the doubleoccurrence of ( .. and ( .. states due to branching of auxiliary roots we obtain 24states having plane-wave degenerate spectrum, which exactly corresponds to the degeneratestring two oscillator spectrum.Below we therefore solve Bethe equations for these states and find their spectrum. Wework at large λ , in the first order in J , and up to the th order in λ (cid:48) = λJ . The procedure isa perturbative expansion in J and λ (cid:48) , parallel to the warm up exercise of the SU (2) × SU (2) recalled above.The order in λ (cid:48) seems to be improvable ad infinitum ; we chose the fourth order dueto two considerations. First, it is order λ (cid:48) where discrepancy between the spectrum ofgauge invariant operators and near plane-wave string energies has been first found forthe AdS / CFT correspondence, and cured with the introduction of the AFS phase fac-tor [42] interpolating between weak and strong coupling. Thus, this feature having been– 19 –ubstantially inherited in the AdS / CFT correspondence, as one can read from the Betheequations (4.1), agreement at λ (cid:48) is such a nontrivial statement that be considered an all-order result. Second, larger values of the order become problematic (but not impossible)on Mathematica .To regularize Bethe equations for those solutions carrying auxiliary roots u i = 0 , u i = ∞ perturbatively, one should add twist parameters (cid:15) , , , as suggested in [49], in the followingway: e i(cid:15) = K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 − /x ,k x +4 ,j − /x ,k x − ,j K ¯4 (cid:89) j =1 − /x ,k x +¯4 ,j − /x ,k x − ¯4 ,j ,e i(cid:15) = K (cid:89) j (cid:54) = k u ,k − u ,j − iu ,k − u ,j + i K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i ,e i(cid:15) = K (cid:89) j =1 u ,k − u ,j + i u ,k − u ,j − i K (cid:89) j =1 x ,k − x +4 ,j x ,k − x − ,j K ¯4 (cid:89) j =1 x ,k − x +¯4 ,j x ,k − x − ¯4 ,j , The Bethe equations must be solved perturbatively for the momenta of the physicalroots, in terms of the solution for the auxiliary roots expressed through the parameters (cid:15) , , .At the end of the procedure, one takes the limit (cid:15) , , → and plugs the solution for themomenta in the dispersion relation (4.5). The spectrum of the boson-boson configurations,up to order λ (cid:48) , is given in the following Table (9). In the table we show the quantity ε ,defined as E = ε + εJ + O (cid:18) J (cid:19) . (4.17)In the “Note” column we show the final form of the lowest Bethe equation (the one for u ) that is being actually solved. The phase σ AF S and spectral variable u ( p ) is meantas function of p , the latter given by (4.10). It is important to realize that these energycorrections are quite different from those for , states (4.16), (4.13), despite thesingularity of the roots. Auxiliary roots going to infinity (in the u plane) do not result infull cancellation of their respective contributions in the equations for u and u ¯4 , since the x ( u ) are different for these solutions. A solution of type with x , , = ∞ would havebeen equivalent to ; however, in our case x = ∞ , x = ∞ , x = 0 which yields asolution of a totally different type due to extra x + /x − factor coming from the right-handside of Bethe equation.Similarly in the next Table (10) below we show the bifermionic part of the two-magnonsector of Bethe Ansatz, ε defined as above.This table is quite remarkable, since all states presented here are also found on thestring side, and the energies coincide up to the highest order done on the Bethe Ansatz side.Given this coincidence, as discussed above, this clearly points to an all-order equivalencefor the finite-size corrections calculated from the Bethe Ansatz and from the string theory,in the limit λ (cid:48) → , J → ∞ , for all the two impurity light bosonic states. This is aremarkable further test of the Bethe Ansatz framework and a significant effort towardsquantum integrability of strings in AdS × CP .– 20 – able 9 . Boson-boson spectrum from Bethe Ansatz state Energy coefficient ε Auxiliary roots Note K K ¯4 K K K − n π λ (cid:48) +32 n π λ (cid:48) − u = (cid:15) , x → ∞ e ip ( J +1) = σ AF S − n π λ (cid:48) +2048 n π λ (cid:48) u = (cid:15) , x → ∞ J + 1 due to extra x − /x + u = (cid:15) , x → from (cid:81) K ( · · · )2 0 1 1 1 − n π λ (cid:48) − // e ip ( J +1) = u + i u − i σ AF S +256 n π λ (cid:48) − n π λ (cid:48) − n π λ (cid:48) +96 n π λ (cid:48) − u = (cid:15) e ipJ = J +8 iπ λ (cid:48) n − iπnJ σ AF S u = (cid:15) − n π λ (cid:48) +6144 n π λ (cid:48) x = √ nπ √ λ (cid:48) − n π λ (cid:48) +32 n π λ (cid:48) − // e ipJ = u + i u − i J +8 iπ λ (cid:48) n − iπnJ σ AF S − n π λ (cid:48) +2048 n π λ (cid:48) To facilitate this comparison and summarize, let us represent the spectrum in a moreconcise form, showing the expansion coefficients in powers of λ (cid:48) and the multiplicities of thestates. For conciseness we do not write out the states ... , since they are fully equivalentto the corresponding states ... . All states ... are twice degenerate to all orders due to n → − n symmetry. Thus each of the states on the right hand side must be duplicated,which yields correct matching of the number of the degrees of freedom. Boson-boson sectoris compared in the Table (11). Analogously, fermion-fermion spectrum comparison is donein the Table (12). The spectacular coincidence of the λ (cid:48) expansions for Bethe energies with the string energiessupposes that it might be exact. This exactness can actually be seen directly in some ofthe cases. In the previous subsection the procedure to solve Bethe equations was to startwith “highest” auxiliary nodes , , , then descend to the physical magnons , ¯4 . Here weact reversely: start with the physical node, the momentum of which is known exactly in λ (cid:48) form the exact string spectrum (cid:15) = 4 π n λ (cid:48) (cid:16) A − π ( A +1) n λ (cid:48) π n λ (cid:48) +1 (cid:17) J , (4.20)where A = 2 , , − , − for the four admissible energy values of our spectrum. We can thususe the highest auxiliary node equation as a test. We have seen for several states of BetheAnsatz (e.g. the (1 , , , , , (2 , , , , states) that the first equation is non-triviallysatisfied in a regular manner, that is, by a systematic improvement of the expansion onecan satisfy the Bethe equation up to all orders.– 21 – able 10 . Fermion-fermion spectrum from Bethe Ansatz state Energy coefficient ε Auxiliary roots K K ¯4 K K K − n π λ (cid:48) + u = ( u + u − +256 n π λ (cid:48) − n π λ (cid:48) − i (cid:112) u − u u + u + 1 (cid:17) u = ( u + u ++ i (cid:112) u − u u + u + 1 (cid:17) x = − i − i n π λ (cid:48) + 2 in π λ (cid:48) x = 2 i + i n π λ (cid:48) − in π λ (cid:48) n π λ (cid:48) − n π λ (cid:48) + // +768 n π λ (cid:48) − n π λ (cid:48) − n π λ (cid:48) +32 n π λ (cid:48) − u = ( u + u ) − n π λ (cid:48) +2048 n π λ (cid:48) x , x solutions of ( x − x ( u + i/ x − x ( − u + i/ x − x ( u − i/ x − x ( − u − i/ = e i(cid:15) − n π λ (cid:48) + // +256 n π λ (cid:48) − n π λ (cid:48) − n π λ (cid:48) +32 n π λ (cid:48) + u = (cid:16) u + u + (cid:15) (cid:17) − n π λ (cid:48) +2048 n π λ (cid:48) x = ( u ( p ) ) (cid:15) √ J √ λ (cid:48) ,u ( p ) = cot (cid:0) p (cid:1) (cid:113) λ sin p x = − ( u ( p ) ) ( − i + (cid:15) )4 √ J √ λ (cid:48) − n π λ (cid:48) + // +256 n π λ (cid:48) − n π λ (cid:48) − n π λ (cid:48) +32 n π λ (cid:48) − x = J(cid:15) √ n π √ λ (cid:48) − n π λ (cid:48) +2048 n π λ (cid:48) x = J(cid:15) √ n π √ λ (cid:48) − n π λ (cid:48) + // +256 n π λ (cid:48) − n π λ (cid:48) The main results of this Paper can be summarized as follows: • Our calculations provide a highly non-trivial test for the validity of the string Hamil-tonian for three and four-particle interaction vertices in a near Penrose limit computedin [46]. • The one-loop correction to the single-magnon dispersion relation, as expected, is thesame for bosonic and fermionic excitation, it is finite and exponentially small in J – 22 – able 11 . Boson-boson spectrum comparison Expansion coefficient Multiplicity Corresponding BA states Corresponding ST states c c c c c K K ¯4 K K K State nr. −
12 12 −
12 12 branch 1 , − −
12 12 − branch 2 , , ,
81 1 1 1 1 branch 1 − −
32 32 − branch 2 , (4.18) Table 12 . Fermion-fermion spectrum comparison
Expansion coefficient Multiplicity Corresponding BA states Corresponding ST states c c c c c K K ¯4 K K K State nr. −
32 32 −
32 32 , −
12 12 −
12 12 , , , , , , ,
222 0 2 1 0 branch 1 branch 2 − −
12 12 − branch 1 , , , , ,
161 1 2 1 0 branch 2 (4.19)for J large. The regularization prescription implied by the cubic Hamiltonian and byconsequent unitarity arguments, gives a vanishing one loop correction to the strong-weak coupling interpolating function h ( λ ) , a = 0 . • In the two-particle sector the finite-size corrections (the /J -corrections) to magnoninteraction energies on the string side and on the Bethe-Ansatz side are exactly thesame up to the fourth order in λ (cid:48) ≡ λJ .The second result in the one-particle sector relies on the argument about the unitarity-preserving property of the regularization. This argument, in view of its very general nature,should be applicable to all Bethe Ansatz states, yet it remains an open problem e.g. howexactly it would work for e.g. the GKP case [72], or spinning string states. However wemay so far claim that a possible reason of the problems arising with the equal-frequencyregularization (linear divergencies; disagreement between strings and the algebraic curve)is a possible unitarity violation by regularization. Thus the problems arising with it maybe, somewhat loosely, called “unitarity anomaly”.– 23 –he significance of the third result is to establish another instance of the “mutualunderstanding” between the conjectured BA at all couplings with strings on AdS × C P in Penrose limit. The BA is asymptotic and thus is not a priori expected to work at strongcoupling and arbitrary length. Yet it works, as established by our third result, not justasymptotically at J → ∞ but also at least at the order /J . And at that order it iscompletely non-trivial that equivalence between the spectrum states holds at the fourthorder in λ (cid:48) .We conjecture that the equivalence is actually an exact one, and extends towards higherorders in J . The full Bethe Ansatz, with all finite size and loop corrections, at both weak and strongcoupling is encoded in the Y-system [31, 73]. For
AdS /CF T this infinite system offunctional equations has recently been shown to be equivalent to the T-system in [74],which is reducible to a finite number of integral equations. Thus of importance would beto test the results for finite-size corrections in strings against the T-system. The Lüscherterms are absent in the asymptotic Bethe Ansatz; string calculations would normally seethem directly; in our case the exponentially-suppressed finite-size corrections look preciselylike the typical Lüscher corrections do. Of extreme interest would be to compare in theone-loop sector the T-system with the direct Lüscher calculations (for a review see [50]for example), and with our string calculation of the finite size correction to the dispersionrelation.It is the strong coupling limit where the Y-system calculation should be easier toperform, since for strong coupling the functional/integral equations become algebraic anda full analytic solution becomes possible. Yet to our knowledge these solutions have sofar been applied to GKP [72] states mostly, and not to BMN. Therefore this should beone of the major lines of further research - to obtain the self-energy finite-size correctionsfor the near-BMN spectrum directly from Y or T system, comparing them with the stringcalculation.Probably the most urgent and straightforward further direction of the present workis, on the grounds of the same techniques, its extension to the computation of the finitesize corrections of string states involving at least one heavy mode. Actually the Penroselimit of the geometry decouples light and heavy modes, having different dispersion relation.Yet they look equally fundamental, both being described by a Fourier series of harmonic-oscillator like modes, such that one might think that the fundamental degrees of freedomof the theory are 8B + 8F. When dealing with finite size corrections, we have extensivelydiscussed how infinite sums appear in the computation of the spectrum, which need to beregularized. The momentum conservation at the cubic light-light-heavy vertex forces thecutoff on the mode numbers of a heavy mode being twice as that of a light one. This is thefirst glimpse of the interpretation of a heavy mode being, rather than fundamental, a boundstate of two light modes, such that indeed the basic degrees of freedom are 4B+4F, as in theBethe Ansatz framework. Actually the Bethe program gives a recipe for building a singlelight heavy oscillator state, being a composite of the fundamental roots. If this picture is– 24 –orrect, the finite size spectrum of light-heavy or heavy-heavy two oscillator states mustmatch the corresponding solutions of the Bethe equations. If this occurs, the puzzle aboutthe interpretation of the heavy mode would be definitely solved.The test of the AdS results at strong coupling has taken place at two loops for GKPstrings, and at 1 loop for the near-BMN limit, yielding perfect agreement to the Y-system.Therefore the second-loop corrections to the self-energies and the first-loop corrections toscattering amplitudes could be interesting to calculate. Our Hamiltonian approach herewould be extremely hard to implement, so we guess that perhaps the continuous Lagrangianfield-theoretical approach [30] might be used provided it is supplemented by a unitarity-preserving regularization.Our regularization for self-energies [44] is in essence equivalent to the one suggested byGromov and Mikhaylov [71], who employ cutoffs, and for the sums consisting of heavy ω H and light ω L modes use the prescription Reg (cid:16)(cid:88) ω H ( n ) + ω L ( n ) (cid:17) → (cid:88) (cid:16) ω H ( n ) + ω L (cid:16) n (cid:17)(cid:17) . (5.1)A similar regularization has been applied in the one- and two-spin BMN sector by Lipsteinand Bandres [75]. An important finding of their paper is that the “equal-frequency” regu-larization leads to a linear divergence in the double-spin BMN string for the algebraic curveresult, whereas the Gromov-Mikhaylov prescription yields all convergent results.Let us mention here that in a very elucidating unpublished Note by Gromov, Mikhaylovand Vieira a very general relation between the regularized one-loop self-energies from al-gebraic curve/Bethe Ansatz on one side and worldsheet string semiclassics on the otherwas derived. Gromov, Mikhaylov and Vieira show in the Note that if equal frequency(“world-sheet”) prescription is used then there are linear divergences in self-energies and in-consistency between higher charges of the integrable system as calculated from the discreteBethe Ansatz; that would mean that the Bethe Ansatz equations must be modified in someway. On the other hand, if the unequal cutoff (“algebraic curve”) prescription is used, thenno linear divergency arises and all charges are the same; Bethe Ansatz remains valid in theform we know it. We emphasize here that the result is stated in the Note by its authorsas absolutely universal, extending thus greatly the double-spin BMN string obtained byBandres and Lipstein [75]; this complies to our universal unitarity argument.A special investigation is due on applicability of our regularization prescription for theone-particle sector beyond the near-BMN limit. This question is especially interesting withregard to the generic GKP strings/twist-2 gauge operators. Different subsectors of thissector include long and short spinning folded strings, rotating circular strings, with s large,very large or not very large (in each case a sophisticated technical definition of “largeness”or “smallness” is present). Certainly the physics is quite different from the BMN case.The S-matrix argument we use here should be clarified in adaptation to different sets ofoscillations, since it was based on a BMN-spectrum. The spectrum of GKP oscillations,as obtained by Alday, Arutyunov and Bykov [69], contains of 6 bosons of zero mass, oneboson of √ mass, one mass 2 boson and 6 mass 1 fermions. It is not clear therefore how We thank Victor Mikhaylov for a clarifying discussion on this note. – 25 –he regularization (“unequal frequency”, “algebraic curve”) suggested by Astolfi, Grignani,Harmark and Orselli in [44], further argued for by Minahan and Zarembo, explored byGromov and Mikhaylov [71] could be applied in its exact form in terms of mode numberson world-sheet. We conjecture that the unitarity argument will work here as well, althoughthe explicit form of the argument based on the knowledge of certain pieces of the S-matrixwould be essentially different.Thus we wish to draw once more the attention to the unitarity issue in the regularizationfor generic sectors calling for further research into this subject. A hint may be a veryinteresting suggestion made by Gromov and Mikhaylov in [71]. Namely the “universal”prescription is to choose equal positions in the x -space for the excitations, where the x ( u ) algebraic curve coordinate. For all known cases, such as AdS and AdS this automaticallyleads to the correct mode structure and an “algebraic-curve” type prescription which satisfiesthe unitarity condition and is divergence-free. Thus a link, probably of a very general nature,must be established between this simple scheme and the world-sheet unitarity conservation. Acknowledgments
A.Z. thanks Kolya Gromov and Dima Volin for their patience in explaining him how theBethe Ansatz works. We thank Benjamin Basso, Diego Bombardelli, Constantin Candu,Sergey Frolov, Troels Harmark, Kristan Jensen, Dmitry Kharzeev, Gregory Korchemsky,Thomas Klose, Igor Klebanov, Marta Orselli, Tristan McLoughlin, Juan Maldacena, VityaMikhaylov, Radu Roiban, Arkady Tseytlin, Ismail Zahed and Alexander Zhiboedov forstimulating conversations. A.Z. thanks the Perimeter Institute, Princeton University andPenn State University for hospitality, and the Organizers of the August 2011 meeting onStrings and Integrability at Perimeter for providing the fruitful atmosphere where an essen-tial part of this project was developed. Special thanks to Fedor Levkovich-Maslyuk for athorough reading of the manuscript and comments on literature. This work was supportedin part by the MIUR-PRIN contract 2009-KHZKRX. The work of A.Z. is supported in partby the RFBR grant 10-01-00836 supported by Ministry of Education and Science of theRussian Federation under the contract 14.740.11.0081.
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