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Advances in High Energy Physics
Volume 2015, Article ID 762481, 21 pages
Research Article

Reciprocity and Self-Tuning Relations without Wrapping

1Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, 40126 Bologna, Italy
2Dipartimento di Fisica dell’Università della Calabria, Arcavacata, Rende, 87036 Cosenza, Italy
3Laboratoire Jean Alexandre Dieudonné, Université Nice Sophia Antipolis, 06100 Nice, France
4Dipartimento di Fisica dell’Università della Calabria and INFN, Gruppo Collegato di Cosenza, Arcavacata, Rende, 87036 Cosenza, Italy

Received 6 August 2015; Accepted 15 October 2015

Academic Editor: Stefano Moretti

Copyright © 2015 Davide Fioravanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.


We consider scalar Wilson operators of = 4 SYM at high spin, , and generic twist in the multicolor limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain “reciprocity” and functional “self-tuning” relations up to all terms (inclusive) at any fixed ’t Hooft coupling . Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order in . On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalous dimension expansions at the next order , at fixed ’t Hooft coupling, in such a way to reestablish the aforementioned relation (which fails otherwise).

1. Introduction, Aims, and Results

One of the major achievements of modern theoretical physics is the so-called AdS/CFT correspondence [13] and its description in terms of integrability tools [419]. In fact, being a strong/weak coupling duality, the nonperturbative, exact, though not necessarily explicit (as a simple example we can mention, just with reference to the present paper, that the following nonlinear integral equation (which governs the spectrum) is not explicitly solvable), nature of integrability is of incomparable value and utility. In particular, the spectrum of anomalous dimensions of composite operators in super Yang-Mills (SYM) theory ought to correspond to the energy spectrum of states in type IIB superstring theory in , and both must be described by an integrable system.

Among the different sectors of multicolor SYM (perturbatively closed under renormalisation), one of the most studied ones is the so-called scalar twist sector. This is spanned by local composite operators of single trace form:where is a (light-cone) covariant derivative acting in all the possible ways on the complex bosonic fields , the trace ensuring gauge invariance. The Lorentz spin of these operators is and coincides with the twist, that is, the classical dimension minus the spin. The AdS/CFT correspondence relates operators (1) to spinning folded closed strings on spacetime, with and angular momenta and , respectively [20, 21].

One of the several reasons for the large interest in these operators is their similarity with twist operators in QCD, where, maybe, the scalars are substituted by fermions, that is, the quarks, or gauge fields: because of integrability in these cases would be dealt with in an analogous manner [22, 23]. Similarities among the two theories give the possibility to believe that QCD could take many advantages of a full all-loop solution of its supersymmetric counterpart. In QCD, in the framework of Partonic Model, the Lorentz spin is the conjugated variable, in the Mellin transform (of the splitting function, for instance, which gives the anomalous dimension), to the Bjorken variable , namely, the fraction of the hadron momentum carried by the single parton (of course, the coupling does run in QCD, unlike what happens in the maximally supersymmetric theory). In this context, two regimes emerge naturally: , governed by the BFKL equations [24], and , corresponding exactly to large values of the Lorentz spin, . Properties of this second (called quasielastic) regime can be deduced by large spin results in three-loop twist 2 QCD calculations. In particular, we can highlight two main features about anomalous dimension of twist operators:(1)The leading term has a logarithmic scale: (2)Subleading terms obey hidden relations, the Moch-Vermaseren-Vogt constraints [25, 26]: in brief, terms proportional to and are completely determined by terms proportional to and . These constraints are related with spacetime reciprocity of deep inelastic scattering and its crossed version of annihilation into hadrons. gauge theory shares at large these features, and, besides, allows us an understanding of their origin and thus possible extension to QCD. In specific, the asymptotic large series of the anomalous dimensions are believed to be constrained by nonperturbative (in , the ’t Hooft multicolor coupling) functional relations that work for any finite value of the twist . To be more precise, conformal symmetry implies that anomalous dimensions of twist operators are functions of the conformal spin: this translates into the following “self-tuning” functional relation [27, 28]:Additionally, this has to be meant asymptotically in the sense that the function (the function in (3) actually depends on the twist of the operator as well)is represented by a series in via the conformal Casimironly. Relation (4) is equivalent to the so-called reciprocity symmetry , but for the Mellin space variable , and an important piece of information is that the function has the form of an upper truncated Laurent series:that is, depends on only through powers of . For twist two and three negative powers of are absent and is a polynomial; for generic twist, however, one has to cope with the infinite Laurent series (6).

Relations (3) and (4), both in QCD and in SYM, are developed, checked in various cases, and discussed in [2737]. Recently, they have been proven, restrictively, to twist two operators, but in a generic conformal field theory, in [38], with some arguments for their validity in nonconformal theories at the end. Clearly, they provide important information on the high spin expansion of anomalous dimension of twist operators. Unlike QCD, in SYM, it is possible to obtain better and more suitable results as we can consider these relations into the framework of integrability. The latter was firstly discovered in the planar limit for the purely bosonic sector at one loop [4]; then it was extended to all the gauge theory sectors and to all loops [510]. In specific, it was found that every composite operator can be thought of as a state of a “spin chain,” whose Hamiltonian is the dilatation operator itself, although the latter does not have an explicit expression of the spin chain form, but for the first few loops. Nevertheless, the spectrum of infinitely long operators has turned out to be exactly described by a set of Asymptotic Bethe Ansatz (ABA) equations [510]. On the other hand, anomalous dimensions of operators with finite quantum numbers depend not only on ABA data but also on finite size “wrapping” corrections [11, 12]. Subsequent progress has shown that a set of Thermodynamic Bethe Ansatz (TBA) equations [1317] or an equivalent -system of functional equations [18], together with certain additional information [19], provides a solid ground for exact (any length, any coupling) predictions on anomalous dimensions of planar SYM.

Despite this impressive progress, we believe that it is still important to define the largest domain of composite operators for which the “simpler” ABA equations give the correct anomalous dimensions, especially in connexion with other well-established relevant equations. In fact, we intended this to be the main aim of this paper and the most natural setting to perform this study to be the reformulation of ABA equations as one (Non)linear Integral Equation (NLIE) [39].

Generically and sketchily, for operators composed of elementary fields, ABA gives the correct perturbative expansion of the anomalous dimension up to loops. Starting from loops, “wrapping” diagrams, which are not taken into account by ABA, start to contribute. In this general framework, the high spin limit of fixed twist operators seems to offer a better scenario. Perturbative (up to six loops) computations [4042] for short (twist two and three) operators show that wrapping diagrams (which enter from four loops on) actually give contributions which in the high spin limit behave as . It is then natural to ask if such property extends to higher (and possibly to all) orders of perturbation theory. In this paper we want to provide evidence in favour of this picture, by using the self-tuning and reciprocity properties. In order to do that, we first rewrite (Section 2) the ABA equations as NLIEs for the counting function. Then, in Section 3, we specialise ourselves to the minimal anomalous dimension state and go to the high spin limit, while keeping the twist finite: upon computing the positions of the external holes and the effect of the nonlinear terms, we write a linear integral equation equivalent to ABA up to the orders , , (inclusive). In Section 4, we use this linear integral equation to compute at the same order of , but at all values of the coupling, the ABA prediction for the minimal anomalous dimension. Then, in Section 5, we show the latter to satisfy the self-tuning and reciprocity relations. Interestingly, we also find that the solution of the linear integral equation respects suitable self-tuning and reciprocity relations (up to this order in ). Finally, we provide some arguments supporting the idea that at high spin wrapping corrections affect twist operators starting from orders , so that self-tuning and reciprocity relations still hold (and likely also a modified (non)linear integral equation).

2. From the ABA to the NLIE

As planned in the Introduction, we start from the ABA equations [510] for the sector of SYM:where being the ’t Hooft coupling. The so-called dressing factor [10, 43, 44] is given bythe functions beingand ,being the expression of the th charge in terms of the rapidity . Operators (1) of twist correspond to zero momentum states of the spin chain described by an even number of real Bethe roots which satisfy (7). For a state described by the set of Bethe roots , , the eigenvalue of the th charge isIn particular (asymptotic) anomalous dimension of (1) isLet us focus (in this section; from Section 2 on, we will restrict to the minimal anomalous dimension state) on states described by positions of roots which are symmetric with respect to the origin. These are in particular zero momentum states. An efficient way to treat states described by solutions to a (possibly large) number of (algebraic) Bethe Ansatz equations consists in writing one nonlinear integral equation completely equivalent to them (cf. [45] and references therein for the idea without holes degree of freedom). The nonlinear integral equation is satisfied by the counting function , which in the case (7) reads aswherewithIt follows from its definition that the counting function is a monotonously decreasing function. In addition, in the limit , sinceone has the asymptotic behaviourThis means that there are real points such that . It is a simple consequence of the definition of that of them coincide with the Bethe roots . For Bethe equations (2) Bethe roots are all real and are all contained in an interval of the real line. The remaining points are called “holes” [39, 4652]; they also are real and they will be denoted as . One should distinguish between “internal” or “small” holes , , which reside inside the interval , and two “external” or “large” holes , with .

We finally remark that anomalous dimension appears (18) in the limit of the counting function. We will come back to this fact in Appendix A.

As we are in presence of holes, we may follow the extension of the idea as developed in [53] and make use of the Cauchy theorem to obtain a simple integral formula (; cf. also [54] for more details on the following formulae):Application of (19) to the derivative of (14) givesWe introduce the notationsand pass to Fourier transforms , keeping in mind thatWe obtain the equationand for the equationwhich is the nonlinear integral equation for the counting function , describing states of the sector. We will find it convenient to introduce the following function:because, in Appendix A, we show that it satisfies the simple relationThe function (25) satisfies the nonlinear equationNow, the introduction of the “magic kernel” [10]the use of the property, valid for ,and the restriction to allow to write the equation for in the alternative way:Equations (30) and (26) are our starting points for studying ABA contributions to anomalous dimension of twist operators. As planned in the Introduction, we will consider the minimal anomalous dimension state, go to the high spin limit, and determine the predictions of ABA for the anomalous dimension up to orders , . We therefore discuss in next section all the simplifications that (30) undergoes in the high spin limit.

3. Ground State and High Spin Limit

In this section we start our study of the minimal anomalous dimension state. For this state the positions of the internal holes are as close as possible to the origin; that is, they satisfy the relationswhile the positions of the two external holes are determined after solving the equationsIt follows that the positions of the Bethe roots are all greater in modulus than the positions of the internal holes; that is, , . For our convenience we order Bethe roots in such a way that if .

In the following we will find useful to integrate over the region in which Bethe roots are contained. It is then very important to make the most convenient choice for the “extrema” of integration, which naturally identify the points which separate the last/first root () from the positive/negative external hole : we choose such thatThen, we perform our analysis of the minimal anomalous dimension state in the high spin limit. We have to remark that in this limit the set of operators (1) has been the object of an extensive activity [10, 39, 4652, 5567]; also in perturbative QCD, see [6873]. In the high spin limit, the position of the internal holes is proportional to , so it is very close to the origin: they will be determined by using (31) in Section 4. On the other hand, in order to estimate the position of the two external or “large” holes, we have to evaluate the counting function near the points , , delimiting the interval in which Bethe roots reside. The result we will find, at their leading orders and ,is proved in next subsection. We have to mention that the same formula (34) was found for twist two in [65], by using results of [74, 75]. However, as far as we have understood, results of [74, 75] are proved only at one and two loops. Therefore, we would like to give a different and more general proof of (34).

3.1. Position of the External Holes

When the spin is large, Bethe roots near the two “extrema” scale with . In the proximity of , it is therefore convenient to rescale the variable of the counting function : we will write , where will stay finite. Analogously, we will define , with finite. From the definitions (14) and (16) of the counting function, we haveWe observe that the only “higher loops” effect is in the last term, proportional to the anomalous dimension. For , where is a Bethe root, we expand the various functions for large and evaluate the sum over the Bethe roots contained in (35) as an integral term plus an “anomaly” [7679]. We obtainwhere ,and where we used the relation [7679]We remark that, in order to obtain the last equality in (38), it is crucial to transform the sum into an integration from to , where satisfies (33). If, for instance, we transform the sum over Bethe roots and holes into an integration from the first to the last root, we obtain an extra term in the last line of (38): specifically,Sticking to formula (38), we remember that for the minimal anomalous dimension state and with our ordering of Bethe roots the value of the counting function on a generic root is given by the simple formulaProperty (40) allows to simplify equation (36) as follows:At the leading order, , we know that the equation to be satisfied, for all , iswhose solution is the well-known [55, 80] densityUsing (43), we give an estimate of the last term in (41),which allows to find the function which satisfies (41):Using the form (45) of the solution, we can determine the position of the extremum through the relationwhere we used (33), which givesWe remark that if we had transformed the sum over Bethe roots and holes into an integration from to according to (39), by repeating all the steps until (47) we would have found the position of the largest root at leading and subleading order: this result iswhich, in particular, allows to give an estimate for :We will use this result for in next subsection.

We now pass to determine the position ,  , of the positive external hole. We first compute (35) for (more precisely, ):The sum over the Bethe roots is evaluated asWe now insert (45) into (51) and use the result, valid for :Inserting the resulting expression for (51) into (50), we eventually arrive at the formulawhich is certainly valid for . Now, the position of the (positive) external holes is fixed by the condition . Imposing that on (53), we findWe observe that such result agrees when with the zeroes of the transfer matrix which one can obtain from expressions contained in [74, 75]. This is an important check for our findings.

3.2. High Spin Limit of Nonlinear Terms

Another important simplification occurring for large spin concerns the nonlinear term (containing ) which appears in (30). In this subsection we extend to all loops the result of [4752]. Some of the results of this section have been already announced, but not completely proved, in [81]. We will fill that gap here: at the end, we are able to show thatThis means that, in our approximation, nonlinearity effects in (30) are under control.

In our equation (30) nonlinearity appears in the following integral:It is convenient to pass to the coordinate space and to defineWe can keep generic, having in mind that the case is relevant for our case (56):whereIn general we split as , whereThen, is evaluated using formula of [82]:Now, we remember that (see (49)); in addition, in the high spin limit we are allowed to consider . Therefore, we conclude that and, consequently,Since we can restrict to , we develop the functions in the integrand for large . We obtainIntegrating by parts we can write downWe then use the fact that and the identityto obtainIn order to perform the integration in (66), we need an estimate of when . In Appendix B we prove thatIntegration in (66) is then performed exactly:Plugging (68) into (66) and using the equality , we obtainPassing now to the kernel , its evaluation in the coordinate space shows the following behaviour:Therefore,where is a cutoff such that for approximation (69) can be used. When , in the first integral, we replace with ; in the second integral we estimate using (70) and by means of (57), using that for large :which therefore imply thatPutting all together we find out that

4. High Spin Results from ABA: Up to Order

Having analysed all the simplifications occurring in the high spin limit, let us come back to (30). We insert formula (34) for the position of the external holes, use relation (55) for the nonlinear term, and work out all the “known” terms. We end up with the following integral equation:where . The particular form of the known terms in (75), together with condition (31) for the internal holes, suggests that expands in (inverse) powers (with we denote terms going to zero faster than times any inverse power of .) of :And, consistently with (76), the condition for the internal holes (31) is solved by the following Ansatz on their positions:For the anomalous dimension , therefore, we have the expansionwhere the scaling functions , appear also in other contexts: for instance, is twice the cusp anomalous dimension of Wilson loops [83, 84].

For our purposes, it is important to remark that the strong coupling limits of [63, 64] and [65, 66] agree with string theory computations. This shows that such functions are actually wrapping independent and, consequently, the anomalous dimension is wrapping independent at its leading orders and .

After these first considerations, we come back to (75): the “known” terms driving the equations for and are contained in the first two lines of (75). The structure of such driving terms implies the following equalities between densities:which translate in terms of anomalous dimensions to the equalities [65, 85]

It is possible to obtain analogous relations for expressed in terms of the , for . The first step is a standard procedure for integral equations with a separable kernel, called Neumann expansion [86], applied to (75) for :This procedure is fully explained in this application in Appendix C. For we obtainwhere and belong to a set of “reduced coefficients,” satisfying the system (C.6), reported also in Appendix C.

These expressions are still quite involved, but they can be significantly simplified. This can be done through the following steps.(i)After introducing the notationwe “invert” relation (31), expressing and in terms of the densities and their derivatives in zero, that is, in terms of the coefficients and . In performing this procedure we use techniques and results of [87]. Then, we plug the obtained expressions for , in (82). Detailed calculations are shown in Appendix D, where we have also listed the full expressions for the first (relations (D.4)).(ii)Then, we use the following relations, proven in Appendix E:With the help of these formulae it is possible to compare the complicated relations (D.4) with analogous results for , found in [87] and reported in Appendix F, ending up with the following simple and compact expressions:

For anomalous dimensions, such relations readRelations (80) and (86) are the prediction of ABA for anomalous dimensions at various orders , . In the next section we will show that they agree with predictions coming from self-tuning and reciprocity relations.

5. Contributions from Functional Relations

Self-tuning and reciprocity relations were summarised in formulae (3), (4), and (5). We remember notations (78) for the high spin expansion of the anomalous dimension:Comparing (87) with (3), (4), and (5), we obtain that the leading terms of should readDeveloping in the same regime