Instanton effects on Wilson-loop correlators: a new comparison with numerical results from the lattice
aa r X i v : . [ h e p - ph ] M a y IFUP–TH/2009–17Revised versionApril 2010
Instanton effects on Wilson–loop correlators: a newcomparison with numerical results from the lattice
Matteo Giordano ∗ and Enrico Meggiolaro † Dipartimento di Fisica, Universit`a di Pisa,and INFN, Sezione di Pisa,Largo Pontecorvo 3, I–56127 Pisa, Italy.
Abstract
Instanton effects on the Euclidean correlation function of two Wilson loops atan angle θ , relevant to soft high–energy dipole–dipole scattering, are calculated inthe Instanton Liquid Model and compared with the existing lattice data. Moreover,the instanton–induced dipole–dipole potential is obtained from the same correlationfunction at θ = 0, and compared with preliminary lattice data. ∗ E–mail: [email protected] † E–mail: [email protected]
Introduction
Since its discovery in 1975 [1], the instanton solution of the Yang–Mills equations hasbeen widely studied, both in its mathematical properties and its phenomenologicalapplications [2, 3, 4, 5, 6, 7, 8]. Many insights have been obtained in QCD throughthe use of instantons (see, e.g., the review [9] and references therein), and even if itis known that they cannot provide the framework for a complete understanding ofstrong interactions, it is interesting to investigate instanton effects as they contributeto the nonperturbative dynamics.Among the various open problems in QCD, soft high–energy hadron–hadronscattering is known to be of nonperturbative nature, and so it is worth studying whatthe consequences are of instantons on the scattering amplitudes. In the approachfrom the first principles of QCD [10], such amplitudes are related to properties ofthe vacuum, namely the correlation functions of certain Wilson–line and Wilson–loop operators. In particular, in the case of meson–meson scattering, the scatteringamplitudes can be reconstructed, after folding with the appropriate mesonic wavefunctions, from the scattering amplitude of two colour dipoles of fixed transversesize; the latter are obtained from the correlation function of two rectangular Wilsonloops, describing (in the considered energy regime) the propagation of the colourdipoles [11, 12, 13, 14, 15]. The physical, Minkowskian correlation functions can bereconstructed from the “corresponding” Euclidean correlation functions [16, 17, 18,19, 20, 21, 22], and so it has then been possible to investigate the problem of soft high–energy scattering with some nonperturbative techniques available in EuclideanQuantum Field Theory [23, 24, 25, 26, 27, 28].In this paper we shall derive a quantitative prediction of instanton effects inthe Euclidean loop–loop correlation function, relevant to the problem of soft high–energy scattering. This problem has already been addressed in Ref. [23], using theso–called
Instanton Liquid Model (ILM) [8], but the result reported in that papercontains a severe divergence, apparently not noticed by the authors. In this paperwe critically repeat the calculation, obtaining a well–defined analytic expression,which can be compared with the lattice data presented and discussed in Ref. [28].In this paper we also calculate the instanton–induced dipole–dipole potential fromthe correlation function of two parallel Wilson loops [29], and we compare the resultswith some preliminary data from the lattice.The detailed plan of this paper is the following. In Section 2 we briefly recall themain features of instantons in Yang–Mills theory, we shortly describe the relevantaspects of the ILM, and we discuss the method we will actually use in our calcula-tions, using as an example the expectation value of a rectangular Wilson loop andthe instanton–induced q ¯ q –potential [6, 30]. In Section 3, after a brief review of howhigh–energy scattering amplitudes can be reconstructed from the correlation func-tion of two rectangular Wilson loops at an angle θ in Euclidean space, we evaluatethis same correlation function and we critically compare the result with the calcu- ation of Ref. [23]. We then compare our quantitative prediction with the latticeresults of Ref. [28]. In Section 4 we calculate the correlation function of two parallelWilson loops, from which we derive the instanton–induced dipole–dipole potential,that we compare with some preliminary data from the lattice. Finally, in Section 5we draw our conclusions. Some technical details are discussed in the Appendices . Instantons (anti–instantons) are self–dual (anti–self–dual) solutions with finite ac-tion of the classical Yang–Mills equations of motion in Euclidean space [1]. We willbe interested only in the solutions with topological charge q = +1 and q = − SU (2) Yang–Mills theory, the instanton solution in regular covariant(Lorenz) gauge reads, for the standard colour orientation, A aµ ( x ; z, I) = − g η aµν ( x − z ) ν ( x − z ) + ρ , A µ = A aµ σ a , a = 1 , , , (2.1)where z µ and ρ are free parameters that correspond to the position of the center andthe radius of the instanton, respectively, η aµν are the self–dual ’t Hooft symbols [5],and σ a are the Pauli matrices. The anti–instanton solution A aµ ( x ; z, ¯I) is obtainedby replacing η aµν → ¯ η aµν in Eq. (2.1), where the latter are the anti–self–dual ’tHooft symbols [5]. Applying a SU (2) transformation on Eq. (2.1), A µ → U A µ U † with U ∈ SU (2), we obtain another solution with a different colour orientation.Instantons in SU ( N c ) Yang–Mills theory can be obtained by embedding the SU (2)solution (2.1) in a SU (2) subgroup of SU ( N c ), and it has been shown that these arethe only solutions with | q | = 1 [7]. We will take the standard–oriented solution tobe the embedding of the (standard) SU (2) solution in the upper–left 2 × N c × N c matrix; the other solutions are obtained by applying an SU ( N c ) colourrotation on this one.Early attempts at a description of the QCD vacuum in terms of an ensemble ofinstantons and anti–instantons adopted the picture of a dilute gas [6], but sufferedfrom severe infrared divergences due to the presence of large–size instantons. Theproposal of [8] was instead that the pseudoparticles form a dilute liquid : the di-luteness of the medium allows for a meaningful description in terms of individualpseudoparticles, while the infrared problem is solved assuming that interactionsstabilise the instanton radius. In particular, this model assumes that the vacuumcan be described as a liquid of instantons and anti–instantons, with equal densities In this paper we will deal only with the Euclidean theory, and so we will understand that the metric isEuclidean and that expectation values have to be taken with respect to the Euclidean functional integral,except where explicitly stated. I = n ¯I = n/
2, and with radius distributions strongly peaked around the sameaverage value: in practice, one performs the calculations using a δ –like distribution,i.e., fixing the radii of the pseudoparticles to a given value ρ . Using the SVZ sumrules [31, 32] to relate vacuum properties with experimental quantities, the phe-nomenological values for the total density and the average radius are estimated tobe [9] n ≃ − , ρ ≃ / packing fraction nV ∼ / V = ( π / ρ is the volume of a 3–sphere of radius ρ , which is approxi-mately the space–time volume occupied by a single pseudoparticle]; this indicatesthat the instanton liquid is indeed fairly dilute . We refer the interested readerto [9] and references therein for a more detailed discussion of the model, both onits theoretical foundations [33, 34] and on the development of numerical methodsof investigation [35]. In the rest of this paper we will make use only of the generalpicture described above to give estimates of the relevant quantities.To calculate the expectation value of an operator O one needs to average overthe ensemble with some density function D = D ( { z } , { U } ), where { z } denotescollectively the coordinates of the pseudoparticles and { U } their orientations incolour space; in principle, such a function should be determined by properly takinginstanton interactions into account. Starting with an ensemble of N I instantonsand N ¯I anti–instantons in a space–time volume V , we obtain the desired resultby taking the “thermodynamic limit” N I , N ¯I , V → ∞ , while keeping the densities n I ≡ N I /V and n ¯I ≡ N ¯I /V fixed: hOi ILM ≡ lim N I , ¯I ,V →∞ ,n I , ¯I fixed Z d N z d N U D ( { z } , { U } ) O ( { z } , { U } ) . (2.2)Here O ( { z } , { U } ) denotes the operator O evaluated in the field configuration gen-erated by the pseudoparticles, and dU is the normalised Haar measure, R dU = 1.Since the medium is dilute, the field configuration is approximately equal to thesuperposition of single (anti–)instanton fields, A µ ( x ) = N X i =1 U ( i ) A µ ( x ; z i , σ i ) U † ( i ) , (2.3)where N = N I + N ¯I is the total pseudoparticle number, z i and U ( i ) are the positionand colour orientation of pseudoparticle i , respectively, and σ i = I , ¯I indicates ifpseudoparticle i is an instanton or an anti–instanton.To perform calculations, we take an approximate density function that describesthe instanton liquid as an ensemble of free particles with some (unspecified) strong These values and the corresponding picture are obtained in the realistic case of three light–quarkflavours ( u , d , s ): when comparing the analytic results with the numerical lattice calculations, we haveto take into account that the latter are performed in the quenched approximation of QCD. For generality, we keep N I and N ¯I fixed but independent in the calculation; we will set N I = N ¯I whenneeded. ore repulsion, which approximately localises the instantons in distinct “cells” ofvolume V cell , i.e., D ( { z } ) = N − X P N Y j =1 χ D j ( z P ( j ) ) , N = N ! V N cell (2.4)where { D j } is a partition in cells of the space–time region D occupied by theensemble, ∪ j D j = D , χ D j is the characteristic function of D j , P denotes a per-mutation of { , . . . , N } , and N is the normalisation; the volume of each cell is R d z χ D j ( z ) = V cell . The coordinates of pseudoparticle j are denoted as z j , andwe take the first N I pseudoparticles to be instantons, and the remaining N ¯I to beanti–instantons, σ k = ( I , ≤ k ≤ N I , ¯I , N I + 1 ≤ k ≤ N. (2.5)The function (2.4) is by no means a precise description of the instanton ensemble:it has the purpose of capturing the main features of the liquid picture, namely thediluteness and the uniform density of the medium, while at the same time keepingthe calculation feasible. Notice that D does not depend on the colour orientationof the pseudoparticles. The precise shape of the cells should not be important, andone can choose it according to the geometry of the problem.For a local operator O local ( x ), only configurations with a single pseudoparticlelocated near the point x have both non–negligible effects and non–negligible measurein configuration space, due to the diluteness of the medium and to the short–rangenature of instanton effects. The evaluation of expectation values is more complicatedfor non–local operators such as Wilson lines and Wilson loops, since they involvethe value of the fields on a curve C . However, by properly subdividing C into smallsegments, one can exploit again the diluteness of the medium and the short–rangenature of instanton effects, in order to consider each segment to be affected only bythe field of a single pseudoparticle.As an example, consider the case of the expectation value of a rectangular Wilsonloop W of temporal extension T and spatial width R (say, in the 1 direction). As it iswell known, one can extract from this expectation value the static quark–antiquarkpotential V q ¯ q using the relation [36, 37] hWi = 1 N c Tr [W ( − ) † W (+) ] = T →∞ K e − T V q ¯ q ( R ) , (2.6)where we have neglected the effect of the transverse “links” connecting the Wilsonlines W ( ± ) , which describe the propagation of the quark and of the antiquark, andwhere K is some proportionality constant. The effect of a single instanton located at z on an infinite Wilson loop, which we denote with w ( z, I), is given by the followingexpression [6], w ( z, I) ≡ − N c ∆( z ) , (2.7) here ∆( z ) = 1 − cos α + cos α − − ˆ n + · ˆ n − sin α + sin α − ,α ± = π k n ± k p k n ± k + ρ , n a ± = − ( ± R δ a − z a ) . (2.8)The result does not depend on the colour orientation of the instanton, and it doesnot change if we replace the instanton with an anti–instanton. In order to calculatethe effect of the instanton liquid, i.e., of the whole ensemble, on the Wilson loop,it is natural to see it as the result of a time series of interactions between thecolour dipole and the pseudoparticles: this suggests dividing space–time into time–slices of thickness δT , and then each slice into cells of volume V cell = V (3)cell δT . Thesituation is illustrated in Fig. 1. Here δT and V (3)cell are determined by the followingrequirements:1. only one pseudoparticle falls in V (3)cell δT on the average, i.e., nV cell = 1;2. the spatial size V (3)cell is large enough to accommodate the whole dipole, and alsolarge enough so that the pseudoparticles which fall outside of the cell wherethe dipole lies will not affect it, being too distant;3. δT is large enough for an (anti–)instanton to fully affect the portions of the twoWilson lines that fall in V cell , i.e., δT can be considered infinite ( δT ≫ ρ ) asfar as the colour rotation induced on the Wilson lines by the pseudoparticle isconcerned, and so one can use the colour rotation angles α ± given in Eq. (2.8).The region of 3–space where an (anti–)instanton can affect the operator is approxi-mately determined by requiring that the pseudoparticle is not too far from the twolong sides (say, more than ρ ), which implies V (3)cell ∼ ( R + 2 ρ )(2 ρ ) ; (2.9)in this way the two segments of the loop lying in a given slice fall in the same cell.With this choice we fulfill condition (2.); imposing condition (1.) we obtain δTρ ∼ ρ R + 2 ρ ) ∼
101 + R ρ , (2.10)which is reliable for R . ρ , so that δT /ρ & R .At this point one can perform the average over the instanton ensemble: since theprocedure is the same as the one adopted in [6], we will not report the calculationhere, and we simply quote the result which will be used in the following: hWi ILM = e − T V q ¯ q ( R ) , V q ¯ q ( R ) = n N c Z d z ∆( z ) . (2.11) his result is the same as the one found in Ref. [30] with different, more sophisticatedmethods. As ∆( z ) is localised around the position in 3–space of the quark andthe antiquark, for large R the Wilson–loop expectation value factorises, and thepotential becomes the sum of two constant and equal terms, which are interpretedas the renormalisation of the quark mass [30]. As has already been recalled in the Introduction, in the soft high–energy regime themesonic scattering amplitudes can be reconstructed from the correlation functionof two Euclidean Wilson loops. In this Section we give first a brief account, for thebenefit of the reader, of the main points of the functional–integral approach to theproblem of elastic meson–meson scattering, referring the interested reader to theoriginal papers [11, 12, 13, 14, 15] and to the book [38]. We shall use the samenotation adopted in Ref. [28], where a more detailed presentation can be found .We then critically repeat the calculation of the relevant correlation function in theILM, and compare the result with the one found in Ref. [23].The elastic scattering amplitudes of two mesons (taken for simplicity with thesame mass m ) in the soft high–energy regime can be reconstructed in two steps. Onefirst evaluates the scattering amplitude of two q ¯ q colour dipoles of fixed transversesizes ~R ⊥ and ~R ⊥ , and fixed longitudinal momentum fractions f and f of the twoquarks in the two dipoles, respectively; the mesonic amplitudes are then obtainedafter folding the dipoles’ amplitudes with the appropriate squared wave functions,describing the interacting mesons. The dipole–dipole amplitudes are given by the2–dimensional Fourier transform, with respect to the transverse distance ~z ⊥ , of thenormalised (connected) correlation function of two rectangular Wilson loops, M ( dd ) ( s, t ; ~R ⊥ , f , ~R ⊥ , f ) ≡ − i s Z d ~z ⊥ e i~q ⊥ · ~z ⊥ C M ( χ ; ~z ⊥ ; 1 , , (3.1)where the arguments “1” and “2” stand for “ ~R ⊥ , f ” and “ ~R ⊥ , f ” respectively, t = −| ~q ⊥ | ( ~q ⊥ being the transferred momentum) and s = 2 m (1 + cosh χ ). Thecorrelation function C M is defined as the limit C M ≡ lim T →∞ G M of the correlation Since no ambiguity can arise, with respect to Ref. [28] we drop the subscript E and “tildes” fromEuclidean quantities in order to avoid a cumbersome notation. unction of two loops of finite length 2 T , G M ( χ ; T ; ~z ⊥ ; 1 , ≡ hW ( T )1 W ( T )2 i M hW ( T )1 i M hW ( T )2 i M − , (3.2)where h . . . i M are averages in the sense of the QCD (Minkowskian) functional in-tegral. Here W ( T )1 , are Minkowskian Wilson loops evaluated along the paths C , ,made up of the classical trajectories of the quarks and antiquarks inside the twomesons, and closed by straight–line paths in the transverse plane at proper times ± T . The partons’ trajectories form a hyperbolic angle χ in the longitudinal plane,and they are located at (1 − f i ) ~R i ⊥ (quark) and − f i ~R i ⊥ (antiquark), i = 1 ,
2, in thetransverse plane.The Euclidean counterpart of Eq. (3.2) is G E ( θ ; T ; ~z ⊥ ; 1 , ≡ hW ( T )1 W ( T )2 i E hW ( T )1 i E hW ( T )2 i E − , (3.3)where now h . . . i E is the average in the sense of the Euclidean QCD functionalintegral. With a little abuse of notation, we denote with the same symbol W ( T )1 , theEuclidean Wilson loops calculated on the following straight–line paths , C : X qµ ( τ ) = z µ + u µ τ + (1 − f ) R µ , X qµ ( τ ) = z µ + u µ τ − f R µ , C : X qµ ( τ ) = u µ τ + (1 − f ) R µ , X qµ ( τ ) = u µ τ − f R µ , (3.4)with τ ∈ [ − T, T ], and closed by straight–line paths in the transverse plane at τ = ± T . Here u = (cid:16) sin θ ,~ ⊥ , cos θ (cid:17) , u = (cid:16) − sin θ ,~ ⊥ , cos θ (cid:17) , (3.5)and R = (0 , ~R ⊥ , R = (0 , ~R ⊥ , z = (0 , ~z ⊥ , T removed as C E ≡ lim T →∞ G E .It has been shown in [18, 19, 22] that the correlation functions in the two theoriesare connected by the analytic–continuation relations G M ( χ ; T ; ~z ⊥ ; 1 ,
2) = G E ( − iχ ; iT ; ~z ⊥ ; 1 , , ∀ χ ∈ R + , G E ( θ ; T ; ~z ⊥ ; 1 ,
2) = G M ( iθ ; − iT ; ~z ⊥ ; 1 , , ∀ θ ∈ (0 , π ) . (3.6) The fourth Euclidean coordinate X is taken to be the “Euclidean time”. The functions on the right–hand side of Eqs. (3.6) and (3.7) are understood as the analytic extensions of the Euclidean and Minkowskian correlation functions, starting from the real intervals (0 , π ) and R + ofthe respective angular variables, with positive real T in both cases, into domains of the complex variables θ (resp. χ ) and T in a two–dimensional complex space. See Ref. [22] for a more detailed discussion. nder certain analyticity hypotheses in the T variable, the following relations areobtained for the correlation functions with the IR cutoff T removed [19, 22]: C M ( χ ; ~z ⊥ ; 1 ,
2) = C E ( − iχ ; ~z ⊥ ; 1 , , ∀ χ ∈ R + , C E ( θ ; ~z ⊥ ; 1 ,
2) = C M ( iθ ; ~z ⊥ ; 1 , , ∀ θ ∈ (0 , π ) . (3.7)We turn now to the calculation of instanton effects on the correlation function C E . As it is explained in Appendix A, we can set f = f = 1 / f i in the following formulas, i.e., we un-derstand C E ( θ ; ~z ⊥ ; ~R ⊥ , ~R ⊥ ) ≡ C E ( θ ; ~z ⊥ ; ~R ⊥ , f = 1 / , ~R ⊥ , f = 1 / W ( T ) i = 1 N c Tr [W (¯ q ) † i W ( q ) i ] , i = 1 , , (3.8)with W (¯ q ) i = P exp (cid:20) − ig Z T − T A µ ( X i ¯ q ( τ )) u iµ dτ (cid:21) , i = 1 , ( q ) i . Since the (long) sides of the two loops have different direc-tions in the longitudinal plane, their relative distance grows as we move away fromthe center of the configuration; as a consequence, only pseudoparticles falling in afinite interaction region have effects on both loops (see Fig. 2), and will thereforecontribute to the connected part of the loop–loop correlator. This region is roughlydetermined by requiring that the distance of the pseudoparticles from both loops inthe transverse and in the longitudinal plane does not exceed ∼ ρ . Then V int = V k V ⊥ ,where k and ⊥ refer to the longitudinal and transverse planes, respectively, with V ⊥ ∼ | ~z ⊥ | + | ~R ⊥ | + | ~R ⊥ | ρ ! | ~R ⊥ | + | ~R ⊥ | ρ ! , (3.10) V k ∼ (2 ρ ) sin θ . (3.11)We now make the following approximation, considering only one instanton or anti–instanton in the interaction region, all the other pseudoparticles interacting withone loop only (or not interacting at all with the loops). Clearly, V k blows upas θ → , π , and so does the number of pseudoparticles in the interaction region nV int ∝ nρ / sin θ ; thus, the one–instanton approximation can be good if the angleis not too close to 0 , π . In this case we can perform the integration over the colourdegrees of freedom independently for the two loops, except for the pseudoparticlein the interaction region, but since this last integration is trivial we obtain simply W ( T )1 W ( T )2 → hY ′ w ( k ) Y ′ w ( k ) i w (0) w (0) , (3.12) here the prime indicates that we exclude the terms corresponding to the in-teraction region, and 0 refers to the interacting pseudoparticle. Here w i ( k ) =1 − (2 /N c )∆ i ( x P ( k ) ), with x P ( k ) being the position of the pseudoparticle lying in the k –th cell [see Eq. (2.7)], and with∆ i ( x ) = 1 − cos α i + cos α i − − ˆ n i + · ˆ n i − sin α i + sin α i − ,α i ± = π k n i ± k p k n i ± k + ρ ,n a ± = η aµν u µ ( z ± R − x ) ν , n a ± = η aµν u µ ( ± R − x ) ν . (3.13)Performing the integration over the pseudoparticle positions, making use of for-mula (B.7) in Appendix B, and dividing by the expectation values of the loops [seeEq. (2.11)] we finally obtain for the normalised connected correlation function with T → ∞C (ILM) E ( θ ; ~z ⊥ ; ~R ⊥ , ~R ⊥ )= 1 + n Z int d x (cid:20)(cid:18) − N c ∆ ( x ) (cid:19) (cid:18) − N c ∆ ( x ) (cid:19) − (cid:21)(cid:20) − n Z int d x N c ∆ ( x ) (cid:21) (cid:20) − n Z int d x N c ∆ ( x ) (cid:21) − , (3.14)where “int” indicates that the integration range is restricted to the interactionregion, since all the other terms cancel between the numerator and denominator.Expanding to first order in n , for consistency with the one–instanton approximation,we finally get C (ILM) E ( θ ; ~z ⊥ ; ~R ⊥ , ~R ⊥ ) = n (cid:18) N c (cid:19) Z d x ∆ ( x )∆ ( x ) , (3.15)having extended the integration range to the whole space–time since the integrandis now rapidly vanishing. Exploiting the properties of the ’t Hooft symbols we find k n ± k = ~z ⊥ ± ~R ⊥ − ~x ⊥ ! + (cid:18) sin (cid:18) θ (cid:19) x − cos (cid:18) θ (cid:19) x (cid:19) , k n ± k = ± ~R ⊥ − ~x ⊥ ! + (cid:18) sin (cid:18) θ (cid:19) x + cos (cid:18) θ (cid:19) x (cid:19) , (3.16)and, moreover, n · n − = ( ~z ⊥ − ~x ⊥ ) − ~R ⊥ ! + (cid:18) sin (cid:18) θ (cid:19) x − cos (cid:18) θ (cid:19) x (cid:19) ,n · n − = ~x ⊥ − ~R ⊥ ! + (cid:18) sin (cid:18) θ (cid:19) x + cos (cid:18) θ (cid:19) x (cid:19) . (3.17) e can then make the dependence on θ explicit by performing a change of variables: x ′ = sin (cid:18) θ (cid:19) x − cos (cid:18) θ (cid:19) x ,x ′ = sin (cid:18) θ (cid:19) x + cos (cid:18) θ (cid:19) x , (3.18)which finally leads to C (ILM) E ( θ ; ~z ⊥ ; ~R ⊥ , ~R ⊥ ) = n (cid:18) N c (cid:19) θ F ( ~z ⊥ , ~R ⊥ , ~R ⊥ ) , (3.19)with F ( ~z ⊥ , ~R ⊥ , ~R ⊥ ) ≡ Z d x ∆ ( x )∆ ( x ) | θ = π/ , (3.20)where the subscript means that the integral is evaluated setting θ = π/ θ as in Ref. [23], but the prefactor n (2 /N c ) F ( ~z ⊥ , ~R ⊥ , ~R ⊥ ) in Eq. (3.19) can now be assessed numerically.In order to compare the analytic expression (3.19) to the lattice data, we have toset the density n and the radius ρ to the appropriate values. The phenomenologicalestimates n = 1 fm − and ρ = 1 / physical (i.e., unquenched )case, while our numerical simulations have been performed in the quenched approx-imation of QCD, neglecting dynamical–fermion effects. We have then preferred touse the values n q = 1 . .
64 fm − for the density and ρ q = 0 .
35 fm for the av-erage size of the pseudoparticles, obtained directly by lattice calculations in thepure–gauge theory [39]. The packing fraction is a factor 1 . . . dilute liquid picture is still reasonable . The ILM prediction obtainedwith these values is shown in Figs. 3–7, together with the results obtained on thelattice in Ref. [28], for the loop configurations “ zzz ” ( ~R ⊥ k ~R ⊥ k ~z ⊥ ) and “ zyy ”( ~R ⊥ k ~R ⊥ ⊥ ~z ⊥ ), with | ~R ⊥ | = | ~R ⊥ | = 0 . We drop the absolute value from the Jacobian | sin θ | , since we are limiting to θ ∈ (0 , π ). The phenomenological estimate of n is obtained assuming that instantons and anti–instantons dom-inate the gluon condensate G = h ( α s /π ) F aµν F aµν i [8, 9], so that n ∝ G . If we assumed that the sameholds in the pure–gauge theory, we would obtain n q /n = G q /G ≃ . G the value obtainedin Ref. [32], and for the quenched gluon condensate G q the result obtained on the lattice in Ref. [40]),i.e., a value for the pseudoparticle density n q considerably larger than the one measured on the lattice. q = 1 .
33 fm − , while the dashed line corresponds to n q = 1 .
64 fm − ; for compari-son, we plot also the prediction obtained using the phenomenological values of n and ρ (solid line). As already noticed in [28], the functional form does not seem to be thecorrect one to properly describe the lattice data, and a second term, proportionalto (cot θ ) , must be added to obtain a good fit. In Table 1 we show for comparisonthe ILM prediction of the prefactor K = n q (2 /N c ) F ( ~z ⊥ , ~R ⊥ , ~R ⊥ ), and the valueobtained with a fit to the lattice data with the fitting functions (see Ref. [28]) f ILM = K ILM θ , f
ILMp = K ILMp θ + K ′ ILMp (cot θ ) , (3.21)where f ILMp is obtained by adding the lowest–order perturbative expression [41,15, 19] to the ILM contribution. Notice, however, that K ′ ILMp can also receivenonperturbative contributions from two–instanton effects [23].The instanton prediction turns out to be more or less of the correct order ofmagnitude in the range of distances considered, at least around θ = π/ | ~z ⊥ | = 0 . In this Section we calculate the normalised correlation function of two (infinite)parallel Wilson loops, which describe the time evolution of two static colour dipoles.The one–instanton approximation makes no sense here, since the interaction regionhas infinite extension in the time direction, and the effect of a whole time–seriesof pseudoparticles on the two dipoles has to be considered. The above–mentionedcorrelation function can be used to extract the dipole–dipole potential V dd by meansof the formula [29] hW W ihW ihW i ≃ T →∞ e − T V dd ( ~d, ~R , ~R ) , (4.1)where ~R and ~R are the sizes of the two dipoles, ~d is the distance between theircenters, and T is the length of the two loops. Neglecting again the transverse onnectors, the loops W , are written as W i = 1 N c Tr [W ( − ) † i W (+) i ] , (4.2)with W ( ± )1 = P exp (cid:20) − ig Z T A ( ut + d ± R / dt (cid:21) , W ( ± )2 = P exp (cid:20) − ig Z T A ( ut ± R / dt (cid:21) , (4.3)where u = ( ~ , R i = ( ~R i , d = ( ~d, V (3)cell to be large enough toaccommodate both the dipoles. Requiring V (3)cell ∼ | ~d | + | ~R | + | ~R | ρ ! | ~R | + | ~R | ρ ! , (4.4)we have to set δT to δTρ ∼ , for | ~d | , | ~R i | ∼ ρ/ , δTρ ∼ . , for | ~d | , | ~R i | ∼ ρ, (4.5)which is still a fairly large value. Note however that we are overestimating thevolume of the region of 3–space where a pseudoparticle can affect the loop, so that δT is actually underestimated. Again, for large distances the two dipoles interactwith different pseudoparticles, and the correlation function is expected to factorise,thus giving a vanishing dipole–dipole potential as | ~d | → ∞ .Now let K be the average number of instantons that interact with the loop. Thisnumber grows linearly with the length T of the loop due to the uniformity of theliquid, and we can take, without loss of generality, T = KδT , since we are interestedin the limit T → ∞ . We divide space–time into cells as described above, labelling D , . . . , D K the cells where the loop lives, and we defineW ( ± )1 ( k ) = P exp " − ig Z kδT ( k − δT A ( ut + d ± R / dt , W ( ± )2 ( k ) = P exp " − ig Z kδT ( k − δT A ( ut ± R / dt , k = 1 , . . . , K, (4.6)and the “two–link” variables T i ( k ) = W ( − ) ∗ i ( k ) ⊗ W (+) i ( k ) , W i ( j ) = Y k = j T i ( k ) , (4.7) hich allow us to write W i = 1 N c [ W i ( K )] ijij = 1 N c f Tr W i ( K ) . (4.8)The integration over the colour degrees of freedom of the pseudoparticles is morecomplicated than in the case of a single loop, although the procedure is the same.Starting from the K –th pseudoparticle, we have to calculate the integral I ≡ Z dU ( K ) f Tr W ( K ) f Tr W ( K )= Z dU ( K )ˆ T ( K ) j k ˇ W ( K − j k ˆ T ( K ) j k ˇ W ( K − j k = I j ′ k ′ j k j ′ k ′ j k ˆ T ( K ) j ′ k ′ | s ˇ W ( K − j k ˆ T ( K ) j ′ k ′ | s ˇ W ( K − j k , (4.9)where the subscript s means that the given quantity has to be evaluated in thefield of a pseudoparticle with standard colour orientation, where we have used thenotationˆ T i ( k ) jk = T i ( k ) ijik , ˇ W i ( k ) jk = W i ( k ) jiki , f Tr W i ( k ) = ˆ T i ( k ) jk ˇ W i ( k − jk , (4.10)and where [42] I j ′ k ′ j k j ′ k ′ j k = Z dU U j j ′ U † k ′ k U j j ′ U † k ′ k = (cid:16) aδ j ′ k ′ δ k ′ k ′ + bδ j ′ k ′ δ j ′ k ′ (cid:17) δ j k δ j k + (cid:16) bδ j ′ k ′ δ k ′ k ′ + aδ j ′ k ′ δ j ′ k ′ (cid:17) δ j k δ j k ,a = 1 N c − , b = − N c a. (4.11)Performing the contractions of colour indices we obtain I = a (cid:2) N c w ( K ) w ( K ) − w ( K ) (cid:3)f Tr W ( K − f Tr W ( K − aN c (cid:2) w ( K ) − w ( K ) w ( K ) (cid:3) W ( K − , (4.12)where w i ( k ) ≡ N c ˆ T i ( k ) j ′ j ′ | s = 1 N c Tr [W ( − ) † i ( k )W (+) i ( k )] | s = 1 − N c ∆ i ( z P ( k ) ) , ∆ i ( z ) = 1 − cos α i + cos α i − − ˆ n i + · ˆ n i − sin α i + sin α i − ,α i ± = π k n i ± k p k n i ± k + ρ , n a ± = − ( d ± R − z ) a , n a ± = − ( ± R − z ) a , (4.13) Here and in the following we adopt the following short–hand notation for quantities which dependon the position and the type of the pseudoparticle in the k –th cell, f ( k ) = f ( z P ( k ) , σ P ( k ) ). nd we have introduced the quantities w ( k ) ≡ N c ˆ T ( k ) j ′ j ′ | s ˆ T ( k ) j ′ j ′ | s = 1 N c Tr [W ( − ) † ( k )W (+)1 ( k )W ( − ) † ( k )W (+)2 ( k )] | s , W ( k ) ≡ ˇ W ( k ) j j ˇ W ( k ) j j . (4.14)The effect of a single instanton on W ( ± ) i ( k ) is given byW ( ± ) i = exp (cid:2) i ˆ n ai ± σ a α i ± (cid:3) I N c − ! , (4.15)where I M is the M -dimensional unit matrix, while the effect of an anti–instanton isobtained by replacing n i ± → − n i ± . The expectation value of the quantity W ≡ N c W ( K ) = 1 N c Tr [W ( − ) † W (+)1 W ( − ) † W (+)2 ] (4.16)can be interpreted as the transition amplitude of an inelastic process , the final state e d e d being the initial one, dd , with the (say) antiquarks in the two dipoles interchanged(see Fig. 8). Over the time T a certain number of such transitions can happen, andthis is at the origin of the new terms in the formulas above.To proceed we also need to calculate the integral I ≡ Z dU ( K ) W ( K ) . (4.17)If we write W ( K ) = ˇ W ( K ) j j ˇ W ( K ) j j = [ T ( K )] j j ′ j k ′ | s [ T ( K )] j j ′ j k ′ | s × U j j ′ U † k ′ k U j j ′ U † k ′ k ˇ W ( K − j k ˇ W ( K − j k , (4.18)we find the same Haar integral as before, and thus, after contracting the colourindices, we find I = aN c (cid:2) w ( K ) − e w ( K ) e w ( K ) (cid:3)f Tr W ( K − f Tr W ( K − a (cid:2) N c e w ( K ) e w ( K ) − w ( K ) (cid:3) W ( K − , (4.19)where we have introduced the quantity w ( k ), w ( k ) ≡ N c [ T ( k )] j j ′ j j ′ | s [ T ( k )] j j ′ j j ′ | s = 1 N c Tr [W (+)1 ( k )W ( − ) † ( k )W (+)2 ( k )W ( − ) † ( k )] | s , (4.20)and the quantities e w i ( k ), which come from the contraction of e w ( k ) e w ( k ) = 1 N c [ T ( k )] j j ′ j j ′ | s [ T ( k )] j j ′ j j ′ | s , (4.21) nd are equal to the value of the Wilson loops (of infinite length) obtained from W i interchanging the antiquarks in the two dipoles, in the field of a single pseudopar-ticle: e w ( k ) ≡ N c Tr [W ( − ) † ( k )W (+)1 ( k )] | s = 1 − N c e ∆ ( z P ( k ) ) , e ∆ ( z ) = 1 − cos α cos α − − ˆ n · ˆ n − sin α sin α − , e w ( k ) ≡ N c Tr [W ( − ) † ( k )W (+)2 ( k )] | s = 1 − N c e ∆ ( z P ( k ) ) , e ∆ ( z ) = 1 − cos α cos α − − ˆ n · ˆ n − sin α sin α − , (4.22)where the functions e ∆ i are again independent of the pseudoparticle species. Also, w ( k ) is related to w ( k ) in the same way.To iterate the process we organise the previous results as follows: Z dU ( K ) f Tr W ( K ) f Tr W ( K ) W ( K ) ! = M ( K ) f Tr W ( K − f Tr W ( K − W ( K − ! , (4.23)where M ( k ) is the matrix M ( k ) = 1 N c − N c w ( k ) w ( k ) − w ( k ) N c (cid:2) w ( k ) − w ( k ) w ( k ) (cid:3) N c (cid:2) w ( k ) − e w ( k ) e w ( k ) (cid:3) N c e w ( k ) e w ( k ) − w ( k ) ! . (4.24)The iteration is now straightforward, and yields Z d K U N c W W N c W ! = Z d K U f Tr W ( K ) f Tr W ( K ) W ( K ) ! = " Y k = K M ( k ) N c N c ! , (4.25)where we used the fact that f Tr W (1) f Tr W (1) W (1) ! (cid:12)(cid:12)(cid:12)(cid:12) s = N c w (1) w (1) N c w (1) ! = M (1) N c N c ! . (4.26)We are left with the integration over the pseudoparticle positions. The procedureis described in Appendix B; using Eq. (B.7) we then obtain Z d N z D ( { z } ) " Y k = K M ( k ) = Y j = K n Z D j d z (cid:16) ν I M ( z, σ I ) + ν ¯I M ( z, σ ¯I ) (cid:17) = Y j = K " I + n Z D j d z (cid:16) ν I ˆ M ( z, σ I ) + ν ¯I ˆ M ( z, σ ¯I ) (cid:17) , (4.27) here ˆ M ≡ M − I falls off to zero as | z | → ∞ , and where ν I , ¯I ≡ n I , ¯I /n . Performingthe trivial integration over the time position, extending the spatial integration tothe whole space, and letting (formally) K → ∞ with T fixed, we obtain Z d N z D ( { z } ) " Y k = K M ( k ) = exp { nT J } , (4.28)where the matrix J is given by J ≡ Z d z (cid:16) ν I ˆ M ( z, σ I ) + ν ¯I ˆ M ( z, σ ¯I ) (cid:17) = 1 N c − N c A − B N c ( B − A ) N c ( e B − e A ) N c e A − e B ! = B e B ! + 1 N c − N c ( A − B ) N c ( B − A ) N c ( e B − e A ) N c ( e A − e B ) ! , (4.29)having defined A = Z d z "(cid:18) N c (cid:19) ∆ ( z )∆ ( z ) − N c ∆ ( z ) − N c ∆ ( z ) , e A = Z d z "(cid:18) N c (cid:19) e ∆ ( z ) e ∆ ( z ) − N c e ∆ ( z ) − N c e ∆ ( z ) ,B = Z d z (cid:20) ν I ( w ( z, I) −
1) + ν ¯I ( w ( z, ¯I) − (cid:21) , e B = Z d z (cid:20) ν I ( w ( z, I) −
1) + ν ¯I ( w ( z, ¯I) − (cid:21) . (4.30)To proceed further with the calculation we now set the pseudoparticle fractions totheir phenomenological values ν I = ν ¯I = 1 /
2: in this case we can show that (seeAppendix B) B = e B = 2 N c Z d z (cid:20)(cid:0) ∆ ( z )∆ ( z ) + e ∆ ( z ) e ∆ ( z ) − ∆ + ( z )∆ − ( z ) (cid:1) − (cid:0) ∆ ( z ) + ∆ ( z ) + e ∆ ( z ) + e ∆ ( z ) − ∆ + ( z ) − ∆ − ( z ) (cid:1)(cid:21) , (4.31)where the quantities∆ + ( z ) = 1 − cos α cos α − ˆ n · ˆ n sin α sin α , ∆ − ( z ) = 1 − cos α − cos α − − ˆ n − · ˆ n − sin α − sin α − , (4.32)are the analogues of e ∆ i with the positions of the quark and the antiquark in the(say) second dipole interchanged. It is now easy to diagonalise J : denoting with = A − B , e X = e A − B , the eigenvalues can be written as λ ± = A + e A N c − (cid:26) X + e X ± N c q ( N c − e X − X ) + ( e X + X ) (cid:27) ;(4.33)since they are different , J is diagonalisable. Denoting by Π ± the correspondingprojectors we can write [ v ≡ (1 , v N c ≡ ( N c , N c )] N c hW W i ILM = v · (cid:0) exp { nT J } v N c (cid:1) = v · h(cid:0) e nT λ + Π + + e nT λ − Π − (cid:1) v N c i , (4.34)and, taking the logarithm, in the large– T limit we findlog hW W i ILM = nT λ + + (subleading terms) . (4.35)In the last passage we have implicitly assumed that the projector Π + is such that v · (Π + v N c ) >
0: we have explicitly verified that this is true in the cases that we haveinvestigated numerically. Note that the correlation function of parallel loops mustbe positive, as a consequence of reflection positivity of the Euclidean theory [43]:this can be easily proved following Ref. [44]. Finally, recalling Eqs. (4.1) and (2.11)we can conclude that V dd ( ~d, ~R , ~R ) = V ( ~d, ~R , ~R ) − V q ¯ q ( | ~R | ) − V q ¯ q ( | ~R | ) , (4.36)where we have set V ≡ − nλ + .In Figs. 9–12 we show the comparison of the instanton–induced dipole–dipolepotential with some preliminary numerical data obtained on the lattice: to ourknowledge, lattice measurements of this quantity are not present in the literature.Also in this case we have performed a quenched SU (3) lattice calculation, with thesame parameters (16 hypercubic lattice, β = 6 .
0) used in Ref. [28], and so wehave to use the quenched instanton density n q and radius ρ q , as explained in theprevious Section. The errors are very large for the lattice data corresponding tothe largest distances and lengths, and a plateau has not been reached yet even at T = 0 . | ~d | = 0 fm and | ~d | = 0 . V dd as obtained with loops of increasing lengths (0 . ÷ . V dd extracted from the largest–length loops is plotted at thecorrect value of the distance, while the data points corresponding to shorter lengthsare slightly shifted, with the length increasing from left to right. The instanton–induced potential and the lattice data have different orders of magnitude for the The only exception is the case e X = X = 0, in which case however the matrix is already in diagonalform. hortest distances. At larger distances there seems to be a better agreement, but theerror bars of the lattice data are very large there, and a better precision is needed toquantify the importance of instanton effects at large distances. However, as alreadynoticed in the previous Section, the rate of decrease with the distance seems to beunderestimated in the ILM, which would lead to an increasing discrepancy betweenthe prediction and the lattice data as the distance between the dipoles increases. In this paper we have considered instanton effects, in the framework of the ILM,on the Wilson–loop correlation function relevant to soft high–energy scattering, andwe have compared the results to the lattice data discussed in Ref. [28]. Using anapproximate density function, we have critically repeated the ILM calculation ofthe relevant correlation function, already performed in Ref. [23], properly takinginto account effects which were neglected in that paper. The analytic dependenceon the angular variable θ is the same as the one found in Ref. [23], which we haveused in Ref. [28] to perform fits to the lattice data. In this paper we have insteadperformed a direct comparison of the ILM quantitative prediction, obtained by eval-uating numerically our analytic expression for the loop–loop correlation function inthe relevant cases. In doing so, we have used the pseudoparticle density n q andradius ρ q appropriate for the quenched approximation of QCD, in which the latticecalculation has been performed. A direct numerical comparison allows us to testnot only the given functional form, but also the quantitative relevance of instantoneffects on the Wilson–loop correlation function.The comparison of the ILM prediction with the lattice data is not satisfactory,although it seems to have the correct shape in the vicinity of θ = 90 ◦ . Moreover,although the ILM prediction seems to be (more or less) of the correct order ofmagnitude, the rate of decrease with the transverse distance seems to be underesti-mated. Here one sees the importance of a quantitative comparison: terms behavingas 1 / sin θ (which are qualitatively good in fitting the data around θ = 90 ◦ ) canhave a different origin than instantons, and indeed it seems that the ILM is notable to explain them from a quantitative point of view. Also, the ILM expressioncannot account for the small but non–zero odderon contribution to the correlationfunction which we have found in the lattice results [28]. As a final remark, notethat, after analytic continuation into Minkowski space–time, the ILM result in theone–instanton approximation gives an exactly zero total cross section, since the for-ward meson–meson scattering amplitude turns out to be purely real. The situationcan change after inclusion of two–instanton effects, which are expected to yield aconstant total cross section at high energy [23]; nevertheless, an analytic expressionfor this contribution is currently unavailable. n this paper we have also derived an analytic expression for the instanton–induced dipole–dipole potential from the correlation function of two parallel Wilsonloops, which we have compared to some preliminary data from the lattice. In thiscase, instanton effects seem to be negligible at short distances, less than or equal tothe size of the dipoles, where there is a large discrepancy with the lattice results.The rate of decrease of the potential with the spatial distance between the dipolespredicted by the ILM seems also in this case to be smaller than the value found onthe lattice; a better accuracy in the numerical calculations is needed to make thisstatement quantitative, but we think that the situation would not change from aqualitative point of view. Dependence of the Wilson–loop correlationfunction on the longitudinal momentum frac-tions
Exploiting the symmetries of the functional integral, one can show that the relevantWilson–loop correlation function for given values of the longitudinal momentumfractions can always be reduced to the case f = f = 1 /
2. We discuss here the caseof the physical, Minkowskian correlation function G M ; the proof in the Euclideancase is perfectly analogous.As far as the dependence on the transverse vectors ~z ⊥ , ~R ⊥ , and ~R ⊥ , and onthe longitudinal momentum fractions f and f is concerned, the invariance of thetheory under translations and spatial rotations implies that G M can depend onlyon the scalar products between the relative positions of quarks and antiquarks intransverse space. The number of independent variables is then six, while ~z ⊥ , ~R i ⊥ and f i form a set of eight variables, so two of them are redundant. Indeed, theloop configuration in the transverse plane is determined by the relative distances ~R ⊥ and ~R ⊥ between quarks and antiquarks inside each meson, and by the relativedistance between (say) quark 1 and quark 2 in the transverse plane, ~X q ⊥ ( τ ) − ~X q ⊥ ( τ ) = ~z ⊥ + (1 − f ) ~R ⊥ − (1 − f ) ~R ⊥ , ∀ τ. (A.1)To determine which configurations are equivalent, we have to solve the equation ~z ′⊥ = ~z ⊥ + ( f ′ − f ) ~R ⊥ − ( f ′ − f ) ~R ⊥ . (A.2)Since a configuration specified by ( ~z ′⊥ , f ′ , f ′ ), with ~z ′⊥ given above, is equivalent tothe configuration ( ~z ⊥ , f , f ) we have G M ( χ ; T ; ~z ′⊥ ; ~R ⊥ , f ′ , ~R ⊥ , f ′ ) = G M ( χ ; T ; ~z ⊥ ; ~R ⊥ , f , ~R ⊥ , f ); (A.3)one can then choose a reference value for f ′ i , for example f ′ = f ′ = 1 /
2, and moveall the dependence on f i inside the dependence on the impact parameter. More-over, since G M enters the dipole–dipole scattering amplitude with its 2–dimensionalFourier transform with respect to the impact parameter, one can write Z d z ⊥ e i~q ⊥ · ~z ⊥ G M ( χ ; T ; ~z ⊥ ; ~R ⊥ , f , ~R ⊥ , f )= Z d z ⊥ e i~q ⊥ · ~z ⊥ G M ( χ ; T ; ~z ⊥ − ( f −
12 ) ~R ⊥ + ( f −
12 ) ~R ⊥ ; ~R ⊥ , , ~R ⊥ ,
12 )= e i~q ⊥ · [( f − ) ~R ⊥ − ( f − ) ~R ⊥ ] Z d z ⊥ e i~q ⊥ · ~z ⊥ G M ( χ ; T ; ~z ⊥ ; ~R ⊥ , , ~R ⊥ ,
12 ) . (A.4)It is then clear that one can consider the case f = f = 1 / ~q ⊥ = 0, the dependence on f i drops, and the integration onthe longitudinal momentum fractions affects only the wave functions. Technical details
B.1 Integration over the position of pseudoparticles
Consider the integral
I ≡ Z d N z D ( { z } ) Y k ∈ I K f k ( z P ( k ) , σ P ( k ) ) , (B.1)where f k are K (possibly distinct) matrices, whose entries are functions of theposition and of the type of the pseudoparticle lying in cell k , and I K is the orderedset of the K relevant cells. We have I = N − Z d N z X P N Y j =1 χ D j ( z P ( j ) ) Y k ∈ I K f k ( z P ( k ) , σ P ( k ) )= N − X P V N − K cell Z d K z Y k ∈ I K χ D k ( z P ( k ) ) f k ( z P ( k ) , σ P ( k ) )= N − X P V N − K cell Y k ∈ I K Z D k d z f k ( z, σ P ( k ) ) , (B.2)which, denoting F k ( σ P ( k ) ) ≡ Z D k d z f k ( z, σ P ( k ) ) , (B.3)becomes I = N − X P V N − K cell Y k ∈ I K F k ( σ P ( k ) ) = N − X S K V N − K cell η ( S K ) Y k ∈ I K F k ( σ P ( k ) ) , (B.4)where P S K denotes the sum over all possible sequences of K pseudoparticles, and η ( S K ) is the number of ways in which a given sequence can be obtained from theensemble, η ( S K ) = ( N − K )! N I !( N I − l I )! N ¯I !( N ¯I − l ¯I )! , (B.5) l I and l ¯I being the number of instantons and anti–instantons in the sequence, respec-tively. In the limit of large N I , N ¯I and V , with the ratios ν I = N I /N , ν ¯I = N ¯I /N ,and n = N/V kept fixed, η ( S K ) → N n N ν l I I ν l ¯I ¯I , (B.6)and so Eq. (B.4) simplifies to (recall that nV cell = 1) I = n K X S K ν l I I ν l ¯I ¯I Y k ∈ I K F k ( σ P ( k ) ) = n K X S K Y k ∈ I K ν σ P ( k ) F k ( σ P ( k ) )= n K Y k ∈ I K X σ =I , ¯I ν σ F k ( σ ) = Y k ∈ I K n X σ =I , ¯I ν σ Z D k d z f k ( z, σ ) . (B.7) .2 The function B for ν I = ν ¯I We prove that the functions B and e B , defined in Eq. (4.30), are equal for ν I = ν ¯I ,and find their explicit form. To do so, note the following about SU (2) matrices.One can always write u ∈ SU (2) asu = u I + i~u · ~σ, u , u i ∈ R , ( u ) + ~u = 1 , (B.8)with u = Tr u and ~u = − i Tr ~σ u, with Tr ≡ (1 / σ i , σ j ] = 2 iǫ ijk σ k , { σ i , σ j } = 2 δ ij I , (B.9)we can write for the product of two unitary matricesu ≡ u u = u I + i~u · ~σ, (B.10)with u = [( u ) ( u ) − ~u · ~u ] , ~u = ~u s + ~u a ,~u s = ( u ) ~u + ( u ) ~u , ~u a = − ( ~u ∧ ~u ) . (B.11)Clearly, if we interchange u and u we obtainv ≡ u u = v I + i~v · ~σ,v = u , ~v = ~u s − ~u a . (B.12)In our case, denoting with w ( ± ) i ( k ) the non–trivial 2 × ( ± ) i ( k ), we have(suppressing the argument k )w ( ± ) i = w ± ) i I + ~w ( ± ) i · ~σ, (B.13)so that w i ≡ w ( − ) † i w (+) i = w i I + ( ~w i ; s + ~w i ; a ) · ~σ, (B.14)where, explicitly, w i = [( w ( − ) i ) ( w (+) i ) + ~w ( − ) i · ~w (+) i ] ,~w i ; s = [( w ( − ) i ) ~w (+) i − ( w (+) i ) ~w ( − ) i ] ,~w i ; a = ~w ( − ) i ∧ ~w (+) i . (B.15)Since the vectors ~w ( ± ) i change sign if we replace an instanton with an anti–instant-on, we have that under this replacement ~w i ; s changes sign, too, while ~w i ; a does not.Letting e w = w ( − ) † w (+)1 and e w = w ( − ) † w (+)2 , and ¯w i = w (+) i w ( − ) † i , we have1 − w = 2 N c { − Tr [w w ] } , − w = 2 N c { − Tr [ e w e w ] } = 2 N c { − Tr [ ¯w ¯w ] } ; (B.16) ow, Tr [w w ] = w w − ( ~w s + ~w a ) · ( ~w s + ~w a ) , Tr [ ¯w ¯w ] = w w − ( ~w s − ~w a ) · ( ~w s − ~w a ) , (B.17)and since we are averaging with equal weights over the pseudoparticle species, weneed to consider only terms which are symmetric under I ↔ ¯I,Tr [w w ] = w w − ( ~w s · ~w s + ~w a · ~w a ) + antisymmetric terms , Tr [ ¯w ¯w ] = w w − ( ~w s · ~w s + ~w a · ~w a ) + antisymmetric terms . (B.18)From its definition, Eq. (4.30), it is now immediate to conclude that B = e B . More-over, one can show that ~w s · ~w s + ~w a · ~w a = [( w (+)1 ) ( w (+)2 ) + ~w (+)1 · ~w (+)2 ][( w ( − )1 ) ( w ( − )2 ) + ~w ( − )1 · ~w ( − )2 ] − [( w ( − )1 ) ( w (+)2 ) + ~w ( − )1 · ~w (+)2 ][( w ( − )2 ) ( w (+)1 ) + ~w ( − )2 · ~w (+)1 ] , (B.19)which introducing w + = w (+) † w (+)2 = w I + i ~w + · ~σ, w − = w ( − ) † w ( − )2 = w − I + i ~w − · ~σ, (B.20)leads toTr [w w ] | symmetric = w w + e w e w − w w − = Tr [ e w e w ] | symmetric , (B.21)and thus to B = Z d z (cid:2)(cid:0) w ( z, I) − (cid:1) + (cid:0) w ( z, ¯I) − (cid:1)(cid:3) = 2 N c Z d z (cid:2)(cid:0) ∆ ∆ + e ∆ e ∆ − ∆ + ∆ − (cid:1) − (cid:0) ∆ + ∆ + e ∆ + e ∆ − ∆ + − ∆ − (cid:1)(cid:3) = e B, (B.22)with ∆ ± given in the text, Eq. (4.32). eferences [1] A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkin, Phys. Lett.B (1975) 85.[2] R. Jackiw and C. Rebbi, Phys. Rev. Lett. (1976) 172.[3] C.G. Callan, R. Dashen and D.J. Gross, Phys. Lett. B (1976) 334.[4] A.M. Polyakov, Nucl. Phys. 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33 fm − and the dashed line corresponds to n q = 1 .
64 fm − ,with ρ q = 0 .
35 fm; the solid line corresponds to the phenomenological values n = 1 fm − and ρ = 1 /
20 40 60 80 100 120 140 160 C zzz ( θ ) θ [ ◦ ] Figure 4: Same comparison as in Fig. 3 for the “ zzz ” configuration at | ~z ⊥ | = 0 .
20 40 60 80 100 120 140 160 C z yy ( θ ) θ [ ◦ ] Figure 5: Same comparison as in Fig. 3 for the “ zyy ” configuration at | ~z ⊥ | = 0 .
20 40 60 80 100 120 140 160 C zzz ( θ ) θ [ ◦ ] Figure 6: Same comparison as in Fig. 3 for the “ zzz ” configuration at | ~z ⊥ | = 0 .
20 40 60 80 100 120 140 160 C z yy ( θ ) θ [ ◦ ] Figure 7: Same comparison as in Fig. 3 for the “ zyy ” configuration at | ~z ⊥ | = 0 . ③❥ ③ ③ ❥③ ❥ ❥ ③❥ ③ ③ ❥③ ❥ ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ ✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡✡ ✡✡✡✡✡✡✡✡ ✡✡✡✡✡✡✡✡ Figure 8: Schematic representation of the operators W W (left) and W (right), cor-responding respectively to the processes dd → dd and dd → e d e d . White (black) circlesrepresent quarks (antiquarks). 34 .00010.0010.010.11101001000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 9: The instanton–induced dipole–dipole potential compared to the lattice data(on a logarithmic scale). The three lines correspond to n q = 1 .
33 fm − (dotted line)and n q = 1 .
64 fm − (dashed line) with ρ q = 0 .
35 fm, and to the phenomenological values n = 1 fm − and ρ = 1 / ~R = ~R are parallel to ~d , with | ~R i | = 0 . .0010.010.1110100100010000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 10: Same comparison as in Fig. 9, but with ~R = − ~R , ~R parallel to ~d , with | ~R i | = 0 ..
35 fm, and to the phenomenological values n = 1 fm − and ρ = 1 / ~R = ~R are parallel to ~d , with | ~R i | = 0 . .0010.010.1110100100010000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 10: Same comparison as in Fig. 9, but with ~R = − ~R , ~R parallel to ~d , with | ~R i | = 0 .. .0010.010.11101001000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 11: Same comparison as in Fig. 9, but with ~R = ~R orthogonal to ~d , with | ~R i | = 0 ..
35 fm, and to the phenomenological values n = 1 fm − and ρ = 1 / ~R = ~R are parallel to ~d , with | ~R i | = 0 . .0010.010.1110100100010000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 10: Same comparison as in Fig. 9, but with ~R = − ~R , ~R parallel to ~d , with | ~R i | = 0 .. .0010.010.11101001000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 11: Same comparison as in Fig. 9, but with ~R = ~R orthogonal to ~d , with | ~R i | = 0 .. .010.1110100100010000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 − V dd ( d ) [ M e V ] d [fm] Figure 12: Same comparison as in Fig. 9, but with ~R = − ~R orthogonal to ~d , with | ~R i | = 0 ..