Classical Integrability of the Zigzag Model

Classical In tegrabilit y of the Zigzag Mo del John C. Donah ue and Sergei Dub o vsky Center for Cosmolo gy and Particle Physics, Dep artment of Physics, New Y ork University, New Y ork, NY, 10003, USA The zigzag model is a relativistic N -b ody system arising in the high energy limit of the w orldsheet scattering in adjoin t tw o-dimensional QCD. W e pro ve classical Liouville in tegrabilit y of this mo del b y pro viding an explicit construction of N charges in in volution. F urthermore, w e also prov e that the system is maximally superintegrable b y constructing N − 1 additional independent c harges. All of these c harges are piecewise linear functions of co ordinates and momenta. The classical time dela ys are determined algebraically from this integrable structure. The resulting S -matrix is the same as in the N -particle subsector of a massless T ¯ T deformed fermion. I. INTR ODUCTION Understanding the mechanism of quark confinemen t con tinues to stand out as an in teresting challenge and as a source of new developmen ts in theoretical and mathe- matical physics. Starting with [1], m uch effort has recen tly b een put in to a study of scattering on the w orldsheet of a single confining string in the t’ Hooft planar limit [2]. An es- p ecially interesting problem is to understand the high energy dynamics on the worldsheet. In this regime one exp ects the confining string to exhibit characteristically gra vitational behavior similarly to that of critical strings [3]. An interesting p ossibilit y is that hard high energy scat- tering on the worldsheet approaches integrable asymp- totics [4]. A concrete version of this prop osal, the Axionic String Ansatz (ASA)[4, 5], iden tifies the corresp onding in tegrable theories as T ¯ T -deformations [6–9] of certain free massless mo dels. This prop osal is motiv ated and supp orted by the recent analysis [4, 10, 11] of lattice mea- suremen ts of flux tubes excitations [12 – 15]. F or D = 3 gluo dynamics it is also supported b y lattice determina- tions of glueball masses and spins [5, 16, 17]. The physical reason for the emergence of integrable dynamics in the high energy worldsheet scattering is the asymptotic free- dom of the underlying gauge theory [18]. Finally , axionic strings also came out recently as a result of the flux tub e S -matrix b ootstrap [19]. A natural pla yground for testing this idea is provided b y adjoin t QCD in D = 2 dimensions ( aQC D 2 ). The sp ectrum of this theory has b een extensively studied in early 90’s [20 – 23] (see, e.g., [24, 25] for more recen t in- teresting works). A study of the worldsheet scattering in the mo del has b een initiated in [26], building up on the tec hniques dev elop ed in prehistoric times [27, 28]. Recen tly , a candidate relativistic N -bo dy system de- scribing the integrable high-energy asymptotics of the w orldsheet theory in aQC D 2 has b een identified [29]. F or reasons which will b ecome clear we call this system the “zigzag mo del”. In [29] we provided partial numerical and analytical evidence for classical integrabilit y of the zigzag mo del. The goal of the presen t paper is to provide a complete proof that the zigzag mo del is in tegrable at the classical level. The rest of the paper is organized as follows. In sec- tion I I we describ e the mo del. In section I I I we describ e a discrete top ological in v ariant presen t in the mo del. This top ological inv arian t ensures that the total n umber of left- and right-mo v ers is conserved in an y scattering ev ent. In section IV we construct N conserv ed c harges in inv olution and 2 N − 1 algebraically independent con- serv ed charges. All of these charges are piecewise linear functions of co ordinates and momen ta. This construction establishes that the zigzag mo del is Liouville in tegrable and, moreov er, maximally sup erin tegrable. W e conclude in section V. In the app endix A we pro vide explicit ex- pressions for integrals of motion for N = 2 and N = 3. I I. DESCRIPTION OF THE MODEL The zigzag mo del describ es a chain of N ordered par- ticles on a line with nearest neighbor in teractions. The structure of its Hamiltonian is very similar to the cele- brated T oda c hain [30]. The difference is that particles in the zigzag mo del are massless, i.e. they alwa ys mov e with unit v elocity . The exp onen tial nearest neighbor T oda p o- ten tial is replaced with a piecewise linear one of the form V ( q ) = q + | q | . (1) The full Hamiltonian then takes the following form H = N X i =1 | p i | + N − 1 X i =1 V ( q i,i +1 ) , (2) where q i,i +1 = q i − q i +1 . Pictorially , this system ma y b e represen ted as a sequence of b eads on a rubber band, see Fig. 1. All particles mov e with v elo cities ± 1 dep ending on the sign of the corre- sp onding momentum. A generic configuration of par- ticles exhibits a n umber of zigzags, which explains the name of the mo del. The only particles exp eriencing a force are those at the zigzag turning p oin ts. Momenta of all other particles stay constant. 2 A) B) C) FIG. 1. Snapshots of time ev olution in the zigzag model. At early times A) all left-mov ers are located on the right and righ t-mov ers are on the left. The interaction p erio d B) pro ceeds through a series of zigzag formation resulting in the momenta exc hanges. A t late times C) all left-mov ers are on the left and right-mo vers are on the right. The top ological inv ariant (11) ensures that the difference b etw een the num ber of left- and right mov ers stays constant at all times, where left(right) is defined w.r.t. to the string worldsheet. This definition is illustrated b y the color co ding, where right-mo v ers are colored blue and left-mo vers red. In the gauge theory language, b eads corresp ond to quarks in the adjoint representation and the rubb er band to the confining string. As a consequence of asymptotic freedom, pro cesses which c hange the n um b er of quarks (partons) are suppressed at high energies. Hence the w orldsheet theory splits into separate sectors lab eled by the num b er of partons N , eac h describ ed b y (2) at the leading order in the high-energy expansion. Similarly to the T oda chain, in addition to the op en zigzag mo del describ ed b y (2), one may also consider its compact version. The latter describ es a closed confining string wound around a compact spatial circle. As men- tioned in [29], there is o verwhelming numerical evidence that the compact zigzag model is also in tegrable. In the presen t paper we restrict to the op en case. An imp ortan t prop erty of the zigzag mo del is that it inherits Poincar ´ e symmetry from the underlying gauge theory . Namely , it is straightforw ard to chec k that the P oisson brack ets b et ween the Hamiltonian H , total mo- men tum P = N X i =1 p i , (3) and the b oost generator J = N X i =1 q i | p i | + 1 2 N − 1 X i =1 ( q i + q i +1 ) V ( q i,i +1 ) (4) giv e rise to the I S O (1 , 1) Poincar ´ e algebra { H , P } = 0 , { J, P } = H , { J, H } = P . (5) As we will see, it is often conv enient to treat momenta p i and co ordinate differences q i,i +1 on equal fo oting. This is achiev ed by introducing a string of v ariables Q a = ( p 1 , q 1 , 2 , p 2 , . . . , q N − 1 ,N , p N ) (6) with a = 1 , . . . , 2 N − 1. Asso ciated with this string there is also a sequence of the corresp onding sign v ariables (“classical bits”) S a = ( s 1 , s 1 , 2 , s 2 , . . . , s N − 1 ,N , s N ) , (7) where s i = sign( p i ) , s i,i +1 = sign( q i,i +1 ) . (8) In what follows w e also often use the notation S 0 = S 2 N = − 1 . (9) With these notations the equations of motion take the follo wing simple form ˙ Q a = S a − 1 − S a +1 . (10) F or an y configuration of Q a ’s at early times one finds a bunc h of right-mo ving particles on the left and a bunc h of left-moving particles on the right, freely approac hing eac h other ( i.e. , no zigzags are present). As left- and righ t-mov ers reach each other and start to collide, the string goes through a sequence of zigzag configurations (see Fig. 1). At late times all zigzags are gone and one finds a bunch of left-mo v ers on the left and a bunch of righ t-mov ers on the right. Classical integrabilit y of the mo del manifests itself in the absence of momen tum exc hange as one compares early and late configurations. Namely , the v alues (and orderings) of all early and late left- and righ t-moving 3 momen ta are the same. Of course, particles momenta do change their v alues at intermediate times. The main goal of this pap er is to prov e in tegrability by constructing a sufficiently large set of conserv ed c harges. Note that any solution to (10) is a piecewise linear function of time. Hence, the zigzag mo del is defnitely in- tegrable (or, b etter to say , solv able) in the broad sense— starting with an y initial v alue it is straightforw ard to find an explicit solution for a later time ev olution. All that one needs to do is to linearly evolv e the system forward in time untill one of the sign differences S a − 1 − S a +1 in the r.h.s. of (10) changes its v alue. This corre- sp ond either to a zigzag formation/annihilation ( i.e. to a collision of tw o particles), or to a sign flip of one of the momenta. Then one contin ues linear evolution with different v alues of the velocities. A Mathematic a solv er implementing this pro cedure can b e downloaded at https://jcdonahue.net/research . Note that this solv er provides an exact rather than a n umerical solution to the equations of motion starting with arbitrary ratio- nal initial conditions. W e will pro ve no w that in addition to b eing in tegrable in this broad sense the zigzag mo del is actually Liouville in tegrable and, moreov er, maximally sup erin tegrable. I II. TOPOLOGICAL CHAR GE Let us start a construction of conserv ed charges in the zigzag mo del b y describing the top ological charge in tro- duced in [29]. Namely , it is straightforw ard to chec k that T 2 N = 1 2 2 N − 1 X a =0 S a S a +1 (11) sta ys constant under the time evolution describ ed by the equations of motion (10). W e refer to T 2 N as topological c harge b ecause it defines a piecewise constant function on the phase space, separating it into dynamically dis- connected top ological sectors. In the asymptotic regions t → ±∞ no zigzags are present, hence all s i,i +1 = − 1 so that one finds T 2 N | t →±∞ = − N X i =1 s i = N L − N R , (12) where N L and N R coun t the num b ers of left- and right- mo vers in the initial and final states. This prov es that scattering do es not c hange N L and N R . T o see the ge- ometrical meaning of T 2 N at in termediate times let us rewrite it in the following form T 2 N = 1 2 N X i =1 s i ( s i − 1 ,i + s i,i +1 ) . (13) W e see that also at in termediate times T 2 N can be inter- preted as a difference in the num b er of left- and righ t- mo vers, provided left- and righ t- is determined w.r.t. to the string worldsheet rather than w.r.t. to the physical space. Particles at the zigzag turning p oints should not b e coun ted at all, see Fig. 1. Interestingly , if one thinks ab out S a ’s as a classical spin sequence, T 2 N is equal to the Ising mo del Hamiltonian. In the construction of the dynamical conserved charges presen ted in section IV we will encounter the following piecewise constan t functions on the phase space, whic h generalize (11), T a = 1 2 a − 1 X b =0 S b S b +1 . (14) Unlik e T 2 N , a general T a ma y change its v alue in the course of ev olution when zigzags form/annihilate or par- ticle momenta flip sign. Indeed using the equations of motion (10) we find that for a < 2 N ˙ T a = 1 2 S a − 1 ˙ S a = (1 − S a − 1 S a +1 ) δ ( Q a ) . (15) In what follo ws we need to know what are the p ossible v alues of T a ≡ T (0 , a ) at fixed N L , N R (or, equiv alen tly , at fixed N , T 2 N ). In general, one can write that T a = a 2 − n f , (16) where n f is the num b er of sign flips in the ( S 0 , . . . , S a ) sequence of bits. In the absence of an y additional restric- tions, the p ossible range of v alues for n f is 0 ≤ n f ≤ a . (17) Restricting to the top ological sector with a fixed N L , N R imp oses an additional constraint n f + ¯ n f = 2 N R , (18) where 0 ≤ ¯ n f ≤ 2 N − a (19) is a n um b er of sign flips in the complementary sequence of bits ( S a , . . . , S 2 N ). Com bining (18) and (19) we obtain that in addition to (17) the range of n f is also constrained to satisfy a − 2 N L ≤ n f ≤ 2 N R . (20) Recalling the relation (16) b et ween n f and T a the in- equalities (17) and (20) imply that T a ma y tak e v alues in the shaded region P shown in Fig. 2. Fig. 2 also illustrates another p oint, whic h will b e im- p ortan t in Section IV. Namely , as follows from (16), the v alue of T a unam biguously determines the v alue of the corresp onding spin S a . This relation is sho wn in Fig. 2, where solid dots corresp ond to S a = 1 and empty ones to S a = − 1. 4 FIG. 2. The shaded P region shows possible v alues of T a at fixed N L , N R . F or solid dots S a = 1, and for empty dots S a = − 1. IV. LINEAR CHARGES Let us turn now to the construction of the dynamical conserv ed c harges in the zigzag model. Giv en that the general solution of (10) is a piecewise linear function of time, it is natural to lo ok for charges whic h are piece- wise linear functions in the phase space. Restricting to translationally in v arian t charges we arrive then at the follo wing ansatz, I = 2 N − 1 X a =1 F a ( S ) Q a . (21) A time deriv ativ e of I con tains a smo oth contribution related to time evolution of Q a ’s and δ -functional con- tributions caused by sign flips in the set of S a ’s. All δ -functional contributions hav e to v anish separately , im- plying that ˙ F a ( S ) ∝ δ ( Q a ) or equiv alen tly that the co ef- ficien t functions F a ( S ) satisfy ∂ b F a ( S )( S b − 1 − S b +1 ) = 0 for a 6 = b . (22) It’s clear b y (15) that T a and S a satisfy this requiremen t. Using the equations of motion (10) it is straightforw ard to c heck that these are the only non trivial solutions to (22) and that the coefficient functions take the follo wing functional form F a ( S ) = F a ( S a , T a , N L , N R ) . (23) In what follows we suppress the N L and N R dep endence of F a , assuming that these are kept fixed and non-zero. In addition, as discussed in Section I I I, the v alue of S a is determined by T a , so in what follows we write simply F a ( T a ). Then the single remaining equation comes from requir- ing that I sta ys constant under a smo oth time evolution of Q a ’s and takes the following form C ≡ 2 N − 1 X a =1 F a ( T a )( S a − 1 − S a +1 ) = 0 . (24) In particular, (24) implies that the v alue of C is inv arian t under the c hange S a → − S a , pro vided the flip of S a is dynamically allow ed. By (10) a flip is allo w ed when S a − 1 + S a +1 = 0 , (25) whic h giv es the constraints C ( S a ) | S a − 1 = − S a +1 = C ( − S a ) | S a − 1 = − S a +1 . (26) These reduce to the following set of equations F a +1 ( T a − 1 ) = S a − 1  F a  T a − 1 − S a − 1 2  − F a  T a − 1 + S a − 1 2  + F a − 1 ( T a − 1 ) . (27) These equations hold for an y a = 1 , . . . 2 N − 1, if one sets F 0 = F 2 N = 0. Roughly sp eaking, (27) provide a set of linear recursion relations whic h determine F a +1 in terms of F a and F a − 1 . This is not exactly the case though, b ecause T a − 1 (whic h appears as an argumen t of F a − 1 in (27)) do es not tak e all v alues whic h T a +1 ma y take, see Fig. 2. As a result (27) leav es F a +1 ( T a +1 ) undetermined 5 at T a +1 = ± a + 1 2 . The structure of these recursion relations is illustrated in Fig. 3, where w e use the grid of p ossible v alues of T a from Fig. 2 with the understanding that there is a num- b er, which is the corresponding v alue of F a ( T a ), assigned to eac h point of the grid. Arrows illustrate the relations b et w een n umbers at different p oints on the grid, as de- termined by (27). Fig. 3 makes it clear that a general solution to (27) is determined by 2 N b oundary v alues F a  ± a +1 2  , sub ject to one linear constraint at the right corner of the shaded rectangular P region, F 2 N = 0 . This leav es us with 2 N − 1 linearly indep endent solu- tions to the recursion relations (27). Not all of these solutions corresp ond to conserved charges, b ecause (27) is the condition for C in (24) to b e constan t, rather than zero. Enforcing C = 0 pro vides an additional linear con- strain t lea ving us with 2 N − 2 translationally inv ariant indep enden t integrals of motion. It is straightforw ard to construct these integrals ex- plicitly . Indeed, let us consider F a ’s which are non-zero only on one of the internal diagonals of the P region as sho wn in Fig. 3 and is equal to the corresp onding S a at eac h of the points on that diagonal. It is easy to see that these provide 2 N − 2 linearly indep enden t solutions to (27). An explicit form ula for the corresponding F a ’s is F L,n L a = S a δ T a ,n L − a 2 (28) for diagonals going from the upp er left side of the P region to the low er righ t (like the violet one in Fig. 3) and F R,n R a = S a δ T a , − n R + a 2 (29) for diagonals going from the lo wer left side to the upp er righ t (like the blue one in Fig. 3). Here the range of v alues for n L , n R is n L ( R ) = 1 , . . . , 2 N L ( R ) − 1 , (30) and δ T a ,n is the Kroneck er sym b ol. As a function of S a ’s the latter can b e written as δ T a ,n = k 6 = n + a 2 ,k = a Y k =0 T a + a 2 − k n + a 2 − k , where n can take any of the v alues − a 2 , − a 2 + 1 , . . . , a 2 . T o see the ph ysical meaning of these solutions let us insp ect the corresp onding functions I L,n L , I R,n R in the infinite past and future, t → ±∞ . This is con v eniently done by using the follo wing in teresting space-time inter- pretation of Fig. 3. Note, that an y particle configuration is naturally represented b y a slice of P . Indeed, an y con- figuration Q a ( t ) leads to a “bit” sequence S a ( t ), which can b e equiv alently represented as a sequence of T a v al- ues, such that T a +1 ( t ) = T a ( t ) ± 1 2 , see Fig. 4. A t early times no zigzags are presen t (i.e., all q i,i +1 = − 1) and all left-mov ers are on the righ t and righ t-mov ers are on the left. Hence, this configuration corresp onds to the v alues of T a ’s at the low er b oundary of the P region. As time ev olv es the slice mov es upw ards monotonically . This motion corresponds to the dynamics of a melting 2D crystal—the ev olution pro ceeds through a series of upw ard jumps of the p oints at the corners of the “melting surface”. At late times t → + ∞ the slice reac hes the upp er b oundary of P . Using this picture we see that at t → ±∞ I L,n L = − Q L n L (31) I R,n R = ( − 1) n R +1 Q R n R , (32) where Q L n L and Q R n R are subsets of Q a corresp onding to left- and right-mo vers at t → ±∞ , i.e. at t → −∞ Q R n R = Q n R (33) Q L n L = Q 2 N R + n L (34) and at t → + ∞ Q R n R = Q 2 N L + n R (35) Q L n L = Q n L . (36) Giv en that solutions of (27) are either in tegrals of mo- tion or linear functions of time, we see that I L,n L , I R,n R are actual integrals of motion (b ecause they stay con- stan t in the asymptotic regions). In particular, these in tegrals contain individual momenta of particles in the asymptotic regions t ± ∞ so this construction prov es that the set of initial and final momen ta are conserved in the course of the collision (as w ell as the ordering of the mo- men ta among left- and right-mo vers). It also prov es the Liouville integrabilit y of the system, because integrals corresp onding to the asymptotic momenta provide us a set of N comm uting conserv ed charges. The remaining ( N − 2) constructed integrals in the asymptotic regions reduce to the co ordinate differences among left-mov ers or right-mo vers. Their existence im- plies that the time delays exp erienced by all left-mov ers are equal to each other and the same is true for the time dela ys experienced by all right-mo v ers. T o find these time delays let us insp ect the last remain- ing independent solution of (27). F or reasons which will b ecome clear soon, w e refer to the corresponding piece- wise linear function on the phase space (21) as ˜ H 1 . The 1 This quantit y is different from the one which was called ˜ H in [29], but has similar prop erties. 6 FIG. 3. The structure of the recursion relation (27) as illustrated by arrows in the ( a, T a ) plane. Blue and violet diagonals corresp ond to righ t and left charges respectively . Boundary conditions for the recursion relations (27) are imp osed at the points along the dashed red lines. FIG. 4. A physical configuration of particles can b e represented as a (red) slice in the ( a, T a ) plane. This snapshot corresp onds to the one in Fig. 1B). Physical time evolution corresp onds to the melting dynamics of this slice. Red dashed arrows show next p ossible changes for the shap e of the slice in the course of the time evolution. corresp onding F a ’s are non-zero at the righ t b oundary of the P region. Unlike for in ternal diagonals, we need to use now b oth upp er and low er parts of the b oundary to satisfy the F 2 N = 0 condition. This results in the follo wing non-v anishing F a ’s for this solution F ˜ H a = S a  δ T a , 2 N L − a 2 − δ T a , − 2 N R + a 2  . (37) Then in the asymptotic regions one finds ˜ H =  Q R − Q L − P L , at t → −∞ Q R − Q L + P R , at t → + ∞ , (38) where P L ( R ) is the total asymptotic left(righ t)-moving momen tum 2 and Q L , Q R are the p ositions of the right- 2 It is defined in such a wa y that P L ( R ) ≥ 0. most left- and right-mo vers in the asymptotic regions. This implies that ˜ H is a linear function of time, rather than a conserved c harge, i.e. ˜ H = 2( t − t 0 ) , (39) where t 0 is a constan t. Eqiv alently , the Poisson brack ets of ˜ H with the Hamiltonian H and momen tum P are { ˜ H , H } = 2 , { P , ˜ H } = 0 , (40) where the latter follows from the translational inv ariance of ˜ H . This allows one to construct a new conserved c harge ˜ P = { J, ˜ H } , (41) whic h is not translationally inv ariant, { H , ˜ P } = 0 , { P, ˜ P } = 2 . (42) 7 Altogether, I L,n L , I R,n R and ˜ P provide a set of (2 N − 1) independent conserved c harges, whic h prov es that the zigzag mo del is maximally sup erin tegrable. T o calculate the time delays, let us ev aluate ˜ P in the asymptotic regions. Using the asymptotic expression (38) we obtain ˜ P =  − Q L − Q R + P L , at t → −∞ − Q L − Q R + P R , at t → + ∞ , (43) Com bining (38), (39), (43) and the conserv ation of ˜ P w e find that ∆ t L ( R ) = P R ( L ) (44) for the time delays ∆ t L ( R ) exp erienced b y left(righ t)- mo ving particles. These time delays correspond to the celebrated sho ck wa v e phase shift [31, 32] confirming that the zigzag mo del describ es the N -particle subsector of a massless T ¯ T -deformed fermion. V. DISCUSSION T o summarize, in this pap er w e presen ted an exhaus- tiv e analysis of the in tegrable structure of the classical zigzag mo del (2). The natural next step is to quan- tize the mo del. Given that the classical time delay (44) repro duces the exact phase shift of a kno wn quantum mo del—a massless T ¯ T deformed fermion—one exp ects the quantization preserving the P oincar´ e symmetry and in tegrability to exist. Note that a close relative of the zigzag mo del appeared in mid 70’s under the name of folded strings [33, 34] 3 . There, exact solv ability of a very similar Hamiltonian w as understo od as a consequence of the map b et ween the corresp onding mechanical solutions and folded string solutions of the tw o-dimensional Nambu–Goto theory . This corresp ondence reinforces the relation of the zigzag mo del with the T ¯ T -deformation, giv en that the latter can b e understo od as arising from the coupling of an unde- formed quantum field theory to t wo-dimensional strings [35, 36]. In this language the zigzag mo del describes dy- namics of a long string, while the early papers [33, 34] studied the short string sector. The relation of folded strings to aQC D 2 w as conjectured in [37]. The analysis of [26, 29] makes this relation precise, b y demonstrat- ing ho w the zigzag mo del arises as a leading high energy appro ximation to the w orldsheet dynamics. The p ossibilit y of a consisten t cov arian t quantization of folded strings remains somewhat con tro versial (see, e.g., [38, 39]). W e think that the connections to the T ¯ T defor- mation and aQC D 2 strongly suggest that such a quan- tization is possible, and should in fact b e one-lo op exact (at least in the long string sector, corresp onding to the 3 W e thank Antal Jevic ki for directing us to these early pap ers. zigzag mo del). Hop efully , a detailed understanding of the classical integrable structure achiev ed in the presen t pap er will help to resolv e this. A cknow le dgements. W e thank Ofer Aharony , Misha F eigin, An tal Jevicki, David Kutasov, Grisha Korchem- sky , Conghuan Luo, Nikita Nekraso v, Sasha Penin and Sla v a Rychk ov for useful discussions. This work is sup- p orted in part by the NSF aw ard PHY-1915219 and by the BSF grant 2018068. App endix: Explicit Integrals for N=2,3 F or illustrativ e purp oses we rep ort here the explicit expressions for the integrals of motion for N = 2 and N = 3 with N L = 1. F or N = 2, asymptotics are defined b y s 1 = 1, s 2 = -1 for t = −∞ and s 1 = -1, s 2 = 1 for t = + ∞ . Our translationally in v ariant integrals are I R, 1 = p 1 2 (1 + s 1 ) + q 1 , 2 2 (1 + s 1 , 2 ) + p 2 8 (3 − s 1 + 3 s 2 − s 1 s 2 + s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 + s 1 s 2 s 1 , 2 ) (A.1) and I L, 1 = p 1 2 ( − 1 + s 1 ) + q 1 , 2 2 (1 + s 1 , 2 ) + p 2 8 ( − 3 − s 1 + 3 s 2 + s 1 s 2 − s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 − s 1 s 2 s 1 , 2 ) . (A.2) As outlined in (31-36), asymptotically I R, 1 =  p 1 , at t → −∞ p 2 , at t → + ∞ , (A.3) and I L, 1 =  − p 2 , at t → −∞ − p 1 , at t → + ∞ . (A.4) F urther, we ha ve ˜ H = q 1 , 2 2 ( s 1 − s 1 s 1 , 2 ) + p 2 4 (3 − s 1 s 2 + s 1 , 2 + s 1 s 2 s 1 , 2 ) (A.5) ˜ P = q 1 2 ( − 1 + s 1 , 2 ) + p 2 4 ( − s 1 + 3 s 2 + s 1 s 1 , 2 + s 2 s 1 , 2 ) − q 2 2 (2 − s 1 s 2 + 2 s 1 , 2 + s 1 s 2 s 1 , 2 ) (A.6) In accordance with (38) and (43), asymptotically we find ˜ H =  q 1 − q 2 − | p 2 | , at t → −∞ q 2 − q 1 + | p 2 | , at t → + ∞ , (A.7) and ˜ P =  − q 1 − q 2 + | p 2 | , at t → −∞ − q 2 − q 1 + | p 2 | , at t → + ∞ . (A.8) 8 F or N = 3, N L = 1 asymptotics are defined b y s 1 = s 2 = 1, s 3 = -1 for t = −∞ and s 1 = -1, s 2 = s 3 = 1 for t = + ∞ . Our translationally in v ariant integrals are I R, 1 = p 1 2 ( s 1 + 1) + q 1 , 2 2 ( s 1 , 2 + 1) + p 2 8 ( s 2 s 1 s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 + s 1 , 2 − s 2 s 1 − s 1 + 3 s 2 + 3) + q 2 , 3 8 ( s 1 s 1 , 2 + s 1 s 1 , 2 s 2 , 3 − s 1 s 2 , 3 + s 2 s 1 , 2 + s 2 s 2 , 3 + s 2 s 1 , 2 s 2 , 3 + 2 s 2 , 3 − s 1 + s 2 + 2) + p 3 32 ( − s 2 s 1 s 1 , 2 − s 2 s 3 s 1 s 1 , 2 + 3 s 3 s 1 s 1 , 2 + 3 s 1 s 1 , 2 − s 2 s 1 s 2 , 3 − s 2 s 3 s 1 s 2 , 3 − s 3 s 1 s 2 , 3 + s 2 s 1 s 1 , 2 s 2 , 3 + s 2 s 3 s 1 s 1 , 2 s 2 , 3 + s 3 s 1 s 1 , 2 s 2 , 3 + s 1 s 1 , 2 s 2 , 3 − s 1 s 2 , 3 + 3 s 2 s 1 , 2 − s 3 s 1 , 2 − s 1 , 2 + 3 s 2 s 2 , 3 + 3 s 2 s 3 s 2 , 3 + 3 s 3 s 2 , 3 + s 2 s 1 , 2 s 2 , 3 + s 2 s 3 s 1 , 2 s 2 , 3 + s 3 s 1 , 2 s 2 , 3 + s 1 , 2 s 2 , 3 + 3 s 2 , 3 + s 2 s 1 + s 2 s 3 s 1 − 3 s 3 s 1 − 3 s 1 + 3 s 2 s 3 s 1 , 2 + s 2 + s 2 s 3 + 5 s 3 + 5) , (A.9) and I R, 2 = q 1 , 2 4 ( s 1 s 1 , 2 + s 1 , 2 − s 1 − 1) + p 2 8 ( − s 2 s 1 s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 − s 1 , 2 + s 2 s 1 − s 1 + 3 s 2 − 3) + q 2 , 3 8 ( − s 1 s 2 s 1 , 2 − s 1 s 2 s 2 , 3 + s 1 s 2 s 1 , 2 s 2 , 3 − s 1 , 2 + s 1 , 2 s 2 , 3 + 3 s 2 , 3 + s 1 s 2 − 3) + p 3 16 ( − s 2 s 1 s 1 , 2 + s 2 s 3 s 1 s 1 , 2 + s 3 s 1 s 1 , 2 − s 1 s 1 , 2 − s 2 s 1 s 2 , 3 + s 2 s 3 s 1 s 2 , 3 − s 3 s 1 s 2 , 3 + s 2 s 1 s 1 , 2 s 2 , 3 − s 2 s 3 s 1 s 1 , 2 s 2 , 3 + s 3 s 1 s 1 , 2 s 2 , 3 − s 1 s 1 , 2 s 2 , 3 + s 1 s 2 , 3 − s 2 s 1 , 2 + s 2 s 3 s 1 , 2 + s 3 s 1 , 2 − s 1 , 2 − s 2 s 2 , 3 + s 2 s 3 s 2 , 3 − s 3 s 2 , 3 − s 2 s 1 , 2 s 2 , 3 + s 2 s 3 s 1 , 2 s 2 , 3 − s 3 s 1 , 2 s 2 , 3 + s 1 , 2 s 2 , 3 + s 2 , 3 + s 2 s 1 − s 2 s 3 s 1 − s 3 s 1 + s 1 − s 2 + s 2 s 3 + 5 s 3 − 5) . (A.10) Equations for I R, 3 and I L, 1 are similar. Corresp ond- ingly we find I R, 1 =  p 1 , at t → −∞ p 2 , at t → + ∞ , (A.11) and I R, 2 =  − q 1 , 2 , at t → −∞ − q 2 , 3 , at t → + ∞ . (A.12) F urther, we ha ve ˜ H = q 1 , 2 4 ( − s 1 s 1 , 2 + s 1 , 2 + s 1 − 1) + p 2 8 ( s 2 s 1 s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 + s 1 , 2 − s 2 s 1 − s 1 + 3 s 2 + 3) + q 2 , 3 16 ( − 3 s 2 s 1 s 1 , 2 − s 1 s 1 , 2 − 3 s 2 s 1 s 2 , 3 + 3 s 2 s 1 s 1 , 2 s 2 , 3 + s 1 s 1 , 2 s 2 , 3 − s 1 s 2 , 3 − s 2 s 1 , 2 − 3 s 1 , 2 − s 2 s 2 , 3 + s 2 s 1 , 2 s 2 , 3 + 3 s 1 , 2 s 2 , 3 + 5 s 2 , 3 + 3 s 2 s 1 + s 1 + s 2 − 5) + p 3 32 ( s 2 s 1 s 1 , 2 + 3 s 2 s 3 s 1 s 1 , 2 + s 3 s 1 s 1 , 2 − 5 s 1 s 1 , 2 + s 2 s 1 s 2 , 3 + 3 s 2 s 3 s 1 s 2 , 3 + s 3 s 1 s 2 , 3 − s 2 s 1 s 1 , 2 s 2 , 3 − 3 s 2 s 3 s 1 s 1 , 2 s 2 , 3 − s 3 s 1 s 1 , 2 s 2 , 3 − 3 s 1 s 1 , 2 s 2 , 3 + 3 s 1 s 2 , 3 − 5 s 2 s 1 , 2 + s 2 s 3 s 1 , 2 + 3 s 3 s 1 , 2 + s 1 , 2 − 5 s 2 s 2 , 3 + s 2 s 3 s 2 , 3 − 5 s 3 s 2 , 3 − 3 s 2 s 1 , 2 s 2 , 3 − s 2 s 3 s 1 , 2 s 2 , 3 − 3 s 3 s 1 , 2 s 2 , 3 − s 1 , 2 s 2 , 3 + s 2 , 3 − s 2 s 1 − 3 s 2 s 3 s 1 − s 3 s 1 + 5 s 1 − 3 s 2 − s 2 s 3 + 5 s 3 + 15) (A.13) ˜ P = q 1 4 ( − s 1 s 1 , 2 + s 1 , 2 + s 1 − 1) + p 2 8 ( s 2 s 1 s 1 , 2 + s 1 s 1 , 2 + s 2 s 1 , 2 + s 1 , 2 − s 2 s 1 − s 1 + 3 s 2 + 3) + q 2 16 ( − s 2 s 1 s 1 , 2 + 5 s 1 s 1 , 2 − s 2 s 1 s 2 , 3 + s 2 s 1 s 1 , 2 s 2 , 3 − s 1 s 1 , 2 s 2 , 3 + s 1 s 2 , 3 + s 2 s 1 , 2 − 5 s 1 , 2 + s 2 s 2 , 3 − s 2 s 1 , 2 s 2 , 3 + s 1 , 2 s 2 , 3 + 7 s 2 , 3 + s 2 s 1 − 5 s 1 − s 2 − 3) + q 3 16 ( − 3 s 2 s 3 s 1 s 1 , 2 − s 3 s 1 s 1 , 2 + 4 s 1 s 1 , 2 − 3 s 2 s 3 s 1 s 2 , 3 − s 3 s 1 s 2 , 3 + 3 s 2 s 3 s 1 s 1 , 2 s 2 , 3 + s 3 s 1 s 1 , 2 s 2 , 3 + 4 s 1 s 1 , 2 s 2 , 3 − 4 s 1 s 2 , 3 + 4 s 2 s 1 , 2 − s 2 s 3 s 1 , 2 − 3 s 3 s 1 , 2 + 4 s 2 s 2 , 3 − s 2 s 3 s 2 , 3 + 5 s 3 s 2 , 3 + 4 s 2 s 1 , 2 s 2 , 3 + s 2 s 3 s 1 , 2 s 2 , 3 + 3 s 3 s 1 , 2 s 2 , 3 − 8 s 2 , 3 + 3 s 2 s 3 s 1 + s 3 s 1 − 4 s 1 + 4 s 2 + s 2 s 3 − 5 s 3 − 8) + p 3 32 (3 s 2 s 1 s 1 , 2 + s 2 s 3 s 1 s 1 , 2 − 5 s 3 s 1 s 1 , 2 + s 1 s 1 , 2 + 3 s 2 s 1 s 2 , 3 + s 2 s 3 s 1 s 2 , 3 + 3 s 3 s 1 s 2 , 3 − 3 s 2 s 1 s 1 , 2 s 2 , 3 − s 2 s 3 s 1 s 1 , 2 s 2 , 3 − 3 s 3 s 1 s 1 , 2 s 2 , 3 − s 1 s 1 , 2 s 2 , 3 + s 1 s 2 , 3 + s 2 s 1 , 2 − 5 s 2 s 3 s 1 , 2 + s 3 s 1 , 2 + 3 s 1 , 2 + s 2 s 2 , 3 − 5 s 2 s 3 s 2 , 3 + s 3 s 2 , 3 − s 2 s 1 , 2 s 2 , 3 − 3 s 2 s 3 s 1 , 2 s 2 , 3 − s 3 s 1 , 2 s 2 , 3 − 3 s 1 , 2 s 2 , 3 − 5 s 2 , 3 − 3 s 2 s 1 − s 2 s 3 s 1 + 5 s 3 s 1 − s 1 − s 2 − 3 s 2 s 3 + 15 s 3 + 5) (A.14) In accordance with (38) and (43), asymptotically we find ˜ H =  q 2 − q 3 − | p 3 | , at t → −∞ q 3 − q 1 + | p 2 | + | p 3 | , at t → + ∞ , (A.15) and ˜ P =  − q 2 − q 3 + | p 3 | , at t → −∞ − q 3 − q 1 + | p 2 | + | p 3 | , at t → + ∞ . 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