Confinement, solitons and the equivalence between the sine-Gordon and massive Thirring models

.

Ž 3.18 .

Notice that for the states of Hc , where Ž3.14. holds true, one has from Ž3.6. that ² E " f : ; ² E " w :. For the choice ec - 0, the kinetic terms for r and f have the same sign.

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An important property of the model under consideration is that it presents another spinor field which is not confined. Indeed, using bosonization rules again, we can introduce a spinor x as

b2

i: xg mEm x :s

2

: Ž Ef . : ,

Ž 3.19 .

:cos Ž bf . : ,

Ž 3.20 .

8p

m : x x :s y

2p

: xg mx :s y

b 2p

e mnEn f .

Ž 3.21 .

The Lagrangian Ž3.8. becomes Žassuming Ž3.10.. 2

L s y 12 Ž Em r . q 12 Em nE mh y

k 8

mc2 e 2 h

q ec i xg mEm x y mx eh x x y

ž

g 2

2 Ž xg mx . ,

/

Ž 3.22 .

with 4p

b

2

s1q

g

p

m ,

mx s

m

kmc .

Ž 3.23 .

Therefore, we get a Thirring model coupled to h plus the free scalar field r . After the reductions Ž3.12. and Ž3.15. the theory becomes the pure massive Thirring model. Notice that x becomes a free massive spinor w7x for b 2 s 4p or k s

(

E mr .

Ž 3.24 .

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Therefore, the spinor x does not contribute to such current and so has zero chirality. Therefore, the fact that it does acquire a mass is not incompatible with the chiral symmetry. The spinor with nonzero chirality is c and it disappears from the spectrum in the confining sector of the theory. The possibility of having chiral symmetry and massive fermions is another remarkable property of theory Ž2.1.. For these reasons the model Ž2.1. constitute an excellent laboratory to test ideas about confinement, the role of solitons in quantum field theories, and duality transformations interchanging the role of solitons and fundamental particles. With that motivation we shall now study the two-soliton solutions and their interactions.

4. Soliton solutions It is well known the relevance of localized classical solutions of nonlinear relativistic field equations to the corresponding quantum theories. In particular, solitons can be associated with quantum extended-particle states. Therefore, we will examine the classical soliton type solutions to get insight into the quantum spectrum of the model, in much the same way of what is already known in the remarkable sine-Gordon model. We argue that, at the classical level, the solutions for the w and c fields share some features of the sine-Gordon and the massive Thirring theories, respectively. We have shown in the previous section, using bosonization methods, that at the quantum level the theory Ž2.1. is equivalent to Ž3.8.. However, the space of classical solutions of the two theories are not the same. One cannot use Ž3.1. – Ž3.3., at the classical level, as change of variables to relate classical solutions of one theory to those of the other. The bosonization rules Ž3.1. – Ž3.3. are purely quantum relations. The one-soliton solutions of Ž2.1. have been calculated in Ref. w9x using the dressing transformation method. Here, we construct the two-soliton solutions using the same methods. We start with the zero curvature representation of the equations of motion of the theory

w Eqq Aq , Eyq Ay x s 0 ,

Ž 4.1 .

where the indices " stand for the light cone variables x "s t " x. For the theory Ž2.1. $ the connections A " live on the slŽ2. affine Kac–Moody algebra Gˆ , and are given in the Appendix A Žsee ŽA.1... Since Ž4.1. imply the connections must be flat, one can write them as Am s yEm TTy1 , with T being a group element obtained by exponentiating Gˆ . In addition, using an integral gradation of Gˆ , i.e. Gˆ s [n Gˆn , n g Z, one can perform a generalized Gauss decomposition of the element Tr Ty1 , with r being a given constant group element. Denote by Gˆ- 0 , Gˆ0 and Gˆ) 0 the subalgebras generated by elements of grades negative, zero and positive, respectively. Then Tr Ty1 s Ž Tr Ty1 . -0 Ž Tr Ty1 . 0 Ž Tr Ty1 . )0 ,

Ž 4.2 .

where Ž Tr Ty1 . - 0 , Ž Tr Ty1 . 0 and Ž Tr Ty1 . ) 0 are elements belonging to subgroups whose algebras are Gˆ- 0 , Gˆ0 and Gˆ) 0 , respectively. One now introduces y1

T r ' Ž Tr Ty1 . )0 T s Ž Tr Ty1 . 0

y1

Ž Tr Ty1 . -0Tr .

Ž 4.3 .

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Such relation defines a transformation on the connections Am s yEm TTy1 ™ Amr s yEm T r T ry1

Ž 4.4 .

which preserves their grading structure. Therefore, if one knows a solution T for the zero curvature one obtains a new solution T r , determined by the constant group element r . We point out that the fact that the transformed element T r can be written in two different ways as in Ž4.3. plays a crucial role in the dressing method. It guarantees that the transformed connection is in the same gauge as the original one, and therefore allows one to translate the dressing transformation Ž4.4. into transformation of the physical fields defining the theory. A quite general procedure to construct soliton solutions in integrable theories, using such dressing transformations, is described in Ref. w11x. It constitutes a generalization of the so-called ‘‘solitonic specialization’’ in the context of the Leznov–Saveliev solution for Toda type models w12–14,20x. One starts by looking for a ‘‘ vacuum’’ solution such that the connections A ", when evaluated on it, belong to an algebra of oscillators, i.e. an abelian Žup to central terms. subalgebra of the Kac–Moody algebra. One then looks for the eigenvectors Vi , in Gˆ , of such oscillators. The solitons belong to the orbits of solutions obtained by the dressing transformation performed by elements of the form r s e V i1 e V i 2 . . . e V i n . For theory Ž2.1. we perform the dressing transformation starting from the vacuum solution

w s c s c˜ s h s 0 ,

n s y 18 mc2 xq xy' n 0 .

The connection evaluated on such solution is given by Žsee ŽA.1.. vac vac Aq s yE2 , Ay s Ey2 q 18 mc2 xq C .

Ž 4.5 . Ž 4.6 .

In addition, one has y1 Avac with Tvac s e xqE 2 eyx yEy2 . Ž 4.7 . " s yE " Tvac Tvac 1 Since E " 2 s 4 mc H " 1 Žsee ŽA.2.. the relevant oscillators are H " n and their algebra is given by ŽA.5.. The eigenvectors of the oscillators are given by n V " Ž z . s Ý zyn E " Ž 4.8 . ngZ

and E2 , V " Ž z . s " 12 mc zV " Ž z . , 1 mc Ey2 , V " Ž z . s " V Ž z. . Ž 4.9 . 2 z " Notice that Vq Ž z . and Vy Ž yz . have the same eigenvalues, and such degeneracy is related to the global UŽ1. symmetry discussed in Ž2.12. w9x. The solutions in the orbit of the vacuum Ž4.5., under the dressing transformations, are given by w9x i t1 eyi w s , eyŽ ny n oy 2 w . s t 0 Ž 4.10 . t0 and mc mc t˜ Rrt 1 t Rrt 0 cs , c˜ s y , Ž 4.11 . 4 i yt Lrt 1 4 i t˜ Lrt 0

( ž

/

( ž /

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where we have introduced the tau-functions

t 0 ' ² lˆ 0 < G < lˆ 0 : ,

t 1 ' ² lˆ 1 < G < lˆ 1 : ,

1 t R ' ² lˆ 0 < Ey G < lˆ 0 : ,

0 t˜ R ' ² lˆ 1 < Eq G < lˆ 1 : ,

0 < ˆ : t L ' ² lˆ 1 < GEy l1 ,

<ˆ : t˜ L ' ² lˆ 0 < GEy1 q l0 ,

Ž 4.12 .

and where y1 G ' Tvac r Tvac s e xqE 2 eyx yEy2 r e xyEy2 eyx qE 2 .

Ž 4.13 .

We have denoted < lˆ 0 : and < lˆ 1 : the highest weight states of the two fundamental $ representations of the affine Kac–Moody algebra slŽ2., respectively, the scalar and spinor ones. They satisfy H 0 < lˆ 0 : s 0 ,

H 0 < lˆ 1 : s < lˆ 1 : ,

C < lˆ j : s < lˆ j : ,

Ž 4.14 .

with j s 0,1, and in addition 0 < ˆ : n < ˆ : Eq l j s H n < lˆ j : s E " lj s 0 ,

for n ) 0.

Ž 4.15 . 0

Notice that if one makes the shift r ™ h r hy1 , with h s e i u H r2 , one gets that G ™ hGhy1 , since h commutes with E " 2 . Therefore, the tau-functions transform as

t0 ™t0 , t1 ™t1 , t˜ L ™ eyi u t˜ L ,

t R ™ e iu t R ,

t L ™ e iu t L ,

t˜ R ™ eyi u t˜ R ,

Ž 4.16 .

which corresponds to the global UŽ1. transformations Ž2.12.. We will be interested in classical solutions satisfying the relations Ž2.21.. In terms of the tau-functions defined above those relations can be written as

t 0 Eq t 1 y t 1 Eq t 0 s 12 mc t Rt˜ R , t 0 Ey t 1 y t 1 Ey t 0 s 12 mc t Lt˜ L .

Ž 4.17 .

Using Ž4.12., Ž4.13., one can write Ž4.17. as the algebraic relations 1 0 ² G : 0 ² H 1 G : 1 y ² G : 1 ² H 1 G : 0 s 2² Ey G : 0 ² Eq G:1 , y1 : ² 0 ² G : 0 ² G Hy1 : 1 y ² G : 1 ² G Hy1 : 0 s 2² GEq 0 GEy : 1 ,

Ž 4.18 .

where we have denoted ² lˆ a < X < lˆ a : ' ² X :a . These equations determine the group elements r that lead to solutions satisfying the relations Ž2.21.. One can use Ž4.10., Ž4.11. to write the equations of motion of the theory Ž2.1. in terms of the tau-functions. The equations of motion Ž2.3., Ž2.4. for w and n imply that Žfor h s 0.

t 0 Eq Ey t 0 y Eq t 0 Ey t 0 s 14 mc2 t Lt˜ R ,

Ž 4.19 .

t 1 Eq Ey t 1 y Eq t 1 Ey t 1 s 14 mc2 t R t˜ L .

Ž 4.20 .

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The equations of motion Ž2.6., Ž2.7. for the Dirac spinors imply Žfor h s 0.

t 0 Ey t R y t R Ey t 0 s y 12 mc t 1t L ,

Ž 4.21 .

t 1 Eq t L y t L Eq t 1 s 12 mc t 0t R ,

Ž 4.22 .

t 1 Ey t˜ R y t˜ R Ey t 1 s 12 mc t 0t˜ L ,

Ž 4.23 .

t 0 Eq t˜ L y t˜ L Eq t 0 s y 12 mc t 1t˜ R .

Ž 4.24 .

The relations Ž4.19. – Ž4.24. are the Hirota bilinear equations w21,22x for the model Ž2.1.. Notice that the first order equations Ž4.17. and Ž4.21. – Ž4.24. imply that the difference of Ž4.19. and Ž4.20. should be satisfied, but not that each one separately should do. The N-soliton solutions are constructed by taking the constant group element r of the dressing transformation to be a product of exponentials of the eigenvectors of E " 2 Žsee Ž4.9... Since Vq Ž z . and Vy Žyz . have the same eigenvalue, we take the exponential of a linear combination of them, i.e. N Žl.

Žl . Žl . r Ny sol ' Ł e V Ž a " , z l . with V Ž aŽ"l . , z l . ' 'i Ž aq Vq Ž z l . q ay Vy Ž yz l . . .

ls1

Ž 4.25 . Žl. Žl. In fact, if for a given l Žor more than one. either aq or ay vanishes one does not obtain N-soliton solutions. For example, in the case of the one-soliton one gets a vanishing solution for w w9x. The fact that the N-soliton solutions need the two degenerate eigenvectors, i.e. Vq Ž z . and Vy Žyz ., is what makes them to carry, besides the topological charge, the ‘‘electric’’ UŽ1. charge Ž2.13.. The corresponding group element Ž4.13. becomes

N Žl . G Ny sol s Ł exp Ž e G Ž z l . V Ž a " , zl . . ,

Ž 4.26 .

ls1

with

ž

G Ž z l . s 12 mc z l xqy

1 zl

/

xy ' g l Ž x y Õ l t .

Ž 4.27 .

and so

ž

g l s 12 mc z l q

1 zl

/

s Ž sign z l .

mc

(1 y Õ

2 l

,

Õl s

1 y z l2 1 q z l2

.

Ž 4.28 .

Notice that one needs to take z l real for the soliton velocities to be smaller than light speed, i.e. < Õ l < F 1. In fact, the quantities z l are related to the rapidities u l of the solitons through zl ' el eul ,

Ž 4.29 .

with e l s "1, and so from Ž4.27.

G Ž u l . s e l mc Ž xcosh u l q tsinh u l . .

Ž 4.30 .

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An important point in the calculations involved in the construction of the soliton solutions is that it is much easier to work with the homogeneous Žor Fubini–Veneziano. vertex operator construction for V " Ž z .. Indeed, in such construction one has that 8 Vq Ž z . Vq Ž z . ™ 0 ,

Vy Ž z . Vy Ž z . ™ 0 ,

as z ™ z ,

Ž 4.31 .

and therefore the exponentials involving V " Ž z . truncate. We are interested in the solutions where the field w is real. Using Ž4.10. one obtains that Žsince yi w s log < t 1 < q i arg t 1 . t0

t0

w s w implies
Ž 4.32 .

and consequently

ž

w s arctan i

t 0)t 1 y t 0t 1) t 0)t 1 q t 0t 1)

/

,

or w s z 0 y z 1 q np ,

Ž 4.33 .

where n is any integer, and where we have denoted ta s
t˜ R t1

s yiec

t R) t 0)

,

t˜ L t0

s iec

t L) t 1)

.

Ž 4.34 .

4.1. The one-soliton solutions The one-soliton solution can easily be calculated by evaluating the matrix elements Ž4.12. with r s e V Ž a " , z ., or alternatively by applying the Hirota method to the equations Ž4.19. – Ž4.24.. The result is

t0s1y

i 4

ay aq e 2 G Ž z . ;

t1s1q

i 4

ay aq e 2 G Ž z . ,

t R s 'i aq z e G Ž z . ;

t˜ R s 'i ay e G Ž z . , ay G Ž z . t L s 'i aq e G Ž z . ; t˜ L s y'i e . Ž 4.35 . z One can check that such solutions also satisfy Ž4.17. without any further restriction. Therefore, the one-soliton solution satisfy the condition Ž2.21. for the equivalence between Noether and topological currents. If one requires that w should be real, then one gets from Ž4.32. and Ž4.35. that Ž ay aq . must be real. Such condition is sufficient to make t 1 s t 0) . Therefore, using Ž2.16. and Ž4.33., one gets that the topological charge of the soliton is Qtopol .s ysign Ž ay aq . Qtopol .s sign Ž ay aq .

8

for z ) 0, for z - 0.

See Eq. Ž14.8.14. on page 309 of Ref. w23x, or Eq. Ž6.2.6. of Ref. w24x for the details.

Ž 4.36 .

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However, if one imposes in addition, the reality condition Ž2.2. on the spinors one gets from Ž4.35. that the parameters must satisfy ) z. ays yec aq

Ž 4.37 .

The topological charge in such case is Q topol .s sign ec ,

for any real z.

Ž 4.38 .

Notice that in general, solitons are transformed into anti-solitons by parity transformations. However, theory Ž2.1. is not invariant under spatial parity, and the CP-like symmetry Ž2.17. is broken when the reality condition Ž2.2. is imposed. In order to have a degenerate vacua and topological solitons we need the factor i in the exponential of the potential term in Ž2.1.. That makes the Lagrangian complex and so it is very unlikely that Ž2.1. has a CPT symmetry. The lack of P, CP and CPT symmetries is what reaffirms the existence of only the soliton or anti-soliton solution for a given choice of ec in Ž2.2.. We then reach an interesting conclusion: without the reality condition Ž2.2. theory Ž2.1. has two Dirac spinors and it also has the soliton and anti-soliton solutions, since according to Ž4.36. both signs of the charge are admissible. However, if we impose the reality condition Ž2.2. theory Ž2.1. looses one Dirac spinor and also one soliton solution, since for a given choice of ec one observes from Ž4.38. that only one topological charge Žcorresponding to soliton or anti-soliton. is permitted. That is indicative of the existence of some sort of duality between solitons and spinor particles. The particular one-soliton solution Ž2.22. satisfies Ž4.37. with ec s 1, z ) 0, u being the phase of aq, i.e. aqs < aq < e i u , and < aq <'z r2 s exp ymc x 0r'1 y Õ 2 . The mass of these one-soliton solutions was evaluated in Ref. w9x and it is given by

ž

/

M s 2 kmc ,

Ž 4.39 .

where k is the coupling constant appearing in the Lagrangian Ž2.1.. 4.2. The two-soliton solutions The two-soliton solutions are calculated through the dressing transformation method taking Žsee Ž4.25.. Ž1.

Ž2.

r 2y sol ' e V Ž a " , z 1 . e V Ž a " , z 2 . and so G 2ysol s e Ž e

G Ž z 1.

V Ž a Ž1. " , z 1 ..

eŽ e

G Ž z 2.

V Ž a Ž2. " , z 2 ..

.

Ž 4.40 . The explicit solution is calculated by evaluating the matrix elements Ž4.12.. Alternatively, one can apply Hirota’s expansion method w21,22x to the Hirota equation Ž4.19. – Ž4.24.. The results one obtains are

t0s1y

i 4

Ž1. Ž1. 2 G Ž z 1 . ay aq e y

i 4

Ž1. Ž2. qay aq . e G Ž z 1 .q G Ž z 2 . y

Ž2. Ž2. 2 G Ž z 2 . ay aq e yi

1 Ž z1 y z 2 .

z1 z 2

Ž z1 q z 2 .

2

Ž aqŽ 1 . ayŽ2.

4

16 Ž z 1 q z 2 . 4

Ž1. Ž1. Ž2. Ž2. 2Ž G Ž z 1 .q G Ž z 2 .. ay aq ay aq e ,

Ž 4.41 .

H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

t1s1q

i

Ž1. Ž1. 2 G Ž z 1 . ay aq e q

4

i 4

i

Ž2. Ž2. 2 G Ž z 2 . ay aq e q

1 Ž z1 y z 2 .

Ž1. Ž2. 2 qay aq z 2 . e G Ž z 1 .q G Ž z 2 . y

Ž z1 q z 2 .

2

623

Ž aqŽ 1 . ayŽ2. z12

4

16 Ž z 1 q z 2 . 4

Ž1. Ž1. Ž2. Ž2. 2Ž G Ž z 1 .q G Ž z 2 .. ay aq ay aq e ,

Ž 4.42 . tR

'i

Ž1. s aq z1

e

4

'i

Ž1. s aq

q

t˜ R

'i

'i

e

Ž z1 q z 2 . G Ž z1 .

Ž2. q aq

i Ž z1 y z 2 .

4

sy

z1 i q

e G Ž z1 . y

2

Ž1. Ž1. Ž2. 2 G Ž z 1 .q G Ž z 2 . ay aq aq e

q

i Ž z1 y z 2 .

Ž 4.43 .

2

4 Ž z1 q z 2 . 2

Ž1. Ž2. Ž2. G Ž z 1 .q2 G Ž z 2 . aq ay aq e

2 Ž1. Ž1. Ž2. 2 G Ž z 1 .q G Ž z 2 . ay aq aq e ,

i Ž z1 y z 2 .

Ž 4.44 .

2

4 Ž z1 q z 2 . 2

Ž1. Ž1. Ž2. 2 G Ž z 1 .q G Ž z 2 . ay aq ay e

2 Ž1. Ž2. Ž2. G Ž z 1 .q2 G Ž z 2 . ay ay aq e ,

4 Ž z1 q z 2 . 2 Ž1. ay

Ž z1 q z 2 .

2

Ž1. Ž2. Ž2. G Ž z 1 .q2 G Ž z 2 . aq ay aq e ,

G Ž z2.

4 Ž z1 q z 2 . 2

i Ž z1 y z 2 .

y

i z 2 Ž z1 y z 2 .

2

2

e

e

G Ž z2.

Ž1. G Ž z 1 . Ž2. G Ž z 2 . s ay e q ay e q

q

t˜ L

Ž2. q aq z2

i z1 Ž z1 y z 2 .

y

tL

G Ž z1 .

Ž2. ay

Ž z1 y z 2 .

z2

e G Ž z2. q

i

Ž z1 y z 2 .

Ž 4.45 . 2

4 z1 Ž z1 q z 2 . 2

Ž1. Ž2. Ž2. G Ž z 1 .q2 G Ž z 2 . ay ay aq e

2

4 z 2 Ž z1 q z 2 . 2

Ž1. Ž1. Ž2. 2 G Ž z 1 .q G Ž z 2 . ay aq ay e .

Ž 4.46 .

These tau-functions satisfy the Hirota equations Ž4.19. – Ž4.24. for any value of the Ž2. constants aŽ1. ", a ", z 1 and z 2 , which can be even complex. It is interesting to notice that these tau-functions also satisfy the equivalence between the Noether and topological currents Ž4.17. Žor equivalently Ž2.21.. without any further restrictions. As we have said the Lorentz transformations in 2 d are x "™ xˆ "' l" 1 x ". The corresponding field transformations can be obtained from Ž2.8., Ž2.9. by taking f Ž xq . s l xq and g Ž xy . s ly1 xy. Therefore, choosing the conformal weight d of the field n to vanish, one conclude from Ž4.10., Ž4.11. that, under Lorentz transformations, the tau-functions transform as

ta Ž x . ™ ta Ž xˆ . s ta Ž x .

for a s 0,1,

Ž 4.47 .

and

t R Ž x . ™ t R Ž xˆ . s ly1 r2t R Ž x . ,

t L Ž x . ™ t L Ž xˆ . s l1r2t L Ž x . ,

Ž 4.48 .

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and similarly for t˜ R r L . One then observes that such transformations can be implemented in the space of solutions by transforming the parameters of the solutions Ž4.41. – Ž4.46. as aŽ"i.™ l. 1r2 aŽ"i. ,

zi ™ l zi ,

i s 1,2 .

Ž 4.49 .

In addition, the global UŽ1. transformations Ž2.12. Žsee Ž4.16.. correspond in the space of parameters to zi ™ zi ,

aŽ"i.™ e " i u aŽ"i. ,

i s 1,2 .

Ž 4.50 .

4.3. The reality conditions We want z 1 and z 2 real in order for the soliton velocities to be smaller than the speed of light Ž c s 1, see Ž4.28... Then, one can easily check that in order for the Ž1. Ž1. Ž2. Ž2. solutions Ž4.41., Ž4.42. to satisfy Ž4.32. one has to have aq ay and aq ay real, and so Ž i. Ž i. < i j i aq s < aq e ,

Ž i. Ž i. < yi j i ay s e i < ay e ,

i s 1,2 ,

with j i real being phases, and e i s "1 being the sign of

Ž 4.51 . Ž i. Ž i. aq ay.

In addition, one needs

Ž1. < < Ž2. < Ž1. < < Ž2. < e 1 z 1 < aq ay s e 2 z 2 < ay aq .

Ž 4.52 .

Using Ž4.29. one observes that Ž4.52. implies

e1 e2 e1 e2 s 1 , e

u 1yu 2

Ž1. < < aq

Ž 4.53 .

Ž2. < Ž1. < < ay s < ay

Ž2. < < aq .

Ž 4.54 .

The relations Ž4.51. and Ž4.52. are the necessary and sufficient conditions for Ž4.32. to be satisfied. However, they are sufficient to make t 1 the complex conjugate of t 0 . Therefore,

w ) s w ™ t 1 s t 0) ™ w s 2 z 0 q np .

Ž 4.55 .

Consequently, the conditions Ž4.34. become

t˜ R s yiec t R) ,

t˜ L s iec t L) .

Ž 4.56 .

One can easily check that the solutions Ž4.41. – Ž4.46. satisfy the conditions t 1 s t 0) Žand so w real. and Ž4.56. if the parameters satisfy Ž z 1 , z 2 being real. Ž i. Ž i.) ay s yec aq zi ,

i s 1,2 .

Ž 4.57 .

Ž i. Ž i. Then it follows that the signs of z i and aq ay are related by

e i s ye i sign ec ,

i s 1,2 ,

Ž 4.58 .

which indeed satisfy Ž4.53.. 4.4. The topological charges For the solutions satisfying the conditions Ž4.51. and Ž4.52. we shall denote Ž i.

Ž i. < < Ž i. < < aq ay ' ey2 g i x 0 ,

i s 1,2 ,

Ž 4.59 .

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with g i defined in Ž4.28.. We then define

GˆŽ z i . ' G Ž z i . y g i x 0Ž i. ,

Ž 4.60 .

with G Ž z i . being defined in Ž4.27.. In order to evaluate the topological charges we shall take the solution in the center of mass reference frame, which corresponds to Žsee Ž4.28. and Ž4.29.. z1 s e 1 e u ,

z 2 s e 2 eyu

™

and g 2 s e 1 e 2 g 1

Õ 1 s yÕ 2

Ž 4.61 .

Then, using Ž4.51., Ž4.54., Ž4.61., and Ž4.49., the tau-function t 0 given in Ž4.41. becomes

Ž e 1 e u y e 2 ey u .

t 0 s 1 y e1 e2

y

e1 e2 16

q

e2 4

Ž e u q e 1 e 2 ey u .

ˆ

2

ˆ

sin Ž j 1 y j 2 . e G Ž z 1 .q G Ž z 2 .

4

Ž e u y e 1 e 2 ey u . 2 ŽGˆ Ž z . q Gˆ Ž z . . e yi 4 Ž e u q e 1 e 2 ey u . 1

ˆ

e2 G Ž z2 . q e1 e2

2

Ž e 1 e u q e 2 ey u . Že

u

yu 2

q e1 e2 e

ž

e1 4

ˆ

e 2 G Ž z1 .

ˆ

ˆ

/

cos Ž j 1 y j 2 . e G Ž z 1 .q G Ž z 2 . .

.

Ž 4.62 .

In addition, one has t 1 s t 0) , when Ž4.53. is also enforced. Therefore, according to Ž2.16. and Ž4.55., the topological charge is Qtopol.s p2 Ž arg t 0 Ž ` . y arg t 0 Ž y` . . . We shall use the prescription that the arg t 0 lies between yp and p . When evaluating the charge one should notice that, on the center of mass reference frame, Gˆ Ž z 1 . q Gˆ Ž z 2 . does not depend upon x if e 1 and e 2 have opposite signs. In addition, for e 1 e 2 s e 1 e 2 s 1 one should pay attention on the sign of the imaginary part of t 0 as it tends to zero, in order to determine if arg t 0 tends to yp or p . In such case, one can write Im t 0 ™ y 12 e 1 1 q

ž

cos Ž j 1 y j 2 .

for e 1 e 2 s e 1 e 2 s 1. Since

cosh u cos Ž j 1 y j 2 . cosh u

/

e 2 mc e 1 x cosh uq PPP

as x ™ e 1`

Ž 4.63 .

G y1, the limit is independent of j i . In fact, the

topological charge, for w real, does not depend upon the phases j i . The results one obtains are Qtopol .s 2 e 1 , Qtopol .s y2 e 1

Ž e 1 s e 2 s -1, e 1e 2 s 1 . or Ž e 1 s -e 2 s -1, e 1e 2 s -1 . , for Ž e 1 s e 2 s 1,e 1e 2 s 1 . or Ž e 1 s -e 2 s 1,e 1e 2 s -1 . . Ž 4.64 .

for

If one now imposes the reality condition Ž2.2. on the spinors, or equivalently Ž4.56. – Ž4.58., one gets Q topol .s 2 sign ec ,

Ž 4.65 .

for any e i and e i satisfying Ž4.58.. We then have a situation similar to the one-soliton solutions, since in that case the imposition of the condition Ž2.2. makes either the soliton or anti-soliton to disappear from the spectrum. Here the same thing happens, since Ž2.2. chooses the sign of ec and then either the charge 2 or y2 solutions disappear. However, a new feature emerges.

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We saw we could have Žwithout Ž2.2.. soliton and anti-soliton solutions with w real. We have not found here a charge zero solution corresponding to the scattering of a soliton and anti-soliton for w real. Such solution exists however, for w complex but asymptotically real as we show below. 4.4.1. Asymptotically real solutions One can easily verify that for the solutions Ž4.41. – Ž4.46. the field w is always real asymptotically, i.e.
t 1 s 1 y e1 e2

y

e1 e2 16

q

e2 4

Ž e 1 e u y e 2 ey u . Že

u

yu 2

q e1 e2 e

ˆ

ˆ

sin Ž j 1 y j 2 . e G Ž z 1 .q G Ž z 2 .

.

4

Ž e u y e 1 e 2 ey u . 2 ŽGˆ Ž z . q Gˆ Ž z . . e qi 4 Ž e u q e 1 e 2 ey u . 1

ˆ

e2 G Ž z2 . q e1 e2

2

Ž e 1 e u q e 2 ey u . Že

u

yu 2

q e1 e2 e

ž

e1 4

ˆ

e 2 G Ž z1 .

ˆ

ˆ

/

cos Ž j 1 y j 2 . e G Ž z 1 .q G Ž z 2 . .

.

Ž 4.66 .

Since
Ž 4.67 .

Therefore, such solution corresponds to the scattering of a soliton and a anti-soliton. 4.5. The breather solutions The space-time dependence of the solutions Ž4.41. – Ž4.46. are given through the exponentials exp Ž g l Ž x y Õ l t . . , l s 1,2. Therefore, in order to have solutions periodic in time one needs g l Õ l to be pure imaginary. Writing z l s < z l < e i u l , one observes from Ž4.28. that for that to happen one needs < z l < s 1, and so g l Õ l s yimc sin u l . In order for the solution to present just one frequency one needs to have sin u 2 s "sin u 1. We do not want z 1 s z 2 because Ž4.41. – Ž4.46. becomes a one-soliton solution. In addition, we do not want z 1 s yz 2 because the solution Ž4.41. – Ž4.46. will present singularities. Therefore, the only possibilities of having solutions periodic in time is to have z1 s e i u ;

z 2 s e eyi u ;

e s "1 .

Ž 4.68 .

Then one gets from Ž4.27. and Ž4.28. that

G Ž z1 . s g x q i v t ,

GŽ z2 . s eŽ g x y i v t . ,

Ž 4.69 .

H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

627

with

g ' mc cos u ,

v ' mc sin u .

Ž 4.70 .

4.5.1. The case e s 1 We are interested in the cases where the field w is real, and so from Ž4.10. we need
)

Ž 4.71 .

and ngZ ,

u q jqŽ1. q jyŽ2. s np ,

Ž 4.72 .

where we have denoted Ž i.

aŽ"i. ' < aŽ"i. < e i j " ,

i s 1,2 .

Ž 4.73 .

It follows that the conditions Ž4.71. and Ž4.72. are sufficient to make t 1 s t 0) , and from Ž4.41. one gets

t 0 s 1 q Ž y1 . q iS

ž

nq 1

Ž y1.

ž

S Ry

1

cos u

sin u

R cos 2u

nq 1

ž

/

Rq

1 R

/

e 2 x mc cos u y S 2 tan4u e 4 x mc cos u

/

y 2cos Ž 2 v t q jqŽ 1 . q jyŽ 1 . . e 2 x mc cos u ,

Ž 4.74 .

where the integer n is the same as in Ž4.72., and we have introduced Ž1. < < Ž1. < S ' 14 < aq ay ;

R'

Ž1. < < aq Ž2. < < aq

.

Ž 4.75 .

Therefore, using Ž2.16. and Ž4.10. one gets that the topological charge of such breather solution is Qtopol .s 2 Ž y1 .

nq 1

,

Ž 4.76 .

with n given by Ž4.72.. Now, if besides the reality of w , one imposes the condition Ž2.2. on the spinors, one gets that the parameters have to satisfy Ž1. Ž2.) i u ay s yec aq e ;

Ž2. Ž1.) yi u ay s yec aq e ,

Ž 4.77 .

which to be compatible with Ž4.71. and Ž4.72., one needs exp Ž i Ž u q jqŽ1. q jyŽ2. . . s ysign ec . Then, the topological charge becomes Qtopol .s 2 sign ec .

Ž 4.78 .

Consequently, we have for the breather solutions an effect similar to what happens in the two-soliton solutions Žsee discussions below Ž4.38. and Ž4.65... For w real we have breathers with topological charges "2. However, if one imposes the reality condition Ž2.2. on the spinors, one gets that only one charge is allowed.

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H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

4.5.2. The case e s y1 One can check that for the choice e s y1 in Ž4.68. one cannot obtain nontrivial solutions such that w is real. However, one does have that w is asymptotically real. It is quite easy to verify that the topological charge vanishes for any choice of the parameters aŽ"i. . So, Qtopol .s 0 ,

for e s -1, and any aŽi. " , i s 1,2.

Ž 4.79 .

4.6. The time delays Solitons are classical solutions that travel with constant speed without dispersion and that keep their form under scattering, the effect of it being only a phase shift or a displacement in its position. We now show that the solitons we have been working with are true solitons and indeed fulfill the above requirements. We shall calculate the so-called time delays of the scattering of two solitons, using the procedures of w25x. We consider two solitons that in the distant past are well apart, then collide near t s 0 and then separate again in the distant future. Therefore, except for the region where the scattering occurs, the solitons are free and so travel with constant velocity. Let us denote the trajectories of one of the solitons before and after the collision as Žsince the velocity is the same. x s Õt q x Ž I .

and

x s Õt q x Ž F . .

Ž 4.80 .

The lateral displacement at fixed time is measured by

DŽ x . ' x Ž F . y x Ž I . .

Ž 4.81 .

The time delay is defined by

DŽ t . ' t Ž F . y t Ž I . s y

DŽ x . Õ

,

Ž 4.82 .

with the intercepts of the trajectories with the time axis being given by t Ž F . s yx Ž F .rÕ and t Ž I . s yx Ž I .rÕ. The lateral displacement and the time delay are not Lorentz invariant, and so we consider the invariant E D Ž x . s yp D Ž t . ,

Ž 4.83 .

with E and p s ÕE being the energy and momentum of the soliton, respectively. Since E is positive it follows DŽ x . has the same sign in any reference frame, with only its strength being frame dependent. The time delay on the other hand may change the sign under Lorentz transformations. One can show that the lateral displacements and time delays for the two solitons participating in the collision have to satisfy w25x E1 D1 Ž x . q E2 D2 Ž x . s 0 ;

p 1 D1 Ž t . q p 2 D2 Ž t . s 0 ,

Ž 4.84 .

with the sub-indices i labeling the quantities associated to particle i. Notice therefore that, since the energies are positive, the lateral displacements have opposite signs. Clearly, in the center of momentum frame Žcomf., where p 1 q p 2 s 0, the time delays are equal, i.e. D1Žcomf . Ž t . s D2Žcomf . Ž t .. Therefore, from Ž4.83. we have that E1Žcomf .D1Žcomf . Ž x .rp 1 s yE2Žcomf .D2Žcomf . Ž x .rp 1 s yD1Žcomf . Ž t . s yD2Žcomf . Ž t .. Consequently, since DŽ x . has the same sign in any frame, it follows that, if particle 1 moves

H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

629

to the right faster then particle 2 Žsuch that p 1 is positive in the comf., then ŽyD1Ž x .., D2 Ž x ., D1Žcomf . Ž t . and D2Žcomf . Ž t . all have the same sign. The physical interpretation of that sign is related to the character Žattractive or repulsive. of the interaction forces. Indeed, if the force is attractive then particle 1 will accelerate as it approaches particle 2 and then decelerate. That means that D1Ž x . is positive and so the common sign negative. Therefore, attractive forces lead to a negative time delay in the center of momentum reference frame, and clearly repulsive forces a positive time delay. These considerations assume that the two particles pass through each other, and there is no reflection. However, when the masses of the two particles are equal there is the possibility of occurring reflection. Let us now evaluate the time delays for the two-soliton solutions given in Ž4.41. – Ž4.46.. We choose the particle 1 to move faster to the right than particle 2, i.e. Õ1 ) Õ 2 , with Õ1 ) 0, and therefore from Ž4.28. and Ž4.29. < z 1 < - < z 2 < or u 1 - u 2 . We then track the soliton 1 in time w25x, i.e. we hold x y Õ1 t fixed as time varies. Then one gets x y Õ 2 t s const.qŽ Õ 1 y Õ 2 . t, with const.s x y Õ 1 t. We then get that, if e 2 s 1, e G 2 Ž z 2 . ™ 0 as t ™ y`, and e G 2 Ž z 2 . ™ ` as t ™ `. For the case e 2 s y1 the limits get interchanged. Therefore, taking e 2 s 1, one can check that the solution Ž4.41. – Ž4.46. becomes, in the limit t ™ y`, the one-soliton solution Ž4.35. with the identifications aŽ1. " ' a " and z 1 ' z. Now, in the limit t ™ `, one can also verify that the ratios t 0rt 1 , t Rrt 0 , t Lrt 1 , t˜ Rrt 1 and t˜ Lrt 0 for the solutions Ž4.41. – Ž4.46. tend to the corresponding ratios for the one-soliton solution Ž4.35. with the same identification of parameters, and with the replacement e G 1Ž z 1 . ™

ž

z1 y z 2 z1 q z 2

2

/

e G 1Ž z 1 . .

Ž 4.85 .

The only difference being the fact that the ratio t 0rt 1 gets a relative minus sign, meaning that the w field gets shifted by p Žsee Ž4.10.. during the scattering process. Then one observes from Ž4.10. – Ž4.11. that the relevant effect of the scattering on the solutions for the fields w , c and c˜ is a lateral displacement of the soliton 1 given by

ž

g 1 Ž x y Õ1 t . ™ g 1 x y Õ1 t q

1

g1

ln

ž

z1 y z 2 z1 q z 2

2

//

.

Ž 4.86 .

If one takes e 2 s y1 instead, one observes that the direction of the arrow in Ž4.86. reverses. Therefore, using Ž4.81., Ž4.28. and Ž4.29. one sees that the lateral displacement for the soliton 1 is given by

D1 Ž x . s y

e1 e2 mc cosh u 1

ln

ž

e Žu 1y u 2 . r2 y e 1 e 2 eyŽ u 1y u 2 .r2 e Ž u 1y u 2 .r2 q e 1 e 2 eyŽ u 1y u 2 .r2

2

/

.

Ž 4.87 .

Since e 1 e 2 s "1, one observes that D1Ž x . is in fact independent of such signs, and so

D1 Ž x . s y

1 mc cosh u 1

ž ž

ln tanh

u1 yu2 2

2

//

.

Ž 4.88 .

H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

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Notice that the solutions Ž4.41. – Ž4.46. are symmetric under the interchange of the indices 1 and 2 of the parameters aŽ"i. and z i . Therefore, if we track the soliton 2, i.e. keep x y Õ 2 t fixed as time varies, but under the same kinematical conditions, i.e. Õ 1 ) Õ 2 , with Õ 1 ) 0, then we obtain that 1

D2 Ž x . s

mc cosh u 2

ž ž

ln tanh

u1 yu2 2

2

//

.

Ž 4.89 .

If we reverse the kinematical conditions, i.e. take Õ 2 ) Õ 1 , with Õ 2 ) 0, then the signs of both Di Ž x ., i s 1,2, reverse. The mass of the one-soliton solutions is given in Ž4.39., and therefore the energies of the solitons are Ei s 2 kmc cosh u i , i s 1,2. Consequently, one sees that the Di Ž x .’s do indeed satisfy Ž4.84., i.e.

ž ž

E1 D1 Ž x . s yE2 D2 Ž x . s ysign Ž Õ 1 y Õ 2 . 2 k ln tanh

u1 yu2 2

2

//

.

Ž 4.90 .

In addition, using Ž4.82., Ž4.88. and Ž4.89., one gets that the time delays are given by Žassuming Õ 1 ) Õ 2 , with Õ 1 ) 0.

D1 Ž t . s y D2 Ž t . s

1 mc sinh u 1 1

mc sinh u 2

2

u1 yu2

ž ž // ž ž //

ln tanh

ln tanh

2

u1 yu2 2

;

2

.

Ž 4.91 .

Notice that the hyperbolic tangent varies from y1 to 1 and therefore the logarithm of its square is always negative, and so from Ž4.88. one sees that D1Ž x . for Õ 1 ) Õ 2 , with Õ 1 ) 0, is positive. Therefore, from the considerations made above we conclude that the forces between the solitons is attractive. In addition, it is independent of their topological charges. In fact, the time delays we have obtained coincide with those of the sine-Gordon theory by identifying k with a positive constant multiplied by the inverse of the square of the sine-Gordon coupling constant. Acknowledgements H.S.B.A. is supported by a Fapesp grant, and L.A.F. is partially supported by CNPq. The authors are grateful to J.F. Gomes, M.A.C. Kneipp, D.I. Olive, J. Sanchez Guillen, ´ ´ G. Sotkov and A.H. Zimerman for many helpful discussions. Appendix A. The zero curvature The equations of motion of theory Ž2.1. can be represented as a zero curvature Ž4.1. with connections given by w9x Aqs yB Ž E2 q F1q . By1 ,

Ays yEy B By1 q Ey2 q F1y ,

Ž A.1 .

where Bse

iw H 0

ž e

ny

i 2

/

w C

eh Q ,

E " 2 ' 14 mc H " 1 ,

Ž A.2 .

H.S. Blas Achic, L.A. Ferreirar Nuclear Physics B 571 [PM] (2000) 607–631

631

and 0 1 F1q s imc Ž c R Eq q c˜R Ey .,

(

˜ 0 F1y s imc Ž c L Ey1 q y c L Ey . ,

(

Ž A.3 .

We have written the Dirac spinors as

cs

cR ; cL

ž /

c˜ s

c˜R

ž / c˜L

,

Ž A.4 .

n and have denoted by H n , E " , D and C the Chevalley basis generators of the sl Ž2. affine Kac–Moody algebra. The commutation relations are

w H m , H n x s 2 m C dmq n ,0 , m

n H , E" m n Eq , Ey m

mqn s "2 E " mqn

sH

,

q m C dmqn ,0 ,

m w D , T x s m T m , T m ' H m , E" .

Ž A.5 . Ž A.6 . Ž A.7 . Ž A.8 .

The generator Q is the grading operator for the principal gradation and given by Q ' 12 H 0 q 2 D. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x

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