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Fermionic and bosonic scattering phases on a topological kink
Volume 134, number
1
PHYSICS
LETTERS A
12 December
FERMIONIC AND BOSONIC SCATTERING PHASES ON A TOPOLOGICAL K.A. SVESHNIKOV
1988
KINK
’
Institute ofNuclear Physics, Moscow University, Moscow I 19899, USSR Received 6 September 1988; accepted Communicated by A.R. Bishop
for publication
3 October
1988
A general approach to the scattering of fundamental fermions and bosom on a topological kink excitation is considered. An asymptotic LSZ-analysis for secondary quantized Heisenberg fields and kink motion is developed and some new results for the phases of one-particle matrix elements for the elastic scattering on the kink are derived.
In the present Letter the scattering of fundamental particles of quantum fields on a classical excitation-like kink or soliton is considered. There are several equivalent approaches for the description of soliton sectors in QFT (for a review see ref. [ 1 ] ). The most preferable one for the scattering problem is the consideration of the Heisenberg fields [ 2-41. In such an approach the asymptotic states are constructed by means of the wellestablished LSZ-formalism [ 5 1. In this way we shall consider the scattering of fermions and bosons on a topological kink in a 1 + l-dimensional QFT-model, described by the lagrangian p=
WV) +
f (a,p)*-
@(id--gp)v.
(1)
The model ( 1) is the relativistic dynamical analogue of the statistical mechanical one of quasi-one-dimensional electron-phonon systems [ 6,7]. The boson field possesses a classical component
v1,(& t> =u
x-vt
(QTZ >
in the form of a single topological kink u( + 03 ) = + U* The spectrum of quantum excitations is defined from the linearized field equations and commutation lations in the vicinity of the classical component (2). In the center-of-mass system one has [ 3,8 ]
[o+v(x)lw& [ia-m(x)
f)=O,
v(x)= u” (u(x)),
re-
(3a)
lv(x, t) =O,
(3b)
[@(x, t), a,@(~‘, t>]_ =i[6(x_x’)-M,‘U’(x)U’(X’)],
(4a)
{vl(X, r>, w+ (X’, r>>+ =6(x-x’
(4b)
),
where MO= Jti u’ *(x) is the classical kink mass, and the rest of the commutators and anticommutators ishes. The quantum component @(x, t) of the boson field obeys a subsidiary condition
s
dx u’ (x)@(x,
Permanent USSR.
address:
t)
=o, Quantum
van-
(5)
Theory and High Energy Physics Division,
Physical
Department,
Moscow University,
Moscow
119899,
47
Volume
I
134, number
PHYSICS
LETTERS
12 December
A
which together with (4a) ensures the exclusion of the translational zero mode from the bosonic spectrum The total energy of the system in this approximation is equal to H=M,+
J
1988
[ 9, lo].
dx{+d,2++@‘2+$V(x)02+W+[-iiCY~,+g/3U(x)]~}.
(6)
The detailed analysis of eqs. ( 3 ) and associated spectral problems including construction of resolvents and Green functions for typical kink configurations may be found in refs. [ 6, I 11. Here we shall consider the general features of the LSZ-asymptotics of the Heisenberg fields. Let us begin with the boson field. This problem has already been considered in refs. [ 2-41, so now we only briefly dwell on the basic features and specifications of relations derived in ref. [ 3 1, which will be necessary for what follows. The solution to (3a) has the form @(x,t)=
7 (2wf)-“2[u/e-‘““~Xx)+af+e’wfl~~((X)],
where {@Xx)} is the orthonormal
(7)
set of eigenfunctions
of the spectral problem
[ -a*/ax*+V(X)]~XX)=0~~AX),
(8)
with the zero mode N u’ (x) excluded due to (5), so of> 0. a, a; a boson of frequency w/,
are the annihilation,
creation
operators
of
[Uf,Uf,]- =o.
[qr,aFl-=6/r,,
The spectrum of the operator (8) consists of a discrete part 0 CO, < mB = Jm, which includes nondegenerate real integrable eigenfunctions e,,(x), and of a continuous spectrum, each eigenvalue of which, ok= (P+m;)“*, is doubly degenerate in + k and @ = $_k. Owing to such features of the spectrum the LSZ-asymptotics of the field 0(x, t) consists of two parts. The discrete modes enter without any changes, the continuous spectrum forms the plane-wave part of the asymptotics, which is nothing but the free quantum field of mass ma, [up,‘(k) The asymptotic up,“t(k)=w-
LSZ-condition lim
i
s
exp( -iw,t+ikx)+h.c.].
(9)
[ 5 ] then gives
dx
The evaluation of this limit is well developed into the following form: if
[ 2,3 1. For our purposes it is convenient
to bring the final result
@k(X)x-t-tc.2 --t [xi(k)e*kx+Y+(k)e-ik”]/J211,
(10)
then u~,“‘(k)=8(+k)[X+(k)u(k)+Y+(-k)u(-k)]+8(Tk)[X_(k)u(k)+Y_(-k)u(-k)]. Now let us note that the normalization s
d-x @(x)$r,(x)
condition
=&
forbids the gk to be the ordinary totic behaviour 48
(11)
quantum-mechanical
scattering
wave functions&(x),
whichnbey
the asymp-
Volume 134, number
1
PHYSICS
Pk(X)x-r +-cc eikr+ r( k)e-‘k: since the scalar product
--t
Pk(X)
A
12 December
1988
t(k)eikx,
x--t+00
for these functions
s
LETTERS
reads
dx&(X)&(X)=2X[d(k-k’)+r(k)d(k+k’)].
-The orthonormal functions @k(x) obey the asymptotic behaviour of the form (lo), where the coeficients X, (k), Y* (k) are connected with the transition and reflection coefficients t(k) and r(k) in the following way,
x-(k)={f[l+lt(k)ll}“2, Y_(k)~e’~~(~){j[
I-
X+(k)=ei~(k){~[l+]t(k)]]}‘/‘,
It(k)I]}
Y+(k)=-e
where &(k)=arg t(k), d,(k) =arg r(k). X+(k), which may be dropped without loss of generality, X*,(k)=X,(-k),
iI6R(k)--6r(k)l{f
[
1_
it(k)
1 I}
112,
(12)
Y*(k) are defined by (12) up to a common and obey the following relations,
phase factor,
Y*,(k)=Y,(--k),
IX_(k)(2+IY_(k)12=IX+(k)12+lY+(k)12=1, X_(k)Y_(k)+X+(k)Y+(k)=X+(k)Y*_(k)+X-(k)Y*,(k)=O.
(13)
By means of ( 13) one may easily verify that the relation tinuous spectrum of bosonic excitations,
t~in(k>,~i+,(k’)l-=[~,“,,~~,(k’)land the transition CL
matrix between
( 11) defines a unitary
transformation
in the con-
=a(k-k’), in- and out-states
(14) (scattering
matrix)
is equal to
Iki,)=ra,,,(k’),a~(k)l-
=t(k)d(k-k’)+{@(k)r(k)-8(
-k)
exp[-2i&(k)]r(k)}
6(k+k’).
(15)
Relations ( 14) and ( 15) ensure the completeness and unitary equivalence of the in- and out-states of the boson field in the continuous spectrum. The coefficients in the r.h.s. of ( 15) are defined without any ambiguity and coincide up to a phase factor with the quantum-mechanical transition and reflection coefficients. In agreement with the general theory [ 5 ] for the contribution of bosons to the hamiltonian (6) one finds from (9 )(11)
However, in the kink sector the construction of the scattering matrix requires also an explicit evaluation of the asymptotics of motion of the kink itself, because the S-matrix must connect the asymptotics both of the quantized component of the field and of the classical one. In the c.m.s. the classical component moves in the presence of radiation due to the recoil effect [3,8,12]. In the first O(M,’ ) approximation [ 8,121 one has ul(x, r)=u(x+d(t))+Wx,
Z),
where the shift d(t)
consists of additive
contributions
Ixdx{t~2+f~‘2+fv(X)~2+W+[-i(ya~+gBU(x)]yl)
of bosons and fermions (x,t)-_liJdxlv+ay(x.f)
>
.
(161,
As well as the asymptotics of the field the asymptotics of the bosonic contribution AB( t) to ( 16) includes additive discrete and continuous parts. Discrete modes lead to oscillations of the kink without changing the position average. In the continuous spectrum a calculation quite similar to that of upnUt,but rather lengthy, gives 49
Volume 134, number 1 d,(t);,"'=o-
PHYSICS LETTERS A
12 December 1988
lim &(t) r-*cc
=&(t)dwrete+
& j
dk
Iklt(k)I{a+(k),a(k)}+T
Iklr(k)(a+(-k),a(k))+l
0
dko(k){]t(k)(
+ &
[a+(k)a”,a(k)-@)&a+(k)]
.0 I
T(sgnk)r*(k)[a+(k)~~a(-k)-a(-k)~~a+((k)] +{a+(k),u(k)}+{8(~k)[X:(k)ac;ix+(k)-Y*_(k)~~Y-(k)] +e(Tk)[X*_(k)&x_(k)-Y*+(k)&Y+(k)]} +{u+(k),u(-k)},(e(~k)[X*,(k)~YY:(k)-Y*-(k)~~X*(k)] +e(Tk)[X*_(k)attY*_(k)-Y:(k)dux*,(k)]}}. After substituting A%t(t)=43(t)discrete
+$i
s
( 11) in this expression +(2M,)-’
(
(17) and some algebra on account of relations
t j kdk{u,+,““‘(k),
( 12 ) and ( 13 ) one finds
u:,“‘(k)}+
dko(k)[u~““t(k)~,&~‘,“‘(k)-u~~‘(k)~~ui’,oU’
(k)l).
(18)
In agreement with conservation laws it follows from ( 18)) that in the c.m.s. of the field the asymptotic velocity and position of the kink are equal to [ - (momentum of quantized component)/&] and [ - (boost of quantized component) /MO] [ 81. The difference in the asymptotics of the kink by transition from t = -co to t = + co is completely due to the dynamics of the quantized component - the operator, which transfers oii, into P”‘, transfers simultaneously din into Aout. Thus for the boson field a nontrivial scattering matrix exists in the kink sector already in the zero approximation in the coupling constant and describes elastic scattering of fundamental bosons on the kink. The oneparticle matrix elements of the S-matrix are defined by formula ( 13). Now let us turn to the fermion field. There also exists a nontrivial S-matrix in this case, which describes elastic scattering of fermions on the kink. But unlike the bosonic case, the asymptotic consideration of the fermion field requires a certain modification of the formalism. Let for definition guo > 0. The solution of (3b) is ~(x, t)=
T [b/e-‘““~j+‘(x)+d:e’““~f-)(x)1,
(19)
where [ -iar3/ax+gSu(x)]x)“(x)=
+ v&“(x),
v,>o,
(20)
and 6:) b, (d,? , d, ) are the creation, annihilation operators of an (anti-)fermion. For each 1 positive- and negative-frequency spinors x{’ ) and xl-) are connected with each other through the charge conjugation XI- ) = Crop{ + ) . In the representation (Y=G~, /3=0, the matrix Cy” reduces to u3, the upper and lower spinor components xl” and xj2’ are related by V/xj’) =
[-ax+gu(x)lXPr
wJ2’ = [a x +gutx> lxf” .
(21)
The eigenvalue spectrum of the Dirac operator (20) consists of a number of discrete modes with frequencies 0 < v,< mF=g%,, and of the continuous part with vi =p* + m $. The continuous spectrum is two-fold degenerate in the wave number p and $ =x.+ 50
Volume 134, number 1
PHYSICS LETTERS A
12 December 1988
It follows from (21), that if xF’(x)
-+ [X$!)(p)eiPx+ Y~)(p)e-iPx]/Jiii, x-.+ca
a=l,
2,
(22)
then Xy)(p)=
-tefidCP)Xy)(p),
Y$‘)(p)=
+e’id(p)Y$2)(p),
(23)
where tg h(p) =plm.
(24)
There is the direct consequence of the behaviour of the potential in the Dirac equation (20), that the scattering data for the upper and lower components differ from each other by a phase, of which the sign depends on the sign of the spatial infinity. Indeed, if, when x + - cu we have in (20) a normal mass term, then, when X-P +co this term changes sign. Such a representation of the anticommutation relations (4b) is inequivalent to the free massive Dirac field [ 13 1. In LSZ-language this fact manifests itself in the following way. If one seeks for the asymptotics of the continuous part of v/ in the form of a free field of mass mF, rl/,,(x, t)=
s
dp[Un)e-i”P’uP(x)
I,
+C(p)eiup’~,(x)
(25)
where
then a straightforward b,,(p) =w-
LSZ-procedure
would give
lim dx up’(x)e’“pl dq b,e-‘“~‘X~+) (x), J t-+03 J
and quite similar to ( IO), ( 11) one would obtain bp,“‘(p)=e(~p)[W+(p)b,+~+~-P)b_,l+e~TP)r~-(P)~p+~-~-P)~-,l~
(26)
where X+(p) = 1 [X!‘)(p)
+e-iacp)Xy)(p)
] =e-iacP’X$2’(p)
,
J?_(p)=j[Xl_‘)(p)+e-i”(P)X’_2’(p)]=-isind(p)X’_2’(p), F+(p)=~[Y~)(p)+e-i”(P)Y’:‘(p)]=cosS(p) P_(p)=f[Y!!)(p)+e-iL’p~Y~2’(p)]=0. -
Y(,2’(p), (27)
It follows from (27) however, that the coefficients f’+, f% obtained in such a way, have nothing to do with the scattering data, because they do not obey ( 12) and ( 13), and so (26) is not a unitary transformation. Nevertheless there are some reasons in the asymptotics of the form (25). Actually the effect of the nontrivial topology of the potential in the Dirac equation (20) results in the appearance of the zero fermionic modes and vacuum polarization [ 6,141 whereas the general features of the continuous spectrum remain unaffected. The problem is that due to nontrivial phase relations (23) there is no possibility to obtain the correct coefficients d?, P, for (26) directly from the asymptotics of ~(x, t). In this case let us turn to the asymptotic behaviour of the quantities quadratic in t&x, t), indeed to the fermionic contributions HF to the hamiltonian (6) and AF to the kink motion ( 16) due to recoil. Another argument for such an approach is that unlike ~(x, t) the operators HF and AF are observables. Therefore it seems preferable to take for the correct coefficients in (26) those, which (a) ensure the unitarity of transformation (26) and (b) diagonalize HF and dF as well as the transformation ( 11) diagonalizes the bosonic contribution (18). 51
Volume 134. number 1 Performing a calculation to X$9, Yy), one finds
From (12), It;“)
(13),
12 December 1988
PHYSICS LETTERSA analogous
to the bosonic
case on account
of relations
( 12 ) and ( 13 ) with respect
(23) one obtains
= Jti2’ 1)
r$‘)=ri2),
X~‘)a”,Xc,)-Y*+c1)a”Y~)=+2i[as(p)/ap]It~’)I+X=’2)a”,Xc,‘,-Y:c2)a”,Y!2), Xg 1)a" y;c 1) -Y;“)a”,X*,c’)=+2i[as(p)/ap]r~(‘)+X*,c2)a”,Y*,c2)-Y;(2)aC)X;(2). P
(29)
It follows from (29) that if = +e+iscp)‘2X’,2’(p),
X,(p)
P,(p)
= +e*is(p)‘2Y~)(p),
then these coefficients form a set of scattering applied in the opposite direction they give ~i;,l=Jt~lq=Jt~2q,
(30)
data, that is they obey relations
( 13 ) and by means of ( 12 1
?p:pr;l)=rj2),
and
(Xv%qpYY) _Y~(~'a",x*,c~')=Wya",8*,-9;a",~~.
f J,
As a result if one (b) in (28) all the Further operations the expression (26)
takes for w*+, 8, the expressions (30), then (a) the transformation (26) is unitary and expressions in terms of tilded coefficients. half-sums in a= 1, 2 reduce to corresponding are quite similar to the bosonic case. Indeed, the diagonalization substitution for (28) is for b,,(p) and a similar expression for d’s,
d~,U’(p)=8(fp)[~+(p)d(p)+~+(-p)d(-p)l+~(Tp)[~-(p)d(p)+~-(-p)d(-p)l. In terms of operators t
d,,,(t)=M~’
(I
(26 ) and (3 1) the continuous
Pdp[b~(p)b,,(p)-d,,(p)d,+,(p)l+i-i
(31)
part of (28 ) is equal to
s
dp v(p)[b,:(p)~b,,(p>-d,,(p)dudt(p)l
>
2
(32) and so the continuous H Far
=
I
dp
v(p)
part of the hamiltonian
tGtpPas(p)
Thus in terms of the operators 52
-&(pM&(p)
is (33)
I.
(26) and (3 1) the asymptotical
momentum-energy
and boost of the con-
Volume 134. number
1
PHYSICS
LETTERS
A
12 December
1988
tinuous part of the fermion field are the same as in the case of a free massive Dirac field. The correct coefficients R, , 8, are the “phase averages” between upper and lower spinor components. As well as in the bosonic case the states in the continuous spectrum are scattering states. Matrix elements of one-fermion scattering on the tilded coefficients. Discrete kink are defined by the formula ( 15 ) with X?, Y+ replaced by the corresponding modes are stationary and describe bound states kink + fermion. To conclude we have considered the general approach to the scattering of fundamental bosons and fermions on a topological kink in I+ 1 dimensions, based on the LSZ-analysis of asymptotical behaviour of the Heisenberg fields and of the kink motion due to recoil effects.
References [ I] R. Rajaraman,
Solitons and instantons (North-Holland, Amsterdam, 1982). Nucl. Phys. B 13 1 ( 1977) 459. [3] K. Sveshnikov, Teor. Mat. Fiz. 55 (1983) 361. [4] V. Tverskoj, Teor. Mat. Fiz. 68 (1986) 338; 70 (1987) 218. [ 51 J.D. Bjorken and SD. Drell, Relativistic quantum theory, Vol. 2. Relativistic quantum fields (McGraw-Hill, New York, 1965). [6] R. Jackiwand J.R. Schrieffer, Nucl. Phys. B 190 [FS3] (1981) 253; D.K. Campbell and A.R. Bishop, Nucl. Phys. B 200 [FS4] ( 1982) 297. [7] A.R. Bishop, D.K. Campbell, P. Kumar and S.E. Trullinger, eds., Springer series in solid state science, Vol. 69. Nonlinearity in condensed matter (Springer, Berlin, 1987). [S] K. Sveshnikov, Teor. Mat. Fiz. 74 (1988) 373. [9]E.Tomboulis,Phys.Rev.D 12 (1975) 1678; J.-L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D 12 ( 1975) 1038; N.-H. Christ and T.-D. Lee, Phys. Rev. D I2 ( 1975 ) 1606. [ lo] O.A. Khrustalev, A.V. Razumov and A.Y. Taranov, Nucl. Phys. B 172 ( 1980) 44. [ 111 R. Blankenbecler and D. Boyanovsky, Phys. Rev. D 3 1 ( 1985) 2089; R.J. Flesh and S.E. Trulllinger, J. Math. Phys. 28 (1987) 1619. [ 121 K. Sveshnikov, Phys. Lett. A 126 (1988) 507; Phys. Lett. B 203 (1988) 283. [ 131 H. Grosse and G. Kamer, Phys. Lett. B 172 (1986) 231; J. Math. Phys. 28 (1987) 371. [ 141 A.J. Niemi and G.W. Semenoff, Phys. Rep. 135 (1986) 99.
[ 210. Steinmann,
53
1
PHYSICS
LETTERS A
12 December
FERMIONIC AND BOSONIC SCATTERING PHASES ON A TOPOLOGICAL K.A. SVESHNIKOV
1988
KINK
’
Institute ofNuclear Physics, Moscow University, Moscow I 19899, USSR Received 6 September 1988; accepted Communicated by A.R. Bishop
for publication
3 October
1988
A general approach to the scattering of fundamental fermions and bosom on a topological kink excitation is considered. An asymptotic LSZ-analysis for secondary quantized Heisenberg fields and kink motion is developed and some new results for the phases of one-particle matrix elements for the elastic scattering on the kink are derived.
In the present Letter the scattering of fundamental particles of quantum fields on a classical excitation-like kink or soliton is considered. There are several equivalent approaches for the description of soliton sectors in QFT (for a review see ref. [ 1 ] ). The most preferable one for the scattering problem is the consideration of the Heisenberg fields [ 2-41. In such an approach the asymptotic states are constructed by means of the wellestablished LSZ-formalism [ 5 1. In this way we shall consider the scattering of fermions and bosons on a topological kink in a 1 + l-dimensional QFT-model, described by the lagrangian p=
WV) +
f (a,p)*-
@(id--gp)v.
(1)
The model ( 1) is the relativistic dynamical analogue of the statistical mechanical one of quasi-one-dimensional electron-phonon systems [ 6,7]. The boson field possesses a classical component
v1,(& t> =u
x-vt
(QTZ >
in the form of a single topological kink u( + 03 ) = + U* The spectrum of quantum excitations is defined from the linearized field equations and commutation lations in the vicinity of the classical component (2). In the center-of-mass system one has [ 3,8 ]
[o+v(x)lw& [ia-m(x)
f)=O,
v(x)= u” (u(x)),
re-
(3a)
lv(x, t) =O,
(3b)
[@(x, t), a,@(~‘, t>]_ =i[6(x_x’)-M,‘U’(x)U’(X’)],
(4a)
{vl(X, r>, w+ (X’, r>>+ =6(x-x’
(4b)
),
where MO= Jti u’ *(x) is the classical kink mass, and the rest of the commutators and anticommutators ishes. The quantum component @(x, t) of the boson field obeys a subsidiary condition
s
dx u’ (x)@(x,
Permanent USSR.
address:
t)
=o, Quantum
van-
(5)
Theory and High Energy Physics Division,
Physical
Department,
Moscow University,
Moscow
119899,
47
Volume
I
134, number
PHYSICS
LETTERS
12 December
A
which together with (4a) ensures the exclusion of the translational zero mode from the bosonic spectrum The total energy of the system in this approximation is equal to H=M,+
J
1988
[ 9, lo].
dx{+d,2++@‘2+$V(x)02+W+[-iiCY~,+g/3U(x)]~}.
(6)
The detailed analysis of eqs. ( 3 ) and associated spectral problems including construction of resolvents and Green functions for typical kink configurations may be found in refs. [ 6, I 11. Here we shall consider the general features of the LSZ-asymptotics of the Heisenberg fields. Let us begin with the boson field. This problem has already been considered in refs. [ 2-41, so now we only briefly dwell on the basic features and specifications of relations derived in ref. [ 3 1, which will be necessary for what follows. The solution to (3a) has the form @(x,t)=
7 (2wf)-“2[u/e-‘““~Xx)+af+e’wfl~~((X)],
where {@Xx)} is the orthonormal
(7)
set of eigenfunctions
of the spectral problem
[ -a*/ax*+V(X)]~XX)=0~~AX),
(8)
with the zero mode N u’ (x) excluded due to (5), so of> 0. a, a; a boson of frequency w/,
are the annihilation,
creation
operators
of
[Uf,Uf,]- =o.
[qr,aFl-=6/r,,
The spectrum of the operator (8) consists of a discrete part 0 CO, < mB = Jm, which includes nondegenerate real integrable eigenfunctions e,,(x), and of a continuous spectrum, each eigenvalue of which, ok= (P+m;)“*, is doubly degenerate in + k and @ = $_k. Owing to such features of the spectrum the LSZ-asymptotics of the field 0(x, t) consists of two parts. The discrete modes enter without any changes, the continuous spectrum forms the plane-wave part of the asymptotics, which is nothing but the free quantum field of mass ma, [up,‘(k) The asymptotic up,“t(k)=w-
LSZ-condition lim
i
s
exp( -iw,t+ikx)+h.c.].
(9)
[ 5 ] then gives
dx
The evaluation of this limit is well developed into the following form: if
[ 2,3 1. For our purposes it is convenient
to bring the final result
@k(X)x-t-tc.2 --t [xi(k)e*kx+Y+(k)e-ik”]/J211,
(10)
then u~,“‘(k)=8(+k)[X+(k)u(k)+Y+(-k)u(-k)]+8(Tk)[X_(k)u(k)+Y_(-k)u(-k)]. Now let us note that the normalization s
d-x @(x)$r,(x)
condition
=&
forbids the gk to be the ordinary totic behaviour 48
(11)
quantum-mechanical
scattering
wave functions&(x),
whichnbey
the asymp-
Volume 134, number
1
PHYSICS
Pk(X)x-r +-cc eikr+ r( k)e-‘k: since the scalar product
--t
Pk(X)
A
12 December
1988
t(k)eikx,
x--t+00
for these functions
s
LETTERS
reads
dx&(X)&(X)=2X[d(k-k’)+r(k)d(k+k’)].
-The orthonormal functions @k(x) obey the asymptotic behaviour of the form (lo), where the coeficients X, (k), Y* (k) are connected with the transition and reflection coefficients t(k) and r(k) in the following way,
x-(k)={f[l+lt(k)ll}“2, Y_(k)~e’~~(~){j[
I-
X+(k)=ei~(k){~[l+]t(k)]]}‘/‘,
It(k)I]}
Y+(k)=-e
where &(k)=arg t(k), d,(k) =arg r(k). X+(k), which may be dropped without loss of generality, X*,(k)=X,(-k),
iI6R(k)--6r(k)l{f
[
1_
it(k)
1 I}
112,
(12)
Y*(k) are defined by (12) up to a common and obey the following relations,
phase factor,
Y*,(k)=Y,(--k),
IX_(k)(2+IY_(k)12=IX+(k)12+lY+(k)12=1, X_(k)Y_(k)+X+(k)Y+(k)=X+(k)Y*_(k)+X-(k)Y*,(k)=O.
(13)
By means of ( 13) one may easily verify that the relation tinuous spectrum of bosonic excitations,
t~in(k>,~i+,(k’)l-=[~,“,,~~,(k’)land the transition CL
matrix between
( 11) defines a unitary
transformation
in the con-
=a(k-k’), in- and out-states
(14) (scattering
matrix)
is equal to
Iki,)=ra,,,(k’),a~(k)l-
=t(k)d(k-k’)+{@(k)r(k)-8(
-k)
exp[-2i&(k)]r(k)}
6(k+k’).
(15)
Relations ( 14) and ( 15) ensure the completeness and unitary equivalence of the in- and out-states of the boson field in the continuous spectrum. The coefficients in the r.h.s. of ( 15) are defined without any ambiguity and coincide up to a phase factor with the quantum-mechanical transition and reflection coefficients. In agreement with the general theory [ 5 ] for the contribution of bosons to the hamiltonian (6) one finds from (9 )(11)
However, in the kink sector the construction of the scattering matrix requires also an explicit evaluation of the asymptotics of motion of the kink itself, because the S-matrix must connect the asymptotics both of the quantized component of the field and of the classical one. In the c.m.s. the classical component moves in the presence of radiation due to the recoil effect [3,8,12]. In the first O(M,’ ) approximation [ 8,121 one has ul(x, r)=u(x+d(t))+Wx,
Z),
where the shift d(t)
consists of additive
contributions
Ixdx{t~2+f~‘2+fv(X)~2+W+[-i(ya~+gBU(x)]yl)
of bosons and fermions (x,t)-_liJdxlv+ay(x.f)
>
.
(161,
As well as the asymptotics of the field the asymptotics of the bosonic contribution AB( t) to ( 16) includes additive discrete and continuous parts. Discrete modes lead to oscillations of the kink without changing the position average. In the continuous spectrum a calculation quite similar to that of upnUt,but rather lengthy, gives 49
Volume 134, number 1 d,(t);,"'=o-
PHYSICS LETTERS A
12 December 1988
lim &(t) r-*cc
=&(t)dwrete+
& j
dk
Iklt(k)I{a+(k),a(k)}+T
Iklr(k)(a+(-k),a(k))+l
0
dko(k){]t(k)(
+ &
[a+(k)a”,a(k)-@)&a+(k)]
.0 I
T(sgnk)r*(k)[a+(k)~~a(-k)-a(-k)~~a+((k)] +{a+(k),u(k)}+{8(~k)[X:(k)ac;ix+(k)-Y*_(k)~~Y-(k)] +e(Tk)[X*_(k)&x_(k)-Y*+(k)&Y+(k)]} +{u+(k),u(-k)},(e(~k)[X*,(k)~YY:(k)-Y*-(k)~~X*(k)] +e(Tk)[X*_(k)attY*_(k)-Y:(k)dux*,(k)]}}. After substituting A%t(t)=43(t)discrete
+$i
s
( 11) in this expression +(2M,)-’
(
(17) and some algebra on account of relations
t j kdk{u,+,““‘(k),
( 12 ) and ( 13 ) one finds
u:,“‘(k)}+
dko(k)[u~““t(k)~,&~‘,“‘(k)-u~~‘(k)~~ui’,oU’
(k)l).
(18)
In agreement with conservation laws it follows from ( 18)) that in the c.m.s. of the field the asymptotic velocity and position of the kink are equal to [ - (momentum of quantized component)/&] and [ - (boost of quantized component) /MO] [ 81. The difference in the asymptotics of the kink by transition from t = -co to t = + co is completely due to the dynamics of the quantized component - the operator, which transfers oii, into P”‘, transfers simultaneously din into Aout. Thus for the boson field a nontrivial scattering matrix exists in the kink sector already in the zero approximation in the coupling constant and describes elastic scattering of fundamental bosons on the kink. The oneparticle matrix elements of the S-matrix are defined by formula ( 13). Now let us turn to the fermion field. There also exists a nontrivial S-matrix in this case, which describes elastic scattering of fermions on the kink. But unlike the bosonic case, the asymptotic consideration of the fermion field requires a certain modification of the formalism. Let for definition guo > 0. The solution of (3b) is ~(x, t)=
T [b/e-‘““~j+‘(x)+d:e’““~f-)(x)1,
(19)
where [ -iar3/ax+gSu(x)]x)“(x)=
+ v&“(x),
v,>o,
(20)
and 6:) b, (d,? , d, ) are the creation, annihilation operators of an (anti-)fermion. For each 1 positive- and negative-frequency spinors x{’ ) and xl-) are connected with each other through the charge conjugation XI- ) = Crop{ + ) . In the representation (Y=G~, /3=0, the matrix Cy” reduces to u3, the upper and lower spinor components xl” and xj2’ are related by V/xj’) =
[-ax+gu(x)lXPr
wJ2’ = [a x +gutx> lxf” .
(21)
The eigenvalue spectrum of the Dirac operator (20) consists of a number of discrete modes with frequencies 0 < v,< mF=g%,, and of the continuous part with vi =p* + m $. The continuous spectrum is two-fold degenerate in the wave number p and $ =x.+ 50
Volume 134, number 1
PHYSICS LETTERS A
12 December 1988
It follows from (21), that if xF’(x)
-+ [X$!)(p)eiPx+ Y~)(p)e-iPx]/Jiii, x-.+ca
a=l,
2,
(22)
then Xy)(p)=
-tefidCP)Xy)(p),
Y$‘)(p)=
+e’id(p)Y$2)(p),
(23)
where tg h(p) =plm.
(24)
There is the direct consequence of the behaviour of the potential in the Dirac equation (20), that the scattering data for the upper and lower components differ from each other by a phase, of which the sign depends on the sign of the spatial infinity. Indeed, if, when x + - cu we have in (20) a normal mass term, then, when X-P +co this term changes sign. Such a representation of the anticommutation relations (4b) is inequivalent to the free massive Dirac field [ 13 1. In LSZ-language this fact manifests itself in the following way. If one seeks for the asymptotics of the continuous part of v/ in the form of a free field of mass mF, rl/,,(x, t)=
s
dp[Un)e-i”P’uP(x)
I,
+C(p)eiup’~,(x)
(25)
where
then a straightforward b,,(p) =w-
LSZ-procedure
would give
lim dx up’(x)e’“pl dq b,e-‘“~‘X~+) (x), J t-+03 J
and quite similar to ( IO), ( 11) one would obtain bp,“‘(p)=e(~p)[W+(p)b,+~+~-P)b_,l+e~TP)r~-(P)~p+~-~-P)~-,l~
(26)
where X+(p) = 1 [X!‘)(p)
+e-iacp)Xy)(p)
] =e-iacP’X$2’(p)
,
J?_(p)=j[Xl_‘)(p)+e-i”(P)X’_2’(p)]=-isind(p)X’_2’(p), F+(p)=~[Y~)(p)+e-i”(P)Y’:‘(p)]=cosS(p) P_(p)=f[Y!!)(p)+e-iL’p~Y~2’(p)]=0. -
Y(,2’(p), (27)
It follows from (27) however, that the coefficients f’+, f% obtained in such a way, have nothing to do with the scattering data, because they do not obey ( 12) and ( 13), and so (26) is not a unitary transformation. Nevertheless there are some reasons in the asymptotics of the form (25). Actually the effect of the nontrivial topology of the potential in the Dirac equation (20) results in the appearance of the zero fermionic modes and vacuum polarization [ 6,141 whereas the general features of the continuous spectrum remain unaffected. The problem is that due to nontrivial phase relations (23) there is no possibility to obtain the correct coefficients d?, P, for (26) directly from the asymptotics of ~(x, t). In this case let us turn to the asymptotic behaviour of the quantities quadratic in t&x, t), indeed to the fermionic contributions HF to the hamiltonian (6) and AF to the kink motion ( 16) due to recoil. Another argument for such an approach is that unlike ~(x, t) the operators HF and AF are observables. Therefore it seems preferable to take for the correct coefficients in (26) those, which (a) ensure the unitarity of transformation (26) and (b) diagonalize HF and dF as well as the transformation ( 11) diagonalizes the bosonic contribution (18). 51
Volume 134. number 1 Performing a calculation to X$9, Yy), one finds
From (12), It;“)
(13),
12 December 1988
PHYSICS LETTERSA analogous
to the bosonic
case on account
of relations
( 12 ) and ( 13 ) with respect
(23) one obtains
= Jti2’ 1)
r$‘)=ri2),
X~‘)a”,Xc,)-Y*+c1)a”Y~)=+2i[as(p)/ap]It~’)I+X=’2)a”,Xc,‘,-Y:c2)a”,Y!2), Xg 1)a" y;c 1) -Y;“)a”,X*,c’)=+2i[as(p)/ap]r~(‘)+X*,c2)a”,Y*,c2)-Y;(2)aC)X;(2). P
(29)
It follows from (29) that if = +e+iscp)‘2X’,2’(p),
X,(p)
P,(p)
= +e*is(p)‘2Y~)(p),
then these coefficients form a set of scattering applied in the opposite direction they give ~i;,l=Jt~lq=Jt~2q,
(30)
data, that is they obey relations
( 13 ) and by means of ( 12 1
?p:pr;l)=rj2),
and
(Xv%qpYY) _Y~(~'a",x*,c~')=Wya",8*,-9;a",~~.
f J,
As a result if one (b) in (28) all the Further operations the expression (26)
takes for w*+, 8, the expressions (30), then (a) the transformation (26) is unitary and expressions in terms of tilded coefficients. half-sums in a= 1, 2 reduce to corresponding are quite similar to the bosonic case. Indeed, the diagonalization substitution for (28) is for b,,(p) and a similar expression for d’s,
d~,U’(p)=8(fp)[~+(p)d(p)+~+(-p)d(-p)l+~(Tp)[~-(p)d(p)+~-(-p)d(-p)l. In terms of operators t
d,,,(t)=M~’
(I
(26 ) and (3 1) the continuous
Pdp[b~(p)b,,(p)-d,,(p)d,+,(p)l+i-i
(31)
part of (28 ) is equal to
s
dp v(p)[b,:(p)~b,,(p>-d,,(p)dudt(p)l
>
2
(32) and so the continuous H Far
=
I
dp
v(p)
part of the hamiltonian
tGtpPas(p)
Thus in terms of the operators 52
-&(pM&(p)
is (33)
I.
(26) and (3 1) the asymptotical
momentum-energy
and boost of the con-
Volume 134. number
1
PHYSICS
LETTERS
A
12 December
1988
tinuous part of the fermion field are the same as in the case of a free massive Dirac field. The correct coefficients R, , 8, are the “phase averages” between upper and lower spinor components. As well as in the bosonic case the states in the continuous spectrum are scattering states. Matrix elements of one-fermion scattering on the tilded coefficients. Discrete kink are defined by the formula ( 15 ) with X?, Y+ replaced by the corresponding modes are stationary and describe bound states kink + fermion. To conclude we have considered the general approach to the scattering of fundamental bosons and fermions on a topological kink in I+ 1 dimensions, based on the LSZ-analysis of asymptotical behaviour of the Heisenberg fields and of the kink motion due to recoil effects.
References [ I] R. Rajaraman,
Solitons and instantons (North-Holland, Amsterdam, 1982). Nucl. Phys. B 13 1 ( 1977) 459. [3] K. Sveshnikov, Teor. Mat. Fiz. 55 (1983) 361. [4] V. Tverskoj, Teor. Mat. Fiz. 68 (1986) 338; 70 (1987) 218. [ 51 J.D. Bjorken and SD. Drell, Relativistic quantum theory, Vol. 2. Relativistic quantum fields (McGraw-Hill, New York, 1965). [6] R. Jackiwand J.R. Schrieffer, Nucl. Phys. B 190 [FS3] (1981) 253; D.K. Campbell and A.R. Bishop, Nucl. Phys. B 200 [FS4] ( 1982) 297. [7] A.R. Bishop, D.K. Campbell, P. Kumar and S.E. Trullinger, eds., Springer series in solid state science, Vol. 69. Nonlinearity in condensed matter (Springer, Berlin, 1987). [S] K. Sveshnikov, Teor. Mat. Fiz. 74 (1988) 373. [9]E.Tomboulis,Phys.Rev.D 12 (1975) 1678; J.-L. Gervais, A. Jevicki and B. Sakita, Phys. Rev. D 12 ( 1975) 1038; N.-H. Christ and T.-D. Lee, Phys. Rev. D I2 ( 1975 ) 1606. [ lo] O.A. Khrustalev, A.V. Razumov and A.Y. Taranov, Nucl. Phys. B 172 ( 1980) 44. [ 111 R. Blankenbecler and D. Boyanovsky, Phys. Rev. D 3 1 ( 1985) 2089; R.J. Flesh and S.E. Trulllinger, J. Math. Phys. 28 (1987) 1619. [ 121 K. Sveshnikov, Phys. Lett. A 126 (1988) 507; Phys. Lett. B 203 (1988) 283. [ 131 H. Grosse and G. Kamer, Phys. Lett. B 172 (1986) 231; J. Math. Phys. 28 (1987) 371. [ 141 A.J. Niemi and G.W. Semenoff, Phys. Rep. 135 (1986) 99.
[ 210. Steinmann,
53