Saturation of the quantum bogomolny bound for supersymmetric solitons in 1 + 1 dimensions

Nuclear Physics B278 (1986) 121-140 North-Holland, Amsterdam

S A T U R A T I O N OF T H E Q U A N T U M B O G O M O L N Y B O U N D FOR S U P E R S Y M M E T R I C S O L I T O N S IN 1 + 1 DIMENSIONS Akihiro UCHIYAMA Institute of Physics, Unwersitv of Tokyo, Komaba, Meguro-ku, Tol~vo153, Japan

Received 3 February 1986

We re-examine the consistency of the supersymmetry algebras in the soliton sector and the saturation of the quantum Bogomolny bound on the basis of the algebras. The quantization in the finite box is consistent with the supersymmetry algebras, but it causes the non-conservation of supercharges and the non-saturation of the quantum Bogomolny bound because of a boundary effect which has been missing in previous works. It is shown that when physical quantities are redefined by a smearing-out procedure the consistency of the algebras among physical quantities and the conservation of supercharges are recovered. As a consequence the quantum Bogomolny bound is saturated to O(h ).

1. Introduction It is well k n o w n that in supersymmetric theories the vacuum energy receives no q u a n t u m correction as a direct consequence of the supersymmetry algebras. In the soliton sector Witten and Olive [1] showed that the supersymmetry algebras include the central charges and that the Bogomolny inequality for the soliton mass was derived from the algebras. They also showed that in the classical level the B o g o m o l n y b o u n d was saturated, which corresponds to the vanishing of the v a c u u m energy. It is, therefore, interesting to examine whether the soliton mass receives no q u a n t u m corrections and whether the q u a n t u m Bogomolny b o u n d is saturated. Kaul and R a j a r a m a n [2] have argued that in general the supersymmetric soliton mass in 1 + 1 dimensions was subject to a correction in the q u a n t u m level when the soliton mass is rewritten with the renormalized parameters. In their one-loop calculation they a d o p t e d the periodic b o u n d a r y condition for fluctuations around the soliton backg r o u n d . However, it has been pointed out by Schonfeld [3] on the basis of the s u p e r s y m m e t r y algebras, that different b o u n d a r y conditions gave different results for the soliton mass; his mass formula included a b o u n d a r y condition dependent term. Unless one adopts proper b o u n d a r y conditions, the soliton mass becomes infinite. Recently, on the other hand, several authors [4 8] have argued that the q u a n t u m B o g o m o l n y b o u n d is saturated. Unfortunately, however, there seems to be no work in which the consistency of all supersymmetry algebras is checked under 0550-3213/86/$03.50~' Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

122

A. Ucho'ama / Quantum Bogomoh~vhound

certain boundary conditions. In a previous work [9] by the present author, through a detailed examination of the algebras in the finite box, the following facts are shown: (i) The periodic boundary condition (PBC) or the anti-periodic boundary condition (APBC) is necessary so that the supersymmetry algebras have no extra boundary terms except for the central charge. (ii) Supercharges are not conserved. (iii) The Bogomolny bound is not saturated in the quantum level. (iv) Owing to the non-conservation of supercharges the soliton mass receives an extra quantum correction, which is a boundary effect and cannot be neglected even in the limit that the size L of the finite box tends to infinity. Apparently these results are not consistent with previous works [2,4 8]. In sect. 2 we will review the calculation of ref. [9] and examine the differences between our results and previous ones. The previous calculations are divided into two types. In type-1 the soliton mass is calculated in the finite box with boundary conditions and in type-2 it is calculated in the open space. Refs. [2], [4], [7] and [8] are in type-l, and refs. [5] and [6] are in type-2. It is easy to understand that the differences between our results and previous ones of type-1 are reduced to a boundary effect which has been missing in previous works. On the other hand, the type-2 results cannot be directly compared with ours. In general, physics in the open space may be inequivalent to that in the finite box. For example, translational invariance holds in the open space, but it does not in the finite box. In addition, it is not clear how to quantize a model without introducing any volume cutoff. In sect. 3, therefore, we will propose a method to calculate the quantum correction of the soliton mass without any ambiguities. In this method a model is quantized in the finite box, which is necessary so that the quantization may be well-defined, while physical quantities are redefined by a smearing-out procedure. By this procedure an extra boundary effect of the finite box vanishes in the L ~ ~ limit and the model can be quantized as if it were directly quantized in the open space. Then it will be shown that the supersymmetry algebras among physical quantities are consistent and that the result which is derived from the modified algebras is consistent with the previous ones in the open space. The same result will also be derived by the direct calculation of the expectation value of the hamiltonian in the soliton background. By the same method as in sect. 3 we can also calculate the physical mass of the ordinary, namely, non-supersymmetric soliton. The result is not consistent with the previous one [10] in the finite box because of a similar boundary effect. This calculation will be discussed in appendix C.

2. Soliton mass calculation in the finite box

In this section we will examine the differences between the results in ref. [9] and the previous ones in the finite box. At first, the calculation in ref. [9] will be reviewed.

A. Ucho'ama / Quantum Bogomoh O' bound

123

In a sufficiently large but finite box whose size is L, we consider the following model [7]:

v2(do) +

- v'(do))¢],

where do is a real scalar field and q~ is a two-component Majorana field. In the quantum theory this model requires the following renormalization which is consistent with supersymmetry: V(do) -+ V~(do) - ~ h ( l + C)V~"(do) + O ( h 2 ) , I

I

f

i

,

V,¢) ~

d"Vr ~=~,) ddo"

(2~r) 2 --d2P p2 - (Vr) z

where *0 is a supersymmetric vacuum and C is a finite constant [2,6] which is determined when the renormalized potential V~ is written in terms of the pole masses of one-loop renormalized Green functions. Then

(vt) The supercurrent and the supercharges are J " = [~do+ iV(do)] T ~ , Q =_ fI./2 d x j 0 =

Qt)

1./2

Q2

"

If V ( , ) takes an appropriate form, which is typically V ( do ) =

-

,2 _

~-

or

V(d0)

=

-

sin do,

the model has a classical soliton solution do~(x), satisfying d d o J d x = V(do~) and d d o J d t = 0 . The classical bosonic soliton ground state will be denoted by Is). Under the transformation generated by Q1, Qll s) = 0 and the pair (Is)Q21s)), on the other hand, forms a superdoublet in the classical level. In the semi-classical perturbation theory with respect to h, the fluctuation ~/h~ and ~/h~ around the soliton do~ can be expanded as 1

7)(x,t)= ~

2 ~-~B ( a ,~)lm( x ) e a°~.,t + a ~rl* ( x ) e' .... t ) ,

+ i ( x , t ) = E f ~ (1c , + , i ( x ) e

-i~r.t

+ c ,t~ , i*( x ) ei~F.t) ,

i = 1,2 ,

A. Uchivama / Quantum BogomohO, bound

124

where TI.,, ~.a and ~.2 satisfy the following equations:

[-a~ + ">

[-~

2 vv2 + v,2]~o,=<~B.,~.,.

+ vv2 + v,~]~o, = ~,,+,,, ,

/i~.,i ~ dx :l.

[ - a ~ - vv~ + v,~]¢,,~ : ~,,~,._,

,

fl¢,,,i ~
V"=- dep"

,

V - V,( O~).

Commutation and anti-commutation relations are

[~.., <~1 = [a+.,, <,+.] = 0,

[<,.,, a~] =8 ..... (
[ ~ ( x ) , 7r(y)] = ih Y~.~Tm(x)~l*.,i(y) = i h S ( x - y ) , m

(~Pi(x), ~j(Y)} = hSij~_,t~i(x)~,* ( y ) = hSi.i6( x - y). n

Using these relations we obtain the supersymmetry algebras taking into account boundary terms, which in general do not coincide with Witten and Olive's ones [1] owing to the extra boundary terms. If we adopt

~%,( I L ) = ~7,.,(- ½L ),

Ol~Tm(½L ) = O?7.,(- ½L ),

~,,,(~L) = ¢,,,(- '.L).

+,,2(_'.L) = +,,2(- .~L)

(PBC),

or

II.,(IL) = -~Tm(- ~L). ~,,,('~L) _ = -¢,,1(-

01~7.,( l L

~L) ,

¢,,2('~L

= -a,..,(-'._L), = -¢,,2(-'._L)

(APBC).

we obtain the Witten and Olive algebras:

Q•= g +'r,

(1)

Q2=H-T,

(2)

(Q1,Qz} = 2 P ,

(3)

A. Uch(vama / QuantumBogomolnybound

125

where l _ f / J 2 d x [(c~0~)2+ (01(~)2-] - V2(~) H = s _ 1./2 --i~Y 1 091~2 -- ifz Ol+ 1 -- i~Yl~Y2Vt(q~) q- i~d2~dlVt((~)],

P = -'2 f " ~

2

dx [2 3o0310+ i+l 01'1 + iq~2 0,+21,

= - P I "/2 dx --

l./2

(4)

V(x) ~x

"

In the following, PBC will be adopted for definiteness. The same results hold for APBC. From the algebras (1) and (2) we obtain the following mass formulas in the rest frame, P Is) = O, M = = - + > - < s l r l s ),

(5)

M = ( s [ H l s ) = (slrls) + ( s l Q 2 l s ) .

(6)

(5) is the Bogomolny inequality. If Q l l s ) = 0, the Bogomolny bound is saturated. (siP(Is> and (slQ21s) can be calculated using the method of refs. [31 and [7]. But the method requires the following assumption: [H, Q,] = 0 ,

i=1,2.

(7)

In the finite box, however, a direct calculation shows that (7) is not correct: [H, Qil=(-1)'-'ih3/2[Vl~l+i]~4=O,

i=1,2,

and [~-, Q,] = --ih3/e[Vl~ki]B4:0,

i=1,2,

where

Hence ~- is not a "central" charge in the quantum level. Because (~j = (i/h)[H, Qi], Qi depends on time. The method of calculation for (sl QZls ) and (s IQ21s ) must be modified by taking into account (~i 4= 0: (sl Q21s) = (sl Q2( t - indeP)lS) + (sl Q2( t - dep)ls).

126

A. Uch(vama / Quantum Bogomo&vbound

Then, by the results of ref. [7] and some straightforward calculations:

1

(1)

0 1 (slQ2ls)=¼h~'.,,. WB,.(~Bm+O~F.)2J[Vlr)m~P'I]B]2 = hB=hV~ ~ - XC_ -~0",


n

(s['r]s) is obtained by a straightforward calculation [4-6]: 01~1~) =

-M~-hV°C.

The soliton mass M obtained from (5) and (6) is

(8)

M = Mc + h Vl°C + hB, M = M c + ½hE~oB, ,

lhZfOFn q- hVl°(I q- C).

-

Ol

(9)

tl

(9) coincides with the direct calculation of (slHls >. However, (8) is not consistent with the result of ref. [2] and Q1]s) 4: 0. This means that the Bogomolny bound is not saturated in the quantum level. The algebras (1) and (2), with (8) and (9), require that the following consistency condition should be satisfied:

½h~ . , % . , - ~h E '~F,, + hVl°I m

=

hB.

(10)

it

In ref. [9] it has been shown that this consistency condition is satisfied. Therefore the algebras (1) and (2) are consistent and the soliton mass M is

( 1),

M = i ~ + h VI°C + h V? ¼ -

(11)

and

Q1 Is) ~ 0, i.e. the soliton mass receives an extra quantum correction and the Bogomolny bound is not saturated in the quantum level. These results are not consistent with previous ones [2,4-8], namely that the soliton mass receives no extra quantum correction and that the Bogomolny bound is saturated even in the quantum level. We will examine this discrepancy. In this section, the results of type-1 will only be compared and those of type-2 will be considered in the next section. In refs. [2] and [4] the soliton mass is directly calculated by the use of the same equality as (9) with PBC. The consistency condition (10) can also be proven in the

127

A. Uch(vama / QuantumBogomo&vbound

L --* oo limit. In this case we must carefully examine all terms that do not vanish in the L --* oo limit [3, 10]. Then nl

II

=

h

lim

a-~

4~"

[adk (8 rr ) Jo

=

k/V1 °.

Because

_,h~,oB.,- ~_h~ 1

1

D1

~/k2 + (Vl°) 2

(O-Tr)~/k2+(V°)2]2-L~'dk

= ~lim ,~ ~ where tango

k

(aO

0 -

rr -

2Vl°/k for k >> 1

~dO - V k/ 2 +(17°) [ , (12) and

8 = 0 f o r k = O.

F,,

11

= li+m ~ -

A2+ (V?) 2 q-grV?- 2£'dk

-

=-hv?i+hvl o

~/k2q_(Vl°) 2

1

,-

Thus, (10) can be derived in the L -* oo limit and the soliton mass is given by (11). In ref. [2] the first term of (12) is missing, which corresponds to the boundary effect B. The result of ref. [4] that the Bogomolny bound is saturated even in the quantum level is derived from the result of ref. [2]. Because of the missing term the result of ref. [4] is not correct. In ref. [8] the soliton mass is calculated directly from with the following boundary condition:

(s]HIs)

,7,,(~L) = , 7 , , ( - ~ L ) ,

(e,-<)~°,('~L)=(e,-V,)~°,(-~L),

~nl(IL) =l~nl(-It),

tPn2({L) = l ~ n 2 ( - I L ) .

iOl

Under this boundary condition, however, the derivative operator in the boson part of the lagrangian is not hermitian and the equation of motion of the boson fluctuation ~1,, is

(-a?+

=.

v v ~ + Vf)nm + E ~,, [ ~,*, a , ~ . , ] . = ,~ o,~ o, , It

which is not equivalent to the field equation from the variational principle. It is not clear whether the model is consistently quantized. In ref. [8] this point is not taken into account. In ref. [7] the calculation is done under the assumption (7) which is not correct in the soliton sector, although the result of ref. [7] shows that (s I = 0 to all orders in h.

Q2(t-indep)ls)

A. Ucho'ama / Quantum Bogomohly bound

128

We have thus shown that the calculation in the finite box can be consistent with the algebras and have corrected previous works by paying proper attention to the boundary effect. 3. Soliton mass calculation in the open space

In sect. 2 the soliton mass has been calculated in the finite box in order to quantize the model without ambiguities. On the other hand, there are some works [5, 6] in the open space, whose results cannot be directly compared with our results in sect. 2. In this section the method of sect. 2 for calculating the soliton mass will be modified. We do not calculate the soliton mass directly in the open space because it is not clear how the quantization can be carried out in a well-defined manner without introducing any volume cutoff. We will propose a method in which the boundary terms do not play crucial roles for physical quantities at least to order h. We modify the rules of calculation in the following way: (i) The model is quantized in the finite box of size L which is very large but finite. (ii) Physical quantities are defined through a smearing procedure. Define a function f , O~

1

=

1 -~(l-d)<=x<=½(l-d),

1, 1

-5(x-

l

12t)+

~(I-d)
½(t+ d)<=x <=1

0,

where L >> l >> d >> 1. A following quantity O is defined as a physical quantity in the soliton background:

O(f") =- f,./2 (9(x)f"(x)dx, L/2

O~

lim lim

lim

(slO(f")ls),

where (_9 is an operator density. The limit operation is carried out in the order as exhibited. The exponent n is to be determined by a consistency requirement. (See the following.) Physical observations should not depend on the boundary shape at infinities. Therefore we first smear out observable quantities in a finite domain before taking the limit L ~ ~ . We then let the smearing-out region expand infinitely. In the following the word "physical" will be used in the sense of (ii). Except for (i) and (ii) the calculation will be done in the same way as in sect. 2.

A. Ucho'ama / Quantum Bogomolnybound

129

We first examine the supersymmetry algebras with the following redefined quantities:

2 dx[(-O,q$+ V(O))~+OoO+2lf, Ol(f) =-.,f L /L/2

Q~(i)- f"/~t./; dx [~o<~V,,- (a,++ v(+))i/. .

H ( f 2 ) =_ ~ fL/2 dx [(aoO) 2 + (Ol@) 2 + V2(gJ) 1./2 _i+l

T(f2

) __ __

alt~ 2 _ i~b20q~l _ iVt(O)t~l~2

fL/2 dx V(dQ)a i O / 2

+

iV,(O)~2~bi] f2

,

" L/2

2 x [2 c~0~010 + i@l 01~ 1 + i~b20qlt~2]/2 p ( f 2 ) - _ f,_L /L/2d

.

It is easy to see by a direct calculation that they satisfy

Q~(f ) = H(f 2) +'r(f2),

(13)

OT(f) = H ( f 2) - ' r ( / 2 ) ,

(14)

{Ql(f), Q2(/)} = 2P(/2),

(15)

at least to order h regardless of any boundary conditions. From the algebras (13) and (14) and (ii), the following mass formulae are obtained in the rest frame: lim lim lim (slH(f2)ls>

M-

d ~

oc

/---' o c

I. ~

2c

-- lim lim lim

d~oc /~oo l.~oc

(-(sl~-(S2)l s) +

(slQ~(f)ls)),

(16)

M -= lim lira lira (slH(f2)ls) d~oc [~oo l.~zc

= lim lim lira ((siT(if)Is) + (slQ~(f)ls}).

(17)

In sect. 2 supercharges are not conserved and this non-conservation induces the extra quantum correction to the soliton mass. Therefore the conservation of our redefined supercharges must also be checked. We must use the hamiltonian H defined by (4) because the model is quantized in the whole space of the finite box and because the correct equation of motion which is equivalent to the one derived by the variational principle can be obtained only if (4) is adopted as the generator of the time translation. Thus we obtain

0,(I)= ~[H,Q,(/)] = f i

[(-o,o+ v(<~))+,+Oo<~¢2] o,f,

~b2(i)= ~[/-/.e2(/)] = fi~;;2[Oo++,-(o,++ v(+))+2]

o,i.

(IS) (19)

A. Ucho'ama / Quantum BogomohO, bound

130

From(18) and(19) we can calculate (s] Q Z ( f ) ( t - d e p ) l s ) a s explained in appendix A. The results are

(slQ2(f)(t-

dep)ls )

L>> l>> d>> 1

-~

0,

for i = 1,2.

This shows that the time dependent terms of Q i ( f ) do not contribute to {slQ2i(f)[ s) in the limit of L >> l >> d >> 1. In other words the supercharges are effectively conserved. It is probable that the non-conservation of the supercharges in sect. 2 is an artifact of a sharp boundary of the finite box. Now we calculate the soliton mass. In the calculation of ( s l Q ~ ( f ) l s) it should be noted that the supercharges do not commute with the hamiltonian. We must take account of all energy states Is,)'s as the intermediate states of {s [Q2 ( f ) is):

(slQ2, ( f ) l s) = E l { s , , I Q , ( f ) l s ) l

2.

tl

The r.h.s, of the above equality is obtained by picking up all the poles at = ~, - ie, - ~, + ie of the following Green function:

-if

d, ( s l T [ Q~( f )( ,), Q~( f )( t') ]ls)e '~°u '')

Except for this comment the calculation of ( s ] Q ~ ( f ) l s ) can be carried out using the method in refs. [4] and [7]. In the limit of L >> l >> d >> 1 the results are the following

(slH(f2)Is)

= M ~ + ~ _'h E w . . , f d x

•.,•.,f, 2

~ ,2 - 5hY'~O~F, , f d x ¢~l~nlf

tH

+ hV°C,

/1

( s l t ( f Z)ls) = - M ~ - h V ° C , (slQ~(f)ls)=

~'* r 2 _ 5~h ~ o F , , f ~ dxq~,,t+.lf 5l~V'~o , , ~ B,,Jf d x ~ '~m"mJ , 2 m

(slQ

(f)ls}=

It

+ . hgo . . .B..f. d. x n , * r 2 -

dx +..d,.ff* 2 + 2hVOC.

ol

I1

Then, without any inconsistency with (16) and (17), we obtain the following soliton mass M M = lim lim lim ( s l H ( f 2 ) l s ) d ~

[~oc

L~oo

lim lim

lim

[M~+~,hE~.,.Idx_

nmnm/* 2 f

tl

+hr°c).

n 2

A. Uchivama / Quantum Bogomohlv bound

131

Using the first formula in appendix B

(20)

M = Mc + h Vl°C. We similarly obtain for lim a + o~ l i m / ~ o0 lira L ~ o~( S[ Q12( f ) [ s ): lim lim lim ( s l Q 1 2 ( f ) l s } = O . d---~

/ ~

(21)

L ~

(20) and (21) show that the physical soliton mass receives no extra quantum correction, and that the Bogomolny bound is physically saturated to order h. Now the result should be compared with the previous ones of type-2 [5, 6]. Our final results are consistent with them. However, some comments about ref. [6] must be made. In ref. [6] it is not clear why the form of boson fluctuations should be the same as that of fermion fluctuations without specifying any boundary conditions and whether the fermion fluctuations are properly normalized. If we admit that the boson fluctuation ~k of ref. [6] is properly normalized, the definition

uk(x) =

i

leads to the following equality:

f dxu~(x)uk,(x)= 2~rS(k

- k ' ) + - -1

2~k~ k,

I~Z( d--S'(q~s(X)))~k,]B, ~X

(22)

where the notations of ref. [6] are used. Since the boundary term does not vanish in general, we then obtain

After the renormalization

1

1

I

(~a//>so,r = ~ E ' ~ . k - ~_E'~Fk+ ~( ~Ss (' ' k

k

x ) ) £ ' I~k(x)[ k

09k

21 B

- ½(B + C ) [ S ' ( ~ P s ( X ) ) ] B . 1 ! It should be noted that the boundary term ~[S ( q ~ s ( x ) ) E !k ( I ~ k ( X ) I 2 / % ) ] B

comes

A. Uch(vama /

132

Quantum Bogomolny bound

from one of (22); if the fermion fluctuations are properly normalized.

(AH)~o,r = ½ECOBk-- 21 ~-,COFk-- ~_(B + C)[S'(cPs(X)) ] B" h

a

Without using (22), on the other hand, we have

(AT)[ol ~- f dx(A3-)~ol= ¼[S'(~s(X))~k ' [~k(x)12] °~k B-- X(B + C)[S'(~s(X))]B. Thus assuming that the fermion fluctuations are properly normalized and the Bogomolny bound is saturated: r r (A H)~o, = ~ E~OBk- 51 E~OFk- ~(B + C)[S'(qOs(X)) ] r~= (AT)~ol, h

k

we are led to

:g0, k

k

B

See ref. [5]. Thus, the assumption PB = OF cannot be consistent with the saturation of the Bogomolny bound without ensuring the vanishing of the boundary term. It seems that PB = OF is assumed in ref. [6] (see also ref. [12]) to obtain the (,~H)~ol = (AT)~ol. If PB = OF and the fermion fluctuations are properly normalized, we have (AH)~ol =

- ½(B + C)[S'(ePs(X)) ] ~,

(AT)sol r

¼[S'(cps(X))~, t

=

k

[~k(x)12] OJk

__ 1 ( B -~- C ) [ S t ( ~ s ( X ) ) ] B

"

B

Therefore the renormalization cannot be carried out and the Bogomolny bound is not saturated. If the fermion fluctuations are not properly normalized and 0B = OF, we have

,o,= r I Is '

I k(x)12 ] -~(B+C)[S'(~s(X))]B=(aT)[o,. k

¢0k

B

A. Uch~vama / Quantum Bogomohlv bound

133

It is not clear, however, whether the quantization can be well-defined in this case. Thus ref. [6] does not seem to be self-consistent.

4. Conclusions and discussions In sect. 2 we have shown the consistency of the supersymmetry algebras in the finite box and at the same time established that the soliton mass receives an extra quantum correction from a boundary effect which has been missing in the previous works. As a consequence the Bogomolny bound is not saturated in the quantum level. In sect. 3, on the other hand, we have shown the following: (i) The boundary effect in the finite box is induced by the sharp boundary. This effect does not seem to be physical because of its vanishing when physical quantities are redefined by a smearing-out procedure. (ii) The supersymmetry algebras among physical quantities seem to be justified regardless of boundary conditions. (iii) The physical supercharges are conserved. (iv) The physical soliton mass receives no extra quantum correction and the Bogomolny bound is physically saturated even in the quantum level. The result (iv) is consistent with the previous results in the open space. A field configuration corresponding to a soliton is not trivial at the space infinities. Because of this fact, the effect of the sharp boundary does not vanish even in the limit of an infinite volume. Therefore we have proposed a method for calculating physical quantities without suffering from boundaries in sect. 3. In fact we can also show that the ordinary, namely, non-supersymmetric soliton mass is not equivalent to one in the finite box which seems to receive a boundary effect if the calculation is done by the method in sect. 3. The detailed calculation will be given in appendix C. At present there seems to be no method in which the quantization in the soliton background is completely well-defined in the open space without paying any attention to the boundary effect. These problems will also be important in 3 + 1 dimensions in order to perform correct calculations. In this case we have a monopole solution which is a much more difficult object on which to carry out explicit calculations. The author would like to thank Professor T. Yoneya for valuable discussions and careful reading of the manuscript. He is also grateful to other members of the institute for helpful discussions.

Appendix A In this appendix we calculate (s I Q2(f)(t_dep) Is). In the following calculation we adopt PBC. By (17) and (18) in sect. 3 and some straightforward calculations

A. Uchivama / Quantum BogomohO, bound

134

(sl Q2(f)(t-dep)ls}'s are the following: ( si Q2( f )( t-dep)is ) 1

=hEmm"4OOBm(OaBmq-O~Fm)2

f

dx[i~%.,71m~.a+ {(01- V1)•m) +.a] alf

2

i 8~02m sin½d(k,,, + k.) =hm.nE 4a;l~m(0OBm+~0F.)2 L 2 A,, ~d(km+kn ) Xcos½{l(k m+ k.) + 3(kin)+ 3( k,,) + ~(0(km) + O( k.)) ) sin{d( km - k.)

×cos¼{O(km)-O(k,,)} +u., ~d(k.,-k,,) x cos~ { t ( ~ m - k.,) + a ( ~ )

- ~(k.,) + ~(O(k.,) - O ( < ) ) }

X cos4i { O(k,,,) + O(k.)} 2, 1

(s I QZ(f)(t-dep)ls) = h Y~ m,. 4~Bm(t~B~ + ~Fn) 2 x fdx[i
v,),l.,} e,,~]

Oil 2

1 8~0~m A sin½d(k.,+k,,) =h m,En40OBm(~Bm+WFn)2 L2 ~d(km+k,,) × sin~ { l(k., + k . ) + 3 ( k . , ) + 3(k,,) - ~_(O(k.,) + O(k,,))}

Xcos~{O(k,,,)-O(k,,)}+B,,

sin~d(k.,- k . ) ½d(km_k,,)

× sin½{l(k..- k . ) + 3(kin) - 3(k,,) - ½(O(km) - O(k,,))}

x oo~l{O(k~) + O(k.)} ~,

A. Uch(vama/ QuantumBogomoh(vbound

135

where kmL + 8 = 2m~r, k,,L + 8 = 2n~r or k , L + 8 + 0 = (2n + 1)~r,

3Vl°k

tan 128(k)

k

( V 1 0 ) 2 2k 2 ,

tanl~0(k) = _V_o , -

and A , and B, satisfy [An]2+ ]Bnl 2 = 1. Now the limit operation as in sect. 3 should be carried out in order to obtain the physical quantities from (s[Q2(f)(td e p ) l s ) ' s . In the limit of L--+ ae and l ~ oe

1 f~ dk f ~ dp

(V~')2

(slQZ(f)(t-dep)ls)--+ d J o~-2-~ - ~ 2~r %(wk 2p/d + wk) 2

× 1+

k 2+ ( k - 2 p / d ) 2 + ( V ° ) 2 sin2p OOkO)k_2p/d + k( k - 2 p / d )

- -p2 ,

i = 1,2.

The integrals are not divergent. Therefore in the limit of d ~ oc

(slQ2( f )( t-dep)ls) --, O.

Appendix B In this appendix we calculate (s [ H ( f to order h: =

2) IS } as

an example. H ( f

h , jo

ax

2) is the

following

j

+ '2hf '/2 dx [(aon): + n(-O~ + vv~ + v?)n L/2 -&b,( O1 + V1)t~2+ i(01 + Vl)~2~,]f 2 _ h~-l./2dx f,./2 rlOl'flfO'f + h i f "/2 1./2 dx ~ 2 ~ , f O l f

- ~ h ( I + C ) f 1"/2 dxVV2f 2. L/2 For definiteness we calculate the case whose potential V(dp)= - ~f~X-(~2 - / , 2 / X ) . The calculation of the sine-Gordon case will be similarly done. We will adopt PBC.

A. Uch(vama / Quantum BogomohO, bound

136

By straightforward calculations we obtain the following in the limit of L >> l

d>>l:

(~l f 1~/2 d x VZf2[s) =

M c,

1./2

(sl - 2?~ S"~) dx v,lia,il~> = o,

<.l'~hf" dx [(do,): + , ( - d e + vv: + re).7 -

L/2

-i+,(a, + v , ) ~ + i(o~ + Vl)~2~l] f~l~)

={_h~B.,f dxn.,n.,f2-~_h~v,,f #;~l

~,,l+,,,S*2

#l

(slh f r/2 dx ~ a,,1Sa,il~) = o, L/2

(slhi f 1/2 dx'4,2"4,lfO~fls)= -h L " r/2

(sl '~_h(l + C) f £ ) 2 d x

~o dk 2~r

~l.t

t/'k~ +

2/, 2

-

t,V('_t,

VV2i21sy = hV~°( l + C).

Therefore we obtain the following ( s l H ( f 2 ) l s ) ; 1 <.IH(I'>> = Mc + Jh Z ~ . . , f dx,m,*S' + .h Z ~.,,f dx < < 1 ,I

2

+ hvoc.

#l

i#1

N o w the following equations are useful:

~e,,2-

i {0 Fn

( a , - v~)+,,1,

fr/2 dx . : (kZ'+l)(d2'++)(I-ld)-(6~/2/Ix)(k~"+2) ~lmTlmf = r/2 (km2 + l)(k2, + + ) L - (6~/2/#x)(k 2, + 2) for

q-

~ , #k,,,L + 3 = 2m~r

137

A. Uch(vama / Quantum Bogomoh O"bound

(k2 + 1)(k2 + 4) (k 2 + 1)(k 2 + 4 ) L - ( 6 V ~ / / ~ ) ( k 2 + 2)

X[(l- ~d)

f"

L/2 - -

L/2

.

dx ~ n l ~ n l f

6v~ k2 + 2 ] I~ (k2+l)(k2+4) -M" '

[

2 =

for

t2+1,t2+4,

f~ l.tk,,L + 3 = 2nqr ,

]

(k 2 + 1)(k 2 + 4 ) L - ( 6 v ~ - / / ~ ) ( k 2 + 2 ) + (47'2-//-t)( k 2 + 1) k .2+ 2 -M. 6v~ , (k 2 + 1)(k,,2 + 4)

X ( 1 - 13d)

for

~fi s i n ~ -~dkn

Mn-

t~kn (~ t~dkn

× cosf~-2/~dk"

~22~tk.L+3+O=(2n+ l)Tr,

T 2 sin(~/½ # l k . + 3 ) + - -

bt2kZd

sin~-~/xdk. ) 8) ~221~dk, cos(~-/zlk, + ,

where tan½8 = 3kn/(2 - k 2) and tan½0 = 5k.. 1 Using these equalities

-

t-ld(

,

-L-

~hEWB'- 2hE¢°F" + hVl°I Ol

'

)

II

1) /~

G

where we use the identity (10). Then the soliton mass M is

M=

lim lim lim d ~

/~'~

L-~o~

=M~+hV°C.

M~+

¼-

+hV°C

138

A. Ucho'ama / Quantum Bogomoh O, bound

Appendix C In this appendix the ordinary soliton mass correction will be calculated by the method in sect. 3. The result is not equivalent to the previous one [10] because of a similar boundary effect as the one in the supersymmetric case. We consider a model described by the following lagrangian: <>~= 1 [ ( 0 0 ~ ) 2 ( 0 1 1 / i )

2 -- V 2 ( ~ ) ] ,

g>0, h>0.

This model has a classical soliton solution, namely, a kink solution: ~c(x) = ~ - t a n k ~ g x , where G satisfies the equations, dOc/dx = V(0c) and d 0 e / d t = 0. By using the same method as in sect. 3 we define the physical mass of this soliton. In the rest frame M = l i m ,~lim L--,~lim(
-

<0lH(f2)10)),

where Is) denotes a soliton background and 10) denotes a vacuum background, and

H(f2) = ½S_L~2[(O~oq$)2 + (0,q>)2 + V2(0)If 2 . The model is quantized in the finite box of the size L with PBC for the fluctuations. In the semi-classical perturbation theory with respect to h, ~/h-H around 0c can be expanded as

~(x,t)= ~ ~ (1a m ~ m ( x ) e

** iWmt '~' +a,dqm(x)e ),

where ~,~ satisfies the equations 2 , [-o2, + vv2 + v?lnm= mn

/2

2

L/2 ~/~ d x = l .

Commutation relations among am'S are assumed as usual:

A. Uch(vama / Quantum Bogomolnv bound

139

Then 1./2

£

.

2

(slH(f~)ls)= ff/2 V~f~dx + 2h~_.,o..f n..n,.f dx L/2

m

--3]-1 ~m ~

L/2

1 fL/2 L/2 dxOdlr~mfOlf.

In this calculation the following equalities which are derived by a straightforward calculation are useful:

,~,.~:~ dx ( k2 + 1)(k 2 + 4 ) ( / - ~d) - (6f2-//~)(k 2 + 2) L/2 = (k2 +1)(k2 +4)L-(6v/2/~)(k2m+ 2) for

~ l ~ k , , L + 3 = 2mvr,

0lllm~mfOlf d x = O, f L/2 * L/2 -

where tan½3 = 3kin/(2 - k~). Then we obtain

(sln(f2)ls)

L>> l>> d>> 1

---,

~

Me+ ½h 3 # + ~h~om m

1

6v~-

-~g.,

L

k~ + 2

/~ ( k a + l ) ( k 2 + 4 ) ~ ° m

__6v~-

l-ld +

3

±d

i2--~L'm /~

k2+2

(k2+l)(k2+4)

(1) ~ m + O ~-5 "

The second term is the contribution of a bound state. The remaining terms come from continuous spectra. The corresponding result for the vacuum sector

( 0 l n ( / 2 ) l 0)

L>>l>>d>>l 1

~

2h~,

t l-

~

~d

,

m

By taking the difference between (sill ( f 2)1s) and (0IH ( f 2)10) ' M ~ M c + ~1h ~ -3/ t - h -

v~-cr

t- d

o~ dk

L

_½h3v,~/~j. r~ 2 ~ ~ 4 ( k 2 + l )

k2t 2

140

A. Ucho'ama / Quantum Bogomohlv bound

In this m o d e l the r e n o r m a l i z a t i o n is required b y the following c o u n t e r t e r m :

3~

I-

- L/2

2

o~

dk

-4~rf

1

~ ~/~+ 2

T a k i n g a c c o u n t of this r e n o r m a l i z a t i o n we o b t a i n the r e n o r m a l i z e d soliton mass in the limit of L > > l > > d > > l 1 3 M = Mc + ~h~/Tg.

This result is n o t equivalent to the previous one [10]: F ~

M= gc +

3

I n o u r result there is not the last term of the previous one, which comes from the t e r m c o r r e s p o n d i n g to the b o u n d a r y effect in the s u p e r s y m m e t r i c case. T h e r e f o r e this e x t r a t e r m seems to b e i n d u c e d b y a sharp b o u n d a r y .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

E. Witten and D. Olive, Phys. Lett. 78B (1978) 97 R.K. Kaul and R. Rajaraman, Phys. Lett. 131B (1983) 357 J.F. Schonfled, Nucl. Phys. B161 (1979) 125 A. Chatterjee and P. Majumdar, Phys. Rev. D30 (1984) 844 C. Imbimbo and S. Mukhi, Nucl. Phys. B247 (1984) 471 H. Yamagishi, Phys. Lett. 147B (1984) 425 A. Uchiyama, Nucl. Phys. B244 (1984) 57 A. Chatterjee and P. Majumdar, SINP/TNP/85/5 A. Uchiyama, Prog. Theor. Phys. 75 (1986) 1214 R. Rajaraman, Solitons and instantons (North-Holland, Amsterdam, 1982) C. Callias, Comm. Math. Phys. 62 (1978) 213 H. Yamagishi, private communication