- Email: [email protected]
Complex saddle points in the double well oscillator
Nuclear Physics B 183 ( 1981 ) 251 - 268 © North-Holland Publishing C o m p a n y
C O M P L E X SADDLE P O I N T S IN T H E D O U B L E WELL O S C I L L A T O R J. L. R I C H A R D and A. R O U E T
Centre de Physique Thborique, CNRS, Marseille, France* Received 3 September 1980
Complex saddle points in the double well anharmonic oscillator are derived. The boundary conditions q o ( T ) = - c p ( - T ) = 1 at finite time T required for the computation of tunneling amplitudes lead to a countable set of saddle points. In the limit T ~ 0c, these saddle points behave as a superposition of instantons and anti-instantons and their action tends to the action associated with the quasi-solutions used in the standard procedure. Saddle points with periodic boundary conditions are also investigated.
1. Introduction
According to our present ideas, quantum chromodynamics can be more or less evaluated in the weak coupling limit (perturbative approximation) and in the strong coupling limit (lattice approximation). Some arguments seem to indicate that the transition between the two regimes could be rather fast [1,2] and might be evaluated using semiclassical techniques [2]. These techniques are, as yet, uncontrolled in field theory, even at a heuristic level. One- and two-dimensional models have been developed to test semiclassical methods which we hope are applicable to the four-dimensional case [3, 4]. They consist in a saddle point approximation. However, only real saddle points are usually taken into account, which may provide an incorrect asymptotic approximation. That could be the reason why quasi-solutions must be taken into account in the calculation of the tunneling amplitude in the double well oscillator to recover the WKB result [3]. Moreover, such a procedure seems difficult to apply to the Yang-Mills case and looks quite untrustable. That is the motivation to consider complex saddle points. In this paper, we look for the complex saddle points cp(t) in the double well oscillator (sect. 2). The boundary conditions cp(T) = --cp(- T) = 1 at finite time T, suited to the calculation of tunneling amplitudes, lead to a large class of saddle points (sect. 3). The limit T ~ oo is considered. The corresponding classical action is computed in sect. 4. In the limit T---~ o0, the value of the action for this class of solutions tends to the value of the action associated with the quasi-solutions in the * Postal address: C N R S - - L U M I N Y - - C a s e Cedex 2, France.
907, Centre de Physique Thborique, F-13288 Marseille
261
252
J.L. Richard, A. Rouet / Double well oscillator
standard procedure. Finally (sect. 5), we consider the constraints provided by periodic boundary conditions, such conditions being required in the computation of the partition function at finite temperature. The solutions are expressed in terms of Weierstrass elliptic functions. Appendix A summarizes some well-known properties of these functions. Further properties, related to their dependence on their periods, are proved in appendix B. 2. General solutions of the equation of motion
The euclidean lagrangian of the double well anharmonic oscillator can be written ~=½qS(t)2+½(cp(t) 2 - 1)2.
(1)
The corresponding equation of motion reads qb(t) = 2(q~(t) 2 - 1)qv(t).
(2)
This equation admits two trivial solutions: ~ ( t ) = + I.
(3)
Let us look for the other solutions. Eq. (2) can be integrated:
t+u=
~ f~o(,)
dq~
.o
j ( rp2 -- 1)2 + c
,
(4)
where c and u denote (complex) integration constants. For vanishing integration constant c, eqs. (4) yield the solutions cp(t) = ___th(t + u).
(5)
For c =/= 0, we are faced with the inversion of an elliptic integral. According to elliptic function theory [5], it is convenient to introduce a new variable q~(t) through
e p ( t ) = a 1-t q ) ( t ) + ~ - ½ a2 ,
(6)
where a denotes one of the four solutions of the equation (a 2 - 1)2 + c = 0.
(7)
253
J.L. Richard, A. Rouet / Double well oscillator
Eq. (4) can now be written + ,~(t)[4~ 3 -(~-c),~-~(,,-~)] ,+,--_f]
l/2dq~.
(8)
The inverse ~,(t) of this elliptic integral is known to be a Weierstrass elliptic function (see appendix A) characterized only by c and therefore independent of the choice of a in eq. (6). Hence the solutions of (4) for non-vanishing c are written a2
1
w(t) = a l + ~ ( t + ul,o,,o') + + - ~ u
2 '
(9)
where 6)(z 1¢o,~o') denotes the Weierstrass function with half-periods ~0,w' connected to c in a way explained below [eq. (13)]. All the (in general complex) solutions of eq. (2) are given by (3), (5), and (9). Let us consider the last ones in more details. Denote by +a,-+-b, (a 2 + b 2 ----2) the four solutions of eq. (7). Solutions (9) can then be written ~(t) = a
~ ( t + ulo~,o;) + k - ½ b 2
(10)
~(t + ul,o,,o') + ~ - ~ a 2 For the sake of comparison with other references, [6], we can write ¢p(t) using the jacobian elliptic function $, namely ~p(t)=bS[a(t+
u +½w + ~o')],
(11) (12)
k = b/a,
where k denotes the modulus of the jacobian S function. According to appendix A, the half-periods w and ~0' satisfy the identities @ ( w l w , o Y ) = e , - !3 ,
(13a)
o~(~ + ~,[w,6o, ) : e2 : _ ~+ ½ab,
(138)
9 ( ~'1 o~, w') : e 3 : -- ~ - ½ab.
(13c)
Hence, from (13) one can compute c : a2b 2 -- 1 in terms of the roots e i (i = 1,2,3). Then, from the homogeneity relations (A. 10) and the identity (A. 13), one gets 1 [ 62(111,½~") c----~[ ~
1] 2 - 1,
(14)
254
J.L. Richard, A. Rouet / Double well oscillator B'I lint
A
f
K -1~
0
1/2
Re'r
Fig. 1. The fundamental domain D F. with ~-= ~0'/to. It is a result of a p p e n d i x B that the c o r r e s p o n d e n c e b e t w e e n c a n d ~is o n e - t o - o n e as soon as the d o m a i n of "r is restricted to the f u n d a m e n t a l d o m a i n D F d e f i n e d b y the inequalities (see fig. 1)
12~-1]~1,
12~-+ 1{> 1,
Im,r > 0 ,
(15)
-½
a2 1
]
~p( t ) = a l + 6~( t + u{~, -7)-÷ ~ _ ½a2 ,
(16)
where a is o n e of the solutions of (7), are p a r a m e t r i z e d b y (i) ~" E D F, which d e t e r m i n e s the h a l f - p e r i o d s o~ a n d o~' through ~o2 -- 362(111,~-), t.d r ~--- T ~ ,
(ii) u, an a r b i t r a r y complex n u m b e r defined m o d u l o 2m~0, 2m'oJ.
(17a) (17b)
255
J. L. Richard, A. Rouet / Double well oscillator
3. Boundary conditions for tunneling amplitudes T o c o m p u t e the tunneling amplitude between the two classical vacua ± 1 in a finite time interval 2T, we must require as b o u n d a r y conditions for the F e y n m a n path integral ~ ( T ) = - - , p ( - T)---- 1.
(18)
If a saddle point method is to be used to approximate such a path integral, b o u n d a r y conditions (18) must be satisfied by the saddle points. In this section we look for the solutions of the equation of motion (2) satisfying the b o u n d a r y conditions (18). Let us first note that the solutions (3) and (5) cannot satisfy (18) for a finite T. Thus we are left with the solutions given in (16). 3.1. THE CONSTRAINT ¢p(T) = - c p ( - T) This constraint leads to the condition
~(T+
ulco,co') = 9 ( - - T +
u + col co,oy).
(19)
Such an equation admits two families of solutions: 2u = co + 2kco + 2k'co',
(20)
2 T = co + 2kco + 2k'co',
(21)
where k and k ' are arbitrary integers. Since u is defined modulo 2mco, 2m'co', eq. (20) admits four solutions u=½co, u = ½ co + co, u = ½ co + co', u = ½ co + co + co'. As noticed before, we can equivalently fix u = ½ co and consider separately the four roots ± a, ~ b in defining (6). The analysis of (21) is more involved: 2 T = (2k + 1)co + 2k'co',
(22)
which, plugged into (17a), leads to the equation
1
2k'
]2
)2
-
(23)
It is proved in appendix B that for any given T,k, k', this equation has a solution for ~- E D F only when [2k + 1[ > 12k' I. The solution is then unique. In the particular case k ' = 0 , one can check that ~- is pure imaginary when 2T>
V'½ ~rl2k + 1[, in which case the integration constant c is real in between - 1
and 0. When 2 T < (see appendix B).
~."~7r 12k + 1I, ~-= ½ + iy, y >½ and c is real and smaller than
1
256
J.L. Richard, A. Rouet / Double well oscillator
Let us remark that (17a) defines oa up to a sign. From now on this sign is assumed to be chosen in such a way that T [eq. (22)] is positive. Let us summarize these results. The constraint ~ ( T ) = - q ~ ( - - T ) yields two families of solutions. One family denoted as (I) consists in four solutions parametrized by r E DF: a 2 -- 1
(24)
% ( t ) = a i 1 + ~y(t+½,~{oa,~, )_~_=,,,~_, ,_2
'
where a, (i = 1,2 ..... 4) are the roots of (7). The other family given in (16) is parametrized by u, and two integers k , k ' such that 12k + 11 > 12k'l, r being defined by (23). This last family will be denoted as (II). 3.2. T H E C O N S T R A I N T
~0(T) = 1
This constraint implies ul,o,
0') = - ½ a 2 - a - { - .
(25)
For solutions (II), this is an equation for u which admits two solutions: if u is a solution, ~0 - u is the other solution according to eq. (22). Therefore, solutions (II) are parametrized by the integers k, k ' to which we associate two solutions depending on the value of u. For family (I), eq. (25) reads ° ~ ( T + ½oa]o0,~0') =
-
½a 2 -
a i -
~.
(26)
This is then an equation to be fulfilled by r. We can better use a consequence of (26). Using (A.12), one gets 6~(2T+ ~l¢0,oa,) = _ 2 ,
(27)
independently of the a i's. For a given T there exists one and only one value of r in D F solving that equation. In fact, let us took at the functions T ( r ) which associate a value for T (in general complex) to every $ E D F. If we note that T ( r ) = T( ?), the study of T ( r ) can be restricted to the domain OABB'O (fig. 1) defined by the straight lines R e r - 0, R e r =½ and the circle ] 2 7 - 1{ = 1. Then, T($) being defined as one of the solutions of (27), one looks at the behaviour of T ( r ) along the contour of this domain. It is a simple exercise to check that there is only one family of functions T ( r ) which can take real values (except zero). In fact these functions are real-valued only for r on the arc OA. Then eq. (27) has solutions only when T
1 1 - 2ix'
x>~½
(28)
J. L. Richard, A, Rouet / Double well oscillator
257
in which case co' is pure imaginary. Indeed, using (17) and (B.5) one gets coz -
--~-3@ ( l l l , } + i x ) , 2r 2
(29)
and co ' z =
~2co2 =
-
-~°~(111,½+
ix).
(30)
As x runs from ½ to + m, o0' runs from 0 to ½i~'. Then the integration constant c given in (14) is seen to be real positive and decreases from + 0o to 0 accordingly. Then, from (7) one gets a2 = 1 + N - g , b 2 = 1 - Nb-.
(31)
Then, ~(zl~o,~o' ) takes its real values on the straight lines R e z = m ( , o - ~ o ' ) and I m z = m%0', m and m' being arbitrary integers. 62(z[~o,co') = 2 can hold only when i l m z = -+-w' so that the constraint 62(2T+ wl~o~o')= - 2 defines T real m o d u l o ~--
CO'.
Using the relations
~(~o~1 ~,,~') = - ~ + ~ b 2, @(½~o+co']co,~o')= - - ~ + ½ a 2,
(32)
the solutions (24) can be written
q~(t) = __+a
~ ( , ) = _+b
e ( , + ~,o I,o,,o') - e(½~ + ~'1,o,,o') '
~ ( , + ~ot,o,,o,) - ~0,ot ~,,o')
(33)
The second ones are manifestly singular at t = 0. The first ones are also singular when T > co - co' which eliminates the indetermination we had on T. Finally let us remark that the plus or minus sign corresponds to the fact that, from our equation, T could be positive or negative. Thus, T being given, there is only one solution for r which provides us with a regular solution of the family (I), namely
ep(t) = a
¢ ( t + {-,o I,o,,o') + ~ - ~b: ~(t +,'_,ol,o,~o,) + 1 la2"
(34)
258
J.L. Richard, A. Rouet / Double well oscillator
Taking the complex conjugate just amounts to exchanging a and b and shifting ½ ~0 into ½ ~0 + o~'. But we have already noticed that these two operations c o m p e n s a t e each other. Therefore, this solution is proved to be real:
(35) Finally, one easily checks that this solution satisfies (36)
W(--t)=--w(t). 3.3. THE LIMIT T---, oo
For sake of comparison with the real quasi-solutions usually considered [3], let us study the behaviour of these solutions as T ~ ~ . Let us begin with the last ones (34). Since ~0 - ~0' > T and ~0' has a finite range, ~o tends to infinity with T and so does x. One then derives the following behaviour ~ 0 - - ~o' "~ 2 T ,
~o' ~---½i~r, a 2 ~-~ 1 + 8ie
2T + O ( e - - 4 r ) ,
.6~(t + ½60}00,~ ' ) '~½-- 4 i e - 2 ' - 2 T + O ( e - 4 r ) ,
(37)
cp(t) "" tht + O ( e - Z r ) ;
(38)
which leads to
that is, the usual instanton solution. Let us now discuss the other solutions. For given integers k, k', eq. (23) shows that as T---> ~ , • ~ 0. Defining r = e(cos0 + / s i n e )
(39)
as e ~ 0, we get e/7"2
.
6~(l]l.r)'-'-~ez (tcosO + sinO)2[1 + 24e -('~/E)(sin°+ic°s°)+ O ( e
2~'/~)]
(40)
f r o m which we deduce .
~o = ~--~e(sin 0 + icosO), ~o' = "r~o: ½irr.
(41)
259
d.L. Richard, A. Rouet / Double well oscillator
Since (2k + t + 2k'~')0~ has to be real, one has 2k' 2k +-----~e.
cos0 --
(42)
As before one then derives the behaviour of a 2 and @(t+u]~o,¢0'). Finally, the solutions can be written in the following way:
2k ~(t)~
p=0
2T - T + ( p + (--)P~(t;--T+P2k+I,
×th
W
t-T+(4k+l-2P)2k+l
+ O(e
2T
1) 2--~---~ )
~i~ ~n~ J
4k+12p 5-k7;]
k'
)]
2T/(2k+l)),
(43)
where
E (t; a;/3) =
f l,
iftE[a,fl[,
0,
(44)
otherwise.
In this formula, 7/= ~ 1 corresponds to the two possible solutions for u. Then one can directly check that the solution parametrized by (~/= - 1, k, - k ' ) is the complex conjugate of the solution (~/= + 1,k,k'). These solutions look similar to the quasi-solutions usually considered. However they do not contain the parameters associated to the position of the instantons and anti-instantons as in the quasi-solutions.
4.
The classical action
Let us now compute the classical action
+'t¢-it
(45)
qv(t) being given by (16), the classical action can be written ,
S=-~c/
rT
J--T
. f2a4(a
at~
t
2 - - 1) 2
---
I~4
II ~ 62(t+ ul~,w' ) + + - ½ a 2.
8a4(a 2 -
+
I~3
1)
+
4a2(3a: - 1) 8a 2 H2 q-~-+
] 11 ' (46)
J.L. Richard, A. Rouet / Double well oscillator
260
Integration can be performed using standard formulae [7], which leads to
S = 2T(16c +-~) + ~g(ZTl~,,~' ) .
(47)
~'(2Tl~0,w') is the Weierstrass' zeta function (see appendix A). Let us discuss the limiting case of the solutions (43). ~(2Tl~, w') then satisfies the equation
~'(2T I'~"~')-
(2k + t l + 2 k ' T ) 2 2k+ 1 +2k"rk, ' 2T g ( l l l ' r ) - iv 2T
(48)
according to the properties of the function ~"(A.16), (A. 17). Using arguments analogous to those which lead to (B.3), (B.4) we prove the identities oo
~(1]1,~-) = ~ r 2 ~- 21q72 E n =
~.(lll,~):
1 1 sin2 n qrr ' ~2
wr2 +iwr_k___ ~ 1 12,r 2 2~" 2"r2 n=! sin2(n~'/"r) '
(49)
which together with (14) allow one to compute (47). More precisely, we get the following limits C~
03~l
e - 4 T / ( 2 k + 1) q7 G ,
~r 2
~'(1[1,~-) ~--+i~-~T. 12~-2
(50)
Then, a simple computation shows that the action tends exponentially to the following limit S "~412k + 11 + O(e-4T/12k+'l),
(51)
which is the value one gets by the standard dilute-gas approximation. 4 is the value of the action for one instanton, ]2k+ lJ is the number of instantons and antiinstantons. 5. Periodical boundary conditions If we are interested in the partition function at finite temperature, the relevant
boundary conditions are qo(T) = q~(-- T).
(52)
J.L. Richard, A. Rouet / Double well oscillator
261
The trivial solutions (3) obviously satisfy this condition. Solution (5) cannot satisfy it. Let us discuss solution (9). The constraint (52) implies the equation
ulto,to') = 9 ( - v + ulto,
(53)
'),
which admits two families of solutions 2u = 2kto + 2k'to',
(54a)
2 T = 2kto + 2k'oa'.
(54b)
These equations are the analogue of (20a) and (20b), (2k + 1) being replaced by 2k. Their resolution proceeds in exactly the same way. We wish to thank R. Stora who pointed out the relevance of complex saddle points, and L. Matveev, E. Mourre and K. Yoshida for enlightening discussions.
Appendix A WEIERSTRASS FUNCTIONS A Weierstrass function ~ ( z Ito, to') is a doubly periodic meromorphic function of order two with double pole at z = 0 and principal part at this pole z-2, such that .62(z Ito, 0a') - z -2 is analytic in a neighbourhood of z = 0 and vanishes at z = 0. If we denote by 2to,2to' the periods of P(zlto, to'), then =_1_1 + ~,, ~(zlto'to') z2
1
1 __ 1 / ( z - - Zmto-- Zm'to')2 (2mo°+2m'to')2 ' l
(A.1)
where Y/denotes summation over all integers m, m' except for m = rn' = 0. Weierstrass functions solve the inversion problem for the elliptic integrals of the type
z=
4s 2 - g 2 s - g 3 ) - l / 2 d s ,
(A.2)
w=
(zlto,to'),
(A.3)
the relation between g2, g3 and to, to' being given by g2 = 60 ~ t (2rnto + 2re'to') -2
(A.4)
g3 = 140~,' (2into + 2re'to') -6
(a.5)
262
J.L. Richard, A. Rouet / Double well oscillator
It is convenient to define e, = @(6)16),to'),
e 2 = 62(6) + ¢o'16),w'), e 3 = 6-fl(w'Io~, to').
(A.6)
We shall only list some of the properties of these functions which we used extensively. First of all, we note the algebraic differential equation
[ eY'( z16),,o')]= = 4162( z16),6)') - e,] ×[@(zlw,6)')-e2][62(z16),6)')-e3],
(A.7)
which from (A.2) also reads
[@'(z16),6)')]z=4[@(z]6),6)')]3-gz@(Z]W,6)')-g3, connecting
(A.8)
g2,g3 to el,e2,e 3 and leading, in particular, to e I + e 2 + e 3 = 0.
(A.9)
F r o m (A. 1), a homogeneity relation follows:
~(Xz Ix6),x6)') = x-~62(z 16),6)').
(A.lO)
F r o m addition theorems it follows that
62(z + 6),16),6)')=ei +
( e , - ej)(e, -- e,) ~(zl6),6)')-
e,
where i,j, k is a permutation of 1,2, 3 and coi = w, 6)2 -~- 6) Jr- 6)', formula are also derived; a m o n g them we note
(A.11) '
103 = 6)'. Duplication
62tt(Z I 6), 6), ) ]2. 62(2zl6),6)' ) = --262(z16),6)' )-4- 262,(zl6),6),)
(A.12)
A last property has been used connecting the Weierstrass function 62(zl6), ~ w ' ) with 62( z 16), 6)'); namely
~(z16),~6)O=@(z[6),6)')+62(z-6)q6),6)O-62(6)q6),6)').
(A.13)
Finally, let us introduce the Weierstrass zeta function which is meromorphic with
263
J.L. Richard, A. Rouet / Double well oscillator
simple poles 1
= z-'
1
z -- 2 m w -- 2 m ' w '
z
l
2 m w + 2 m ' w ' q (2m¢0 + 2rn'to') 2 "
(A.14) As one can check, (A.15)
=
The Weierstrass zeta functions are not periodic but satisfy the following identities ~'(z + 2 m w + 2 m ' w ' I w,w') -- ~'(zlw,w' ) + 2m~'(w[w,~0') + 2m'~'(~0'] w, w'),
(A.16) (A.17)
assuming Im(¢0'/~0) > O.
Appendix B THE FUNCTION ~( 1]1,r) The dependence of the Weierstrass function on the argument z, with fixed periods, is under control. T o study its dependence on the half periods w, ¢o' it is thus enough to consider @(z ]~o, ¢o') for a given argument z, for instance z -----w, which because of the homogeneity property (A.10) a m o u n t s to consider the function ~(1]1,~'), r = w ' / ~ o . In this appendix we derive the properties of this function which have been used in the paper. B. 1. SERIES REPRESENTATION OF '~(11l,r) F r o m the definition (A.1) we get the equation
@(lll,r) = 1 + E t ~
1 L [1
=l+
~ ,,,4:o
~veo m
- 2m-
2nr
1 (1 - 2 m ) 2
]2
1} (2m + 2nr )2
1 (2m) 2
(1 - 2m - 2 n r ) 2
(2m + 2 n r ) 2 ]"
(B.1)
264
J. L. Richard, A. Rouet / Double well oscillator
We can perform the summation over m: the first two terms sum up into ~ ~r2. In order to sum up the last term
(_),. I
(m + 2nr) 2 we consider the contour integral
1 (f) dz
2~ri
1
7r
(z + 2n'r) 2 sinTrz
which obviously vanishes for an infinite contour ( ] z [ ~ oo). So does the sum of the residues ( - 1)m (m + 2n~') 2
2 cos2nrrr sin 2 2 n rrr
O.
(B.2)
We have thus proved the identity 6)(lll,r)={~r2_~r2
~]
cos2n~rr
.vao.sin2 2nrrr
(B.3)
Analogous techniques provide us with the equivalent formulae: ~r 2
6~(l[1,r) -
12r2 -
~(lll,r)=+
-
+7
COS(//'/7"/'/')
,~>o sin2( n~r/r ) '
~r2 +~r2 E ( - ) " 1 n>0 sin2 n ~'r ' -/72
9(lll,r )-
•17.2
12r2
7r2
1 ( - - ) " sin2(n~r/Zr) •
2'r2 n > 0
(B.4)
An immediate consequence of these relations is the identity: °~(lll,~) = -B.2. T H E F U N D A M E N T A L
1
lllll
2r 2 \
1 ' -~T ) "
(B.5)
DOMAIN
F r o m the expression (B.3) of 9(111,r) we deduce the identities
e(lll,-
) = @(Ill,,),
~ ( l [ 1 , r + p ) --- 9 ( l l l , r ) ,
V integer p,
(B.6)
J.L. Richard, A. Rouet / Double well oscillator
265
which allow one to restrict ~ to the d o m a i n D defined by Imz>0,
< Re~ ~<~.
(B.7)
Moreover the functions ~ ( z Io~, 60') are k n o w n to be invariant with respect to unimodular transformations ~(zlw,60' ) = ~(zl~2, f~') ¢~, 3 a , f l , y , 6
~2'
=
7
o~' '
integers such that
aS--fly=l.
(B.8)
If we require the additional constraint that a and 6 are odd and that 13 is even, it is a straightforward exercise to check the identity 6f(tol w,oJ ) = ~(~21 ~2, ~2') .
(B.9)
r can thus be defined up to a conformal m a p p i n g r'-- "y+Sr a +/3"r '
a and 6 odd, 13 even, a6 - ely = 1.
(B.10)
For any such • the identity
~(11l,~.) : ~ 2
(B.I1)
still holds. We are going to prove that such conformal transformations allow one to restrict the domain D to the fundamental domain D v C D defined by the supplementary constraints
1~-½1~½, 1~-+½l > ½ .
(B.12)
Let us consider • C D - D F and let us construct a conformal transformation m a p p i n g r on T' belonging to D v. We first state the following lemma: If ~- satisfies the following inequalities: Im'r>O,
,+~
1
< 181'
(B.13)
266
J.L. Richard, A. Rouet / Double well oscillator
,r+2a+l
>/
-V-
it is straightforward to prove that ~-'= (7 + inequalities:
1
2{/3{ '
8~)/(a + fir) satisfies the following
Imp-' > 0,
-
>l
l'
8-e 8+e ----fl-- ~< R e r ' ~ < - - f f - ,
e = sgn/3.
(B.14)
For any r ~ D - D F, 3a, fl,'y, 8 satisfying (B. 13) since the plane can be covered using /3 =- ___2", a = 2k + 1 (k = 0, 1. . . . . 2 n-2 - 1). Using iteratively the lemma we transform r E D - D F into ~-' lying outside and above a circle of center ½8 and radius ½.6 being odd, such a circle can be translated to the circle of center +__½ and radius ½. After this translation, r ' E D F. That D F is really fundamental is obvious from the following remark: r E D F lies outside any circle of center a//3 and radius 1//3, so that r ' = (~, + 8~')/(a + fir) lies inside the circle of radius 1//3 and center 8//3 which does not belong to D F. B.3, A B A S I C T H E O R E M
W e are now in a position to prove the following theorem. Theorem: The equation (a + fl~')2p(lll,r ) = R 2
(B.15)
( R 2 being some given positive real number) admits one and only one solution ~" C D v if {al > ]fl{, no solution otherwise. Let us first remark that changing ~- into - ~ and a into - a amounts to complex conjugate ( a +/3~-)162(1 ]1, ~-), so that we m a y restrict our analysis to D r A ( R e r > 0) =D + " In order to determine the n u m b e r of solutions of (B.15) we calculate 6 / & r ( ~ +/3r)2°~(111 ,'r) (~ +/3~)2°~(111 ,~) - R E along the b o u n d a r y O 2 A B B ' O 1 0 2 of D~-. ( I m O 1 ----ImO2 ---~0, I m B = ImB'---~
267
J.L. Richard, A. Rouet / Double well oscillator
+ ~ ) . This integral is equal to the number of times (a +/3~-)2~(111,~ ") winds R 2 along this path. Let us firstly evaluate .~(111, ~') along the circuit, using the identities (B.3), (B.4). A l o n g A B , • =½ + ix, x ~ [½, + ~ ] . ~ ( l [ 1 , r ) is real and increases from zero in A to ~ r z in B. Along BB', ~(lll,~- ) is constant equal to ~ Trz. Along B ' O 1, r = ix, x E [ + oo,0], c~(1]l,~-) is real and increases from ~ ct 2 in B' to + ~ i n O 1. Along 0102, r = ix + ~ x 2, ~ E [0, 1], x --~ 0 and ~(111,~" ) ~ -Tr 2/12"r 2 so that +
+
q7 2
+
X2
Along OeA, ~"= 1/(1 - 2ix), x ~ [ + ~ , ½] and
6f(l[l,~.) .
. 1 0 ~. ( l l l ., 1
1 )
-
2ix) 2~(111,
1 +
ix)
.
If/~ vanishes (which implies a = 1), the theorem is trivially satisfied: 6~(1]1, ~') is real positive only on ABB'O and increases from zero in A to ~ ~r2 in B, B', to + oo in O. For non-vanishing 13 we check that (a +/~-)2@(111, z): Vanishes in A. May be real positive once on O10 2. This occurs if and only if aft and a 2 + aft have opposite sign, which implies ]a I > Ifll, aft < O. Cannot be real positive anywhere else on the boundary. As a result the positive real axis is winded once if and only if ]a] > ] B ] , aft < 0. Since the case Re~ < 0 is obtained by changing a into - a , we have thus proved the theorem. Let us end this appendix by noticing that the curves on which ~- satisfies (ct + fl~r)26)(lil,'r) real positive are labelled by a / f t . For a / f l --~ ~ these curves tend to the half straight lines ABB'O (which is the curve for fl = 0, a = 1). For a / f l ~ 1, they tend to the arc of circle OA. All these curves start in O and end in A. On a given curve ( a + fl~-)2@(l[1,~') decreases from + ~ (in O) to zero (in A) (fig. 2).
Fig. 2. Curves for which (a + flr)2P(l[ l,'r) is real positive.
268
J.L. Richard, A. Rouet / Double well oscillator
References [1] K.G. Wilson, Mont6 Carlo calculations for the lattice gauge theory, Lab. Nucl. Studies Cornell University (1979) [2] C. Callan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717 [3] E. Gildener and A. Patrascioiu, Phys. Rev. D16 51977) 423; D. Olive, S. Sciuto and R.J. Crewther, Riv. Nuovo Cim. 8 (1979) [4] R, Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DI0 (1974) 4114, 4130, 4138: D l l (1975) 3424; B. Berg and M. Li~scher, Comm. Math. Phys. 69 (1979) 57 [5] A, Erdelyi ct al., Bateman manuscript project, Higher transcendental functions, vol. 2, (McGraw-Hill, 1953); J. Tannery and J. Molk, Elements de la theorie des fonctions elliptiques (Chelsea Pub. Co., Bronx, 1972; A.I. Markushevich, Theory of functions of a complex variable, (Prentice Hall, 1967). [6] B.J. Harrington, Phys. Rev. D18 (1978) 2982 [7] P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists (SpringerVerlag, 1971)
C O M P L E X SADDLE P O I N T S IN T H E D O U B L E WELL O S C I L L A T O R J. L. R I C H A R D and A. R O U E T
Centre de Physique Thborique, CNRS, Marseille, France* Received 3 September 1980
Complex saddle points in the double well anharmonic oscillator are derived. The boundary conditions q o ( T ) = - c p ( - T ) = 1 at finite time T required for the computation of tunneling amplitudes lead to a countable set of saddle points. In the limit T ~ 0c, these saddle points behave as a superposition of instantons and anti-instantons and their action tends to the action associated with the quasi-solutions used in the standard procedure. Saddle points with periodic boundary conditions are also investigated.
1. Introduction
According to our present ideas, quantum chromodynamics can be more or less evaluated in the weak coupling limit (perturbative approximation) and in the strong coupling limit (lattice approximation). Some arguments seem to indicate that the transition between the two regimes could be rather fast [1,2] and might be evaluated using semiclassical techniques [2]. These techniques are, as yet, uncontrolled in field theory, even at a heuristic level. One- and two-dimensional models have been developed to test semiclassical methods which we hope are applicable to the four-dimensional case [3, 4]. They consist in a saddle point approximation. However, only real saddle points are usually taken into account, which may provide an incorrect asymptotic approximation. That could be the reason why quasi-solutions must be taken into account in the calculation of the tunneling amplitude in the double well oscillator to recover the WKB result [3]. Moreover, such a procedure seems difficult to apply to the Yang-Mills case and looks quite untrustable. That is the motivation to consider complex saddle points. In this paper, we look for the complex saddle points cp(t) in the double well oscillator (sect. 2). The boundary conditions cp(T) = --cp(- T) = 1 at finite time T, suited to the calculation of tunneling amplitudes, lead to a large class of saddle points (sect. 3). The limit T ~ oo is considered. The corresponding classical action is computed in sect. 4. In the limit T---~ o0, the value of the action for this class of solutions tends to the value of the action associated with the quasi-solutions in the * Postal address: C N R S - - L U M I N Y - - C a s e Cedex 2, France.
907, Centre de Physique Thborique, F-13288 Marseille
261
252
J.L. Richard, A. Rouet / Double well oscillator
standard procedure. Finally (sect. 5), we consider the constraints provided by periodic boundary conditions, such conditions being required in the computation of the partition function at finite temperature. The solutions are expressed in terms of Weierstrass elliptic functions. Appendix A summarizes some well-known properties of these functions. Further properties, related to their dependence on their periods, are proved in appendix B. 2. General solutions of the equation of motion
The euclidean lagrangian of the double well anharmonic oscillator can be written ~=½qS(t)2+½(cp(t) 2 - 1)2.
(1)
The corresponding equation of motion reads qb(t) = 2(q~(t) 2 - 1)qv(t).
(2)
This equation admits two trivial solutions: ~ ( t ) = + I.
(3)
Let us look for the other solutions. Eq. (2) can be integrated:
t+u=
~ f~o(,)
dq~
.o
j ( rp2 -- 1)2 + c
,
(4)
where c and u denote (complex) integration constants. For vanishing integration constant c, eqs. (4) yield the solutions cp(t) = ___th(t + u).
(5)
For c =/= 0, we are faced with the inversion of an elliptic integral. According to elliptic function theory [5], it is convenient to introduce a new variable q~(t) through
e p ( t ) = a 1-t q ) ( t ) + ~ - ½ a2 ,
(6)
where a denotes one of the four solutions of the equation (a 2 - 1)2 + c = 0.
(7)
253
J.L. Richard, A. Rouet / Double well oscillator
Eq. (4) can now be written + ,~(t)[4~ 3 -(~-c),~-~(,,-~)] ,+,--_f]
l/2dq~.
(8)
The inverse ~,(t) of this elliptic integral is known to be a Weierstrass elliptic function (see appendix A) characterized only by c and therefore independent of the choice of a in eq. (6). Hence the solutions of (4) for non-vanishing c are written a2
1
w(t) = a l + ~ ( t + ul,o,,o') + + - ~ u
2 '
(9)
where 6)(z 1¢o,~o') denotes the Weierstrass function with half-periods ~0,w' connected to c in a way explained below [eq. (13)]. All the (in general complex) solutions of eq. (2) are given by (3), (5), and (9). Let us consider the last ones in more details. Denote by +a,-+-b, (a 2 + b 2 ----2) the four solutions of eq. (7). Solutions (9) can then be written ~(t) = a
~ ( t + ulo~,o;) + k - ½ b 2
(10)
~(t + ul,o,,o') + ~ - ~ a 2 For the sake of comparison with other references, [6], we can write ¢p(t) using the jacobian elliptic function $, namely ~p(t)=bS[a(t+
u +½w + ~o')],
(11) (12)
k = b/a,
where k denotes the modulus of the jacobian S function. According to appendix A, the half-periods w and ~0' satisfy the identities @ ( w l w , o Y ) = e , - !3 ,
(13a)
o~(~ + ~,[w,6o, ) : e2 : _ ~+ ½ab,
(138)
9 ( ~'1 o~, w') : e 3 : -- ~ - ½ab.
(13c)
Hence, from (13) one can compute c : a2b 2 -- 1 in terms of the roots e i (i = 1,2,3). Then, from the homogeneity relations (A. 10) and the identity (A. 13), one gets 1 [ 62(111,½~") c----~[ ~
1] 2 - 1,
(14)
254
J.L. Richard, A. Rouet / Double well oscillator B'I lint
A
f
K -1~
0
1/2
Re'r
Fig. 1. The fundamental domain D F. with ~-= ~0'/to. It is a result of a p p e n d i x B that the c o r r e s p o n d e n c e b e t w e e n c a n d ~is o n e - t o - o n e as soon as the d o m a i n of "r is restricted to the f u n d a m e n t a l d o m a i n D F d e f i n e d b y the inequalities (see fig. 1)
12~-1]~1,
12~-+ 1{> 1,
Im,r > 0 ,
(15)
-½
a2 1
]
~p( t ) = a l + 6~( t + u{~, -7)-÷ ~ _ ½a2 ,
(16)
where a is o n e of the solutions of (7), are p a r a m e t r i z e d b y (i) ~" E D F, which d e t e r m i n e s the h a l f - p e r i o d s o~ a n d o~' through ~o2 -- 362(111,~-), t.d r ~--- T ~ ,
(ii) u, an a r b i t r a r y complex n u m b e r defined m o d u l o 2m~0, 2m'oJ.
(17a) (17b)
255
J. L. Richard, A. Rouet / Double well oscillator
3. Boundary conditions for tunneling amplitudes T o c o m p u t e the tunneling amplitude between the two classical vacua ± 1 in a finite time interval 2T, we must require as b o u n d a r y conditions for the F e y n m a n path integral ~ ( T ) = - - , p ( - T)---- 1.
(18)
If a saddle point method is to be used to approximate such a path integral, b o u n d a r y conditions (18) must be satisfied by the saddle points. In this section we look for the solutions of the equation of motion (2) satisfying the b o u n d a r y conditions (18). Let us first note that the solutions (3) and (5) cannot satisfy (18) for a finite T. Thus we are left with the solutions given in (16). 3.1. THE CONSTRAINT ¢p(T) = - c p ( - T) This constraint leads to the condition
~(T+
ulco,co') = 9 ( - - T +
u + col co,oy).
(19)
Such an equation admits two families of solutions: 2u = co + 2kco + 2k'co',
(20)
2 T = co + 2kco + 2k'co',
(21)
where k and k ' are arbitrary integers. Since u is defined modulo 2mco, 2m'co', eq. (20) admits four solutions u=½co, u = ½ co + co, u = ½ co + co', u = ½ co + co + co'. As noticed before, we can equivalently fix u = ½ co and consider separately the four roots ± a, ~ b in defining (6). The analysis of (21) is more involved: 2 T = (2k + 1)co + 2k'co',
(22)
which, plugged into (17a), leads to the equation
1
2k'
]2
)2
-
(23)
It is proved in appendix B that for any given T,k, k', this equation has a solution for ~- E D F only when [2k + 1[ > 12k' I. The solution is then unique. In the particular case k ' = 0 , one can check that ~- is pure imaginary when 2T>
V'½ ~rl2k + 1[, in which case the integration constant c is real in between - 1
and 0. When 2 T < (see appendix B).
~."~7r 12k + 1I, ~-= ½ + iy, y >½ and c is real and smaller than
1
256
J.L. Richard, A. Rouet / Double well oscillator
Let us remark that (17a) defines oa up to a sign. From now on this sign is assumed to be chosen in such a way that T [eq. (22)] is positive. Let us summarize these results. The constraint ~ ( T ) = - q ~ ( - - T ) yields two families of solutions. One family denoted as (I) consists in four solutions parametrized by r E DF: a 2 -- 1
(24)
% ( t ) = a i 1 + ~y(t+½,~{oa,~, )_~_=,,,~_, ,_2
'
where a, (i = 1,2 ..... 4) are the roots of (7). The other family given in (16) is parametrized by u, and two integers k , k ' such that 12k + 11 > 12k'l, r being defined by (23). This last family will be denoted as (II). 3.2. T H E C O N S T R A I N T
~0(T) = 1
This constraint implies ul,o,
0') = - ½ a 2 - a - { - .
(25)
For solutions (II), this is an equation for u which admits two solutions: if u is a solution, ~0 - u is the other solution according to eq. (22). Therefore, solutions (II) are parametrized by the integers k, k ' to which we associate two solutions depending on the value of u. For family (I), eq. (25) reads ° ~ ( T + ½oa]o0,~0') =
-
½a 2 -
a i -
~.
(26)
This is then an equation to be fulfilled by r. We can better use a consequence of (26). Using (A.12), one gets 6~(2T+ ~l¢0,oa,) = _ 2 ,
(27)
independently of the a i's. For a given T there exists one and only one value of r in D F solving that equation. In fact, let us took at the functions T ( r ) which associate a value for T (in general complex) to every $ E D F. If we note that T ( r ) = T( ?), the study of T ( r ) can be restricted to the domain OABB'O (fig. 1) defined by the straight lines R e r - 0, R e r =½ and the circle ] 2 7 - 1{ = 1. Then, T($) being defined as one of the solutions of (27), one looks at the behaviour of T ( r ) along the contour of this domain. It is a simple exercise to check that there is only one family of functions T ( r ) which can take real values (except zero). In fact these functions are real-valued only for r on the arc OA. Then eq. (27) has solutions only when T
1 1 - 2ix'
x>~½
(28)
J. L. Richard, A, Rouet / Double well oscillator
257
in which case co' is pure imaginary. Indeed, using (17) and (B.5) one gets coz -
--~-3@ ( l l l , } + i x ) , 2r 2
(29)
and co ' z =
~2co2 =
-
-~°~(111,½+
ix).
(30)
As x runs from ½ to + m, o0' runs from 0 to ½i~'. Then the integration constant c given in (14) is seen to be real positive and decreases from + 0o to 0 accordingly. Then, from (7) one gets a2 = 1 + N - g , b 2 = 1 - Nb-.
(31)
Then, ~(zl~o,~o' ) takes its real values on the straight lines R e z = m ( , o - ~ o ' ) and I m z = m%0', m and m' being arbitrary integers. 62(z[~o,co') = 2 can hold only when i l m z = -+-w' so that the constraint 62(2T+ wl~o~o')= - 2 defines T real m o d u l o ~--
CO'.
Using the relations
~(~o~1 ~,,~') = - ~ + ~ b 2, @(½~o+co']co,~o')= - - ~ + ½ a 2,
(32)
the solutions (24) can be written
q~(t) = __+a
~ ( , ) = _+b
e ( , + ~,o I,o,,o') - e(½~ + ~'1,o,,o') '
~ ( , + ~ot,o,,o,) - ~0,ot ~,,o')
(33)
The second ones are manifestly singular at t = 0. The first ones are also singular when T > co - co' which eliminates the indetermination we had on T. Finally let us remark that the plus or minus sign corresponds to the fact that, from our equation, T could be positive or negative. Thus, T being given, there is only one solution for r which provides us with a regular solution of the family (I), namely
ep(t) = a
¢ ( t + {-,o I,o,,o') + ~ - ~b: ~(t +,'_,ol,o,~o,) + 1 la2"
(34)
258
J.L. Richard, A. Rouet / Double well oscillator
Taking the complex conjugate just amounts to exchanging a and b and shifting ½ ~0 into ½ ~0 + o~'. But we have already noticed that these two operations c o m p e n s a t e each other. Therefore, this solution is proved to be real:
(35) Finally, one easily checks that this solution satisfies (36)
W(--t)=--w(t). 3.3. THE LIMIT T---, oo
For sake of comparison with the real quasi-solutions usually considered [3], let us study the behaviour of these solutions as T ~ ~ . Let us begin with the last ones (34). Since ~0 - ~0' > T and ~0' has a finite range, ~o tends to infinity with T and so does x. One then derives the following behaviour ~ 0 - - ~o' "~ 2 T ,
~o' ~---½i~r, a 2 ~-~ 1 + 8ie
2T + O ( e - - 4 r ) ,
.6~(t + ½60}00,~ ' ) '~½-- 4 i e - 2 ' - 2 T + O ( e - 4 r ) ,
(37)
cp(t) "" tht + O ( e - Z r ) ;
(38)
which leads to
that is, the usual instanton solution. Let us now discuss the other solutions. For given integers k, k', eq. (23) shows that as T---> ~ , • ~ 0. Defining r = e(cos0 + / s i n e )
(39)
as e ~ 0, we get e/7"2
.
6~(l]l.r)'-'-~ez (tcosO + sinO)2[1 + 24e -('~/E)(sin°+ic°s°)+ O ( e
2~'/~)]
(40)
f r o m which we deduce .
~o = ~--~e(sin 0 + icosO), ~o' = "r~o: ½irr.
(41)
259
d.L. Richard, A. Rouet / Double well oscillator
Since (2k + t + 2k'~')0~ has to be real, one has 2k' 2k +-----~e.
cos0 --
(42)
As before one then derives the behaviour of a 2 and @(t+u]~o,¢0'). Finally, the solutions can be written in the following way:
2k ~(t)~
p=0
2T - T + ( p + (--)P~(t;--T+P2k+I,
×th
W
t-T+(4k+l-2P)2k+l
+ O(e
2T
1) 2--~---~ )
~i~ ~n~ J
4k+12p 5-k7;]
k'
)]
2T/(2k+l)),
(43)
where
E (t; a;/3) =
f l,
iftE[a,fl[,
0,
(44)
otherwise.
In this formula, 7/= ~ 1 corresponds to the two possible solutions for u. Then one can directly check that the solution parametrized by (~/= - 1, k, - k ' ) is the complex conjugate of the solution (~/= + 1,k,k'). These solutions look similar to the quasi-solutions usually considered. However they do not contain the parameters associated to the position of the instantons and anti-instantons as in the quasi-solutions.
4.
The classical action
Let us now compute the classical action
+'t¢-it
(45)
qv(t) being given by (16), the classical action can be written ,
S=-~c/
rT
J--T
. f2a4(a
at~
t
2 - - 1) 2
---
I~4
II ~ 62(t+ ul~,w' ) + + - ½ a 2.
8a4(a 2 -
+
I~3
1)
+
4a2(3a: - 1) 8a 2 H2 q-~-+
] 11 ' (46)
J.L. Richard, A. Rouet / Double well oscillator
260
Integration can be performed using standard formulae [7], which leads to
S = 2T(16c +-~) + ~g(ZTl~,,~' ) .
(47)
~'(2Tl~0,w') is the Weierstrass' zeta function (see appendix A). Let us discuss the limiting case of the solutions (43). ~(2Tl~, w') then satisfies the equation
~'(2T I'~"~')-
(2k + t l + 2 k ' T ) 2 2k+ 1 +2k"rk, ' 2T g ( l l l ' r ) - iv 2T
(48)
according to the properties of the function ~"(A.16), (A. 17). Using arguments analogous to those which lead to (B.3), (B.4) we prove the identities oo
~(1]1,~-) = ~ r 2 ~- 21q72 E n =
~.(lll,~):
1 1 sin2 n qrr ' ~2
wr2 +iwr_k___ ~ 1 12,r 2 2~" 2"r2 n=! sin2(n~'/"r) '
(49)
which together with (14) allow one to compute (47). More precisely, we get the following limits C~
03~l
e - 4 T / ( 2 k + 1) q7 G ,
~r 2
~'(1[1,~-) ~--+i~-~T. 12~-2
(50)
Then, a simple computation shows that the action tends exponentially to the following limit S "~412k + 11 + O(e-4T/12k+'l),
(51)
which is the value one gets by the standard dilute-gas approximation. 4 is the value of the action for one instanton, ]2k+ lJ is the number of instantons and antiinstantons. 5. Periodical boundary conditions If we are interested in the partition function at finite temperature, the relevant
boundary conditions are qo(T) = q~(-- T).
(52)
J.L. Richard, A. Rouet / Double well oscillator
261
The trivial solutions (3) obviously satisfy this condition. Solution (5) cannot satisfy it. Let us discuss solution (9). The constraint (52) implies the equation
ulto,to') = 9 ( - v + ulto,
(53)
'),
which admits two families of solutions 2u = 2kto + 2k'to',
(54a)
2 T = 2kto + 2k'oa'.
(54b)
These equations are the analogue of (20a) and (20b), (2k + 1) being replaced by 2k. Their resolution proceeds in exactly the same way. We wish to thank R. Stora who pointed out the relevance of complex saddle points, and L. Matveev, E. Mourre and K. Yoshida for enlightening discussions.
Appendix A WEIERSTRASS FUNCTIONS A Weierstrass function ~ ( z Ito, to') is a doubly periodic meromorphic function of order two with double pole at z = 0 and principal part at this pole z-2, such that .62(z Ito, 0a') - z -2 is analytic in a neighbourhood of z = 0 and vanishes at z = 0. If we denote by 2to,2to' the periods of P(zlto, to'), then =_1_1 + ~,, ~(zlto'to') z2
1
1 __ 1 / ( z - - Zmto-- Zm'to')2 (2mo°+2m'to')2 ' l
(A.1)
where Y/denotes summation over all integers m, m' except for m = rn' = 0. Weierstrass functions solve the inversion problem for the elliptic integrals of the type
z=
4s 2 - g 2 s - g 3 ) - l / 2 d s ,
(A.2)
w=
(zlto,to'),
(A.3)
the relation between g2, g3 and to, to' being given by g2 = 60 ~ t (2rnto + 2re'to') -2
(A.4)
g3 = 140~,' (2into + 2re'to') -6
(a.5)
262
J.L. Richard, A. Rouet / Double well oscillator
It is convenient to define e, = @(6)16),to'),
e 2 = 62(6) + ¢o'16),w'), e 3 = 6-fl(w'Io~, to').
(A.6)
We shall only list some of the properties of these functions which we used extensively. First of all, we note the algebraic differential equation
[ eY'( z16),,o')]= = 4162( z16),6)') - e,] ×[@(zlw,6)')-e2][62(z16),6)')-e3],
(A.7)
which from (A.2) also reads
[@'(z16),6)')]z=4[@(z]6),6)')]3-gz@(Z]W,6)')-g3, connecting
(A.8)
g2,g3 to el,e2,e 3 and leading, in particular, to e I + e 2 + e 3 = 0.
(A.9)
F r o m (A. 1), a homogeneity relation follows:
~(Xz Ix6),x6)') = x-~62(z 16),6)').
(A.lO)
F r o m addition theorems it follows that
62(z + 6),16),6)')=ei +
( e , - ej)(e, -- e,) ~(zl6),6)')-
e,
where i,j, k is a permutation of 1,2, 3 and coi = w, 6)2 -~- 6) Jr- 6)', formula are also derived; a m o n g them we note
(A.11) '
103 = 6)'. Duplication
62tt(Z I 6), 6), ) ]2. 62(2zl6),6)' ) = --262(z16),6)' )-4- 262,(zl6),6),)
(A.12)
A last property has been used connecting the Weierstrass function 62(zl6), ~ w ' ) with 62( z 16), 6)'); namely
~(z16),~6)O=@(z[6),6)')+62(z-6)q6),6)O-62(6)q6),6)').
(A.13)
Finally, let us introduce the Weierstrass zeta function which is meromorphic with
263
J.L. Richard, A. Rouet / Double well oscillator
simple poles 1
= z-'
1
z -- 2 m w -- 2 m ' w '
z
l
2 m w + 2 m ' w ' q (2m¢0 + 2rn'to') 2 "
(A.14) As one can check, (A.15)
=
The Weierstrass zeta functions are not periodic but satisfy the following identities ~'(z + 2 m w + 2 m ' w ' I w,w') -- ~'(zlw,w' ) + 2m~'(w[w,~0') + 2m'~'(~0'] w, w'),
(A.16) (A.17)
assuming Im(¢0'/~0) > O.
Appendix B THE FUNCTION ~( 1]1,r) The dependence of the Weierstrass function on the argument z, with fixed periods, is under control. T o study its dependence on the half periods w, ¢o' it is thus enough to consider @(z ]~o, ¢o') for a given argument z, for instance z -----w, which because of the homogeneity property (A.10) a m o u n t s to consider the function ~(1]1,~'), r = w ' / ~ o . In this appendix we derive the properties of this function which have been used in the paper. B. 1. SERIES REPRESENTATION OF '~(11l,r) F r o m the definition (A.1) we get the equation
@(lll,r) = 1 + E t ~
1 L [1
=l+
~ ,,,4:o
~veo m
- 2m-
2nr
1 (1 - 2 m ) 2
]2
1} (2m + 2nr )2
1 (2m) 2
(1 - 2m - 2 n r ) 2
(2m + 2 n r ) 2 ]"
(B.1)
264
J. L. Richard, A. Rouet / Double well oscillator
We can perform the summation over m: the first two terms sum up into ~ ~r2. In order to sum up the last term
(_),. I
(m + 2nr) 2 we consider the contour integral
1 (f) dz
2~ri
1
7r
(z + 2n'r) 2 sinTrz
which obviously vanishes for an infinite contour ( ] z [ ~ oo). So does the sum of the residues ( - 1)m (m + 2n~') 2
2 cos2nrrr sin 2 2 n rrr
O.
(B.2)
We have thus proved the identity 6)(lll,r)={~r2_~r2
~]
cos2n~rr
.vao.sin2 2nrrr
(B.3)
Analogous techniques provide us with the equivalent formulae: ~r 2
6~(l[1,r) -
12r2 -
~(lll,r)=+
-
+7
COS(//'/7"/'/')
,~>o sin2( n~r/r ) '
~r2 +~r2 E ( - ) " 1 n>0 sin2 n ~'r ' -/72
9(lll,r )-
•17.2
12r2
7r2
1 ( - - ) " sin2(n~r/Zr) •
2'r2 n > 0
(B.4)
An immediate consequence of these relations is the identity: °~(lll,~) = -B.2. T H E F U N D A M E N T A L
1
lllll
2r 2 \
1 ' -~T ) "
(B.5)
DOMAIN
F r o m the expression (B.3) of 9(111,r) we deduce the identities
e(lll,-
) = @(Ill,,),
~ ( l [ 1 , r + p ) --- 9 ( l l l , r ) ,
V integer p,
(B.6)
J.L. Richard, A. Rouet / Double well oscillator
265
which allow one to restrict ~ to the d o m a i n D defined by Imz>0,
< Re~ ~<~.
(B.7)
Moreover the functions ~ ( z Io~, 60') are k n o w n to be invariant with respect to unimodular transformations ~(zlw,60' ) = ~(zl~2, f~') ¢~, 3 a , f l , y , 6
~2'
=
7
o~' '
integers such that
aS--fly=l.
(B.8)
If we require the additional constraint that a and 6 are odd and that 13 is even, it is a straightforward exercise to check the identity 6f(tol w,oJ ) = ~(~21 ~2, ~2') .
(B.9)
r can thus be defined up to a conformal m a p p i n g r'-- "y+Sr a +/3"r '
a and 6 odd, 13 even, a6 - ely = 1.
(B.10)
For any such • the identity
~(11l,~.) : ~ 2
(B.I1)
still holds. We are going to prove that such conformal transformations allow one to restrict the domain D to the fundamental domain D v C D defined by the supplementary constraints
1~-½1~½, 1~-+½l > ½ .
(B.12)
Let us consider • C D - D F and let us construct a conformal transformation m a p p i n g r on T' belonging to D v. We first state the following lemma: If ~- satisfies the following inequalities: Im'r>O,
,+~
1
< 181'
(B.13)
266
J.L. Richard, A. Rouet / Double well oscillator
,r+2a+l
>/
-V-
it is straightforward to prove that ~-'= (7 + inequalities:
1
2{/3{ '
8~)/(a + fir) satisfies the following
Imp-' > 0,
-
>l
l'
8-e 8+e ----fl-- ~< R e r ' ~ < - - f f - ,
e = sgn/3.
(B.14)
For any r ~ D - D F, 3a, fl,'y, 8 satisfying (B. 13) since the plane can be covered using /3 =- ___2", a = 2k + 1 (k = 0, 1. . . . . 2 n-2 - 1). Using iteratively the lemma we transform r E D - D F into ~-' lying outside and above a circle of center ½8 and radius ½.6 being odd, such a circle can be translated to the circle of center +__½ and radius ½. After this translation, r ' E D F. That D F is really fundamental is obvious from the following remark: r E D F lies outside any circle of center a//3 and radius 1//3, so that r ' = (~, + 8~')/(a + fir) lies inside the circle of radius 1//3 and center 8//3 which does not belong to D F. B.3, A B A S I C T H E O R E M
W e are now in a position to prove the following theorem. Theorem: The equation (a + fl~')2p(lll,r ) = R 2
(B.15)
( R 2 being some given positive real number) admits one and only one solution ~" C D v if {al > ]fl{, no solution otherwise. Let us first remark that changing ~- into - ~ and a into - a amounts to complex conjugate ( a +/3~-)162(1 ]1, ~-), so that we m a y restrict our analysis to D r A ( R e r > 0) =D + " In order to determine the n u m b e r of solutions of (B.15) we calculate 6 / & r ( ~ +/3r)2°~(111 ,'r) (~ +/3~)2°~(111 ,~) - R E along the b o u n d a r y O 2 A B B ' O 1 0 2 of D~-. ( I m O 1 ----ImO2 ---~0, I m B = ImB'---~
267
J.L. Richard, A. Rouet / Double well oscillator
+ ~ ) . This integral is equal to the number of times (a +/3~-)2~(111,~ ") winds R 2 along this path. Let us firstly evaluate .~(111, ~') along the circuit, using the identities (B.3), (B.4). A l o n g A B , • =½ + ix, x ~ [½, + ~ ] . ~ ( l [ 1 , r ) is real and increases from zero in A to ~ r z in B. Along BB', ~(lll,~- ) is constant equal to ~ Trz. Along B ' O 1, r = ix, x E [ + oo,0], c~(1]l,~-) is real and increases from ~ ct 2 in B' to + ~ i n O 1. Along 0102, r = ix + ~ x 2, ~ E [0, 1], x --~ 0 and ~(111,~" ) ~ -Tr 2/12"r 2 so that +
+
q7 2
+
X2
Along OeA, ~"= 1/(1 - 2ix), x ~ [ + ~ , ½] and
6f(l[l,~.) .
. 1 0 ~. ( l l l ., 1
1 )
-
2ix) 2~(111,
1 +
ix)
.
If/~ vanishes (which implies a = 1), the theorem is trivially satisfied: 6~(1]1, ~') is real positive only on ABB'O and increases from zero in A to ~ ~r2 in B, B', to + oo in O. For non-vanishing 13 we check that (a +/~-)2@(111, z): Vanishes in A. May be real positive once on O10 2. This occurs if and only if aft and a 2 + aft have opposite sign, which implies ]a I > Ifll, aft < O. Cannot be real positive anywhere else on the boundary. As a result the positive real axis is winded once if and only if ]a] > ] B ] , aft < 0. Since the case Re~ < 0 is obtained by changing a into - a , we have thus proved the theorem. Let us end this appendix by noticing that the curves on which ~- satisfies (ct + fl~r)26)(lil,'r) real positive are labelled by a / f t . For a / f l --~ ~ these curves tend to the half straight lines ABB'O (which is the curve for fl = 0, a = 1). For a / f l ~ 1, they tend to the arc of circle OA. All these curves start in O and end in A. On a given curve ( a + fl~-)2@(l[1,~') decreases from + ~ (in O) to zero (in A) (fig. 2).
Fig. 2. Curves for which (a + flr)2P(l[ l,'r) is real positive.
268
J.L. Richard, A. Rouet / Double well oscillator
References [1] K.G. Wilson, Mont6 Carlo calculations for the lattice gauge theory, Lab. Nucl. Studies Cornell University (1979) [2] C. Callan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717 [3] E. Gildener and A. Patrascioiu, Phys. Rev. D16 51977) 423; D. Olive, S. Sciuto and R.J. Crewther, Riv. Nuovo Cim. 8 (1979) [4] R, Dashen, B. Hasslacher and A. Neveu, Phys. Rev. DI0 (1974) 4114, 4130, 4138: D l l (1975) 3424; B. Berg and M. Li~scher, Comm. Math. Phys. 69 (1979) 57 [5] A, Erdelyi ct al., Bateman manuscript project, Higher transcendental functions, vol. 2, (McGraw-Hill, 1953); J. Tannery and J. Molk, Elements de la theorie des fonctions elliptiques (Chelsea Pub. Co., Bronx, 1972; A.I. Markushevich, Theory of functions of a complex variable, (Prentice Hall, 1967). [6] B.J. Harrington, Phys. Rev. D18 (1978) 2982 [7] P.F. Byrd and M.D. Friedman, Handbook of elliptic integrals for engineers and scientists (SpringerVerlag, 1971)