The temperature dependent effective potential in Kaluza-Klein theory

ANNALS

OF PHYSICS

161, 121-151

(1985)

The Temperature Dependent Effective in Kaiuza-Klein Theory DANNY School

BIRMINGHAM* of Muthematics,

AND SIDDHARTHA Trinil!

College,

Dublin,

Potential

SEN Ireland

Received April 25, 19B4

The temperature dependent part of the effective potential to order T’Ifor the Kaluza-Klein theory is calculated for several examples using quantum mechanical rather than tield theoretic methods. 0 198s Academx Press, lnc

1. INTR~~~CTI~N

Quantum effects have recently been studied in the five-dimensional Kaluza-Klein theory. The one-loop effective potential as a function of the distance, L = 2&( g55)“2 around the compact tifth dimension has been computed both at zero temperature [l, 4, 51 and at finite temperature [2, 31. Appelquist and Chodos [ 1 ] computed the effective potential at zero temperature using path integral methods. They found the potential dropped to - ao as L + 0 indicating a collapse of the fifth dimension. They interpreted this result as a Casimir-induced collapse [6]. Subsequently, Rohrlich [5] showed that the same result is obtained when the vacuum energy density is computed using straightforward quantum mechanical methods. This is as one would expect since the effective potential is exactly the zero-point energy density. Rubin and Roth [2, 31 calculated the effective potential at finite temperature using the zeta function regularisation method. They found the effective potential had minima at L = 0 and L = co, which indicated a two-mode instability of the Kaluza-Klein vacuum. Their calculation was performed for both boson and fermion fields. Here we shall compute the effective potential at finite temperature for boson and fermion fields using only quantum mechanics. As a check on the efficiency of the calculational aspects of this approach we have repeated the calculations of Rubin and Roth [2,3] and Appelquist Chodos and Myers [4-J. We then proceed to five new calculations. We consider massless fermions, massive fermions and massive * Supported by Department of Education Postgraduate Award.

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bosons in a Kaluza-Klein theory in 4+ ti dimensions with the structure fl!t4xs’x ... x S’. Finally we consider massless twisted boson and fermion fields in the five-dimensional theory.

2.

THE

METHOD

Let us assume that space-time has the Kaluza-Klein structure AI4 x ,S’, with metric gPE,= diag( - 1, 1, 1, 1, gx5). We shall take gS5= 1. The circumference of the compact lifth dimension is L = 2nR(gz5) ‘/2 = 2rcR, where the tifth coordinate x5 runs from 0 to 2nR. Let us suppose that a massless scalar held defmed in this space satisfies the wave equation [7] @ll

•,5,4-~)=0 where

Replacing 4(x) by its Fourier series with respect to the compact dimension x5 we have

with x’ = (f, x) E M4. We see from this that the operator ( - 8:) plays the role of a mass operator for the 4-dimensional component fields c$‘~‘(x’), each of which has one degree of freedom. Replacing d(nl(.xi) its Fourier transform one gets (2.4) with H an integer for the frequency modes of the system. For each CD(~,n) the associated tower of energy states is E(L a) = (f + $) k.o(k, n),

l=O, 1, 2 ,... .

(2.5)

The /-values are restricted by (a)

1=0,1,2

(b)

l=O, 1

,..., cc

(Bosons),

(2.6)

(Fermions).

(2.71

If now instead of the scalar field d(x) with one degree of freedom we want to con-

EFFECTIVE

123

POTENTIAL

sider a field $J,K) with N degrees of freedom we proceed as above. We find the zero-mode #$))(.x~) describes N massless degrees of freedom and &$‘(.x’) describes N massive degrees of freedom. We shall evaluate the effective potential for the field 4 with one degree of freedom. To obtain the corresponding quantity for bN the result is multiplied by N. Similarly we may have a massive field with one degree of freedom defined in M’ x s’ and satisfying the equation (no1 - M2) (h(x) = 0. The corresponding

(2.8 1

frequency modes are

Other cases we shall consider are those of twisted fields detined to be antiperiodic the x5-coordinate, i.e.,

in

with

Finally we will consider effective potentials in D-dimensional space-time with d toroidally compact dimensions, i.e., M4 x s’ x ... x s’ with d s’ factors. Frequencies in this space are of the form

We begin with the partition function (2.13)

(2.14)

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and [e- u/2w4k~~( 1 + e-ww~k.n’)]

.z=fl

(Fermions).

(2.15)

k,n We detine the free energy by F= -1 ln Z. b

(2.16)

Thus (2.17) (Fermions).

(2.18)

Using the standard replacement (2.19) we have the expressions for the free energy density ln[l TeCfifiC”Ck,fl’] (2.20) where the upper and lower signs refer to the bosonic and fermionic cases, respectively. The first term of (2.20) is the usual T= 0 result while the second term gives the temperature dependent free energy density. It is worth noting at this point that (2.20) is equivalent to the expression for the finite temperature effective potential obtained by Dolan and Jackiw [8]. We now set #r= c = 1. 3. APPLICATIONS A. Effective Potential for a Single A4assless Bosonic Degree of Freedom Inserting (2.4) in (2.20) we have (3.1) The n = 0 term gives ( - 7c2/90/14) and on integration x4 dx ,f

by parts we get

(x2 + n2f12/R2)- li2 ?I=1 Le(x2 + P&~*/R~+‘~ - 11.

(3.2)

EFFECTIVE

125

POTENTIAL

Use the successive substitutions y* =x2 + n2fi2jR2,

(3.3)

2 = WVR

to give (3.4) where x0 = p/R. Tbe integral in (3.4) has a branch cut from I- 1,l 1 and simpIe poles at z = 2mzi/nxo with m = 0, k 1, *2,.... Consider the contour C, which is suitably deformed so as to avoid the cut (Fig. 1). Now = 0 = (-xi)

x Residues (m = I, 2, 3,...)

if C

+ ( - 7r72). Residue (m = 0) + P j,m + P jl+ where P denotes the principal value of the integral. We have

FIG. 1. Contour C for evaluation of integral in (3.4).

P Jo1

(3.5)

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In the Appendix we will show that (3.7) Thus (3.4) becomes (3.8) Next we define

(3.9) and regard V(R, /I) as the analytic continuation of V(R, /?; s) to S= 0. The continuation variable 3 can be introduced in Eq. (3.4) so that divergent formal sums like (3.8) never appear. We now introduce the Epstein zeta function [9] of order 2, defined by Z2(u, b; r) =

i! [(mj2 n,,,,= r

+ tbn121 ‘,

r> 1

(3.10)

where the prime on the sum indicates the n = vz = 0 term is omitted. Z2(a, b; r) converges for r > 1. For r c 1 it is defined by analytic continuation via the reflection formula ~~r~(r)Z2(~,b;r)=(ub)~‘~~~“~~‘~(l-r)Z2(u~’,b~’;

l-r).

(3.11)

Now Z2(u, !j; .Y- 3/2)=

if [(unz)2+ (bn)2]p”p3’2b ,7l,l? = x x x

+ 2 x

[(hn)2] -1.5 32).

(3.12)

??=I Using (3.12) in (3.9) we can write (3.13)

EFFECTIVE

127

POTENTIAL

We now use the reflection formula (3.11) with r = (3 - 3/2) and analytically tinue to s = 0. We get

con-

(3.14) This can be written as

-&C(5)+!$$2 which yields the asymptotic

f ~,~=, [(F)*

+n2]-512

(3.15)

behaviour whence l(5) is the Riemann zeta function

II91

This result is in agreement with that of Rubin and Roth [2], with the identification 5flV(L, b) = IJL, /I). The factor of 5 here refers to the fact that F(L, fi) in [2] is delined for a field with N = 5 degrees of freedom. B. Massless Fermions Inserting (2.4) in (2.20) we have (3.18) The n = 0 term gives ( - rc2/90b4)(7/8) and integration

by parts gives

In this case the simple poles occur at with

m= -+1 , +3 - >...

and residue (3.20)

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Thus we get [lo] L

f

1

37cR3fl ~=, ~= 1,3,5,,.,

[(~)2+n2]3’2.

(3.21)

[(e)2+n2]3’2.

(3.22)

Now

=z,

;I

[(F)2+n2]3’2-!l

$I

Using (3.22) in (3.21) we can write

with a continuation

to complex s. Use of the reflection formula gives

(3.24)

We can immediately

write down the asymptotic behaviour of (3.24):

(3.26)

Again this agrees with the result in [3]. C. Effective

Potential

per Degree

of Freedom

for

Massive

Bosom (3.27)

where .

(3.28)

EFFECTIVE

129

POTENTIAL

We treat the n = 0 separately: (3.29)

With the now standard substitutions

we can reduce this to (3.30)

Performing the contour integral in the usual way yields

- & Jl [(g2+*q3’2-$$

(3.31)

The other terms in the potential give

(3.32) We have [lo] f [M2 + (czn)2+ (bm)2]3” ??.,?I -=-x cc =l? x [M2+(Un)2]3’2+2 Pl=l

+4

f

[kf2+(h)2]3’2

>>I= 1

[M2 + (un)‘+ f e7.m =L

(bm)2]3’2 + kf3.

(3.33)

Using this we can write [lo]

The sums here do not converge but may be detined by analytic continuation the use of the formula [3]

with

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K is a modified Bessel function, and the prime on the sum means the term PZ,= n2= ... =nP=O is omitted. Using this formula with r = -3, p = 2, a, = L/2x, a2 = b/2x we get

+ m421”2~

~5,2wL-v4

n,m= - % ((M/2)[(Ln)2

+ (pm)2]“2)5’2.

(3.36)

Il. Effective Potential per Degree of Freedom for Massive Fermions

(3.37)

with (3.38)

The n = 0 term yields [IO] -7

l&f4 cc (z2 - 1)3’2 dz 67c- j , (epMz + 1) E&g,

[(y)2+A4q3J2--&f,

[(y2+A4y2.

(3.39)

Evaluating the other terms in the usual way we get [ 10 J

WW~=~ ~,~=, f [M2+(q2+(y)2]312 -&f, [M2+(y)2+(y2]312 +&Z,

[ki2+(g2]3’2--&~,

[A42+(y)2]3’2.

(3.40)

EFFECTIVE

13t

POTENTIAL

Using (3.33) we get YL

P, w = i&

m,m;ex LM2 + (y2+

-j&fx

(y)2]312

(3.41)

[M+(y2+(y2]3’2

Llw = -m+

Llw 16,312

z, K&f[(Ln)~ ((M/2)[(Ln)* Ix rim = -cc

x, ~5,2u4UW2 -- LIW 32dJ2 n,mx= -cc wp)[(Ln)*

+

+ (2p7l)y) + (2@l)yy

uw21”*)

+ (@z)2]1’2)s’2

(3.42) by (3.35).

E. Massive Twisted BOSCVIS

(3.43) with (3.44) IVe note the following

[lo] (3,45)

Thus we have

Proceeding in the standard way we get [lo]

(3.47)

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Using (3.33) we have [lo]

which yields on using (3.35)

F. A4assive Twisted Fermions

(3.50)

with (3.51)

Again using (3.45) and proceeding in the usual manner we get [ 101

--+& -&fl

[A42+(F)2+(T)2]3’2 [iw+(~)2+(g2]3’2

(3.52)

EFFECTIVE

133

POTENTIAL

Using (3.33) this can be written as [lo]

(3.53)

LM5

+@E

m, ~5,2@4lw~~* == -am((M/2)[(2Ln)2+ l7.m

K, &z(M[t=d’+ -- LM5 16x5’* n,*=’ -cc ((M/2)[(2Ln)2 =, ~5,2ww4* -- LMs 167~~~’n,m= x -cc ((M/2)[(Ln)* LM5 + 32x512

=‘, K5,2(“[(Ld2+ = ((M/2)r(LG)* n,m= -cc

+ ww211’*~ (2@7qyp2 @‘d211’2) + (/h~)~]~‘*)~‘~

+ cv~~211’2~ + (2flw~)~]~‘*)~‘* ~~m~2~1’2~

+ (pz)2]“2)5’2

(3.54) by (3.35).

Equations (3.36), (3.42), (3.49), (3.54) are in agreement with the results in Ref. [3]. We now calculate effective potentials in D-dimensional space-time with d toroidally compact dimensions. G. Massless Bosons The frequencies are m(k,n ,,..., ad)=

iI*

... +z 4

L2+z2+ 1

,

(3.55)

d

(3.56) The term nl=n2=

-.. = nd= 0 gives ( -n2/90f14). We get [lo]

(3.57)

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where the summation convention is used and the prime on the sum indicates the H,zn2z ... = Hi = 0 term is omitted. Now the Epstein zeta function of order (LI’+ 1) is detined by z ci+l u,h,...,bi~-C

3 2)

(3.58) f

The corresponding

1’ [(mz)2+(b,tz,)2+

... +(bdnd)2]-‘~m-3’2’.

reflection formula is

*~~~~312)~(~-~)~~+,(~,b,,...,b~;~-~) z(&l...bd)-‘n-

t!++;)+!+

S+- ;

(3.59)

We have

(3.60)

(3.61) Using the reflection formula one can write this as

(3.62) Dl2

Now the number of degrees of freedom for a second rank symmetric tensor in D-dimensions is

(3.63)

EFFECTIVE

135

POTENTIAL

Thus we have for the effective potential at finite temperature for a second rank symmetric tensor in D dimensions with d toroidally compact dimensions VL

1 ,...,

Ld,p)=

-f

(D-3)r

7cDqLy’Ld) (3.64)

x

[(flm)2+

(Lini)y?

This agrees with the result obtained in [4] with finite temperature as an extra compact dimension.

being regarded

H. Massfess Fermions in D-Dimensions (3.65) with (3.66) The ni = 0 term gives [lo] We have

( - n2/90fi4)(7/8).

(3.67)

Using (3.60) we get

(3.68)

Using the reflection formula gives

(3.69) [(2j?m)2 + (Ljni)2] pD’2.

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I. Massive Fermions

with

nT 112 cu(k,n,,M)=

The term n, = ... =nd=O

c*+A4*++

C

,)

.

(3.71)

gives [IO] (3.72)

Evaluating the remaining terms gives QL

#&Ml

Using f [M* + (aini)* + (bm)2]3’2 n<.m= -a x’ f [M2 + (aini)2 + (bm)2]3’2 + h13 tl,= -cc In=,

=2 + we get

f’ [M2 + (aini)2]3’2 + 2 f [M2 +- (bm)2]3’2 “,= -cc m=l

(3.74)

EFFECTIVE

Using (3.35) with r = -3, p=d+ V&,

p, A4)=(L1

‘..LJ)

137

POTENTIAL

1 we get I14D7rD’z~c-ml 2(2rc)D

nm

MD(Ll ...&) + (27ry)

1

(3.76)

where

21=$ [&n;y + (jnl)2]“2,

(3.77)

z2 =;

(3.78)

[(L,ni)2 + (2prn)y.

J. Massive Bosom (3.79) with (3.80) The n, = n2 = . . . = nd = 0 term gives [lo]

- --&J, [M2+(y2]3~2--&. The remaining terms yield

(3.8f)

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Using (3.74), we get the following: (3.83) Using (3.35), we get

(3.84)

where (3.85) Finally we consider massless twisted fields in the five-dimensional

theory

K. Massless Twisted Bosons (3.86) with (3.87) Using (3.45) we have

(3.88)

This we can write as [lo]

EFFECTIVE

Performing the contour integration

139

POTENTIAL

and using (3.12) we get

(3.90)

Use of the reflection formula (3.11) gives

(3.91)

This has the asymptotic behaviour J-P>

(3.92)

L%j.k

(3.93)

L. h4assless Twisted Fermions

Using (3.45) and proceeding in the usual way we get

Evaluating

the contour integral and using (3.22) and (3.12) we get

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Using the reflection formula (3.11) gives

This has the asymptotic behaviour (3.98) (3.99)

4. CONCLUSION

The temperature-dependent part of the effective potential to 0(h) has been computed in this paper using quantum mechanical methods, The method can be applied to more general situations, e.g., where the compact dimensions are different from a product of circles Si. In our calculations we have used the same c~(k, n) for both the boson and fermion cases. The difference between the boson and fermion case involved the constraint on occupation numbers and the degeneracy factor. It should be apparent that in our approach problems involving ghosts and tachyonic modes do not arise and it is amusing to see how the simple idea of zero-point fluctuations leads to a correct computation of the one-loop effective potential for both T= 0 and T#O.

We conclude with a discussion of the physical implications of the paper. The physical interpretation of the one-loop effective potential seen by studying the potential in various asymptotic regions. In obtains information about the possible modes of stability for the vacuum.

results in this is most easily this way one Kaluza-Klein

Case 1. Massless bosom in 5-dimensions.

To obtain the asymptotic behaviour (3.16), (3.17) we proceed as follows: (i) Fix pw co (i.e., T-0) with L//?e 1. Then the leading term in the mode sum (4.1) is given by taking m = 0. Because of

EFFECTIVE

141

POTENTIAL

the prime on the sum the integer n can now no longer be zero. Thus we get the asymptotic behaviour (4.2) Thus we see that for low temperature the effective potential leads to a contiguration in which L + 0. (ii) Fix p-0 (i.e., T-m) with L/p$l. Then the leading term is given by taking a = 0. We get (4.3) Here the potential tells us that in the asymptotic region LB fl, we require L + cc in order that V( L, fi) is allowed its minimum value of - co. The asymptotic analysis used here for the massless bosonic case goes through in exactly the same way to the massless fermion, massless twisted fermion and massless twisted boson cases. This yields the results (3.25) (3.26), (3.92), (3.93) (3.98) and (3.99). A point worth noting in regard to the asymptotic behaviour of these four cases is that massless bosons behave in exactly the same way as massless twisted fermions. Also massless fermions behave in the same way as massless twisted bosons. For this reason it is sufftcient to restrict attention to the un-twisted cases. Case 2. Massless bosom in D-dimensioru. Gl v(Li,p)= Fix p-0

(i.e., T-a)

-;(L,...Ld)

9

f’ [(pm)‘+ m.n,= -XI

(Lini)2] pDj2.

(4.4)

such that Li$j3. Then

Then the leading term in the sum is given by taking n, = . . . = nd = 0. This gives the asymptotic behaviour

We see therefore that for high temperature the only possible stable configuration one in which L, ,..., LA all expand to infinity.

is

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(ii) Fix p-co (i.e., T-0) such that L&/3 and let Lj=ujLd with rxi constants. Then m = 0 gives the leading term

(i= l,..., d- 1)

(4.7) where

SO

(4.9) In order for V(Ll, /I) to attain Ld + 0, i.e., we seem to require However, consider the following (iii) Fix p-co (i.e., T-0) Then

its minimum values of -cc we seem to require that all d compact dimensions contract to zero. case. such that Ld
The leading term is obtained by taking m = n, = . . . = nd- 1 = 0 so that (4.11) Now comparing (4.11) with (4.9) we see that the fall-off in (4.11) is stronger, i.e., LifcD-” compared to Ld“(DPd). Thus (iii) is more stable than (ii) and therefore the potential will choose (iii). We see then that in the zero-temperature limit we are led to a conliguration in which only one of the compact dimensions becomes small while the remaining (d- I) expand to intinity [4]. Case 3. Massless fermions in D-dimensions. v(Li,p)=

-gyg +v

(L,...LJ (Ll ... Ld)

f’ [(/?m)2+ (Lyzi)2]p”‘2 ??I,“,=-cc f’ [(2/lm)2 + (Lgi)2]-D’2. n7,nr= -m

(4.12)

EFFECTIVE

POTENTIAL

143

(i) Fix p-0 (i.e., 7-m co) S.&I f/rat Ligp. Then the leading term is obtained by taking ni = 0. We get

where

+-+o. Therefore at high temperature we require a contiguration where Ll ,..., Ld all expand to infmity as in the boson case. (ii) Fix /I-KI (i.e., T-0) such that Ld$fl and let L,=aiLt, (i= I,..., d-l) with IX; constants. Then m = 0 gives the leading term

with j-(aj) given by (4.8). Thus (4.17) Then V(L,, /?) attains its minimum value of zero when Ld expands to infinity. Thus all the d compact dimensions become large. (iii) Fix b-co (i.e., T-0) such that Ld fl) attain its minimum value of zero we would like Ld + co and Ll,..., Ldp, -+ 0. However, the asymptotic region we are studying here is 595/l6l/l-IO

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Ld< L, ,..., Lc, -, . Therefore if V(Li, /?) were to attain its minimum the asymptotic limits would be broken. So this case leads to no firm conclusions as to the evolution of the compact dimensions. (iv) Fix /I-m (i.e., T-0) such that L14Ld ,..., Ldp, < Ld. Then the leading term is the nz = 0 term. We get

(4.19)

We have the relation (4.20) Using (4.20), (4.19) becomes

(4.21)

(4.22) For u-0,

i.e., u-‘-m,

we take only the H = 0 term, i.e,, (4.23)

Using (4.23) in (4.21) we get

(4.24)

where ~2~can no longer be zero since we have taken nl = ... = ndp, = 0. Using (4.20) we can reduce (4.24) to (4.25)

EFFECTIVE

145

POTENTIAL

Now in this case V(Li, fi) is allowed to attain its minimum value of zero by letting Ld-+ oo. We notice that the fall-off in this case is identica1 to that m Case (ii). However, in the present case we see that the potential (4,25) is independent of L 1,..., Ldp I . Thus we may choose them to be compact and small, whiie at the same time allowing Ld+ m. In order that v(Li, j?) attains its minimum value. We see therefore that the fermionic case is qualitatively different from the bosonic case in that the possibility exists for {LI- 1) dimensions to be chosen to be compact with only one dimension expanding to infinity. Case 4.

Massive bosons in D-dimensions. D/2 vtLi*

Pt Ml

zz

-

fLl

. . . L‘,) MDf( -D/2)

’ ZGp

(4.26)

with z=$

(i) Then

Fix /?- EI with L&l/M@ z=~[(ML~H,)~+

- #&/%/~

[(Linl)2+

(flm12]“‘.

(4.27)

L, ,..., L(,- ,6/j’.

*.. + (M&-

,nd- ,)2 + (MLgzd)’

+ (A4pm)‘]“*

to leading order.

(4.28)

We also have

(e-MLJ)fld-O(l)

for

MLc,+ 1

we get 4L

Lt W

KDj2(MLdnd)

((M/2) Lgzd)D”’

(4.29)

This is of the same form as (4.11) as we would expect since the asymptotic hmit we are considering here is l/M large, i.e., M-O.

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(ii) l/M+ Ld4 Ll ,..., Ld-,
... L,J ~

(M;#‘+

‘12’

(4.31)

We need only take the nd= 1 term giving

We note here that we are disregarding the first part of (4.26) as explained in 131. (iii) l/M
Case 5. Massive fermions in D-dimensions. The corresponding are

asymptotic limits

(4.34) (4.35) (4.36)

APPENDIX

A

We need to evaluate the integral os (z* - 1 )3’2 dz .I, (ew - 1) ’

X0=-. P R

We first consider the contour C, as shown in Fig. 2.

EFFECTIVE

POTENTIAL

147

\ FIG. 2. Contour C, for evaluation of integral in (A.1).

Then

4c,

(z’- 1)3’2 dz (PO= - 1 ) = 0 = ( -zi)

+

x Residues (m = 1, 2, 3,...) . Residue (m = 0) + P JAB+ P jDE + P jFG

where the symbol P denotes the principal value of the integral. It is easy to show that JBCD and jGHA both vanish. On DE arg z= 7r/2 so 2 z reinI z ir giving

On AB arg z = 0 giving with z =.x (A.4)

On FG(z-l)=pe’=

with p~(l,O)

giving

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Thus we have P

mY(x2- 1)3’2LIx 1, (PY - 1 ) = (nil 1 Residues (m = 1, 2,...)

We will now show that (A.7)

and

Consider the contour C2 as shown in Fig. 3 and the integral (z2 - 1 )3’2 dz #-=~ (@.yo=- 1) .

FIG,

3.

Contour

C2 for evaluation

of integral

in (A.7).

EFFECTIVE

149

POTENTIAL

On AB

On CD Z=

re-

W2 =

-ir,

rE

(0,

a).

Thus (A.10)

(A.12) Let (A.13) Then we have just shown F(Hx~) = -F( -PZX,,), i.e., F is an odd function of (xxO). But F(nxo) is also an even function as can be shown: (A.14) Let mar = y then cc

FWol=P

j.

dy

m

(y2+

(r~x~)~)~‘~ (&? - 1)

(A.15)

which is manifestly even. Thus F is both even and odd and hence can only be zerc We now wish to show that (A.16 Consider $cj [(z* - 1)3’2 dz/(PYo’

- l)],

where C3 is the contour shown in Fig. 4.

150

BIRMINGHAM

FIG.

4.

Contour

AND

C3 for evaluation

SEN

of integral

in (A.8).

Then

+fq +PJ +j . EF

HA

FGH

f BcD and JFGH are easily seen to be zero. We have just shown that P lH.+, is zero. So PJEF is therefore also zero. We are left with

(A.18) on AB

on DE

(A.19)

EFFECTIVE

15t

POTENTIAL

giving (A.20) Thus (A.21) We have thus shown tha: from our original contour C,, the integral we require, namely, P j? [(z* - 1)3’2 &/(en-“” - 1)] is given solely by the pole contributions at 2 = 2mrci/nx,,, with the rest of the contour contributing zero. The same analysis used here for the bosonic integral goes through for the fermionic case.

REFERENCES APPELQUIS-I AND A. CHODOS, Hqs. Rec. Lea. 50 (1983). 141. A. RUBIN AND B. D. ROTH, Nd. Phys. B 226 (1983). 444. A. RUBIN AND B. D. ROTH, Phys. Left. B 127 (1983). 55. APPELQUIST, A. CHODOS,AND E. MYERS, Phys. ML B 127 (1983), 5. D. ROHRLICH, Phys. Rev. D 29 (1983), 330. 4. H. B. G. CASIMIR, Proc. K. Ned. Akad. Wef. Ser. BSI (1948), 793.

1. 2. 3. 4.

T. M. M. T.

51.

7, This equation in fact follows from the usual Kaluza-Klein ansatz by keeping the dependence on the compact dimension .I-‘. See, for example, A. SALAM AND J. STRATHDEE, Ann. Ph.w. (N. Y.) 141 (1982),

8. 9. 10. I I.

316.

L. DOLAN AND R. JACKIW, Phyx Rtw D 9 (1974), 3320. “Encyclopedic Dictionary of Mathematics,” Vol. II, p. 1378, MIT Press, Cambridge, Mass., 1977. See remarks after Eq. (3.9). E. WHITTAKERAND G. WATSON,“A Course in Modern Analysis,” (Cambridge Univ. Press, London, 1962,)