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Gravitational instability in the higher-derivative Kaluza-Klein theory
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
GRAVITATIONAL INSTABILITY IN THE HIGHER-DERIVATIVE KALUZA-KLEIN THEORY
W. ISHIZUKA and Y. KIKUCHI Department of Physics, Tohoku University, Sendai 980, Japan Received 21 October 1983
We study the one-loop effective potential in the higher-derivative theory. The similar instability found by Appelquist and Chodos emerges in a simple Kaluza-Klein model.
The Kaluza-Klein (KK) theory is a version of grand unification theories [ 1]. It starts with general relativity in higher-dimensional space-time, and then reproduces the usual four-dimensional relativity as well as appropriate gauge theories by means of dimensional reduction with suitable boundary conditions. The essential point in this approach is that extra dimensions are not taken as intermediate devices for obtaining the realistic theory but as truly physical entities. Recently Appelquist and Chodos (AC) [2] have succeeded in computing the one-loop effective potential in these apparently non-renormalizable theories. Then the dynamical stability or instability of the Kaluza-Klein vacuum has become an interesting subject. AC discovered, in a simple (4+ 1)-dimensional KK theory, that the effective potential tends to minus infinity as the length of the fifth dimension L5 shrinks to zero. They interpreted this instability as the gravitational analogue of the Casimir effect. Later , Rubin and Roth [3] have taken the finite temperature effect into the analysis of AC, and found that two types of instabilities emerge. Meanwhile Tsokos [4] has shown that inclusion of the five-dimensional fermion field can possibly cure the AC instability provided that the fermionic degrees of freedom exceed those of boson fields. As is stressed in ref. [2], the loop expansion is expected to be reliable at regions where the circumference of the compactified fifth dimension is much larger than the Planck length (Lp). This is because the short distance behavior of the theory becomes unreliable in the Einstein-Hilbert action approach. One way to inquire into the behavior of the effective potential at shorter distances, is to consider the higher-derivative theory [5]. In this letter, we compute the additional contribution induced by higher-derivative terms to the one-loop effective potential. We start with the following action in five-dimensional space-time,
S = f(dx)5(2nLs)-l(det gAB) 1/2 [(167rG)-lR + otR 2 + 16R2B + TR2BcD] ,
(1)
where the indices A, B, C and D run over 1..... 5. 2nL5 is the volume of the compactified dimension, a,/3 and 7 are the dimensionless parameters to be determined later. The R are the curvatures in five dimensions. The classical analysis of the model was carried out by Wetterich [6] in which he showed that the structure M4 ® S1 could be taken as a classical background geometry. Since we are interested in the gravitational effect brought about by the higher-derivative terms on the vacuum stability, let us parameterize the metric tensor in the following form, gAB = 1,9(~b)
01
v K~
'
(2)
where K = (16riG) 1/2, and ~ is the scalar field which is supposed to develop the vacuum expectation value ~bC. The 35
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
Weyl factor W(q~) is adjusted as W(q~) = ¢-1/3 in order to reproduce ordinary four-dimensional Einstein gravity. Next we introduce the quantum fields as guy = Buy + K h ~ v ,
¢ = ¢C + ¢' •
(3)
For quantizing the theory in the path-integral formulation, we take the following gauge conditions [2,4,7] : gA5, 5 = 0 ,
guy, v = 0 .
(4)
The one-loop effective potential is computed in almost the same way as in ref. [2]. In the calculation at oneloop order, it suffices to expand the action functional in powers of the quantum fields and keep the terms quadratic in them. Each of the terms in the right-hand side of eq. (1) yields the following expression (hereafter we set K = 1): R = ( 6 ~ b b - t ( 0 u ¢ ' ) 2 -- ~huv [Z]MUVe'~he~ + .... R 2 = (9~b4C/3)-l(m~b,)2 - 5 2 ¢-4/3[~ ¢' [¢C[-q + (05)2] huU+ ¢~4/3h~ [~bCE] + (05)2] 2 hW + "", C 2 -4]3 R A2 B -_ 6¢C (/-10)' 2 + (4¢4C/3)- lhtau ([¢C V] + (05) 2] 2NUU'~# + [] 2MUUa#} h~#
--
~q~,4/315]¢' [2¢C1-1 _ (05) 2] h i + ..,,
~ ".#r~ ~ "t~/a u + ... , R A2B C D = ~q~C4/3(Vq¢') 2 + (aC4/3hlau[OC[-1 +(05)2] 2 h u ~ - ~5wC .~-1/3 rm
(5)
where M uvc4s = 8tav~5c~ -- 6u08 v~ ,
N uvc'~ = BuY8 ~ + 6u~8 vt3 .
(6)
Remember that the Weyl factor W(~b) = ¢-1/3 was introduced in order that the mixing terms between hu~, and ~b' in the expression of R vanish, and each be a physically meaningful field. In general, the higher-derivative terms also give rise to these mixing terms, however, as is easily seen from (5), they can be eliminated by the following choice of parameters: 13= 4 ~ ,
~' = --3c~.
(7)
In other words, when quantum effects are concerned, the ratios among these coefficients must be unique so long as our model stays physically meaningful. This is a rather restrictive constraint compared to those met in the classical argument studied in ref. [6], and we are left with only one parameter c~. We note that the terms quadratic in the scalar field ¢' in R can be made independent of ¢C by scaling as ¢' -+ ¢C~' so that their contribution to the effective potential can be absorbed in the normalization [2]. While those in R 2 , R 2 B and R2BCD remain and give rise to the divergent contribution which, however, should be regarded as a cosmological constant to be subtracted. So we need concentrate on the gravitational degrees of freedom only. The ghosts corresponding to the gauge conditions (4) do not contribute either [2]. The terms given in the RHS of eq. (5) are summed up under the constraint eq. (7) to yield ,1 -~huvMU~'c'~ {[] + ¢~1(05)2 -8U¢2C/3 [m + ¢~1(05)212 } h ~ .
(8)
We note that the tensor structure MUV~Ohas become an overall factor, which is due to the choice of the gauge (4). This makes the following computations remarkably simpler. The effective potential can now be readily read off from (8). Following AC, we normalize eq. (8) at ~C = 1 (in fact ¢C = K) to obtain #1 R, Rtav and Ruvo43 are not all independent in four dimensions. We can use the following identity, x/~(R 2 - 4R~v + R~vo43) = (total derivative), to eliminate one of them.
36
Volume 139B, number 1,2
PHYSICS LETTERS
q~71/3 Veff(q~C) -- Veff(~bC = 1) = (47rL5) - 1 ~ n
(dk/27r) 4 log
3 May 1984
5
J
f(dk/2~r) 4 log( -L2 - 8~l/3(¢Ck2L2 + n 2)
+ (41rL5) - 1
(9)
7.7-
.
The first term on the RHS is just the expression derived by AC except for an overall factor. This difference is due to the fact that we have employed the four-dimensional gauge condition gu~,, v = 0 which was not needed in the analysis of AC. Of course the gauge fixing does not affect any physically meaningful consequences. Turning to the second term on the RHS of eq. (9), we see that another constraint on the parameter a, a < 0, must be satisfied unless the argument of the logarithm could take negative values which results in an imaginary part of the effective potential. In what follows, a is taken to be negative. Let us proceed to the evaluation of the RHS of eq. (9). The summation over n can be replaced by an integral using the usual contour technique to give
f(dk/2rO s log[(Ock2 + k2s)l(k2+ks2)] + (2rrLsS)-1
f(dy121r)4 f
o~+ie
dz[exp(-2rriz)- 1] -1
-o.+ie
× log [(~bCy 2 +
+(2~rL~) -1
z2)l(.v2+ z2)] + 1
f(dy/2~)4 f -
log {[1 - 8a¢~ 1/3 (~ck2 + k2)] / [1 _ 8a(k 2 + k2)] )
oo+ie
dz[exp(-2~iz)-
1] -1 l o g ~ [ 1 - 8 a # ~ l / 3 ( # c y 2 + z 2 ) ] / [ 1 - 8 a ( y 2 + z 2 ) ] } .
(10)
~+ie
The first and the third terms which are both divergent give the cosmological constant again, and are to be subtracted from the effective potential. The second term has already been calculated in ref. [2] to be given as (2¢r~blc/3L5)-5 ~,
~ = -(15/4~r 2) ~'(5) ~ - 0 . 3 9 4 .
(11)
Apparently this expression shows up the monotonous decrease of the effective potential as ¢C goes to zero, which indicates the instability. The fourth piece gives the non-trivial contribution which originates from the higher-derivative terms. After a little manipulation, the ¢c-dependent part can be written as
(2rrLs)-sf (dx/2~r)4 q~2 log{1
- e x p [ - ( x 2 + U)I/2]},
(12)
where
u= -0r2/2~)
~blc/3L2 .
(13)
We note that the terms linear in 6(5)(0) log ~bC have been discarded since they are to be absorbed in the measure [21. We can approximately evaluate (12) when U ~ 1 (L5 "~ 1) to see the behavior of the effective potential at shorter distances. Expansion in powers of U of(12) yields (2rrgpl~3L5)-5~ + rrZ(zrr~b~/3L5)-5 [2~'(3) U+ (U2/4) log U+ O(/.I4)] .
(14)
It is interesting to note that the first term is the same as (11) which originates from x/gR. This term is leading for small U, and signals out the same type of instability as was found by AC. Since the higher-derivative terms will dominate over the term v ~ R at shorter distances, we are led to the conclusion that the instability is again generated in the higher-derivative theory and the size of the fifth dimension would continue to shrink to zero. In the ordinary four-dimensional picture, the inclusion of terms such as R 2 37
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
means the addition of the Yukawa-type potential [5]. So it is probable that for the choice of the parameters such as (6), this potential works attractively, which leads again to the attractive Casimir effect resulting in instability. Therefore at present, we have to include something into the theory which produces a repulsive Casimir force. The fermionic degree of freedom studied by Tsokos is one of the candidates. However, the original philosophy of the KK theory is to derive everything from the geometry, so some natural reasoning to add the kinetic term of the fermion is waited for. In this connection, supersymmetric theories, especially those formulated in superspace [8], may provide some useful guides. The authors thank Dr. T. Moriya for useful discussions.
References [1] Th. Kaluza, Sitzungsber. Press. Akad. Wiss. Berlin, Math-Phys. KI. (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [2] T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141; Phys. Rev. D28 (1983) 772. [3] M.A. Rubin and B.D. Roth, University of Texas preprint UTTG-2-83; Phys. Lett. 127B (1983) 55. [4] K. Tsokos, Phys. Lett. 126B (1983) 451. [5] K.S. Stelle, Phys. Rev. D16 (1977) 953. [6] C. Wetterich, Phys. Lett. l13B (1982) 377. [7] A. Salam and J.D. Strathdee, Ann. Phys. 141 (1982) 316. [8] See e.g.P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249.
38
PHYSICS LETTERS
3 May 1984
GRAVITATIONAL INSTABILITY IN THE HIGHER-DERIVATIVE KALUZA-KLEIN THEORY
W. ISHIZUKA and Y. KIKUCHI Department of Physics, Tohoku University, Sendai 980, Japan Received 21 October 1983
We study the one-loop effective potential in the higher-derivative theory. The similar instability found by Appelquist and Chodos emerges in a simple Kaluza-Klein model.
The Kaluza-Klein (KK) theory is a version of grand unification theories [ 1]. It starts with general relativity in higher-dimensional space-time, and then reproduces the usual four-dimensional relativity as well as appropriate gauge theories by means of dimensional reduction with suitable boundary conditions. The essential point in this approach is that extra dimensions are not taken as intermediate devices for obtaining the realistic theory but as truly physical entities. Recently Appelquist and Chodos (AC) [2] have succeeded in computing the one-loop effective potential in these apparently non-renormalizable theories. Then the dynamical stability or instability of the Kaluza-Klein vacuum has become an interesting subject. AC discovered, in a simple (4+ 1)-dimensional KK theory, that the effective potential tends to minus infinity as the length of the fifth dimension L5 shrinks to zero. They interpreted this instability as the gravitational analogue of the Casimir effect. Later , Rubin and Roth [3] have taken the finite temperature effect into the analysis of AC, and found that two types of instabilities emerge. Meanwhile Tsokos [4] has shown that inclusion of the five-dimensional fermion field can possibly cure the AC instability provided that the fermionic degrees of freedom exceed those of boson fields. As is stressed in ref. [2], the loop expansion is expected to be reliable at regions where the circumference of the compactified fifth dimension is much larger than the Planck length (Lp). This is because the short distance behavior of the theory becomes unreliable in the Einstein-Hilbert action approach. One way to inquire into the behavior of the effective potential at shorter distances, is to consider the higher-derivative theory [5]. In this letter, we compute the additional contribution induced by higher-derivative terms to the one-loop effective potential. We start with the following action in five-dimensional space-time,
S = f(dx)5(2nLs)-l(det gAB) 1/2 [(167rG)-lR + otR 2 + 16R2B + TR2BcD] ,
(1)
where the indices A, B, C and D run over 1..... 5. 2nL5 is the volume of the compactified dimension, a,/3 and 7 are the dimensionless parameters to be determined later. The R are the curvatures in five dimensions. The classical analysis of the model was carried out by Wetterich [6] in which he showed that the structure M4 ® S1 could be taken as a classical background geometry. Since we are interested in the gravitational effect brought about by the higher-derivative terms on the vacuum stability, let us parameterize the metric tensor in the following form, gAB = 1,9(~b)
01
v K~
'
(2)
where K = (16riG) 1/2, and ~ is the scalar field which is supposed to develop the vacuum expectation value ~bC. The 35
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
Weyl factor W(q~) is adjusted as W(q~) = ¢-1/3 in order to reproduce ordinary four-dimensional Einstein gravity. Next we introduce the quantum fields as guy = Buy + K h ~ v ,
¢ = ¢C + ¢' •
(3)
For quantizing the theory in the path-integral formulation, we take the following gauge conditions [2,4,7] : gA5, 5 = 0 ,
guy, v = 0 .
(4)
The one-loop effective potential is computed in almost the same way as in ref. [2]. In the calculation at oneloop order, it suffices to expand the action functional in powers of the quantum fields and keep the terms quadratic in them. Each of the terms in the right-hand side of eq. (1) yields the following expression (hereafter we set K = 1): R = ( 6 ~ b b - t ( 0 u ¢ ' ) 2 -- ~huv [Z]MUVe'~he~ + .... R 2 = (9~b4C/3)-l(m~b,)2 - 5 2 ¢-4/3[~ ¢' [¢C[-q + (05)2] huU+ ¢~4/3h~ [~bCE] + (05)2] 2 hW + "", C 2 -4]3 R A2 B -_ 6¢C (/-10)' 2 + (4¢4C/3)- lhtau ([¢C V] + (05) 2] 2NUU'~# + [] 2MUUa#} h~#
--
~q~,4/315]¢' [2¢C1-1 _ (05) 2] h i + ..,,
~ ".#r~ ~ "t~/a u + ... , R A2B C D = ~q~C4/3(Vq¢') 2 + (aC4/3hlau[OC[-1 +(05)2] 2 h u ~ - ~5wC .~-1/3 rm
(5)
where M uvc4s = 8tav~5c~ -- 6u08 v~ ,
N uvc'~ = BuY8 ~ + 6u~8 vt3 .
(6)
Remember that the Weyl factor W(~b) = ¢-1/3 was introduced in order that the mixing terms between hu~, and ~b' in the expression of R vanish, and each be a physically meaningful field. In general, the higher-derivative terms also give rise to these mixing terms, however, as is easily seen from (5), they can be eliminated by the following choice of parameters: 13= 4 ~ ,
~' = --3c~.
(7)
In other words, when quantum effects are concerned, the ratios among these coefficients must be unique so long as our model stays physically meaningful. This is a rather restrictive constraint compared to those met in the classical argument studied in ref. [6], and we are left with only one parameter c~. We note that the terms quadratic in the scalar field ¢' in R can be made independent of ¢C by scaling as ¢' -+ ¢C~' so that their contribution to the effective potential can be absorbed in the normalization [2]. While those in R 2 , R 2 B and R2BCD remain and give rise to the divergent contribution which, however, should be regarded as a cosmological constant to be subtracted. So we need concentrate on the gravitational degrees of freedom only. The ghosts corresponding to the gauge conditions (4) do not contribute either [2]. The terms given in the RHS of eq. (5) are summed up under the constraint eq. (7) to yield ,1 -~huvMU~'c'~ {[] + ¢~1(05)2 -8U¢2C/3 [m + ¢~1(05)212 } h ~ .
(8)
We note that the tensor structure MUV~Ohas become an overall factor, which is due to the choice of the gauge (4). This makes the following computations remarkably simpler. The effective potential can now be readily read off from (8). Following AC, we normalize eq. (8) at ~C = 1 (in fact ¢C = K) to obtain #1 R, Rtav and Ruvo43 are not all independent in four dimensions. We can use the following identity, x/~(R 2 - 4R~v + R~vo43) = (total derivative), to eliminate one of them.
36
Volume 139B, number 1,2
PHYSICS LETTERS
q~71/3 Veff(q~C) -- Veff(~bC = 1) = (47rL5) - 1 ~ n
(dk/27r) 4 log
3 May 1984
5
J
f(dk/2~r) 4 log( -L2 - 8~l/3(¢Ck2L2 + n 2)
+ (41rL5) - 1
(9)
7.7-
.
The first term on the RHS is just the expression derived by AC except for an overall factor. This difference is due to the fact that we have employed the four-dimensional gauge condition gu~,, v = 0 which was not needed in the analysis of AC. Of course the gauge fixing does not affect any physically meaningful consequences. Turning to the second term on the RHS of eq. (9), we see that another constraint on the parameter a, a < 0, must be satisfied unless the argument of the logarithm could take negative values which results in an imaginary part of the effective potential. In what follows, a is taken to be negative. Let us proceed to the evaluation of the RHS of eq. (9). The summation over n can be replaced by an integral using the usual contour technique to give
f(dk/2rO s log[(Ock2 + k2s)l(k2+ks2)] + (2rrLsS)-1
f(dy121r)4 f
o~+ie
dz[exp(-2rriz)- 1] -1
-o.+ie
× log [(~bCy 2 +
+(2~rL~) -1
z2)l(.v2+ z2)] + 1
f(dy/2~)4 f -
log {[1 - 8a¢~ 1/3 (~ck2 + k2)] / [1 _ 8a(k 2 + k2)] )
oo+ie
dz[exp(-2~iz)-
1] -1 l o g ~ [ 1 - 8 a # ~ l / 3 ( # c y 2 + z 2 ) ] / [ 1 - 8 a ( y 2 + z 2 ) ] } .
(10)
~+ie
The first and the third terms which are both divergent give the cosmological constant again, and are to be subtracted from the effective potential. The second term has already been calculated in ref. [2] to be given as (2¢r~blc/3L5)-5 ~,
~ = -(15/4~r 2) ~'(5) ~ - 0 . 3 9 4 .
(11)
Apparently this expression shows up the monotonous decrease of the effective potential as ¢C goes to zero, which indicates the instability. The fourth piece gives the non-trivial contribution which originates from the higher-derivative terms. After a little manipulation, the ¢c-dependent part can be written as
(2rrLs)-sf (dx/2~r)4 q~2 log{1
- e x p [ - ( x 2 + U)I/2]},
(12)
where
u= -0r2/2~)
~blc/3L2 .
(13)
We note that the terms linear in 6(5)(0) log ~bC have been discarded since they are to be absorbed in the measure [21. We can approximately evaluate (12) when U ~ 1 (L5 "~ 1) to see the behavior of the effective potential at shorter distances. Expansion in powers of U of(12) yields (2rrgpl~3L5)-5~ + rrZ(zrr~b~/3L5)-5 [2~'(3) U+ (U2/4) log U+ O(/.I4)] .
(14)
It is interesting to note that the first term is the same as (11) which originates from x/gR. This term is leading for small U, and signals out the same type of instability as was found by AC. Since the higher-derivative terms will dominate over the term v ~ R at shorter distances, we are led to the conclusion that the instability is again generated in the higher-derivative theory and the size of the fifth dimension would continue to shrink to zero. In the ordinary four-dimensional picture, the inclusion of terms such as R 2 37
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
means the addition of the Yukawa-type potential [5]. So it is probable that for the choice of the parameters such as (6), this potential works attractively, which leads again to the attractive Casimir effect resulting in instability. Therefore at present, we have to include something into the theory which produces a repulsive Casimir force. The fermionic degree of freedom studied by Tsokos is one of the candidates. However, the original philosophy of the KK theory is to derive everything from the geometry, so some natural reasoning to add the kinetic term of the fermion is waited for. In this connection, supersymmetric theories, especially those formulated in superspace [8], may provide some useful guides. The authors thank Dr. T. Moriya for useful discussions.
References [1] Th. Kaluza, Sitzungsber. Press. Akad. Wiss. Berlin, Math-Phys. KI. (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [2] T. Appelquist and A. Chodos, Phys. Rev. Lett. 50 (1983) 141; Phys. Rev. D28 (1983) 772. [3] M.A. Rubin and B.D. Roth, University of Texas preprint UTTG-2-83; Phys. Lett. 127B (1983) 55. [4] K. Tsokos, Phys. Lett. 126B (1983) 451. [5] K.S. Stelle, Phys. Rev. D16 (1977) 953. [6] C. Wetterich, Phys. Lett. l13B (1982) 377. [7] A. Salam and J.D. Strathdee, Ann. Phys. 141 (1982) 316. [8] See e.g.P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249.
38