IL NUOVO CIMENTO VOL. 92 A, N. 4 21 Aprile 1986

Quantum Corrections in Kaluza-Klein Theory (*).

1~. DELBOURGO a n d R. O. WE]~E~

Department o] Physics, University o/ Tasmania - Hobart, Australia 7005

(ricevuto il 3 0 t t o b r e 1985)

Summary. - - Wc have examined some of the one-loop quantum correc- ctions to the five-dimensional Kaluza-Klein model, which are due to the infinite number of massive spin-2 excitations, in order to discover how the tower of levels can affect the classical results. I t turns out that quantities which vanish classically, such as (9} and the mass of the scalar field ~, receive finite contributions which are determined by the Planck radius. On the other hand, amplitudes such as W-W scattering which are one-loop infinite, even when massive modes are disregarded, do not have their divergences ameliorated by the inclusion of the tower. We would conclude that a higher-dimensional model should be considered seriously as a candidate for a physical, quantized theory if (at the very least) the zero-mode t runcat ion is already renormalizable, for only then are the entrained higher modes likely to produce finite corrections.

PACS. 04.50. - Unified field theories and other theories of gravitation. PACS. 04.60. - Quantum theory of gravitation.

1 . - I n t r o d u c t i o n .

I t is on ly n a t u r a l t h a t the or ig inal K l e i n - K a l u z a t h e o r y shou ld h a v e led

to so m a n y gene ra l i za t ion a n d d e v e l o p m e n t s of la te . The t h e o r y provides

an a t t r a c t i v e m a y of u n i f y i n g g r a v i t y a n d i n t e r n a l s y m m e t r i e s w i t h i n a h igher-

d imens iona l f r amework even t h o u g h the chiral m a t t e r fields (1) do n o t fit in

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) ]~. WITT:EN: -Fermion quantum numbers in Kaluza-~Klein theory, Princeton preprint (October 1983). This paper is a veritable treasure-trove of original references in the subject. P. H. FRAMPTON: Chiral ]ermions in Kaluza-Kteiu theories, I F P 228-UNC, talk presented at Pifth Workshop on Grand Uni]ication (Brown University, April 1984).

347

$4S I~. DELBOD-2G0 and ~. o. WEBER

so na tura l ly within the scheme of things. Much of the research in this area has been directed towards unders tanding the na ture of the compactif ication to M4× (internal space) and exploring the relat ive radius scales (3) in the com-

pactification. Any serious s tudy of such theories has to contend with the higher mass

modes of the (~ gravi tat ional ~> and (possibly) other fields which come via the harmonic expansions over the compact internal co-ordinates. In the early days it was common to disregard the massive excitat ions for low-energy physics (low relat ive to the Planck scale) and to concentra te on the light four-dimen- sional sector in the hope tha t the observed part icle spectrum would emerge na tura l ly f rom a simple model. Equivalent ly , the neglect of the ex t ra di- mensions was accompanied by the thesis t ha t it is impossible to probe the minuscule region in practice, except insofar as the s t ructure of the internal space is reflected in the internal symmet ry group classification of the low mass states. While this a t t i tude is perfect ly correct at a semi-classical level, i t is quite untenable when we examine the quan tum effects. In tha t case we have to take seriously the internal d imens ions- - they have to be in tegrated o v e r - - and sum over all the vi r tual states in the model. I f there were only a finite number of massive modes, there might still be some justification for neglecting them at low energy, bu t in real i ty there are an infinite number and it is over- optimist ic to suppose tha t they lead to no observable effects whatever , especially

af ter quantizat ion. The purpose of this paper is to examine the one-loop quan tum effects in

Kaluza-Kle in theory and to look for the consequences of the infinite spin-2 tower of particles (3) on low-energy phenomena. We focus on the original five-dimensional construct , despite its obvious flaws in the ma t t e r field sector, since we are just seeking to ex t rac t general features tha t m a y prove re levant for other, more complicated, bu t more realistic models (4). In sect. 2 we see how the exchange of the massive modes modifies the full five-dimensional p ropaga tor at small and large distances; af ter all i t is integrals over the prop- agator t ha t govern the quan tum loop corrections. Following the covar iant quant iza t ion of the model in sect. 3, we ex t rac t the resulting F e y n m a n rules for the zero modes and the excitat ions in the theory. These are convenient ly summarized in an appendix. Such rules are needed for the nex t sections, when we compute the quantum-mechanical contr ibutions to certain ampli tudes

associated with the scalar field ~0 incorporated in the metr ic ; specifically the

(3) P. CANDELAS and S. WEINBERG: Nuel. Phys. B, 237, 397 (1984); D. BAILIN and A. LovE: Phys. Lett. B, 144, 359 (1984). (s) A. SALA~ and J. ST]~ATHDE~,: Ann. Phys. (N.lr.), 131, 316 (1982). (4) S. RANDJBAER-DAEMI, A. SALAM and J. ST~ATHDXX: .LYUcl. Phys. B, 214, 491 (1983); Phys. Left. B, 132, 56 (1984); C. WETT~,~ICH: Phys..Dett. B, 110, 379 (1982). Z. F. EzAwA and I. G. Koa: Phys. Lett. B, 142, 157 (1984).

QUANTUM CORRECTIONS IN KALUZA-KLEIN THEORY 3 ~ 9

corrections to the classical ampl i tudes for (~}, ( ~ } and ( T ( ~ ¢ ~ ) } . The first of these corresponds to reshif t ing the scalar field b y an addi t ional amount , the second leads to a mass for ~ which breaks the scale invariance, and the th i rd corresponds to an incipient nonrenormal izable effect associated with the

compac t fifth dimension which does nothing to improve the infinity caused

b y the quaut ized zero-mode corrections.

2 . - T h e f i v e - d i m e n s i o n a l p r o p a g a t o r .

After p~rametr iz ing the met r ic (5) in the usual way (1,~), viz.

(1) ( ~ M ~ = q _ i ~ g ~ - - A ~ A ~ , - - A ~ q J ) \ --A~,

one finds t h a t the convent ional f ive-dimensional Lagrangian,

(2) ~ = - - (K5)-2 V~GR~); K~ = K 2 " 2 ~ R = 32~2RG~

reduces to

7= 1GM~GPsGQS~G~ M~,5 ~PQ,5"~' - - G~p, sG~vQ,5) 4-

4- ~ ~ ~,~v, ~Q,~ 4- --..,~MP,~r~O,~) •

This represents normal Einstein g rav i t a t ion and Maxwell e lect rodynamics

Sheory combined with a scalar field, plus a tower (~) of mass ive spin-2 particles. This is mos t easily unders tood by defining the q u a n t u m fields via expansion abou t fiat space:

(4) g ~ : ~ . ~ ~ h.~, ~ : 1 4 - ] ,

expanding harmonica l ly over the fifth co-ordinate,

: ht,~(x ) exp [inx~/R],

(5) Our notation is to use upper-case Latin letters for 5-dimensional components, lower-case Greek for 4-dimensional indices, and lower case Latin for ordinary 3-space.

is the Minkowski (time-like) metric, and the coupling factors, K = (16~G~)i for gravitation and e for charge, are absorbed into the fields h and A, respectively. (6) T. APPF~LQUIST and A. C~ODOS: Phys. Rev. D, 28, 772 (1983).

350 R. D]~LBOIffRGO and R. o. W]~B]~R

and set t l ing on the gauge choice (e)

(5) ~ugu~ = 0 , ~sAu ---- ~5~v ---- 0,

for the massless modes. Actual ly we shall have more to say abou t the quest ion

of gauge fixing and the a t t e n d a n t ghosts in the nex t section. Fo r the present , we note t h a t the harmonic expansion of the Lagrang ian leads to a succession

of h e a v y modes wi th masses M . - ~ n /R entirely fixed b y the radius of the compac t fifth dimension. The free mass ive propagators , coming f rom the quadra t ic (in h) pa r t of the Lagrangian ,

(6) K~ £f = [h"~'Qhu~,o- 2h""qh~,q,~ + 2h"~,. hQo,~ - - hQ~,u hJn']/4 - -

h ~ + [ ~,u - - h.~,~]Au,5 - -3~ "A~dl2q~ ÷ 3[(~/)~/~ 2 - h~,~d/cf]14~f

are the usual ones:

(7) A~,,o~(p) = [(P~,~P~ + P u ~ P , ~ ) / 2 - Pu~P~/3]/(p 2 - M~);

Pu. ~ ~lu, - - Pu P,/M~ .

Once we realiso t h a t the f ive-dimensional p ropaga to r is the sum over the dis-

crete modes wi th Four ier coefficient exp [in~5/R], i t is simple to see t h a t

= _ cost _ ) . (8) Au~Q~(x) ~ [Pu~D~ + Du~P~ 2Du~DQo/3] 4 ~ .K~

A t this junc ture we have excluded the (gauge-dependent) zero mode f rom the

sum and set

(9) D ~ -~ ~1~. + aua./M2, with p~ - - - - x ~ .

Three types of summat ion present themselves :

[1 or t t ~ O ~ / n ~ or R ~ u ~ / n "] n l ~ ( n r / ~ ) 4 ~ r R cos (nx~[R) ,

with with 0 <x~ < 2nR. They are far f rom tr ivial to car ry out, bu t in the l imit

of small or large four-dimensional separa t ion r we can m a k e good progress. Consult appendix A. Fo r r<< R we obta in as the contr ibut ion f rom the tower :

(]0) 4z~ Au.q~ --> [~luorba -[- rluarl~o-- 2~u~o~/3]X°(x) +

Q]SANT~TM CORR]~CTIONS IN KALlJZA-KLI~,IN T~I]~OttY ~ 1

where the S teusors and Z~(x) ~re defined in tha t appendix. As such, (10) is not especially useful one should contract over external wave functions or mo- menta in order to recover physically interest ing quantit ies. We can however ven ture the s ta tement tha t the most significant (singular) t e rm comes from the last par t of (10) and is of order Rd/r ~ times a polynomial of degree 4 in xJR. The gravi ton contribution, of order 1/r 2, which must be separately included is relat ively innocuous in this limit and can be safely neglected.

A similar analysis applies to r >>R. Again see appendix A. The tower

of spin-two states yields here

(11) 4 z ~ A , , ~ ~ [(v.~v~ + v,. ' ,~)/ '- ' -- , ~V~ . /3]~R/ (x~ + r*)~..~ n / r ~ .

Now the supplementary gravi ton contr ibut ion dominates the long-range ef- fects and the infinite sum over the massive modes is no longer impor tant . Such conclusions accord perfect ly with the s tandard classifieal, low-energy expecta- tions.

The significance of the above results becomes more t ransparent when we evaluate some scattering process, where the tensor indices contrac t out. Le t us, therefore~ consider ~ scalar field ~b in the full Kaluza-Klein gravi ta t ional metric, having its own propagat ing modes. The ext ra action for this field is

(1~) 8O

d

Since it is the exchange of the spin-2 tower tha t we wish to s tudy, it is enough to oxp~nd (12) to first order in the (~ gr~viton ~) field, whereupon the re levant interact ion Lagrangian is

(13) ze = h"' ~ , . ¢,~/2 -- ~. ¢ , . h , ' /4 .

This has the associated F e y n m a n ver tex (fig. 1)

(14) Fu,(p, p ' ) ---- [(p --P')u(P --P') , -- (P + P')u(P ÷ P'), + 2p'p'vu,]14.

When we subst i tu te the mode expansion,

q~(x) = ~ exp [inx~/R] q~(')(x) ~rt

Fig. 1. - The qs-O-h vertex.

P 1.

p !

q

3 ~ 2 R. D]~LBO'U-RGO and R. o. W~B]~R

into (13) and integrate over the fifth eo-ordinat% the to ta l mode number of the legs in the ver tex must vanish.

I t helps then to consider a scat ter ing process (fig. 2) where the initial scalars are in the ground s ta te n ~ 0 and the final scalars have equal and opposite charge n. The ampli tude for the react ion gives rise to the promised contrac t ion:

(15) /~(o,o;,,-,~(p, k; p ' , ~') -~ K2 F ~ ( p , p ' ) A,,q,(q) Q~(k, k') .

p pr (0 ) • , (n)

l (n)

(0) • ~ ( - n ) k k I

Fig. 2. - ¢(o)~(o)~ ~b(,)~(,) scattering through h (") exchange.

Note the impor tan t point t ha t the ~(") have the same masses M . ~ n / R as the heavy spin-2 fields h("L Inser t ing (14) into (15), we arrive after some work

a t the ampl i tude

T ~ [(p ~ p ' ) ' ( k -~ k') ~- M~] 2 ~_ (polynomial in q2/M~ of degree 3). M~. q~-- M~

Proceeding to the centre of mass f rame where the momenta take on the values

k ' = ( E , - - p ' ) , p ' = ( E , p ' ) , k ~-- ( E , - - p ) , p ---- (E, p ) ,

the high-energy limit, E ~ > > ( p - p,)2, gives

(16a) Ty)(q) --> 2E2.K2[(q 2 + M~) ,

with the Four ier t ransform

(16b) T(~")(r) --> - - E 2 K s exp [-- M , r ] / ~ r

typica l of a ¥ u k a w a potential , bu t now multiplied b y an energy factor, as expected for a gravi tat ional interaction. We m a y convert the answer (16) into a five-dimensional stat ic potent ia l by reinstat ing the relat ive xs-dependence

through

exp E- M.,]eos( °xs) = [exp + x )lR] + 1 1

exp [r/R] cos(x~/R) - - 1 ~- exp [-- n(r - - ix6)/R]]/2 -~ exp [2r]/~] - - 2 exp [r]l~] cos(x6//t ) ~- 1"

QUANTSYM: COR~]~]CTIO~S IN KALTJZA-KL:EIN TI-I~OI~Y 3~3

This gives

(]7) V~(r , :~'~) - - - - (KE) ~ cxp [r /R] cos(x~/ /~)- ]

z r exp [2r /R] - - 2 exp [r /R] c o s ( x s / R ) + 1

In the long-range limit,

V~(r , Xs) -+ ( K E ) 2 exp [ - - r /R] cos(xs /R) /y~r ,

we see tha t the potent ia l due to the exchanged tower is suppressed relat ive to the gravi tat ional potent ia l (KE)2/27~r, as suggested earlier. In the other ex- treme, r --> 0, the tower potent ia l cancels the pure gravi ty t e rm and the delta- funct ion cont r ibu t ions- -connec ted with the thi rd-order q~ pieces of the ex- c h a n g e - c o m p l e t e l y dominate the scattering, again in agreement with the previous discussion.

Summarizing this section, we can s ta te tha t the influence of the whole tower on classical ampli tudes or potentials is negligible at long-range, bu t is ve ry impor tan t for the short-range, Planck limit. Our nex t task is to inves- t igate the quan tum effects of the tower~ when it becomes necessary to in tegrate over all distances; the consequences are far f rom obvious.

3. - Quantizat ion o f the theory.

An a t t rac t ive proposal for quantising the Kaluza-Klein model has been pu t forward by CHO~)OS and APPELQUIST (8). They suggest adding the gauge- fixing t e rm

(AMGM~)2/2~ , where A M : (Av , fl~5) ,

and compensating i t appropr ia te ly by fictitious field terms. Taking the limit --> ~ , the constraint

(18) A ~ G ~ + fl~sGs~. = 0

is implied. Hence the n = 0 (xs-independent) modes yield A z G ~ = 0 or

A z A ~ = 0 and A ~ g ~ = 0

provided tha t A does not act like a derivative. Th a t is, provided one is dealing with an axial gauge for the vector and tensor fields in the problem. For the other modes they suggest taking fl -~ c~ as well, whereupon

~GsN = 0 ,

meaning tha t A , and ~0 lose their x5-dependence and t ru ly represent a massless photon and se~l~r. Thus the only xs-dependence resides in g~, leading to the

3 5 4 R. D ~ L B O U R G O and R. O . W ~ B ~ R

tower of mass ive excited gravi tons found b y SALAM and STRATHI)EE~ aS indeed

we assumed in sect. 2. Unfor tuna te ly , the above gauge-fixing procedure is not so useful if A acts

like a differential operator , as we would prefer. W e shall, therefore, p rov ide

an a l te rna t ive scheme, more suited to our needs. W h a t we are real ly hanker ing

for are gauge fields which obey

aM/'£( 0 ) ~ ----- 0 and a s Gs~ 0

s imul taneously ; because the fo rmer condition leads to a Loren tz - type gauge for the massless gravi ton and the l a t t e r ensures t h a t Au and ~0 have no mass ive

excitat ions. We will~ therefore, adop t the gauge-fixing t e rm

(19)

where M ~1 , ~ N ~ M ~ (~(m~n)lO A~ ~("'") = (6~ a~ + ~,~ '.5 ,~5 ~

(20) A g ~(°'") = (a~U + a~a ~) a(°,"~/2.

This then leads to the fictitious field addi t ion

(21a)

~ g f -- ~ IAM'N(m'n)gY~'(n)\910 k z J-/., 'O,M~ ! /zaO~

for m, n # 0 ,

wi th

(21b) n¢~,.)-- (G~,e -~ Gn;eOu -~ G~e~M)5 ("'")

Combining (20) and (21) we remain with the ghost Lagrangian

(22) . ~ , h o . = --~"5)LC°)(GL",N ÷ G~,.aL + G~b.)co ~(°)-

I n (22) one m a y discern t h a t - (~) and - (~) fields will mix (7) for 1 # 0 bu t not tO 5 CO/x

for 1 ---- 0. Despi te mixing, in m o m e n t u m space it is easy to inver t the to bilinears

and arr ive a t the ghost p ropaga tors

<o4°>(k) ~i° ' ( - - k)> = ilk ~ ,

(22a) <co(~'(k ) co~'(-- k ) > -~ i(Vu, - - ku k,[2 k2) /k ~

k.,l .] (<oJ~'(k)w(:'(--k)><o;~'(klo~(:'(--k)>]= {--V,I2M'. M' i

(22b) n # O : \ < w s (k)w, (-- (") '"' ~--ku/2M,, :l12M~/" ~"' ("' k)><co5 (k) % (-- k)>/

(7) The same phenomenon occurs in ref. (6). In fact for the zero modes, our prop- agators identically coincide with theirs. Minor differences arise for the higher modes.

QUANTUM OORRECTIOlgB I ~ KALUZA-K.L]~IN T H E O R Y ~

I t should be recognized tha t Lagrangian (21) incorporates interactions between the ghosts and the metr ic fields in order to ensure the an]rar i ty of the theory. For the remainder of this paper we shall only be concerned with the quan tum effects on the scalar ] in G and i t will be enough for us to ex t rac t the first order in ] coupling to the ghosts, namely

{~.u f.~(o) ~ (~, ,.,~(o), i (23) ~ / o , (~"~')~o(/'~'//3 + ,~ ~, , , ~ ~ , , ,/13 +

+ (~'co,'°))(~.o~(/.))1/3 + 2(~,~(:')(~,~(:))//3 +

[2(3 o~5 )o~. a ] + 2(~ o,. )(~.~o5 )] + nOO

+ 4(~ ~ ~' : , )(~ ~o':,) i + (~, ~,'(-,)(~ ~o~-,) ] .

This then leads to the F e y n m a n rules for the masslcss and massive ghosts which are contained in appendix B and which are drawn in fig. 3a), b).

We shall make use of these almost at once.

ca(O) ~(0) ta(n) -- -']b"-- -- -- 1 -- - -- -.~-- . . . .

s,~ k -.'*--s,~ (~) k

f f a) ~)

Fig. 3. - a) The m~°)-eD (°)-] vertex; b) the m(")-(D (-)-] vertex.

k, (~)

n4:0

4. - C o r r e c t i o n s to (fP).

We began with the (classical) assumption tha t the scalar field had an ex- pectat ion value of one or t ha t the (, quan tum ~> field was ] - - - - ~ - 1. In fact this scale has to be changed by the quan tum corrections to / induced by the tadpole graphs of fig. 4. Such graphs have a dangerous l / k 2 pole in the I-line which can be avoided only if we suppose tha t the self-energy effects (to be confirmed later) produce a self-mass

~ ----. H ( k = O)

in the ]-field. In tha t event, we m a y evaluate (~p) b y summing over all the gray-

w(n) h (n)

- @ • I I $

\ / / /

a) b)

Fig . 4. - a) G h o s t c o n t r i b u t i o n s t o ( / ~ ; b) s p i n - 2 c o n t r i b u t i o n s to {]~.

25 - I I N u o v o Ct~nenlo A .

35~ R, D]~LBO~RG0 and ~. o. W~BER

iron excitat ions in the tensor loop and summing over the ghost tower as well. The ghost-] interactions were spelled out in (23) and the h-h-] ~ couplings

arise through the interact ion

(24) £#ah, = - - (hi'V,6 ht,,,5 - - h%,s h',,5) /4q~ .

Sewing the vertices (see appendix B) with tho closed-loop propagators we obta in the correct ion terms

~ 2 2l

(25) (f~ -~ i ~ - 7 ~ j ( ( 2 u ) ~ , , d~'q rM2~A.~,,-'"'t.~--Au"u,,("'(q)}.t ~ . . ,,., +

, /~ (n) ~ (n} 4M, A55/3 -~ + 2 M , , q Au , / 3 + ~/.'A..(-'/3]

where the two-index propagators refer to the ghost and can be read off f rom (22). W e might also point out t ha t the calculation is carried out in the f ramework of dimensional regalar izat ion so tha t our integrals are being cont inued to 21-dimensions before going to the limit l = 2. Since the zero modes give zero integrals by the usual rules, the expression for ( ] ) simplifies to

• X2 ~ 0 f d'~q (26) (1) = ~ ~ ~ ((2/Jr- ]) [1-- (l -- 1) M'. l (q ~ - M.')] +

.Ks ~ f d,,q n ' l t t ' + ( z - q~13~)) = ~ ~ (1 - ~)(2~ + 1) . . _ (2~) . q ~ - n~l ~ "

Proceeding to 1 -~ 2,

(27) ( D = ~Lm~ (2Z + 1)(1 - - l ) r ( 1 - - l ) K ~ - " ~ n " / ( 8 " ) ~ ~ =

= lira [F(2 - - l) ¢(-- 2/)] 5K~/32~ ~ R ' # ~ ---- 5 ! $(5) K ' / ( 3 2 ~ * ) ~ R ' t t ~

since ~'(-- 4) = 4 ! $(5)/32~ 4 and $(5) -~ 1 + 2 -5 + 3 -5 -}- ... ~ 1.2. I t is interest ing t ha t the sum over the massive tower provides a finite answer

in the form of a Riemann zeta funct ion and t h a t the sum over the ghost tower gives zero (in our version of gauge-fixing and even in the version of (e)). The final answer (27) depends on the induced mass, the Newtonian constant and the P lanek volume. Since dimensional reasoning tells us t h a t p ~ K ~ / R 4,

the vacuum expecta t ion value undergoes a small shift (dimensionless in our

units). We imagine t ha t the same phenomenon will occur in more sensible Kaluza-Kle in models, with the shift of ~ depending on the relat ive size scales

which characterize the internal space.

5. - Correct ions to the sca lar m a s s .

The classical equations tell us tha t ~v is a massless field. Since we know tha t quan tum renormalizations break the scale invariance, we can ant ic ipate

QUANTUM CORRECTIONS IN KALUZA-KLEIN THEORY 357

that pl will develop a mass through its interactions with the other fields in the metric. We, therefore, study the vacuum expectation value (ff) to determine the magnitude of the induced mass. There are four diagrams to be calculated in this context at one-loop level, as shown in fig. 5. As we shall evaluate the

Fig. 5. - a) Two-particle tower contributions to <f f ) , b ) tadpole tower contributions to <ff>, c) two-partiole ghost contributions to <ff>, d) tadpole ghost contributions to <ff>-

various contributions in the zero-momentum limit, they only involve the massive modes of the ghosts and the tensors. (The massless modes give zero in the infra-red.)

The three- and four-point vertices for fhh, fo6, ffhh and ffww are annotated in appendix B and are

Altogether, the self-energy reads

We need not worry about the ghost interactions because they are polynomials in the momenta and are multiplied into ghost propagators which are also polynomials in the internal momentum q2 times poles at q2 = 0. Such integrals vanish in dimensional regularization. Hence the last two terms in (28) may be safely dropped. The first term or two-graviton diagram (fig. 5a) ) , dimension- ally continued, yields

358 :R. D:ELBOURGO and R. o. w ~ B ~

U p o n carrying out the integrat ions wi th M~ ~ n / R ,

llEa ~ --_ 8 R - 3 ~ ( - - 2 1 ) I ' ( 2 - - / ) ( 8 / 6 - 20/4 -{- 42l a -}- 83/3 ~- 1 - - 11) (4~ ) , ( f z - - 1)3 (z - - 1)

Taking l -> 2, a zero f rom the zeta funct ion cancels the pole f rom the gs ,mma

funct ion to leave (8),

(30) /-~'~ --4 238.5 !~(5)/(6~) 3 ( 2 # R ) ' .

The tadpole d iagram (fig. 5b)) yields a similar answer~

• ~ d~q M~(q3/M~-- l) + =

= - - ( 2 / + 1) F ( 2 - - / ) ~(-- 2 / ) / ( 4 ~ ) , R ~ , ~ 2 .5 ! ~ ( 5 ) / ( 4 ~ ) ~ (2 ~J ~) ' .

Since//Eo~ = HEa~ = O, the net result is

(32) / / (k ---- 0) = - - #3 = 970.5 ! ~(5)/(12~) 3 ( 2 ~ R ) ' .

This means t h a t the q u a n t u m corrections genera te a nonzero mass p a t one- loop order. The fac t t h a t i t is t achyonic is not especially worr isome for we

know t h a t the higher-order loops will a l ter the precise n u m b e r ob ta ined above. Howeve r this is the mass t h a t should be used in (27) to es t imate the magn i tude

of ( ] ) a t the same level of approx imat ion .

6. - S c a t t e r i n g correc t ions .

Our final exercise will be to de termine the effects of the spin-2 tower on

]-1 scat ter ing. This is where the nonrenormal izabi l i ty of the model s tr ikes

mos t severely. I n the previous cases we were dealing with quant i t ies which

d isappeared in the absence of the tower and which underwent finite correc-

tions. Fo r the first t ime now, we are hav ing to calculate a process which is

intr insical ly d ivergent even when the mass ive exci ta t ions are disregarded.

The quest ion then arises as to whether the sum over the modes improves or

worsens the si tuation. Unfo r tuna te ly our answer favours the l a t t e r a l te rnat ive . There are four dist inct contr ibut ions to f-] scat tering, depicted in fig. 6,

to which one should add the eight crossed counterpar ts . I t m a y be wor th r emark ing t h a t there are no analogues of the g raph of fig. 7 which involve the

(8) Here we apply ~(1-- ~) ---- 2(2~)-3~(z)F(z) cos (~z/2).

Q~/ANTUM CO!~RECTXO1WS II~T YLAL~ZA-]KL:EIN THEOI%Y ~59

I (~'~(n) I (n)/ i(--n) (n) (-~) I k ~ k' ' k'--- 1 " ,

• 4 ..... ~ "q¢

a) ~) c) a)

Fig. 6. - a) Box graph for <]]1/> with tower loop, b) seagull graph for <1]]]> with tower loop, o) box graph for <1/]1> with ghost loop, d) seagull graph for </f/I> with ghost loop.

°hi l Fig. 7. - Two-vector or two-graviton contributions to <////>. These cannot involve exchange of modes n # 0.

tower - - those graphs can only involve zero-mode exchanges. Again we make use of the F e y n m a n rules of appendix B and we have to realize tha t at zero external m o m e n t u m the ghost loop contributions lead to zero integrals. Note always t ha t there are only two momen tu m factors for the vertices h-h-l". The complete scat ter ing ampli tude is a highly complicated affair. Ra the r t han evaluate i t in complete detail, we shall concentra te on the dominant con- t r ibut ion arising from the t e rm

( n ) ~ 4 2 A~,,qo(q) 2q~,q~qoq,/3M,(q - - M~,)

in the tensor propagator . Also, for simplicity, we shall look at elastic forward scattering where p ---- p ' , k ---- k' and we will of course stick to the mass shell p2 __ k~ = 0 .

Then diagram 6a) has the dominant behaviour,

[ 5 ~ , ~ d d q (p .q)4(k .q) , [p .q 4- q2]~[--k.q 4- q~]2 (33) T't'~,~ - - i ~ \qM~] J (2=)' (q2_ M~)2(q2 4- 2p.q - - M,~)(q 2 - 2k.q - - i ~ ) "

I ts crossed counterpar t (legs p and p ' interchanged) gives

( 5 ~4~ ddq ( p ' q ) ' ( k ' q ) 4 [ - - p ' q 4- q~]S[--k-q 4- q~]~ (38') T,:~ _ i ~,qM:] ,) (~=) (q2 _ M~)(q~ _ 2p .q - - M~)(q ~ - - 2k .q - - M~) n

and the final crossing (k' +-~p' in the original figure) gives

(33") T"~,J,-~--i ~ \ q ~ j .] ( ~ ) 4 (p .q)2(p .q _ p . k ) 2 ( k . q ) ~ ( k . q ÷ k.p)2.

(q~ 4- p .q)(q~-- q. k )(q~ -4- 2p.q - - k .q - - p . k )(q 2 4- p .q - - 2k .q - - p . k ) 2 2 , ) (q~-- M~)(q , , - -_k .q - - M~)(q ~ 4- 2q .p - - M~)((q 4- k - - p ) ~ - - M~)

360 R. DELBOURGO a n d R. o . W E B E R

The leading par t of the result is of order ( k . p ) ~° and i t can be ex t rac ted b y expanding each of the denominators in the manner

(q -t- 2p.q -- M,)-~ = (q~-- M,) -1 ~ [--2p.q/(q ~ - M,)],.

After considerable labour, and with some assistance from the compute r package REDUCE, we find tha t the high-energy behaviour adds up to

,.~ 20001B~e[ (27~R)te (P'k)~° (~-~__l ) (34) Tt°t 3 ! ] 6 ! (4:~) 2 ,.-,~

where Ble comes from the sum over massive modes,

(n~) ~-1° as l -+ 2 .

The contr ibut ions f rom fig. 6b) are of lower order in (p. k) and have, therefore, been omi t ted above.

The final ampli tude (34) spells disaster f rom the point of view of renor- malizat ion and perhaps should have been expected from the outset : snmmlng over the massive tower does nothing to cure the incipient renormalizat ion disease of the original ground-s ta te theory. Thus if quant ized Kaluza-Kle in models are to be accorded full physical respectabi l i ty i t will be necessary to ameliorate the high-energy problems of the light mass sector b y modifying the action to include terms R ~ at the ve ry least, or in some other way.

* $ *

We wish to t hank Dr. G. THOMPSON for m a n y per t inent comments on the

gauge-fixing procedure in five dimensions.

APPENDIX _A_

Summing over the massive modes.

Refer to formula (8) in the t ex t where we are faced with sums over modified Bessel functions. In the l imit of small z = M ~ r we ma y approximate

K I ( z ) by z -1 •

Q U A N T U M C O R R E C T I O N S I N K A L ~ J Z A - K L E I N T t t I ~ O R Y 361

Consequently, the three basic sums will reduce to

(A.1) i ) r - ~ c o s ( n x ~ l R ) : e x p [ i n x J R ] - - I 12r2=[27~RS(x~)--l]12r~=X"(x), 1

(A.2) ii)~v~z ~cos(nx~/R) [6 I =__ x~(x)_~,, ~ n ~ - = ( ~ r - ~ ) 2x~ -t- ~tR~ [

r .I (A.3) iii) ~,,~e~,~ ~1 c°s(nx~lJ~)r~n a - - (~"~v3e~zr-2) i ~ ]2/~2 + ] 2 R a 48R'J i

The terms on the r ight are special cases of the general formula (")

co

2 ~ n -~" cos(2uny) = (-- 1)~+~(2~)~'B2~(y)/('2r)!

and the tensors in (A.1) to (A.3) are

(~.4)

(A.5)

~- 384x~ xv x~ x./r ~"

sui tably i a t e rp re ted as distr ibutions (10) when t -+ 0, so as to produce delta- functions (or thei r derivatives) af ter contract ion over indices.

Contrariwise, if z = zg, r is large (the conjugate mo me n t u m is thus small in re la t ion to the Pl~nek mass), we may neglect the ~3/M~ te rms in the prop- ag~tor and simplify the sum to

Using an integral representa t ion of K1, the square brakers on the r ight of (A.6) can be re-expressed as

f dt ~ 1 2( t2 -t- r~)~ -:: exp [in(t -~ ~rs)/R ] - - r-- 7 = -~ ~Ri[ (x 5 -~ 2 n ~ . R ) 2 -~ r2] - t - - 1]r ~ =

co

[~R(x~ -~ r~) -'# - - 1/r 2] + (2~R) -~ ~(n~ ~- (r/'2~R)2)-# --+ 7~R/(x~ + r~) "-# 1

(9) W, MAGNUS, F. 0BERttETTINGER and R. P. So,I : Eormulas a~d Theorems ]or the Special Funvtions oJ Mathematical Physics (Springer-Verlag, New York, N.Y., 1966), p. 468. (lo) I. M. GEUFA~D and D. SHILOV: Generalized 2'uqwtions, Voh 4 (Academic Press, New York, N. Y., I964).

3 6 2 R . D ~ L B O U R G O a n d R . O. W~B~.I~

upon approximat ing the sum as an integral in the l imit as r -+ oo. Thus the sum over the massive modes provides eq. (~1) quoted in the text .

A P P E N D I X B

Feynm~n rules for scalar-field interactions.

The rules for the propagators of the physical fields and of the ghosts are to be found in (7), (22a) and (22b). They will not be repeated here. We will instead provide the vertex rules involvong the scalar / which s tem from (23) and (24).

The three-point/-0)-5 vertices are of two varieties: when the mode number is zero, there is no mi~ing between w5 and ~o~ components; otherwise there is. The actual factors can be extracted from (23) af ter going to momen tum space and they should be read in conjunction with fig. 3.

F ,o ,~ k') = k~(k - - k')~/3 + k~k./3 + k . k ' ~ , [ 3 , Y \'~,

r:;,(k, ~ ' ) = 2~.~'/3 , ( B . 1 )

(n) ~ z~(n)/Z" 1 F~,,(k, k') = M,,n~,,/3 , ~ ,~,.~, k') = 2M,,(k - - k ' )d3 O (n) (~) ~ " F{. (k, k') 2M,,k,/3, = I"~ (k, k') = 4M*.,/3 n >

I t is to be noted tha t these polynomials in the momenta are eventual ly multi- plied by the propagators (22a), (22b) so t ha t if one is working out Green's functions a t zero external momentum, the ensuing intagrals will give zero by the rules of dimensional continuation. ~or t h a t reason we shall refrain from exposing the rules for the f-]-~o-~ vertices t ha t come from (22). We m a y dispense wi th them.

The f-f-h" interactions develop through the expansion

9 -~ = (1 + 1)-~ = 1 - - f + I ' - - . . .

applied to (26) as well as to

(B.2) ~ ~at.t(h , 9 ) = (h%h..,~ + 2h~'ah.,: + 3h~'ah..,.)9~,/129 +

+ (h 2 terms)a'9,.9,. /3692 .

Only the first par t is relevant to fig. 6a) and i t gives the vertex rule (all momenta are incoming and a delta-function expressing momentum conservation is understood) :

- - (~.qp~q~ + ~?~opoqa + ~lmpoq. + V~apoq. +

+ ~.qp~,ra + ~.ap.rQ + ~ e p . r a + ~.¢p.re)]24].

QUANTUM COI~I~CTIO]~S I~ KALUZA-KL~IN TH]~O~Y 363

F o r the ca l cu l a t i ons in sect. 5, i t is suff icient to go to t he z e r o - m o m e n t u m l i m i t of (]3.3). T h a t v e r t e x is q u o t e d i n t h e t ex t . Howeve r , i n sect. 6 the fu l l expres s ion is needed .

• R I A S S U N T 0 (*)

Sono state esaminate alcune delle correzioni quantiche ad un loop nel modello di Kaluza-Klein a 5 4imensioni, che sono dovute al numero infinite di eccitazioni a spin-2 dotate di massa per scoprire come la torre di livelli pub condizionare i r isultat i classici. Risulta the le quantit~ the si annul lano classicamente, come (~0} e la massa del eampo scalare ~o, ricevono contributi finiti che sono determinati dal raggio di Planck. D'al t ra parte, ampiezze come lo scattering ~-~0 the sono infinite ad un loop, anehe quando si scartano i modi dotati di massa, non hanno divergenze migliorate dall ' inclusione della torre. Concludiamo ehe un modcllo a pid dimensioni non si dovrebbe considerare seriamente come candidate per una teoria fisica quantizzata sc (come minimo) il tron- camento del mode zero b gi£ rinormalizzabile, perch6 solo allora b probabile che i modi di ordine superiorc producano correzioni finite.

(*) Trad~z ione a eura della Redazione.

Pe31oMe He IIOJ'IytIOHO.