Splitting the superstring vacuum degeneracy

Volume 174, number 1 PHYSICS LETTERS B 26 June 1986

S P L I T T I N G T H E S U P E R S T R I N G V A C U U M D E G E N E R A C Y

Mark E V A N S and Burt A. O V R U T 1

Department of Phystcs, The Rockefeller Umversttv, New YorA, NY IO021, USA

Received 14 November 1985, revised manuscript received 9 April 1986

A general method Js presented for calculating the energy sphnmg of gauge field vacua to the one-loop level on compact. multiply-connected manifolds Two exphclt examples are given m which non-zero gauge vacua have the lowest energy, and the gauge group is spontaneously broken

It has been shown [ 1] that the superstrlng is anom-

aly-free if and only ff the internal gauge group IS E 8 X

E 8 or Spin (32)/Z 2. The spacetlme in which the super-

string is defined can compactify to M 4 X K, where M 4 is MlnkowskI space van K are st,x-dimensional, RIccv

flat, Kahler manifolds called Calabl-Yau spaces [2].

When the background gauge field is identified with

the spin connection, the gauge group is reduced to E 6

X E 8 or 0(26). There are many Inequlvalent compact-

fflCatlons, and It is presently not known which one is

chosen in superstrIng theory. Calabl-Yau manifolds

can be diwded into two categones, those that are sim-

ply- connected and those with non-trivial fundamen-

tal group. Typically, the simply-connected spaces lead

to a large number of quark and lepton families, and

therefore appear to be phenomenologIcally unsuccess-

ful Mult@y-connected spaces, however, can have

many fewer quark and lepton families. For example,

multiply-connected Calabi-Yau manifolds with three

[3] and four families have been constructed. Such

manifolds are obtained as follows.

Let K be a simply-connected Calabl-Yau manifold

and let H be a finite group which acts freely on K

Then K' = K/H is a multiply-connected Calabl-Yau

manifold with

~r 1 (K/H) = H . (1)

' Work supported in part under the Department of Energ 5 Contract Number DE-AC0281er40033B 000

i On leave of absence from the Umverslty of Pennsylvama, Phlladelphm, PA 19104, USA

0370-2693/86/$ 03 50 © Elsevier Science Pubhshers B V

(North-Holland Physics Publishing Division)

Henceforth, we assume that superstrlng theory selects

a multiply-connected manifold of this type.

Vacuum gauge fields, Bua , are those fields which

satisfy Fu~ a = 0 For example, Bua = 0 is a vacuum

field. On simply-connected manifolds all gauge field

vacua are globally gauge equivalent to Bua = 0. How-

ever, as pointed out by Hosotanl and Toms [4], inul-

tlply-connected manifolds admit non-zero vacuum

gauge fields which cannot be made to vanish globally

by a gauge transformation Associated with Bu a, and

any closed curve % IS the path-ordered Walson loop

(2)

where the ir a are the Lie algebra generators of the

gauge group U B IS lnvarlant under continuous defor-

mations of 7. Thus, on a simply-connected manifold

U B = 1. However, on a nmltlply-connected manifold

U B 4= 1 necessarily. Specifically, consider the Calabl

Yau manifold K' = K/H. Let Bua be a vacuum gauge

field on K' and let c'dg = { U B } Then, as an abstract

group 9f C H Furthermore, denoting the gauge group

E 6 X E 8 or 0(26) by G, c~ C G. If ~ is the subgroup

of G that commutes with ~ then G IS spontaneously broken to ~ . That is, if

[q , ~ 1 = 0 (3)

the n

G -+ ~ . (4)

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We emphasize that there may be may be many lnequiv- alent vacuum gauge fields, B(Z) a, on K'. Associated with eachB(0, a are the groupsC'd(0) and ~ (0 In general, ~ (0 =~ ~q) . It follows that there may be many breaking patterns for G [15]. At tree level, the energy of gauge field Au a on K' is

E~ ° ) = f d6xx/g~Fu~aF a"~ . (5)

Therefore,E(0)~ = 0 for all/ Hence, vacua B 0) a / ~ t , ! t.t have degenerate energy at tree level and the theory cannot choose between them. Can radiative corrections break this vacuum degeneracy? The radlatlvely correc- ted energy of B(t)ua is given by

exp(--EB(0) = Z [B 0) ] , (6)

where Z [B(0] is the vacuum-to-vacuum amphtude in the presence of the background field B(1)ua. Since there is no symmetry connecting B(i)p a with B(I),, a, Z [B(0] 4: Z [B (1) ] and, therefore, EB(O ,~ EB 0)" lqence, we expect radiative corrections to break the vacuum degeneracy. The gauge vacuum with lowest energy IS the correct ground state of the theory. In this paper we do the following.

(1) We give a general method for calculating EB( 0 to the one-loop level on compact, multiply-connected manifolds.

(2) We construct on an explicit manifold ($3/Z2 with gauge group SU(3)) the inequwalent gauge field vacua B(Oa (B(1) = 0, B (2) 4= 0), and calculate the associated groups c'df(z), ~ (0 ( ~ ( 1 ) = 1, ~(1)= SU(3) and ~ ( 2 ) = Z2" ~(2) = SU(2) X U(1)). Using the method in (1) we calculate the EB( 0 to the one-loop level, and show that the tree-level vacuum degeneracy IS hfted. We fend that B(Oua 4= 0 can have the lowest energy and, hence, that the gauge group is spontaneously broken to a smaller group (SU(3) ~ SU(2) × U(1)).

(3) We give the results for another manifold (S3/Za)

and gauge group (SU(2)). All our analysis will be performed for a field theory

rather than a full string theory, to which, it has been argued [6], the field theory IS a poor approxlmanon. One may nevertheless hope that general features of the field-theory results survive in the string theory, and the fact that the field-theory results are finite (see be- low) suggests that perhaps this hope is not entirely vain

First we present a method for calculating EB( 0 Let M be a n-damensmnal, compact, stmply-con-

nected manifold with metric tensor guu and vamshing torsion Choose gauge group G and let the quantum fields on M be (a) gauge fIelds,Aua (b) ghosts, c a (c) complex scalars, ~b / and (d) fermions, flA. The gauge fields and ghosts transform as adjolnt representanons and the scalars and fermlons as arbitrary (In general reducible) representations of G. Let H be a finite group that acts freely on M. Then M' = M/H is an n- dimensional, compact, multiply-connected manifold. We first show how to calculate Z[B(O] to the one- loop level on M, and then how this result can be mod- Ified to obtain EB( 0 on M'. Let Bp a be a gauge field vacuum on M and expand

A , a = Bua + Qu a . (7)

Then [7]

Z[B] = f [dQu a] [dc ?a] [dc a] [dO ?t ] [d~b t] [ d r ?A]

× [d~A] exp(-- f dnxx/g.t2), (8)

where

.(9 =./2y M +-QGF +'QGT +-~q~ +.Off . (9)

The first term, f~YM, is given by

" ~ Y M = (1/4e2)Fuv a ]A Fa#VlA (10)

where

Fuva]A = C-l),uAv a - Q) vAu a + cabcApbAv e (1 1)

and ~ u xs the covarlant derwative on M. Define the background gauge-covarlant derivative by

DlBuab = CD ugab -- l(TC)abBu c , (12)

where (Te) ab = - w cab. Then

Fur a [ A = D [Buab Qv b -- D] B v ab Qu b

+ cabcQubQvC (13)

The first two terms contribute to the one-loop level. Henceforth, drop all terms (such as cabeQubQve ) which contribute only to the two-loop level or higher The Wilson loop associated with Bua is

UB(x)=Pexp(- , ~ TCBuC(v)dyU). (14)

X 0 ,"[

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UB(X ) is independent of path 7 since nl(M ) = 1. Wilson loops have the property that

CD UB ab = --IUBad ( Tc)db B S . (15)

Therefore

D I B uab Q v b = U B lad @ u ( U Bde Q ve ) . (16)

Defining

QB laa = UBab Qub (17)

It follows that

F~vaIA = UBlab(@,QB v b - c-O vQB la b) • (18)

Integrating by parts, we find

.t2yM = (1/2e2)QB ua(-gUVC] + <0 vcl)la)Q B v a (19)

where [] = @la@la. Note that B j has been absorbed Into QB laa by simply gauging it away, something we can do on the simply connected manifold M. The re- roaming terms in eq. (9) can be calculated m a sum- lar way. For an arbitrary field ~, define

~B = UB* (20)

Then Bla c~ can be completely absorbed into the ~/'B' Working in the covariant c~-gauge, and keeping only those terms contributing to the one-loop level, we find that

.12 = (1/2e2)QB u a [-guvD + R lay

+ (1 -- 1/a)"~la@ v] QB v a + c~3a(M~)CB a

+ dp;t(--[~)~B i + ~ B A (i ~;ff) ~IB A .

Energy E B is expected to be gauge independent. Henceforth, take a = 1. Since o is quadratic m all fields we can do the path integral. The result IS that

Z[B] = Z Y M Z G T Z o Z ~ ,

where

Z y M ° : H [det(-gUVr~v+RUV)] 1/2 dim G

ZGTOC [-[ de t (_ [ ]S ) , dim G

Zo~x 1-1 [det( -DS)] -1 , dun rep

H [det(-cq F +~R)11/2 Z ~ oc dun rep

(21)

(22)

, ( 2 3 )

(24)

(2S)

(26)

Subscripts V, S, and F mdmate that the laplaclans act on vectors, scalars, and fermlons, respectively. One now solves for the elgenvalues of the operator arguments of the determinants. Having found these, Z [B] on M IS easily calculated

However, we are interested in calculating Z [B], not on M, but on M' = M/H Ttus can be done as follows.

(1) Restrict BuS to be lnvaraant under H This is necessary to insure that Bua IS defined on M' as well as on M.

(2) Construct on M the elgenfunctlons, gB, of the operator arguments of the determinants, and define functions g by

e = UB 1 (5 B . (27)

(3) The expression for Z [B] on M' is formally Iden- tical to that o n M (eqs (22)-(26)) . However, on M' the determinants are products of only those eagen- values whose corresponding ~ functions on M are H Invaraant (that is, whose g functions are defined on M' as well as on M).

Rules (1) - (3) allow a complete determination of Z [B] on M'. The energy E B on M' can be calculated using eq. (6).

We would like to use this formahsm to calculate the sphtting of the gauge field vacumn degeneracy on Calabi-Yau manifolds K' = K/H. Unfortunately, evaluation of the elgenmodes of the relevant operators on K IS very difficult since the metric tensor is unknown. Instead, we consider the simplest multiply-connected manifold which admits globally defined spinor fields, M' = S3/Zn (S 1 with gauge group SU(N) has been studied by HosotanI [4] However, he found no SU(N) breaking on this manifold).

Let M = S 3. The Z [B] is given by eq. (22) where

/ Y M cc I"I [det(gUU(-Dv + 2)) 1 1/2 , (28) dam G

ZGTCC I-I det(--g3S) , (29) dun G

Z~cr l-I [det(-Vls)] 1 , (30) dun rep

Z¢ = H [det (_Dv +~)11/2 (31) dun rep

These determinants can be calculated since S 3 admits

65


SO(4) as its isometry group. Evaluating [] on S 3 we find that

3 = 2 j 2 3 - [ ] v + 2 = 2j2 , - [ ] S = 2J2 , --[]F + ~ + ~, , (32)

where j 2 as one of the Caslmlr invarlants of SO(4). Clearly Z [B] can be determined if we can find the scalar, spinor, and vector harmonics of SO(4) on S 3 . A full discussion o f these harmonic functions will be given elsewhere [8]. Here we present only the highest- weight scalar harmonics

f l ( $ , 0, q~)B : Xl(Sm ~)l( sm O) ledo , (33)

where l = 0, 1, 2 . . . . and N l are normalization con- stants. In general, it as necessary to know the entire irreducible multlplet . However, for the discussion in this paper, it suffices to know the degeneracy of the multiplet and its j 2 elgenvalue. These are

degeneracy = (l + 1) 2 , 2J 2 = l(l + 2 ) . (34)

Z [B] can now be calculated on S 3. However, we are interested in calculating Z [B], not on S 3 , but on M' = S3/Zn To proceed further we must take an explicit example. Consider Z 2 = ( 1, - 1 ). The action o f Z 2 on S 3 IS defined so that - 1 takes any point on S 3 to its antipodal point. That is

- 1 ( ~ , 0, ~) = (~ - ~, 7r - 0, ~ + ~ ) . (35)

Note that the - 1 action on S 3 has no fixed points. Therefore, Z 2 acts freely on S 3, and $3/Z2 is a manifold with

~1 ($3/Z2) = Z2 " (36)

Take gauge group G = SU(3). We want to find two vacuum gauge fields B (1)u a , B(2)ua on S 3 with the following properties.

(1) Both B (I)u a, B(2)ua are invariant under the Z 2 action. Hence, they are defined on $3/Z2 as well as on S 3 .

(2) On manifold $3/Z2

~ ( 1 ) = 1 , ~ ( 2 ) = Z 2 • (37)

In general, B ua = ( B j , Bo a , B4~a). Take

B (1)taa = (0 , 0, 0 ) . ( 3 8 )

B(1 ) a clearly satisfies proper ty (1). Since for any SU(3) representation

UB(a)(Zr -- ~, 7r -- 0, ~ + 7r) = 1 (39)

proper ty (2) is also satisfied. Now take

B(2)ua = (0, 0, 263a) . (40)

B(2)ua satisfies proper ty (1). For the 3 and 3- represen. tations of SU(3) we find that

Ua(z )(rr - ~, rr - O, ¢ + ~r)

- t - 1

= (41)

1

U8(2) has similar properties in any other representation of SU(3). It follows that B(2)ua satisfies property (2). Note that

~(1) = SU(3), ~(2) = SU(2) X U(1 ) . (42)

We now must find, for each vacuum B(Oua , the functions g (defined in eq. (27)) on S 3 which are lnvarlant under Z 2 (an this example the gB are the SO(4) scalar, splnor, and vector harmonics). We present the results for the scalar harmonics only. First, note that for the highest weight

f/(rr - ~, rr -- 0, ~ + rOB = ( - 1 ) 9 } ( ~ , 0, ~ )B. (43)

It can be shown that all (l + 1) 2 functions in the Irre- ducible S0(4) multlplet generated from f l transform in the same manner. (This IS not true for other group actions, for example Z3, on $3). Henceforth, take scalars and fermaons to be m a 3 or 3 of SU(3). For vacuum B (1)u a we fred, using eq (39), that

f } ( ~ - ~ , ~ - 0, ~ + ~r)

= ( - 1 ) l ( 1 1 1) f z l (~ ,O,O)B , (44)

where i = 1...3, and f l denotes any function in the irreducible SO(4) multlplet. It follows that the Z 2 invariant scalar harmonics on S 3 are SU(3) triplets with l = 2n (n = 0, 1 ,2 , .. ). Now consider vacuum B(2) a ~t " We find, using eq. (41), that

f ' t ( z r - ~ , 7r - 0, ~ + 7r)

(1 r) = ( _ l ) t - 1 f ' l ( ~ , 0, ~ ) e , (45) 1

where i = 1 ...3, and f / d e n o t e s any function in the Ir-

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reducible SO(4) multlplet. It follows that the Z 2 m- variant scalar harmonics on S 3 are (a) SU(2) doublets with l = 2n + 1 (n = 0, 1,2, ..) and (b) SU(2) smglets with l = 2n (n = 0, 1, 2 , . . ) . Using these results (along with the results for spmor and vector harmonics) one can calculate Z [B 0)] on M' = S3/Z2 . It is clear from eqs. (44) and (45) that

Z[B (1)] 4 :Z[B (2)] . (46)

For gauge and ghost fields only we find that

= ~ [ -2(2n + 1) 2 In ~1 ~ E B ~ 2 ~ n l

+ 8(n + 1)21n ~2 + 4(2n + 1)(2n + 3)ln ~3

- 16n(n + 1)ln ~4]

4

n = l c~=l

where

(47)

~1 = 4n(n + 1), ~2 = (2n + 1) (2n + 3) ,

~'3 = (2n + 2) 2 , ~4 = (2n + I) 2 . (48)

Note that eq. (47) is primitively positive definite. The actual evaluation of the energy splitting.requires some care m the manipulation of divergent series. We regu- late equation (47) using the Pauh-Villars method [9] as follows. Introduce R additional fields with masses M l and alternating metric e z such that (e 0 = 1,M o = 0)

R R

e, = l~O e l M ' = .. = O , t=0 =

R

23 e l l nM t = - l n M . (49) t=l

Then eq. (47) becomes

(EB(1 ) - EB(Z))reg

A 4 R

= h m 2 3 2 3 2 3 e t f a ( n , ~o, +Mz2) • (47') A - ' ~ n = l c~=l .,=0

Eq. (49) guarantees that the hmlt A -+ ~ is finite. A tedious calculation shows that (47') also remains finite and positive as the regulator mas M -+ ~.

We conclude that gauge and ghost contributions to

the radlatwe corrections lower the energy of the broken vacuum relative to the unbroeken vacuum. Hence

su(3) --, su(2) x u (1 ) .

Adding fermlons and scalars we find the following. (1) Fermlon contributions to EB(1) - E B (2) vanish

to the one-loop level. (2) Scalars tend to restore the SU(3) gauge sym-

metry. The contribution to E B ( 1 ) -- KB(2) from massless scalars is infrared divergent, presumably m&catlng that some sort of Coleman-Wemberg mechanism [10] is operating.

As a second exphclt example consider Z3, M' = S3/Z3 , and gauge group G = SU(2). Here, we simply state the results. As always, there is the trivial vacuum B(1 ). a = 0. However, there are now two non-trwlal vacua B(2), a, B (3)u a whose energies are degenerate by charge-conjugation invanance. Note that

~ ( 1 ) = S U ( 2 ) , ~ ( 2 ) = ~ ( 3 ) = U ( 1 ) . ( 5 0 )

In this example we find the following (1) The gauge and ghost field contributions to E B ( I )

- EBO ) are negatwe definite. Therefore, these fields favor the trivial vacuum and unbroken gauge symmetry

(2) Fermlons contribute to the one-loop level. Their contribution to E B ( i ) - EBO ) is posmve definite. Therefore. for sufficiently large number of fermlons the non-trwial vacua have lowest energy and

SU(2) -+ U(1) .

(3) Scalars tend to favor the trwlal vacuum and restore SU(3) gauge symmetry (again, massless scalars suffer from infrared dwergences). These calculanons will be described in a greater detail elsewhere [8].

The models we have considered are not supersym- metric. Plausibly, Bose and Fermi mgenvalues cancel when supersymmetry is unbroken and the degen-

eracy is not lifted, so that our formulation is appro- priate only after supersymmetry breaking. However, we remark that the non-renormahzatlon theorem [11] is proved only for chlral, not vector, superfields.

In summary, we have described a method for de- termining the gauge group symmetry breaking in superstring theory. Furthermore, we presented concrete examples In which non-zero gauge field vacua have the lowest energy, and the gauge group is spontaneously broken.

67


We wou ld hke to t h a n k J. Weeks and N K u h n for

m a t h e m a t i c a l advice.

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Documents

Splitting the superstring vacuum degeneracy