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Gauge group breaking by Wilson loops
Volume 200, number 3
PHYSICS LETTERS B
14 January 1988
GAUGE GROUP BREAKING BY WILSON L O O P S A.T. DAVIES and A. M c L A C H L A N Department of Physws and Astronomy, The Umverstty, Glasgow G I 2 8QQ, Scotland, UK
Received 10 October 1987
We examine the breaking of gauge symmetries b~ Wdson loops in the Hosotam-Toms model by determining the background gauge field which mmlmlses the one-loop effective potentml for massless D~rac fermlons For anti-periodic fermlons, all gauge groups remain unbroken For periodic fermions, the groups G2, F4 and E8 are broken by quantum corrections due to fermions in any ~rreduclble representatmn, whereas E6, E7 and the classical groups only break if the fermmn representation is in the same congruency class as the adjomt
The p r o h f e r a t i o n m recent years of gauge theorxes on spacetimes o f d i m e n s m n greater than four has stimulated interest m a new m e c h a m s m for gauge s y m m e t r y breakmg, variously referred to as fluxbreaking or the H o s o t a m m e c h a n i s m . Th~s was qualitatively i n v o k e d by Candelas et al and W i t t e n [ 1 ] in considering c o m p a c n f i c a n o n and s y m m e t r y breaking o f t e n - d i m e n s i o n a l C h a p h n e - M a n t o n theory, a n d m o r e realisncally by G r e e n e et al. [2]. It has also been a p p l i e d to orbifold c o m p a c t i f i c a t l o n o f the heterotic string [ 3 ]. In the C a l a b l - Y a u models, e m b e d d i n g the spin connection o f the internal m a n ifold, K, in the gauge group hnks the ten- to four-dimensional reduction to s y m m e t r y reduction, Le. E s × E s breaking to Es×E6. Requiring K to be muln p l y connected takes a n o t h e r step along th~s path it decreases the large geometrically d e t e r m i n e d n u m ber o f fermion generations and at the same time offers the poss~bihty o f further breaking the gauge group, along directions which d e p e n d on the choice o f global v a c u u m on K. As m the A h a r a n o v - B o h m case [4] gauge i n e q m v a l e n t potenttals, Au, can correspond to the same field strength Fi,~= 0 and lead to energetically degenerate & s t m c t vacua on K. Ambitiously, one might hope that q u a n t u m corrections will break this degeneracy and d e t e r m i n e a u m q u e v a c u u m with smaller residual gauge group. The effective p o t e n t m l [5] offers a f o r m a h s m for Supported by the Science and Engineering Research Council
such calculations but, m practice, d y n a m i c a l calculations on muln-dlmens~onal curved m a m f o l d s such as C a l a b l - Y a u are not feasible. Irrespective o f the fate o f the particular models alluded to, the symmetry breaking m e c h a m s m itself is likely to r e m a i n in the repertoire, and thus merits study in simpler, if less realistic, models. The m o d e l o f H o s o t a m [6] often quoted in this context ~s an unsansfactory p a r a & g m , since an his examples the S U ( N ) gauge s y m m e t r y remains unbroken. An example m which the broken s y m m e t r y v a c u u m has lower energy than the symmetric one has been p r o w d e d by Evans and Ovrut [7]. They descrtbe a general m e t h o d for calculating one-loop effective potentmls but, due to technical complexities, their analysis as applied only to S U ( 3 ) on $3/Z2 and S U ( 2 ) on $3/Z3. In both cases, s y m m e t r y breaking can he achieved by gauge bosons and fermions, but scalar loops favour the original symmetry. Recently, Lee et al. [8 ] have used this model to clarify the relation between Wilson loops and broken s y m m e t r y vacua. We first o u t h n e the flux breaking m e c h a m s m [ 1 ], then exhibit the one-loop effective potential for our model. We start with a simply connected m a n i f o l d o f the form M × Ks, where M is some spacetime and 1Q has a freely acting discrete symmetry group F . The quotient space M × K , where K = K J F , ts then a m u l n p l y connected manifold with fundamental group zr~ ( M × K ) =n~ ( K ) = F . G i v e n a Y a n g - M i l l s sym-
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305
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metry group G on M × Ks, our problem is to determine the residual gauge symmetry G' on M × K. The flux-breaking mechanism can be described in two equivalent ways, depending on whether one considers fields defined on the manifold M × K or its covering space M × K~. Witten [ 1 ] has given a general account from this latter v~ewpolnt. In this approach, points x and yx on K~ related by the actmn o f F are identified and fields q) in an ~rreduclble representation of G which, in order to be definable on K must be invariant under the actmn of elements of F, are required to be constant up to a constant gauge transformation
14 January 1988
favoured residual gauge group G'. For this simple case, where Ks is the real line and F = Z, the additive group of integers, it is more convenient to work with fields defined directly on S ~ rather than on the covering space R I. This implies that Bose fields should be periodic in the x4 coordinate ~(X 4 " ~ ) = 0(X4),
(3)
For consistency, we require Urn= U~.Uo. The Ur form a discrete subgroup of the gauge group denoted by Fn, which is homomorphlc to F. As a consequence of this embedding, only those elements of G which commute with all the elements of FG are acceptable gauge transformations on M × K. Such a homomorphlsm is explicitly implemented by the set of Wilson operators
where/~ is the c~rcumference of the circle. For fermlon fields, due to the existence of lneqmvalent spin connections on this manifold [9], we can impose either peno&c or antx-perio&c boundary condihons on the fermion fields. The information that our gauge theory is defined on M × K not M × K s is now incorporated not via the U), factor in the perlodlc~ty conditmn but by the explicit appearance of a background gauge field B~,. Whereas on R3XR l any B~ with vamshing field strength is gauge equivalent to zero, this is not so on R3X S 1, where the best that can be done by a sequence of gauge transformations [ 10] is to reduce a general B,(x) to B , = ( 0 , 0, 0, B4), where B4 is a constant matrix in the Cartan subalgebra of G, i.e. it xs of the form
Ur=Pexp( f B,,(y) dyU),
B4 = E n~aa'
c1,~(~,x) = U'¢h eJ'( x).
(1)
N
(2)
where c(7) is a path from x to ~x on K~ which maps to a non-trivial closed loop on K, and Bu is a background gauge field with vamshing field strength. For a given B~,,this maps F to its not necessarily faithful gauge group representation F~. Different choices of B~, may then result m different FG, and hence in different residual symmetry groups G'. The symmetry will remain unbroken if FG consists of the ~dentity only, which corresponds to the zero background case. I f F n lies m the centre C of e~ther G or its covering group, the algebra will be unbroken, but the corresponding group may change. In applications to E6 breaking where K is a Calabi-Yau manifold [ 2 ], different embeddmgs of the discrete group are found and one which offers a promising route to the standard model ~s selected. No dynamical principle is invoked to select the operators U,,,. In the much simpler case of R3XS l, we will determine the background fields which mimmise the effective potential and hence find the energetically 306
(4)
a=l
c(/)
where the H" are commuting generators, and N = rank(G). The constant field B4 is not umque, being gauge equivalent to B; via the gauge transformation
g(x)=go exp[x4(B'4-B4)].
(5)
For this to be a valid gauge transformation on S t we must have g(fl) = g ( 0 ) , 1.e.
exp[fl(B;-B4)]
= 1.
(6)
We therefore consider the lagrangian L=
1 Uv 4g2C2(ad ) Tr(F~,F~,,)+,=~ O,?~D~u,,
(7)
where
F,~ =O~,A,-O~A, + [Az,,A~], Da~, =0 u + Tf~A~,.
(8)
Nv is the number of generations of Dirac fermions m some irreducible representation R of G. We have chosen a euchdean metric, the coupling constant has
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PHYSICS LETTERSB
been absorbed into the annherm~tian
A~, and
the generators are
[T ~, T t'] =ff, bT~,
(9)
and in R are normahsed such that Tr(T~.T~) = -
[d(R)/d(ad)]
C2(R) 6 ~
14 January 1988
d(R) states of R, and N = r a n k ( G ) . 2j IS a simultaneous eigenvector of all H a m the Cartan subalgebra, the exgenvalue of H a being - t 2 ] , where, as H a is annhermman, 2~ is real, and, by suitable normalisation of H a, is an integer. In this (Dynkin) basis, B 4 m the representation R is diagonal, with elements • a a
= - 1 2 ( R ) 6~",
(10)
__
(B4)/~ = -1B42j ~j~ - - (i/fl) O,k ( J , k = l .... ,d(R)},
where d(R) is the dimensmn of the representation R, d(ad) is the dimension of the adjomt representation, and C2 is the quadratic Casimir for R. The quannty I2 is known as the Dynkin index, and will be relevant when discussing the stability of certain backgrounds later in the paper. We then expand the gauge fields about a classical constant background field of the form g~ven above:
A,,=B,,+Q~,,
(15)
where O is a real matrix. The gauge freedom of eq. (6) amounts to shifts of 2rr in ~gjs, and thus constant fields with B~ equal modulo 2Jr/fl are gauge eqmvalent. Using the techniques of Gross et al. [ 14], we obtain a final expression for the potential 1 //d(ad)
(11)
and obtain the partinon function [ 11 ]
d(R)
-2Nv ~
)
cos n[(~gR),--fi] ,
(16)
Z=J- d [ ~ ~u Q ?c]
-a ab b ~ l -e"D2(B)~/'c/'+ AY y. qJ, 7~Dt,(B ) ~, l=l
}J
,
(12)
where we have used the background gauge fixing condition D,, (B) Q/, =0,
(13)
and taken the gauge-fixing parameter to be unity. As we are only interested in the one-loop contribution, we just retain the terms which are quadratic in the quantum fields. The one-loop effective potentml can then be expressed m terms of functional determinants [ 12 ]
V,(B)- [ & k
- J (2n)4
U= exp(flB4) = e x p ( - i O ) .
{½In deti-6¢,.D2(B) ~'1
- I n det[ - D 2 (B) a~']--NF In
where 3 is the fermlon phase which ~s either 0 (periodic) or ~ (antipenodic). Our procedure now is to mimmize I/1 as a function of the N parameters B~, and then find the residual symmetry group G'. The determination of G' in the presence of a non-zero background can be discussed in different languages. From the point of view of the spacetime M, in this case R 3, 84 is hke a constant adjoint Higgs field, giving mass to some gauge bosons vm terms F,4F,4m the lagrangian, and to some fermions via terms ~74B4~u. In th~s, it has to be remembered that some apparent mass terms are gauge equivalent to zero as B4 is multi-valued. Alternatively one can study symmetry breaking m terms of Wilson loops U", where n is the winding number round S ~, and
det[7~,D~,( B ) at~] }. (14)
In order to evaluate this, we require the eigenvalues of B~, in both the adjoint and fermaon representanons. These are easily calculated in terms of the Weyl weights [13] 2j= (2),..., 2N), where j labels the
(17)
In this, one may take UR=exp(--itgR) m any representation R, but ~t ~s convement to look at the fundamental matrix Ur. (This is quite independent of the choice of fermmn representatmn used in the potential). Noting the number of UR elements taking the same value gives the block-dmgonal form of the commuting subgroup and so, by comparing with the known branching rules of the appropriate represen307
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tatlon [ 15 ], G ' may be identified. If UR is a multiple of the identity then the symmetry is unbroken, modulo the discrete centre. With various choices of fermlon representations, the minima o f V1 were found by computer for the exceptional groups and for low-rank groups in the four classical series. Each classical group series had a characteristic form of effective potential, so the resuits for higher-rank groups could be obtained by extrapolation. The point o f departure in dxscussing our results ~s the case of gauge fields only, i.e. N F = 0 in (16). Obviously V1 has an absolute m i n i m u m at B ~ = 0 ( m o d 2zc/fl) and degenerate m i n i m a at any nonzero B4 which also gives @ad=0. These B4 satisfy B~oL~ = 0 ( m o d 2rc/fl), where a~ are the roots of the algebra. There are no non-trivial solutions for G2, F4 or Es, but for the other algebras, a number of nonzero fields exist (see table 1 ). By construction, Uad= 1 for these fields but there are other representations in which
( UR)s, = e x p ( - i B ~ 2 ~ ) ~ 1.
(18)
This happens when R is in a different congruency class [ 13 ] from the adjoint. Two representations are defined to be in the same congruency class if their highest weights differ by a linear combination o f roots, and since weights in a single R may be obtained by subtracting simple roots from the h~ghest
14 January 1988
weight, ( UR)j¢ for a given B4 in table 1 just depends on the class. (Congruency class is just triahty for S U ( 3 ) and E6, and for S O ( 2 N + 1 ) distinguishes spinor from vector representations.) We can therefore write
UR=Iexp(-I27rq),
0~
where I is the group identity, and the parameter q depends on the choice of B4 and the class of the representation, q is zero either for B~ = 0 or for R in the adjolnt class. When fermion loops are included, the symmetric configurations remain stationary points of the potential, but not always minima. We give in detail the results for fundamental representation fermlons, and in the ensuing discussion more general features will emerge. With antiperlodic fermlons and all NF > 0, the results were undramatic. As Nv was increased, the B ~ = 0 m i n i m u m deepened and the other Nv=O m i n i m a of table 1 became shallower (except for S O ( 2 N + 1), where the degeneracy remained) so the symmetry remained unbroken. Fig. 1 shows how for 62, which had no such degeneracy, local minima at B4= (0, + ~) disappear as Nv is increased. We quote B~ in convenient units of~z/fl, and use its periodicity to reduce it into the interval ( - 1, 1 ]. More varied and interesting behavxour was found
Table 1 Non-zero background fields corresponding to unbroken symmetry
308
(19)
Group
Symmetric fields
SU(3) SU(4) SU(5) etc
+_2/3(1,- 1) (1,0,1), _+1/2(-1,2,1) +_2/5(1,2,-2,- 1) ,+_2/5(2,- 1,1,-2)
SO ( 2N+ 1)
(o,0, ,oA)
SP(2N)
(1,o,l, ,O,l,O) Neven (1,0,1, ,1,O,I)Nodd
SO(2N)
(0,0, ,0,1,1), (1,0,1, ,0,1,0), (1,0,1, ,1,0,0,1) Neven (0,0, ,0,1,1),_+1/2(2,0,2, ,2,1,-l)Nodd
E6
+_2/3(1,- 1,0,1,- 1,0)
E7
(0,0,0,1,0,1,1)
Volume 200, number 3
PHYSICS LETTERS B
14 January 1988
V 2E~
~0 0 /
B42
..,.'/
-10 i"
-20 / ~
...., ...-''"
/
/
J
J
¢
/
/
/ NF
/ /
-50
/,/
/"/
/
/ /
//
-30 -40
/
i
J
--0
N F -- .1.
""
...............
.-
........
NF
--;2
NF
--3
-G0
Fig 1 Effective potential for G 2 as a funcnon of B] with B4~= 0, showing the evolunon of the periodic ferm~on number. There is a reflecnon symmetry an the Vaxis for periodic fermlons (table 2). In all cases, the mintma occur for non-zero Bg, but the s y m m e t r y only breaks for G2, F4, E8 and S O ( 2 N + 1). Insight mto this is g a m e d by evaluating the derivatives o f Vj at the s y m m e t r i c m i n i m a . The c o n d i t i o n for such a configuranon to r e m a i n a local m i n i m u m when N F > 0 is I2(ad)
>2Nvlz(R) ( 1 - 6 q + 6 q 2 ) ,
(20)
where I 2 ( R ) is the D y n k m index o f R a n d q is the p a r a m e t e r defined in (19). This criterion generallses that found by Toms [ 16] for p e r t u r b a n v e stability o f the Bg = 0 case, for which q = 0 . The first conclu-
B4 = 0
mlmmum with increasing antlo
slon from (20) is that the zero m i n i m u m is unstable for large enough Nv. C o m p u t e r search shows that with any n u m b e r o f f u n d a m e n t a l fermlons, the global m i n i m u m lies elsewhere. Fig. 2 illustrates this for G2, showing that B~ = 0 remains a local m i n i m u m up to Nv=2 (which is the cntical n u m b e r in this case), but, for any Nv, the global m i n i m a are at B4= (0, +__2 ). To identify G ' in the G2 example, we evaluate Ur at B4 = (0, ~ ). The f u n d a m e n t a l weights are (0,1 ), ( 1 , - 1), ( - 1,2), (0,0), ( 1 , - 2 ) , ( - 1,1) and ( 0 , - 1) so
Ur=diag(0), 09, 09, 0) 2,
0)2, 0)2
1 ),
(21)
Table 2 Breaking patterns for fundamental periodic fermlons Representations in bold type correspond to fermions which remain massless after compact~ficatton Group
Residual group
Branching
SU(N+ 1) SO(2N+ 1) SP(2N) SO(2N) G2 F4 E6 E7 E8
SU(N+ t) SO(2N) SP(2N) SO(2N) SU(3) SO(9) E6 E7 SU(6) ×SU(3) ×U(1 )
( N + I ) ~ ( N + l) (2N+I)~(2N) +1 (2N) -, ( 2N) (2N) -, (2N) 7-,3+3"+1 26+16+9+1 27-~27 56-,56 248---,(15",3") + (15,3) +2(6,3*) +2(6*,3) + 2(20,1 ) +2(1,1 ) + (l,l) + (35,1) + (1,8)
309
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PHYSICS LETTERS B
14 January 1988
v 40
30
20
.,
B42
-20 NF
-0
NF
-&
.................
NF"
-2
........
NF"
-3
Fig 2 As fig l, but for periodic fermmns. The minimum at B] = -~ is the global minimum of the effective potential, and corresponds to residual symmetry SU ( 3 ) The local m l m m u m at zero disappears at Nv = 2, which is the critical number for G2.
where co is a cube root of unity. In this case, F G = Z 3. Any matrix commuting with these elements of Z3 has block-dmgonal form 3 × 3, 3 × 3 and 1 × 1, so a 7 of G2 branches into multiplets of the form 3, 3 and I under G ' . This corresponds to the breaking G 2 ~ SU(3) with brachlng 7 ~ 3 + 3* + 1. Alternatively, one can construct the 14-dimensional adjoint representation matrix Oad. The diagonal element • a a iflB4aj, where % is the jth root of the algebra of G, may be interpreted as the mass acquired by the jth gauge boson on compactificatlon. In the G2 example, 8 of the bosons remain massless while the other 6 fall into two groups of 3 with different non-zero masses, again re&citing G2---*SU(3). A similar analysis shows that F4 breaks to SO(9) and that E8 breaks to SU(6) XSU(3) x U ( 1 ). Examination of (20) for q ¢ 0, which can occur for the groups of table 1 when R is not m the adjomt class, shows that the right-hand side ~s negative for at least one symmetric minimum, which is therefore stable for all NF, and moreover is found by computer to give a global minimum. Although the gauge algebra is unbroken, the constant background B4 gives equal masses to the fermions, as was found by Ho-
310
sotani, so that the global gauge group is actually the adjoint group, e.g. SU(3)/Z3. The result that only for S O ( 2 N + 1 ) of the groups m table 1 does the fundamental cause breaking of the algebra is now seen to be a consequence of the fact that it is the only one of these groups for which the fundamental is in the adjoint class. Thus q=0, in which case all the symmetric minima are unstable. This points the way to successful breaking of the gauge group: xt is necessary to have periodic fermmns in an adjoint congruency class representation. We have verified this in several cases, e.g. fermlons in the 10 of SU(3) break the symmetry to SU(2) X U(1) and fermions in the adjoxnt of E6 break it to [SU(3)] 3. On the other hand, fermaons in other classes, e.g. the 16 spinor of SO(10) or the 10 of SU(5) do not give symmetry breaking. A similar analysis for anti-periodic fermlons shows that q = 0 minima are always stable, and therefore the zero background is stable for fermions in all representations. For adjoint class representations, all the Nv=O degenerate m i m m a deepen and remain degenerate as Nv increases. Considerable flexibility in the breaking pattern can be achieved by taking combinations of fermion rep-
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resentations. One interesting example involves placing one generation of periodic fermaons in the adjoant of E6 and then including an arbitrary n u m b e r of 27's of a n t l p e n o d i c fermlons. Provided the n u m b e r of antipertodic families is less than 4, the symmetry breaks to S U ( 3 ) × S U ( 3 ) × S U ( 2 ) × U ( 1 ), with the fermions in the 27 which remain massless after compactification transforming as (3,3",1) u n d e r S U ( 3 ) × S U ( 3 ) × S U ( 2 ) . The fermions from the 78 which survive compactificatlon transform as (8, 1, 1) + (1, 8, 1 ) + ( 1 , 1 , 3 ) + ( 1 , 1 , 1 ) . For four or more generat~ons, the n u m e n c a l results show that B g = 0 becomes the global m i n i m u m , leaving the E6 symmetry unbroken. Although the result that adjoint fermlons are conducive to symmetry breaking ~s superficially encouraging for the use of flux-breakang in supersymmetric models, one generation of M a j o r a n a adj o i n t fermaons corresponds to Nv=½ in (16), and therefore gives vanishing one-loop potential. It would be of interest to find an exphctt dynamical example in which Wilson loops break gauge symmetry but preserve supersymmetry. To s u m m a n s e : the "fragile" groups G2, F4 and E8 are broken by periodac ferm~ons m any representation, the other groups are more robust, but will break ffthe fermions are an a representation which is in the same congruency class as the adjomt. The key feature necessary for symmetry breaking is that the fermxon loops destructively interfere wath all the gauge loop m i n i m a , a n d this occurs only for fermion representauons in the adjolnt congruency class. We therefore speculate that this dependence on congruency class may extend beyond our p a m c u l a r form
14 January 1988
of effectlve potential and be relevant at higher loops or on other manifolds. We would lake to thank D. McMullan, R.G. Moorhouse and D,G. Sutherland for helpful discussions.
References [ 1] P Candelas, G Horowltz, A Strommger and E Wltten, Nucl Phys B 258 (1985) 46, E Wltten, Nucl Phys. B 258 (1985) 75 [2] B R Greene, K H Karkhn,PJ Mlronand G.G Ross, Nucl Phys B 278 (1986) 667, B 292 (1987) 606, Oxford University preprlnt 11/87. [3] L Dixon, J A Harvey, C Vafa and E Wnten, Nucl Phys B261 (1985) 678, B274 (1985) 285, L E Ib~ifiez,H P Ntlles and F Quevedo, Phys Lett B 187 (1987) 25, CERN preprmt TH4673/87 [4] Y Aharonov and D Bohm, Phys Rev 115 (1959) 485, R G Chambers, Phys Rev Lett. 5 (1960) 3 [5] E g C Itzykson and J-B Zuber, Quantum field theory (McGraw-Hill, New York, 1986) [6] Y. Hosotani, Phys Lett B 126 (1983) 309 [7] M EvansandB A Ovrut, Phys Lett B 174 (1986) 63 [8] K Lee, R. Holman and E W. Kolb, Fermllab preprmt 87/105-A [9] C J. Isham, Proc R Soc London A 364 (1978) 59 [10] N Bataklsand G Lazarldes, Phys Rev D 18 (1978) 4710 [I 1] D J Toms, Phys Rev D 27 (1983) 1803 [12] R Jacklw, Phys Rev D 9 (1974) 1686 [13] R Slansky,Phys Rep 79 (1981) 1 [ 14] D J Gross, R D Plsarskl and L G Yaffe, Rev Mod Phys 53 (1981) 43 [ 15 ] W G McKay and J Patera, Tables of dimensions, indices and branching rules for representations of simple De algebras, Lecture Notes in Pure and Apphed Mathematics (Dekker, New York, 1981 ) [16] DJ Toms, Phys Len B 126 (1983) 445
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PHYSICS LETTERS B
14 January 1988
GAUGE GROUP BREAKING BY WILSON L O O P S A.T. DAVIES and A. M c L A C H L A N Department of Physws and Astronomy, The Umverstty, Glasgow G I 2 8QQ, Scotland, UK
Received 10 October 1987
We examine the breaking of gauge symmetries b~ Wdson loops in the Hosotam-Toms model by determining the background gauge field which mmlmlses the one-loop effective potentml for massless D~rac fermlons For anti-periodic fermlons, all gauge groups remain unbroken For periodic fermions, the groups G2, F4 and E8 are broken by quantum corrections due to fermions in any ~rreduclble representatmn, whereas E6, E7 and the classical groups only break if the fermmn representation is in the same congruency class as the adjomt
The p r o h f e r a t i o n m recent years of gauge theorxes on spacetimes o f d i m e n s m n greater than four has stimulated interest m a new m e c h a m s m for gauge s y m m e t r y breakmg, variously referred to as fluxbreaking or the H o s o t a m m e c h a n i s m . Th~s was qualitatively i n v o k e d by Candelas et al and W i t t e n [ 1 ] in considering c o m p a c n f i c a n o n and s y m m e t r y breaking o f t e n - d i m e n s i o n a l C h a p h n e - M a n t o n theory, a n d m o r e realisncally by G r e e n e et al. [2]. It has also been a p p l i e d to orbifold c o m p a c t i f i c a t l o n o f the heterotic string [ 3 ]. In the C a l a b l - Y a u models, e m b e d d i n g the spin connection o f the internal m a n ifold, K, in the gauge group hnks the ten- to four-dimensional reduction to s y m m e t r y reduction, Le. E s × E s breaking to Es×E6. Requiring K to be muln p l y connected takes a n o t h e r step along th~s path it decreases the large geometrically d e t e r m i n e d n u m ber o f fermion generations and at the same time offers the poss~bihty o f further breaking the gauge group, along directions which d e p e n d on the choice o f global v a c u u m on K. As m the A h a r a n o v - B o h m case [4] gauge i n e q m v a l e n t potenttals, Au, can correspond to the same field strength Fi,~= 0 and lead to energetically degenerate & s t m c t vacua on K. Ambitiously, one might hope that q u a n t u m corrections will break this degeneracy and d e t e r m i n e a u m q u e v a c u u m with smaller residual gauge group. The effective p o t e n t m l [5] offers a f o r m a h s m for Supported by the Science and Engineering Research Council
such calculations but, m practice, d y n a m i c a l calculations on muln-dlmens~onal curved m a m f o l d s such as C a l a b l - Y a u are not feasible. Irrespective o f the fate o f the particular models alluded to, the symmetry breaking m e c h a m s m itself is likely to r e m a i n in the repertoire, and thus merits study in simpler, if less realistic, models. The m o d e l o f H o s o t a m [6] often quoted in this context ~s an unsansfactory p a r a & g m , since an his examples the S U ( N ) gauge s y m m e t r y remains unbroken. An example m which the broken s y m m e t r y v a c u u m has lower energy than the symmetric one has been p r o w d e d by Evans and Ovrut [7]. They descrtbe a general m e t h o d for calculating one-loop effective potentmls but, due to technical complexities, their analysis as applied only to S U ( 3 ) on $3/Z2 and S U ( 2 ) on $3/Z3. In both cases, s y m m e t r y breaking can he achieved by gauge bosons and fermions, but scalar loops favour the original symmetry. Recently, Lee et al. [8 ] have used this model to clarify the relation between Wilson loops and broken s y m m e t r y vacua. We first o u t h n e the flux breaking m e c h a m s m [ 1 ], then exhibit the one-loop effective potential for our model. We start with a simply connected m a n i f o l d o f the form M × Ks, where M is some spacetime and 1Q has a freely acting discrete symmetry group F . The quotient space M × K , where K = K J F , ts then a m u l n p l y connected manifold with fundamental group zr~ ( M × K ) =n~ ( K ) = F . G i v e n a Y a n g - M i l l s sym-
0370-2693/88/$ 03.50 © Elsevier Science Pubhshers B.V. ( N o r t h - H o l l a n d Physics Pubhshing D i v i s i o n )
305
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metry group G on M × Ks, our problem is to determine the residual gauge symmetry G' on M × K. The flux-breaking mechanism can be described in two equivalent ways, depending on whether one considers fields defined on the manifold M × K or its covering space M × K~. Witten [ 1 ] has given a general account from this latter v~ewpolnt. In this approach, points x and yx on K~ related by the actmn o f F are identified and fields q) in an ~rreduclble representation of G which, in order to be definable on K must be invariant under the actmn of elements of F, are required to be constant up to a constant gauge transformation
14 January 1988
favoured residual gauge group G'. For this simple case, where Ks is the real line and F = Z, the additive group of integers, it is more convenient to work with fields defined directly on S ~ rather than on the covering space R I. This implies that Bose fields should be periodic in the x4 coordinate ~(X 4 " ~ ) = 0(X4),
(3)
For consistency, we require Urn= U~.Uo. The Ur form a discrete subgroup of the gauge group denoted by Fn, which is homomorphlc to F. As a consequence of this embedding, only those elements of G which commute with all the elements of FG are acceptable gauge transformations on M × K. Such a homomorphlsm is explicitly implemented by the set of Wilson operators
where/~ is the c~rcumference of the circle. For fermlon fields, due to the existence of lneqmvalent spin connections on this manifold [9], we can impose either peno&c or antx-perio&c boundary condihons on the fermion fields. The information that our gauge theory is defined on M × K not M × K s is now incorporated not via the U), factor in the perlodlc~ty conditmn but by the explicit appearance of a background gauge field B~,. Whereas on R3XR l any B~ with vamshing field strength is gauge equivalent to zero, this is not so on R3X S 1, where the best that can be done by a sequence of gauge transformations [ 10] is to reduce a general B,(x) to B , = ( 0 , 0, 0, B4), where B4 is a constant matrix in the Cartan subalgebra of G, i.e. it xs of the form
Ur=Pexp( f B,,(y) dyU),
B4 = E n~aa'
c1,~(~,x) = U'¢h eJ'( x).
(1)
N
(2)
where c(7) is a path from x to ~x on K~ which maps to a non-trivial closed loop on K, and Bu is a background gauge field with vamshing field strength. For a given B~,,this maps F to its not necessarily faithful gauge group representation F~. Different choices of B~, may then result m different FG, and hence in different residual symmetry groups G'. The symmetry will remain unbroken if FG consists of the ~dentity only, which corresponds to the zero background case. I f F n lies m the centre C of e~ther G or its covering group, the algebra will be unbroken, but the corresponding group may change. In applications to E6 breaking where K is a Calabi-Yau manifold [ 2 ], different embeddmgs of the discrete group are found and one which offers a promising route to the standard model ~s selected. No dynamical principle is invoked to select the operators U,,,. In the much simpler case of R3XS l, we will determine the background fields which mimmise the effective potential and hence find the energetically 306
(4)
a=l
c(/)
where the H" are commuting generators, and N = rank(G). The constant field B4 is not umque, being gauge equivalent to B; via the gauge transformation
g(x)=go exp[x4(B'4-B4)].
(5)
For this to be a valid gauge transformation on S t we must have g(fl) = g ( 0 ) , 1.e.
exp[fl(B;-B4)]
= 1.
(6)
We therefore consider the lagrangian L=
1 Uv 4g2C2(ad ) Tr(F~,F~,,)+,=~ O,?~D~u,,
(7)
where
F,~ =O~,A,-O~A, + [Az,,A~], Da~, =0 u + Tf~A~,.
(8)
Nv is the number of generations of Dirac fermions m some irreducible representation R of G. We have chosen a euchdean metric, the coupling constant has
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been absorbed into the annherm~tian
A~, and
the generators are
[T ~, T t'] =ff, bT~,
(9)
and in R are normahsed such that Tr(T~.T~) = -
[d(R)/d(ad)]
C2(R) 6 ~
14 January 1988
d(R) states of R, and N = r a n k ( G ) . 2j IS a simultaneous eigenvector of all H a m the Cartan subalgebra, the exgenvalue of H a being - t 2 ] , where, as H a is annhermman, 2~ is real, and, by suitable normalisation of H a, is an integer. In this (Dynkin) basis, B 4 m the representation R is diagonal, with elements • a a
= - 1 2 ( R ) 6~",
(10)
__
(B4)/~ = -1B42j ~j~ - - (i/fl) O,k ( J , k = l .... ,d(R)},
where d(R) is the dimensmn of the representation R, d(ad) is the dimension of the adjomt representation, and C2 is the quadratic Casimir for R. The quannty I2 is known as the Dynkin index, and will be relevant when discussing the stability of certain backgrounds later in the paper. We then expand the gauge fields about a classical constant background field of the form g~ven above:
A,,=B,,+Q~,,
(15)
where O is a real matrix. The gauge freedom of eq. (6) amounts to shifts of 2rr in ~gjs, and thus constant fields with B~ equal modulo 2Jr/fl are gauge eqmvalent. Using the techniques of Gross et al. [ 14], we obtain a final expression for the potential 1 //d(ad)
(11)
and obtain the partinon function [ 11 ]
d(R)
-2Nv ~
)
cos n[(~gR),--fi] ,
(16)
Z=J- d [ ~ ~u Q ?c]
-a ab b ~ l -e"D2(B)~/'c/'+ AY y. qJ, 7~Dt,(B ) ~, l=l
}J
,
(12)
where we have used the background gauge fixing condition D,, (B) Q/, =0,
(13)
and taken the gauge-fixing parameter to be unity. As we are only interested in the one-loop contribution, we just retain the terms which are quadratic in the quantum fields. The one-loop effective potentml can then be expressed m terms of functional determinants [ 12 ]
V,(B)- [ & k
- J (2n)4
U= exp(flB4) = e x p ( - i O ) .
{½In deti-6¢,.D2(B) ~'1
- I n det[ - D 2 (B) a~']--NF In
where 3 is the fermlon phase which ~s either 0 (periodic) or ~ (antipenodic). Our procedure now is to mimmize I/1 as a function of the N parameters B~, and then find the residual symmetry group G'. The determination of G' in the presence of a non-zero background can be discussed in different languages. From the point of view of the spacetime M, in this case R 3, 84 is hke a constant adjoint Higgs field, giving mass to some gauge bosons vm terms F,4F,4m the lagrangian, and to some fermions via terms ~74B4~u. In th~s, it has to be remembered that some apparent mass terms are gauge equivalent to zero as B4 is multi-valued. Alternatively one can study symmetry breaking m terms of Wilson loops U", where n is the winding number round S ~, and
det[7~,D~,( B ) at~] }. (14)
In order to evaluate this, we require the eigenvalues of B~, in both the adjoint and fermaon representanons. These are easily calculated in terms of the Weyl weights [13] 2j= (2),..., 2N), where j labels the
(17)
In this, one may take UR=exp(--itgR) m any representation R, but ~t ~s convement to look at the fundamental matrix Ur. (This is quite independent of the choice of fermmn representatmn used in the potential). Noting the number of UR elements taking the same value gives the block-dmgonal form of the commuting subgroup and so, by comparing with the known branching rules of the appropriate represen307
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tatlon [ 15 ], G ' may be identified. If UR is a multiple of the identity then the symmetry is unbroken, modulo the discrete centre. With various choices of fermlon representations, the minima o f V1 were found by computer for the exceptional groups and for low-rank groups in the four classical series. Each classical group series had a characteristic form of effective potential, so the resuits for higher-rank groups could be obtained by extrapolation. The point o f departure in dxscussing our results ~s the case of gauge fields only, i.e. N F = 0 in (16). Obviously V1 has an absolute m i n i m u m at B ~ = 0 ( m o d 2zc/fl) and degenerate m i n i m a at any nonzero B4 which also gives @ad=0. These B4 satisfy B~oL~ = 0 ( m o d 2rc/fl), where a~ are the roots of the algebra. There are no non-trivial solutions for G2, F4 or Es, but for the other algebras, a number of nonzero fields exist (see table 1 ). By construction, Uad= 1 for these fields but there are other representations in which
( UR)s, = e x p ( - i B ~ 2 ~ ) ~ 1.
(18)
This happens when R is in a different congruency class [ 13 ] from the adjoint. Two representations are defined to be in the same congruency class if their highest weights differ by a linear combination o f roots, and since weights in a single R may be obtained by subtracting simple roots from the h~ghest
14 January 1988
weight, ( UR)j¢ for a given B4 in table 1 just depends on the class. (Congruency class is just triahty for S U ( 3 ) and E6, and for S O ( 2 N + 1 ) distinguishes spinor from vector representations.) We can therefore write
UR=Iexp(-I27rq),
0~
where I is the group identity, and the parameter q depends on the choice of B4 and the class of the representation, q is zero either for B~ = 0 or for R in the adjolnt class. When fermion loops are included, the symmetric configurations remain stationary points of the potential, but not always minima. We give in detail the results for fundamental representation fermlons, and in the ensuing discussion more general features will emerge. With antiperlodic fermlons and all NF > 0, the results were undramatic. As Nv was increased, the B ~ = 0 m i n i m u m deepened and the other Nv=O m i n i m a of table 1 became shallower (except for S O ( 2 N + 1), where the degeneracy remained) so the symmetry remained unbroken. Fig. 1 shows how for 62, which had no such degeneracy, local minima at B4= (0, + ~) disappear as Nv is increased. We quote B~ in convenient units of~z/fl, and use its periodicity to reduce it into the interval ( - 1, 1 ]. More varied and interesting behavxour was found
Table 1 Non-zero background fields corresponding to unbroken symmetry
308
(19)
Group
Symmetric fields
SU(3) SU(4) SU(5) etc
+_2/3(1,- 1) (1,0,1), _+1/2(-1,2,1) +_2/5(1,2,-2,- 1) ,+_2/5(2,- 1,1,-2)
SO ( 2N+ 1)
(o,0, ,oA)
SP(2N)
(1,o,l, ,O,l,O) Neven (1,0,1, ,1,O,I)Nodd
SO(2N)
(0,0, ,0,1,1), (1,0,1, ,0,1,0), (1,0,1, ,1,0,0,1) Neven (0,0, ,0,1,1),_+1/2(2,0,2, ,2,1,-l)Nodd
E6
+_2/3(1,- 1,0,1,- 1,0)
E7
(0,0,0,1,0,1,1)
Volume 200, number 3
PHYSICS LETTERS B
14 January 1988
V 2E~
~0 0 /
B42
..,.'/
-10 i"
-20 / ~
...., ...-''"
/
/
J
J
¢
/
/
/ NF
/ /
-50
/,/
/"/
/
/ /
//
-30 -40
/
i
J
--0
N F -- .1.
""
...............
.-
........
NF
--;2
NF
--3
-G0
Fig 1 Effective potential for G 2 as a funcnon of B] with B4~= 0, showing the evolunon of the periodic ferm~on number. There is a reflecnon symmetry an the Vaxis for periodic fermlons (table 2). In all cases, the mintma occur for non-zero Bg, but the s y m m e t r y only breaks for G2, F4, E8 and S O ( 2 N + 1). Insight mto this is g a m e d by evaluating the derivatives o f Vj at the s y m m e t r i c m i n i m a . The c o n d i t i o n for such a configuranon to r e m a i n a local m i n i m u m when N F > 0 is I2(ad)
>2Nvlz(R) ( 1 - 6 q + 6 q 2 ) ,
(20)
where I 2 ( R ) is the D y n k m index o f R a n d q is the p a r a m e t e r defined in (19). This criterion generallses that found by Toms [ 16] for p e r t u r b a n v e stability o f the Bg = 0 case, for which q = 0 . The first conclu-
B4 = 0
mlmmum with increasing antlo
slon from (20) is that the zero m i n i m u m is unstable for large enough Nv. C o m p u t e r search shows that with any n u m b e r o f f u n d a m e n t a l fermlons, the global m i n i m u m lies elsewhere. Fig. 2 illustrates this for G2, showing that B~ = 0 remains a local m i n i m u m up to Nv=2 (which is the cntical n u m b e r in this case), but, for any Nv, the global m i n i m a are at B4= (0, +__2 ). To identify G ' in the G2 example, we evaluate Ur at B4 = (0, ~ ). The f u n d a m e n t a l weights are (0,1 ), ( 1 , - 1), ( - 1,2), (0,0), ( 1 , - 2 ) , ( - 1,1) and ( 0 , - 1) so
Ur=diag(0), 09, 09, 0) 2,
0)2, 0)2
1 ),
(21)
Table 2 Breaking patterns for fundamental periodic fermlons Representations in bold type correspond to fermions which remain massless after compact~ficatton Group
Residual group
Branching
SU(N+ 1) SO(2N+ 1) SP(2N) SO(2N) G2 F4 E6 E7 E8
SU(N+ t) SO(2N) SP(2N) SO(2N) SU(3) SO(9) E6 E7 SU(6) ×SU(3) ×U(1 )
( N + I ) ~ ( N + l) (2N+I)~(2N) +1 (2N) -, ( 2N) (2N) -, (2N) 7-,3+3"+1 26+16+9+1 27-~27 56-,56 248---,(15",3") + (15,3) +2(6,3*) +2(6*,3) + 2(20,1 ) +2(1,1 ) + (l,l) + (35,1) + (1,8)
309
Volume 200, number 3
PHYSICS LETTERS B
14 January 1988
v 40
30
20
.,
B42
-20 NF
-0
NF
-&
.................
NF"
-2
........
NF"
-3
Fig 2 As fig l, but for periodic fermmns. The minimum at B] = -~ is the global minimum of the effective potential, and corresponds to residual symmetry SU ( 3 ) The local m l m m u m at zero disappears at Nv = 2, which is the critical number for G2.
where co is a cube root of unity. In this case, F G = Z 3. Any matrix commuting with these elements of Z3 has block-dmgonal form 3 × 3, 3 × 3 and 1 × 1, so a 7 of G2 branches into multiplets of the form 3, 3 and I under G ' . This corresponds to the breaking G 2 ~ SU(3) with brachlng 7 ~ 3 + 3* + 1. Alternatively, one can construct the 14-dimensional adjoint representation matrix Oad. The diagonal element • a a iflB4aj, where % is the jth root of the algebra of G, may be interpreted as the mass acquired by the jth gauge boson on compactificatlon. In the G2 example, 8 of the bosons remain massless while the other 6 fall into two groups of 3 with different non-zero masses, again re&citing G2---*SU(3). A similar analysis shows that F4 breaks to SO(9) and that E8 breaks to SU(6) XSU(3) x U ( 1 ). Examination of (20) for q ¢ 0, which can occur for the groups of table 1 when R is not m the adjomt class, shows that the right-hand side ~s negative for at least one symmetric minimum, which is therefore stable for all NF, and moreover is found by computer to give a global minimum. Although the gauge algebra is unbroken, the constant background B4 gives equal masses to the fermions, as was found by Ho-
310
sotani, so that the global gauge group is actually the adjoint group, e.g. SU(3)/Z3. The result that only for S O ( 2 N + 1 ) of the groups m table 1 does the fundamental cause breaking of the algebra is now seen to be a consequence of the fact that it is the only one of these groups for which the fundamental is in the adjoint class. Thus q=0, in which case all the symmetric minima are unstable. This points the way to successful breaking of the gauge group: xt is necessary to have periodic fermmns in an adjoint congruency class representation. We have verified this in several cases, e.g. fermlons in the 10 of SU(3) break the symmetry to SU(2) X U(1) and fermions in the adjoxnt of E6 break it to [SU(3)] 3. On the other hand, fermaons in other classes, e.g. the 16 spinor of SO(10) or the 10 of SU(5) do not give symmetry breaking. A similar analysis for anti-periodic fermlons shows that q = 0 minima are always stable, and therefore the zero background is stable for fermions in all representations. For adjoint class representations, all the Nv=O degenerate m i m m a deepen and remain degenerate as Nv increases. Considerable flexibility in the breaking pattern can be achieved by taking combinations of fermion rep-
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PHYSICS LETTERS B
resentations. One interesting example involves placing one generation of periodic fermaons in the adjoant of E6 and then including an arbitrary n u m b e r of 27's of a n t l p e n o d i c fermlons. Provided the n u m b e r of antipertodic families is less than 4, the symmetry breaks to S U ( 3 ) × S U ( 3 ) × S U ( 2 ) × U ( 1 ), with the fermions in the 27 which remain massless after compactification transforming as (3,3",1) u n d e r S U ( 3 ) × S U ( 3 ) × S U ( 2 ) . The fermions from the 78 which survive compactificatlon transform as (8, 1, 1) + (1, 8, 1 ) + ( 1 , 1 , 3 ) + ( 1 , 1 , 1 ) . For four or more generat~ons, the n u m e n c a l results show that B g = 0 becomes the global m i n i m u m , leaving the E6 symmetry unbroken. Although the result that adjoint fermlons are conducive to symmetry breaking ~s superficially encouraging for the use of flux-breakang in supersymmetric models, one generation of M a j o r a n a adj o i n t fermaons corresponds to Nv=½ in (16), and therefore gives vanishing one-loop potential. It would be of interest to find an exphctt dynamical example in which Wilson loops break gauge symmetry but preserve supersymmetry. To s u m m a n s e : the "fragile" groups G2, F4 and E8 are broken by periodac ferm~ons m any representation, the other groups are more robust, but will break ffthe fermions are an a representation which is in the same congruency class as the adjomt. The key feature necessary for symmetry breaking is that the fermxon loops destructively interfere wath all the gauge loop m i n i m a , a n d this occurs only for fermion representauons in the adjolnt congruency class. We therefore speculate that this dependence on congruency class may extend beyond our p a m c u l a r form
14 January 1988
of effectlve potential and be relevant at higher loops or on other manifolds. We would lake to thank D. McMullan, R.G. Moorhouse and D,G. Sutherland for helpful discussions.
References [ 1] P Candelas, G Horowltz, A Strommger and E Wltten, Nucl Phys B 258 (1985) 46, E Wltten, Nucl Phys. B 258 (1985) 75 [2] B R Greene, K H Karkhn,PJ Mlronand G.G Ross, Nucl Phys B 278 (1986) 667, B 292 (1987) 606, Oxford University preprlnt 11/87. [3] L Dixon, J A Harvey, C Vafa and E Wnten, Nucl Phys B261 (1985) 678, B274 (1985) 285, L E Ib~ifiez,H P Ntlles and F Quevedo, Phys Lett B 187 (1987) 25, CERN preprmt TH4673/87 [4] Y Aharonov and D Bohm, Phys Rev 115 (1959) 485, R G Chambers, Phys Rev Lett. 5 (1960) 3 [5] E g C Itzykson and J-B Zuber, Quantum field theory (McGraw-Hill, New York, 1986) [6] Y. Hosotani, Phys Lett B 126 (1983) 309 [7] M EvansandB A Ovrut, Phys Lett B 174 (1986) 63 [8] K Lee, R. Holman and E W. Kolb, Fermllab preprmt 87/105-A [9] C J. Isham, Proc R Soc London A 364 (1978) 59 [10] N Bataklsand G Lazarldes, Phys Rev D 18 (1978) 4710 [I 1] D J Toms, Phys Rev D 27 (1983) 1803 [12] R Jacklw, Phys Rev D 9 (1974) 1686 [13] R Slansky,Phys Rep 79 (1981) 1 [ 14] D J Gross, R D Plsarskl and L G Yaffe, Rev Mod Phys 53 (1981) 43 [ 15 ] W G McKay and J Patera, Tables of dimensions, indices and branching rules for representations of simple De algebras, Lecture Notes in Pure and Apphed Mathematics (Dekker, New York, 1981 ) [16] DJ Toms, Phys Len B 126 (1983) 445
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