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One loop flux breaking by massive fields
Volume 234, number 1,2
PHYSICS LETTERS B
4 January 1990
O N E L O O P F L U X B R E A K I N G BY M A S S I V E F I E L D S Mark B U R G E S S and D a v i d J. T O M S Department o f Physics, University o f Newcastle upon T)'ne, Newcastle upon Tyne NE I 7R U, UK
Received 12 August 1989; revised manuscript received 25 October 1989
We consider the effect of fields with non zero masses on flux breaking to first order in perturbation theory. It is found that non zero mass leads to greater stability of the unbroken symmetry configurations, for adjoint class fermions, and never assists the symmetry breaking mechanism. We also show that, in the massive case, there is a critical length associated with the compact dimension, above which stability is assured.
1. I n t r o d u c t i o n - dR
Tr(TaTh) = T The m e t h o d o f flux breaking o f gauge s y m m e t r i e s has attracted much interest in recent literature. This stems from the observation that, on multiply connected spacetimes, gauge fields can acquire non zero, constant c o m p o n e n t s which are not gauge equivalent to zero [ 1,2 ]. The p a r a m e t r i z a t i o n of the m e t h o d in terms of Wilson loops has resulted in a considerable simplification o f the formalism. Following a certain a m o u n t o f confusion in the literature [ 3 ], the mechanism has been investigated for massless fields by Davies and McLachlan [ 4 - 7 ]. Related work has also been carried out by D o w k e r and J a d h a v [8], and Higuchi and Parker [9]. The full extension o f the m e t h o d to fields of any mass is considerably m o r e difficult, but has a n u m b e r o f clear features which are not difficult to unravel, notably the emergence o f a critical periodicity length in the c o m p a c t dimension, which will stabilize a given theory [ 2 ]. O u r aim here is to briefly reveal those general features.
2. N o t a t i o n a n d c o n v e n t i o n s
Consider a non abelian gauge theory generated by [ T a, T t' ] =j" ~'~'cT< ,
( 1)
where a, b, c = I, ..., N with N the d i m e n s i o n o f the gauge group. The T ~ satisfy'
C2(GR)6~b
(2)
in an arbitrary basis. T ~ is the generator o f the matrix representation GR o f G, and is a dR × dR matrix. In the adjoint representation dadj is N. C2(GR) is the second o r d e r Casimir invariant of the representation GR. O u r conventions imply that the generators are anti-hermitian, i.e. T a r = - T a. Working on locally flat spacetime with topology Rn × S 1, with a euclidean metric du, and #, v = O, ..., n, we aim to compute the one loop effective potential for a theory involving fermions a n d / o r scalar fields, using the background field m e t h o d in the "background field" class o f gauges [ 2]. The gauge field m a y be written
~,+A,,
(3)
where -~u = T , ia ~ A-i, (i = 1, ..., rank ( G ) ) is the classical background field; a linear c o m b i n a t i o n o f the generators o f the Cartan sub-algebra o f G. It is convenient to work in a basis in which these matrices are diagonal, in which case these generators are conventionally denoted by H; i . e . ~ , - - H a'd j A-i~. A ~ _- T a~a j A l, is a q u a n t u m field and is a linear c o m b i n a t i o n of all the generators. In the basis which diagonalizes the H, only the subset H o f T a r e diagonal matrices. To preserve the symmetries o f the lagrangian at the classical level, we define A~ to be a spacetime constant. This ensures that the field tensor vanishes clas-
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sically. The background field can then be written A~, = H~dj (0, 0, ..., U) where 2 ~are constants which lie in the S z direction, the other components having been gauged to zero. The action with group and spinor indices suppressed is consequently of the form [ 10]
/=l
4 January 1990
theory by transforming the quantum fields covariantly as follows:
q)t-~URq)~, T~--,Uw~t,
(12,13)
7 A~-~ LadjAu U a*d j
(14,15)
,
r/--, Uadj ~] ,
We fix /JR SO that the background field is eliminated explicitly from the action, i.e. Dy~O~. This requires
(4) S,i~¢= ~ f dVx (~Pt)'UDuTt+~PtmTt) , /=1
SVM =
4C2-1 ( Gadj )
-1
f dVxTr(P'"P,,~)
f
SGF= 2ozCT(G~dj) dV~Tr(13~'A~)2'
(5)
(6)
UR(x)=Pexp(g i A~,H~ dz~')
(16)
XO
in the representation GR. After this transformation, the action is reduced to that of a comparatively trivial vacuum energy calculation,
(7) 4. Boundary conditions
Sgho,, = f dV~ O(x) [ -~)2-A"Oa - (OUA.) ]r/(x)
,
(8) with covariant derivatives D u =Ou +g(Au
+Au),
(9)
D~,=O. +gA~,
(10)
and flu, =F,~ +g[-'{u, A~] + gtA,,
A,,] ,
( 11 )
with/tz and A, in the appropriate matrix representation ~. Note that ff~,~=0. @ is a complex scalar field, which transforms like a vector with dR components under the action of the group. ~Pj and TJ are d~ component Dirac fermions; f/and ~/are the ghost fields. All of these fields are pure quantum variables of integration. For generality we allow N~ generations of scalar multiplets and Nr generations of the Dirac fermion.
The price paid for this simplification of the action is a modification of the boundary conditions for the quantum fields. Initially the gauge fields must satisfy periodic boundary conditions on S ~, likewise the ghosts since their role is to quell the overcounting of gauge field configurations in the path integral. The matter fields need not be periodic prior to their transformation. We allow a general phase change around S ~. For periodicity L in S ~ ~2
cP(X+ L ) =exp(2zfiS)q~(X).
After the Wilson loop transformation, each group component of a given quantum field will pick up an extra phase from the multiplication by a Wilson line. lfwe denote the Wilson loop simply by
UR(L)=exp(2giOR),
It is of considerable utility to reparametrize the ~ The masses in (4) and (5) need not be the same, but since we shall not discuss mixed fermion/scalar theories here, extra notation is avoided.
98
(18)
where OR is a diagonal matrix, ~ t ~ e x p [ 2rci(OR +J1) ] q~t,
3. Wilson loops
(17)
(19)
Thus in momentum space, we have momenta k, (l+ 0 + J ) where [~Z and 0 is one component of the OR matrix.
(2n/L)
~2 Again delta need not be the same for scalars and fermions.
Volume 234, number 1,2
PHYSICS LETTERS B
calculate (22) analytically, though for even n, a sensible expansion in terms of polylogarithm functions can readily be given by expanding (u 2- v2) ~/2 and noting that
5. Evaluation of V m The one loop effective potential is given by ~s I dV~ V~I>= + N s Tr ln( -- ['-]scalar + r n 2)
_ ½Nf2 ~(,+l)/2j Tr In ( - [~ fermion+
i
=
~
P!(27~)m-(P+l)l/m
,,=o
m!
l,
×Lip+,_m(exp[-2n(/2+i0)])
+ ½( n + 1 ) Tr ln( - Dgaug~) + ( 1 / a ) t e r m (20)
where Tr is the functional trace and the trace over group indices. We employ the technique of Ford [ 11 ] in calculating (20) % Defining v=mL/2n we have, for each element 0 of 6)R in the group trace, a term Trln [-Qo+
updu
exp[2n(u+iO) ] - 1
m2 )
- Y r In( - ~gho~) ,
4 January 1990
(2n/2/L) 2] fdl~
(p>0).
(24)
The expansion for odd n proves unfruitful. (See Appendix. ) Recalling that the Wilson loops are diagonal matrices in our chosen basis, V ~~~may be written with one term of the form (21) for each diagonal element of the Wilson loop matrix. For example, for fundamental scalar fields transforming under su (2), the Wilson loops are of the form Uf = diag [ exp ( - 2~ix), exp (2nix)
],
n/2
4I'(-~n)(-L5 )
and
sin(½rlzc)f_n/2(/2,~) (21)
Uadj = diag[ 1, exp( -- 2zdy), exp (2niy) ] , giving V ¢11 in the form
(see eq. ( 2 6 ) ) , where
[
f-~/2 ( v,
O) = R e i
(25)
V~)= L F(
d u ( u 2 - / 2 2 ) ,,/2
v
× [f-~/2(
× { e x p [ 2 7 r ( u + i 0 ) ] - - 1} -j ,
\n/2
v, x+ d) +f-,,/2 ( v, - x + d) ]
(22) +
and we have dropped an irrelevant (infinite) constant. It is easy to show that
F(n+ 1)
f_,/2 (0, 0) = ~(27r),
× [f_,~/2(O, O) +f_n/2(O, - Y ) + f _ , / 2 ( 0 , y) ] • (26)
Re El,+ ~[ e x p ( - 2 z t i 0 ) ] (23)
in the massless limit, recovering the results of Davies and McLachlan [4,5]. (Lip(z) is the polylogarithm function, adopted in ref. [ 5 ].) (21 ) and (22) conceal zeros and singularities, tbr even n but well defined, regular limits are easily found for the results we require. In the massive case it is not generally feasible to ns Working to general oL we pick up terms of order 1/a al one loop. These are cancelled by terms arising from the functional measure, making V~I ) regular and gauge parameter independent. ~4 A massive generalization of the methods used in ref. [4] has undesirable features in the massless limit.
The first term comes from the scalars, the latter from the gauge + ghost fields.
6. Classically symmetric minima Davies and McLachlan have shown that for any group G, generated by a semi-simple Lie algebra, the turning points o f the effective potential, to one loop, give the symmetry preserving configurations of the background field [ 5 ]. This follows from the fact that the Wilson loops are multiples of the identity matrix when the symmetry is preserved, which corresponds to setting all values of 0 to a uniform constant, for a given field. This results in 99
d V ~1~ -a-
symmetric background
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PHYSICS LETTERS B
Volume 234, number 1,2
o c T r ( H a) ,
(27)
which is zero for any semi simple Lie algebra. Generally, there is no unique 0 which preserves the symmetry. A value which has proven its importance is 0 = 0, corresponding to fields in an adjoint congruence class, or to A , = 0 in any class. This issue has been addressed at length in ref. [ 5 ], where it is shown that matter fields in an adjoint congruence class are required to break the classical symmetry, and that fields in other representation classes map to non zero values of-4¢, without breaking the symmetries. This only results in an addition to the effective mass of the form ~Tu,g"~u. We define a stable theory as one in which neither of these phenomena occurs. As an example, we focus our attention on adjoint class, periodic fermions, 0 = 6 = 0 , and look for the symmetric m i n i m u m at .g~=0. Differentiating the potential twice with respect to 2 and evaluating at 0 = 0 , we obtain a condition for the stability of the ~it,= 0 case:
tain (28). The primes denote differentiation with respect to 0. When 0 = 0 , n = 3 the RHS of (28) is unity and we recover the result of ref, [2]. For general n, with v = 0 we recover the result of ref. [51. For all other v the RHS is less than unity, falling offapproximately as e x p ( - 2 7 t v ) . Since v=mL/27t we can remark generally on the behaviour of (28). For L>> m -~ (u>> 1 ), the RHS tends to zero, thus the theory becomes stable irrespective of group of numbers of fermions. We therefore expect a critical length scale for fixed masses at which the theory changes from being stable to unstable. This critical length scale was first noted in ref. [21. Moreover, since the RHS < 1, the fermion mass always tends to stabilize the A u = 0 minimum. Technically the simplest example which verifies this is Kaluza-Klein theory R4X S I, for which we obtain, for arbitrary groups, 3
f ' - 2 ( / , I , O) = -- ~ Re Li3 (exp [ - 2 r t ( v + 61/
- ~-3 Re Li2(exp[ - 2 ~ ( v + i 0 ) ] )
NCa(G~d~)(n-I)2 -{'+t)/2 l~ 't~n/2 ( lJ, 0_~ > - - - . (28) dR Cz ( GR ) Nf [f "n/2 ( O, The modulus sign refers to the fact that we have used the negative sign o f f L~/2 ( V, 0) explicitly to ob-
10It 2
--
7C
Re Lil (exp[ -- 2 ~ ( v + i 0 )
ABSOLUTE VALUE 0 F f " ( u , 0 ) / f " ( 0 , 0 )
for n = 3
0.8 o
o-0.6
':" 0.t,
0.2
0t.~.
016
0.8
1.0
11.2
11
v = mL/2r~
Fig. 1. Graph illustrating the fall-off off '23/2( v, 0 ) / f "3/2(O, O) as a function of v.
100
])
- 20 u 3 Re Lio (exp [ - 27r(u + i0) ] ) ,
1.0
012
i0) ] )
(29)
Volume 234, number 1,2
PHYS1CS LETTERS B
4 January 1990
8. Conclusions
and
-3
,f"-2(0, 0 ) = ~
~R(3) ,
(30)
where ~R is the Riemann zeta function. On ~73×S ~, taking f 312(u, 0 + d ) , with cases 0 = 0 , d = 0 , ½, corresponding to .4u = 0 with periodic and antiperiodic fermions, we can change variables in (22) to obtain
fQ3/2(V, +--_) exp ( - 2sty)
=-+
[ (In y ) 2 (27rp)2] 3/2 (y__+1 )dy (2g)2( 1Ty)3
o
(31) where, in f( _+ ), + refers to untwisted (periodic) fields and - refers to twisted (antiperiodic) fields. This form is suitable for numerical investigations, should they be of further interest (see fig. 1 ). For an arbitrary gauge group, we can determine the critical length for the stable theory (to one loop). Consider, for example, a single generation of adjoint fermions on ~3×S~. Setting GR= Gadj, d R = N i n (28), the LHS becomes ~. Referring to fig. 1, we read off v ~ 0 . 2 , giving Lcrit ~ 1.26 m - t .
We have shown that the effect of fermion masses is to stabilize the unbroken symmetric configurations. By considering results analogous to (28) it is not difficult to see that this result carries through in the case of twisted scalar fields also, and has no effect on twisted fermions or untwisted scalars (which cannot disturb the symmetries o f the lagrangian anyway). Cases which have critical stability in the massless limit are stable for non zero mass. It is clear why this should be so: in order for matter fields to destabilize the classical minima o f the potential, they must interfere destructively with the gauge ( + ghost) fields. For symmetry breaking we require the matter field contribution to be large with respect to the gauge contribution and of the opposite sign. The presence of masses always tends to reduce the magnitude o f the matter field contribution. Our results have concentrated on the stability o f the/~, = 0 background field state. It would be interesting to examine the role o f congruence classes (nality) in massive theories to determine the extent to which realistic masses inhibit symmetry breaking in the cases exposed by Davies and McLachlan. We defer a discussion o f this and other features o f flux breaking until a later date.
7. SU(L) For SU (L) there are simple geometric expressions for the group invariants. C2(Gaaj) =L, N=L 2- 1 and with matter fields in the fundamental representation dr=L and C2(Gf)=(La-1)/2L. Thus setting 0 = J = 0 , for periodic fermions, we see that
2 L ( n - 1)2 -(n+')12
N,.
>
If "_ni2( v, O)l - - - -
If"-n/2(0,0) I
.
(32)
In particular, when n = 3 ( E 3 × S I ) the RHS is just L/Nf. Thus for S U ( 2 ) with two generations of fundamental fermions, the stability of A~ teeters on the brink of indecision when the mass is zero. For m non zero, stability is recovered. In the massless case it would be necessary to go to two loops to determine the stability of the theory. This calculation is given in ref. [ 12].
Appendix Here we derive the results for eqs. (23) and (24). Consider (22) for v = 0 . Expanding the denominator as a binomial series {exp [2~r ( u + i 0 ) ] - 1 } -~
= ~ exp[-27r(l+l)(u+iO)] /=0
(u>0),
recognizing the subsequent integration of u as the gamma function, we have F(n+ 1)
f_./~(o, 0)--Re (y~)~z~ ,~o ~;
exp[ - 2~zi(l+ 1
(-7~5 ~
)ol
If we define the function Lip(z) -= ~,~_~ zn/n p (which is the polylogarithm function cited by Davies and 101
Volume 234, number 1,2
PHYSICS LETTERS B
M c L a c h l a n ) , this gives (28). (The sum exists for Izl ~< 1,p>~2 or [zl < 1, a l l p . ) F o r even n, u ¢ 0 we can e x p a n d (u 2 - v2) ~/2, giving a series o f integrals o f the form o f (29). These can be evaluated partially in terms o f the incomplete g a m m a function, see ref. [ 13 ]: F(ce, x)-= i e x p ( - t )
t'~-Zdt.
X
4 January 1990
for p < 0. This suggests that for ~ 3 × S ~we might consider expanding
(u 2 - ~,2) 3/2= u3( 1-~v2/u2+...). However, this series does not converge, since each term contains O ( v 2 ) , thus we resort to numerical m e t h o d s for these cases. Forf';3/2( v, +_), eq. ( 3 1 ) , first take the real part in (22), then let x = e x p ( 2 n u ) . N o w letting x = 1/y gives ( 31 ).
Consider i Un - l du I= ~ exp[2~-~;--~) 1- 1
Acknowledgement
Proceeding as for the massless case we o b t a i n
F(n, 2re(l+ 1 )v) e x p [ - 2re(l+ 1 )i0]
I=
l=o/-"
( l + 1)~(2rc) n
Expressing F a s a series,
References
F(n, x ) = ( n - 1 )! e x p ( - x )
rn! m=0
( n = l , 2 .... ) , we obtain
, - i ( n - 1 ) ! e x p [ - 2 n ( l + l )(v+iO) ] v m (2~Z)"-m(l+l)"-mm!
1 = ,=o~m=o~ Therefore
n-~ ( n - l ) !
l=y~
m=0
(2~z)'-'~v~Li,,_~(exp[-2rc(v+iO)]) m!
Note that m < n is always satisfied. Derivatives o f the polylogarithm are particularly simple. We have, from the defining relationship, d Liv(exp ( c 0 ) ) = c Lip_, (exp ( c 0 ) ) . N o t e that a similar expression to ( 2 9 ) can be d e r i v e d
102
M.B. is s u p p o r t e d by the Science a n d Engineering Research Council and would like to thank P. W o o d and M. H i n d m a r s c h for helpful discussions.
[ 1] Y. Hosotani, Phys. Lett. B 126 (1983) 309. [2] D.J. Toms, Phys. Lett. B 126 (1983) 445. [3l M. Evans and B.A. Ovrut, Phys. Lett. B 174 (1986) 63; F. Freire, J.C. Romao and A. Barroso, Phys. Lett. B 206 (1988) 491; K. Shiraishi, Prog. Theor. Phys. 77 ( 1987 ) 1253; 78 (1987 ) 535. [4] A.T. Davies and A. McLachlan, Phys. Lett. B 200 (1988) 305. [5 ] A.T. Davies and A. McLachlan, Nucl. Phys. B 317 (1988) 237. [6] A. McLachlan, Phys. Lett. B 222 (1989) 372. [ 71 A. McLachlan, University of Dublin preprint. [ 8 ] J.S. Dowker and S. Jadhav, Phys. Rev. D 39 (1989) 1196, 2368. [ 9 ] A. Higuchi and L. Parker, Phys. Rev. D 37 (1988 ) 2853. [ 10] D.J. Toms, Phys. Rev. D 27 (1983) 1803. [ 11 ] L.H. Ford, Phys. Rev. D 21 (1983) 933. [ 12 ] M. Burgessand D.J. Toms, University Newcastle upon Tyne preprint NCL89-TP 17. [ 13 ] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products (Academic Press, New York, 1986).
PHYSICS LETTERS B
4 January 1990
O N E L O O P F L U X B R E A K I N G BY M A S S I V E F I E L D S Mark B U R G E S S and D a v i d J. T O M S Department o f Physics, University o f Newcastle upon T)'ne, Newcastle upon Tyne NE I 7R U, UK
Received 12 August 1989; revised manuscript received 25 October 1989
We consider the effect of fields with non zero masses on flux breaking to first order in perturbation theory. It is found that non zero mass leads to greater stability of the unbroken symmetry configurations, for adjoint class fermions, and never assists the symmetry breaking mechanism. We also show that, in the massive case, there is a critical length associated with the compact dimension, above which stability is assured.
1. I n t r o d u c t i o n - dR
Tr(TaTh) = T The m e t h o d o f flux breaking o f gauge s y m m e t r i e s has attracted much interest in recent literature. This stems from the observation that, on multiply connected spacetimes, gauge fields can acquire non zero, constant c o m p o n e n t s which are not gauge equivalent to zero [ 1,2 ]. The p a r a m e t r i z a t i o n of the m e t h o d in terms of Wilson loops has resulted in a considerable simplification o f the formalism. Following a certain a m o u n t o f confusion in the literature [ 3 ], the mechanism has been investigated for massless fields by Davies and McLachlan [ 4 - 7 ]. Related work has also been carried out by D o w k e r and J a d h a v [8], and Higuchi and Parker [9]. The full extension o f the m e t h o d to fields of any mass is considerably m o r e difficult, but has a n u m b e r o f clear features which are not difficult to unravel, notably the emergence o f a critical periodicity length in the c o m p a c t dimension, which will stabilize a given theory [ 2 ]. O u r aim here is to briefly reveal those general features.
2. N o t a t i o n a n d c o n v e n t i o n s
Consider a non abelian gauge theory generated by [ T a, T t' ] =j" ~'~'cT< ,
( 1)
where a, b, c = I, ..., N with N the d i m e n s i o n o f the gauge group. The T ~ satisfy'
C2(GR)6~b
(2)
in an arbitrary basis. T ~ is the generator o f the matrix representation GR o f G, and is a dR × dR matrix. In the adjoint representation dadj is N. C2(GR) is the second o r d e r Casimir invariant of the representation GR. O u r conventions imply that the generators are anti-hermitian, i.e. T a r = - T a. Working on locally flat spacetime with topology Rn × S 1, with a euclidean metric du, and #, v = O, ..., n, we aim to compute the one loop effective potential for a theory involving fermions a n d / o r scalar fields, using the background field m e t h o d in the "background field" class o f gauges [ 2]. The gauge field m a y be written
~,+A,,
(3)
where -~u = T , ia ~ A-i, (i = 1, ..., rank ( G ) ) is the classical background field; a linear c o m b i n a t i o n o f the generators o f the Cartan sub-algebra o f G. It is convenient to work in a basis in which these matrices are diagonal, in which case these generators are conventionally denoted by H; i . e . ~ , - - H a'd j A-i~. A ~ _- T a~a j A l, is a q u a n t u m field and is a linear c o m b i n a t i o n of all the generators. In the basis which diagonalizes the H, only the subset H o f T a r e diagonal matrices. To preserve the symmetries o f the lagrangian at the classical level, we define A~ to be a spacetime constant. This ensures that the field tensor vanishes clas-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
97
Volume 234, number 1,2
PHYSICS LETTERS B
sically. The background field can then be written A~, = H~dj (0, 0, ..., U) where 2 ~are constants which lie in the S z direction, the other components having been gauged to zero. The action with group and spinor indices suppressed is consequently of the form [ 10]
/=l
4 January 1990
theory by transforming the quantum fields covariantly as follows:
q)t-~URq)~, T~--,Uw~t,
(12,13)
7 A~-~ LadjAu U a*d j
(14,15)
,
r/--, Uadj ~] ,
We fix /JR SO that the background field is eliminated explicitly from the action, i.e. Dy~O~. This requires
(4) S,i~¢= ~ f dVx (~Pt)'UDuTt+~PtmTt) , /=1
SVM =
4C2-1 ( Gadj )
-1
f dVxTr(P'"P,,~)
f
SGF= 2ozCT(G~dj) dV~Tr(13~'A~)2'
(5)
(6)
UR(x)=Pexp(g i A~,H~ dz~')
(16)
XO
in the representation GR. After this transformation, the action is reduced to that of a comparatively trivial vacuum energy calculation,
(7) 4. Boundary conditions
Sgho,, = f dV~ O(x) [ -~)2-A"Oa - (OUA.) ]r/(x)
,
(8) with covariant derivatives D u =Ou +g(Au
+Au),
(9)
D~,=O. +gA~,
(10)
and flu, =F,~ +g[-'{u, A~] + gtA,,
A,,] ,
( 11 )
with/tz and A, in the appropriate matrix representation ~. Note that ff~,~=0. @ is a complex scalar field, which transforms like a vector with dR components under the action of the group. ~Pj and TJ are d~ component Dirac fermions; f/and ~/are the ghost fields. All of these fields are pure quantum variables of integration. For generality we allow N~ generations of scalar multiplets and Nr generations of the Dirac fermion.
The price paid for this simplification of the action is a modification of the boundary conditions for the quantum fields. Initially the gauge fields must satisfy periodic boundary conditions on S ~, likewise the ghosts since their role is to quell the overcounting of gauge field configurations in the path integral. The matter fields need not be periodic prior to their transformation. We allow a general phase change around S ~. For periodicity L in S ~ ~2
cP(X+ L ) =exp(2zfiS)q~(X).
After the Wilson loop transformation, each group component of a given quantum field will pick up an extra phase from the multiplication by a Wilson line. lfwe denote the Wilson loop simply by
UR(L)=exp(2giOR),
It is of considerable utility to reparametrize the ~ The masses in (4) and (5) need not be the same, but since we shall not discuss mixed fermion/scalar theories here, extra notation is avoided.
98
(18)
where OR is a diagonal matrix, ~ t ~ e x p [ 2rci(OR +J1) ] q~t,
3. Wilson loops
(17)
(19)
Thus in momentum space, we have momenta k, (l+ 0 + J ) where [~Z and 0 is one component of the OR matrix.
(2n/L)
~2 Again delta need not be the same for scalars and fermions.
Volume 234, number 1,2
PHYSICS LETTERS B
calculate (22) analytically, though for even n, a sensible expansion in terms of polylogarithm functions can readily be given by expanding (u 2- v2) ~/2 and noting that
5. Evaluation of V m The one loop effective potential is given by ~s I dV~ V~I>= + N s Tr ln( -- ['-]scalar + r n 2)
_ ½Nf2 ~(,+l)/2j Tr In ( - [~ fermion+
i
=
~
P!(27~)m-(P+l)l/m
,,=o
m!
l,
×Lip+,_m(exp[-2n(/2+i0)])
+ ½( n + 1 ) Tr ln( - Dgaug~) + ( 1 / a ) t e r m (20)
where Tr is the functional trace and the trace over group indices. We employ the technique of Ford [ 11 ] in calculating (20) % Defining v=mL/2n we have, for each element 0 of 6)R in the group trace, a term Trln [-Qo+
updu
exp[2n(u+iO) ] - 1
m2 )
- Y r In( - ~gho~) ,
4 January 1990
(2n/2/L) 2] fdl~
(p>0).
(24)
The expansion for odd n proves unfruitful. (See Appendix. ) Recalling that the Wilson loops are diagonal matrices in our chosen basis, V ~~~may be written with one term of the form (21) for each diagonal element of the Wilson loop matrix. For example, for fundamental scalar fields transforming under su (2), the Wilson loops are of the form Uf = diag [ exp ( - 2~ix), exp (2nix)
],
n/2
4I'(-~n)(-L5 )
and
sin(½rlzc)f_n/2(/2,~) (21)
Uadj = diag[ 1, exp( -- 2zdy), exp (2niy) ] , giving V ¢11 in the form
(see eq. ( 2 6 ) ) , where
[
f-~/2 ( v,
O) = R e i
(25)
V~)= L F(
d u ( u 2 - / 2 2 ) ,,/2
v
× [f-~/2(
× { e x p [ 2 7 r ( u + i 0 ) ] - - 1} -j ,
\n/2
v, x+ d) +f-,,/2 ( v, - x + d) ]
(22) +
and we have dropped an irrelevant (infinite) constant. It is easy to show that
F(n+ 1)
f_,/2 (0, 0) = ~(27r),
× [f_,~/2(O, O) +f_n/2(O, - Y ) + f _ , / 2 ( 0 , y) ] • (26)
Re El,+ ~[ e x p ( - 2 z t i 0 ) ] (23)
in the massless limit, recovering the results of Davies and McLachlan [4,5]. (Lip(z) is the polylogarithm function, adopted in ref. [ 5 ].) (21 ) and (22) conceal zeros and singularities, tbr even n but well defined, regular limits are easily found for the results we require. In the massive case it is not generally feasible to ns Working to general oL we pick up terms of order 1/a al one loop. These are cancelled by terms arising from the functional measure, making V~I ) regular and gauge parameter independent. ~4 A massive generalization of the methods used in ref. [4] has undesirable features in the massless limit.
The first term comes from the scalars, the latter from the gauge + ghost fields.
6. Classically symmetric minima Davies and McLachlan have shown that for any group G, generated by a semi-simple Lie algebra, the turning points o f the effective potential, to one loop, give the symmetry preserving configurations of the background field [ 5 ]. This follows from the fact that the Wilson loops are multiples of the identity matrix when the symmetry is preserved, which corresponds to setting all values of 0 to a uniform constant, for a given field. This results in 99
d V ~1~ -a-
symmetric background
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Volume 234, number 1,2
o c T r ( H a) ,
(27)
which is zero for any semi simple Lie algebra. Generally, there is no unique 0 which preserves the symmetry. A value which has proven its importance is 0 = 0, corresponding to fields in an adjoint congruence class, or to A , = 0 in any class. This issue has been addressed at length in ref. [ 5 ], where it is shown that matter fields in an adjoint congruence class are required to break the classical symmetry, and that fields in other representation classes map to non zero values of-4¢, without breaking the symmetries. This only results in an addition to the effective mass of the form ~Tu,g"~u. We define a stable theory as one in which neither of these phenomena occurs. As an example, we focus our attention on adjoint class, periodic fermions, 0 = 6 = 0 , and look for the symmetric m i n i m u m at .g~=0. Differentiating the potential twice with respect to 2 and evaluating at 0 = 0 , we obtain a condition for the stability of the ~it,= 0 case:
tain (28). The primes denote differentiation with respect to 0. When 0 = 0 , n = 3 the RHS of (28) is unity and we recover the result of ref, [2]. For general n, with v = 0 we recover the result of ref. [51. For all other v the RHS is less than unity, falling offapproximately as e x p ( - 2 7 t v ) . Since v=mL/27t we can remark generally on the behaviour of (28). For L>> m -~ (u>> 1 ), the RHS tends to zero, thus the theory becomes stable irrespective of group of numbers of fermions. We therefore expect a critical length scale for fixed masses at which the theory changes from being stable to unstable. This critical length scale was first noted in ref. [21. Moreover, since the RHS < 1, the fermion mass always tends to stabilize the A u = 0 minimum. Technically the simplest example which verifies this is Kaluza-Klein theory R4X S I, for which we obtain, for arbitrary groups, 3
f ' - 2 ( / , I , O) = -- ~ Re Li3 (exp [ - 2 r t ( v + 61/
- ~-3 Re Li2(exp[ - 2 ~ ( v + i 0 ) ] )
NCa(G~d~)(n-I)2 -{'+t)/2 l~ 't~n/2 ( lJ, 0_~ > - - - . (28) dR Cz ( GR ) Nf [f "n/2 ( O, The modulus sign refers to the fact that we have used the negative sign o f f L~/2 ( V, 0) explicitly to ob-
10It 2
--
7C
Re Lil (exp[ -- 2 ~ ( v + i 0 )
ABSOLUTE VALUE 0 F f " ( u , 0 ) / f " ( 0 , 0 )
for n = 3
0.8 o
o-0.6
':" 0.t,
0.2
0t.~.
016
0.8
1.0
11.2
11
v = mL/2r~
Fig. 1. Graph illustrating the fall-off off '23/2( v, 0 ) / f "3/2(O, O) as a function of v.
100
])
- 20 u 3 Re Lio (exp [ - 27r(u + i0) ] ) ,
1.0
012
i0) ] )
(29)
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4 January 1990
8. Conclusions
and
-3
,f"-2(0, 0 ) = ~
~R(3) ,
(30)
where ~R is the Riemann zeta function. On ~73×S ~, taking f 312(u, 0 + d ) , with cases 0 = 0 , d = 0 , ½, corresponding to .4u = 0 with periodic and antiperiodic fermions, we can change variables in (22) to obtain
fQ3/2(V, +--_) exp ( - 2sty)
=-+
[ (In y ) 2 (27rp)2] 3/2 (y__+1 )dy (2g)2( 1Ty)3
o
(31) where, in f( _+ ), + refers to untwisted (periodic) fields and - refers to twisted (antiperiodic) fields. This form is suitable for numerical investigations, should they be of further interest (see fig. 1 ). For an arbitrary gauge group, we can determine the critical length for the stable theory (to one loop). Consider, for example, a single generation of adjoint fermions on ~3×S~. Setting GR= Gadj, d R = N i n (28), the LHS becomes ~. Referring to fig. 1, we read off v ~ 0 . 2 , giving Lcrit ~ 1.26 m - t .
We have shown that the effect of fermion masses is to stabilize the unbroken symmetric configurations. By considering results analogous to (28) it is not difficult to see that this result carries through in the case of twisted scalar fields also, and has no effect on twisted fermions or untwisted scalars (which cannot disturb the symmetries o f the lagrangian anyway). Cases which have critical stability in the massless limit are stable for non zero mass. It is clear why this should be so: in order for matter fields to destabilize the classical minima o f the potential, they must interfere destructively with the gauge ( + ghost) fields. For symmetry breaking we require the matter field contribution to be large with respect to the gauge contribution and of the opposite sign. The presence of masses always tends to reduce the magnitude o f the matter field contribution. Our results have concentrated on the stability o f the/~, = 0 background field state. It would be interesting to examine the role o f congruence classes (nality) in massive theories to determine the extent to which realistic masses inhibit symmetry breaking in the cases exposed by Davies and McLachlan. We defer a discussion o f this and other features o f flux breaking until a later date.
7. SU(L) For SU (L) there are simple geometric expressions for the group invariants. C2(Gaaj) =L, N=L 2- 1 and with matter fields in the fundamental representation dr=L and C2(Gf)=(La-1)/2L. Thus setting 0 = J = 0 , for periodic fermions, we see that
2 L ( n - 1)2 -(n+')12
N,.
>
If "_ni2( v, O)l - - - -
If"-n/2(0,0) I
.
(32)
In particular, when n = 3 ( E 3 × S I ) the RHS is just L/Nf. Thus for S U ( 2 ) with two generations of fundamental fermions, the stability of A~ teeters on the brink of indecision when the mass is zero. For m non zero, stability is recovered. In the massless case it would be necessary to go to two loops to determine the stability of the theory. This calculation is given in ref. [ 12].
Appendix Here we derive the results for eqs. (23) and (24). Consider (22) for v = 0 . Expanding the denominator as a binomial series {exp [2~r ( u + i 0 ) ] - 1 } -~
= ~ exp[-27r(l+l)(u+iO)] /=0
(u>0),
recognizing the subsequent integration of u as the gamma function, we have F(n+ 1)
f_./~(o, 0)--Re (y~)~z~ ,~o ~;
exp[ - 2~zi(l+ 1
(-7~5 ~
)ol
If we define the function Lip(z) -= ~,~_~ zn/n p (which is the polylogarithm function cited by Davies and 101
Volume 234, number 1,2
PHYSICS LETTERS B
M c L a c h l a n ) , this gives (28). (The sum exists for Izl ~< 1,p>~2 or [zl < 1, a l l p . ) F o r even n, u ¢ 0 we can e x p a n d (u 2 - v2) ~/2, giving a series o f integrals o f the form o f (29). These can be evaluated partially in terms o f the incomplete g a m m a function, see ref. [ 13 ]: F(ce, x)-= i e x p ( - t )
t'~-Zdt.
X
4 January 1990
for p < 0. This suggests that for ~ 3 × S ~we might consider expanding
(u 2 - ~,2) 3/2= u3( 1-~v2/u2+...). However, this series does not converge, since each term contains O ( v 2 ) , thus we resort to numerical m e t h o d s for these cases. Forf';3/2( v, +_), eq. ( 3 1 ) , first take the real part in (22), then let x = e x p ( 2 n u ) . N o w letting x = 1/y gives ( 31 ).
Consider i Un - l du I= ~ exp[2~-~;--~) 1- 1
Acknowledgement
Proceeding as for the massless case we o b t a i n
F(n, 2re(l+ 1 )v) e x p [ - 2re(l+ 1 )i0]
I=
l=o/-"
( l + 1)~(2rc) n
Expressing F a s a series,
References
F(n, x ) = ( n - 1 )! e x p ( - x )
rn! m=0
( n = l , 2 .... ) , we obtain
, - i ( n - 1 ) ! e x p [ - 2 n ( l + l )(v+iO) ] v m (2~Z)"-m(l+l)"-mm!
1 = ,=o~m=o~ Therefore
n-~ ( n - l ) !
l=y~
m=0
(2~z)'-'~v~Li,,_~(exp[-2rc(v+iO)]) m!
Note that m < n is always satisfied. Derivatives o f the polylogarithm are particularly simple. We have, from the defining relationship, d Liv(exp ( c 0 ) ) = c Lip_, (exp ( c 0 ) ) . N o t e that a similar expression to ( 2 9 ) can be d e r i v e d
102
M.B. is s u p p o r t e d by the Science a n d Engineering Research Council and would like to thank P. W o o d and M. H i n d m a r s c h for helpful discussions.
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