- Email: [email protected]
Unification of effective field theories
Nuclear Physics B 179 (1981 ) 477- 49 I © North-Holland Publishing Company
UNIFICATION O F EFFECTIVE FIELD T H E O R I E S * Sally DAWSON and Howard GEORGI l
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 14 October 1980
We consider the renormalization of the coupling constants in theories with extended gauge hierarchies. An effective field theory approach is used to include an interesting class of higher-order effects in the renorma!iT~tion group formulas. We calculate these corrections for all possible breakdowns of O(10) to SU(3) x SU(2) × U(I).
Unified gauge theories of the weak, electromagnetic, and strong interactions have received much theoretical attention recently. Many simple models based on the unifying groups SU(5) and O(10) have been proposed [1]. These models have only two levels of symmetry b r e a k i n g - - t h e unifying group is broken down to SU(3) x SU(2) x U(1) at a mass scale around 1014 GeV and SU(3) x SU(2) x U(1) is further broken down to SU(3) X U(1)E M near the mass of the W-boson (~, 80 GeV). In this picture, the region between 80 GeV and 1014 GeV is a desert with no interesting structure. In a previous paper [2], we have suggested that it is interesting to consider the effects of allowing the theory to have a hierarchy of arbitrary gauge symmetries. In these models, the relationship between sin 2 Ow and M (the grand unification scale) is interesting phenomenologically. Experimentally sin2 0 w ~ 0.23 and the lifetime of the proton is greater than 103° years, (~1,~ In this note, we generalize our treatment of coupling constant renormalization in these models by including an interesting class of higher-order effects. We include those corrections to the coupling constant renormaliza,tion dictated by the structure of the gauge symmetries and neglect those effects dependent on the specifics of the scalar and fermion structures of a given model. We begin by reviewing our earlier results to set the notation. Suppose that a simple gauge group G is broken down to SU(3) x SU(2) x U(1) in N steps. The theory is described by N + 1 masses/~x, x -- 0 to N, where we have assumed that/~x-1 << #x. In the region (region x ) between/~x-i and #x, there is an effective gauge theory S x. S x is the subgroup of G which is left unbroken in steps x through N of the hierarchy. S T M is broken down to S ~ at a mass/x x. The gauge
M4/M~).
* Research supported in part by the National Science Foundation under grant no. PHY 77-22864 and
Ithe Alfred P. Sloan Foundation. 477
S. Dawson, H. Georgi / Unification
478
bosons associated with S T M but not with S x get masses of the order of/~x- The subgroup S x is a product: SX = 1-[s~,
where s~ is either a simple non-abelian subgroup or a U(I) group. The generators of s aX are T,~Xi. Because of the nested gauge structure, we can express the generators of S x as linear combinations of the generators of S y for any y > x: T~ --- Z r" xa iyBj -ry "BJ"
(1)
#J
In ref. [2], we defined xy 2 e ~ = Y, IC2,~jl
(2)
J
and showed that [assuming only one U(1) subgroup] E
l'~xy (-,xy -- R R D x Y ~"'aivk""fljTk -- "aflvij" a y
"
(3)
k
The p xy may be interpreted as the probability that the s~ subgroup of S x "comes from" the s~ subgroup of S y. If g ~ ( E ) is the gauge coupling constant for the s~ subgroup renormalized at E(/~x-! < E < #x), then
1
gaX(//,x_ 1) 2
-
-
1
g(/J,N) 2
N xy y + ~,, In /~y ~, P~/~ 2bo/3 .
y--x
#y--I
(4)
fl
b~'~ is the constant which appears in the fl function for g,~x ( f l ~x( g a x) = b o ax g a,,3 + O(g~5)), and g is the guage coupling constant of the unifying group G. This is valid to second order in the gauge coupling constants and in the approximation of sharp transitions between the different regions [3]. In this paper we present a consistent method for calculating the corrections to the renormalization group formula of eq.
(4). Our calculations utilize the effective field theory techniques used recently by Weinberg [4]. In each region S x, physical processes are described in terms of functional integrals of the action I x. The action is a function of the "light" fields in s~, (call them ~x), and the "heavy" fields, ~x+ 1, which have obtained their masses in the breakdown of the S x+l symmetry, (the masses of the ~x+l are thus of the
S. Dawson, H. Georgi / Unification
479
same order of magnitude as #x). Following Weinberg, we define an effective action in each stage of the symmetry breaking: e " ~ " * ~ ' = f ( d q , x+l)eiz~t '(,~÷',0").
(5)
In doing this, we must use a background field gauge [5] in which the gauge-fixing term breaks the Sx+~ symmetry, but leaves the Sx symmetry unbroken. The effective gauge coupling is defined by modified minimal subtraction in the effective theory. This definition combines the convenience of minimal subtraction with the utility of momentum-space subtraction. At each stage of the symmetry breaking, we calculate the one-loop integrals with two of the W~, (the gauge fields of s~), as the external particles. The q,x+l are the internal particles of the loops and consist of heavy scalars, heavy ghosts and those vector mesons which get a mass from the vacuum expectation value of the Higgs scalars used to break Sx+m to S ~. After these integrations, the remaining effective theory contains only particles whose masses are much less t h a n / ~ . The one-loop integrals yield corrections to the boundary conditions between the g~(#~) and the g~+ '(t~x) [61. A consistent approach is to do the appropriate one-loop integrals at the boundary between ST M and Sx and then integrate the coupling constants from/~x to/~xusing the second-order beta functions. At ttx_ ~, we compute the boundary conditions between the g~-l(/~x_l) and g~(/~-l) by doing the one-loop integrals with the ~ as the internal particles and the WX7 1 as the external particles. This procedure is repeated at every threshold between the unifying group and SU(3) × U(1). W e integrate o u t the h e a v y p a r t i c l e s at e a c h stage of the s y m m e t r y b r e a k i n g b y d o i n g the integrals of fig. 1. T h e integrals a r e p e r f o r m e d using d i m e n s i o n a l
c•x÷l Ca)
K._J
7
(b) k
k
Fig. 1. Feynman diagrams needed to calculate ?~(/tx). W~ are the gauge fields of s~,. ~x+l contains the vector mesons which obtain masses in the breakdown of Sx+1 to Sx, the heavy ghost fields, and the heavy scalar fields. (The masses of the q,x+l are approximately equal to/~x.)
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S. Dawson, H. Georgi / Unification
regularization and the MS renormalization scheme [7]. In this approach, at every stage in the symmetry breaking the non-renormalizable heavy gauge couplings automatically decouple in the beta functions from the light gauge couplings. After calculating the effects of the diagrams in fig. 1 on the W~ propagators, we need never consider the heavy vector mesons (and the heavy scalars and ghosts) again. Their entire effect on the theory (to this order) is taken into account by using the results from fig. 1 to redefine the boundary conditions on the coupling constants between region x and region x + 1. It has been noted that this approach must be generalized to include higher-order effects consistently, but to this order it is adequate [8]. To describe the calculation and the results, it is convenient to introduce some notation. We must explicitly parametrize the breakdown of SX+m---~Sx. We do this by rewriting the gauge fields of region x + 1, W~ ÷ l, in a more convenient basis. We rewrite the W~ +l in terms of fields which are mass eigenstates (ignoring the breaking of SX). In addition to the massless (in the x region) gauge fields W~, there are massive fields V~a,.. The meaning of the subscripts is as follows: R = (G) labels the S x representation under which the fields transform, a = 1,2 . . . . is an additional label (if necessary) to distinguish between different sets of fields transforming under the same S x representation. All the vectors with the same R and a have mass MRa (ignoring S ~ breaking), m labels the particular state within a given representation. We will feel free to choose the basis in the representation space at our convenience and will often suppress the m dependence. Wx couples to the V ~ through the appropriate representation matrices ( T,r°),, ,,,. The calculation will be described in appendix A. Here we give the results for the corrected boundary conditions, keeping contributions proportional to logs of heavy particle masses. We find 1
1
- = E Pftl ~+t - k~, g2(/xx) 2 B gt~+'(#x) 2
(6)
where h'~--
11 E I n ( M R a / I ~ : ) T r ( T , . ' Q 24rr 2 R,a
2
(7)
In what follows, we will assume that we have spontaneous symmetry breakdown due to the vacuum expectation value ( V E V ) of a Higgs field. From the structure of the spontaneous symmetry breaking, we then know that the vector mass matrix has the form: ( M 2 v ) .iOj ---g ~ x+ 1g ox+ i t~l~i ~x+ --
x+l = g¢~ gflx + lIIaioj,
1(¢5' Tfl~+l(d'~5) (8)
S. Dawson, H. Georgi / Unification
481
where ( ~ ) is the Higgs VEV and T_,~; x+l are the generators of S T M on the scalar field representation. This matrix has eigenvectors with zero eigenvalue,
(
)
=
ck#j, +TgU ,
(9)
corresponding to the W~'s. The non-zero eigenvalues are the M2a's. We want to concentrate on the dependence of eq. (8) on the coupling constants x4-1 ga . We expect the non-zero eigenvalues of II which contribute to S x÷l breaking all to be of the order of ( ~ ) , but the coupling constants g~÷ l may be very different for different ft. Thus in eq. (7), when we sum over appropriately weighted logs of MRa, we will keep only the g~+l dependence. This is non-trivial only because the matrix II is singular. In appendix B we derive the desired result x x} ~,~'~2 ( ~ ~ . , ,l~X,x+lhx+l ~ ~o~ lng~ +1 - b o~lng= ,
where we have taken /t x of the order of the constants b ~ are defined by
Y ~Ji):i, )
(lO)
S x÷l
breaking VEV, <@). The
--Tfqr OOa,
(11)
where fo~/{' are the structure constants of s x in the adjoint representation. This completes our discussion of boundary conditions. We next show that the fl-function corrections can be cast into a similar form. When there are no fermions, the beta functions for the different coupling constants decouple from each other: X
X
oo~g ~ + o,~g~ + O ( g ~ 7 ) .
(12)
Eq. (12) can be integrated to find g ~ ( / G _ , ) -2 = g X ( # x ) - 2 + 2b~,,ln(l~x/IG_,)
+ (b~,Jb&)ln[(g;(I.G_,)-2+(b~,,/b~))/(g;(p~)-2+(b~a/b~)=))] .~ gX(#~) -2 + 2 b ~ l n ( P x / t G _ , ) + 2( b;,~/b~) ln[ g~(Px) / g ~ ( P x - , ) ] .
(13)
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S. Dawson, H . Georgi
/ Unification
By using induction and eqs. (6) and (7), it is a simple matter to derive the following form: N
g~(ttx_l) -2 =g0,,,) - : - Y, Y e:gx 0,,) y--x
fl
N
+ X E 2P2~{boY~ln(#y/l~y-l) y~x
fl
+ (bY/3/bffa) In [ g~(t~y ) Ig~(#y_ l)
Now we can use the expression for form:
]}"
(14)
hY~(~y) of eq. (10) to put eq. (14) in a simpler
N
gX(/~x_l)-2=g(/~,v)-2+
~ y=x
×
Y~2Pf~ fl
{bY0ln(/~y/l~y_,) + [
b~ + (bYt~lb~t~)]ln [ g~(I,,ly)lg~(I,~y_l)]}
+ 2b~lng~(l~x_l) -- 2bolng(/~N) •
(15)
In this approximation the effects on the renormalization of the coupling constants arising from the corrections to the boundary conditions of the coupling constants, [eq. (10)] have the same functional dependence on the coupling constants as do the terms coming from the g25 terms in the beta functions. (b 0 and b t are the coefficients of the beta function of the unifying group: fl(g) = bog 3 + bigS.) Finally, we can set x = 1 and/~0 = Mw, ( $1 ~- SU(3) × SU(2) × U(1)). However, #~v is not a free parameter since we have already taken/~N to be approximately equal to the S N+I breaking VEV. To the order in which we are working, M
l~N~--~g(b~N )' where M is the superheavy vector meson mass and g(/~N) is the coupling constant of the unifying group. Using the renormalization group formula of eq. (13),
b' ) I n ~( gg(-/ Z g(M)-2=g(~tN)-2 + 2b°ln-~+ 2 ( -~o ~ N] ) / .
(16)
S. Dawson, H. Georgi / Unification
483
Substituting eq. (16) into eq. (15), we obtain our final result: N
g~(Mw)-Z=g(M)-Z+ Y, y=l
X
Y~2P2~ /3
bot~ln(/~y/#y_,) +
-b~',
ln[ga(izy)/gh(#y_ ,
- 2 X P~;bo~ In g(M) + 2 b~ In g~( M w ) ,
(17)
where #~v = M , / ~ = Mw, and #'i (i :/: N, i :/: 0) is the S i+ l breaking VEV. At this point, we think it is important to discuss the philosophy behind the approximations we have made. We have considered only the corrections to the coupling constants for which a given gauge theory makes definite predictions. Our calculations are independent of the Higgs and fermion structures. The matching functions, A~, which occur in the boundary condition of eq. (6) have, however, a weak dependence on the Higgs and heavy fermion structure of the theory [4]. We neglect these effects not only because they are smaller than the terms we have included, but also because we can only guess their exact form. By the same logic, we have also neglected contributions to the beta functions from light fermions. However, all of the effects which we have ignored tend to decrease sin20w(Mw) for a fixed value of the superheavy mass scale [6]. Thus, our results may be viewed as a reasonable upper bound to the corrections to the renormalization of the coupling constants occurring in this order of perturbation theory. We will now illustrate our techniques in a model in which the strong, weak, and electromagnetic interactions are unified in an O(10) group at very large energies. We will consider three possible breakdowns of O(10): (i)
O(10)--->SU(4) × SU(2) x SU(2)--,SU(3) × SU(2) x U(1) ;
(ii)
O(IO)-->SU(4) x SU(2) x U(1)---~SU(3) x SU(2) × U(1) ;
(iii)
O(10)--->SU(3) × SU(2) × SU(2) x U(1)-->SU(3) x SU(2) × U(1).
(18)
The lowest-order results have been worked out previously by Georgi and Nanopoulos [9]. We calculate sin 2 0 w ( M w ) as a function of M (the superheavy vector mass) using as inputs as(Mw) = 0.15, a ( M w ) -~-~, - l and M w = 8 0 GeV [10]. We then use eq. (17) to write renormalization group equations for each of the coupling constants
484
S. Dawson, H. Georgi / Unification I
I
I
SU(4)x SU(2) x SU(2)
1@9t
/ /
1017 --
Mass (GeV) lOm
10~3
10H [
,
I
.20
.24
,
I
,
.28
sinZOw( Mw) Fig. 2. O(10)--,SU(4) X SU(2) X SU(2)--~SU(3) x SU(2) x U(1). M is the superheavy vector m e s o n mass a n d M l is the mass scale of the intermediate symmetry breaking. The solid lines axe the lowest-order values [eq. (4)] a n d the dotted lines are the values with the second-order corrections of the text [eq. (17)] included.
in the SU(3)× SU(2)× U(1) region. To solve these equations, we substitute the lowest-order results of eq. (4) into the lng~(/~y) 2 terms in eq. (17) to obtain N I
g,,(Mw)
--2
-2+
=g(M)
E E P2~{2b~ln(l~'y/IZ'y-l) fl y ~ l
-
IN N 2~ P~p boplng(M) + 2b~.lng~(Mw).
(19)
#
Results are presented in figs. 2 - 4 for each of the breakdowns of O(113) listed in eq. 0 8 ) . The net effect of the second-order corrections which we have calculated is that for a fixed value of sin2 0 w ( M w ) the superheavy mass scale is smaller than the lowest-order value.
485
S. Dawson, H. Georgi / Unification I
I
I
SU(3) x SU(2) x SU(2) x SU(I) 1019
lOr7 Mass (GeV) 1015
1013
10I1
I
.20
,
I
24
,
L
.28
,
sin28w(Mw) Fig. 3. O(10)-+SU(3)X SU(2) x SU(2) x U(1)---~SU(3)X SU(2) x U(I). I
I
I
SU(4) x SU(2) x SU(1) 1019t
101rt Mass (GeV)
lo,°S NN\ 1 0 I~ -
10' f
Mx I
.20
,
I
.24
,
I
.88
sinZOw(Mw) Fig. 4. O(10)....~SU(4)x SU(2)× U(I)....~SU(3) × SU(2) × U(1).
S. Dawson, H. Georgi / Unification
486
Conclusion In this paper, we have presented a systematic method for calculating corrections to the lowest-order renormalization group equations in a theory with a complicated gauge hierarchy. However, perhaps our most important result was seen in the context of an O(10) model--these corrections are potentially quite significant, leading us to believe that they must be taken into account in any realistic calculation. Appendix A We use 't Hooft's dimensional regularization [8] definition of the coupling constants:
[
1
g,,P'x
'
(A.1)
4 +
Here D is the dimension of space-time, "/E is the Euler constant, and B2 is the coefficient of the beta function for g 2 ( # x ) . In this language, the boundary condition between the S ~ and S T M regions which is given in ref. [2] is,
[]2 ~x.2-n/2
= E P~;'+'
oar-x
[112 o x + 1,02--D/2 ~B V'x
fl
.
(A.2)
Since we break the S T M symmetry with a gauge-fixing functional which preserves the S x symmetry, the gauge-field lagrangian in the S x region is 1
x ~auge ~-
. .la)F,~i~,F . . .
a~(1
(A.3)
ia, ,
ai
where x
X
X
X
X,a
X
X
Fdi,, = 3F,W:i, - O,W;i~, + g , fijk W~j~,Wdk,"
The 12 contain all of the effects of the heavy particles, g,~'+1, on the theory. In order to be able to canonically normalize the gauge-field terms in the lagrangian, we define the renormalized coupling constant g2(#x)R to be, g~(#x)R = ( 1 -- i~(#x))- l/2gX(#~).
(A .4)
We follow Weinberg's notation [4] and further define lim l:(/~x)= [ )~(#x) + ) ~ = ( # x ) ( ~ _ 4 D--~
+ ½"rE- lnX/~
)]gX(#x)2.(A.5)
S. Dawson, H. Georgi / Unification
487
Substituting eqs. (A.1), (A.4) and (A.5) into eq. (A.2), we obtain, after much algebra, the corrected boundary condition
1
-- Z
x 2 ga(P'x)R
px~x+l
,e
1 gj+'(p,x) 2
Xx(/.tx)"
(A.6)
(We now drop the subscript R.) The techniques for evaluating ~ ( / t x ) from the Feynman diagrams of fig. 1 are described in refs. [4, 6]. For a simple non-abelian group G breaking to a product of simple or non-abelian groups, they find the result: ~x(/~x ) = _
l___~l~ Tr(T:.)21n(MR~//~x), 24 ~r2 R,~
(A.7)
where we have translated their result into our notation. When there is a product of groups, II,,s~ +1, which breaks to II~s x, the identical result is obtained for X~,,(/~). This can be proven by writing W~ + i as the approximate sum of the W~ and V~,,, finding the Feynman rules, and evaluating the graphs of fig. 1. This is a tedious process and, except for the notation, our method is identical to that detailed in ref. [61.
Appendix B Did you ever notice that in the SU(2) x U(1) theory the Z mass is a constant times the V E V G f 1/2 times the product of the SU(2) and U(1) couplings divided by e? In this appendix, we generalize this curious fact in order to derive eq. (I0) from eqs. (6) and (7). The reader who cannot abide any more notation is advised to skip from eq. (6) directly to eq. (17) for a more qualitative discussion. We first go to a Cartan basis for the T~, and restrict our attention to generators in the Cartan subgroup, H~i, which can be simultaneously diagonalized. The mass eigenstates correspond to definite HXi eigenvalues called weights ~ i . We can take the m label in V~,~ to be the weight to. Now take T,.r~ in eq. (7) to be a Caftan generator. Then we can write eq. (7) as follows:
~'= a
11 Z(toai)2 y~ ~ 24~r2 ~ R~,~ a
ln(MRa/l~x).
(B.I)
The second sum runs over all representations containing the weight ~. It is possible to show (and we will do it explicitly below) that ~, Y, l n ( M R a / # x ) = ~ n ~ + l ( w ) lng~ +' -- n~(~o) lng~ + - . . RD~0
a
fl
,
(B.2)
S. Dawson, H. Georgi / Unification
488
where the remaining terms are independent of g. The integer n~ + I(to)(nxa(t0)) is the degeneracy of the weight to in the adjoint representation. Note that the generators of S x+' can also be labeled by weights according to [H~X.,qr'x+l] -to
"tpa~ J -
(B.3)
'7"' x + l ai'Bao "
The weights of the adjoint representation s~ are the "roots" (which are not degenerate for to ~ 0). Thus n~(to) = 0 or 1. Inserting eq. (B.2) into eq. (B. 1) we obtain
hx __
11 ~ ( t o a i ) 2 ( ~ n ~ + l ( t o ) l n g ~ + , _ n x ( t o ) l n g X } 24¢r2 p
"
(B .4)
In the second term we can use the relation
Y. (too,):n;(to) = Y, t\ Jijk / w
(B.5)
j,k
in the adjoint representation. In the first term, we use the relation
E (toai)2n~+l(to) -- PXl~X+l E ( fi;:l'l~) 2 , ,,,
(B.6)
jk
Combining eqs. (B.4)-(B.6) with eq. (11) yields eq. (10). All that remains is to derive eq. (B.2). Indeed, it was this amusing relation that first piqued our interest in these corrections. The masses M~a for R D to are the non-zero eigenvalues of the matrix 2
M~a#b
x+l
=
g~
x+lI~x+l/--\
gp
~1~,~,~q g ) ,
T,X+l/,~\~ p,,o,
\v/i •
(B.7)
If nX(to) = 0, the matrix Mda,b 2 is non-singular. Then eq. (B.2) is completely trivial because
M 2 = GHG,
(B.8)
where x+l
x+l
l~otafl b = ( r~a u ( dp ) , Tl~bw ( dp ) ) ,
(B.9)
and G is a diagonal matrix,
G~a#b -- g~x + 16,~t~6,,b,
(B.10)
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S. Dawson, H. Georgi / Unification
Thus we find E E ln(mRa/#x)----~ RDoo a
1 Trln(M2/i z2) = Trln G +i l Trln(II2/#]).
(B.11)
This is clearly equivalent to eq. (B.2) if <#~)-----#x so that T r l n ( H 2 / # 2) can be neglected. If n~(to) = l, the above derivation breaks down. The sum over MR,, includes only the non-zero eigenvalues of M 2. The sum is not equal to i T r l n ( M 2 / # ] ) which is just as well because the trace doesn't exist. To deal with this case, we must introduce yet more notation. Since nX(to) -- 1, there is an s~ generator with weight ~o. Call it EX~0. It satisfies EXw = E
(B.12)
t"'x'x+ITx+I
fl,a
The C ' s are simply the /',x,x+ -~iOj 1 rewritten in a different notation. Thus for example
E [('x'x÷l~2-ox'x÷l " a , B a ] -- " aB Q
(B.13)
•
The normalized eigenvector of M 2 with zero eigenvalue is e9 a _= g~,~a x ( , x ' ~~+ l l g ~ + 1 •
(B. 14)
The matrices M E, G and H act in a space of dimension d---
with the directions labeled by fl and a. We define a completely antisymmetric tensor in this space in the usual way: e~,al~:2...#: ~ = _+ 1.
(B.15)
Then for any matrix A in the space, we can define the cofactor matrix A ¢ as
"gibi X eYbY2b2...YabaAfl2a2y2b2. • • ABaaaYaba.
(B.16)
S. Dawson,H.Georgi/ Unification
490
NOW consider the cofactor of M 2. Because M 2 has a zero eigenvalue, M 2c must have the form
( M2C)#avbm [ RDtoI-[~ M2c]e#aevb.
(B.17)
O n the other hand, it follows f r o m eq. (B.7) that
Tr(GM2~G)in
If we calculate have the desired result
two ways, using eqs. (B.17), (B.18) a n d (B.14) we
(gX)2[~,~M2cJ=[~(g~+l)n~÷'(~')]2Tr(II¢).
(B.19)
Since Tr YIc-~ ( t ~ 2(d- 5), eq. (B.19) is the exponentiated version of eq. (B.2), for nX(o~)---- 1. We c a n n o t resist closing this appendix with some examples of eq. (B.19). As we implied earlier, one example is familiar to all. I n the SU(2) x U(1) theory g[ is the U(1) coupling is the SU(2) coupling g. gO is e. All the n ( ~ ) ' s are equal to one. Thus eq. (B.19) implies that M z is proportional to which it is. A n even more surprising example is the case S U ( 2 ) x S U ( 2 ) x SU(2) b r o k e n d o w n to the diagonal SU(2). I n this case there are two triplets of massive vector bosons with masses M s a n d M 2. If the gauge couplings of the u n b r o k e n theory are gi, i = 1 to 3, to gauge coupling g of the diagonal SU(2) is
g',g~
gg'/e,
__1= ~ _1 g2
. g2"
(B.20)
Here no matter h o w the s y m m e t r y is broken, eq. (B. 19) implies that M l M 2 is equal to times a function of the VEV's.
g~g2g3/g
References {1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438; H. Georgi, H. Quirm and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451 [2] S. Dawson and H. Georgi, Phys. Rev. Lett. 43 (1979) 821 [3] T.W. Applequist and J. Carazzone, Phys. Rev. D1 i (1975) 2856 [4] S. Weinberg, Phys. Lett. 91B (1980) 51 [5] G. 't Hooft and M. Veltman, Ann. Inst. Poincar~ 20 (1974) 69; B.S. deWitt, Phys. Rev. 162 (1967) 1195; 1239; Phys. Reports 19 (1975) 295 [6] L. Hall, Nucl. Phys. B178 (1981) 175 [7] G. "t Hooft, Nucl. Phys. B61 (1973) 455; B82 (1973) 444
S. Dawson, H. Georgi / Unification [8] B. Ovrut and H.J. Schnitzer, Brandeis preprints (1979, 1980); T. Hagiwara and N. Nakazawa, Brandeis preprint (1980); T. Hagiwara, Phys. Lett. 90B 0980) 305 [9] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B159 (1979) 16 [10] D.A. Ross, Nucl. Phys. Bl40 0978) l; W. Marciano, Phys. Rev. D20 (1979) 274; T.J. Goldman and D.A. Ross, Cal Tech preprint 68-704 (1979)
491
UNIFICATION O F EFFECTIVE FIELD T H E O R I E S * Sally DAWSON and Howard GEORGI l
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 14 October 1980
We consider the renormalization of the coupling constants in theories with extended gauge hierarchies. An effective field theory approach is used to include an interesting class of higher-order effects in the renorma!iT~tion group formulas. We calculate these corrections for all possible breakdowns of O(10) to SU(3) x SU(2) × U(I).
Unified gauge theories of the weak, electromagnetic, and strong interactions have received much theoretical attention recently. Many simple models based on the unifying groups SU(5) and O(10) have been proposed [1]. These models have only two levels of symmetry b r e a k i n g - - t h e unifying group is broken down to SU(3) x SU(2) x U(1) at a mass scale around 1014 GeV and SU(3) x SU(2) x U(1) is further broken down to SU(3) X U(1)E M near the mass of the W-boson (~, 80 GeV). In this picture, the region between 80 GeV and 1014 GeV is a desert with no interesting structure. In a previous paper [2], we have suggested that it is interesting to consider the effects of allowing the theory to have a hierarchy of arbitrary gauge symmetries. In these models, the relationship between sin 2 Ow and M (the grand unification scale) is interesting phenomenologically. Experimentally sin2 0 w ~ 0.23 and the lifetime of the proton is greater than 103° years, (~1,~ In this note, we generalize our treatment of coupling constant renormalization in these models by including an interesting class of higher-order effects. We include those corrections to the coupling constant renormaliza,tion dictated by the structure of the gauge symmetries and neglect those effects dependent on the specifics of the scalar and fermion structures of a given model. We begin by reviewing our earlier results to set the notation. Suppose that a simple gauge group G is broken down to SU(3) x SU(2) x U(1) in N steps. The theory is described by N + 1 masses/~x, x -- 0 to N, where we have assumed that/~x-1 << #x. In the region (region x ) between/~x-i and #x, there is an effective gauge theory S x. S x is the subgroup of G which is left unbroken in steps x through N of the hierarchy. S T M is broken down to S ~ at a mass/x x. The gauge
M4/M~).
* Research supported in part by the National Science Foundation under grant no. PHY 77-22864 and
Ithe Alfred P. Sloan Foundation. 477
S. Dawson, H. Georgi / Unification
478
bosons associated with S T M but not with S x get masses of the order of/~x- The subgroup S x is a product: SX = 1-[s~,
where s~ is either a simple non-abelian subgroup or a U(I) group. The generators of s aX are T,~Xi. Because of the nested gauge structure, we can express the generators of S x as linear combinations of the generators of S y for any y > x: T~ --- Z r" xa iyBj -ry "BJ"
(1)
#J
In ref. [2], we defined xy 2 e ~ = Y, IC2,~jl
(2)
J
and showed that [assuming only one U(1) subgroup] E
l'~xy (-,xy -- R R D x Y ~"'aivk""fljTk -- "aflvij" a y
"
(3)
k
The p xy may be interpreted as the probability that the s~ subgroup of S x "comes from" the s~ subgroup of S y. If g ~ ( E ) is the gauge coupling constant for the s~ subgroup renormalized at E(/~x-! < E < #x), then
1
gaX(//,x_ 1) 2
-
-
1
g(/J,N) 2
N xy y + ~,, In /~y ~, P~/~ 2bo/3 .
y--x
#y--I
(4)
fl
b~'~ is the constant which appears in the fl function for g,~x ( f l ~x( g a x) = b o ax g a,,3 + O(g~5)), and g is the guage coupling constant of the unifying group G. This is valid to second order in the gauge coupling constants and in the approximation of sharp transitions between the different regions [3]. In this paper we present a consistent method for calculating the corrections to the renormalization group formula of eq.
(4). Our calculations utilize the effective field theory techniques used recently by Weinberg [4]. In each region S x, physical processes are described in terms of functional integrals of the action I x. The action is a function of the "light" fields in s~, (call them ~x), and the "heavy" fields, ~x+ 1, which have obtained their masses in the breakdown of the S x+l symmetry, (the masses of the ~x+l are thus of the
S. Dawson, H. Georgi / Unification
479
same order of magnitude as #x). Following Weinberg, we define an effective action in each stage of the symmetry breaking: e " ~ " * ~ ' = f ( d q , x+l)eiz~t '(,~÷',0").
(5)
In doing this, we must use a background field gauge [5] in which the gauge-fixing term breaks the Sx+~ symmetry, but leaves the Sx symmetry unbroken. The effective gauge coupling is defined by modified minimal subtraction in the effective theory. This definition combines the convenience of minimal subtraction with the utility of momentum-space subtraction. At each stage of the symmetry breaking, we calculate the one-loop integrals with two of the W~, (the gauge fields of s~), as the external particles. The q,x+l are the internal particles of the loops and consist of heavy scalars, heavy ghosts and those vector mesons which get a mass from the vacuum expectation value of the Higgs scalars used to break Sx+m to S ~. After these integrations, the remaining effective theory contains only particles whose masses are much less t h a n / ~ . The one-loop integrals yield corrections to the boundary conditions between the g~(#~) and the g~+ '(t~x) [61. A consistent approach is to do the appropriate one-loop integrals at the boundary between ST M and Sx and then integrate the coupling constants from/~x to/~xusing the second-order beta functions. At ttx_ ~, we compute the boundary conditions between the g~-l(/~x_l) and g~(/~-l) by doing the one-loop integrals with the ~ as the internal particles and the WX7 1 as the external particles. This procedure is repeated at every threshold between the unifying group and SU(3) × U(1). W e integrate o u t the h e a v y p a r t i c l e s at e a c h stage of the s y m m e t r y b r e a k i n g b y d o i n g the integrals of fig. 1. T h e integrals a r e p e r f o r m e d using d i m e n s i o n a l
c•x÷l Ca)
K._J
7
(b) k
k
Fig. 1. Feynman diagrams needed to calculate ?~(/tx). W~ are the gauge fields of s~,. ~x+l contains the vector mesons which obtain masses in the breakdown of Sx+1 to Sx, the heavy ghost fields, and the heavy scalar fields. (The masses of the q,x+l are approximately equal to/~x.)
480
S. Dawson, H. Georgi / Unification
regularization and the MS renormalization scheme [7]. In this approach, at every stage in the symmetry breaking the non-renormalizable heavy gauge couplings automatically decouple in the beta functions from the light gauge couplings. After calculating the effects of the diagrams in fig. 1 on the W~ propagators, we need never consider the heavy vector mesons (and the heavy scalars and ghosts) again. Their entire effect on the theory (to this order) is taken into account by using the results from fig. 1 to redefine the boundary conditions on the coupling constants between region x and region x + 1. It has been noted that this approach must be generalized to include higher-order effects consistently, but to this order it is adequate [8]. To describe the calculation and the results, it is convenient to introduce some notation. We must explicitly parametrize the breakdown of SX+m---~Sx. We do this by rewriting the gauge fields of region x + 1, W~ ÷ l, in a more convenient basis. We rewrite the W~ +l in terms of fields which are mass eigenstates (ignoring the breaking of SX). In addition to the massless (in the x region) gauge fields W~, there are massive fields V~a,.. The meaning of the subscripts is as follows: R = (G) labels the S x representation under which the fields transform, a = 1,2 . . . . is an additional label (if necessary) to distinguish between different sets of fields transforming under the same S x representation. All the vectors with the same R and a have mass MRa (ignoring S ~ breaking), m labels the particular state within a given representation. We will feel free to choose the basis in the representation space at our convenience and will often suppress the m dependence. Wx couples to the V ~ through the appropriate representation matrices ( T,r°),, ,,,. The calculation will be described in appendix A. Here we give the results for the corrected boundary conditions, keeping contributions proportional to logs of heavy particle masses. We find 1
1
- = E Pftl ~+t - k~, g2(/xx) 2 B gt~+'(#x) 2
(6)
where h'~--
11 E I n ( M R a / I ~ : ) T r ( T , . ' Q 24rr 2 R,a
2
(7)
In what follows, we will assume that we have spontaneous symmetry breakdown due to the vacuum expectation value ( V E V ) of a Higgs field. From the structure of the spontaneous symmetry breaking, we then know that the vector mass matrix has the form: ( M 2 v ) .iOj ---g ~ x+ 1g ox+ i t~l~i ~x+ --
x+l = g¢~ gflx + lIIaioj,
1(¢5' Tfl~+l(d'~5) (8)
S. Dawson, H. Georgi / Unification
481
where ( ~ ) is the Higgs VEV and T_,~; x+l are the generators of S T M on the scalar field representation. This matrix has eigenvectors with zero eigenvalue,
(
)
=
ck#j, +TgU ,
(9)
corresponding to the W~'s. The non-zero eigenvalues are the M2a's. We want to concentrate on the dependence of eq. (8) on the coupling constants x4-1 ga . We expect the non-zero eigenvalues of II which contribute to S x÷l breaking all to be of the order of ( ~ ) , but the coupling constants g~÷ l may be very different for different ft. Thus in eq. (7), when we sum over appropriately weighted logs of MRa, we will keep only the g~+l dependence. This is non-trivial only because the matrix II is singular. In appendix B we derive the desired result x x} ~,~'~2 ( ~ ~ . , ,l~X,x+lhx+l ~ ~o~ lng~ +1 - b o~lng= ,
where we have taken /t x of the order of the constants b ~ are defined by
Y ~Ji):i, )
(lO)
S x÷l
breaking VEV, <@). The
--Tfqr OOa,
(11)
where fo~/{' are the structure constants of s x in the adjoint representation. This completes our discussion of boundary conditions. We next show that the fl-function corrections can be cast into a similar form. When there are no fermions, the beta functions for the different coupling constants decouple from each other: X
X
oo~g ~ + o,~g~ + O ( g ~ 7 ) .
(12)
Eq. (12) can be integrated to find g ~ ( / G _ , ) -2 = g X ( # x ) - 2 + 2b~,,ln(l~x/IG_,)
+ (b~,Jb&)ln[(g;(I.G_,)-2+(b~,,/b~))/(g;(p~)-2+(b~a/b~)=))] .~ gX(#~) -2 + 2 b ~ l n ( P x / t G _ , ) + 2( b;,~/b~) ln[ g~(Px) / g ~ ( P x - , ) ] .
(13)
482
S. Dawson, H . Georgi
/ Unification
By using induction and eqs. (6) and (7), it is a simple matter to derive the following form: N
g~(ttx_l) -2 =g0,,,) - : - Y, Y e:gx 0,,) y--x
fl
N
+ X E 2P2~{boY~ln(#y/l~y-l) y~x
fl
+ (bY/3/bffa) In [ g~(t~y ) Ig~(#y_ l)
Now we can use the expression for form:
]}"
(14)
hY~(~y) of eq. (10) to put eq. (14) in a simpler
N
gX(/~x_l)-2=g(/~,v)-2+
~ y=x
×
Y~2Pf~ fl
{bY0ln(/~y/l~y_,) + [
b~ + (bYt~lb~t~)]ln [ g~(I,,ly)lg~(I,~y_l)]}
+ 2b~lng~(l~x_l) -- 2bolng(/~N) •
(15)
In this approximation the effects on the renormalization of the coupling constants arising from the corrections to the boundary conditions of the coupling constants, [eq. (10)] have the same functional dependence on the coupling constants as do the terms coming from the g25 terms in the beta functions. (b 0 and b t are the coefficients of the beta function of the unifying group: fl(g) = bog 3 + bigS.) Finally, we can set x = 1 and/~0 = Mw, ( $1 ~- SU(3) × SU(2) × U(1)). However, #~v is not a free parameter since we have already taken/~N to be approximately equal to the S N+I breaking VEV. To the order in which we are working, M
l~N~--~g(b~N )' where M is the superheavy vector meson mass and g(/~N) is the coupling constant of the unifying group. Using the renormalization group formula of eq. (13),
b' ) I n ~( gg(-/ Z g(M)-2=g(~tN)-2 + 2b°ln-~+ 2 ( -~o ~ N] ) / .
(16)
S. Dawson, H. Georgi / Unification
483
Substituting eq. (16) into eq. (15), we obtain our final result: N
g~(Mw)-Z=g(M)-Z+ Y, y=l
X
Y~2P2~ /3
bot~ln(/~y/#y_,) +
-b~',
ln[ga(izy)/gh(#y_ ,
- 2 X P~;bo~ In g(M) + 2 b~ In g~( M w ) ,
(17)
where #~v = M , / ~ = Mw, and #'i (i :/: N, i :/: 0) is the S i+ l breaking VEV. At this point, we think it is important to discuss the philosophy behind the approximations we have made. We have considered only the corrections to the coupling constants for which a given gauge theory makes definite predictions. Our calculations are independent of the Higgs and fermion structures. The matching functions, A~, which occur in the boundary condition of eq. (6) have, however, a weak dependence on the Higgs and heavy fermion structure of the theory [4]. We neglect these effects not only because they are smaller than the terms we have included, but also because we can only guess their exact form. By the same logic, we have also neglected contributions to the beta functions from light fermions. However, all of the effects which we have ignored tend to decrease sin20w(Mw) for a fixed value of the superheavy mass scale [6]. Thus, our results may be viewed as a reasonable upper bound to the corrections to the renormalization of the coupling constants occurring in this order of perturbation theory. We will now illustrate our techniques in a model in which the strong, weak, and electromagnetic interactions are unified in an O(10) group at very large energies. We will consider three possible breakdowns of O(10): (i)
O(10)--->SU(4) × SU(2) x SU(2)--,SU(3) × SU(2) x U(1) ;
(ii)
O(IO)-->SU(4) x SU(2) x U(1)---~SU(3) x SU(2) × U(1) ;
(iii)
O(10)--->SU(3) × SU(2) × SU(2) x U(1)-->SU(3) x SU(2) × U(1).
(18)
The lowest-order results have been worked out previously by Georgi and Nanopoulos [9]. We calculate sin 2 0 w ( M w ) as a function of M (the superheavy vector mass) using as inputs as(Mw) = 0.15, a ( M w ) -~-~, - l and M w = 8 0 GeV [10]. We then use eq. (17) to write renormalization group equations for each of the coupling constants
484
S. Dawson, H. Georgi / Unification I
I
I
SU(4)x SU(2) x SU(2)
1@9t
/ /
1017 --
Mass (GeV) lOm
10~3
10H [
,
I
.20
.24
,
I
,
.28
sinZOw( Mw) Fig. 2. O(10)--,SU(4) X SU(2) X SU(2)--~SU(3) x SU(2) x U(1). M is the superheavy vector m e s o n mass a n d M l is the mass scale of the intermediate symmetry breaking. The solid lines axe the lowest-order values [eq. (4)] a n d the dotted lines are the values with the second-order corrections of the text [eq. (17)] included.
in the SU(3)× SU(2)× U(1) region. To solve these equations, we substitute the lowest-order results of eq. (4) into the lng~(/~y) 2 terms in eq. (17) to obtain N I
g,,(Mw)
--2
-2+
=g(M)
E E P2~{2b~ln(l~'y/IZ'y-l) fl y ~ l
-
IN N 2~ P~p boplng(M) + 2b~.lng~(Mw).
(19)
#
Results are presented in figs. 2 - 4 for each of the breakdowns of O(113) listed in eq. 0 8 ) . The net effect of the second-order corrections which we have calculated is that for a fixed value of sin2 0 w ( M w ) the superheavy mass scale is smaller than the lowest-order value.
485
S. Dawson, H. Georgi / Unification I
I
I
SU(3) x SU(2) x SU(2) x SU(I) 1019
lOr7 Mass (GeV) 1015
1013
10I1
I
.20
,
I
24
,
L
.28
,
sin28w(Mw) Fig. 3. O(10)-+SU(3)X SU(2) x SU(2) x U(1)---~SU(3)X SU(2) x U(I). I
I
I
SU(4) x SU(2) x SU(1) 1019t
101rt Mass (GeV)
lo,°S NN\ 1 0 I~ -
10' f
Mx I
.20
,
I
.24
,
I
.88
sinZOw(Mw) Fig. 4. O(10)....~SU(4)x SU(2)× U(I)....~SU(3) × SU(2) × U(1).
S. Dawson, H. Georgi / Unification
486
Conclusion In this paper, we have presented a systematic method for calculating corrections to the lowest-order renormalization group equations in a theory with a complicated gauge hierarchy. However, perhaps our most important result was seen in the context of an O(10) model--these corrections are potentially quite significant, leading us to believe that they must be taken into account in any realistic calculation. Appendix A We use 't Hooft's dimensional regularization [8] definition of the coupling constants:
[
1
g,,P'x
'
(A.1)
4 +
Here D is the dimension of space-time, "/E is the Euler constant, and B2 is the coefficient of the beta function for g 2 ( # x ) . In this language, the boundary condition between the S ~ and S T M regions which is given in ref. [2] is,
[]2 ~x.2-n/2
= E P~;'+'
oar-x
[112 o x + 1,02--D/2 ~B V'x
fl
.
(A.2)
Since we break the S T M symmetry with a gauge-fixing functional which preserves the S x symmetry, the gauge-field lagrangian in the S x region is 1
x ~auge ~-
. .la)F,~i~,F . . .
a~(1
(A.3)
ia, ,
ai
where x
X
X
X
X,a
X
X
Fdi,, = 3F,W:i, - O,W;i~, + g , fijk W~j~,Wdk,"
The 12 contain all of the effects of the heavy particles, g,~'+1, on the theory. In order to be able to canonically normalize the gauge-field terms in the lagrangian, we define the renormalized coupling constant g2(#x)R to be, g~(#x)R = ( 1 -- i~(#x))- l/2gX(#~).
(A .4)
We follow Weinberg's notation [4] and further define lim l:(/~x)= [ )~(#x) + ) ~ = ( # x ) ( ~ _ 4 D--~
+ ½"rE- lnX/~
)]gX(#x)2.(A.5)
S. Dawson, H. Georgi / Unification
487
Substituting eqs. (A.1), (A.4) and (A.5) into eq. (A.2), we obtain, after much algebra, the corrected boundary condition
1
-- Z
x 2 ga(P'x)R
px~x+l
,e
1 gj+'(p,x) 2
Xx(/.tx)"
(A.6)
(We now drop the subscript R.) The techniques for evaluating ~ ( / t x ) from the Feynman diagrams of fig. 1 are described in refs. [4, 6]. For a simple non-abelian group G breaking to a product of simple or non-abelian groups, they find the result: ~x(/~x ) = _
l___~l~ Tr(T:.)21n(MR~//~x), 24 ~r2 R,~
(A.7)
where we have translated their result into our notation. When there is a product of groups, II,,s~ +1, which breaks to II~s x, the identical result is obtained for X~,,(/~). This can be proven by writing W~ + i as the approximate sum of the W~ and V~,,, finding the Feynman rules, and evaluating the graphs of fig. 1. This is a tedious process and, except for the notation, our method is identical to that detailed in ref. [61.
Appendix B Did you ever notice that in the SU(2) x U(1) theory the Z mass is a constant times the V E V G f 1/2 times the product of the SU(2) and U(1) couplings divided by e? In this appendix, we generalize this curious fact in order to derive eq. (I0) from eqs. (6) and (7). The reader who cannot abide any more notation is advised to skip from eq. (6) directly to eq. (17) for a more qualitative discussion. We first go to a Cartan basis for the T~, and restrict our attention to generators in the Cartan subgroup, H~i, which can be simultaneously diagonalized. The mass eigenstates correspond to definite HXi eigenvalues called weights ~ i . We can take the m label in V~,~ to be the weight to. Now take T,.r~ in eq. (7) to be a Caftan generator. Then we can write eq. (7) as follows:
~'= a
11 Z(toai)2 y~ ~ 24~r2 ~ R~,~ a
ln(MRa/l~x).
(B.I)
The second sum runs over all representations containing the weight ~. It is possible to show (and we will do it explicitly below) that ~, Y, l n ( M R a / # x ) = ~ n ~ + l ( w ) lng~ +' -- n~(~o) lng~ + - . . RD~0
a
fl
,
(B.2)
S. Dawson, H. Georgi / Unification
488
where the remaining terms are independent of g. The integer n~ + I(to)(nxa(t0)) is the degeneracy of the weight to in the adjoint representation. Note that the generators of S x+' can also be labeled by weights according to [H~X.,qr'x+l] -to
"tpa~ J -
(B.3)
'7"' x + l ai'Bao "
The weights of the adjoint representation s~ are the "roots" (which are not degenerate for to ~ 0). Thus n~(to) = 0 or 1. Inserting eq. (B.2) into eq. (B. 1) we obtain
hx __
11 ~ ( t o a i ) 2 ( ~ n ~ + l ( t o ) l n g ~ + , _ n x ( t o ) l n g X } 24¢r2 p
"
(B .4)
In the second term we can use the relation
Y. (too,):n;(to) = Y, t\ Jijk / w
(B.5)
j,k
in the adjoint representation. In the first term, we use the relation
E (toai)2n~+l(to) -- PXl~X+l E ( fi;:l'l~) 2 , ,,,
(B.6)
jk
Combining eqs. (B.4)-(B.6) with eq. (11) yields eq. (10). All that remains is to derive eq. (B.2). Indeed, it was this amusing relation that first piqued our interest in these corrections. The masses M~a for R D to are the non-zero eigenvalues of the matrix 2
M~a#b
x+l
=
g~
x+lI~x+l/--\
gp
~1~,~,~q g ) ,
T,X+l/,~\~ p,,o,
\v/i •
(B.7)
If nX(to) = 0, the matrix Mda,b 2 is non-singular. Then eq. (B.2) is completely trivial because
M 2 = GHG,
(B.8)
where x+l
x+l
l~otafl b = ( r~a u ( dp ) , Tl~bw ( dp ) ) ,
(B.9)
and G is a diagonal matrix,
G~a#b -- g~x + 16,~t~6,,b,
(B.10)
489
S. Dawson, H. Georgi / Unification
Thus we find E E ln(mRa/#x)----~ RDoo a
1 Trln(M2/i z2) = Trln G +i l Trln(II2/#]).
(B.11)
This is clearly equivalent to eq. (B.2) if <#~)-----#x so that T r l n ( H 2 / # 2) can be neglected. If n~(to) = l, the above derivation breaks down. The sum over MR,, includes only the non-zero eigenvalues of M 2. The sum is not equal to i T r l n ( M 2 / # ] ) which is just as well because the trace doesn't exist. To deal with this case, we must introduce yet more notation. Since nX(to) -- 1, there is an s~ generator with weight ~o. Call it EX~0. It satisfies EXw = E
(B.12)
t"'x'x+ITx+I
fl,a
The C ' s are simply the /',x,x+ -~iOj 1 rewritten in a different notation. Thus for example
E [('x'x÷l~2-ox'x÷l " a , B a ] -- " aB Q
(B.13)
•
The normalized eigenvector of M 2 with zero eigenvalue is e9 a _= g~,~a x ( , x ' ~~+ l l g ~ + 1 •
(B. 14)
The matrices M E, G and H act in a space of dimension d---
with the directions labeled by fl and a. We define a completely antisymmetric tensor in this space in the usual way: e~,al~:2...#: ~ = _+ 1.
(B.15)
Then for any matrix A in the space, we can define the cofactor matrix A ¢ as
"gibi X eYbY2b2...YabaAfl2a2y2b2. • • ABaaaYaba.
(B.16)
S. Dawson,H.Georgi/ Unification
490
NOW consider the cofactor of M 2. Because M 2 has a zero eigenvalue, M 2c must have the form
( M2C)#avbm [ RDtoI-[~ M2c]e#aevb.
(B.17)
O n the other hand, it follows f r o m eq. (B.7) that
Tr(GM2~G)in
If we calculate have the desired result
two ways, using eqs. (B.17), (B.18) a n d (B.14) we
(gX)2[~,~M2cJ=[~(g~+l)n~÷'(~')]2Tr(II¢).
(B.19)
Since Tr YIc-~ ( t ~ 2(d- 5), eq. (B.19) is the exponentiated version of eq. (B.2), for nX(o~)---- 1. We c a n n o t resist closing this appendix with some examples of eq. (B.19). As we implied earlier, one example is familiar to all. I n the SU(2) x U(1) theory g[ is the U(1) coupling is the SU(2) coupling g. gO is e. All the n ( ~ ) ' s are equal to one. Thus eq. (B.19) implies that M z is proportional to which it is. A n even more surprising example is the case S U ( 2 ) x S U ( 2 ) x SU(2) b r o k e n d o w n to the diagonal SU(2). I n this case there are two triplets of massive vector bosons with masses M s a n d M 2. If the gauge couplings of the u n b r o k e n theory are gi, i = 1 to 3, to gauge coupling g of the diagonal SU(2) is
g',g~
gg'/e,
__1= ~ _1 g2
. g2"
(B.20)
Here no matter h o w the s y m m e t r y is broken, eq. (B. 19) implies that M l M 2 is equal to times a function of the VEV's.
g~g2g3/g
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