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Extended-natural gauge models and complementarity
Nuclear Physics B219 (1983) 116-124 © North-Holland Publishing Company
EXTENDED-NATURAL GAUGE MODELS AND COMPLEMENTARITY H. NAKAHARA Institute o[ Physics, University o[ Tokyo, Komaba, Japan Received 14 December 1982 We propose a new condition "extended naturalness" which should be satisfied by any physically sensible gauge theory. In order that a theory is extended-natural, all discrete quantities observed in low energies must be stable against variations of discrete parameters of the theory defined in large mass scales. We find that SU(N) gauge models become extended-natural when we choose appropriate fermion representations. Further, if N is a multiple of 8, the models turn out to be good examples of complementary gauge theories.
1. Introduction "Naturalness" is one of the criteria in adopting gauge models of elementary particles [1]. It has extensively been used to criticize the standard model with Higgs scalars and also to uphold the technicolor models or the supersymmetric models [2]. There are several ways to describe naturalness, a m o n g which is the following: suppose all the particles and interactions (except gravity) observed in low e n e r g i e s - a few G e V - - a r e described by an effective theory (let us call this an E - t h e o r y for short) of a renormalizable gauge theory (an F-theory for short) in the mass scale as large as the Planck mass. W e call the theory natural if all the p a r a m e t e r s of the E - t h e o r y are stable against variations of those of the F-theory. For example, a theory which contains quadratic divergences is unnatural because the mass parameters of the E - t h e o r y are unstable [2]. Usually this stability condition is imposed on continuous parameters. We propose to impose it also on discrete parameters. W e call this property "extended naturalness". As an example, consider the case in which the F-theory is S U ( N ) gauge theory and the E - t h e o r y is SU(M) gauge theory. In order that the theory is extended-natural, M should be stable under the variation of N. In this paper we show how one can use extended naturalness to choose the gauge model. We present an explicit model satisfying this condition. In sect. 2 we list five conditions including the requirement of extended-naturalness which we impose on the gauge theory. We find that all these conditions are satisfied if we choose an S U ( N ) gauge model with fermions in the following representation: eight fundamental representations plus one antisymmetric second rank tensor plus one conjugate representation of a symmetric second rank tensor. In the next two 116
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117
sections we discuss the E-theory of the model in two schemes. In sect. 3 we assume confinement and apply 't Hooft's anomaly equations for massless composite fermions [1]. In sect. 4 we adopt the tumbling scheme of the gauge model [5]. We find that the U(1) symmetry which is left unbroken is the same in both schemes. Further the fermion spectra in the E-theories obtained in both schemes are equivalent to each other when N is a multiple of 8. This is an example of what is called "complementarity" [3]. 2. An extended-natural gauge model
We begin this section with listing the conditions which should be satisfied by any physically sensible gauge theory. The F-theory must be anomaly-free (condition (i)), and asymptotically free (condition (ii)). Its fermion spectrum must be complex (condition (iii)). The theory must be natural (condition (iv)) and extended-natural (condition (v)). Condition (v) is the one which we have introduced in this paper and will turn out to be quite powerful. A gauge theory must not contain elementary scalar fields in order to be natural because of quadratic divergence [2]. (We do not consider supersymmetric theories in this paper.) The F-theory is specified by a gauge group G and a fermion representation R. By condition (iii), G must be either SU(N) or SO(4N + 2) with R being a spinor representation, or an exceptional group. We do not discuss the exceptional groups here because there is no discrete parameter to vary and so there is no room to apply extended naturalness. The dimension of the spinor representation of SO (4N + 2) grows exponentially as N increases causing the theory asymptotically non-free if N is larger than a certain value N*. This fact conflicts with conditions (ii) and (v), because breaking of asymptotic freedom would cause a radical change in the E-theory. For SU(N), the dimension of R grows in some power of N. The F-theory remains asymptotically free for any N if dim R ~
(1)
There are four irreducible representations which satisfy the inequality (1); the fundamental representation ([1]) of N dimensions, the antisymmetric tensor of rank 2 ([2]) of ½ N ( N - 1 ) dimensions, the symmetric tensor of rank 2 ((2)) of ½N(N + 1) dimensions, and the adjoint representation (adj) of N 2 - 1 dimensions. Their anomaly indices are 1 (by definition), N - 4 , N + 4 , and 0 respectively. According to condition (iii) we must not choose the real combinations of irreducible representations such as adj or [1] +[i]. (/~ is for the conjugate representation of R.) One can find three combinations of these irreducible representations that are complex and anomaly-free
(a)
(~.) + (N + 4)1-11,
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H. Nakahara / Extended-natural gauge models
(b)
[2] + (N - 4)[1],
(c)
(2) + [2] + 8[1],
(2)
Repetition of the same representations leads to a global symmetry; SU(N +4) for (a), S U ( N - 4) for (b) and SU(8) for (c). Here, let us make an assumption. Naturalness, which states that we cannot observe the parameters of the F-theory at low energies, suggests that the F-gauge theory is somehow hidden at low energies due to confinement or due to tumbling down to the violation of the original gauge group. For this reason we assume that the symmetry group of the E-theory is not a part of the gauge group of the F-theory but its global symmetry group. A part, at least, of the symmetry we can see in low energies is gaugeized. But we do not argue how the global symmetry of the F-theory, or a part of it, pretends gauge symmetry in the E-theory. Here we give our attention only to symmetry group and fermion spectrum. Of the three possible choices for fermion representation given in eqs. (2), we choose the last one (c) making use of condition (v), extended naturalness. The global symmetry groups SU(N + 4) and S U ( N - 4 ) of (a) and (b) Which would be symmetries of the E-theory depend explicitly on N, the discrete parameter of the F-theory, while SU(8) of (c) does not. In this way we have arrived at the unique choice of the gauge model which satisfies all the conditions given above. In the next two sections we discuss the E-theory of the model.
3. Effective theory of the extended-natural gauge model (I) The extendedly natural gauge model we discovered in the previous section contains the gauge bosons and three multiplets of left-handed spinors in the F-theory: 0~b = _0b~,
x
ai
,
a, b = 1, 2 . . . . . N ,
i=1,2 .... ,8.
(3)
The indiced a and b are for the gauge group SU(N), and i is for the global symmetry group of the F-theory which is SU(8) × U(1) x U(1). The U(1)A symmetry generated by the charge Oh given by 8
QA = Y. i=1
n(x')+(N+2)n(~)+(N-2)n(O),
(4)
is broken explicitly by the SU(N) anomaly. (n is the number operator of each
H. Nakahara / Extended-natural gauge models
119
multiplet.) We choose the basis Q1, Q~ of anomaly-free U(1)x U(1) symmetry; O l = ~, n ( x i ) - 2n(~) + 2 n ( ~ ) ,
i
Q2 = N E n (X i) + (2N - 8)n (~) - (2N + 8)n (~). i
(5)
First we assume that the SU(N) gauge group is unbroken and confinement occurs in the E-theory. The fermions r/appearing in the E-theory must be SU(N) singlet. We can construct the r/'s as follows: [4] n lira: X aieac [(~)m-1]~ X bj,
(6)
where rn indicates that the fermion bilinear in the adjoint representation is inserted m - 1 times. In the S U ( 8 ) x U ( l h x U(1)2 global symmetry n,, is ((2) or [2], 0, 4 N - 1 6 m +8). Unless there are some selection rules, it is very plausible that it becomes harder for the composite fermion r/m to be realized in the E-theory as m gets larger because of its exoticity. So we assume that K singlets r / l , . . . , ~K are realized in order of simplicity. 't Hooft's condition consists of anomaly equations of three types: (i) [SU(8)] 3, (ii) [SU(8)]2×U(1), and (iii) [U(1)] 3. First we discuss the case n°"s are antisymmetric in i and/', i.e. [2] representations of SU(8). In the F-theory, the anomaly of type (i) is N (fermion X i in [1] of SU(8) appears N times; each carries the anomaly index 1), while the anomaly in the E-theory is 4K (fermion 17ij in [2] appears K times; each carries 4). The anomaly equation is saturated if 4K = N .
(7)
The anomaly of type (ii) for U(1) whose charge is S given by S = AOl +BQ2,
is ½(A + B N ) N in the F-theory and ~2 B N 2 in the E-theory considering eq. (7). The anomaly equation becomes then A = 2NB. (8) The equation for the anomaly of type (iii) for U(1)s appears in rather complicated form, still the anomalies in the F- and E-theories turn out to match due to eqs. (7) and (8). All the anomaly equations are saturated provided that eqs. (7) and (8) are satisfied. We thus conclude that the global symmetry SU(8) x U(1)s given by S = B ( 2 N Q 1 + Q2),
(9)
may remain unbroken in the E-theory to be the guardian symmetry of the massless fermions r/provided eq. (7) is satisfied. (Again we have no idea about the way this symmetry is realized as the gauge symmetry in the E-theory.) Quantum numbers of ~,, is given in table 1.
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H. Nakahara / Extended-natural gauge models TABLE 1 Quantum numbers of the fermions in the F-theory and the gauge singlet composite fermions
~: x *lm
SU(N)
SU(8)
O1
[21 (2) [1] 1
1 1 [1] [2]
2 -2 1 0
O5
S (B = 1)
-2N-8 2N-8 N 4(N+2-4m)
2N-8 -2N-8 3N 4(N+2-4m)
In the case of ,/,1= ~71,, one cannot saturate the anomaly equations with U(1) symmetries. 4. Eflective theory of the extended-natural gauge model (II) Next we apply the tumbling procedure [5]. There are two attractive channels in which the difference of the Casimir operators is of order N, [2Ix ( 2 ) + A d j . ,
4C= -1(N2-4),
(2) x [ 1 ] ~ [ i ] ,
AC = - l ( N 2 + N - 2 ) .
(10)
The latter is MAC. We assume the condensate
¢, ~ =
(11)
<¢~bx °') # o ,
forms. We assume further as in ref. E3] that
e~ = 00oL.
(12)
The gauge group SU(N) tumbles down to S U ( N - 8 ) . No generators of the original global symmetry are conserved, but another global symmetry appears. The broken subgroup SU(8) of SU(N) plus the original global SU(8) give rise to a new conserved global SU(8) in just the same way as the chiral symmetry SU(2)L X SU(2)a of two-flavor QCD is broken to SU(2)L+a by the diagonal quark mass. We pick up two conserved combinations Y and Z from three broken U(1) generators, Or, 02 and E given by E = ((N-8)0x 1(8 )
0 -8xl(N-8)
)
(13) '
where 1 (k) is a unit matrix of k x k. Y = (N-8)O1-E,
Z = (N-8)O2+(3N-8)E.
(14)
Massive Dirac spinors are formed by combining ½(Xij +X jl) and X '~i with ~:~jand se~i (i, / = 1 . . . . . 8, and a = 9 . . . . . N). Other fermions remain massless. Their quantum numbers are listed in table 2.
121
H. Nakahara / Extended-natural gauge models TABLE 2 Quantum numbers of the fermions in the tumbled theory. Old ~(x ~i-x ~)
New
SU(N- 8)
SU(8)
~"
1 1 [1]
[2] (2) [1] (2) [i] 1 [2] [1] 1
½(x '~ + x j')
x ~,~
1
0'1
[i] (2.) 1 [1] [2]
~e"0 sr* X'"~ t#'"b
Y
Z
O~.
S (B = 1)
0 0 N
4(N - 8)(N - 2) 4(N- 8)(N- 2) N2-32N+64
0
4(N - 2)
0
-4(N-8)(N-2)
-2 0 1 2
-2(N - 8) - 8 4(N - 6) 3(N- 8) 2(N-8)-8
- N - N 2+32N-64 - 2 N 2(N2+ 12N-32) 0 4(N- 8)(N- 6) N N2-48N+ 192 2N -2(N 2+ 20N- 96)
In the first column are given the old names of the fermions in the F-theory. In the second are given the new names in the tumbled theory. The fermion spectrum of the tumbled theory is very similar to that of the original F-theory. G a u g e non-singlet fermions are expressed in exactly the same Young tableaux in both theories. In fact 0 ~i, tk ~B and ~e~ are in [1], [2], and (2) representations of S U ( N - 8) respectively, which correspond to the fermions of the F-theory, X a~, 0 ab and seab. To emphasize the similarity let us denote these fermions of the tumbled theory by the same symbols (but with primes) as those of the fermions of the F-theory; X '~i, ~b'"b, and se'b. The tumbled theory looks as if it was the original F-theory with the S U ( N - 8 ) gauge group, so that it would tumble down again in the same way. There are, however, some other differences between these two theories. In the tumbled theory appear the massless gauge singlet fermions ½(X'j - X j'~) and 0 'j, to which we give new names ~,ij and ~'*~( The appearance of massless singlets is an inevitable consequence of a gauge theory with anomalies of global symmetry, and has no influence upon tumbling. Also two conserved U(1) charges Y and Z in the tumbled theory are different from the U(1) charges Q~ and Q~ which are defined by eqs. (5) but with primes for fermions and with N - 8 in place of N. This discrepancy causes some troubles when we consider the U(1) charges of gauge singlets, Fortunately we can determine the basis of U(1) charges which takes the same form in the tumbled theory as in the F-theory when it is modified by combining with the charge E as in eqs. (14). Let us search for the U(1) charge P whose modification P * coincides with P ' which is defined by the same equation as P but with primes and N - 8 . In the following we show that there are two independent solutions for P which form the basis of anomaly-free U(1) charges. Let f be a fermion of the F-theory and its U(1) charge be P(f). In the tumbled theory the modified charge P * can be constructed as a linear combination of P and E. Particularly, if we express P on the basis given in eqs. (5) as
P =A(N)Q1 +B(N)Q2,
(15)
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H. Nakahara / Extended-natural gauge models
we can write P* as P* = C ( N ) [ A ( N ) Y
+B(N)Z],
(16)
where C ( N ) is a normalization constant. On the other hand P' is defined by eq. (15) with primes and N - 8, P' =A(N-8)Q~
+B(N-8)Q~.
(17)
Now we apply eqs. (16) and (17) to fermions X, ~ and 0, and set P * and P ' equal to each other. P*O(') = C [ A Y (X') + B Z (X')] = C A N + C B (N 2 - 4 8 N + 192) = P'(X') = A ' O l (X') +B'Q'2 0(') = A ' + ( N - 8 ) B ' , P*(~')
= P'(~') P*(~b')
= P'(O)
= -2CAN
(18)
+ 2 C B (N 2 + 12N - 32)
= - 2 A ' + 2(N - 12)B',
(19)
= 2 C A N - 2 C B ( N 2 + 2 0 N - 96)
= 2 A ' - 2 ( N - 4)B'.
(20)
H e r e we used A, B and C for A ( N ) , B ( N ) and C ( N ) and A ' and B ' for A ( N - 8 ) and B (N - 8). From eqs. (19) and (20) we get (N- 8)CB = B'.
(21)
T o satisfy this equation with B in a finite form, either B = 0,
(22)
or
( N - 8 ) C = 1,
B --B'
(23)
must hold. The U(1) charge P obtained by using eq. (22) coincides with Q1 when we set C and A equal to unity. On the other hand substituting eq. (23) into (18), we get A = 2NB,
(24)
P = B (2NO1 + Q 2 ) .
(25)
which leads to P given by
Here B is independent of N. The U(1) charge P obtained in eq. (25) is nothing other than S that was defined by eq. (9) as the unbroken U(1) charge allowed by 't Hooft's condition. These two solutions Q1 and S form the basis of anomaly-free U(1) charges. Quantum numbers of massless fermions concerned with the new basis are also listed in table 2.
H. Nakahara / Extended-natural gauge models
123
With this basis we can pursue a series of successive tumblings occurring in our model. Finally the theory will tumble down completely. All the massless fermions left will be gauge singlets. This final stage should be our E-theory. We have several varieties of the E-theory according to the value of N. First we discuss the case N = 8L which is most interesting in relation to complementary gauge theories. The F-theory tumbles down L - 1 times in the way we described above, but the last stage of the series is a little different from the preceding ones. The condensate $ forms according to eq. (11) as usual, breaking the gauge symmetry SU(8) completely and breaking also U(1)I since Ol(~) = - 1 ,
(26)
S ( $ ) = S ( ~ ) + S ( x ) = 2 4 - 2 4 = 0.
(27)
but not U(1)s because
The symmetry U(1)I disappears here, because there is no U(1) generator E to combine with 01 forming a conserved generator. The remaining symmetry is global SU(8)xU(1)s and the remaining massless fermions are ~" and ~'* which appear every time a tumbling occurs. Their quantum numbers in SU(8)xU(1)s are the following: ~1' = ([2], 8 ( 4 / - 1)), ~'*~J= ([2], 8 ( 4 l - 3)),
l = 1,..., L,
(28)
where l indicates that the fermions appear when the lth tumbling occurs. This should be compared with the result obtained in sect. 3. The fermions in the E-theory which saturate 't Hooft's equations are 77~'s given by eq. (6). (See also table 1.) It turns out that the r/'s and ~"s are related to each other as = ?7 2 L - 2 l +
1
77 2 L - - 2 / + 2
•
(29)
This is an example of the complementary gauge theories introduced by Dimopoulos et al. [3]. The spectrum of the E-theory is the same whether confinement or tumbling is assumed. This correspondence fails in other cases. If N = 8L + 4, the symmetry remaining at the end of the series of tumblings is SU(4)x SU(4)x U(1)r global symmetry where T is a linear combination of S and a U(1) generator F of the broken global symmetry SU(8) given by F = (1(04)
_ 10(4)) .
(30)
But 't Hooft's condition does not single out a U(1) symmetry corresponding to T. In the cases other than the above two, we can find the symmetry group of the E-theory in the same way. If N = 8L + J where J equals to 2, 3, 5, 6, or 7, the
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H. Nakahara / Extended-natural gauge models
symmetry of the E-theory is SU(8- J)X SU(J)x U(1)t x U(1)s and when J = 1 it is SU(8) x U(1)~ x U(1)s. These cases conflict with 't Hooft's conditions especially with eq. (7). 5. Conclusion Our model was determined by assuming that the symmetry group of the E-theory is just the global symmetry group of the F-theory. The analysis given above shows however that the global symmetry group cannot survive as a whole but only in part at low energies. Further we have shown that the symmetry group of the E-theory would change according to the size of the gauge group. We can still consider the SU(8L +d) theory with J staying constant as an extended natural theory, since the symmetry group of the E-theory does not change when we vary the discrete parameter L. When the theory is complementary we have some information about the fermion spectrum. In SU(8L) theory, the fermions in the E-theory are ~'1. . . . ~L and (* . . . . . ~* each of which is in 28 dimension. The discrete parameter L appears in the number of massless fermions. We must give masses to fermions. One can do this for instance, by introducing a small dimensionless coupling constant g. Then the mass m k given to (k or (* must have the form mk =
(a certain function of g) x Ak
where A k is the mass scale at which the kth tumbling occurs. At low energies only a few lightest ~'s are observable. The fermion spectrum thus may be insensitive to the parameter L. The author would like to thank Professor Y. Fujii and Dr. S. Wada for helpful suggestions. References [1] G. 't Hooft, in Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum Press, NY, 1980) [2] L. Susskind Phys. Rev. D20 (1979) 2619 M. Veltman Acta Phys. Pol. B12 (1981) 437 [3] S. Dimopoulos, S. Raby and L. Susskind, Nucl. Phys. B173 (1980) 107 [4] J. Preskill DPF 1981; 572 [5l S. Raby, S. Dimopoulos and L. Susskind, Nucl. Phys. B169 (1980) 373
EXTENDED-NATURAL GAUGE MODELS AND COMPLEMENTARITY H. NAKAHARA Institute o[ Physics, University o[ Tokyo, Komaba, Japan Received 14 December 1982 We propose a new condition "extended naturalness" which should be satisfied by any physically sensible gauge theory. In order that a theory is extended-natural, all discrete quantities observed in low energies must be stable against variations of discrete parameters of the theory defined in large mass scales. We find that SU(N) gauge models become extended-natural when we choose appropriate fermion representations. Further, if N is a multiple of 8, the models turn out to be good examples of complementary gauge theories.
1. Introduction "Naturalness" is one of the criteria in adopting gauge models of elementary particles [1]. It has extensively been used to criticize the standard model with Higgs scalars and also to uphold the technicolor models or the supersymmetric models [2]. There are several ways to describe naturalness, a m o n g which is the following: suppose all the particles and interactions (except gravity) observed in low e n e r g i e s - a few G e V - - a r e described by an effective theory (let us call this an E - t h e o r y for short) of a renormalizable gauge theory (an F-theory for short) in the mass scale as large as the Planck mass. W e call the theory natural if all the p a r a m e t e r s of the E - t h e o r y are stable against variations of those of the F-theory. For example, a theory which contains quadratic divergences is unnatural because the mass parameters of the E - t h e o r y are unstable [2]. Usually this stability condition is imposed on continuous parameters. We propose to impose it also on discrete parameters. W e call this property "extended naturalness". As an example, consider the case in which the F-theory is S U ( N ) gauge theory and the E - t h e o r y is SU(M) gauge theory. In order that the theory is extended-natural, M should be stable under the variation of N. In this paper we show how one can use extended naturalness to choose the gauge model. We present an explicit model satisfying this condition. In sect. 2 we list five conditions including the requirement of extended-naturalness which we impose on the gauge theory. We find that all these conditions are satisfied if we choose an S U ( N ) gauge model with fermions in the following representation: eight fundamental representations plus one antisymmetric second rank tensor plus one conjugate representation of a symmetric second rank tensor. In the next two 116
H. Nakahara / Extended-natural gauge models
117
sections we discuss the E-theory of the model in two schemes. In sect. 3 we assume confinement and apply 't Hooft's anomaly equations for massless composite fermions [1]. In sect. 4 we adopt the tumbling scheme of the gauge model [5]. We find that the U(1) symmetry which is left unbroken is the same in both schemes. Further the fermion spectra in the E-theories obtained in both schemes are equivalent to each other when N is a multiple of 8. This is an example of what is called "complementarity" [3]. 2. An extended-natural gauge model
We begin this section with listing the conditions which should be satisfied by any physically sensible gauge theory. The F-theory must be anomaly-free (condition (i)), and asymptotically free (condition (ii)). Its fermion spectrum must be complex (condition (iii)). The theory must be natural (condition (iv)) and extended-natural (condition (v)). Condition (v) is the one which we have introduced in this paper and will turn out to be quite powerful. A gauge theory must not contain elementary scalar fields in order to be natural because of quadratic divergence [2]. (We do not consider supersymmetric theories in this paper.) The F-theory is specified by a gauge group G and a fermion representation R. By condition (iii), G must be either SU(N) or SO(4N + 2) with R being a spinor representation, or an exceptional group. We do not discuss the exceptional groups here because there is no discrete parameter to vary and so there is no room to apply extended naturalness. The dimension of the spinor representation of SO (4N + 2) grows exponentially as N increases causing the theory asymptotically non-free if N is larger than a certain value N*. This fact conflicts with conditions (ii) and (v), because breaking of asymptotic freedom would cause a radical change in the E-theory. For SU(N), the dimension of R grows in some power of N. The F-theory remains asymptotically free for any N if dim R ~
(1)
There are four irreducible representations which satisfy the inequality (1); the fundamental representation ([1]) of N dimensions, the antisymmetric tensor of rank 2 ([2]) of ½ N ( N - 1 ) dimensions, the symmetric tensor of rank 2 ((2)) of ½N(N + 1) dimensions, and the adjoint representation (adj) of N 2 - 1 dimensions. Their anomaly indices are 1 (by definition), N - 4 , N + 4 , and 0 respectively. According to condition (iii) we must not choose the real combinations of irreducible representations such as adj or [1] +[i]. (/~ is for the conjugate representation of R.) One can find three combinations of these irreducible representations that are complex and anomaly-free
(a)
(~.) + (N + 4)1-11,
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H. Nakahara / Extended-natural gauge models
(b)
[2] + (N - 4)[1],
(c)
(2) + [2] + 8[1],
(2)
Repetition of the same representations leads to a global symmetry; SU(N +4) for (a), S U ( N - 4) for (b) and SU(8) for (c). Here, let us make an assumption. Naturalness, which states that we cannot observe the parameters of the F-theory at low energies, suggests that the F-gauge theory is somehow hidden at low energies due to confinement or due to tumbling down to the violation of the original gauge group. For this reason we assume that the symmetry group of the E-theory is not a part of the gauge group of the F-theory but its global symmetry group. A part, at least, of the symmetry we can see in low energies is gaugeized. But we do not argue how the global symmetry of the F-theory, or a part of it, pretends gauge symmetry in the E-theory. Here we give our attention only to symmetry group and fermion spectrum. Of the three possible choices for fermion representation given in eqs. (2), we choose the last one (c) making use of condition (v), extended naturalness. The global symmetry groups SU(N + 4) and S U ( N - 4 ) of (a) and (b) Which would be symmetries of the E-theory depend explicitly on N, the discrete parameter of the F-theory, while SU(8) of (c) does not. In this way we have arrived at the unique choice of the gauge model which satisfies all the conditions given above. In the next two sections we discuss the E-theory of the model.
3. Effective theory of the extended-natural gauge model (I) The extendedly natural gauge model we discovered in the previous section contains the gauge bosons and three multiplets of left-handed spinors in the F-theory: 0~b = _0b~,
x
ai
,
a, b = 1, 2 . . . . . N ,
i=1,2 .... ,8.
(3)
The indiced a and b are for the gauge group SU(N), and i is for the global symmetry group of the F-theory which is SU(8) × U(1) x U(1). The U(1)A symmetry generated by the charge Oh given by 8
QA = Y. i=1
n(x')+(N+2)n(~)+(N-2)n(O),
(4)
is broken explicitly by the SU(N) anomaly. (n is the number operator of each
H. Nakahara / Extended-natural gauge models
119
multiplet.) We choose the basis Q1, Q~ of anomaly-free U(1)x U(1) symmetry; O l = ~, n ( x i ) - 2n(~) + 2 n ( ~ ) ,
i
Q2 = N E n (X i) + (2N - 8)n (~) - (2N + 8)n (~). i
(5)
First we assume that the SU(N) gauge group is unbroken and confinement occurs in the E-theory. The fermions r/appearing in the E-theory must be SU(N) singlet. We can construct the r/'s as follows: [4] n lira: X aieac [(~)m-1]~ X bj,
(6)
where rn indicates that the fermion bilinear in the adjoint representation is inserted m - 1 times. In the S U ( 8 ) x U ( l h x U(1)2 global symmetry n,, is ((2) or [2], 0, 4 N - 1 6 m +8). Unless there are some selection rules, it is very plausible that it becomes harder for the composite fermion r/m to be realized in the E-theory as m gets larger because of its exoticity. So we assume that K singlets r / l , . . . , ~K are realized in order of simplicity. 't Hooft's condition consists of anomaly equations of three types: (i) [SU(8)] 3, (ii) [SU(8)]2×U(1), and (iii) [U(1)] 3. First we discuss the case n°"s are antisymmetric in i and/', i.e. [2] representations of SU(8). In the F-theory, the anomaly of type (i) is N (fermion X i in [1] of SU(8) appears N times; each carries the anomaly index 1), while the anomaly in the E-theory is 4K (fermion 17ij in [2] appears K times; each carries 4). The anomaly equation is saturated if 4K = N .
(7)
The anomaly of type (ii) for U(1) whose charge is S given by S = AOl +BQ2,
is ½(A + B N ) N in the F-theory and ~2 B N 2 in the E-theory considering eq. (7). The anomaly equation becomes then A = 2NB. (8) The equation for the anomaly of type (iii) for U(1)s appears in rather complicated form, still the anomalies in the F- and E-theories turn out to match due to eqs. (7) and (8). All the anomaly equations are saturated provided that eqs. (7) and (8) are satisfied. We thus conclude that the global symmetry SU(8) x U(1)s given by S = B ( 2 N Q 1 + Q2),
(9)
may remain unbroken in the E-theory to be the guardian symmetry of the massless fermions r/provided eq. (7) is satisfied. (Again we have no idea about the way this symmetry is realized as the gauge symmetry in the E-theory.) Quantum numbers of ~,, is given in table 1.
120
H. Nakahara / Extended-natural gauge models TABLE 1 Quantum numbers of the fermions in the F-theory and the gauge singlet composite fermions
~: x *lm
SU(N)
SU(8)
O1
[21 (2) [1] 1
1 1 [1] [2]
2 -2 1 0
O5
S (B = 1)
-2N-8 2N-8 N 4(N+2-4m)
2N-8 -2N-8 3N 4(N+2-4m)
In the case of ,/,1= ~71,, one cannot saturate the anomaly equations with U(1) symmetries. 4. Eflective theory of the extended-natural gauge model (II) Next we apply the tumbling procedure [5]. There are two attractive channels in which the difference of the Casimir operators is of order N, [2Ix ( 2 ) + A d j . ,
4C= -1(N2-4),
(2) x [ 1 ] ~ [ i ] ,
AC = - l ( N 2 + N - 2 ) .
(10)
The latter is MAC. We assume the condensate
¢, ~ =
(11)
<¢~bx °') # o ,
forms. We assume further as in ref. E3] that
e~ = 00oL.
(12)
The gauge group SU(N) tumbles down to S U ( N - 8 ) . No generators of the original global symmetry are conserved, but another global symmetry appears. The broken subgroup SU(8) of SU(N) plus the original global SU(8) give rise to a new conserved global SU(8) in just the same way as the chiral symmetry SU(2)L X SU(2)a of two-flavor QCD is broken to SU(2)L+a by the diagonal quark mass. We pick up two conserved combinations Y and Z from three broken U(1) generators, Or, 02 and E given by E = ((N-8)0x 1(8 )
0 -8xl(N-8)
)
(13) '
where 1 (k) is a unit matrix of k x k. Y = (N-8)O1-E,
Z = (N-8)O2+(3N-8)E.
(14)
Massive Dirac spinors are formed by combining ½(Xij +X jl) and X '~i with ~:~jand se~i (i, / = 1 . . . . . 8, and a = 9 . . . . . N). Other fermions remain massless. Their quantum numbers are listed in table 2.
121
H. Nakahara / Extended-natural gauge models TABLE 2 Quantum numbers of the fermions in the tumbled theory. Old ~(x ~i-x ~)
New
SU(N- 8)
SU(8)
~"
1 1 [1]
[2] (2) [1] (2) [i] 1 [2] [1] 1
½(x '~ + x j')
x ~,~
1
0'1
[i] (2.) 1 [1] [2]
~e"0 sr* X'"~ t#'"b
Y
Z
O~.
S (B = 1)
0 0 N
4(N - 8)(N - 2) 4(N- 8)(N- 2) N2-32N+64
0
4(N - 2)
0
-4(N-8)(N-2)
-2 0 1 2
-2(N - 8) - 8 4(N - 6) 3(N- 8) 2(N-8)-8
- N - N 2+32N-64 - 2 N 2(N2+ 12N-32) 0 4(N- 8)(N- 6) N N2-48N+ 192 2N -2(N 2+ 20N- 96)
In the first column are given the old names of the fermions in the F-theory. In the second are given the new names in the tumbled theory. The fermion spectrum of the tumbled theory is very similar to that of the original F-theory. G a u g e non-singlet fermions are expressed in exactly the same Young tableaux in both theories. In fact 0 ~i, tk ~B and ~e~ are in [1], [2], and (2) representations of S U ( N - 8) respectively, which correspond to the fermions of the F-theory, X a~, 0 ab and seab. To emphasize the similarity let us denote these fermions of the tumbled theory by the same symbols (but with primes) as those of the fermions of the F-theory; X '~i, ~b'"b, and se'b. The tumbled theory looks as if it was the original F-theory with the S U ( N - 8 ) gauge group, so that it would tumble down again in the same way. There are, however, some other differences between these two theories. In the tumbled theory appear the massless gauge singlet fermions ½(X'j - X j'~) and 0 'j, to which we give new names ~,ij and ~'*~( The appearance of massless singlets is an inevitable consequence of a gauge theory with anomalies of global symmetry, and has no influence upon tumbling. Also two conserved U(1) charges Y and Z in the tumbled theory are different from the U(1) charges Q~ and Q~ which are defined by eqs. (5) but with primes for fermions and with N - 8 in place of N. This discrepancy causes some troubles when we consider the U(1) charges of gauge singlets, Fortunately we can determine the basis of U(1) charges which takes the same form in the tumbled theory as in the F-theory when it is modified by combining with the charge E as in eqs. (14). Let us search for the U(1) charge P whose modification P * coincides with P ' which is defined by the same equation as P but with primes and N - 8 . In the following we show that there are two independent solutions for P which form the basis of anomaly-free U(1) charges. Let f be a fermion of the F-theory and its U(1) charge be P(f). In the tumbled theory the modified charge P * can be constructed as a linear combination of P and E. Particularly, if we express P on the basis given in eqs. (5) as
P =A(N)Q1 +B(N)Q2,
(15)
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H. Nakahara / Extended-natural gauge models
we can write P* as P* = C ( N ) [ A ( N ) Y
+B(N)Z],
(16)
where C ( N ) is a normalization constant. On the other hand P' is defined by eq. (15) with primes and N - 8, P' =A(N-8)Q~
+B(N-8)Q~.
(17)
Now we apply eqs. (16) and (17) to fermions X, ~ and 0, and set P * and P ' equal to each other. P*O(') = C [ A Y (X') + B Z (X')] = C A N + C B (N 2 - 4 8 N + 192) = P'(X') = A ' O l (X') +B'Q'2 0(') = A ' + ( N - 8 ) B ' , P*(~')
= P'(~') P*(~b')
= P'(O)
= -2CAN
(18)
+ 2 C B (N 2 + 12N - 32)
= - 2 A ' + 2(N - 12)B',
(19)
= 2 C A N - 2 C B ( N 2 + 2 0 N - 96)
= 2 A ' - 2 ( N - 4)B'.
(20)
H e r e we used A, B and C for A ( N ) , B ( N ) and C ( N ) and A ' and B ' for A ( N - 8 ) and B (N - 8). From eqs. (19) and (20) we get (N- 8)CB = B'.
(21)
T o satisfy this equation with B in a finite form, either B = 0,
(22)
or
( N - 8 ) C = 1,
B --B'
(23)
must hold. The U(1) charge P obtained by using eq. (22) coincides with Q1 when we set C and A equal to unity. On the other hand substituting eq. (23) into (18), we get A = 2NB,
(24)
P = B (2NO1 + Q 2 ) .
(25)
which leads to P given by
Here B is independent of N. The U(1) charge P obtained in eq. (25) is nothing other than S that was defined by eq. (9) as the unbroken U(1) charge allowed by 't Hooft's condition. These two solutions Q1 and S form the basis of anomaly-free U(1) charges. Quantum numbers of massless fermions concerned with the new basis are also listed in table 2.
H. Nakahara / Extended-natural gauge models
123
With this basis we can pursue a series of successive tumblings occurring in our model. Finally the theory will tumble down completely. All the massless fermions left will be gauge singlets. This final stage should be our E-theory. We have several varieties of the E-theory according to the value of N. First we discuss the case N = 8L which is most interesting in relation to complementary gauge theories. The F-theory tumbles down L - 1 times in the way we described above, but the last stage of the series is a little different from the preceding ones. The condensate $ forms according to eq. (11) as usual, breaking the gauge symmetry SU(8) completely and breaking also U(1)I since Ol(~) = - 1 ,
(26)
S ( $ ) = S ( ~ ) + S ( x ) = 2 4 - 2 4 = 0.
(27)
but not U(1)s because
The symmetry U(1)I disappears here, because there is no U(1) generator E to combine with 01 forming a conserved generator. The remaining symmetry is global SU(8)xU(1)s and the remaining massless fermions are ~" and ~'* which appear every time a tumbling occurs. Their quantum numbers in SU(8)xU(1)s are the following: ~1' = ([2], 8 ( 4 / - 1)), ~'*~J= ([2], 8 ( 4 l - 3)),
l = 1,..., L,
(28)
where l indicates that the fermions appear when the lth tumbling occurs. This should be compared with the result obtained in sect. 3. The fermions in the E-theory which saturate 't Hooft's equations are 77~'s given by eq. (6). (See also table 1.) It turns out that the r/'s and ~"s are related to each other as = ?7 2 L - 2 l +
1
77 2 L - - 2 / + 2
•
(29)
This is an example of the complementary gauge theories introduced by Dimopoulos et al. [3]. The spectrum of the E-theory is the same whether confinement or tumbling is assumed. This correspondence fails in other cases. If N = 8L + 4, the symmetry remaining at the end of the series of tumblings is SU(4)x SU(4)x U(1)r global symmetry where T is a linear combination of S and a U(1) generator F of the broken global symmetry SU(8) given by F = (1(04)
_ 10(4)) .
(30)
But 't Hooft's condition does not single out a U(1) symmetry corresponding to T. In the cases other than the above two, we can find the symmetry group of the E-theory in the same way. If N = 8L + J where J equals to 2, 3, 5, 6, or 7, the
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H. Nakahara / Extended-natural gauge models
symmetry of the E-theory is SU(8- J)X SU(J)x U(1)t x U(1)s and when J = 1 it is SU(8) x U(1)~ x U(1)s. These cases conflict with 't Hooft's conditions especially with eq. (7). 5. Conclusion Our model was determined by assuming that the symmetry group of the E-theory is just the global symmetry group of the F-theory. The analysis given above shows however that the global symmetry group cannot survive as a whole but only in part at low energies. Further we have shown that the symmetry group of the E-theory would change according to the size of the gauge group. We can still consider the SU(8L +d) theory with J staying constant as an extended natural theory, since the symmetry group of the E-theory does not change when we vary the discrete parameter L. When the theory is complementary we have some information about the fermion spectrum. In SU(8L) theory, the fermions in the E-theory are ~'1. . . . ~L and (* . . . . . ~* each of which is in 28 dimension. The discrete parameter L appears in the number of massless fermions. We must give masses to fermions. One can do this for instance, by introducing a small dimensionless coupling constant g. Then the mass m k given to (k or (* must have the form mk =
(a certain function of g) x Ak
where A k is the mass scale at which the kth tumbling occurs. At low energies only a few lightest ~'s are observable. The fermion spectrum thus may be insensitive to the parameter L. The author would like to thank Professor Y. Fujii and Dr. S. Wada for helpful suggestions. References [1] G. 't Hooft, in Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum Press, NY, 1980) [2] L. Susskind Phys. Rev. D20 (1979) 2619 M. Veltman Acta Phys. Pol. B12 (1981) 437 [3] S. Dimopoulos, S. Raby and L. Susskind, Nucl. Phys. B173 (1980) 107 [4] J. Preskill DPF 1981; 572 [5l S. Raby, S. Dimopoulos and L. Susskind, Nucl. Phys. B169 (1980) 373