Constraints on moose-model building

Volume 167B, number 2

PHYSICS LETTERS

6 February 1986

CONSTRAINTS ON MOOSE-MODEL BUILDING Ann E. N E L S O N 1 Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 18 November 1985

A problem is pointed out in a class of models of composite fermions and a composite Higgs. These theories seem to automatically contain a Peccei-Quinn symmetry; a U(1) symmetrywhich is exact up to a color anomaly. Such a symmetryis acceptable only if it results in an invisible axion. In one attempt at a realistic model SU(2)w must be embedded in a larger group if the axion is to be invisible.

The two major theoretical and experimental challenges still confronting the standard model are to understand the SU(2)w X U(1)y breaking mechanism and to understand flavor. While the breaking of SU(2)w X U(1)y can be accomplished in several ways, a fundamental I-Iiggsdoublet seems to be the easiest way of producing the fermion masses. Unfortunately, in fundamental Higgs models the ferrnion mass matrices are completely arbitrary, with such obvious structure as the mass hierarchy and the smallness of the Kobayashi-Maskawa mixing angles unexplained. The most popular alternative to a fundamental I-Iiggs is a new force, called technicolor [1], which gets strong at the weak scale and dynamically breaks the weak interactions. However, technicolor will not give mass to the light fermions without additional interactions at some higher scale called extended technicolor [2]. In order to produce the observed pattern of quark and lepton masses the extended technicolor interactions must be extremely complicated. Placing a form of GIM [3] mechanism in extended technicolor, which is necessary to suppress flavor changing neutral currents, renders the models even more elaborate [4], and no fully realistic example exists. Like technicolor, the ultracolor force of composite Higgs models [5] breaks SU(2) X U(1) dynamically, but ultracolor is assumed to become strong well above the weak scale. The I-Iiggsmay be thought of as a composite SU(2)w X U(1)y doublet pseudoi Junior Fellow, Harvard Societyof Fellows. 200

Goldstone boson, with a VEV which is much smaller than its binding scale. In analogy with extended technicolor, we could construct extended ultracolor, but flavor-changing neutral currents could be suppressed by simply making the ultracolor scale sufficiently large. There is no reason, however, to expect these models to be much more appealing than extended technicolor. An attractive way of coupling the composite Higgs is to make the light fermions composite as well, as suggested by Georgi [6]. He has recently discovered a class of simple examples with reasonable ultracolor dynamics [7]. These models, affectionately called mooses, contain several strong QCD-like groups, which are assumed to all form QCD-like condensates (the "moose calculus") at the ultracolor scale. The global symmetries of the moose, G, are broken to some group G' by weakly gauging some subgroup of G. Usually one is able to deduce unambiguously the approximate symmetries, which we call H, left unbroken by the ultracolor dynamics. The overlap of G' and H, known as the weak alignment, generally can be computed up to the signs and magnitudes of some unknown strong parameters [ 7 - 9 ] . Mooses can be found with a composite Higgs, a flavor hierarchy, and many other realistic features. The purpose of this paper, however, is to point out that there is a Peccei-Quinn symmetry [10] in all known mooses and come up with a set of constraints. The basic rules for moose model building are very simple. The ultracolor group is taken to be a direct 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 167B, number 2

PHYSICS LETTERS

product of SU(n), SO(m), and Sp(2k) groups, and all fermions are in the fundamental (or antifundamental) representation of one or more groups ,1. The moose notation below is a simple way of representing such a model. SU(n)

~ an SU(n) group with fermions in an n(h).

(~) so(m) - Sp(2k) - -

su(/)

an SO(m) group with fermions in an m. an Sp(2k) group with fermions in a 2k. an (approximate) global symmetry group, soq).

For instance, a model based on the gauge group SU(n) X SU(m) with fermions in the representation m(n, 1) + (~, m) + n(l, m) can be written as +I

SU(m)

-I

~ SU(n)-,-SU(m)

+I

~ SU(n).

(1)

Numbers above the lines represent fermion charges under a U(1) global symmetry. According to the moose calculus, if SU(n) and SU(m) get strong at the same scale, they both produce QCD-like condensates ,2, breaking SU(m)gauge × SU(m)globa I and SU(n)gauge × SU(n)globa 1 to the diagonal global subgroups. Thus an SU(m)globa1 × SU(n)globa I symmetry remains unbroken. The U(1) also is unbroken, and a set of fermions transforming as ( ~ , n, +1) are required by the global symmetries to remain massless. This model cannot work as a realistic theory of massless composite fermions because there is an unbroken U(1) symmetry which commutes with all the original global symmetry generators. While SU(m)globa1 X SU(n)globa 1 can be broken by gauging some subgroup, the U(1) symmetry cannot (neglecting the contribution of weak instantons, which can weakly break the U(1) to a discrete subgroup). Therefore, the model always contains exactly massless fermions. A better model can be constructed by adding an orthogonal strong group, as in the moose below.

,1 Following standard practice, all ferrnions are taken to be left-handed. ,2 By moose calculus I simply mean to assume that each group produces the bilinear condensate it would ff the other groups were not gauged. Vafa-Witten [11 ] arguments can then be used to determine the symmetry breaking produced by each group.

6 February 1986 +1

SU(m) +

SU(n)

-1 +n(n+k) ~ SO(m) ) SU(n + k). (2)

+

SO(k) The global symmetry breaking pattern is, if we assume the moose calculus ,2 : SU(m) × SU(n + k) × U(1)

-, so(m) x su(n) x SO(g) x u(1)', with a set of massless fermions transforming as (m, n, 1, 1). Here H contains a U(1) factor, but it is a linear combination of the original U(1) and an SU(n + k) generator. It is possible to explicitly break SU(n + k) and so the surviving U(1) need not be exact. In this case with an appropriate subgroup of SU(n + k) gauged there need not be any exactly massless fermions. There will however be a very light pseudo-Goldstone boson, because there is a spontaneously broken U(1)which is exact to weak gauge anomalies. On the basis of these and other examples, I make the following conjecture. Any moose for which the moose calculus predicts an unbroken chiral global symmetry has a global U(1) symmetry with U(1) 3 and U(1)SU(n)~obal anomalies. Since only nonabelian symmetries can be explicitly broken by a weak gauge group, this U(1) is exact up to weak gauge anomalies. Therefore the anomaly matching condition [12] predicts either a very light pseudo-Goldstone boson or exactly massless fermions, even when the global symmetries are weakly broken. The possibility that a chiral U(1) symmetry is realized in the low energy theory is not ruled out, provided that the U(1) does not have a color or electromagnetic anomaly. Then the neutrinos could conceivably saturate the anomaly condition. However, no one has discovered a way of embedding SU(3)celor in the moose global symmetries without having a Peccei-Quinn color anomalous U(1). Thus all known mooses have either exactly massless colored particles or an axion [13]. Visible axions are ruled out [14], therefore the Peccei-Quinn symmetry breaking scale must be very high [15], between 107 and 1012 GeV [16]. Furthermore a weak alignment must be possible which conserves SU(2)w X U(1)y but breaks U(1)pQ. Then it is possible to avoid phenomenological axion prob201

lems by making the ultracolor scale very large. As an example of the constraints axions place on the weak gauge group let us examine the following model, known as the H-moose. This model has been proposed and studied by Georgi as a candidate for the real world of three families of quarks and leptons.

SU(6 +m)

SU(6 +m)

t

+

SU(4)L

t

su(6) L

SU(4)L

> SU(m)

> SU(4)R

+

any really elegant mechanism for getting rid of the antineutrinos we will examine a slightly simpler model containing only three families of quarks. The analysis of the Peccei-Quinn breaking is not affected by this simplification. The global symmetry breaking pattern is the same, with SU(4) replaced by SU(3). In the following H moose, +6/(6+m)

(3)

su(6) R

SU(4)R

As explained in ref. [7], the global symmetry breaking pattern is SU(6 + m)L X SU(6 + m)R X SU(4)L X SU(4)R X U(1)A X U(1)B × U(1)C SU(m) X SU(6)L X SU(6)R X SU(4) X U(1)A X U(1)B X U(1)C. The massless fermions transform under the unbroken symmetry as (1,6, 1, 4, +1, +1,0) ÷ (1, 1,6, 4, +1, 1,0) ,s. It is possible to choose a weak gauge group which breaks SU(6)L X SU(6) R X U(1)~ in such a way that all the fermions get mass. The unbroken SU(4) is identified as a Pati--Salam group [16], containing color and lepton number. SU(2)W, U(1)y, and some additional "flavor" interactions are embedded in SU(6 + m)L and SU(6 + m)R. The weak alignment is assumed to break the additional weak interactions but approximately conserve SU(2) w X U(1)y. As it stands this model has the disadvantage of having the antineutrinos treated the same way as the positively charged leptons by the ultracolor dynamics. It is possible to append still more strong groups [7] to give the antineutrino mass at the ultracolor scale, and so explain the lightness of the observed neutrinos. However, because there is not yet -

,3 Note that my U(1)k and U(1)~ are a different linear combination of the U(1)'s than the U(1))t and U(1)~ in the Les Houeheslectures [7 ]. 202

6 February 1986

PHYSICS LETTERS

Volume 167B, number 2

SU(6+m)R

SU(6+m) L + SU(3)

-1

+

~ SO(m)

+6/(6+m)

~ SU(3)

+

+

-1

SO(6) L +1 +

SU(6) R +

+1

SU(3JL

SU(3JR

, (4)

the diagonal sum of SU(3)L × SU(3)R is gauged as color. While there are actually three global U(1) symmetries only the anomalous one is indicated in the moose diagram. This Peccei-Quinn symmetry is broken by the SU(3) condensates, however, it is possible for a linear combination of SU(6 + m)L X SU(6 + m)R X U(1)pQ generators to remain unbroken. I will now show that unless SU(2)w is embedded in a larger weak gauge group that the low energy theory contains either a standard axion or massless ouarks. Let us use a (12 + 2m)-dirnensional matrix to represent the space of SU(6 + m)L × SU(6 + m)R X U(1)pQ × U(1)B generators. We will neglect discussion of the other nonanomalous U(1) and other global symmetries since what happens to them does not affect the Peccei-Quirm symmetry. The generators 6+m 6+m

6+m{Ta 6+m { 0

:0 ' 0

_;.,110 0, 1:,(5 ) -1

all belong to G, while only the subset

0

Volume 167B, number 2 6

m

6

m{O

0

0

6{0

0

0

m{0

0

0

0

0

0

PHYSICS LETTERS the generators of G" are of the form

m

0 '

0

6 February 1986

0

0

0

0

0

0

0

0

0

0 0

0

i1,

0

6

2i ]

6

2i /

x

x

0

0

0

0

2i x

x

0

0

0

0

]

0

x

0

0

0,

6{0

0

0

x

0

x

6

0

(8)

0

0

2i( 0

0

0

0

x

0

o

o

i[0

o

0

x

0

x

and commute with the color anomalous generator

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

-/a*

1

0

0

0

0

0

1

0

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0;

0

1

0

0

0

-1

0

0 0

-1

0

0 TI'Q = 0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

-1

0

0

0

0

0

0

(6)

-1

belong to H. Here the Ta are SU(m + 6) generators, the t a are SU(6) generators and the la are SU(m) generators. The color anomaly is proportional to the trace of the generators. In order to give mass to the massless quarks, we gauge a subgroup G" of SU(6+m)L × SU(6+m)R , which contains SU(2)W X U(1)y and some additional flavor interactions. G' contains both G" and those ~enerators of G which commute with

Gft.

If SU(2)W is not unified in a larger group, then we may write the decomposition of SU(6 + m)L X SU(6 + m)R under G" = SU(2) × f as (6 + r n , 1 ) ~ ( 2 , i

+ 3)+(1,j),

(1,6 + m ) ~ (1,j + 6) + (2, i).

(7)

Here f contains all other weak interactions including hypercharge and the second entry stands for any representation of f with that dimension. We choose 2i +/" = m so that it is possible for the weak alignment to conserve SU(2)W. With this SU(2)w preserving alignment there are massless quarks with the usual quantum numbers under SU(3)C × SU(2) w, 3(3,2) + 6(3, 1). We assume that some of the flavor interactions will slightly destabilize the SU(2)w conserving vacuum, producing small masses for the W's and quarks. In the basis where the vacuum conserves SU(2)w ,

0

(9)

1

which is conserved by the same vacuum. Therefore T~Q is a Peccei-Quinn symmetry generator which cannot be spontaneously broken above the SU(2)W breaking scale. This model contains either massless quarks or a Weinberg-Wilczek visible axiom The H moose cannot be realistic unless SU(2)w is embedded in a larger group. For clarity, let us restate the argument for this unwanted U(1) in a different way. The representation (7) of G" is reducible, so there are at least two new global symmetries, including U(1)~ O. A weak alignment of (4) which preserves SU(3)C X SU(2)W breaks the four SU(3) groups down to the diagonal sum, which is color. The ultracolor groups SU(6)L × SU(6)R × SU(m) are broken to SU(2)w (ignoring hypercharge and other generators of f) in such a way that (6, 1, 1)--, 3(2),

(1,6, 1) ~ 6(1),

(1,1, m) -+ i(2) +](1).

(i0)

Now under SU(3)c X SU(2) w × U(1)~Q the ferrnions of (4) transform as

203

Volume 167B, number 2 3 ( 3 , 2 , * 1 ) + 6(3, 1,+1)

PHYSICS LETTERS (11)

+ 2i(3,2,+1) + 2j(3,1,-1) + 2i(],2,-1) + 2](3, 1,+1) + 3 ( 3 , 2 , + 1 ) + 3 ( 3 , 2 , - 1 ) + 6(3, 1,+1) + 6 ( 3 , 1 , - 1 ) .

(12)

The fermions (11) are the massless quarks, and (12) are the massive fermions. The U(1)~Q charge is not broken spontaneously b y any dynamical mass for the fermions which conserves SU(3)c × SU(2) w . Furthermore, it commutes with all the gauge interactions and so is explicitly broken only b y weak anomalies. The H-moose has a visible PQ symmetry unless SU(2) w is unified with some o f the flavor interactions.

Conclusion. The presence o f axions in this intriguing class o f composite models orovides two constraints. The first is that the ultracolor scale must be very large, between 107 and 1012 GeV, and the second is that the ultracolor interactions must break the P e c c e i Quinn symmetry at the ultracolor scale b u t conserve SU(2)w x U(1)y. A fine tuning o f the SU(2) w X U ( 1 ) y gauge couplings against the gauge interactions which destabilize the SU(2) w × U ( 1 ) y conserving vacuum is necessary to orovide such a large hierarchy between the weak and the ultracolor scales. Also, in the potentially realistic H-moose SU(2) w must be embedded in a larger group. Perhaps unifying SU(2) w with some flavor group will solve b o t h problems - naturally motivating the time tuning as well as breaking the P e c c e i - Q u i n n symmetry at a large scale. I thank Howard Georgi and David Kosower for not believing m y arguments until they were correct. This research is supported in part b y the National Science F o u n d a t i o n under Grant No. PHY-82-15249.

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[31 S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [4 ] S. Dimopoulos, H. Georgi and S. Raby, Phys. Lett. 127B (1983) 101 ; S.C. Chao and K. Lane, Phys. Lett. 159B (1985) 135. [5] D. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 183; S. Dimopoulos, H. Georgi and D. Kaplan, Phys. Lett. 136B (1981) 187; P. Galison, H. Georgi and D.Kaplan, Phys. Lett. 143B (1984) 152; H. Georgi and D. Kaplan, Phys. Lett. 145B (1984) 216; M. Dugan, H. Georgi and D. Kaplan, Nuel. Phys. B254 (1985) 299; T. Banks, Nucl. Phys. B243 (1984) 125; S. Dimopoulos and J. Preskill, Nucl. Phys. B199 (1982) 206. [6] H. Georgi, Phys. Lett. 151B (1985) 57. [7] H. Georgi, Harvard preprint HUTP-85/A060 (1985); Lectures Les Houches Summer School for Theoretical Physics (1985). [ 81 D. Kosower, in preparation. [9] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247; M. Peskin, Nuel. Phys. B175 (1980) 197; J. PreskiU, Nuel. Phys. B177 (1981) 21. [10] R. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440; Phys. Rev. D16 (1977) 1791. [11] C. Vafa and E. Witten, Nucl. Phys. B234 (1984) 173; D.A. Kosower, Phys. Lett. 144B (1984) 215. [12] G. 't Hooft, in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980); S. Coleman and B. Grossman, Nucl. Phys. B203 (1982) 205. [13] S. Weinberg, Phys. Rev. Lett. 40 (1977) 223; F. Wilczek, Phys. Rev. 40 (1977) 279. [14] T.W. Donnelly et al., Phys. Rev. D18 (1978) 1607. [15] J. Kim, Phys. Rev. Lett. 43 (1979) 103; M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B104 (1981) 199. [16] N. Iwamoto, Phys. Rev. Lett. 53 (1984) 1198; D.A. Dicus, E.W. Kolb, V.L. Teplitz and R.V. Wagoner, Phys. Rev. D22 (1980) 839; M. Fukugita, S. Watamura and M. Yoshimura, Phys. Rev. Lett. 48 (i982) 1522; J. Preskill, M.B. Wise and F. Wilezek, Phys. Lett. 120B (1983) 127; L.F. Abbott and P. Sikivie, Phys. Lett. 120B (1983) 133; M. Dine and W. Fisehler, Phys. Lett. 120B (1983) 137; D. Kaplan, Nucl. Phys. B260 (1985) 215. [17] J.C. Pati and A. Salam, Phys. Rev. D10 (1974) 275.