Constraints on Higgs couplings from axionatics

Volume 81B, number 1

PHYSICS LETTERS

29 January 1979

CONSTRAINTS ON HIGGS COUPLINGS FROM AXIONATICS ~r P. RAMOND 1 and G.G. ROSS

California Institute of Technology, Pasadena, CA 91125, USA Received 29 September 1978

The dependence of the axion mass on the number and coupling of the Iliggs bosons is emphasized. In particular, by requiring a unification of Yukawa and gauge couplings, the axion mass can be enhanced by a factor of the W-boson mass over a light quark mass. We present a Higgs extension of the Weinberg-Salam model in which this idea is realized. We discuss the limitations of current algebra techniques for deducing axion properties

The computation of instanton [1 ] effects in QCD leads to a strong interaction term OFu~F~v which apparently violates P and CP. The dimensionless constant 0 is left undetermined. While it could be chosen sufficiently small [O(10 -11)] to satisfy the existing limit on the neutron electric dipole moment, there is no existing explanation for such a choice in as much as 0 will suffer renormalization, usually infinite. A natural answer to this puzzle was proposed by Peccei and Quinn [2] ( P - Q ) who introduced a global chiral invarlance in the non-QCD part of the Lagrangian, which they then used to rotate away any apparent 0 effect. But, if no quark mass vanishes, this symmetry must be spontaneously broken in addition to being explicitly broken by instanton effects. The would-be Goldstone boson [3,4] produced by this spontaneous breaking of the P - Q symmetry acquires a mass from explicit instanton corrections [5]. It was argued this mass should be very small - several keV [3,4,6]. Yet the existence of such a low mass boson appears to be ruled out by experiment [7]. Here we will show that the mass of the axion is very model-dependent. In our view the absence of a light axion should be regarded as an important hint about the Higgs sector, the least known part of Weak Interaction theories. We find the predictions for the Work supported in part by the U.S. Department of Energy under Contract No. EY76-C-03-0068. I Robert Andrews Millikan Senior Research Fellow.

axion mass change drastically if we abandon the usual assumption that the large variation in masses (of quarks, leptons and vector bosons) are due to widely varying couplings to scalars with one universal vacuum expectation value (V.E.V.). For instance, taking Yukawa and gauge couplings to be of the same order of magnitude (as is expected for example in supersymmetric theories) we find the predictions for the axion mass may be enhanced relative to the usual estimates by a factor Mw/rn q where M w is the mass of the W-boson and mq is a light quark mass. A simple extension of the Weinberg-Salam model with this property is constructed. We start with the Weinberg-Salam gauge group with fermions in left-handed doublets ~kL and righthanded singlets u R, d R. So far as the P - Q chiral invarlance is concerned, the important terms in the Lagrangian are those involving the Yukawa couplings, Z?y, and the scalar self-couplings, £ ~ . The former is written in matrix form as ~ y = . ~ O i u -~bLI~ui u R + i ~. ¢ ~ L I ' d i d R +h.c.,

(1)

1

where F ui and 1-'di a r e N X N matrices for 2 N flavors; flavor and SU2L indices have been suppressed in ~L, d R and u R. The index i runs over the Higgs doublets 4~u and q~d which have (weak) hypercharge - 1 and +I respectively. Besides fermion number Z?y is invariant under several phase transformations performed on the fermions 61

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and scalars simultaneously. One of these corresponds to the (gauged) weak hypercharge. A second one we take to be the chiral P - Q symmetry. We assume that .t?~ contains terms that explicitly break the remaining transformations. The instanton breaks the chiral P - Q symmetry and a redefinition of the fermion fields can be used to set 0 = 0 in the 't Hooft term. Peccei and Quinn argue that this situation persists after spontaneous symmetry breaking as the V.E.V.'s giving mass to the quarks pick up the phase 0 due to instanton corrections to .67~. When this phase is eliminated from the fermion mass matrix by redefining the fermion states the term OFu~,FUVis simultaneously eliminated from the Lagrangian. The price for this is the appearance of a pseudo-Goldstone boson. Writing the neutral Higgs components as

jS,s u = jS,h u _ xfi 17u')'5 u 1 - Ydl 7~x')'5dl ,

caO=(oa(x)+~)expli(aa(x)+rla)},

Mw = ~2 V|/-~va. i,a l vq. t

t \

v~

a= u,d, (2)

where va exp(irlT) are the V.E.V.'s of ~70(x), there will appear a field h(x) not given a mass by the instanton-uncorrected ~oo and orthogonal to the scalar component which will generate the longitudinal component of the Z. h(x) may be written as a linear combination of the canonical fields aU(x) and a~(x):

h(x): ;Or%:(x),

(3)

(6)

where u 1(d l) are the up (down) light quark fields and x + y = Tr(S u + S d) -- X. PCAC methods [3,6], fixing x - y so that the axion does not couple directly to the pion, estimate the mass of the axion as

mh = 2 flrmrr (1-+~z) X/z

X,

(7)

with z given in terms of the value of light quark bilinear operators in the vacuum [6] m~ [(filLexp (ihr~)UlR)0 + h.c.]

z rnT[(alLexp(ihr~)dlR)O + h.c.]

(8)

With this Higgs structure, the mass of the W boson is (9)

We can now discuss the expected axion mass in models with various Higgs structures. In the example discussed by Peccei and Quinn, there are two Higgs doublets. Then ru=v~/v,

r~=v~/v,

where

M w = (g/x~2)v

and v = X/Vld~ + v~2.

Thus where mh:2J;mn

__Z--#r~r a : 1.

~

IvY+ v~]

~vi~E L od v~[ "

(10)

l~a

By identifying the divergence of the current with the coefficient of h(x) in the Lagrangian we may construct the naive P - Q current describing the coupling ofh(x). j5,h = /.~

~lTu75SUu + ~Tu75Sdd

+ a /.t h, (4) where (u)i and (d)i are mass eigenstates. S u and S d are matrices with matrix elements

(S,~)iS_ 1

(rak)o r~..

(5)

(m a + m ; )

Ju5,h is not

conserved due to the anomaly. We may de'fine a current which is conserved in the chiral SU 2 limit by

62

Provided vuI iv dI is not vastly different from 1, the axion will be much lighter than the pion. In this model the u, c (d, s) quarks get their mass from v~ (vd), so there is no obvious reason for v~/v d to have an extreme value The reason the axion is so light is due to the fact that the W bosch and quark both get their mass from the same vacuum breaking effect so that the Yukawa coupling is of order mq X/~-E, whereas the gauge coupling is of order e. What happens in the other extreme where we choose Yukawa and gauge couplings to be of comparable magnitude and ascribe the widely differing masses to very different V.E.V.'s? l e t us consider a simple extension of the model discussed above in which there is at least one further Higgs doublet, ¢~, coupling to heavy

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quarks but not to u, c, d, or s. For the moment we assume the original doublets, 0~, 01d, do not couple to heavy quarks. From eq. (10) we find

m h - ~j---+-~) m , f ,

a,i

.

(11)

The dominant contribution to rn h comes from the minimum value of(oa/ra), whereas from eq. (9) the dominant contribution to M w comes from the maximum value o f v a. If the v's are very dissimilar, the estimates for m h2 may be enhanced. In particular, if the d and s quarks get their masses from v d then, assuming Yukawa couplings are o f order g, m d ~ gvld and

29 January 1979

sive Higgs and small dimensionless couplings. This leads us to consider more complicated scalar couplings the existence of which were, in any case, forced upon us by the necessity of giving mass to the extra would-be Goldstone bosons in our model. As an example of a workable scheme consider our extension of the Weinberg-Salam model with three Higgs doublets 0~, 0~, and 0~. We also include a complex Higgs singlet S..6?o is chosen as ~0 = ~ m a 2 [0al2 + Xq[0t'a]4 + ~.~ijab ]Oia 1210b[2

a,i

t

a,I

+,n tsl 2 + v l s l 4 +

x olsl210 l 2 +

u+N 0 ,d,

all

2v~ "m r~ mh - ( i - ~ z ) JTr lrm~dg

+,,,el

(12) Mw

-(?~z)

fTrmlrrd 2X/~ GI: m ~ '

where r d2 < 1. Compared with eq. (10) we see the estimate for rn h is enhanced by a factor ((rdl/md)Mw). This could change the axion mass estimate to ~1 GeV, where we have used m d evaluated at a distance scale

of o(1/M w). Of course all this rests on the assumption that it is possible to generate such a pattern of symmetry breaking via a reasonable scalar potential and that the rotation coefficient r d is not extremely small. In the original Weinberg-Salam model the Higgs doublet gets its mass through a term in/2~ of the form -m2~b+0 + ;k(0+~b)2 . The V.E.V. is

(O0} = m/~/~. This form can easily generate spontaneous symmetry breaking for doublets with V.E.V.'s o f O(Mw) by choosing ~ = O(g 2) and m = O(Mw). However, this form is unacceptable for V.E.V.'s much smaller than this for then we must either choose m ,~M w or ~k>~g2. The residual Higgs scalars have mass of order m and if m "~M w they give unacceptably large AS = 1 weak transitions in the example given above. If we try to let >~g2 we must abandon perturbation theory in ~. and this is not particularly appealing to us. Rather we look for a Z?, which can give a small V.E.V. with very mas-

v2+h.c.

,e2,el

(13)

All other terms may be forbidden by discrete symmetries. We choose all dimensionless couplings to be of the same order. Then there is a range of the parameters for which 0~ develops a large V.E.V. much larger than that of 4~u, 0 d, and S. We find, dropping terms of

o(o~-') ' u2

u2

u2

u

v 2 ~ - - - m 2 m 2 /2~, 2 u2

v ! ~ md'2m's2/k 2 d2

,

o1 =m~2m~2/k 2

(14)

2 ... md'2rn,21k 2 Os l si where u'=

ml

d '2

m1

=ml

u= d2

uu + 2xl)v~ =

+(~'12

) d u . u2

= m I + ..1202

(15)

xSU U2

m's2 = m2s + A2 u2 " With all Xs = O(g 2) and m~ = O(Mw), 0 u2gets a large V.E.V. and generates most o f the W boson mass. 0~, 0dl and S have V.E.V. s whose size is given by ratios of dimensionfull parameters, eq. (14), and these may be adjusted to be small relative to (0~}. Moreover, this can be done even while making my', m d' and m's very large. As a result all the Higgs scalars remaining after spontaneous symmetry breakdown, with the exception of the axion, may have large masses/> M w and transitions mediated by these scalars, such as AS = 1 or AS 63

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= 2 transitions, may be arbitrarily suppressed. The axion in this model is h ~ ( vli da d

Vsas)/O~---i d2 + 0 2

(16)

so

rld = x/v~2 + Vs2"

(17)

Thus this example achieves our goal of a hierarchy of V.E.V.'s even though all dimensionless variables are of the same order. The axion, for reasonable values of v~/Vs, has the desired mass enhancement, eq. (12). As our model stands there is no mixing between light and heavy quarks and there will be one absolutely stable heavy quark. To avoid this we may include in a mixing term ~, (H) I-",, (L) ~ dlWL "~R

L) are where ~(LH) are the heavy quarks doublets and U(R the light quark singlets. This induces mixing between top light and heavy quarks of order m(qL)/mOqI) if F' is chosen of O(g). Note that since the V.E.V. of S is small, it could have been an I = 1 triplet of SU 2 without affecting the dominant I = ~breaking relation between M w and M z. This coupling can take place in the SU 5 model where the triplet comes from the Higgs representation that gives superheavy breaking [8]. We have argued that models with several V.E.V.'s which vary greatly in size may have an axion of mass much larger than previously supposed. P.C.A.C. estimates of the mass give a value o f O(1 GeV). However we expect large instanton corrections to the P.C.A.C. estimate. Baluni [9] has computed these for the light axion and finds contributions three to four times the naive P.C.A.C. estimate. Corrections of this size would mean the axion in our model could be several GeV. Of course with a mass as large as this the P.C.A.C. method fails, but alternative estimates using the effective Lagrangian of 't Hooft [5] also give masses in the GeV range. How would one see such an object? It couples to light quarks with electromagnetic strength and should be produced, if energetically possible, in the decay of states containing light quarks. The main decay mode will be into 31r's and one should look for

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29 January 1979

a resonance in this channel, ttowever, the signal will be O(t~) relative to the allowed strong decays of the original state and it will be difficult to see. The axion in our example couples to strange quarks more strongly than to down quarks but this is not a necessary feature. If it does couple strongly to strange quarks it could be seen in the radiative decay ¢ ~ 7 + h. The absence of a monochromatic photon in this decay suggests either the axion couples more weakly than g, or it is more massive than the ¢. In our example the axion couples to charmed quarks with strength (mu/mc)g but again this is not a general feature. If it couples to charmed quarks it should be produced in the decay ¢ ~ 3' ÷ h. The absence of such a signal here suggests the coupling of the axion is less than g/l 0 or it is heavier than the 4. We should also consider leptonic couplings of the axion. For instance, the axion could mediate the decay r/-/a+/a--, if it coupled to muons. The absence of a significant signal means the muon coupling of the axion is greatly suppressed. However, the axion need not couple to leptons when the P - Q symmetry need only be defined as quarks. Further, there is no anomaly-based agreement forcing it to couple to (light) leptons.

References [1] A.A. Belavin et al., Phys. Lett. 59B (1975) 85. [2] R.D. Peccei and II.R. Quinn, Phys. Rev. Lett. 38 (1977) 40; Phys. Rev. DI6 (1977) 1791. [3] S. Weinberg, Phys. Rev. Lett. 40 (1978) 22.

[41 F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [5] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. DI4 (1976) 3432. [6] W.A. Bardeen and S.-H.H. Tye, Phys. Lett. 74B (1978) 229; J. Kandaswamy, Per Salomonson and J. Schechter, Phys. Lett. 74B (1978) 377; Phys. Rev. DI7 (1978) 3051. [7] P. Alibran et al., Phys. Lett. 74B (1978) 134; T. Hause et al., Phys. Lett. 74B (1978) 139; P.C. Besetti et al., Phys. Lett. 74B (1978) 143; J. Ellis and M.K. Galliard, Phys. Lett. 74B (1978) 374; E. BeUotti, E. Fiorini and L. Zanotti, Phys. Lett. 76B (1978) 223; T.W. Donnelly et al., Stanford Preprint ITP-598 (1978). [8] A.J. Buras et al., Nucl. Phys. B135 (1978) 66. [9] V. Baluni, Phys. Rev. Lett. 40 (1978) 1358.