Minimal composite Higgs systems

Volume 95B, number 2 PHYSICS LETTERS 22 September 1980

MINIMAL COMPOSITE HIGGS SYSTEMS

P. DI VECCHIA Institut fur Theorie der Elementarteilchen, Freie Universitiit, Berlin, Germany

and

G. VENEZIANO CERN, Geneva, Switzerland

Recewed 10 June 1980

Extending to teehmcolour models recent results on the chiral, large N c limit of QCD, we argue that minimal composite Hlggs systems must contain an r/like Higgs particle, whose peculiar properties follow from current algebra and large N arguments only. By contrast, the usual scalar Higgs is a model-independent entity without clear experimental signature.

1. The presence of elementary Higgs fields in the standard Glashow-Weinberg-Salam (GWS) model is generally regarded as one of its less attractive features: it is, for instance, "unnatural" in the sense recently defined by 't Hooft [1].

A simple alternative to elementary Hlggs scalars is to regard [2] them as bound states of a new kind of "quarks" and "gluons", which interact strongly accord- hag to a QCD-like lagrangian, Z?QC, D , in which the colour gauge symmetry SU(N) (N = N c = 3) is replaced by a new one (techni-prime-hyper-meta-heavy-colour in the literature) taken to be, for simplicity, SU(N') . In order to get the right value of m w (or o f GF) , this technicolour (TC) interaction is assumed to become strong at a scale A', some three orders of magnitude larger * 1 than A = Acolour -~ 0.5 GeV.

The dynamical Higgs mechanism originates in TC schemes from gauging a spontaneously broken chiral symmetry o f QC'D. It is important therefore to start our discussion from a consistent picture of chiral symmetry breaking in QCD-like theories. One such picture has recently emerged [3,4] within the framework of

,1 According to grand-umfication ideas, this occurs naturally fiN' > N = 3, in which case the 1IN expansion arguments, we are going to make, should work better for QC'D than for QCD.

the 1IN c expansion of QCD: in particular the famous U(1) problem [5] can be brought under control, a very reasonable mass spectrum and interaction pattern for the nine low-lying pseudoscalars is obtained and a consistent 0-dependence of physical quantities is derived [6].

The above results are neatly summarized by an effectiCe lagrangian [ 7 - 9 ] containing the fight degrees of freedom of QCD in the combined chiral, large N limit. There is, as usual, a great deal o f arbitrariness in the choice of Z?eff; yet, the quantities we shall be con- cerned with follow just from current algebra and large N expansion arguments and will not depend on the actual choice made.

In this paper a minimal composite-Higgs model is thus constructed out of a single doublet of hght techniquarks. After combining QCD, QC'D and electroweak interactions ~ la GWS, an almost conventional scheme results. There is however an interesting novelty: whereas the properties of the usual scalar Higgs meson appear to be model dependent (to the extent that such a particle even disappears in the non linear version presented here), one predicts the existence of a composite ~/-Higgs particle (the U(1) particle of QC'D, denoted hereafter by ~H) whose peculiar properties can be reliably comput ed via current algebra and 1IN' expansion techniques. The most amazing property of ~TH will turn out to be

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its narrow width (F/M of order 10 . 4 to be contrasted with 1/3 to 1/2 for a typical technicolour scalar Higgs).

2. We shall first recall the form of./~eQfCf D proposed in refs. [7] and [8] for the large N chiral limit of QCD. In its most economical version, where only physical light fields appear, it reads:

./gQCD _ ./gQCD + ~oQCD + ./gQCD eff - inv anom glass '

o Q C D = 1 Tr (0 u V+0~ V) inv

~?QCD = -(af~/2N) [0 - ½ i Tr log V/V + ] o a n o m

22Qm CD = ½fTrTr(MV+M+V+). (1)

The mesonic field V(x) is here an f X f matrix ( fbe ing the number of quark flavours) subject to the constraint V + V= VV + = f2 I. We then write:

V = f~r exp (if~l'cflri) ;

Tr(r,T/)=Si], i,/= 1,2, . . . f 2 , (2)

with f,r "" 56 MeV the pion decay constant and ~r i the f 2 pseudoscalar fields. Since Vii transfoCms like i~R ~L under (global) chiral U ( f ) X U ( f ) transformations:

V(x) -->A V(x)Bt ; A, B @ V ( f ) , (3)

eq. (2) implies (Vii) =fTrSi] i.e. the spontaneous breaking of U ( f ) X U ( f ) down to SUrf)vector. Furthermore, an explicit breaking t e rm ~Qt~D has been included to mass represent the effect of quark masses and the U(1) axial subgroup is also broken explicitly by/2anom. Such a term does also include [8], through the parameter a, the effect of Kogut-Susskind (KS) poles so that, in the chiral limit (/2mass ~ 0), the model has ( f 2 _ 1) massless Goldstone bosons and one massive S U ( f ) singlet particle with [4]:

2 = faiN (4) msinglet

Through anomalous Ward identities expanded in l/N, the parameter a can be related [3,4] to a quantity of the theory without quarks (pure Yang-Mills) i.e.

f d4x T(Q(x) Q(o)) IN__,Y'M'oo = ia f2 /N (5)

with Q(x) the usual topological charge density. Since Q(x) is a total divergence, a 4 :0 implies KS poles. To leading order in l/N, a = kA 2 with k (in principle) a calculable constant.

Fitting pseudoscalar masses with ~mass taken into account, one finds [4]:

a -~ 0 .7-0 .75 GeV 2 . (6)

Finally, in eq. (1), 0 is the QCD vacuum angle [10] and M is a real-diagonal mass matrix:

Mil = p2~i/ . (7)

3. We now consider a rescaled version of eq. (1) for a new strong interaction, QC'D, characterized by a scale A' with:

A ' = p A , p ~ l . (8)

Besides p, the other free parameters of QC'D in the chiral limit are N ' and the number of techniflavours f ' . The rest is fixed from rescaling and large N counting so that ~?QC'D is obtained from .t2 QCD by the replace- me nt s:

f~rof~=p(N'/3)l/2frt; a~a'=p2a, (9)

and by "priming" all fields and parameters (V--> V' , O -~ 0' etc.).

At this stage, in the absence of Z?mass, we have two decoupled worlds with global invariance under:

G gl°bal = S U 0 e) @ SU( f ) ® S U r f ' ) @ S V ( f ' ) , (10)

which is spontaneously broken by f~r,f~-¢ O. In order to construct a model containing a minimal

number of fight particles (in particular, no other massless particle in the chiral limit) we choose f ' = 2. Since our next step is the gauging of an SU(2~ × U(1) sub-

of global ~CD QC'D group G , we need to replace ~mass + "/~mass by a term which preserves such group e.g.

_ t ~ P

./gQCD+QC'Dmass ''> Z?mass =f~rf~ Tr [XV t V+ h.c.] , (11)

where V' is an f b y f m a t r i x made o f f / 2 two by two blocks equal to V' and X is a real diagonal .2 coupling matrix:

hi~ = hiSi/. (12)

~?mass does indeed preserve the GWS subgroup of

,2 It is easy to see, however, that the most general form of -~mass compatible with SU(2) × U(1) invanance allows for arbitrary Cabibbo angles and Kobayashi-Maskawa phases. However, for the purposes of this paper we can set to zero all these angles as well as 0 and 0'.

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G gl°bal given by two-by-two block diagonal matrices with the following matrices in each block:

SU(2)L: A = A ' E S U ( 2 ) ; B = B ' = 1 , (13)

U(1): A = e itr~/2 , A '= e ioW2 ,

B = eir3~/x/2e i°~/2 , B' = eiraf3/x/-2e i°'0/2 .

Since we want SU(2)L × U(1) to break down to U(1)Q of electric charge, we see immediately that o, or' are determined by the requirement that:

0) r3/x/~+ o = Q = i.e. o = i / 6 ,

(14)

0 , ½. r 3 / x / ~ + o ' = Q ' = ( Qu' , 1 l i e o = Q n , -

Qd = Qu- ! " " Gauging this GWS group we arrive at our final lagrangaan:

.tgQCD+QC'D+GWS = ,/9

_ t->QCD-~ --_+ + i ~ 1 - - t i n v (OuV auV g2AuV-'glVBu'T3/~¢/2)

+ .~QC'D(~mv u V'-+~u V'+ig~A, u V ' - ig lV 'Bur3 /x /~ )

+ Z?mass- 1Fi F i _ ¼GuvG, v I.t v I.t V

oGWS (15) + "/~WEM anomaly + "" leptons "

Here g l , g2 are the two usual gauge couplings of SU(2)L × U(1) and

A u = Ai r i / v ~ ,

Fiuv = auAiv - avAiu + oz~ei]kAJ--u--v a k ' (16)

Guy = ~uBv - ~vBu •

In eq. (15)"~u and ~ 3 stand for obvious block diagonal f × f matrices.

Besides the mass term (11) we have also added electroweak anomalies and a leptonic term (see below). The latter is supposed to free the model from the dangerous anomalies of gauged axial currents. This is easily achiev- ed with I/2 standard leptonic doublets if

l/2 =f/2 + ( 2 Q u , - 1)N' =f/2 + 2tr'N' , (17)

which is an integer if "technibaryons" (made out of N ' techniquarks) have integer charge (e.g. Qu' = ( N ' - i)/ N' , 2o'N' = ( N ' - 2)).

4. We now discuss the spectrum of the model defined by eq. (15). It is convenient to perform a "gauge

transformation" on the vector fields A/u by def'ming:

w. = V* A V- O/g2)Vt

= -O/g2)U*DuU, (18)

with

U = ( U t ) -1 = l+i~-iX ̀ + ...,

r " ' + " like__21 ] 1 ~-¢2,-1 L]n~ri l~r = rr~ k) (19) xi=(f;2+ ji.j

where 7r} 1) = zri, 7r} 2) is the analog of 7r i for the 2nd generation of quarks etc. We also introduce the usual combinations:

Z~u= 2 2 -1/2 3 (gl + g2) [g2 Wu - gl B~ ] , (20)

el ( g 2 + g 2 ) - l / 2 3 AIs = [gl W~ +g2Bu] .

One then finds that W -+ = (W 1 + iW2)/X/~ and Z 0 fields acquire a mass term ~ la Higgs through the following piece of Z?:

2 , 1 -2~,2 (W~)2 + ~ (gl

~OHlgg s = ~g2 ~' .= (21)

F2 = 2(f~ 2 x 2

giving * 3 :

1 roW+_ = ~g2 F, m z 0 = m w / c o s 0 w ,

cos 0 w = g2/(,g~ + g2)1/2 . (22)

Since 8GF/X/2 -= g2mw2 we find:

F = X/~-(f~ 2 + ~ff2)l /2 = (G F V~-)-l/2 = 246 GeV .(23)

Consequently, from eq. (9):

f'Tr/fTr = o(U' /3) 1/2"~ 2.6X 103 ,

O = 2.6 × 103 (3]N')l/2 . (24)

At this point one can rewrite (21) in terms of the original fields and observe that exactly the X i combinations of eq. (19) have been eaten up (having lost their kinetic term) by becoming the longitudinal components of W ± and Z 0.

• 3 Notice that we are able to preserve the standard relation m w = m Z cos 0 w even ff we allow, from .~mass, mu ~ rod. In the spirit of ref. [ 11 ], we can say that our teehnieolour condensate conserves isospin, whereas extended teehnieolour interactions (which we do not analyze here) may not do so.

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We now turn to "Qmass" Because of its lnvariance under SU(2) × U(1) (broken spontaneously to U(1)), ~rnass gives a mass matrix with three eigenvectors which are just the X i combinations of eq. (19), eaten up in the Higgs phenomenon ,4 . The rest of the spectrum is easy to describe even if not easy to determine analyti- cally. It consists of

(a) Flavourful pseudo-Goldstone bosons of QCD which are unaffected (i.e. unmixed) by QC'D (e.g. fis, ~c etc.). They have masses given by

2 = (x~ + x y ~ 2 , (2s) #at3

m agreement with standard formulae [8] if we identify /a 2 of eq. (7) with 2X,, 6 2.

(b) Flavourful pseudo-Goldstone bosons which are the surviving (i.e. not eaten up) mixtures of QCD and QC'D states. A typical example are the physical rr -+ eigenstates (which are orthogonal to X ~ of eq. (19)) given by:

+ , + '+ , 2 + rrph = rr + -- (frr/f~r)7 r- O((frr/f~r) rra) . (26)

We conclude that, in the above two sectors, physics is just conventional QCD once the X + slates are ab- sorbed.

(c) There are, finally, flavourless pseudo~calars. Besides the eaten-up combination X 3 , we find the usual QCD flavourless pseudoscalars (with a tiny admix- ture of fi'u' and d'd ') . One of them becomes, in the chiral limit ( X - + 0), the U(1) partmle of QCD of mass given as in eq. (4). For a general mass (or coupling) matrix the situation is as discussed in refs. [4,8] up to O(flr/f~) corrections.

There is, however, a new state in this sector: it is (again up to a small mixing) the U(1) particle of QC'D, i.e.

r I

~/H ----- (~ru + 7rh)/X/'2+ O((flr/f~r) (~u +/r~)) . (27)

In perfect analogy with eq. (4) its (mass) 2 is:

m2(r/H) = 2a ' /N ' . (28)

We stress that this particle xs a singlet of SU(2) X U(1) and that, as such, it does not enter directly in the GWS Higgs phenomenon.

.4 Omass is also mvariant under a subgroup of UA(1) X U h,(1) and, as a consequence, one finds that only a linear combination of 0 and 0' affects physms (we can set 0 '= 0). However, unless something more is done for the techniquark mass terms, (essentially) the same CP violating interaction discussed in ref. [6] originates if 0 =~ 0.

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Its existence, however, is necessary in the 1IN' expansion of QC'D or, equivalently, m any techniquark model. Furthermore, as shown in our last paragraph, its properties can be reliably computed from the effec- tive lagrangian approach and bear resemblance to those of either an r/or a Hlggs particle according to the process considered.

5. We now discuss the detailed properties of the r/H particle. Using eqs. (24) and (29) the mass of r/H is given by

m(r/H) = (2a' /N') 1/2 = p ( 2 a / N ' ) 1/2 = 5.4/N' TeV. (29)

There is no "hght" (1.e. m < mw) Higgs particle in this model. The analog of the a particle of QCD (0 ++ broad rrTr structure at about 700 MeV) would have a mass [1 ]:

3 2 m (OH) "" 0.7 p GeV ~ _ r ~ TeV > m (r/H) for N'~> 4 .

V l v (30) As in QCD, this particle would have a very large width, this time into W+W - and 2Z 0. By going into a non linear "o-model" we have sent m (OH) to infinity* s. One can easily check, however, that in a linear version with m(aH) ~ 1 TeV one gets P(o H -+ W+W - , 2Z 0) "" 0.3 TeV.

By contrast, r/H has a hard time decaying just like her QCD fellows 7), r/'. There is no first order decay of 77 H -+ vector mesons but there is one via the electroweak anomaly (same as for rr 0, r/0 -* 23'). This turns out to be one o f the main decay channels of r/H together with lepton pairs and q~ pairs (i.e. jets).

The decay into pairs of vector mesons can be evaluated using the standard analysis of electroweak anomalies [13]. We found the following amplitudes:

i H "+ 27 H -+ 2Z0

H -+ Z0T (31)

H ~ W+W-

/ 3(1 + 4° '2) ' 1

~4 [1 + 3(1 + 40 2) sin40 -- 3 sin20 ]/sin220 ]

= I \ 3 1 2 ( l + 4 o , 2 ) s i n 2 0 _ l ] / s i n 2 0 / '

\ ( s in20) -1 /

,s In a recent investigation [12] it has been shown that low energy phenomenology depends only weakly (i.e. logarith- mieaUy) on the mass of the scalar Higgs particle, hence "low energy" phenomena depend very little on re(OH).


where o' is defined in eq. (14) and:

a N ' I - 6X/~-rrf'~r euvpoPluelvP2pe2o,

sin20 = sin20w -~ 0 .23 .

With this value of sin20 w and taking for instance o' = 1/6 = o one gets:

+ 73 0 12 , F ( r / H ~ W W - ) = ~ M e V , F ( r /H~Z 7 ) = ~ r M e v ,

(32) 15

P(r/H--+ 27) = ~ MeV, P(r /H~ 2Z 0) = ~ v MeV,

provided that N ' is small enough so that m (r/H) 2m(Z 0) (e.g. N ' = 4, 5).

Notice the small width obtained. For N ' = 4, the total width into vector mesons is of order ,6 30 MeV: we can say that 77 H behaves rather like rr o, r/, 77' in this respect !

Before saying that we are dealing with a narrow ob- ject we have to search for other possibly large decay modes. We found the only appreciable ones to be those induced by the mass generating Higgs couplings to quarks and leptons. For instance, if all ,7 the mass ofleptons comes from coupling ~ to V ' , one inhe- rits for r/H a coupling:

1 rn~_ .6?nH~+ ~- : - ~ ~-~ ~('/5 ~ r / H . (33)

We see here the r/H behaving like a bona fide Higgs particle !

From (33) we get a decay rate:

l_J__[m~ 2 F0?H -+£+£-) = 16rr ~f~ ] m(~/H) ' (34)

i.e. negligible (compared to eq. (32)) widths into light leptons and

11.5 r(r/H -~ r+r - ) = - y - MeV. (35)

For q, ~jets the situation is analogous with an obvious

,6 Even smaller widths are predicted into W+W-3, etc. ,7 It is important that this is so to a high precision if one wants

f~r to be the decay constant of ~r ~ tw. In this case, as first notmed by Weinstein [14], the decay proceeds via the small physical pion content of zr' and the ~r'~tv coupling of magnitude muller.

extra factor of three for colour. For a b-quark of mass 5 GeV one gets:

;-b-+ , 260 Y'(nH -+ 19 jets) --~-~r- MeV. (36)

We conclude that, if the sequence o f quarks does stop at about 10 GeV, the ~TH particle will have a total width of about 100 MeV mainly into t t , bb jets, but also with appreciable branching ratios into r+r - or pairs of vector mesons.

All this may sound exciting since the r/H will have some clear experimental signature. Unfortunately, however, we have to conclude on a more pessimistic note for Hlggs hunters.

Unlike what happens [15] in TC models w i t h f ' > 2 (hence with technipions) our minimal model has a sort of desert between mw,zo and 1 TeV. But even with this type of energies available, production of r/H is not easy (for reasons related to its small width).

The least impossible way to produce r/H appears to be that of Wilczek [16] i.e. 2-gluon fusion in hadron hadron collisions (~p energy doubler at Fermilab?) or heavy cl~ fusion (which is essentially the same). We have not made a detailed study of this possibility, but there seems to be little hope for a reasonably large productmn cross section.

One of us (GV) wishes to acknowledge fruitful dis- cussions with D. Amati, R. Barbieri, J. Ihopoulos and P. Sikivie.

References

[1] G. 't Hooft, Lecture given at the 1979 Carg~se Summer Institute.

[2] S. Wemberg, Phys. Rev. D19 (1979) 1272; L. Susskind, Phys. Rev. D20 (1979) 2619; S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237; see also E. Elchten and K. Lane, Phys. Lett. 90B (1980) 125.

[3] E. Witten, Nucl. Phys. B156 (1979) 269. [4] G. Venezmno, Nucl. Phys. B159 (1979) 213;

P. D1 Vecchia, Phys. Lett. 85B (1979) 357; see also P. Nath and R. Arnowitt, Northeastern University preprlnt NUB-2417 (1979).

[5] For a recent assessment see: R.J. Crevcther, CERN preprint TH 2791 (1979) and Bern preprint (1980).

[6] R.J. Crewther, P. D1 Vecchia, G. Veneziano and E. Witten, Phys. Lett. 88B (1979) 123; M.A. Shifman, A.I. Vainstein and V.I. Zaeharov, preprint ITEP-64 (1979).

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[7] C. Rosenzweig, J. Schechter and G. Trahern, Syracuse preprint SU-4217-148(1979).

[8] P. Di Vecchla and G. Veneziano, CERN preprint TH.2814 (1980).

[9] E. Witten, Harvard preprlnt, HUTP-80/A005 (1980). [10] R. Jacklw and C. Rebbi, Phys. Lett. 37 (1976) 172;

C. Callan, R. Dashen and D.J. Gross, Phys. Lett. 63B (1976) 334.

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[12] T. Appelquist and R. Shankar, Nucl. Phys. B158 (1979) 317; T. Appelquist and C. Bernhard, Yale preprint YTP 80-01 (1980);

A.C. Longhltano, Yale preprint YTP80-04 (1980); see also M. Veltman, Acta Phys. Pol. B8 (1978) 475, Phys. Lett. 70B (1977) 253.

[13] See, for instance, B. Zumino, 1972 Carg~se Lectures in Physics, ed. M. Levy, Vol. 7, p. 450.

[14] M. Wemstein, Phys. Rev. D7 (1973) 1854. [15] M.A.B. B6g, H.D. Politzer and P. Ramond, Phys. Rev.

Lett. 43 (1979) 1701; M.E. Peskin, private communication.

[16] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304; see also H. Georgi, S.L. Glashow, M. Machacek and D.V. Nanopoulos, Phys. Rev. Lett. 40 (1978) 692.

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Minimal composite Higgs systems