PHYSICAL REVIEW D VOLUME 46, NUMBER 1 1 1 DECEMBER 1992
Effective couplings of axigluons and technicolor pseudo Goldstone bosons
Xin-zhou Li East China Institute for Theoretical Physics, 130 Mei Long Road, Shanghai 200237, China
Jian-zu Zhang China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, China
and East China Institute for Theoretical Physics, 130 Mei Long Road, Shanghai 200237, China* (Received 25 March 1991; revised manuscript received 19 June 1992)
The anomaly-free chiral-color models in the technicolor scheme are discussed. The effective couplings of the low-lying technicolor pseudo Goldstone bosons (PGBs) to the gauge bosons, y , w',z', gluons, and axigluons are constructed in the effective chiral Lagrangian framework. The model suggests a rich phenomenology and shows the new axigluon-PGB coupling G, G, PP which contributes to technion pair production via axigluon fusion and should be accessible to the next generation of accelerators and the Superconducting Super Collider.
PACS numberk): 12.10.Dm, 12.50.Lr, 14.80.Er, 14.80.Pb
The recent and forthcoming precision electroweak measurements at the SLAC Linear Collider (SLC), and CERN e + e - collider (LEP), and Fermilab Tevatron col- lider are providing a sensitive probe for possible new physics at and above the Fermi scale MF-250 GeV which could potentially reveal effects outside the minimal standard model. As is well known, the low-energy phe- nomena is well described by a Yang-Mills theory with the gauge group
This defines the minimal standard model of particle phys- ics which is in good agreement with present experiments ii.e., for ds 5 100 GeV). What is not so clear is what lies beyond this theory. One such proposed extension of the standard model is offered by chiral-color theory [I]. There is a possibility that the SU(31, of the strong in- teraction originates in the chiral-color gauge group SU(3),,@ SU( 3 ),,. The gauge group of strong and elec- troweak interactions above M F is taken to be
and the SUi3), group is identified with the diagonal sub- group of SU(3),l s SUi 3 ),,. Therefore the chiral-color theory behaves like the standard model after spontaneous breakdown of the chiral-color and electroweak sym- metries below M,. However, the chiral-color theory ex- hibits nonstandard phenomenology above MF . There are many different implementations of chiral-color [l -41 theory. The most important model-independent predic- tion of chiral-color theory is a massive color octet of gauge bosons, the axigluons g A . The chiral-color model
ailin in^ address.
can be examined [2,3,5,6] in the technicolor (TC) scheme [7,8]. The range of the axigluon mass M A is fixed by the T C dynamical symmetry-breaking scheme, 280 < M A < 970 GeV . The absence of anomalies in chiral-color theory plays an essential role in constraining the quark spectrum in this theory. For chiral-color mod- els in the T C scheme, both the ordinary and T C fermions must be free of triangle anomalies with respect to SUi3),l@ SUi 3 I,,@ Ui 1 1,. The anomaly-free chiral-color models based on the gauge interaction (1.2) are not as unique as the standard model. In this paper we show some chiral-color models in the T C scheme which are free of anomalies with respect to the gauge group (1.2). In these models there is a rich spectrum of the T C pseudo Goldstone bosons. Some of the low-lying TC bosons may have masses much less than 1 TeV. We argue that chiral-symmetry low-energy-theorem techniques can be expected to work for the chiral-color model in T C phys- ics [2,9,10]. It is quite interesting to consider the new vertices involving axigluons and T C pseudo Goldstone bosons which, in addition to the pure gauge-boson anom- aly vertices [2,10], show the difference between the chiral-color models ( 1.2) and the standard model i 1.1)
11. EFFECTIVE COUPLINGS OF THE AXIGLUONS
AND THE TC PSEUDO GOLDSTONE BOSONS
We consider a T C model with one family of T C fermions [2,3] which have the following SU(3),l@ SU( 3 ),,@ SUi 2 1, @ U( 1 ). representations. T C quarks:
i u/,D/),: i3,1,2,f ), , (2.la)
UL: ( 1 , 3 , 1 , + ) R , D L : i1,3,1,-+)R .
T C leptons:
46 5092 @ 1992 The American Physical Society
EFFECTIVE COUPLINGS OF AXIGLUONS AND TECHNICOLOR . .
It is convenient to discuss the triangle anomalies in terms of SU(3),.@ SU( 3 ),,a U ( 1 ), rather than the SU(3),,@ SU( 3 I,,@ U ( 1 I,, gauge interactions. This is be- cause the weak hypercharge quantum number Y is the same for all members in a representation, while the elec- tric charge of the members can vary according to their weak-isospin quantum numbers. In the above we assume the T C interactions to be an SU(NTc) vector-gauge theory, t = 1,2,. . . ,NTc ( N T c > 3 ). The indices 1 and r refer to the left- and right-handed chiral-color model, while the indices L and R refer to the helicity. This is an extension of the Farhi-Susskind model [ l l ] . The T C fer- mions have triangle anomalies due to the SU(3),,@ SU(3 I,,@ U ( 1 1 , interactions.
Let the general coupling between the gauge boson and fermion be
where ra may involve y5 and satisfies the Lie algebra
where the fabc are the structure constants of the underly- ing Lie algebra. The triangle graph with three-gauge- boson vertices has an anomaly given by [12,13]
It is convenient to write ra in terms of left- and right- handed couplings
where T ; are Hermitian matrices and satisfy the same commutation relations as (2.3), i.e.,
Then (2.4) can be rewritten as
The anomalies are of the type [SU(3),,13 - [SU(3),,I3 and Y [ S U ( ~ ),, 1 2 - Y [ SU( 3 ), , I2 . All other anomalies either cancel between quarks and leptons or are zero. Let the gauge fields belonging to SU(3),,, SU(3),,, SU(2),, and U ( l ) , be denoted by G I , G,, W L , and Y. The nonvanish- ing anomaly coefficients are
(3,1,2, f )L T C quarks:
( 1,3,1, I R T C quarks:
(1 ,3 ,1 , - ) )R T C quarks:
(1,1,2, - 1 ), T C leptons:
( 1,1,1,-2)R TC leptons: A,( y3)=8
The chiral-color anomalies in the T C quark sector can be canceled by that in the ordinary quark sector [2,3]. This is based on the fact that the anomaly is mass independent [14,15]; i.e., anomalies due to a fermion with a large sepa- ration in masses can cancel against each other, which is not prohibited by the decoupling theorem . Thus, for the ordinary-fermion sector, we assume that there are n generations and each generation has chiral-color repre- sentations opposite to that of T C quarks, i.e., for quarks,
and for leptons,
If we relate the number n of ordinary quark generations to the rank NTC of the TC gauge group, n =NTC( =4), the cancellation of the chiral-color anomalies of the type [su(~) , , ]~-[ S U ( ~ ) , , ] ~ and Y [ S U ( ~ ) , , ] ~ - Y [ S U ( ~ ) , , ] ~ is achieved between the T C and ordinary quarks.
The general form of the low-energy chiral Lagrangians of T C theory is expected to be a valid phenomenological framework for new strong physics between 1 TeV and around the Fermi scale M F . This low-energy effective chiral Lagrangian allows us to describe interactions among low-lying T C pseudo Goldstone bosons and the gauge bosons y , w', z O , gluons, and axigluons. It is useful to represent (2.1) as the eight-component left- and right-handed fermion fields
Above the T C condensate scale AT,- 1 TeV, the chiral- color and electroweak interactions are negligible, and the T C fermions have an SU(8), 0 SU( 8 ), global chiral sym- metry. Below ATc, the T C interaction becomes strong and T C fermion condensate appears, ( qLqR )# 0, which breaks SU(8),@ SU(8 ), into SU(81, + R . Then there ap- pear 63 Goldstone bosons. Among them, three color- singlet weak-triplet Goldstone bosons n f + n ~ = & y 5 7 i ~ + Q y 5 ~ i ~ (i=1,2,3) are absorbed by W' and Z O to give W' and Z O masses, and the eight color-octet weak-singlet neutral Goldstone bosons n$=Dy5haQ ( a = 1,2, ..., 8 ) are absorbed by the axi- gluons to give the axigluon masses, where Q denotes the T C quarks, Q = ( U,D), and L denotes the T C leptons, L = ( N , El . Among the rest, there are four color-singlet T C pseudo Goldstone bosons (technions): the isotriplet #= 0 y s ~ i ~ - 3&y ,T'L ( i = 1,2,3 ) and the isosinglet ~ 0 = ~ y 5 ~ - 3 ~ y , ~ . They may have masses much less than 1 TeV (a rough estimation is 2 GeV < mpo, m,,. < 40
5094 XIN-ZHOU LI AND JIAN-ZU ZHANG 46
GeV, 8 < m,+ < 40 GeV ) and have unusual produc- tion and decay characteristics [ l l ] . Because of their ex- pected low masses, we are motivated to study their cou- plings to gauge bosons.
Now we discuss the low-energy effective Lagrangian for technions which takes the form [18,10]
where U( 4 ) - ( $L qR ) is a phenomenological field which is related to the matrix-valued T C pseudoscalar mesons 4. The phenomenological reasonable form for U ( 4 ) is
U ( d ) = exp(2i4/FT) . (2.12)
In (2.1 1) and (2.121, the technion decay constant FT is the anologue of the pion decay constant f ,, but it is related to the scale of a T C force-induced condensate and can be determined by its relationship to the weak W' (and z O ) boson masses
where g is the SU(2) gauge coupling and r is the number of the doublet of technicolor fermions. In order to keep the local gauge invariance, a,U(4) in (2.1 1) should be re- placed by the covariant derivative.
For the gauge group (1.2), the left- and right-handed gauge fields are
For the gauge group (1.21, the left- and right-handed gauge fields are
where Wf, ( i=1 ,2 ,3 ) and B, are the SU(2) ,@U(l ) , gauge bosons and gl, g,, g and g ' are, respectively, the SU(3)c1, SU(3),,, SU(2),, and U(1) gauge couplings. In the asymmetric chiral-color models, the left- and right- handed gluons GI, G, are related to the gluons G, and ax- igluons G A by the relations
where 8 is the mixing angle. Consider the chiral La-
grangian of quarks in the fundamental representation. We obtain an axial-vector coupling of the gluon to the quarks proportional to gl sine-g, cose, which is strongly suppressed by parity conservation and should be set to zero. Thus we have 
where g, is the SU(31, gauge coupling. In (2.151, Aa, T', YL, and YR are 8 X 8 matrices [lo]:
In (2.18)-(2.201, T, /2 and h a 2 are, respectively, the gen- erators of SU(2) and SU(3), I, ( n =2,3,4) is a n X n unit matrix, and 0, is a 2 X 2 zero matrix.
For technions, the matrix-valued TC pseudoscalars 4 are
where ro=Iz g, is normalized to tr(gag8 4" is related to the physical technions P*,', and PO' by the re- lations
Substituting (2.14) into (2.1 I), using (2.121, (2.15)-(2.22), and the relations
(where e is the unit electric charge and Qw is the Wein- berg angle) up to the lowest order, we obtain
EFFECTIVE COUPLINGS OF AXIGLUONS AND TECHNICOLOR . . .
e +- sine, [ (PO~,P--P-~,PO)W~+ +( -pOa,p- + ~ + a , ~ O ) w p - ]
In (2.25) the first three terms are, respectively, the mass TeV. In order to meet the lower bound M A ,, > 280 terms of the axigluon, W' and zO, which show the GeV, the mixing angle 6 between gluon and axigluon has correct relation M w = M z c o d , and keep the photon a maximum 6,,,-20" . and the gluon massless. The fifth term is the coupling of We cannot make a quantitative estimation of the technions to the axigluons. The rest terms are the u ( p p + G A G A +P+P-) for lack of knowledge of the ax- couplings of the technions to y , w', and zO, some of igluon structure functions fG , (x ,Q2) . The G A G A P P them have already been obtained in the literature [191, vertices in the low-energy chiral ~~~~~~~i~~ but (2.25) shows, in a systematic way, the complete re- represent the contribution of the loop diagrams; thus, the suits of the coupling of technions to gauge bosOns in the corresponding processes are of higher order in a,, but normal piece of the chiral Lagrangian. they may still be important. The gluons G, and axi-
111. TECHNION PAIR PRODUCTION VIA AXIGLUON FUSION
The new GAG, PP terms contribute to technion pair production via axigluon fusion, which may be used as a new source to look for possible new physics beyond the standard model at future hadron colliders such as the Su- perconducting Super Collider (SSC) and the CERN Large Hadron Collider (LHC).
The physical production rate of technions at proton collisions is given by integration over axigluon structure functions:
with 3=7s, x ~ , ~ = + [ ( x ~ + ~ T ) ~ / ~ ~ x ] , x 1 x 2 = 7 , and x l -x2 =x. From (2.25) the cross section a(?) is
where mp is technion mass which can be neglected at the SSC energy d 2 = 4 0 TeV and the LHC energy d y = 16
gluons GA are related to the left- and right-handed gluons GI and G, by the mixing angle 6. Thus we can reasonably guess that the axigluon structure functions have a similar qualitative behavior as gluon structure functions. As is well known, the gluon-gluon luminosity is much higher than the quark-antiquark luminosity for small values of x at hadron colliders. If the axigluon- axigluon luminosity has such a similar behavior, we may expect that the processes of the axigluon fusion are the main mechanism for technion pair production, which is similar to vector boson pair production via gluon fusion 1201.
IV. CASE OF A MODEL WITH EXOTIC TC FERMIONS
Now we consider a model in which the chiral-color anomaly cancellation is achieved by the introduction of exotic TC fermions. From (2.8) it follows that the anomalies due to SU(2)L@ U( 1 ), electroweak interac- tions cancel between the T C quarks and T C leptons. The surviving anomalies are due to chiral-color and weak hy- percharge interactions which are of the typ; [ s u ( ~ ) , , ] ~ - [ SU( 3 ),,13 and Y[SU( 3 12- Y [ SU( 3 I,, ] . We introduce exotic quarks 8,6, which transform under SU( 3 e SU( 3 ),,e SU( 2 )L @ U( 1 ) as 
5096 XIN-ZHOU LI AND JIAN-ZU ZHANG
Their nonvanishing anomaly coefficients are
Equations (2.8) and (4.2) show that the anomaly cancella- tion is achieved among T C quarks (2.1) and exotic T C quarks (4.1 ).
It is useful to represent (2.1) and (4.1) as 14-component left- and right-handed fermion fields:
For gauge group (1.21, the left- and right-handed gauge fields are
+ ~ W ; ( X ) T ; / 2 + g ' ~ , ( x ) ~ L /2 (4.4)
-- G;, + G;, + W,,, + BL, ,
In the above, 0, (n = 2 , 8 ) are n X n zero matrices and
In this model there are 195 Goldstone bosons. Among then 11 are absorbed by the gauge bosons. In the rest there are 26 color-singlets, including 3 isotriplets p',', 4 isodoublets ( B y 5 B , b y , U ) and ( v y 5 d , D y 5 d 1, and the corresponding antiparticles, and, in addition, P"=P?, 4
isosinglets: P; - Ey ,a, P: - Ey5B, P! --Ey,a( - P ! ) and p 5 - B y 5 d . The generalization of (2.25) of inclusion ( i =2,3,4,5 ) is straightforward.
In summary, the low-energy chiral Lagrangian pro- vides us with a reas...