VOLUME68, NUMBER 19 PHYSICAL REVIEW LETTERS 11 MAY 1992
Colored Pseudo Goldstone Bosons and Gauge Boson Pairs
R. Sekhar Chivukula, Mitchell Golden, and M. V. Ramana Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215
(Received 17 January 1992) If the electroweak symmetry breaking sector contains colored particles weighing a few hundred GeV,
then they will be copiously produced at a hadron supercollider. Colored technipions can rescatter into pairs of gauge bosons. As proposed by Bagger, Dawson, and Valencia, this leads to gauge boson pair rates far larger than in the standard model. In this Letter we reconsider this mechanism, and illustrate it in a model in which the rates can be reliably calculated. The observation of both an enhanced rate of gauge-boson-pair events and colored particles would be a signal that the colored particles were pseudo Goldstone bosons of symmetry breaking.
PACS numbers: 13.85.Qk, ll.15.Pg, 12.15.Cc, 14.80.Gt
Technicolor [l] remains an intriguing possibility for electroweak symmetry breaking. Typically, the techni-hadrons which are lowest in mass are the technipions. A technicolor model must have at least three technipions, because they become the longitudinal components of the W and Z. In general there may be others. In models where there are colored technifermions, there are colored technipions. The one-family model , for example, has 63 technipions, including some which transform as (8,3) under SU(3)coiorXSU(2)weak.
Colored technipions are easy to produce at a hadron supercollider. Interestingly, in technicolor models a pair of colored technipions can rescatter into the colorless ones. For example, in the one-family model the reaction (8,3)(8,3) (1,3)(1,3) can occur. Since the former par-ticles are easy to produce, and the latter are the longitu-dinal components of the gauge bosons, there will be a larger rate for gauge-boson-pair events at the Supercon-ducting Super Collider (SSC) or the CERN Large Had-ron Collider (LHC). This mechanism was recently pro-posed by Bagger, Dawson, and Valencia  to test elec-troweak symmetry breaking at a hadron supercollider.
We are left with an interesting possibility. The colored technipions are produced in large numbers at the SSC and LHC, and should be observable even in the worst case scenario in which they decay exclusively into light quarks and flavor tagging is useless . However, be-cause they are produced strongly, there is no obvious way to connect them to the symmetry breaking sector, and indeed a skeptic might argue that the colored scalars are unrelated to it. (On the other hand, the generic expecta-tion is that the technipions will mostly decay to the heavi-est fermions. Observation that the colored scalar decayed in this way would be an argument that it had something to do with flavor physics, and hence symmetry breaking.) It is the combination of their discovery with the observa-tion of a large number of gauge-boson-pair events which permits us to argue that the colored scalars are pseudo Goldstone bosons of the symmetry breaking sector.
Since they are approximate Goldstone bosons, the low-energy behavior of the technipions can be described by chiral Lagrangian techniques . Consider the lowest-
order, two-derivative, chiral Lagrangian for the one-family model: X2 = ( /
2 /4)tr[(D^) t(Z)^)] , where I =exp(2inaTa/f). Here Ta are generators of SU(8), and na are the 63 technipion fields. The derivatives above are gauge covariant, using the proper imbedding of the color and electroweak groups into the chiral S U ( 8 ) L X S U ( 8 ) / ? . We may fix / by noting that since there are four elec-troweak doublets condensing in this modelthree quarks and one leptonwe have Mw*=g2f2- From this we deduce that f = v/2, where v ~ 250 GeV.
This lowest-order chiral Lagrangian contains terms which allow the computation of the gg ZZ process in which we are interested. The computation is facilitated by the use of the equivalence theorem , which states that at energies large compared to the W mass, the am-plitude for a process containing external longitudinal gauge bosons is equal to the amplitude for the process with the longitudinal gauge bosons replaced by their swallowed unphysical Goldstone bosons. It is impossible to construct a gauge and chiral invariant four-derivative counterterm for the coupling of the gluons to the un-colored swallowed technipions, so the one-loop calculation of the gluon-gluon to the longitudinal ZZ state must be finite. This is the computation presented by Bagger, Dawson, and Valencia .
To how high an energy scale can we trust the calcula-tions using JL-p. Quite general considerations  show that in a theory in which the symmetry breaking pattern is S U ( A 0 L X S U ( A 0 * ^ S U O V ) K , the scale at which the calculations fail must fall as 1/V/V". Consider nanb
Kcnd scattering. We can calculate the SU(7V)|/-singlet s-wave amplitude using JLi :
where s is the usual Mandelstam variable. This ampli-tude grows bigger than 1 in magnitude at a scale A^/lnf/^fN, In the one-family model, this is a mere 440 GeV. Thus, the lowest-order chiral Lagrangian must fail at a very low energy, at a scale which would be easily reachable by the SSC or LHC.
It is now easy to see what goes wrong when we calcu-late gg ZZ using JLi. Consider the computation of the
1992 The American Physical Society 2883
VOLUME 68, N U M B E R 19 P H Y S I C A L R E V I E W L E T T E R S 11 M A Y 1992
imaginary part of the amplitude. (The amplitude given in  has no imaginary part because that paper con-sidered the case of massless colored technipions. Colored technipions are expected to have a mass in the hundreds of GeV.) Only when the two gluons produce a pair of on-shell technipions can the diagram be cut in such a way as to give a physical intermediate state, and therefore the imaginary part of the amplitude comes from the produc-tion of on-shell colored technipions which rescatter in-to the swallowed Goldstone bosons. However, as we have noted, this rescattering will violate unitarity. The SU(8)|/-singlet spin-0 part fails first, at 440 GeV. We therefore conclude that the lowest-order chiral Lagrang-ian calculation of gg ZZ displays bad high-energy be-havior.
The failure of the chiral Lagrangian to address the scattering of technipions in the one-family model at the requisite energies leads us to consider a simplerbut calculablemodel which resembles technicolor. Our toy model of the electroweak symmetry breaking sector is based on an O(TV) linear a model solved in the limit of large N . It has both exact Goldstone bosons (which will represent the longitudinal components of the W and Z) and colored pseudo Goldstone bosons. To this end let N =j + n and consider the Lagrangian density 
X = l ( ^ ) 2 + l ( / ) y f ) 2 _ 1 2^2
2 _ L . 2 ^ 2 (2)
where 0 and yr are j - and ^-component real vector fields. This theory has an approximate Oij + n) symmetry which is broken to O ( y ) x O 0 i ) as long as iio^nov If Ho* is negative and less than no, one of the components of 0 gets a vacuum expectation value (VEV), breaking the approximate O(TV) symmetry to OiN 1). With this choice of parameters, the exact Oij) symmetry is broken to Oij 1) and the theory has j \ massless Goldstone bosons and, at tree level, one massive Higgs boson. The Oin) symmetry is unbroken, and there are n degenerate massive pseudo Goldstone bosons. We will consider this model in the limit that j,n-+ with j/n held fixed.
The scalar sector of the standard one-doublet Higgs model has a global 0 ( 4 ) ~ S U ( 2 ) x S U ( 2 ) symmetry, where 4 of 0 ( 4 ) transforms as one complex scalar dou-blet of the S U ( 2 ) ^ X U ( 1 ) K electroweak gauge interac-tions. We will model the 0 ( 4 ) of the standard model by the Oij) of the Oij + n) model solved in the large j and n limit. Of course, 7 = 4 is not particularly large. Nonetheless, the resulting model will have the same qual-itative features, and we can investigate the theory at moderate to strong coupling .
We have gauged an SU(3) r subgroup of Oin), SO the y/ fields are colored. [Technically, this gauging of the color symmetry for ip and not breaks the OiN) by a
dimension-four operator. We are free to ignore this if we regard as as small compared to X and accordingly neglect diagrams with loops involving gluons.] We have chosen y/ to be three color octets, analogous to the (8,3) of the one-family model. Our choice corresponds to n =24 .
A simple trick  for the solution of this theory to leading order in \/N involves introducing a new field #, and modifying the Lagrangian to
x-*x+ 1 J V 2 XQ XQ
VOLUME 68, NUMBER 19 PHYSICAL REVIEW LETTERS 11 MAY 1992
agator, Dxx(s)y including all radiative corrections coming from loops of 0's and v 's. Postponing for the moment the evaluation of Dxx(s), we see that this process is very easy to calculate, since the gy/y/ vertex is just the ordinary cou-pling of a gauge boson to a scalar, and the couplings of the x t 00 ad a r e both just /. We find that the sum of the diagrams in Fig. 1 is
\6x2 g' iiV 2p$p\ 2n*C*I(s,m2)Dxx(s). (5)
Here p\ and p2 are the momenta of the two incoming gluons; their polarization vectors are associated with the indices p and v, respectively. The number of octets in y/ is n%\ as we noted above, we have chosen wg ~ 3 . The fac-tor C% denotes the Casimir operator of an SU(3)r octet, which is 3. The variable s is 2p\-p2, and the function lis,m^) is a Feynman parameter integral
I(sym2) = Vdx f %dy 2
XyS . J o Jo m$-xys-ie
two subtractions are sufficient to