Volume 169B, number 1 PHYSICS LETTERS 20 March 1986

THE W E S S - Z U M I N O LAGRANGIAN AND COLORED T E C H N I - P S E U D O - G O L D S T O N E B O S O N S

Douglas W. McKAY

Department of Physics and Astronono'. Umversttv of Kansas. l.awrence. KS 66045, USA

and

Bing-Lin Y O U N G

Ames Laboratory and Department of Physics. Iowa State Universt(v. Ames, IA 5(X)I I, USA

Received 21 October 1985

The construction of the Wess-Zumino type effective action is discussed for color octet techni-pion and techni-eta fields interacting with the light gauge bosons - gluon, photon. W * and Z. The explicit effective lagrangian for the one-pseudoscalar meson sector is displayed. F(r/---, GWW). I'(,/---, G G y ) and I'(r/--, GGZ) are compared to I ' ( r / ~ GZ) to illustrate the predictive content of the lagrangian.

Although hints of new physics in the first round of reports on CERN collider data analysis ,1 have, except for possibly monojects, [2,3] largely disappeared, ex- pectations that striking signatures of new physics might be seen at higher energies have been raised. Faced with prospects of new physics in the multi-TeV region, one would like to have some framework, such as that of effective lagrangians, which summarizes the contents of theoretical "new physics" ideas whose ef- fects turn up in this energy region. Such lagrangians are useful tools for surveying testable consequences of theoretical notions. In particular a lagrangian involv- ing heavy, new particles and the light gauge bosons (gluon, photon, W ~ and Z) is of considerable value. The anomaly - related effective lagrangians of the Wess-Zumino type [5,4] are notable for their tight constraints among particle couplings. We develop here the anomalous effective lagrangian in the technicolor scheme ,2 involving colored pseudo-Goldstone bosons and the light gauge bosons. The colored techni-pseudo-

,1 Reviewed for example in ref. [1 ]. ~:2 For a review see ref. [6].

0370-2693/86/5 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Goldstone bosons are among the new physics possibil- ities currently entertained. Our approach affords a full description of the interactions exemplified by the triangle graph anomaly calculations in early technicol- or discussions [7-9] .

A few remarks are in order concerning the use of low energy techniques in the TeV energy region. In spontaneously broken chiral theories, which impose low energy theorems among pseudo-Goldstone boson amplitudes, the energy scale is set by the scale of the spontaneous global chiral symmetry breaking, com- pared to which the pseudo-Goldstone boson masses are small. In general, "low energy" means the energy regime below the symmetry breaking scale. Therefore the chiral symmetry low energy theorem techniques can be expected to work in technicolor physics involv- ing the decays of pseudo-Goldstone-bosons. These bosons have masses on the order of a couple of hun- dred GeV, small compared to the breaking, or con- densation, scale, which is of the order of 1 TeV. The calculations of n o -+ 9'3' [ 10] and K~3 [ 11 ] using, re- spectively, anomalous and normal current algebra are classic examples of these low energy theorem tech-

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Volume 169B, number 1 PHYSICS LETTERS 20 March 1986

niques applied in the region below the QCD condensa- tion scale. There has been a recent renewal of interest in the problem of constructing Wess-Zumino effective lagrangians [5 ,12-15 ] which incorporate all o f the anomaly content o f chiral symmetry low energy the- orems, and in the problem of extracting low energy

QCD phenomenological predictions from them [ 13, 16]. The predictions have been quite successful on the whole.

In extending the Wess-Zumino approach to a 1 TeV mass scale, our considerations are guided by the often discussed one-family technicolor model [6], where the condensation scale is about 1 TeV and the decay constant, determined by the weak-boson masses, is in the range below 300 GeV. However, since colored pseudo-Goldstone-bosons are expected to occur in any technicolor theory and in compositeness schemes as well, the problem of anomaly-related Wess-Zumino lagrangians for such objects is more general than the specific class of models referred to here.

We focus on the techni-pion and techni-eta, both color octets, which are flavor triplet and singlet, re- spectively, and which are expected to have masses in the 2 0 0 - 3 0 0 GeV range. The production and two- body decay modes of colored particles have been dis- cussed by several authors [ 7 - 9 ] emphasizing striking decay features [17] and recently reviewed in surveys of prospects for new physics in the multi-TeV region [18,1]. However, to our knowledge a systematic ap- proach to the anomaly-driven interactions leading to multiparticle f'mal states for pseudo-Goldstone boson decays has not been presented before ,3.

We are primarily interested here in obtaining the features of a single pseudoscalar boson interacting with the light gauge fields. We discuss below the develop- ment o f the terms in the effective action using the fermion loop approach [ 15 ] and we obtain all the anomaly-related couplings in the one pseudoscalar sec- tor. This sector involves up to four gauge fields. Some of the three-body modes are of significant strength while others are severely suppressed. These features provide new phenomenological test of the technicolor interpretation o f possible new physics.

Consider a non-linear transforming pseudoscalar field

*3 For an effective lagrangian treatment of colored vector bosons, see ref. [19].

where h a, a = I, 2 ..... 8 are color SU(3) Gell-Mann matrices and r i, i = 0, I, 2, 3, are flavor SU(2) Pauli matrices with r 0 = 1. We will denote the techni-pions and e t a s b y ~ a - a - -~i , i = 1,2, 3, and r~ a = q~, respec- tively. A color (3,3) and flavor (2,2) can be written as

M = exp(2i ¢r),5/FT) ,

where F T is the analog off~ r and is related to the mass o f a techniforce-induced condensation. In standard examples, F T ~ 100-200 GeV as determined by its relationship to the weak W and Z boson masses*4. We adopt the following intraction lagrangian of heavy quarks, Q, where Q and Q bind to form the ¢ bosons and other states as a result of the Q interaction with technicolor gauge fields, where the gauging of an SU(3)c X SUE(2), × U y ( l ) subgroup of the global chiral groups is displayed ,s :

L = O(i~ - /aM +g31~ +g2 WL

+½gl YLI~L+½gl YRI~L+½gl YRI~R)Q, (1)

where G are normal gluons and W and B the SUL(2 ) X U(I) weak bosons, with

1 - i ' ( 1 ":s), • WL -

A sum over the Q hypercolor indices is understood,

YL = (2Qu - 1), YR = (2 Qu - 1) + 7-3,

and Qu is the heavy up, U, quark charge ,6. The techni- quarks Q are chosen to be in the fundamental repre- sentation of SU(3) and SU(2), as in many standard technicolor treatments [6]. After integrating out the Q fields in the functional integral, one can express the resulting fermion loop contributions as an effective lagrangian ,s ;

.4 With our normalization F T is 1 ].4t2 times that in the litera- ture, ref. [61.

• s For a discussion of this approach, which starts at an inter- mediate stage between the QCD lagrangian and the com- pletely gluon and quark "integrated" low energy lagrangian of composite Goldstone bosons and vector mesons [4,5}, see ref. [ 15 ].

• 6 Denoting the companion heavy neutrino and electron charges by QN and QN - 1, respectively, anomaly cancella- tion requites 3Qu - QN - 2 = 0.

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Volume 169B, number 1 PHYSICS LETTERS 20 March 1986

oo

Lerr= iTr 23 2 {(iO - U) -I n=l rl

X [p(M - 1 ) - g 3 6 --g2WL -- ½gl YL~L 1

- ~'gl YRIBR]) n, (2)

and the trace includes integrating over space-t ime variables and taking traces of Dirac and internal sym- metry matrices.

To extract the anomalous action, we keep only those terms which are proportional to euvxo and have the fewest possible derivatives. Furthermore, we are interested here in terms linear in the pseudo-Goldstone field. Defining

v~, = % + ½ ~u

with

GU - a ½ ~a , -g3Gu

~ -g2w,,- ~ ~ ½ ,~a +~[(2e,, -1) 1 +~3] X (A~ m + tan OwZO),

g',, ~ = o w) z °) = (W u, W u, - (1/cos

and defining

AU = ½g2W ~ ½r a ,

we can express our result for a single P vertex, where P = sin(2O/FT) , as

L~-iNTceu~xpfd4x Tr [ p ( a u V ~ _ iVuV~) 8rr 2

X (bhVp - iVxV p + 2iAxAp) ] . (3)

The interaction preserves the gluon and photon invari- ance. The three-point interactions in eq. (3) reproduce all o f the triangle anomaly results [6,7]. Spelling out the single pseudoscalar vertices, we have

- - -

16rr2F T

{rla rg2 dabb' ( oxGbo ' + ½ g3 fb'c'Cl'Gf Go d') X t 3

+ 8ba2eg3 (2Q u - 1) (~xA~ m + tan 0w~xZo) ]

0,a em + 2eg 3 11 (axA p - cot 20 w ~xZo)

+gzg3 (II+'b OxWo - + I I - ' b a~W~)

+ ieg2g 3 ( I I - ' b w ~ " _ II+,bW~ -)

X (A~ rn + tan 0wZo) ) .

iNTc

(4)

Here r/and rr refer to the SU(2) singlet and triplet of color-octet pseudo-Goldstones, respectively. Recall that eqs. (3) and (4) contain only those contact inter- actions with the fewest derivatives, in the spirit o f a low energy expansion. Note that a cancellation occurs between the AxA p and VxV p terms in eq. (3). Conse- quently there is an absence o f contact terms for the lr0W+W-G and , /W+W-G processes, which leads to a suppression of 7/--* GW+W - and 7r 0 ~ GW+W - decays. We have computed ~ --* GW÷W - , fig. I, and it is very small. We comment on this again below. From the kinetic energy lagrangian of the pseudoscalars, we get interactions with EW bosons

LEW = - i e A~m(n+aurt - _ aun+rr-)

+ ie cos 2 0w Z0 (Tr+0UTr - - 0un+rr - )

- ½ ig2W~ (rr- aUrr 0 _ aurr-rr 0)

- ½ ig2(rr0aUrr + - Ourr0rr+),

where a sum over color indices is undestood. The inter action with the gluon takes the form L G = - g 3 f abc

W +

W -

Fig. 1. Feynman diagram for the decay r? --* GW÷W -. Both -y and Z ° intermediate states contribute. Note that there is no contact term for this amplitude. The color octet techni-eta is designated ~ in the figure.

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Volume 169B, number 1 PHYSICS LETTERS 20 March 1986

X OunaTr b G c and a sum over flavor labels is understood. To illustrate the phenomenological implication o f our scheme, we choose the processes 7/-+ GWW, r / ~ ZGG and 7? "-* "),GG as examples o f striking jet physics in p~ final states, p~ production mechanisms for colored pseudoscalars have been discussed by various authors [7-9 ,17] and reviewed in a recent paper on new phys- ics [18]. The r/production by two-gluon fusion is promising [9,18]. We wish to compare the decay rates into three bodies to that o f the ZG two-body mode, which would have a very distictive two-lepton plus jet signature. The latter has already been covered in earli- er work [6]. The three-body final state GWW, which is very small in our scheme, would produce a jet plus lepton pair plus missing transverse energy. The ZGG mode, which has an appreciable branching ratio com- pared to the ZG mode, could give two jets plus lepton pair, merging to the single-jet signal for one soft gluon. The 3,GG process is computationaUy almost identical to the ZGG one, and we include it in order to study phase space and coupling dependence of the rates. The appropriate Feynman graphs are shown in figs. 1 and 2. The contact term in fig. 2a is necessary to ensure the gauge invariance with respect to the gluons, and this is built into out construction of the effective lagrangian. There are both infrared and collinear di-

G 2 ------

P t

G2

(o) (b)

_ _ ~ . ~ ~ ~ Gz ~ / ~ 2 G t

)',Z " Z

( c ) ( d )

Fig. 2. Diagrams for the decays r/--* GG7 and GGZ.

vergences in the gluon momenta in the amplitudes of fig. 2, and we introduce a cutoff in the invariant mass of the gluon pair, M(GG), to regulate these divergences. Note that this is a reasonable cutoff, as all gluonic jets have finite masses. In fig. 3 we plot the ratios P0? ~ GGZ)/F(n ~ GZ) r ( n + GZ) and P(~ ~ GGT)/F(~ -+ GZ), respectively, as functions of the two gluon in- variant mass M(GG). Two ~ mass values, 150 GeV and 250 GeV, are chosen for illustration. The 07 ~ GGZ)/ 07 ~ GZ) plots show strong dependence on M(GG), falling by an order of magnitude from 0.76 to 0.067 asM(GG) ranges from 47 GeV to 110 GeV in the case M n = 250 GeV. The sharp suppression of the ratio in the M n = 150 GeV case illustrates the importance of the Z-mass in limiting phase space when M n is only 1 . 5 - 2 . 5 times M z. We want to stress that the GGZ

mode is appreciable compared to GZ up to substantial

5 O O

OJ °~oo

t

c. .

i | i l , l I t t i l

• " "~m~= , ~ ' ~ v ~ - . . . GeV " ~

Z,m9 = 150 GeV

\ \

\ \

\ \ \ \ \

l i l l l I A I I | I I I ~ ~ ~ I10

- GLUON INVARIAN,T MASS CUT

Fig. 3. Decay rates of r(n -+ GGT)/F(n ~ GZ) and F(n GGZ)/I'(rl ~ GZ) versus the two-gluon invariant mass cutoff in GeV. 3"he three-body modes are integrated over all allowed two-gluon inva.riant mass above the cutoff. Mass values Mr/= 250 GeV (solid curves) and M n = 150 GeV (dashed curves) are considered for each mode.

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Volume 169B, number 1 PHYSICS LETTERS 20 March 1986

values o f the cutoffM(GG). Therefore GGZ should be clearly distinguishable from GZ experimentally.

The ( r / ~ GG.y)/(r/--> GZ) ratio is much larger than the corresponding (7/--> GGZ) one, as illustrated in fig. 3. For the lower cutof fM(GG) ranging between 43 GeV and 110 GeV, the ratio is seen to fall from 4.7 to 0.5 in the M n = 250 GeV case. The phase space ef- fects caused by the Z mass are exemplified again by the M n = 150 GeV case, where the ratio (77 -+ GG.r)/ ( r /~ GZ) is higher than for M n = 250 GeV at the low end of the M(GG) range because o f the suppression of F(rt -+ GZ) due to M z ~e 0 at this relatively small value ofMn. The three-body 77 ~ GG is larger than

or comparable to the 71 -+ GZ for M(GG) values up to 75 GeV or so. Of course we are not applying experi- mentally realistic cuts nor addressing detailed ques- tions o f discriminating among final state jet signatures, but we feel that the sizeable ratios shown in fig. 3 and the sharp suppression of some modes like 77 -+ WWG (typically ~ I 0 -6 branching ratio)indicate some po- tentially useful diagnostics. As in the cases o f two-jet (gg or qF:t) and jet plus weak boson (g + W, Z, 7) de- cays o f techni-eta and techni-Tr [18], the pp and p~ production cross section times branching ratio o f the two,e l plus weak boson decay signal (ggZ or ggT) which we discussed above is small compared to the standard model estimates o f the background, [ 18,20] and distinction between gluon and quark jets [91 or be- tween different quark flavors [18] will be necessary to reduce background , 7 For example, combining the single-r/production cross section by gluon fusion [18] the 0.3% branching ratio o f r/--> gZ and F(r/--> ggZ)/F(r/-+ gZ) with mn = 250 GeV and Mgg = 80 GeV from fig. 3, we estimate o(p~ ~ 72 + X ) ' BR(r/ ggZ) = lpb at x/s = 20 TeV, whereas a recent calcula- tion o f o(p~ ~ ggW + X) [20] where Mgg = M w = 80 GeV, in the standard model leads us to a background estimate o(p~ ~ ggZ + X) ~ 2ng , s .

Pair production of technibosons by gluon fusion is a possible application o f our analysis o f multi-body decays modes, when combined with other decay mode signatures, to test the technicolor interpretation o f a

,7 Recent progress in statically separating quark and gluon jets by the UA1 group is described in DiLella's review of ~p physics at the Kyoto Photon-lepton conference.

*a We have estimated a factor 2/3 in going from W + two jets to Z + two jets by comparing W + X and Z + X in p~ as discussed in ref. [ 18 ].

new physics signal. In the cases pp ~ rr + ~ + X or *7 + rl + X, say, one could look for events with striking back-to-back jet activity. If heavy flavors could be tagged, then a signature for the rr0 channel could be t t on one side and ggZ or gZ on the other side, lead- ing to two jets opposite two jets plus missing momen- tum. Similarly, the rr+rr - channel could leave a sig- nature tb on one side and W- g or W-gg on the other side. The W-gg mode is free from the competing background from heavy quark, Q, to light quark plus W(Q ~ qW), which also gives jet plus W. As in the r/ -+ Zgg case, Ir ~ Wgg is comparable to 7r --> Wg and gives an extra handle on techni-Tr decays.

In summary we have developed the Wess-Zumino lagrangian containing the anomaly-driven techni- Goldstone boson contact interactions necessary to compute 7r and rt decay to multi-gauge boson final states. Illustrating our study with several r/decays, we have concluded that detailed multi-gauge boson fea- tures o f the final states in colored techni-pion and techni-eta decays might serve as useful discriminators to determine whether or not new physics signals are interpretable as technicolor signals.

Tiffs work is supported in part by the Department o f Energy under contract numbers W-7405-Eng-82, No. KA-01-O1 and No. DE-ACO2-79ER 0528 and Grant No. DE-FG02-85ER 40214. This work was begun while we were visitors at Fermilab in the sum- mer of 1984, and we thank H. Thacker, W. Bardeen and the rest o f the Theory Group for their hospitality and support.

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[8] J. Ellis, M.K. Gaillard, D.V. Nanopoulos and P. Sikivie, Nucl. Phys. B182 (19~,1) 529.

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