Zero-parameter calculation of the low energy Nambu-Goldstone boson elastic scattering in quantum chromodynamics and in technicolor-like theories

Physics Letters B 273 ( 1991 ) 292-300 North-Holland PHYSICS LETTERS B

Zero-parameter calculation of the low energy Nambu-Goldstone boson elastic scattering in quantum chromodynamics and in technicolor-like theories

T r a n N. T r u o n g Centre de Physique Thdorique de l'Ecole Polytechnique, F-91128 Palaiseau, France

Received 26 July 1991

Within the framework of the quantum chromodynamics, the low energy pion-pion elastic scattering is calculated without introducing free parameters apart from the number of colors and the pion decay constant. The basic assumptions are the spontaneous chiral symmetry breaking, the derivative expansion of the QCD effective action, and the elastic unitarity condition for the partial wave amplitudes. Good agreement with the experimental data is obtained. Using the same method in either the technicolor model or the Nambu-Jona-Lasinio fourth family model, the elastic WW scattering is calculated without introducing further parameters.

The s tandard model is verified to a great accuracy by recent exper iments at LEP. The symmetry breaking of the electroweak interact ion is, however, least understood because its effect is insensit ive to these low energy experiments. Next generat ion experiments at LHC and SSC will certainly provide valuable informat ion on the Higgs sector. I f the Higgs boson is sufficiently heavy, the study of the longitudinal vector boson scattering can provide valuable informat ion on the spontaneous symmetry breaking mechanism of the s tandard model. The theoret ical study of this process is a challenge to theoret icians because the s tandard per turbat ive field theoret ical methods have to be modif ied due to the strong interact ion of the heavy Higgs sector.

The purpose of this note is twofold: We ul t imately want to study the strongly interact ing Higgs problem, be it an elementary field or a composi te object such as the technicolor scheme, which is an extension of the quantum chromodynamics ( Q C D ) theory of the hadron physics, or a consequence of the fourth family scheme which is recently discussed in order to avoid the fine tuning problem associated with the top quark scenario for the Higgs boson. To check our calculation scheme, we are forced to answer the corre-

Bitnet address: PTHTNT @ FRPOLY 11.

sponding QCD problem at low energy where experimental data are plentiful: Is it possible to calculate the low energy p ion -p ion scattering problem up to 1 GeV within the f ramework of the QCD without in- t roducing extra free parameters except the number of color Arc and the pion decay constant f ? The purpose of this note is to take up this challenge to calculate the low energy p ion-p ion scattering problem and then generalise it to W W scattering for LHC and SSC physics.

There is now little doubt that QCD is the underly- ing theory of the strong interaction. For hard pro- cesses such as the calculation of the hadronic width of the Z, the accuracy of the theoretical study is im- pressive, of the order of a few percent. For soft pro- cesses such as the hadron spectra, form factors, the numerical approach of the latt ice gauge is encourag- ing. It is doubtful that, in the foreseeable future, the lattice gauge calculation can provide detai led informat ion on the p ion -p ion scattering problem due to the euclidean nature of the calculation.

This work is inspired by the pioneering work o f Lehmann [ 1,2 ] who a t tempted to calculate the low energy two-pion problem using chiral per turbat ion theory for the non-l inear a model ( N L a M ) lagrangian, with the nucleon loop as the driving force to generate the low energy dynamics. Because of the use

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Volume 273, number 3 PHYSICS LETTERS B 19 December 1991

of the non-renormalisable NLaM, two undetermined parameters are needed to regularise the theory at the one-loop level. To the extent that the Nambu-Gold- stone boson loop can be neglected, he showed that the nucleon loop can generate the p resonance. The work of Lehmann was later extended by Jhung and Willey [ 3 ] who considered the NLaM as a large a mass limit of the linear a-model (LaM) [4], an approach suggested previously by Bessis and Zinn-Jus- tin [5 ]. The work of Jhung and Willey contains es- sentially also two parameters, the a mass and the axial coupling constant gA as adjustable parameters. An excellent fit to the low energy data was obtained with m , = 1.3 GeV and gA= 1.31 instead of 1 as given by the chiral lagrangian. In this article, we calculate mo to be 1.2 GeV and show that the number of color in QCD, No=3 gives rise naturally to the value g] = ( 1.31 )4 = 3 as needed in the work of Lehmann [ 1,2 ] and Jhung and Willey [ 3 ].

In the following we provide an analytical approach to the QCD problem, using the derivative expansion of the QCD effective action and assuming that the quark-gluon interaction gives rise to the spontaneous chiral symmetry breaking. Within this framework, the one-loop pion-pion scattering amplitude is calculated perturbatively [ 1,3,5,6] and then it is re- summed by the Pad6 method in order to impose the exact elastic unitarity condition instead of the usual perturbative unitarity. The elastic unitarity condition is crucial if one wants to use the perturbative calculation not only near the physical threshold, but also at higher energy where interactions are strong and are dominated by resonances. It is hopeful that the success of this calculation, together with the success of the previous pion form factor calculation [7] will convince the readers of the importance of the unitarisation which is neglected in many recent articles about WW scatterings.

It is assumed here that the quark-gluon interaction in QCD is such that the chiral symmetry is sponta- neously broken which leads to the following O(p 2) non-linear model (NLaM) chiral lagrangian (p being the pion momenta):

Lo =¼ Tr(0~, U0~ U t ) , ( 1 )

where U= exp [ina (x)Za/f], ~ is the pion field,f= 93 MeV the pion decay constant, and Za are the usual Pauli SU (2) matrices. It is assumed furthermore that

the low energy pion-pion interaction near the two- pion threshold can be adequately described by in- cluding the O(p 4) terms. There are, in general, two such terms:

L~ = ~2E Tr( [0u UU*, O~ UU* ] 2)

+½G [Tr(0u U0uU*) ] 2 . (2)

The derivative expansion of the QCD effective action at large Arc, integrating out the quark field gives [8]

E=Nc/12n 2, G=Nc/48n 2. (3)

These results are just the nucleon loop calculation by Lehmann [1,2] and Jhung and Willey [3] adapted to QCD with gA = 1 and with the color factor N~ taken into account.

Writing the pion-pion scattering amplitude as T(s, t, u) =A(s, t, U)~ab~Sca+ .... the tree level amplitude computed from eqs. ( 1 ), (2) and using eq. (3) is

A ( s , t , u ) = ( s / f 2)

+ (Nc/48x2f 4) ( - - S 2 + 1 2 + U 2 ) . (4)

Eq. (4) is also valid in the large Nc limit because the loop effect due to the Nambu-Goldstone boson intermediate states is suppressed by a factor of 1/Nc, compared with the fermion loop contribution and hence can be neglected, the low energy scattering amplitudes in this limit are just polynomials.

It is our fundamental assumption here that the fermion (quark) loop generates the low energy dynamics. Our main task is therefore to construct a pion- pion elastic scattering amplitude which has the correct analytic property, satisfies the elastic unitarity relation, and in the large Nc limit, must satisfy the low energy theorems as given by eqs. ( 2 ) - (4). Be- cause the large N~ limit is likely not to be unique, we must be guided by the conventional field theoretic perturbative scheme in the construction of the non- perturbative solution. We can either use the NLaM or the LaM as our basic lagrangians, because they have the correct chiral properties. The NLaM, being non renormalisable, introduces an extra parameter for the S-wave amplitudes which we cannot determine.

We discuss first the LaM approach which is renormalisable and hence there is only one parameter in the problem, the a mass, m~. It will be shown below

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that m,, can be determined from eq. (3), in the large Arc limit, and because mo is quite large, m,,= 12nf/ N ~ 1.2 GeV, we can use the large m~ limit of the I_~M for the scattering amplitude which simplifies consid- erably the calculation. Because m,, is large, it plays only a role of giving a scale to the logarithm term and hence the calculated scattering amplitudes depend insensitively on its value and on the 1/N~ approximation.

The result for the one-loop amplitude, in the chiral and large m~ mass limit is

A(s, t, u) = ( s / f 2) + ( 1/16n2f 4)

1 2 X [ - ~ s l o g ( - s / m Z o ) - ~ t ( t - u ) l o g ( - t / m 2)

- ~ u ( u - t ) l o g ( - t / m 2) +c,s2 +cztu+ ½Nc

X ( - s2+tZ+u 2)]+s2/ f2m 2, (5)

where c , = ( 3 x / ~ ) n / 2 - z ~ , c2 =4, with No=3 for QCD and where we add the quark loop contribution to the one-loop result of the large mo limit of the L~M without fermions[3,9,10], m. is defined by refs. [9,10 ] as the position where the real part of the in- verse ~ propagator vanishes. Because mo is not sufficiently large, we must also take into account the O(p 4) term coming from expanding the ~ propagator as represented by the last term on the RHS ofeq. (5). It is legitimate for use to use eq. (3) in (5) because the O(p 4) terms in the QCD effective action constitute the convergent one-loop amplitude. There is no free parameter in eq. (5) except mo which is to be determined below, using the 1/N~ approach.

There is, however, a problem with eq. (5) because it does not satisfy our basic assumption that only the derivative expansion of the effective action eq. (3) gives rise to the O(p 4) term. To see this, it is suffi- cient to calculate the tree amplitude of the x°x°~x°x ° process, using the Lc~M and also taking into account the quark loop contribution to the O(p 4) terms,

T(x°Tr°-~ x°x ° )

=(s2+t2+u2)[1/ ( f2m~)+N~/48~2f 4] . (6a)

The first term in square brackets on the RHS of eq. (6a) comes from the expansion of the ~ propagator, the second term from the fermion loop contribution which has the Nc factor. Eq. (6a) can be considered as the large Nc limit of a more complete one-loop amplitude eq. (5), where the boson loop contribution,

which does not have the Arc factor, is neglected. (It should be noticed t h a t f 2 is of the order of Nc.) Sim- ilarly we can calculate the same amplitude, using eq. (4),

T ( x ° x ° ~ x ° x °) = (s2+t2+u 2) (Nc/48~2f4). (6b)

Comparing eq. (6a) with (6b), it is seen that con- sistency is only obtained if mZ~>>48~zf2/Nc, which is not the case as will be shown below. Hence we have to adopt the prescription of subtracting out the O (p4) term s2 / fZm 2 from the expression ofA (s, t, u) calculated from the L~M lagrangian to avoid the double counting problem. It should be pointed out that, in a different motivation and assumption, Bessis and Zinn-Justin [ 5 ], and later Jhung and Willey [ 3 ] gave the same prescription of neglecting terms of the order 1/rn~ in their study of the NL~M, using the large m~ limit of the L(~M as a regulator.

It is useful to carry out this analysis further to calculate the scale parameter m~. Explicitly for this purpose we introduce the P field in the LcM lagrangian, anticipating that the quark loop approach will produce the P resonance and that the KSRF relation [ 11 ] for the p width will be shown below to be valid independent of whether the 1/mR term is taken into account or not. We have [ 12 ]

A(s, t, u) = ( s / f 2) 2 2 [ m o / ( m o - s ) ]+ (1 /2 f 2)

X [ t ( s - u ) / ( m 2 - t ) + u ( s - t ) / ( m 2 - u ) ] , (7)

where the Weinberg chiral-invariant P field lagrangian was used [ 13 ]. It should be noticed that there is no quark loop contribution in eq. (7), because taking it into account would be double counting. Expanding the P and c propagators for s, t, u << m~, mZwe have

A(s, t, u)= ( s / f z) +s2/f2m 2

+ ( -2s2 + u2 +t2) /2 f2m 2. (8)

Comparing this equation with eq. (4) we arrive at the following relations:

m 2 =48~2f2/Nc, (9a)

m 2 =24~2fZ/Nc . (9b)

These results should be considered to be valid in the large Nc limit. In this limit, resonances would appear as poles and because f 2 is of the order Arc, both m o

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and m~ are finite. Eq. (9b) was previously given by Aitchison, Fraser and Miron and others [ 14] under the assumption that the KSRF relation holds. Under some assumptions, the relation m~ =2m~ was previously derived by Hung, Pham and Truong and Aitchison, Fraser and Miron [ 14 ]. Eq. (9a) will only be used to set the scale for the logarithm term, the radiative correction for eq. (9b) will be given below.

This discussion leads to the conclusion that a correct phenomenological description of the low energy p ion-pion problem can be done either with the NL(~M as discussed in refs. [ 1-3,15 ], or with the full L~M, but the p field must be included by hand in the lagrangian. The failure o f the pure LoM or the pure p dominance model done in the early 70's [ 1,3,5,6 ] can then be easily understood [ 12 ].

In the following, we do not use the p lagrangian in combination with the L~M model leading to eq. (7), because it is not renormalisable. Instead we want to generate the p meson dynamically and therefore we want to use the result ofeq. (5) with the scale parameter m~ given by eq. (9a). The large Nc limit is only used to determine the scale mo of the logarithm term. The final result for the p ion-p ion scattering amplitude depends weakly on the value of the G mass and hence weakly on the validity of the 1/Nc expansion.

After projecting out the eigenstate of the isospin and the partial wave amplitudesfzj, dropping the last term in eq. (5), and using eq. (9a) for the scale of the logarithm term, we have

foo = (s /16z~f2){1 + (s /16z~zf 2)

× [ - ~ log ( s N c / 4 8 g 2 f 2) +

_[_~Cl __~C21 + ~N~+iTr]} , (10a)

f ~ = (s /96zr f2){1 + (s /167rz f 2)

x [ - 3 _ (c , + c : ) + -:3 N¢ + liTr] }, ( 1 0 b )

f2o = ( - - s /32zr f 2){ 1 -- (s/16z~2f 2 )

× [ - ~ l o g ( s N d 4 8 z t 2 f 2) -~ 1~8

-]- 2Cl --C2 "~- -~Nc+ ½izr ] }. (10c)

It should be noticed that the P-wave amplitude is independent of the scale mo while the two S-wave amplitudes do depend on this scale.

As emphasized above and in the previous works [1,3,5,6], it is important to unitarise these partial

wave amplitudes since we deal with the strong interaction. For an energy up to 1 GeV, it is reasonable to neglect the inelastic effect due to the four-pion intermediate state in the unitarity relation as it is of the order O (p 8) (the square of the four-pion production amplitude is of the order p4, the four-body phase space factor is proportional t o p4) or the terms of the order O (p4) due to KI~ intermediate state because of their high mass; this assumption is also shown to be valid using the experimental data. We can use any unitarisation scheme as long as it does not introduce a spurious singularity which destroys causality. For our purpose, it is most convenient to use the Pad6 method which consists of writing the diagonal [1,1] Pad6 approximant:

f [l"l l l j = (fljtree) 2 / (fijtree--fijl°°P ) ,

f t l,~lj satisfies the elastic unitarity, I m ( f t l,mlij ) = I f t 1,1 l j [ 2. The n~ phase shifts calculated from this unitarisation scheme are

tan ~oo = (s /167cf2){ 1 - ( s / 1 6 ~ 2 f 2)

× [ - ~ 8 1 o g ( s N ~ / 4 8 n 2 f 2 ) - O . 5 2 7 + ~ N ~ ] } -~ ,

( l l a )

tan ~11 = ( s /96zr f 2) [ 1 - (s /16z~zf 2)

X ( - 0 . 2 1 8 + ~ N c ) ] -1 , ( l l b )

tan ~2o = - (s /327rf2){1 - (s /167r2f 2)

× [ ~ l o g ( s N c / 4 8 7 r 2 f z ) + O . 4 0 1 - 2 N 3 c]) - l ,

( l l c )

where we have used the numerical values for c1 and c2 in eqs. ( 11 ).

Let us first discuss the more reliable P-wave solution which is independent of m,, the scale of the logarithm term. This simple result, as given by eq. ( 1 lb) , is due to the cancellation of the logarithmic term contribution to the right-hand cut (unitarity cut) and that from the left-hand cut (exchange cut). This simpli- fication exists only in the chiral limit. By giving a small mass to the Nambu-Golds tone boson we can separate these two cuts and show that the Pad6 unitarisation scheme does not introduce any spurious singularity which violates causality. (It is known in the literature that the standard K-matrix unitarisation scheme for the P-wave amplitude eq. (10b), does

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Volume 273, n u m b e r 3 PHYSICS LETTERS B 19 D e c e m b e r 1991

not lead to a resonant solution, this solution violates however causality by introducing a pair of complex conjugate poles in the physical sheet and very near to the physical region, and should therefore be re- j ec ted) . F rom eq. ( 1 l b ) , we get

m ~ = ( 1 6 n 2 f 2 ) ( - O . 2 1 8 + 2 N ¢ / 3 ) -~ , (12a)

which is numerical ly equal to 870 MeV with Nc = 3 as given by QCD, this value can be favorably compared with the exper imental value of 775 MeV. Eq. (12a) differs from eq. (9b) by the numerical constant - 0 . 2 1 8 which is due to the radiat ive correction. The p width satisfies the K S R F relat ion [ 11 ],

Fp = m 3/96z~ f 2 (12b)

and is reasonably well satisfied by the exper imental data, using the exper imental value for the p mass. F rom eqs. (10b) and ( 11 b) , it should be clear that the val idi ty of the K S R F relation is independent o f whether a O ( p 4) term due to a heavy field such as the o is added to the RHS of eqs. (10b) , ( 1 l b ) . This is so because we would have

tan 6,1 = (S/967~f 2) [1 +s/m2o

- (s /161rz f 2) ( - 0 . 2 1 8 + 2Nc) 1 - ' , (13)

hence the K S R F is still val id as long as the P-wave phase shift passes through 90 °, but the value of m o is changed. We were therefore jus t i f ied in writ ing down eq. (7) .

The two S-wave ampli tudes are shown in figs. 1 and 2, where we have mul t ip l ied the RHS of eqs. (1 l a ) and ( 1 l c ) by the phase space fac torp(s ) = ( 1 - 4 m ~ 2 / s) ~/2 where m~ is the pion mass, to take into account the threshold effect. It can be seen that the agreement with the experimental data is good. Unlike the P-wave solution, the two S-wave ampl i tudes have a pair of complex conjugate poles. The I = 2 pole is sufficiently far away but the I = 0 pole is closer at s = 16/27[ 2 ( - - 1.22 + i 0.48 ) in the upper half plane which makes the calculated 6oo phase shift unrel iable above the P mass (this pole can be removed by mult iplying the Pad6 ampl i tude by an appropr ia te factor which is normal ised to unity at s = 0) .

There is, however, another problem with the I - - 0 S-wave ampli tude. If one takes instead the large Arc l imit in eq. ( 1 l a ) , one would discover that the 6oo phase shift passes through 90 ° only at m . 2 = 967r2f2/

I00

8o t

40

2O

, , , , , I i i ~ J

200 300 400 500 600 700 800

4s(MeV)

Fig. I. Comparison between the theoretical prediction and the experimental data for the S-wave I= 0 pion-pion phase shift 6oo (in degrees) as a function of the di-pion energy x/s (in MeV); the solid curve is the zero parameter theoretical prediction by the LaM, the dashed curve is the one parameter calculation of the NLaM, using m,,= 1.2 GeV as the scale of the logarithm.

-80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-50

-40

0 -30 0,/

- 20

- 10 ~ ~ - -

200 300 400 500 600 700 800

4s(MeV)

Fig. 2. Comparison between the theoretical prediction and the experimental data for the S-wave 1= 2 pion-pion phase shift 62o (in degrees) as a function of the di-pion energy x/s (in MeV); the solid curve is the zero parameter theoretical prediction by the LoM, the dashed curve is the one parameter calculation of the NLoM, using rn~= 1.2 GeV as the scale of the logarithm.

Nc, which is a factor 2 larger than that given by eq. (9a) . The discrepancy between the values mo de- duced from the propagator and m* de termined by the part ia l wave ampl i tude is expected for a wide resonance (for m , = 1.2 GeV, the o width is 4.5 t imes its mass! ). Which value of mass is to be used for the scale of the logari thm? Eq. (5) instructs us to use m , instead of m*. We must be prepared however to accept

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that there is a considerable uncertainty in determin- ing m, because the large Arc limit is far from being valid for the I = 0 S-wave, as can be seen from eq. ( 1 la) ; the situation is better for the 1=2 state as can be seen from eq. ( 11 c). To have a feeling, let us use m~ for the scale of the logarithm instead, the modification for the phase shifts is not large, 600 increases by 20% and 620 decreases by 10% which are accept- able.

We now want to discuss the NLcM approach. The one-loop NL~M calculation was first done by Lehmann [ 1,2 ] and was later reproduced by many authors [ 1,3,5,6 ]. In the following, we discuss the one loop calculation in the language of the dimensional regularisation. The Pad6 partial wave amplitudes are given by

foo(S) = (S/16~zf 2)

× ( 1 -- ( s / 4 f2 ) { -- 4ER(v) + ~GR(U)

+ ( 4 n ) - 2 [ U - - ~ l o g ( s / v Z ) + i 4 n ] } ) - ' , (14a)

f l l (s) = ( s /96n f2 ) { l -- ( s / f 2) [3ER(v)

- G n ( v ) + ( 4 n ) - 2 ( - ~ 8 + - ~ i n ) ] } - ' , (14b)

fzo(S) = ( - s / 3 2 n f 2 ) ( 1 + ( s / 2 f2 ) {4Gn( u)

2 + gEn(u) + (4n) -2[ ~ _ .~log(s/u2) +in] }) -1

(14c)

where En(u) and GR(u) are the renormalised parameters which depend on a scale u; the physical quantities are however scale independent. There are two parameters in this model which are related to the divergences of the pion loop integrals. For further discussions see for example ref. [ 15 ]. Similarly to the calculation of Lehmann [ 1,2], there is no logarithmic term in the P-wave equation. It is then simple to see that the combination 3 E n ( u ) - G n ( u ) is independent of the scale u and that the p mass is given by

2 2 3 m p = f / [ - ~ E n ( u ) - G n ( u ) - l / 1 8 ( 4 n ) 2] (15)

and is independent of the scale u. In general En(u) and Gn(u) receive a contribution

from the quark and the pion loops. In the large N~ limit, we can neglect the pion loop contribution which has no N~ factor and takes only the quark loop into account. Because the quark loop calculation is a convergent integral, their contribution to En(u) and

GR(v) is independent of the scale and just given by eq. (3). Using these results in eq. ( 15 ) we now have

mp 2=24g2f 2/ (Nc-- l-!~) , (16)

which is numerically equal to 835 MeV; the P width satisfies the KSRF relation [ 11 ] as in the case of the LaM approach. Eq. (16) was previously given by Lehmann [ 1,2 ] and Jhung and Willey [ 3 ] with the factor Nc replaced by g4, using the assumption on the minimal growth of the one-loop amplitude. Within the validity of our approach, this assumption is not necessary. The radiative correction due to the pion loop contributes very little to the P mass as can be seen from eq. (16) (i.e. ~ compared with Nc due to the fermion loop). The P-wave solution is therefore similar to that obtained from the L~M calculation.

In the large Arc limit, En (u) and GR (U) of eqs. (14) are given by eq. (3), and are independent of the scale u. The S-wave amplitudes then depend on the scale u due to the approximation. We must consider u as one undetermined parameter. To have a feeling, let us as- sume that this scale u is given by the scale of the logarithm term of the L~M, u= 1.2 GeV as given by eq. (9a), then the S-wave phase shifts are in good agreement with the LoM calculation as can be seen in figs. 1 and 2. (An almost identical fit to the LaM prediction is obtained if we use the scale u = 1 GeV. )

An equality can be derived for a linear combination of the S-wave phase shifts such that the logarithm term cancels out. It is given by

cot 600 - ~cot 620

= 16nf2s- '{9 + ( s / f 2) [3En( u ) - GR(v) ]

+3s(256n2f2) -1}

= 1 6 n f 2 s - t [ 9 + s ( 2 4 n 2 f 2 ) - ' ( N c + 9 ) ] . (17)

Notice that the v scale independent combination ¼ER (v) -- GR (v) appears again in eq. ( 17 ). A similar relation involving S and P waves was previously given by Lehmann. Because this equality is dominated at low energy by the O(p 2) term, the presence of the O(p 4) term, which is proportional to Arc, can only be tested at a moderate energy where the theory is still reliable and the experimental data is reasonably good. We choose this energy v/s=0.7 GeV, where ~oo = 65 ° +5 ° and 62o= 14 ° +2 °. Using these values to evaluate eq. (17), we have the LHS equal to 3 + 0.38

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compared with the value 2.70 of the RHS; the equality is therefore satisfied. Without the quark loop contribution, the RHS is only 2.0 and in disagreement with the data. This analysis supports the evidence for the contribution of the O(p 4) term as given by the derivative expansion of the QCD effective action. (Using eq. ( 11 ), a similar equality can also be derived for the LcyM; one would obtain the same expression as that given by eq. (17) except that the factor Nc + 9 in parentheses on the RHS is replaced by Arc-0.04 and hence, to a good accuracy, there is no difference between the two models. )

Before discussing the application of this technique to the technicolor-like theory, it is useful to discuss to what extent it is possible to use the derivative expansion of the effective action for the low energy pion- pion scattering problem, i.e. how well can one justify the technique of integrating out the quark fields, because the up and down quarks are much lighter compared with nucleon masses? The answer to this problem lies in the quark confinement hypothesis, i.e. at the low energy region where this calculation is car- ried out, the quark-antiquark pair cannot annihilate into any other channel than the two-pion states. They effectively play the role of the heavier fermions and hence can be integrated out below the energy region where the inelastic effect becomes important.

It is useful to summarise now our main results. Both the Lt~M and NL~M with the quark loop contribution, using the derivative expansion of the effective action, produce the p resonance with the observed mass, its width satisfies the KSRF relation. This prediction is the most reliable one. The L~M prediction of the I = 0 and I = 2 S-wave phase shifts with our prescription is in good agreement with the experimental data; the I = 2 prediction is more reliable. The NL~M prediction of the I - -0 and I = 2 S-wave phase shifts depend on one parameter, the scale of the logarithm term, which we cannot determine in this model. An equality is derived for the linear combination of the two S-wave phase shifts which are independent of this parameter, and is shown to agree with the experimental data. An improvement in the determination of E and G either by the analytical approximation or by the future lattice gauge calculation (which does not require 1/N~ expansion) can be used in eqs. ( 11 ) of the L~M to get a more accurate prediction of the low energy pion-pion interaction.

We now want to extend our analysis to the Higgs problem. Using the equivalence theorem [ 16,17 ] which states that, when the center of mass WW energy is sufficiently large compared with the W mass, the study of the longitudinal components of WW scattering is simplified to the problem of Nambu- Goldstone boson scattering [ 18,19 ]. The above con- sideration can be straightforwardly generalised to take care of this situation. The principal modification is to replacef by v= (N/~GF) - 1 /2 = 246 GeV and mo by the Higgs boson mass. There are however some com- plications which require us to study separately the situation where the Higgs boson is an elementary field or a composite object such as in the technicolor scheme.

If the Higgs boson is an elementary field, we can add little to what is already known in the literature, that is, the contribution of the top and light quarks and leptons are negligible in comparison to the scattering amplitudes [20]. If there are heavier leptons and quarks whose masses are heavier or comparable to the Higgs boson mass itself, we must calculate the scattering amplitudes using the L~M lagrangian and add to it the contribution of the heavy fermions which is obtained either by calculating the box diagram or by integrating out these heavy fields in the same line as discussed above. As can be seen from eq. ( 13 ), the existence of the P-wave resonance depends on the in- terplay between the number of species Nc of heavy fermions and the Higgs mass. More precisely, the P- wave resonance exists if Nc >/0.327 + 24~zzvZ/m 2. It follows that the lighter the elementary Higgs boson is, the larger the number of species of fermions is re- quired to generate the P-wave resonance. An elementary Higgs boson without the presence of heavy fermions cannot generate a vector meson resonance (see also ref. [ 9 ] ). When the Higgs boson is very light, less than 1 TeV, eqs. (5) and (13) are no longer ap- plicable, we must refer to a previous treatment to show that a P-wave resonance cannot exist in the ab- sence of the heavy fermions [ 19 ].

Our QCD result can be straightforwardly generalised to include the technicolor or technicolor-like scheme. The Nambu-Jona-Lasinio model [ 21 ] with the four-fermion interaction coming from a possible heavy fourth family is more difficult to handle. As a first approximation one can use the L~M approach with the ~ mass being roughly twice the fermion mass.

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If the fermion mass is about 1 TeV or larger one can use eq. ( 5 ) to describe WW scattering in this model. The corresponding P-wave phase shifts are given by eq. (13). Depending on the Higgs mass or the fermion mass a P-wave resonance may or may not exist.

Because the current analysis of the LEP data gives a top quark mass of about 130 GeV, its contribution to a heavy Higgs sector, in which we are interested in this article, is completely negligible. It could play an important role for a light Higgs scenario in the framework of the Nambu-Jona-Lasinio model as recently discussed by many authors [ 22 ]. In these models the P-wave resonance could only exist at the cut-off scale of the four-fermion interaction which is 10 ~5 GeV.

We have emphasized in the last few paragraphs the question of the P-wave resonance in WW scattering as an indication for new physics at LHC and SSC because it is a clear signal, free from the background problem, unlike the experimental measurement of the I = 0 S-wave.

It is a pleasure to thank Professor H. Lehmann and Professor R. Willey for useful discussions. Part of this work was done during the "Puzzles on the Electro- weak Scale" meeting in Warsaw. The author would like to thank Professor Stefan Pokorski for his invi- tation and hospitality.

Note added: After completion of this work, the author became aware of a recent article on a similar subject by Veltman and Veltman [23]. The treatment by these authors follows that of H. Lehmann where the nucleon loop is supposed to be the origin of the vector meson resonance. A more complete treatment is given for the large mo limit of the Lt~M.

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Documents

Zero-parameter calculation of the low energy Nambu-Goldstone boson elastic scattering in quantum chromodynamics and in technicolor-like theories