Physics Letters B 313 ( 1993 ) 165-170
    PHYSICS LETTERS B
North-Holland

    Strongly interacting W's and Z's and the elastic unitarity constraint
    S.R. B e a n e i a n d S. V a r m a 2
    Center for Parttcle Phystcs and Department of Physics, Umverstty of Texas at Austm, Austin, TX 78 712, USA

    Received 17 April 1993, rewsed manuscript received 8 June 1993
    Editor. H Georgl

    By employing the dictum that axiomatic principles are devoid ofpre&ctlve power, we find that the elastic unitarity constraint,
    apphed to strong WLWL scattering, does not alter the assumed spectrum of lntermedmte states In other words, ffone integrates
    out the heavy degrees of freedom in an effective theory, then ~mposmgelastic umtarlty on the resultant low-energy theory effectively integrates back in the original heavy degrees of freedom, but otherwise leaves the spectrum unchanged. As a case study, we
    consider lntermedmte states involving a heavy Hlggs and heavy fermlons ofa hypothehcal fourth generation doublet In contrast
    to recent studies, we find no p-wave resonance.

1. Introduction

    Long ago, it was o f interest to d e t e r m i n e whether a strongly interacting linear sigma m o d e l ( L S M ) , in all its
a p p a r e n t simplicity, could prove capable o f generating the complex hadronlc spectrum [ 1 ]. In recent times, the
familiar i s o m o r p h i s m between the pions a n d the longitudinal m o d e s o f the standard model gauge bosons, m a d e
precise through the equivalence theorem, has led to a resurgence o f interest in the strongly interacting LSM [ 2 4]. M o t i v a t i o n rests on the u n d e r s t a n d m g that W L W L scattering provides a unique probe o f the m e c h a n i s m
responsible for electroweak s y m m e t r y breaking [ 5 ]. Evidently, triviality b o u n d s [ 6 ] p r o v i d e at best a narrow
w i n d o w within which a strongly interacting Htggs sector could exist. Nevertheless, a strongly interacting Higgs
sector r e m a i n s an interesting posstblhty. The onset o f strong interactions is, of course, signaled by the breakdown o f p e r t u r b a t i v e unltarlty. Thus, there have been m a n y a t t e m p t to make meaningful statements about the
electroweak s y m m e t r y breaking sector by i m p o s i n g elastic unitarity as a constraint, an a p p r o a c h known as unitarization. It is the purpose o f this letter to show that if unitarlzatlon is carried out properly, nothing is learned
about the physical spectrum o f the s y m m e t r y breaking sector.
    U n l t a r i z a t i o n o f a scattering a m p l i t u d e requires trading crossing s y m m e t r y for elastic unitarity, in a nonunique way. Beyond requiring that our unitary a p p r o x i m a t i o n have good analytlclty properties, and be consistent to a certain degree with chlral s y m m e t r y and crossing, we impose an additional constraint: In sync with
current lore [ 7 ], we ensure that unitartty per se yields no predictive power, a point o f view clearly orthogonal to
S-matrix theory (in the b o o t s t r a p sense). We consider the d e r i v a t i o n o f the well-known K S R F relation [8 ] as a
heuristic tool to m a k e clear exactly what we mean by thts constraint.
    In order to stress the generality o f our statements, we m a k e use o f the effective lagrangian approach. As will
be m a d e clear, this also allows us to m a i n t a i n a m o d i f i e d power-counting scheme by decomposing the scattering
a m p l i t u d e according to its crossing properties. As a case study, we saturate the u n d e r m i n e d constants o f chiral
p e r t u r b a t i o n theory with c o n t r i b u t i o n s from l n t e r m e d t a t e states involving a heavy Higgs and heavy degenerate

1 E-mad address: sbeane@utpapa.ph utexas.edu
2 E-mall address" varmlnt@utpapa ph utexas.edu.

0370-2693/93/$ 06.00 © 1993 Elsevier Science Pubhshers B V All rights reserved.    165

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fermions of a hypothetical fourth generation doublet. As expected, the only nearby pole o f the full amplitude is
seen to be the physical Hlggs pole.
   In refs. [ 3,4] the method of Pad6 approxlmants ,1 is use to show that, for a large fermlon mass, it is possible
to dynamically generate a p-wave resonance. The S parameter bound can then serve to exclude a heavy fourth
generation of fermions [4]. We will argue that the p-wave resonance generated by the Pad6 method is unphyslcal, and owes its existence to the incorrect presumption that elastic unitarity should be imposed for the purpose
of making predictions.
   The plan o f the remainder of the letter is as follows. In section 2 we obtain the chiral perturbation series to
order E 4 by way o f a chiral Lagrangian. In section 3 we present our unltarlzation scheme. We then present our
case study in section 4, and contrast our results with other recent work. Finally, in section 5 we present a summary and conclusion.

2. Effective Lagrangian

  Exploitation o f the model-independent low-energy structure of the theory is essential to our approach. Assuming a custodial S U ( 2 ) symmetry, the most general effective Lagrangian including terms with four derivatives is
given by #2

Lf = ~vETr(OuXOUX*) + ~CI Tr(OuXOuX*)Tr(O~XO~X*) + ~C2 Tr(O~,XO~X*)Tr(OUXO~X*) •    (2.1)

The Goldstone boson fields ( w +, w - , and z) are contained within the field variable X = exp [ i ( , . v) /v]. Ct and
C2 are undetermined constants which characterize the underlying theory at low energies. In general, there are
contributions to Ct and C2 from all heavy degrees o f freedom, as well as continuum contributions arising from
Goldstone boson loops. To order s 2, the relevant partial wave amplitudes of definite custodial lsospin obtained
from this lagrangian are given by

ao(s) =-aoo(S)
   =OtoS{l_~_~log(72)_6(2Ct+C2)]_°toS[710g(S~zeL
    '8 ~,/t 2,1- .to8 t_ ~2t (4C + 5C2) ] } ,    (2.2a)

ai(s)-alt(S)
   =otts{l _ ~-~log(~2 )-12C2 ]+ -°t--~Ilog(~ ) - ½+12 ( C2 -4Cl ) l} ,    (2.2b)

a2(s) =-a2o(S)
   =   ot2s{l__~_S[log(_~)_12C21_ot2s[,,.    {s'~    25   4    +7C2)]} ,    (2.2c)

where Oto- 1 / 16gv 2, a t - 1/96~zv z, and a2 - - 1 /32~zv 2. EAch curly bracket consists of three terms, correspondmg to the low-energy theorem, and the O (s 2) contributions in the direct- and the crossed-channel respectively.
Note that we have been careful to preserve the crossmg properttes of the undetermmed coefficients [ 10 ]. This
decomposition will prove an important ingredient in what follows.

#1 Although Veltman and Veltman make use of an "effective range" approximation, their amphtude is equivalent to the [ 1,1 ] Pad6
   approxlmant.
a2 The coefficients, normalized m this way, are ofO( 1) i n the sense of naive &menslonal analysis [9]

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3. Unitarization

  Our umtarizatlon scheme corresponds to a simple bubble-sum with amplitude given by

t'(s)=V'(s)/(1-1f~    s'V'(s')ds')-s-le    (3.1)

In order to avoid double counting of graphs, the effective potential Is, consists of all two-particle irreducible
diagrams in the chiral expanston, with the exception of those contact graphs which are related to dtrect-channel
resonance exchange ~3. This amplitude can be thought of as arising from the solution of a relativistic L l p m a n n Schwlnger equation [ 11 ]. The leading order effective potential yields

    a,(s)    (3.2)
t,(s) = 1 + ( a,sl~r) [log( -sl/~ 2) + R,(/~ 2) ]
The R, are obtained by expanding eq. (3.2), and matching against the dtrect-channel piece of the chiral expansion. By inspection o f e q . (2.2) we find R 0 = - 6 ( 2 C 1 + C 2 ) and R2=R1 = - 12C2 (the "complementarity" between the I = 1 and I = 2 channels that follows from R2 = R 1 is investigated elsewhere in detail [ 10,12 ] ).
   The approximation of neglecting crossed-channel contributions clearly works best near an s-channel pole. Eq.
(3.2) is therefore ideally suited to the task of investigating the existence of resonances for definite values of C~
and Cz. More importantly, we find that if one wants to play the umtarizaUon game and not be burdened with a
large number of undetermined parameters, then the constraint that S-matrix elements should not be uniquely
determined by elastic unttartty alone appears to require one to make this approximation. That axiomatic constraints hke unttaraty and causality do not uniquely determine S-matrix elements was an important lesson learned
with the advent of QCD. A priori, there are an infinite number of S-matrices consistent with the most general
phystcal principles [ 7 ]. (For example, in the context o f a non-abelian gauge field theory, changing gauge group
and fermion content certainly does not affect the umtarity of the theory.) In the present context, we translate
this dictum into the constraint that predictions should always be accounted for by means independent of the
unitarization scheme being used. For example, we see in eq. (3.2) that if t~ is resonant, the width of the resonance is automatically fixed to the weak scale analogue of the KSRF relation [ 13 ]. However, we need not worry.
This prediction is not a consequence of imposing elastic umtarity, but rather of neglecting the left-hand cut. This
is easily seen by going to higher order in the effective potential. To order s 2 in the effecttve potential, we find

    oqs+3,s2[log(s/I.t 2) + B, (/~ 2) ]
t,(s) =
    1 + (o~,s/n) [log( -sl/~ 2) + R,(/~ 2) ] + (3,/2n)s2{ [log( -silt 2) ] 2 + 2B,(/~2)log ( -sl/~ 2) + M , ( # 2 ) } '
    (3.3)
where 30 = ( - 7 / 1 8 7 0 (Oto) 2, 3, = (1/~z) (c~,)2, and 3 2 - ( - 1 1/9n) (o~2) 2 (see eq. (2 2).) The B, are the lowenergy constants associated with heavy particle exchange in the crossed-channel. The M, are undetermined constants that appear at two-loop order in the chiral expansion. We see that it is by neglecting the contribution to
the imaginary part of the inverse amplitude involving BI that we are able to predict the K S R F relation. Therefore, the predictive power of eq. (3.2) is not a result of imposing unitarlty, but rather a result of neglecting a
class of graphs associated with heavy particle exchanges in the crossed-channel, which are manifest at O (s 2) in
the chiral expansion. It is important to note that the above does not constitute a new derivation of the KSRF
relation. In fact, all justifications of the KSRF relation, including the original current algebra derivation [ 8 ],
require the tacit assumption that the left-hand cut of the I = 1 scattering amplitude is effectively absent [ 14 ]. We
find it powerful evidence in favor o f our approach that, by ensuring that predictive power come from a source
other than elastic umtarlty, we arrive at a well-known derivation of the KSRF relation.

~3 Eq (3 1) is therefore of N/D form in the sense that Is, contains the left-hand cut. The right-hand cut is generated by the dispersion
   integral in the denominator function
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4. Case study. Intermediate states involving the standard model Higgs and heavy fermions of a hypothetical
fourth generation doublet

   The contributions to the low-energy constants C~ and C2 arising from intermediate states involving the Higgs
boson and degenerate ~4 heavy fermions of a fourth generation doublet have been calculated perturbatively in
ref. [ 15 ] using an on-shell subtractxon scheme. They are given by #5

    1 F / 97r - ~ ) - ~ l o g    ~    + 2 ; ' r 2 ~-2"~,    ~[--(~)--½log(~-~H)3,    (4.1)

and

CFI = -   1 ~ [ ½+6 ( 2 + ~
    4 --a j "    log[1-ax(1-x)Jdx ) ] , c{=Nc12,
    __    (4.2)

where a=M}/M}. Note that for definiteness we use values of the low-energy constants extracted from perturbation theory. However, we stress that we could equally well consider the most general couplings of fields with
any quantum numbers to the Goldstone bosons, and estimate the values of these couplings using naive dimensional analysis. The uncertainty associated with a change of the C, of O(1 ) should certainly not exceed the
inherent uncertainty that accompanies any unitarization scheme. In fact, we find that our basic conclusions are
insensttive to natural changes in scale. For example, we can replace the Cff by the values that obtain from
coupling a scalar to the Goldstone bosons in the most general way [ 16 ]. In this case there is an undetermined
parameter that can be related to the scalar width. If, instead of choosing the perturbatlve standard model value
for the width, we choose one-half of that value, as is the case when the existence of a narrow p-wave resonance
is assumed [ 13 ], our results are unaffected.
   Inspection ofeq. (4.2) reveals that the I = 1 and I = 2 amplitudes are independent of the heavy fermion mass.
Only the I = 0 amplitude has non-logarithmic contributtons that depend on the Higgs and fermion masses. This
is gratifying because it is these terms that are responsible for the "binding" which yields resonance behaviour.
Nevertheless, this result is not surprising; the values of CI and C2 given in eq. (4.2) are the low-energy manifestation of a scalar-dominated theory. UnitarIzatlon simply restores the basic properties of the assumed underlying theory.
   In fig. 1 we schematically depict the complex s-plane. With a rather conservative choice of cutoff, given by
A=47w_~ (3 TeV), and with MH=Mf=1 TeV, we see that the only pole in the theory is the "physical" Higgs
boson. In fig. 2 we display the partial wave amplitudes of definite custodial isospm for values of the tree Higgs
mass of 0.75 TeV and 1 TeV. For values of Myabove 250 GeV, the fermlonic contrtbutions to the I = 0 amplitude
amount to a negligible renormalization of the physical Higgs mass, and so we neglect them in the graph. The
complementary character of the non-resonant I = 1 and I = 2 amplitudes is clearly evident. We also display the
Pad6 prediction for the I = 1 amplitude, with MH=0.75 TeV and My= 1 TeV.
   What is the source of discrepancy between our method and the Pad6 method? The method of Pad6 approximants, as applied in refs. [ 1-4 ], also predicts the KSRF relation in the I = 1 channel, and yet the O (s 2) crossedchannel contributions are Included.Since the neglect of crossed-channel contributions can no longer serve as the
source of predicttve power, it follows that the crossed-channel contributions necessarily appear in the wrong
place. These misplaced contributions appear m the real part of the inverse amphtude and therefore, in the
current context, it is quite understandable that an unphys~cal pole is present. The moral of this story ~s that the
Pad6 method, which ts ideally stated to problems in potential theory, should probably not be applied to problems where crossing symmetry ts important.

   The heavy fermlons are taken to be degenerate m order to avoid introducing lsospm breaking terms.
,5 We have corrected the error m C~(#) mref. [ 15]

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    ½
    2    2
    =/3 Toy/

    X
    I=1

    Fig. 1 Schematic depiction of the complex s-plane for characterlStiC values of the input parameters, Mn=My= 1 TeV The only
    pole below the cutoff is the physical Hlggs pole

    1    02    '    t   •    '

    08
    0 175    ibl    i    f

    06    0 125
    01
  Ito(s)l 0 4    It ~(s)l 0 075

    0 05
    02
    0 025
    0    0
    0    05    1    15    2    05    1    15    2
    qS (TeV)    4s (TeV)

    08'
    (c)

    O4
 It2(s)l    Fig. 2 (a) I = 0 s-wave amplitude. The dashed line corresponds
    to M n = 0 75 TeV and the solid line to M,~= 1 TeV (b) I = 1 p0.2    wave amplitude The dashed hne corresponds to MH=O75 TeV
    and the solid hne to Mrs= 1 TeV The dotted line corresponds to
    J    ,!    the Pad6 method prediction for M n = 0 75 TeV and My= 1 TeV
    0'5    ' i    1'5    2    (c) I = 2 s-wave amphtude The dashed line corresponds to
    qs (TeV)    M~/=0 75 TeV and the sohd hne to M~v= 1 TeV

5. Conclusion

    I n t h i s l e t t e r we a r g u e d t h a t i n a c o n s i s t e n t u n i t a r y a p p r o x i m a t i o n , u n i q u e p r e d i c t i o n s o f S - m a t r i x e l e m e n t s
s h o u l d b e a c c o u n t e d f o r I n d e p e n d e n t o f t h e s p e c i f i c u n i t a r l z a t l o n s c h e m e used. T h i s c o n s t r a i n t follows f r o m t h e
w e l l - k n o w n fact t h a t a x i o m a t i c p r i n o p l e s like u n i t a n t y , w h i l e p o s s e s s i n g c o n s t r a i n i n g p o w e r , are d e v m d o f
p r e d i c t i v e p o w e r . W e c o n s t r u c t e d a u n i t a r y a p p r o x i m a t i o n for s t r o n g WLWL s c a t t e r i n g t h a t satisfies t h i s c o n s t r a i n t , a n d yet p o s s e s s e s p r e d i c t i v e p o w e r as a r e s u l t o f n e g l e c t i n g classes o f g r a p h s a s s o c i a t e d w i t h c r o s s e d c h a n n e l p h y s i c s , w h i c h m c e r t a i n cases c a n b e a s s u m e d small. I n th~s s e n s e o u r a n a l y s i s is s o m e w h a t in t h e s p i r i t

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of large N. For example, the O ( N ) model is exactly solvable to leading order in 1/Nprecisely because left-hand
cut contributions first appear at O ( 1 / N 2) [ 17 ]. In order to test the implications of our unitary parametrizatlon,
we considered the specific case of m t e r m e d m t e states involving a Hlggs and heavy fermions of a hypothetical
fourth g e n e r a u o n doublet. We investigated the singularity structure of our unitary amplitudes, and found that
the only nearby pole corresponds to the physical H~ggs. We then showed that the Pad6 method prediction of a
p-wave resonance can be understood based on the fact that the [ 1,1 ] Pad6 approximant does not satisfy the
constraint o u t h n e d above. That is, certain predictions of the [ 1,1 ] a p p r o x l m a n t were shown to be artifacts of
the unitarlzation scheme.
  Our conclusxons are not surprising. The effective field theory viewpoint implies that one gets out essentially
what one puts in. Once we saturate the low-energy constants of chiral perturbation theory with contributxons
from a scalar-dominated underlying theory, reformation regarding the mtermedtate-energy spectrum is, m a
sense, exhausted. The elastic unitarlty constraint does not, a n d should not, change the character of the assumed
underlying theory, albeit a strongly interacting one, e.g., by inducing a p r o m i n e n t vector contribution.

Acknowledgement

  We are grateful to D. Chao, C.B. Chiu, D. Dicus, E.C.G. Sudarshan, S. Thomas, and U. van Kolck for useful
discussions a n d suggestions. We also thank M. Veltman and R.S. Wllley for fruitful correspondence. This work
was supported m part by the D e p a r t m e n t of Energy G r a n t No. DEFG05-85ER40200.

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