One-loop corrections and unitarity effects in gauge boson scattering

Volume 228, number 4

PHYSICS LETTERS B

28 September 1989

O N E - L O O P C O R R E C T I O N S AND UNITARITY EFFECTS IN GAUGE BOSON SCATTERING Duane A. DICUS Center for Particle Theory, University of Texaz~,Austin, TX 78712, USA

and Wayne W. REPKO Department of Physics and Astronomy, Michigan State University, East Lansing, M148824, USA Received 27 June 1989

We examine the effects of including one-loop corrections in the K matrix unitarization of the two-channel system consisting of the s t a t e s W ~ W ~ and ZLZt.. o o F o r a Higgs-boson mass (ran) of 0.5 TeV, inclusion of the one-loop corrections produces a very small change in both the perturbative and unitarized amplitudes. At mrs= 1.0 TeV, the one-loop corrections to the Born amplitudes are substantial: 20 - 30%. The corresponding unitarized amplitudes are very similar to their unitarized Born term counterparts. In general, the one-loop corrections to the unitarized amplitudes becomes important for values of x/s which are large compared to mH.

Despite the numerous successes of the standard model, the precise nature of its electro-weak symmetry brcaking mechanism remains unclear. Determining whether the symmetry breaking is due to the Higgs mechanism [ 1 ] or to some dynamical scheme [2] is one of the main goals for proposed high energy colliders. Even in the simplest Higgs scenario, the Higgsboson (H °) mass (mH) is essentially unconstrained by the theory. Should it happen that mH> 2mzo, the processes H°~W+W - ,

(la)

H ° ~ Z°Z ° ,

( 1b)

would be the dominant Higgs-boson decay modes. Under these circumstances, it should be possible to search for Higgs-boson resonant behavior in pair events associated with gauge boson scattering [ 3,4 ]. Another important consequence of a "heavy" H ° involves its self coupling. The strength of this coupling is given by ~" - ,' g 2m H2 / m w2, wheregis the weak coupling constant. When mH is sufficiently large, 2, which by the Goldstone boson equivalence theorem [ 5 ] characterizes the dominant H°-gauge boson cou-

piing, becomes strong. Early investigations of gauge boson scattering in the Standard Model by Dicus and Mathur [ 6 ] and Lee, Quigg and Thacker [ 7 ] showed that gauge boson scattering Born amplitudes violate the unitarity bound for mH>~ 1.0-1.4 TeV in the limit that the center of mass energy x/~--,oo. While not completely rigorous bounds, these results together with subsequent one-loop calculations [8-11 ] indicate that higher-order corrections are no longer negligible when mH exceeds 1 TeV. The underlying theory is unitary, though not order-by-order in perturbation theory. Given this, it is of interest to examine the sensitivity of unitarized gauge boson scattering amplitudes to changes in mH. This has been done [ 12 ] using the Born amplitudes for the two-channel system consisting ofW~" W£ and o o The subscript L denotes a longitudinally poZLZL. larized W +- or Z °. Purely longitudinal states are known to describe gauge boson scattering amplitudes to the leading order in the center-of-mass energy [ 13 ] ~. Our purpose here is to asscss the effect of in~ See ref. [14] for an analysis of the effect of transverse polarizations.

503

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PHYSICS LETTERSB

eluding one loop corrections in the unitarization procedure. We unitarize the S matrix by introducing the K matrix as l+~iK"

(2)

The matrix K is hermitian and it can be expressed order by order in perturbation theory in terms of the Feynman-Dyson expansion of the S matrix [ 15 ]. Through fourth order, the relation is K(2)=iS

K~4)=½i(S~4)-SI4)*).

(2) ,

(3)

I f S is expressed as S = 1 +i,T, and the resulting integral equation for Y- in terms of K is projected into partial waves, the unitarized Jth partial wave amplitude, tv, is related to perturbative partial wave amplitude ag as

ts = ( 1 - i a s ) - ' a v ,

(4)

The elements of aj consist of the partial wave projections of various Born amplitudes plus the real parts of their one-loop corrections [ 15 ]. Complete expressions for the one-loop corrections to the processes W?) WtT .

.

-.o~.o

---~LIL

(5a)

l

Z',' Z°L~ W{ Wi. ,

(5b)

w~- we- -.-,wd-w c ,

(5c)

Z°LZ]'"--,Z°Z°L,

( 5d )

have been given by Dawson and Willenbrock [ 11 ]. Their results were obtained in a scalar theory by making use of the Goldstone boson equivalence theorem. Because of a residual rotational symmetry in the equivalent Goldstone boson lagrangian, the amplitudes for the processes enumerated above are related as ~t't(Wl ~.WE ~ZLZL) . o =A(s, t, U),

(6a)

.//t'(Z°.Z° ~Wi~ WC )=A(s, t, u),

(6b)

(6c) 0___+ 0

~q(ZLZL

0

ZLZL)=A(s,t,u)+A(t,s,u)+A(u,t,s), (6d)

504

1

'f

dzA(s, t, u) ,

(7a)

dzA(t,s,u),

(7b)

-1 1

b(s) = 3--~ -I

where z=cos(O). With these definitions, the s-wave projections of the amplitudes in cqs. (6) are

ao(W + WE-+Z°ZI' ) = a ( s ) ,

(8a)

ao(Z° z ° - , w + w c ) = a ( s ) ,

(8b)

ao(W~ W,- --,Wg Wt7 ) = a ( s ) + b ( s ) ,

(8c)

a0(Z°Z~. --'LL'-~ j~'°v°~=a(s)+2b(s) .

(8d)

After including the appropriate factors associated with the identity of the Z°'s, the matrix ao for the twochannel system takes the form

(a(s)+t,(s) a(s)/,/~ a°=~,a(s)/,~,~

~a(s)+b(s)/"

(9)

In the Born approximation, A(s, t, u) is

A,,orn(S, t, u) = - 2 2 ( ~ ) ,

(10)

and the explicit expressions for a(s) and b(s) in ref. [ 12 ] follow from eqs. (7). At the one-loop level, the expression forA (s, t, u) consists of contributions from six types of diagrams [ 1 1 ]. The partial wave projections for these corrections were evaluated numerically. Note that the a(s) contains Higgs-boson pole terms in both the Born and one-loop contributions. We do not include a Higgs-boson width in these terms. The width is generated by the unitarization procedure. The eigen-amplitudes ofao are ( 3 a + b ) and b, and the matrix to is expressible in terms ofao as ao - i det (ao) to- , det ( 1 - iao)

( 11 )

The elements of to are

J,'(Wx~ WE --, Wd W~7 ) =A(s, t, u) + A(t, s, u), 0

where A (s, l, u) is symmetric in its last two variables. To discuss the s-wave amplitudes, it is convenient to indroduce a(s) and b(s) defined by a(s)= ~

S = l-½iK

28 September 1989

(a+b)-i(~a+b)b to(Wit wf_ --,w~ wit )= [ l _ i ( 3 a + b ) ] ( l _ i b ) , (12a)

Volume 228, n u m b e r 4

to(W~- WF

PHYSICS LETTERS B

28 S e p t e m b e r 1989

s-wave

~o ~,o,)

--¢- L L / , i .

=

WL+W~-~WL~-W~A m p l i t u d e

m n = 1.0 TeV

[1-i(~a+b)l(1-ib)' (12b)

(½a+b)-i(~a+b)b [1-i(3a+b)](l-ib)

to(ZOZO--,z~zo) =

1.0

"

+.a

(12c) We show the absolute value of the unitarized s-wave Wi~ Wfi elastic amplitude in figs. 1-3 together with the corresponding perturbative amplitude (without the addition o f a Higgs width). In each case, the Born contribution is plotted in dashes and the total contribution as a solid line. It is evident from fig. 1 that for mH=0.5 TcV the inclusion of the one-loop corrections makes very. little difference to the amplitude. This is particularly true in the vicinity of the Higgs peak. For m . = 1.0 TeV, the one-loop correction is a substantial addition to the Born amplitude. Note, however, that the unitarized version of the total amplitude is not very different than the unitarized Born /-. amplitude. The differences begin to appear as x/s xncreases beyond the peak. This behavior persists for m H = 1.4 TeV, where the perturbative amplitude violates unitarity for all values of x/~ above the peak.

0.5

0.0

WL+WL-*Wt+W~.A m p l i t u d e

o..t,,

ri

,t,

,

,

, , I

600

, , I , .

800

' , , , I , , ,

1000 1200 (GeV)

1400

1600

Fig. 2. Same as fig. 1 with m H = 1.0 TeV.

s-wave

W~/~-~Wt+W~ A m p l i t u d e

m n = 1.4 TeV

\

m_

\ \ \ \

,L v

s-wave

/ / ~,'/

/,

o

m H = 0.5 TeV

...,

.

.

o . o -

--

0 500

, 1000

,

I , 1500

2000

2500

v~s (CeV) Fig. 3. Same as fig, 1 with rn H = 1.4 TeV.

o

0.2

0.0

,

[

400

.

.

.

.

600 (GeV)

I

,

000

Fig. 1. The absolute value o f the partial wave a m p l i t u d e to(Wi~ Wt7 - , W ~ W f i ) is plotted as a function of the ccnter-ofmass energy ,4's for m , = 0 . 5 TeV. The solid lines are the perturbative ( w i t h o u t the a d d i t i o n o f a Higgs w i d t h ) and u n i t a r i z e d a m p l i t u d e s including one loop corrections. The d a s h e d lines are thc c o r r e s p o n d i n g Born a m p l i t u d e s .

In figs. 4, 5 wc show the amplitudes for the processes W~W~- - ; , Z L oZ oL and Z LoZ L - o- - ~ L ~~,o-~o LZ.~L for m n = l . 0 TeV. Again, we find a substantial one-loop perturbative contribution which translates into a small correction to the unitarized amplitude. Our results confirm that for mH>~ 1.0 TcV the oneloop corrections to perturbative gauge boson scattering amplitudes arc significant. However, by using K matrix unitarization, we find that the changes in the unitary amplitudes produced by the higher order terms are quite small. This is particularly true near the H ° peak and for values of x/s accessible at SSC 505

Volume 228, number 4

PHYSICS LETTERS B

-0 0 WL+~L~ZLZL

s-wave

1.5

'''

I 'l'

Amplitude m~ = 1.0 TeV

' "

-~-r'''

III °N~ ,L

i''

'4

~\\

1.0

v

.oo

I~ ->

0.5

0.0 600

800

tO00 1200 (GeV)

1400

1600

Fig. 4. The absolute value of the partial wave amplitude v/2to(Wt~ W E ->Z°Z~,) is plottcd as a function of the center-ofmass energy v/S for mH=1.0 TeV. The solid lines are the perturbative (without the addition of a Higgs width) and unitarizcd amplitudes including one loop corrections. The dashed lines are the corresponding Born amplitudes.

0 0 0 0 ZLZL~ZLZL

s-wave 1.5

m

,

,

,

[

Amphtude m H = 1.0 TeV

,

i!:

1.0

'

'~1\'

~i iI

oL o%

'

'

I

'

'

'

28 September 1989

o n e - l o o p c o r r e c t i o n s do yield a substantially larger u n i t a r i z e d a m p l i t u d e . T h e b e g i n n i n g o f the d e v i a t i o n f r o m the u n i t a r i z e d B o r n a m p l i t u d e can be seen in the figures. F o r mH = 1.0 T e V the l i m i t i n g values for x/~>> mH o f the a m p l i t u d e s in eqs. ( 1 2 a ) - ( 1 2 c ) are 0.75, 0.03, and 0.64 w h e n the o n e - l o o p c o r r e c t i o n s are included. T h e u n i t a r i z e d Born a m p l i t u d e s , in the s a m e limit, are 0.27, 0.03, a n d 0.17. In a d d i t i o n to p r o v i d i n g s o m e insight into the effect o f h i g h e r o r d e r corrections, the K m a t r i x app r o a c h p r o v i d e s a c o n v c n i e n t m e t h o d for i n c l u d i n g the H ° width. By m u l t i p l y i n g the n u m e r a t o r a n d den o m i n a t o r o f any o f the e x p r e s s i o n s in eqs. ( 1 2 ) by (s-m2), o n e can see that the w i d t h in this f o r m a l ism is g i v e n by the residue o f 3a(s) at the p o l e s=m~ (b(s) is regular at s=m2). We h a v e v e r i f i e d that the e x p r c s s i o n o f Fll0 o b t a i n e d in refs. [8,11 ] is r e p r o d u c e d in this way. We h a v e b e n e f i t t e d f r o m c o n v e r s a t i o n s w i t h Charles Chiu, J o n P u m p l i n , a n d Scott W i l l c n b r o c k . C o m p u t i n g resources were p r o v i d e d in part by the U n i v c r s i t y o f Texas C e n t e r for H i g h P e r f o r m a n c e C o m p u t i n g . T h i s research was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t P H Y 86-05967 and by the U S D e p a r t m e n t o f Energy und e r c o n t r a c t no. D E - F G 0 2 - 8 5 E R 4 0 2 0 9 .

\

J R e f e r e n c e s

S

0.5

//

z

/y Y

0.0 600

,

,

,

800

"/

~__, .... I . . . . . . . . . . I , 1000 1200 ~fs" (GeV)

t _.t__.k.L. J

1400

i.

1600

Fig. 5. The absolute value of the partial wave amplitude 2t0 (ZLZ~--'ZLZL) 0 o o o is plotted as a function of the center-of-mass energy \/s for mH = 1.0 TeV. The solid lines are the perturbative (without the addition of a Higgs width) and unitarized amplitudes including one loop corrections. The dashed lines are the corresponding Born amplitudes. energies. T h e u n i t a r i z e d a m p l i t u d e i n c l u d i n g o n e loop c o r r e c t i o n s does not yield a substantially larger W - p a i r fusion cross section than the u n i t a r i z e d Born a m p l i t u d e . At m u c h higher energies, h o w e v e r , the 506

[ 1] P.W. Higgs, Phys. Rev. leu. 12 (1964) 132; F. Englert and R. Brout, Phys. Rev. Left. 13 (1964) 321; G.S. Guralnik, C.R. Hagan and T.W.B. Kibble, Phys. Rev. Lctt. 15 (1965) 585; T.W.B. Kibble, Phys. Rev. 155 (1967) 1554. [2] See for example E. Fahri and L. Susskind, Phys. Rcp. 74 (1981) 277. [3] M. Veltman, Acta Phys. Pol. B 8 (1977) 475. [4] M.J. Duncan, G.L. Kane and W.W. Repko, Nucl. Phys. B 272 (1986) 517. [5] M.S. Chanowitz and M.K. Gaillard, Nucl. Phys. B 261 (1985) 379. [ 6 ] D.A. Dicus and V.S. Mathur, Phys. Rev. D 7 ( 1973 ) 3111. [7] B.W. Lee, C. Quigg and H.B. Thacker, Phys. Rev. D 16 (1977) 1519. [8] W.J. Marciano and S.S.D. Willenbrock, Phys. Rev. D 37 ( 1988 ) 2509. [9] S. Dawson and S.S.D. Willenbrock, Phys. Rev. Lett. 62 (1989) 1232.

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PHYSICS LETTERS B

[10]M. Veltman and F. Yndurain, University of Michigan preprint. [ I I ] S . Dawson and S. Willenbrock, Brookhaven National Laborator3, preprint BNL 42767 (May, 1989). [12]W.W. Repko and C.J. Suchyta 11I, Phys. Rev. Left. 62 (1989) 859.

28 September 1989

[ 13 ] J.M, Cornwall, D.N. Levin and G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145;D 11 (1975) 972(E). [ 14] C. Bilchak, M. Kuroda and D. Schildknecht, Nucl. Phys. B 299 (1988) 7. [15]S.N. Gupta, Quantum elcctrodynamics (Gordon and Breach, New York, 1981 ) pp. 191-198.

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