Pair production of W and Z bosons and the goldstone boson equivalence theorem

ANNALS OF PHYSlCS 186, 15-42 (1988)

Pair Production of W and Z Bosons and the Goldstone Boson Equivalence Theorem

SCOTT S. D . W I L L E N B R O C K

Physics Department, University of Wisconsin, Madison, Wisconsin 53706

Received August 20, 1987

W and Z boson pairs are the signals of the production of many types of new, heavy particles, e.g., heavy Higgs bosons, new Z' bosons, heavy quark and lepton pairs, and heavy gaugino pairs. I show that the W and Z bosons which result from the decay of these heavy particles are predominantly longitudinally polarized, while those produced from background processes are primarily transversely polarized. I study W and Z boson pair production from quark-antiquark annihilation and present compact formulae for the cross sections to produce vector bosons of definite polarization. I also discuss W and Z boson pair production from gluon fusion via a virtual quark loop. Use is made throughout of the equivalence theorem, which relates the properties of longitudinally polarized vector bosons to the corresponding Goldstone bosons which are absorbed via the Higgs mechanism. I also discuss some subtleties related to the equivalence theorem which have not been previously addressed. © 1988 Academic Press, Inc.

1. INTRODUCTION

W and Z boson pairs will serve as the signals of the production of many types of new, heavy particles at future high-energy colliders. Some examples are

1. Production of a heavy Higgs boson [1, 2]. If it is kinematically allowed, the Higgs boson decays predominantly to W and Z boson pairs [3].

2. Production of a heavy Z ' boson associated with an additional U(1) group present at low energies in many grand unified theories [4]. This particle can decay to W l~oson pairs via its mixing with the ordinary Z boson [5] .

3. Pair production of fourth-generation quarks and leptons [2 , 6 ] . If kinematically allowed, these particles will decay to real W bosons, yielding a W boson pair signal. This is also true of a heavy top quark (m t + m b > Mw).

4. Pair production of exotic fermions such as those found in many grand unified theories [7] . If kinematically allowed, these particles will decay to real W and Z bosons, yielding a W or Z boson pair (or a WZ) signal.

5. Production of heavy quarkonia formed from long-lived fourth-generation quarks [8] . The r / (0 -+) and q s ( 1 - - ) states are the most copiously produced. The r/ state decays only rarely into W and Z boson pairs. The ff width, on the other hand, is dominated by decays to W boson pairs, and also decays rarely to Z boson pairs.

15 0003-4916/88 $7.50

Copyright © 1988 by Academic Press, Inc. 595/186/1-2 All rights of reproduction in any form reserved.

16 SCOTT S. D. WILLENBROCK

6. Pair production of gauginos (supersymmetric partners of W and Z bosons) [9]. If kinematically allowed, the gauginos will decay to real W and Z bosons. The gauginos themselves may come from the production and decay of heavy gluino and squark pairs [10].

These examples serve to illustrate a general rule. A heavy boson may be singly produced, and may decay to W or Z boson pairs. Heavy fermions, on the other hand, are usually pair-produced, but since each fermion will only decay to a single W or Z boson, the signal is again W or Z boson pairs. Since the weak interaction is responsible for the decay of all known fundamental particles, it is not surprising that new heavy particles should decay to W and Z bosons.

Of course, there are other processes which produce W and Z boson pairs, and these tend to obscure the signal of new particle production. This is particularly problematic in a hadron collider. In this paper we concentrate on the proposed Superconducting Super Collider (SSC), although our remarks will be valid for any hadron collider capable of pair-producing W and Z bosons.

The principal W and Z boson pair production background process is quark- antiquark annihilation, as depicted in Figs. 1 and 2. The W boson pair process has been studied recently by Duncan, Kane, and Repko [11], and the Z boson pair process by Duncan [12], in the context of Higgs boson production. They emphasized that the Higgs boson decays predominantly to longitudinally polarized W and Z bosons, while the quark-antiquark annihilation process produces mostly transversely polarized gauge bosons [13, 14]. This fact was also used by Gunion, Kunzst, and Soldate [15] in their work on Higgs boson decays to W boson pairs.

One may garner some intuition for this result from the "equivalence theorem" [3, 16-18], studied in detail recently by Chanowitz and Gaillard [19]. This theorem relates longitudinally polarized gauge bosons to the corresponding Goldstone bosons (unphysical scalar bosons) which are "eaten" via the Higgs mechanism. The Goldstone bosons couple to matter in a manner similar to the Higgs boson, and hence couple more strongly to heavy particles. This explains why heavy Higgs bosons decay predominantly to longitudinally polarized gauge bosons.

We can use the equivalence theorem to conclude that the W and Z bosons which result from the decay of any heavy particle are predominantly longitudinally polarized. Hence the signal for the production of the heavy particles listed earlier is longitudinally polarized W and Z boson pairs.

We can also use the equivalence theorem to understand why quark-antiquark annihilation is a negligible source of longitudinally polarized W and Z bosons. In

FIG. 1. Feynman diagrams for Z boson pair production from quark-antiquark annihilation.

PAIR PRODUCTION OF W AND Z BOSONS 17

q w. 7", Z

FIG. 2. Feynman diagrams for W boson pair production from quark-antiquark annihilation.

the case of Z boson pair production (Fig. 1) only light particles are involved, so the Goldstone boson couplings are suppressed. The situation is a bit more subtle for W boson pair production (Fig. 2) since the Goldstone bosons couple to the s-channel photon and Z boson with unsuppressed gauge strength. Nevertheless, the equivalence theorem will be useful in understanding the suppression of longitudinally polarized W boson production.

Another source of background W and Z boson pairs is gluon fusion via quark loops, as shown in Fig. 3. Ignoring heavy flavors, the W and Z bosons couple only to light particles, and we can use the equivalence theorem to conclude that gluon fusion is a negligible source of longitudinally polarized W and Z boson pairs.

Hence we see that the signals of new, heavy particle production and the backgrounds to these processes differ in a universal way, namely the polarization of the W and Z bosons. These polarization states will manifest themselves in the angular distribution of the decay fermions of the W and Z bosons. Longitudinally polarized gauge bosons tend to decay perpendicular to their direction of motion while transversely polarized gauge bosons decay parallel [11]. Of course one can infer the polarization state of the W and Z bosons only in a statistical sense, not on a case-by-case basis.

In the remainder of this paper we will expand on the ideas introduced in this section. In Section 2 we discuss the equivalence theorem. In Section 3 we calculate the pair production of Z bosons from quark-antiquark annihilation and compare the result with our expectations based on the equivalence theorem. Section 4 does the same for W boson pairs. Section 5 is devoted to a discussion of some subtleties per- taining to the equivalence theorem. In Section 6 we consider W and Z boson pair production from gluon fusion. Finally, in Section 7, we discuss other sources of W and Z boson pairs, and W Z boson production from quark-antiquark annihilation. There are also two appendixes which support the main text.

Some of the results in this paper are not new, and will be referenced at the

i iii FIG. 3. Pair production of W and Z bosons from gluon fusion via a massless quark loop. There are

six diagrams, corresponding to the 3! permutations of the external legs in the figure.


appropriate place. Both the pair production of W and Z bosons and the equivalence theorem are topics which have received a good deal of attention in the literature. I would like to think of this paper as an attempt to relate these two topics and, in the process, to enrich our understanding of each [20].

2. THE EQUIVALENCE THEOREM

The equivalence theorem [16-19] states that amplitudes (S-matrix elements) involving external longitudinally polarized gauge bosons are equivalent (up to a phase) to the same amplitudes with the external gauge bosons replaced by the corresponding Goldstone bosons, plus corrections of order M/E, where M and E denote the gauge boson mass and energy, respectively. Thus, at high energies (E>> M), the longitudinal components of the gauge bosons remember that they were Goldstone bosons before the Higgs mechanism occurred. Even at high energies, however, the gauge symmetry is still broken, as evidenced by the presence of Goldstone bosons. Equivalently, the longitudinal components of the gauge bosons do not decouple in the high-energy limit.

To be more precise, we must define what we mean by the Goldstone bosons. These are defined as the unphysical scalars present in the R e gauge [19, 21]. In the limit ~ ~ oo (Landau, or R gauge), the unphysical scalars are massless, and are indeed Goldstone bosons. The equivalence theorem is actually valid for any value of the parameter 4, however. We will continue to refer to the unphysical scalars as Goldstone bosons, although this is only manifestly true in Landau gauge.

Since the Goldstone bosons are part of the Higgs doublet field, their couplings to matter are similar to those of the physical Higgs field. The couplings of the Goldstone bosons, which we will denote as w +, ZL, are given in Fig. 4 (Ref. [17]). These couplings may be divided into four types. Figure 4a shows vector-scalar- scalar couplings of gauge strength, i.e., proportional to gauge couplings with no enhancement or suppression factors. Figure 4b shows vector-vector-scalar couplings of suppressed gauge strength. These are suppressed at high energies (E>Mw) relative to tri-vector couplings by Mw/E. Figure 4c shows couplings to fermions of Yukawa strength, i.e., proportional to gmf/Mw, where mf is the fermion mass. Figure 4d depicts scalar interactions of Higgs strength, i.e., proportional to gm~/Mw. The latter two types of couplings are familiar to us from the couplings of the physical Higgs boson. The couplings of gauge and suppressed gauge strength are less familiar since there are no such couplings of the physical Higgs scalar to gauge bosons.

As we noted in the Introduction, the equivalence theorem explains why the W and Z bosons which result from heavy particle decay are primarily longitudinally polarized. On the basis of the couplings given in Fig. 4, we see that the decay width of fermions to longitudinal W bosons is enhanced by (mf/Mw) 2. The decay width of heavy Higgs bosons to W and Z boson pairs is enhanced by (mH/Mw) 4. Similarly, the decay width of a new heavy Z' boson to W pairs is enhanced by (Mz,/Mw) 4.


A ,v,.,......< Z r.,.....v,..~ (a) P2"-w~ P2"w~

l I -2x w ,~ -ie(pl-P2) ~" - Ig 2--'~-_xw (pf-p2)

A , ¢ . ~ . . , . ~ ~ W-V Z ~ . . . , . ~ . ~ ~ W-~

(b) WL -eMwg#~ gMzxwg ~v

ZL------ WL~---- (C) f f2

mf -g / (2 ,V2 Mw) g~-M-w~ ~'~ × [r~,(~-&)-r%(~+~#]

/ / / z L / / / w L

H - - - - ~ . . .~ H - - - - < " " " w~ (d) ~ZL

i m H 2

FIG. 4. Feynman rules for the coupling of Goldstone bosons (unphysical scalar bosons of the R¢ gauge) to other particles: (a) vector-scalar-scalar couplings, (b) vector-vector-scalar couplings, (c) couplings to fermions, and (d) Higgs-scalar-scalar couplings. All particles are taken to be entering the vertices. The arrows on the fermion lines indicate the flow of fermion number.

The equivalence theorem also tells us that the coupling of longitudinal gauge bosons to light particles is suppressed. Since the equivalence theorem is valid only to order M/E, where M and E denote the gauge boson mass and energy, we expect the suppression to be some power of M/E. The theorem does not tell us what this power is, but, as we show in the next two sections, it can be derived fairly easily.

The equivalence theorem is valid for amplitudes, and is not valid for each Feynman diagram individually. Indeed, it is well known that Feynman diagrams involving external longitudinally polarized gauge bosons have bad high-energy behavior, growing like s "/2, where n is the number of longitudinal gauge bosons and s is the square of the center-of-mass energy. The equivalent Goldstone boson diagrams, however, are well behaved. Hence the gauge theory cancellations which restore unitarity to amplitudes involving longitudinally polarized gauge bosons are not present in the equivalent Goldstone boson theory-- the amplitude is manifestly unitary.

Thus we see that we cannot in principle apply the equivalence theorem to individual Feynman diagrams. Nevertheless, it is tempting to try to do so. As we demonstrate in the next two sections, one can actually apply the theorem to individual diagrams if care is taken to first cancel the badly behaved terms (growing with s) in the amplitude.


3. PAIR PRODUCTION OF Z BOSONS

In this section we derive the cross section for Z boson pair production from quark-antiquark annihilation (Fig. 1), using the equivalence theorem as a guide. We will treat separately the cross section for two longitudinally polarized Z bosons, one longitudinal and one transverse, and two transverse Z bosons.

Let us begin by discussing the longitudinal polarization vector. A spin one particle of mass M and four-momentum

p~' = (E, O, O, p)

has a longitudinal (helicity zero) polarization vector

1 ~¢. = ~ (p, o, o, E).

At high gauge boson energy, e~. is almost proportional to p~; this suggests the decomposition

p// e[ = - ~ + v", (3.1)

where v ~ is (9(M/E). It is the first term, p"/M, which is responsible for the badly behaved terms (growing with s) in each Feynman diagram, as we discussed at the end of the last section.

The Z bosons in Fig. 1 are coupled to vector and axial-vector currents. The vector current is conserved; so is the axial-vector current, since we are neglecting fermion masses. Therefore, the first term in the decomposition of the longitudinal polarization vector (3.1) vanishes by current conservation. Since the surviving term is (9(M/E), we see that each longitudinal Z boson suppresses the amplitude by one power of Mz/Ez.

This result is an extension of the equivalence theorem, which tells us only that an amplitude with longitudinal Z bosons coupled to massless fermions vanishes to leading order in Mz/Ez. The theorem does not tell us the first non-vanishing power of Mz/Ez. We now see that the leading term in an expansion of the amplitude in powers of Mz/Ez receives one power of Mz/E z from each longitudinal Z boson. This is shown figuratively in Fig. 5, where each suppressed coupling, of C(Mz/Ez), is circled.

Note that this point of view is rather different from the conventional one. Usually one thinks of longitudinal gauge bosons as having enhanced interactions (leading to terms which grow with energy), which cancel only after all diagrams have been summed. The equivalence theorem teaches us to think of longitudinal gauge bosons as Goldstone bosons, which interact weakly with ordinary matter. The conventional attitude is deficient in that it only explains the cancellation of terms which grow with s. As we have seen, there are also constant terms and terms which


~ . x , X 1 /ZL

(a) / ~ / \,,

(b) q ~ ~zc q ~ zu

i ,-.zr ~ .--'PI---) \ Z z

~ . f ~-'-. Z. r r : t / 'Z T

FIG. 5. Feynman diagrams in the equivalent Goldstone boson theory for Z boson pair production from quark-antiquark annihilation: (a) two longitudinally polarized Z bosons, (b) one transverse and one longitudinal, and (c) two transverse Z bosons. The circled vertices are suppressed by M z / E z.

decrease with powers of s which may cancel. This is best understood in terms of the expansion of the longitudinal polarization vector in powers of M/E, Eq. (3.1).

We can take this point of view all the way to the bank and actually use it to calculate the Z boson pair-production cross section. It is not hard to show that the longitudinal Z boson polarization vectors may be written in terms of the Z boson four-momenta (Pl, P2) in the center-of-mass system. The expression is

(1 + f12) p~ 2Mz ~ L = 2fl M z sfl p~' (3.2)

where fl= ( l - 4 M 2 / s ) m is the velocity of the Z bosons in the center-of-mass system. The same expression holds for e~L after p~ and P2 are interchanged. Since the first term vanishes by current conservation, we need only keep the second term, which greatly simplifies the calculation. Furthermore, since the expression is written in terms of four-vectors rather than components, Lorentz covariance is manifest throughout the calculation.

Below we present the spin- and color-averaged cross section for two longitudinal Z bosons, one longitudinal and one transverse, and two transverse Z bosons. These expressions were first given by Duncan [12, 22], and we reproduce them here for completeness.

(a) qcl -'4" Z L Z L

The differential cross section for two longitudinal Z bosons is

(1 7 da 1 ~ ~2 1 1 (u t -M4z ) - ~ (3.3) d - - ~ = 3 6 4 x ~ ( l _ x w ) 2 ( c [ + c ~ ) y 4 f l 4 s 2


where xw = sin 20w, cL = 2T3L -- 2eqXw, and CR = --2eqXw, where T3L = +½ and eq is the quark electric charge. Note the explicit factor of 7 - 4 = (Mz/Ez) 4, where

E z = a/~/2.

(b) q~ ~ Zt .Zv The cross section for one longitudinal and one transverse Z boson may be

calculated either by using an explicit transverse polarization vector or by summing over all polarizations for one of the Z bosons [23] and then subtracting the cross section for two longitudinal Z bosons given above. We employ the latter technique because it maintains Lorentz covariance. The result is

da I n ~2 1 1 1 ~ d t : 3 a x ~ ( 1 - - - x w ) 2 ( c 4 +c4)-~-fi z 7 L[~2(u2+t2-2M4z)(1-+l]2\u t /

- ( u Z + t e + 4 u t + 4 M 2 z u + 4 M 2 z t + 2 M 4 ) - . (3.4)

Note the explicit factor of 7 - 2 = (Mz/Ez) 2.

(c) qFI ~ ZTZT Finally, we calculate the cross section for two transversely polarized Z bosons.

One may either use explicit transverse polarization vectors or calculate the cross section summed over all Z boson polarizations [24] and then subtract the longitudinal-longitudinal cross section and twice the longitudinal-transverse cross section. We use the latter method and obtain

da 1 n ~2 4 1 (u t -M4z) d-7 3 1 6 x Z w ( l Z x w ) 2 ( c 4 + c R ) f l 4 s4

Note that the second term is suppressed by ? 4. Although we did not show it, this term corresponds to the cross section for Z boson pairs with the same helicities and the first term corresponds to opposite helicities. This may be verified by inspection of the helicity amplitudes given in Appendix A of Ref. [ i 1 ] or in Appendix D of Ref. [25 ]. The suppression of the equal-helicity amplitude is related to the fact that in QED photon pairs are always produced with opposite helicities in quark- antiquark collisions. This is not a consequence of angular momentum conservation. It also has nothing to do with the equivalence theorem.

Note that all of the cross sections are proportional to (c~ + c~). The subscripts refer to the helicity or chirality of the incident quarks. If the incoming quark is left- handed (right-handed), the cross section is proportional to 4 4 CL(CR). The helicity of the incident antiquark is opposite that of the quark, since gauge interactions conserve the helicity of massless fermions.


I [ I pp--ZZ

1(~2 ~ " ' ~ = 4 0 T e V ly1<2.5

LT

16 5 _

Jd zoo 400 6oo BOO 1000

Mzz (GeV)

dc~ lo -3 dMzz

(pb/GeV) i(~ 4

FIG. 6. Invariant mass distribution for Z boson pair production from quark-an t iquark annihilation at the SSC. L (longitudinal) and T (transverse) refer to the polarization of the Z bosons.

We can use these formulae to calculate the hadronic cross section for Z boson pairs of various helicity combinations. The invariant mass distributions are shown in Fig. 6 for the SSC. We have used set 2 of the distribution functions of E H L Q (Ref. [2 ] ) and have imposed a rapidity cut on the Z bosons of [Yl < 2.5. The cross section for one longitudinal and one transverse Z boson (LT) is suppressed by 7-2 = ( 4 M 2 / M Z z ) ' and the cross section for two longitudinal (LL) by V 4, relative to the doubly transverse (TT) cross section.

Note that the longitudinal-transverse (LT) cross section is apparently suppressed further than just by 7 -2 . This is due to the fact that the cross section is not peaked in the forward-backward direction. One usually expects such peaking from t- and u-channel diagrams due to the t- and u-channel propagators. However, we find from Eq. (3.4) that there is no enhancement for small t (forward direction) in the high-energy limit; the terms within the brackets approach a constant. The same is true in the backward direction, i.e., small u. This is not a consequence of angular momentum conservation, since both the initial and final states have J~ = _ 1 in the forward or backward (z) direction [25].

4. PAIR PRODUCTION OF W BOSONS

We now go on to discuss W boson pair production from quark-ant iquark annihilation (Fig. 2). It is more complicated than Z boson pair production due to the presence of a non-Abelian vertex.


We expand the W boson longitudinal polarization vector as in Eq. (3.2),

(1 + f12) p~ 2Mw ~L= 2fl Mw sfl p~' (4.1)

with fl = ( 1 - 4M~/s) ~/2. It is again the first term, proportional to p~/M, which is responsible for the badly behaved terms (growing with s) in each Feynman diagram. The fermionic vector and axial-vector currents to which the W boson couples are conserved. However, the W boson couples to a non-conserved current in the three-gauge-boson vertex. Hence we cannot conclude that the p~/Mw term vanishes by current conservation.

Nevertheless, we know that the theory is unitary, so the terms which grow with s must cancel to yield a well-behaved cross section. We will still use Eq. (4.1) for the longitudinal polarization vector, although we cannot discard the first term as we did for Z bosons.

The three-gauge-boson vertex induces interactions between Goldstone bosons and gauge bosons (see Figs. 4a and 4b). As we shall see, this will make our discussion of the production of longitudinal W bosons qualitatively different from that of longitudinal Z bosons. We will even be able to use the equivalence theorem to actually calculate the high-energy behavior of the doubly longitudinal amplitude.

We now present the spin- and color-averaged cross section for two longitudinal W bosons, one longitudinal and one transverse, and two transverse W bosons. These cross sections were first given by Duncan, Kane, and Repko [11 ], although in a somewhat more complicated form. In particular, their expressions contain terms which grow with s and hence are not manifestly unitary. The expressions we give below are manifestly unitary as a result of using Eq. (4.1) for the longitudinal polarization vector. A manifestly unitary expression for the cross section summed over all W boson helicities is given in Ref. [2].

(a) q ~ W d - W E

The Feynman diagrams in the equivalent Goldstone boson theory are shown in Fig. 7a. The coupling of the Goldstone bosons to massless quarks vanishes, so there is no contribution from the t-channel diagram to leading order in Mw/Ew. The s-channel diagrams are non-vanishing, however--the Goldstone bosons couple to the ~ and Z bosons with electroweak strength. Hence we learn that the cross section for two longitudinal W bosons is unsuppressed at high energies, unlike the corresponding cross section for Z bosons, which is suppressed by 7 -4. We also learn that at high energy the cross section is dominated by the s-channel diagrams, so there is no forward peaking from the t-channel diagram.

We can actually use the equivalence theorem to calculate the high-energy behavior of the amplitude. The couplings of the Goldstone bosons to gauge bosons are given in Fig. 4a. A short calculation gives [14, 17]

da 1 rt ~2 ut & = 3 16X~v s4 (a~+a~) ( s> M2) , (4.2)


w: " / t / q /.//'wL

(a)

~, ~ w,- " wL

(b) ~ ~ W T ~- ~ W T

(C) q ~ WT

~ ~ w ; ~-'-w~

FIG. 7. Feynman diagrams in the equivalent Goldstone boson theory for W boson pair production from quark-antiquark annihilation: (a) two longitudinally polarized W bosons, (b) one transverse and one longitudinal, and (c) two transverse W bosons. The circled vertices are suppressed by M w / E w.

where

a R = 2 [eql Xw/(1--Xw),

aL = aR + (1 -- 2Xw)/(1 -- Xw).

The subscripts L, R denote the helicity of the incident quark. The scattering angle is between the incident quark and the W boson with the same sign charge (e.g., u and W +, d and W-).

We now proceed to calculate the cross section exactly. We use Eq. (4.1) for the longitudinal polarization vectors. The only terms which grow with s are from (p~/Mw)(p~/Mw) ~,~. These terms, which arise in both the s- and t-channel diagrams, cancel among themselves. The remaining terms are all well behaved at high energy, and lead to the cross section

,,] d~=l ~ ~2 1 ut~_.M4w[ 1 (b~+b~)q 2 1 l b l ~ + ~ . ~ (4.3) dt 3 16x2w/34 S 2 L(s-MZz) 2 72s-M2z t

where

be : ]eql x w/32(3 -/32)/(1 - Xw),

br = bR + (1 -- 2x w)/(1 -- Xw) + 2/72,

and the scattering angle is between the incident quark and the W boson of the same sign charge. The subscripts L and R refer to the helicity of the incoming quark. If the quark is left-handed, drop the b~ term in Eq. (4.3); if it is right-handed, keep only the b 2 term. The expression is asymmetric in bE and be because only left- handed quarks couple to the W bosons in the t-channel diagram.


This expression for the cross section is not only simple; it is also easy to compare with the equivalent Goldstone boson theory. The first term is entirely s-channel, and corresponds to the equivalent Goldstone boson expression (4.2) in the high- energy limit. The last term comes entirely from the t-channel diagram, and the middle term is from s- and t-channel interference. Note that the t-channel amplitude is suppressed by 7-2=4M2w/S. This is because the Goldstone boson coupling to massless quarks in Fig. 7a is suppressed by Mw/Ew, and Ew = x/~/2. The factor of 7-2 reflects the fact that there are two suppressed couplings.

Note that the above discussion uses the equivalence theorem diagram by diagram, rather than for the entire amplitude. Strictly speaking, this is not valid, as we discussed at the end of Section 2. However, once we have canceled the terms which grow with s in the amplitude, the remaining terms can be grouped according to the corresponding Goldstone boson diagrams. Recall that the Goldstone boson diagrams are manifestly well behaved at high energies, so no cancellations are necessary there.

Let me underscore what I think is the true value of the equivalence theorem in this context. It is not that it enables us to calculate things we could not calculate before. Furthermore, although Eq. (4.1) for the longitudinal polarization vector makes the calculation of the amplitude somewhat simpler, this is also not the most important aspect of the equivalence theorem. Its true value is that it allows us to understand the qualitative nature of amplitudes involving longitudinal vector bosons. For longitudinal W boson pair production it tells us that the high-energy cross section is entirely s-channel, and that the t-channel amplitude is suppressed by 7 -2. Although these facts could be uncovered by direct calculation [26], the equivalence theorem provides us with insight which could not be obtained otherwise. It is a wonderful tool for extracting qualitative information.

(b) q~ --, W E W~

The equivalent Goldstone boson diagrams for the production of one longitudinal and one transverse W boson are shown in Fig. 7b. The t-channel diagram is suppressed by Mw/Ew since the Goldstone boson is coupled to a massless fermion. The s-channel diagram is also suppressed by Mw/Ew. This is because the Goldstone boson coupling to two gauge bosons is proportional to Mw (see Fig. 4b). Since the corresponding three-gauge-boson vertex is linear in the gauge- boson momenta, the Goldstone boson coupling is suppressed by Mw/Ew. Hence the equivalence theorem tells us that the amplitude for one longitudinal and one transverse W boson is suppressed by Mw/Ew.

We cannot actually use the equivalence theorem to calculate the high-energy behavior of the amplitude, since the amplitude vanishes in this limit. It is interesting to note, however, that the s-channel amplitude in the Goldstone boson theory corresponds to replacing the longitudinal polarization vector by p~/Mw. However, the t-channel part of the amplitude cannot be calculated in the Goldstone boson theory, so we must consider this to be simply an interesting fact which cannot be explained by the equivalence theorem.


The amplitude is calculated by using Eq. (4.1) for the longitudinal polarization vector and canceling the terms which grow with s. We then square the amplitude, sum over all polarizations of the other W boson, and subtract the doubly longitudinal cross section (4.3). The result for qq ~ W { W7 r + WE W~- (which is twice q~ --* W~ W~- ) is

da 1 z~ oc 2 1 1 1 [ 1 d-'7 = 3 16 X~v f14 y2 s 2 (s - MZz) 2 [s2f12 - 2(ut - M4)](d~. + d~)

1 1 4 - -

s--MZz t [sfl2(t - M2w) + (ut -- M ~ ) ] dr

4[ , ]] - t ~ stfl 2 + ~ ( l + f l z ) ( u t - M 4) , (4.4)

where

dR = 2 leql XwB2/(1- xw),

dL= dR + (3 -- 4Xw)/(1 -- Xw),

and the scattering angle is between the incident quark and the W boson with the same sign charge. The subscripts L and R refer to the helicity of the incoming quark, as we discussed earlier.

Note that the cross section contains a suppression factor of ? - 2 = 4 M 2 / s , as anticipated by the equivalence theorem. The first term is from the s-channel diagram, the last from the t-channel, and the middle from s- and t-channel interference. Note that there is no forward peaking, as we might have expected from the t-channel diagram. In the high-energy limit the numerator of the last term is st + ut ~ - t 2, which cancels the t 2 in the denominator [25]. Recall that we obser- ved the same lack of peaking in the case of one longitudinal and one transverse Z boson. This is surprising, since it is not a consequence of angular momentum conservation, as we discussed earlier.

(c) q ~ - o W ~ W T

Finally, we calculate the cross section for two transversely polarized W bosons. We obtain this by subtracting the longitudinal-longitudinal cross section and twice the longitudinal-transverse cross section from the helicity summed cross section in Ref. [2]. The result is

da l x ~2 1 ut _~¢14 [ ~ t 2 d--t" = 3 4 x 2 f14 - - ( u2 + - 2 M 4 )

s 2 (4.5)

28 S C O T T S. D. W I L L E N B R O C K

where d E and d R were given previously, and the scattering angle is again defined between the incoming quark and the W boson of the same sign charge.

Note that the last three terms are suppressed by ~-4. Although we did not show it, these terms correspond to the W bosons having the same helicity. Recall that we found the same suppression for equal-helicity Z bosons. This suppression is related to the fact that in QCD (or any massless, non-Abelian gauge theory) the gluons are always pair-produced with opposite helicities in quark-antiquark annihilation. It is not a consequence of angular momentum conservation, nor is it related to the equivalence theorem.

The first term in the cross section (4.5) corresponds to W bosons with opposite helicities. The component of angular momentum along the direction of motion of the W bosons is +2, and thus the smallest partial wave which contributes to the amplitude is J = 2. Since the s-channel diagrams are entirely J = l, they do not contribute. Hence the entire amplitude comes from the t-channel diagram, resulting in the t -2 factor in the cross section [25].

Thus, at high W boson energies, the doubly transverse cross section comes entirely from the t-channel diagram. Recall, however, that the doubly longitudinal cross section is entirely s-channel in the high-energy limit. Thus, in a very real sense, longitudinal and transverse W bosons are produced by different processes.

We now use these formulae to calculate the hadronic cross sections for W boson pairs of various helicity combinations. The invariant mass distributions are shown in Fig. 8 for the SSC. We have used set 2 of the distribution functions of EHLQ (Ref. [2]) , and have imposed a rapidity cut on the W bosons of lyl < 2.5. Since the cross section for one longitudinal and one transverse W boson (LT) is suppressed

1o-I

do" 10 -2 dMww

(pb/GeV)

io-3~

id 4

1 I I

PP - W + w - TeV lyl< 2.5

TT

IG 5 _

,6 6 ] 200 400

I I 600 800 1000

Mww (GeV)

FIG. 8. Invariant mass distribution for W boson pair production from quark-antiquark annihilation at the SSC. L (longitudinal) and T (transverse) refer to the polarization of the W bosons.


by ?-2 2 2 = (4M ~v/M ww), it dies off at large invariant mass. It is further suppressed by the lack of forward peaking, as discussed earlier.

We also see the somewhat surprising result that the doubly longitudinal cross section is suppressed with respect to the doubly transverse. This is not a kinematic suppression, since both cross sections are unsuppressed in the high-energy limit. This suppression is partially due to the high-energy doubly longitudinal cross section (4.2), which is entirely s-channel and therefore has no forward peaking. This is not sufficient to explain the entire suppression, however.

Some insight into this puzzle may be gained by comparing the numerical coef- ficients of the two cross sections. The ratio of the doubly longitudinal cross section to the doubly transverse cross section at 0 = ~/2 (t = u) and at high energies is

1 1 ( a [ + a 2 ) = 16(1 - X w ) 2 [(4 leql X w + 1 -2Xw)2+ (1-2xw)2]. (4.6)

The size of this ratio is dependent on two parameters, l eq[ and Xw. If Xw ~ 1, the ratio blows up. This corresponds to a large hypercharge coupling constant g' in the standard model, which would correspond to the weak neutral current being much stronger than the charged current. Since X w= 0.23 rather than unity, we conclude that the weak neutral current is weaker than the charged current, which is well known.

The ratio also blows up if [eql is large. However, for eq = 2 and - 3 (and Xw= 0.23) the ratios are about 1/6 and 1/10, respectively. Hence we also conclude that the ratio is small because the quark charges are relatively small.

Recall that the high-energy doubly transverse cross section is entirely t-channel, and is therefore mediated by weak currents. We are thus led to conclude that the suppression of the doubly longitudinal cross section with respect to the doubly transverse is due to the weakness of the weak neutral current and the electro- magnetic current with respect to the charged current.

This result is clearly manifest in the equivalent Goldstone boson theory. The amplitude consists of s-channel photon and Z boson diagrams (Fig. 7a). If the quark charge is large, the former is enhanced, whereas if Xw~ l, the latter blows up (due to the factor of (1-Xw) 1/2 occurring at both the fermionic and scalar vertices). Such an interpretation is not as straightforward in the full theory since there are cancellations which must be performed before it is possible to isolate the dependence of the amplitude on eq and X w. The equivalence theorem once again provides us with the qualitative information we seek.

5. MORE ON THE EQUIVALENCE THEOREM

Now that we have had some experience with the equivalence theorem, I would like to discuss some subtleties which have not been previously addressed.

Recall that the equivalence theorem is accurate only to order M/E, where M and


E are the gauge boson mass and energy, respectively. Since the energy E is not a Lorentz-invariant quantity, one might wonder if the theorem is only valid in frames of reference where M/E is small. Recall, however, that the cross sections we calculated in the preceding sections depend only on s, t, and u, and not on the individual energies of the gauge bosons. So it is clear that M/E is not the relevant quantity in a discussion of the validity of the equivalence theorem.

The discussion above is further complicated by the fact that the helicity of a massive particle is also not a Lorentz-invariant concept. To be sure, boosting a massive particle along its direction of motion does preserve its helicity (modulo a flip of the helicity + 1 states if the particles's momentum is reversed), but a boost in the transverse direction does not.

The cross sections we derived in the preceding sections were calculated in the center-of-mass frame, and are therefore invariant under boosts along the common line of motion of the outgoing gauge bosons, since these boots preserve the vector boson helicities [27]. Since we found the equivalence theorem to be valid in the center-of-mass frame, it is valid in any of these frames, regardless of the individual energies of the gauge bosons. It is even valid in the rest frame of one of the gauge bosons, in which case M/E = 1, which is certainly not a small number. Thus we see that the requirement that M/E be small in order for the equivalence theorem to be valid is certainly not correct; rather, it is necessary that M/x/~ be small.

Actually, there is another condition which must be satisfied. Since some of the amplitudes are peaked in the forward (or backward) direction, they can gain an enhancement at small angles which invalidates the equivalence theorem. For example, consider the doubly longitudinal W boson pair cross section, Eq. (4.3). The t-channel amplitude is suppressed by ~,-2, yet if Itl ~ s7 -2 the t-channel amplitude is (9(1), i.e., the same order as the unsuppressed s-channel amplitude. Thus we need the additional condition Itl >> s7 -2, or M / ~ I small.

Thus the correct statement, at least for the processes we have considered, is that the equivalence theorem is valid if M/x/-s, M/v/~], and M/,v/~l are small, regardless of the individual gauge boson energies One can speculate that the generalization of this to multiparticle amplitudes is that all invariants must be large compared to M, but I have not proven this.

Let us now discuss boosts perpendicular to the direction of motion of the gauge bosons, which do not preserve helicity. As an extreme example, consider the following boosts on a longitudinal gauge boson. First, boost to the particle's rest frame. Then boost in a direction perpendicular to the first boost. The.resulting state is purely transverse. This demonstrates that what one observer sees as a longitudinal gauge boson, another sees as transverse. On the other hand, the equivalent Goldstone boson amplitude is Lorentz invariant, so there is a real ambiguity regarding the proper frame in which the equivalence theorem applies. Again, the condition that M/E be small is irrelevant, since the gauge boson may be arbitrarily energetic in either observer's frame, yet it is longitudinal in one and transverse in the other.

The only way I see to resolve this ambiguity is to agree that the equivalence


theorem treats the gauge bosons as being kinematically massless. This is somewhat subtle, since a massless vector particle does not have a longitudinal polarization state. Nevertheless, the longitudinal state of a massive vector particle survives in the limit M/E ~ O. It is well known that the helicity of a massless particle is a Lorentz- invariant concept, so if we treat the gauge bosons as being kinematically massless, we have a frame-independent definition of a longitudina ! gauge boson. This is discussed in more detail in Appendix B.

To see how this resolves the preceding longitudinal-transverse ambiguity, recall that the first boost of the longitudinal gauge boson was to its rest frame. If we treat the gauge boson as being kinematically massless, this boost is impossible. No matter what boost we perform, the gauge boson remains light-like and longitudinal.

The final point I would like to discuss is the relationship between the equivalence theorem and the expansion of the longitudinal polarization vector in power of M/E, Eqs. (3.1) and (4.1). It may seem that the Goldstone boson amplitudes are equal to the corresponding longitudinal gauge boson amplitudes with the polarization vectors ~ replaced with p"/M. In fact this is true for Z bosons, but it is not always true for W bosons. In particular, if the longitudinal W boson is coupled to a three- gauge-boson vertex, it may be necessary to keep the next-order term in the expansion of the polarization vector. An example of this is the calculation of the doubly longitudinal W boson cross section, Eq. (4.3) (Ref. [19]).

6. GLUON FUSION

We can also apply the equivalence theorem to a less-well-known source of W and Z boson pairs--gluon fusion via a virtual quark loop, as shown in Fig. 3. At first sight this process is a negligible source of gauge boson pairs. If the diagram in Fig. 3 is dissected vertically, the right-hand side is just the quark-antiquark annihilation process discussed in Sections 3 and 4. The addition of a closed loop and two strong coupling constants then suppresses the gluon-fusion amplitude relative to quark-antiquark annihilation by oq/2n. However, the gluon-gluon luminosity at the SSC is nearly 100 times larger than the quark-antiquark luminosity [2]. Furthermore, the fermion loop receives coherent contributions from all flavors of quarks. The result is that the ratio of the gluon-fusion process to quark-antiquark annihilation is naively (cq/2n)2x 100xn~. Using ~ = 0 . 1 and nf = 6, we find that the ratio is approximately unity. Hence gluon fusion may be an important source of gauge boson pairs at the SSC.

The box diagram is difficult to calculate, and a complete analysis of it does not yet exist. There is a related result, however, which supports the importance of gluon fusion as a source of W and Z boson pairs at the SSC. Ametller, Gava, Paver, and Treleani [28] have shown that gluon fusion is an equally important source of photon pairs at the SSC as quark-antiquark annihilation. They have also studied Z7 production from gluon fusion. They find that this process is slightly suppressed

595/186/1-3


relative to quark-antiquark annihilation, but that this is a result of the small vector coupling of the Z boson to quarks.

If the W and Z bosons are longitudinally polarized, we may use the equivalence theorem to extract the high-energy behavior of the gluon-fusion amplitude. Let us consider the case where both gauge bosons are longitudinal. The equivalent Goldstone boson diagram is shown in Fig. 9. Since we are neglecting the mass of the quark in the loop, each Goldstone boson coupling is suppressed by M/E. Hence the amplitude for two longitudinal gauge bosons is suppressed by MZ/s. Similarly, the amplitude for one longitudinal and one transverse gauge boson is suppressed by

One may worry about applying the equivalence theorem to loop diagrams [3, 14, 16, 17], since all previous examples in the literature have been for tree diagrams. However, the proof of Chanowitz and Gaillard [19] and subsequent work 1-18] tell us that the theorem is valid to all orders in the gauge couplings. As an example, we demonstrate in Appendix A the validity of the equivalence theorem for the gluon- fusion process (Fig. 3), for non-zero mass quarks.

For massless quarks, it is simple to derive the suppression factors of M/E associated with the Goldstone boson couplings. As before, we decompose the longitudinal polarization vector into two terms,

e[ = p"/M + v ~, (6. i )

where v ~ is (9(M/E). For zero-mass quarks the first term vanishes due to current conservation, leaving the second term, which is suppressed by M/E.

This argument is certainly sufficient for longitudinal Z bosons, which couple only to the conserved fermionic currents. As we saw in Section 4, however, the W boson couples to a non-conserved current in the three-gauge-boson vertex. This vertex contributes to the gluon-fusion process as shown in Fig. 10a. (There is no contribution from an s-channel photon since Furry's theorem forbids a gluon-gluon- photon coupling.) We cannot immediately conclude that the p~/M term vanishes for W bosons, due to this diagram.

A little thought shows that this diagram in fact vanishes. The virtual Z boson in the diagram has both spin-one and spin-zero parts. Yang's theorem forbids the coupling of two gluons to a colorless spin-one state, so only the spin-zero part can contribute. The spin-zero part is just the Goldstone boson associated with the Z

g ~ ~ -" w,*, zt-

c j f ~-~ ~" ~" "- w[, z L

FIG. 9. Feynman diagrams in the equivalent Goldstone boson theory for W and Z boson pair production from gluon fusion via a massless quark loop. There are fix diagrams, corresponding to the 3! permutations of the external legs in the figure. The circled vertices are suppressed by M/E.


boson. Since this Goldstone boson does not couple to W bosons, the spin-zero contribution vanishes [29].

One may also worry about the contribution from the s-channel Higgs boson in Fig. 10b. Since we are neglecting the quark mass, however, this diagram also vanishes, due to the coupling of the Higgs boson to the quarks, which is proportional to the quark mass.

The result is that the only non-vanishing diagrams for W boson pair production from gluon fusion via massless quarks are the box diagrams of Fig. 3. Since the massless fermionic currents are conserved, the p~'/M term vanishes for W bosons, just as it does for Z bosons. We therefore conclude that the coupling of longitudinal W and Z bosons to the massless quark loop is suppressed by M/E. The box diagram with light quarks is thus a feeble source of longitudinal gauge bosons.

Thus far we have discussed only the gluon-fusion process with light quarks in the loop. On the scale of the W and Z bosons, the first five quarks are indeed light. The top quark is not light, however. In this case we must also include the contribution from the s-channel Higgs boson, Fig. 10b.

The top-quark contribution to the gluon-fusion process is qualitatively very different from the contribution from light quarks. Whereas the Goldstone boson coupling to light quarks is suppressed by M/E, it is unsuppressed for the top quark, and proportional to mt/M. Hence the box diagram with a top quark is a non- negligible source of longitudinal gauge bosons. The same is true of the s-channel Higgs boson diagram with a top quark in the triangle. So we conclude that gluon fusion via a top-quark loop is a large source of longitudinal W and Z bosons.

The proper attitude to take is that gluon fusion with a top quark is not a background, but rather a signal, namely a signal of the Higgs boson. We noted in Section 1 that a heavy Higgs boson decays predominantly to longitudinal W and Z bosons. If the Higgs boson is produced via gluon fusion, then Fig. 10b represents this production and decay. If we go off the Higgs mass shell, i.e., consider s :/: m 2, then this diagram is no longer enhanced by the Higgs resonance, and is the same order as the box diagrams, Fig. 3. So we should consider the box diagram with a

(a) =0

q

(b) g ' ~ H ~- '~ "w+'Z g . _ ~ ---- ~-W-, Z

FIG. 10. Feynman diagrams which potentially contribute to the pair production of W and Z bosons from gluon fusion. Diagram (a) vanishes, as discussed in the text. Diagram (b) contributes if the quark in the loop is massive.

34 S C O T T S . D . W I L L E N B R O C K

top quark to be a non-resonant contribution to the production of a Higgs boson which decays to W and Z boson pairs. This point of view is supported by the calculations in Ref. [-30].

Thus we see that the box diagram and the s-channel Higgs boson diagram are related if they contain a top quark. However, the contribution from light quarks is totally unrelated to the Higgs boson diagram, since the Higgs boson does not couple to massless quarks. So we should think of the top quark, the Goldstone bosons, and the Higgs boson as interacting "strongly" with each other, but the light quarks and Goldstone bosons as having suppressed interactions.

The box diagram with light quarks (and also heavy quarks) can also produce transverse W and Z bosons, of course. This is where gluon fusion is potentially the most dangerous source of background W and Z boson pairs. Since each flavor of quark contributes coherently to the amplitude, the cross section could be large, as we discussed at the beginning of this section. There are no suppression factors coming from the polarization vectors as there are for longitudinal gauge bosons. This background could potentially obscure the Higgs boson signal, and is therefore of the utmost importance. It could be reduced by using the polarization of the gauge bosons, which is, of course, one of the main themes of this paper.

The equivalence theorem can be used to qualitatively understand other box diagrams as well. For example, consider Z7 production from gluon fusion [28] (Fig. l la). If the Z boson is longitudinal, we may use the equivalence theorem to extract the high-energy behavior of the amplitude. The equivalent Goldstone boson diagram is shown in Fig. 1 lb. Since Furry's theorem forbids an odd number of vector couplings on a fermion loop, this diagram vanishes, even for non-zero quark masses. This means that the amplitude vanishes to leading order in Mz/E z. This can be seen in the results of Ref. [28], where the longitudinal Z amplitudes, Eqs. (A.9)-(A.12), contain explicit factors of Mz/xfs.

This result may also be understood in terms of the decomposition of the longitudinal polarization vector, Eq. (6.1). Furry's theorem tells us that only the

(o)

(b)

O Y

FIG. 11. (a) Z~, production from gluon fusion via a quark loop. Only the vector coupling of the Z boson contributes, due to Furry's theorem. (b) The same diagram in the equivalent Goldstone boson theory. Diagram (b) vanishes as explained in the text.


vector coupling of the Z boson contributes. Since the vector current is conserved even for non-zero quark masses, the first term, proportional to p~'/Mz, vanishes The remaining term is 6(Mz/Ez).

7. W W SCATTERING; W, Z PLUS TWO JETS; AND W Z BOSON PRODUCTION

We have seen that the production and decay of many types of new, heavy particles result in a final state containing longitudinally polarized W and Z boson pairs. In contrast, ordinary background processes tend to produce W and Z boson pairs which are transversely polarized. We have understood these results by invoking the equivalence theorem, which relates the properties of longitudinal gauge bosons to the corresponding Goldstone bosons.

The two background processes which we have studied are W and Z boson pair production from quark-antiquark annihilation (Figs. 1 and 2) and from gluon fusion via a virtual quark loop (Fig. 3). Another source of W and Z boson pairs is WW (and ZZ) scattering, in which the gauge bosons are initially radiated from incoming fermions [11, 12, 19, 31]. This produces both transverse and longitudinal W and Z boson pairs. The longitudinal W and Z bosons should not be regarded as backgrounds, however. Rather, they are associated with the signal for the Higgs boson, which decays to longitudinal W and Z bosons if it is sufficiently heavy. If the Higgs boson is produced via WW fusion, then we see that the longitudinal W and Z bosons which come from WW scattering are simply non-resonant contributions to the Higgs boson production and decay amplitude. This is the same attitude which we espoused in Section 6 for the pair production of longitudinal W and Z bosons from gluon fusion.

Furthermore, if the Higgs boson is very heavy (m H > 1 TeV), or if it simply does not exist, then longitudinal W and Z boson pair production from WW scattering will serve as a signal of the weak interaction becoming strong at the TeV energy scale. The equivalence theorem is useful in this situation since the longitudinal vector bosons may be treated as the "pions" of the strongly coupled weak interaction, and the technology of the strong interactions applied [3, 19].

Another source of background W and Z boson pairs which deserves mention is actually a fake. It is the production of a W or Z boson in association with two jets which have an invariant mass near the W or Z boson mass. This process has been studied for W boson pairs [15, 32], where it is a particularly severe background since one may not be able to avoid it by using only leptonic decay modes of the W bosons, due to the loss of neutrinos.

Since the W boson in this process comes from a massless fermion, we may use the equivalence theorem to conclude that it is predominantly transversely polarized. This information was used in Ref. [15] to reduce this background somewhat. It is difficult to infer the angular distribution of the two-jet system, so one cannot say a priori to what extent it may be distinguished from a longitudinal vector boson.


However, the calculations in Ref. [15] show that it is indeed possible to make this separation by imposing PT cuts on the jets.

Although we have concentrated on the pair production of W and Z bosons in this paper, our results are somewhat more generally applicable. The equivalence theorem tells us that W and Z boson pairs produced from quark-antiquark annihilation in association with any number of jets will be primarily transversely polarized.

The conclusions of this paper are not valid for WZ boson production. Recall that the cross section for the production of longitudinal W boson pairs from quark- antiquark annihilation, Eqs. (4.2) and (4.3), was numerically small due to the weakness of the s-channel 7 and Z boson couplings to the initial quarks. WZ production is mediated by an s-channel W boson, however, which couples with full strength to the initial fermions. The W-boson-scalar-scalar vertex (analogous to Fig. 4a) is also of unsuppressed electroweak strength, -ig/2. Thus, using the equivalence theorem, we find that the high-energy spin- and color-averaged cross section for qcl ~ W~ ZL is

da 1 ~ ~2 ut d--~ = 3 8X2wS4 ( s ~ M ~ ) . (7.1)

We would like to compare this with the high-energy limit of the doubly transverse cross section. In the high-energy limit only the doubly longitudinal and the opposite-helicity transverse amplitudes survive--the longitudinal transverse amplitude and the same-helicity transverse amplitude are kinematically suppressed, just as they were for I4" and Z boson pairs. Thus the high-energy limit of the opposite-helicity transverse cross section is obtained by subtracting Eq. (7.1) from the helicity-summed cross section given in Ref. [2]. The spin- and color-averaged cross section for qfTj ~ W~ZT in the high-energy limit is thus

do ut (o u + --d-~=38xZw(l_xw) s4 c~ ~u2 t2 j

(s>> M2), (7.2)

where i= up, down and CL is given after Eq. (3.3). The scattering angle is defined between the incoming quark and the W boson. Note that there is no s-channel contribution to the cross section since the gauge bosons have opposite helicities, and hence J~> 2.

The high-energy doubly transverse cross section (7.2) receives contributions from the t- and u-channel diagrams, both of which involve the coupling of the Z boson to fermions. Since this coupling is numerically suppressed, we expect the doubly transverse cross section to be smaller than the doubly longitudinal cross section. The ratio of the doubly longitudinal cross section (7.1) to the doubly transverse cross section (7.2) at 0 = ~/2 (t = u) is

( ( l~2xw) ( c [+ c~)2 ) ' = 9 ( 1 - x w ) (7.3) 8 x~


For Xw= 0.23, this ratio is quite large, roughly 16. Hence the doubly longitudinal WZ boson cross section dominates at high energies, rather than being suppressed, as is the case for the doubly longitudinal W and Z boson pair cross sections.

Fortunately, the signal for the production of most new, heavy particles, such as those listed in Section 1, is W and Z boson pairs, not WZ bosons. Thus the conclusions of this paper are not affected by the large percentage of longitudinal WZ bosons produced via quark-antiquark annihilation.

APPENDIX A

In this appendix I demonstrate the equivalence theorem for the process gg --. Z L Z L, shown in Fig. 3. My purpose is to demonstrate the theorem for a loop diagram; all previous examples have been for tree diagrams [3, 14, 16, 17].

There are six box diagrams in all, but it is easy to show that they form three pairs. A typical diagram, with momenta labeled, is shown in Fig. 12a. The other two diagrams are obtained by switching p~ ~ p~, and p~ ~ p~.

The equivalence theorem is proven by expanding the longitudinal polarization vectors to leading order in M/E, i.e.,

#{L = p~/Mz, e~e = p~/Mz.

It is straightforward to show (using techniques we will mention shortly) that terms involving the vector coupling of the Z bosons vanish. This is due to conservation of the vector current. We are left with

~ - - 2 A d4k IT r Pl. P2v F.VOa= _M_Tzf (_~n)4

where

Mz Mz I I

I I X 7 ° 7P j ~ 2 - - m P275 [f + p2 + p 3 - r n ~ +

1 1 1

1 1 1 x~ / ° /~+p l - -m 'P '75+Tr~-ZmP175~- -~ 1 - m

X T a T ° , t + ~#2 + j~3-- m P275 t + ~#3-- m (A.I)

g 2

4,o 0.)


,b~ k~-H-~ z

FIG. 12. (a) A typical box diagram for Z boson pair production from gluon fusion, with the momenta of the Orticles labeled. (b) The s-channel Higgs boson contribution to the process depicted in (a).

contains the coupling constants, and the factor of 2 is from the aforementioned pairing of diagrams. We write

#1 = (~ - m) -- (]~ - #1 + m) + 2m

#1 = (/~ + ,01 - m ) - - (/{ + m ) + 2 m ( A . 2 )

#I = ( ~ - r n ) - (~--#1 + m ) + 2 m

in the first, second, and third terms, respectively. The terms in parentheses cancel the denominators adjacent to them in Eq. (A.1). After performing various shifts of the integration variable k and using the cyclic property of the trace, all of these terms vanish, leaving only the terms proportional to 2m in Eq. (A.2). This corresponds to the fact that the axial-vector current is conserved in the limit of zero fermion mass. We then write

#2= ( J ~ + p 2 - m ) - (/~+m)+Zm (A.3)

# z = (~ - - m ) - (/~ - #2 + m ) + 2m

#z=(¢ + #2 + l~3-m)-(l~ + #3 + m)+ 2m

in the first, second, and third terms, respectively. The terms proportional to 2m correspond to the box diagram with Goldstone bosons replacing the longitudinal Z bosons--we have succeeded in replacing both #, and #2 with 2m inthe numerator of Eq. (A.I). Using the same procedure as before, we find that all but two of the terms in parentheses in Eq. (A.3) cancel, leaving only the term

4mA ( d4k 1 I I (A.4) 2 M-----ffj(-~)4Trl~+#4_mT~--~TPt~_#3_ m.

Now consider the s-channel Higgs boson diagram shown in Fig. 12b with the momenta labeled. The same diagram with the gluon lines crossed may be similarly


labeled, so we simply multiply by 2. Using e l = p' /Mz for the longitudinal polarization vectors yields an expression identical to Eq. (A.4) above, except for the Higgs propagator. Adding the two gives back Eq. (A.4) times a factor (to leading order in M2z/S)

s - m ~ 1

s - m ~ s - m ~

This corresponds to the s-channel Higgs boson diagram with Goldstone bosons replacing the longitudinal Z bosons. Hence our proof of the equivalence theorem is complete.

Note that the theorem is not valid diagram by diagram, only for the whole amplitude. It is not even valid separately for the box diagrams and the s-channel Higgs boson diagrams, except in the limit of massless quarks.

APPENDIX B

In this appendix I show that a longitudinal vector boson is a Lorentz-invariant concept in the limit M/E ~ O, where M and E are the gauge boson mass and energy, respectively.

A massive vector boson with four momentum

p"= (E, 0,0, p) (B.1)

has a longitudinal polarization vector

1 ~ [ = ~ ( p , 0,0, E). (S.2)

Consider a boost of velocity 13' in the y direction. The boosted momentum and polarization vector are

p'"= (~,'E, O, 13'7'E, p) (B.3)

1 , ~'" = ~t (7 P, 0, fl'7'P, E), (B.4)

where ? '= (1-13'2) m, as usual. The boosted polarization vector is no longer purely longitudinal, but is a linear combination of longitudinal and transverse pieces:

t# t,u a'" = fLeL +fTaT • (B.5)

Since only the longitudinal polarization vector has a non-zero time-like component, we may write

a ' ° = fLa~_ °. (B.6)

40 SCOTT S. D. WlLLENBROCK

Using this equat ion, a long with e l 0 = ( l / M ) [ P ' I (by definition), we find

7'p fL = (p2 + fl, 2 7, 2E2)m" (B.7)

We may rewrite this, using E = 7M, p = fl7M, as

f L = ( 1 fl;2~ 1/2 + f l - -~ / . ( B . 8 )

We interpret fL as the fract ion of the longitudinal polar izat ion vector which remains longitudinal after a t ransverse boos t of velocity fl'. Clearly fL must be < 1, which is manifest in the above expression.

Strictly speaking, a massless vector particle has no longitudinal component . In fact, the definition of the longitudinal polar iza t ion vector, Eq. (B.2), blows up as M ~ 0. However , f• is well behaved in the massless limit (which corresponds to fl --. 1, 7 ~ ~ ) , and is equal to unity. This means that a t ransverse boost induces no transverse c o m p o n e n t - - t h e polar izat ion vector remains purely longitudinal in all frames. Hence a longitudinal vector boson is a Lorentz- invar iant concept in the massless limit.

A simpler way to derive this result is to expand the longitudinal polar izat ion vector (B.2) in powers of M/E. To leading order, one finds

p/.t ~ = ~ . (B.9)

Since this is a Lorentz-covar ian t s tatement, it is true in all reference frames, and we recover our previous result.

It is not surprising that a longitudinal vector particle is a Lorentz- invar iant concept in the limit M/E--', O. It is well known that the helicity + 1 states of a massless vector particle are Lorentz- invar iant . We have just extended this to the longitudinal (helicity zero) states as well.

ACKNOWLEDGMENTS

I am grateful for many enlightening conversations with D. Zeppenfeld and X. Tata. This work has evolved over a long period of time, and I have benefitted from conversations with many people, especially V. Barger, C. Burgess, M. Chanowitz, S. Dawson, D. Dicus, M. Drees, M. Duncan, L. Durand, C. Goebel, K. Hikasa, G. Kane, F. Olness, R. Phillips, J. Pumplin, W. Repko, and M. Soldate. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, and in part by the U.S. Department of Energy under Contract DE-AC02-76ER00881.

PAIR PRODUCTION OF W AND Z BOSUNS 41

Note added in proof. The process gg --+ ZZ, discussed in Section 6, has recently been calculated in D. A. Dicus, C. Kao, and W. W. Repko, Phys. Rev. D 36 (1987), 1570.

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Pair production of W and Z bosons and the goldstone boson equivalence theorem