Decays of intermediate vector bosons, radiative corrections and QCD jets

Nuclear Physics B166 (1980) 4 6 0 - 4 9 2 © North-Holland Publishing C o m p a n y

DECAYS OF INTERMEDIATE VECTOR BOSONS, RADIATIVE CORRECTIONS AND QCD JETS David ALBERT, William J. M A R C I A N O and Daniel WYLER * The Rockefeller University, New York, N.Y. 10021, USA

Zohreh PARSA Department of Physics, New Jersey Institute o/' Technology, NewarK, New Jersey 07102, USA Received 24 October 1979

We investigate decay properties of the intermediate vector bosons W ~ and Z ~. Q E D and Q C D radiative corrections to leptonic and hadronic decay modes are calculated. Implications of the results for decay widths, branching ratios, determination of the n u m b e r of neutrino species, e-tx-~- universality and properties of hadronic jets produced in W ~ and Z ° decays are examined.

1. Introduction It is almost universally believed that weak charged and neutral current interactions are mediated by the massive intermediate vector bosons (IVB's) W ~ and Z°; so their discovery by the next generation of pp, Op and e+e - colliding beam facilities is anxiously anticipated. Indeed, these bosons play such a fundamental role in unified gauge theories of weak and electromagnetic interactions such as the phenomenologically successful Weinberg-Salam (WS) SU(2)x U(1) model [1] that a failure to find them would force a drastic revision in our present perception of weak interactions. In the standard WS model, the mass of the W ~ is predicted to be [2] ** Mw-~ 38.5/sin 0w G e V ,

(1.1)

and if only Higgs isodoublets are used to break the gauge symmetry then the Z°'s mass must be [2] M z = Mw/cos 0w = 77.1/sin 20w G e V ,

(1.2)

* Address after Oct. 1, 1979, Universitat Bonn-Physikalisches Institut Nussallee 1 2 - 5 3 - B o n n , W. Germany. ** The estimates in (1.1) and (1.2) incorporate effects of radiative corrections and are therefore about 3.2% larger than the lowest-order values usually quoted in the literature. 460

D. Albert et al. / Decays o/intermediate ~'ector bosons

461

where 0w is the weak mixing angle *. So the current experimental world average

[3] sin 2 0w = 0.23 + 0.02

(1.3)

implies Mw ---77-84 GeV and Mz-~ 89-95 GeV; values within reach of proposed accelerator facilities. Optimistically anticipating that the W ± and Z ° will be discovered in the near future and that high luminosity accelerators such as Isabelle [4] and LEP [5] will allow precise determinations of their properties, considerable labor has already been expended studying their theoretically predicted production rates and decay modes [6, 7]. In this paper we further examine predicted properties of the W ± and Z ° within the framework of the standard WS model ** and quantum chromodynamics (QCD) [8], the candidate theory of strong interactions. Our analysis extends and refines the results of earlier work on this subject in that we include QED and QCD radiative corrections to the lowest-order predictions. As we shall see, these corrections have interesting implications for predicted properties of the W ± and Z °. The topics which we will consider are effects of radiative corrections on decay widths and branching ratios, determination of the number of neutrino species via a measurement of the Z°'s total width, e-~-T universality and properties of hadronic jets produced by W ± and Z ° decays. The remainder of this paper is organized as follows; in sect. 2 we review some properties of the W ± and Z ° such as their couplings to leptons and quarks, lowest order partial decay rates and total widths. In sect. 3 we apply the results of our radiative corrections calculations to the topics described above and examine their implications. Finally, in sect. 4 we summarize and discuss our findings. Technical details of our calculations are given in the appendices.

2. Lowest-order decay rates We begin by considering the lowest-order decay of any generic spin-1 intermediate vector boson, denoted by B (subsequently we consider B--W ± or Z °) into a pair of spin-½ fermions fl + f2 which may be either leptons or quarks. Using a general Bflf2 coupling of the form (see fig. 1) *** Bflf2 coupling:

-igy u

a +2by5 '

* In this paper we define the renormalized mixing angle by the relationship cos 0w

M w / Mz. "* Our results are general enough to be easily applied to other weak interaction gauge theories. *** We employ the notation and conventions of ref. [33].

(2.1 a)

D. Albert et al. / Decays of intermediate vectorbosons

462

f~~f2 i

. , ,a*b1~,

-ig~pt~j

i B Fig. 1. General coupling between a generic intermediate vector boson B = W~ or Z °, fermion f~ and antifermion f2. where g satisfies

g2/8M2 = x/~Gv ~-8.25 x 10 -6 GeV -z ,

(2.1b)

Mw = mass of W ~ bosons and a, b are arbitrary, one finds that the lowest-order decay rate (in the B's rest frame) for B - * f l + f 2 is

j2m2"l/2

2

\ ml 2l F°(B-+fl+f2) = g 4 ~ B [ ( 1 Mzum2 M~)m2"2-4-h-~

2

[(a2+b2)(1 - ml

Xl_

k

2M 2

m2

1{ ml2

m 2,~ ~2,

2M .

.2x mlm2"]

t o g a - o ),--772-[ , MB (2.2)

where ml, m2 and MB are the masses of fl, fz and B, respectively. This result was obtained by squaring the amplitude in fig. 1, summing over final-state polarizations, averaging over initial-spin states of the B and performing the phase-space integra2 tions. For the cases of interest to us, M 2 >>mx,2, so we can set ml = m2 = 0 (to a very good approximation), then (2.2) simplifies to

g 2Mn (a z +

Fo(B ~ fl +fz) = 4---~~ .

b2),

for m l = m 2 = 0 .

(2.3)

The decay rates in eqs. (2.2), (2.3) can also be obtained from the one-loop fermion contribution to the IVB's two-point function (see fig. 2), i.e., the polarization tensor 1-I]~(q) found from all one-particle irreducible corrections to tlae IVB's propagator can be written as l-l~v(q) = A(q2)g~ + C(q2)q~,q~,

(2.4) fl

f~ Fig. 2. One-loop fermion contribution to the B's two-point function.

463

D. Albert et al. / Decays of intermediate vector bosons

and once A(q 2) is known, the total decay width F(B ~ all) is obtained via MBF(B ~ all) = Im A (q 2) Iq2=M 2

(2.5)

.

Partial decay widths into specific channels [such as those in (2.2) and (2.3)] are obtained from the subset of contributions to A(q 2) which contain the final-state particles as intermediate configurations (see fig. 2). We will have occasion to use the formalism in (2.5) when we discuss radiative corrections in sect. 3. Specializing now to the standard WS SU(2)× U(1) model with six leptons and six flavors of quarks with each quark coming in three colors as specified by QCD, one finds for the couplings of the W - to leptons and quarks the following values for a and b in (2.1a): Wdie coupling: Wq_1/3Ct2/3

a = - b = ~/21--,( = e,/z or r,

coupling:

(2.6a)

a = - b = x/~(KM),

(2.6b)

where (KM) is the Kobayashi-Maskawa [9] matrix element given by q-1/3\q2/3

fi

C

(2.7)

d

cl

S1C2

S1S2

S

--SLC3

CLC2C3 -- $2S3 e i~

¢1S2C3 + C2S3 e i6

b

-sis3

CLC2S3 + $2C3 e i8

C1S2S3 --

ci = cos 0i,

sg = sin 0~,

C2C3 e i6

with 0i, i = 1, 2, 3 the generalized Cabibbo mixing angles and 8 a CP-violating phase. Because of the Glashow-Iliopoulos-Maiani (GIM) mechanism [10], all couplings of the Z ° to fermions are diagonal in lowest order and the values of a and b in (2.1a) for B = Z ° are given by*: Zff couplings:

ae

m

2 cos 8w ( 1 - 4 sin 2 0w), 1

a~=-bv~

m

be = 2 cos 0w ~= e,

orT~

2cos0w' (2.8)

* In general, the neutral current couplings of any fermion with third c o m p o n e n t of weak isospin t~ and t3n for its left and right-handed components and electric charge Qf are given in the WS model by (2.1a) with af = (t R + t~ - 2Qf sin 2 0w)/COS 0w, bf = (t3a - t ~ ) / c o s 0w; the quantities in (2.8) follow from these general formulas.

D. Albert et al. / Decays of intermediate vector bosons

464

aqe/3

bq2/3 -

2cos1 ow(l_~sineOw),

aq 1/3 _ 2 c o--1 s 0 w (1 4_3sin • 20w),

-1 2cosOw'

qe/3=u,c, o r t ,

1 bq 1/3 2 c o s 0 w '

q_l/3=d, sorb,

where 0w is the weak mixing angle in (1.3). Using the couplings in (2.6)-(2.8), we can estimate (in lowest order) the partial decay widths of the W ± and Z ° predicted by the WS model. From the general formula in (2.3), (i.e., neglecting lepton and quark masses), we find the following rates *'**' W ± decays: 2

Fo(W- ~ ( + r e )

,

( - - e , tt or r,

(2.9a)

2M

F o ( W - - q-1/3 + C12/3)= ~ I K M I

e,

[see (2.7)]

(2.9b)

Z ° decays: F°(Z° ~ ( + {) -

g2Mz 1 96~r cos 2 0w (1 - 4 sin s 0w + 8 sin 4 0w),

2M

Fo(Z °-> ve+~e)

g z 96~'cos 2 0 w '

(2.10a)

I ( = e, Iz or z, (2.10b)

2M Fo( Z° ~ q2/3 + c12/3) 32~'c0s g ze0w (1 - sss m• 2 -0w+~32 sin 4 0w),

q2/3 = u, C, t, (2.10C)

Fo(Z° ~ q-1/3 + Ol 1/3) - 327rgZMz0w(1-4sina0w+Ssin40W)cos a ,

q 1/3 = d , s , b ,

(2.10d)

where the subscript zero on F reminds us that these are lowest-order predictions. In the above expressions involving quarks [(2.9b), (2.10c,d)], color has been summed over which introduced an extra factor of 3. Next, summing over the 6 quark flavors, (u, c, t, d, s, b) we obtain the total hadronic decay widths of the W :~ and Z °

3geMw Fo(W- ~ hadrons) = - - , (2.1 la) 167r * We note that the as yet undiscovered top quark's mass may be large enough to significantly alter the predictions in (2.9b) and (2.10c). ** We always give rates for W decays; the decay rates for W ÷ into charge conjugated final states are the same as these.

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D. A l b e r t et al. / D e c a y s o f intermediate vector bosons

3 ( 1 - 2 sin 2 0w + ~o sin 4 0w) Fo(Z° ~ hadrons)= 16~-g2Mzow cos 2

(2.1 lb)

Of course, this simple procedure assumes that the lowest-order decay rates into "free" colored quarks are not significantly modified by the dynamics of strong interaction hadronization effects (the process by which colored quarks are converted into uncolored physical hadrons); we examine this assumption and give the perturbative QCD corrections to these rates in sect. 3. Adding the leptonic decay widths in (2.9) and (2.10) to the total hadronic decay widths in (2.11), we find that for six flavors of quarks and leptons (as we have assumed) the predicted lowest-order total decay widths of the W ± and Z ° are *'**

g2Mw 4~r '

Fo(W -0 all) -

gZMz

Fo(Z°~all)

(2.12a) • 2

• 4

12gc---~sZ0w(3-6sm 0w+8sln

0w).

(2.12b)

From (2.9)-(2.12), the individual leptonic branching ratios of the W ± and Z ° are predicted in lowest order to be (for l = e, tx or z) B(W ~ d + i e ) -

F(W ~ ( + v e ) 1 F(W ~all) ° 1 2 '

=" F(Z°~ ( + *7) 1 - 4 sin2 0w+8 sin4 0w B ( Z ° ~ ( + e)-= F ( ~ ~l) - 2 4 - 4 8 sin 2 0w+64 sin 4 0w' F(Z °-~ re+ 17e) 1 . F(ZO__,all) - 2 4 - 4 8 s i n Z 0 w + 6 4 s i n 4 0 w

B(Z°~ve+ie) =-

(2.13) (2.14a)

(2.14b)

It is very important that the branching ratios into electrons and muons be large enough to provide clear experimental signals for W ~ and Z ° production. Another set of useful ratios is Rw ~-F(W ~ hadr°ns)=9, F(W --, ( + •,) Rz-=

F(Z°~hadrons) 1 8 - 3 6 sin 2 0w+40 sin 4 0w F(zO_~(+[) - l _ 4 s i n 2 0 w + 8 s i n 4 0 w ,

(2.15)

(2.16)

* If there are actually N > 6 sequential flavors of "light" quarks and leptons (mr < Mw), then the results in (2.11) and (2.12) are multiplied by ~N. However, in that case our nelgect of fermion mass effects for the new heavier flavors would be a bad approximation which we would need to rectify via (2.2). ** If additional light Higgs scalars are appended to the WS model, then the W ± and Z ° could also decay into these scalars in lowest order. We do not consider such a possibility in this paper. Z ° ~ &o+ $o (two identical scalars) is forbidden by spin-statistics.

466

D. Albert et al. / Decays of intermediate vector bosons

with ( - - e, # or r. These quantities are decay analogs of R -= ~r(e+e - ~ hadrons)/~r(e+e - -~ # +/x-) which is measured in e+e - annihilation. To numerically illustrate the magnitudes of the quantities just described, we employ the experimental range 0.21 ~
Table 1 WS model predictions for Mw and Mz along with lowest-order decay widths and branching ratios of the W- and Z ° in (2.9)-(2.16) corresponding to values of sin 2 0w between 0.21 and 0.25 sin 2 0w Mw(GeV) Mz(GeV) Fo(W ~ t'+ ~Te) Fo(W ~ hadrons) Fo(W- ~ all) B (W- ~ ~+ ~7~) Rw Fo(Z°-* C+ t7) Fo(Z° ~ ve+~e) Fo(Z°-~ hadrons) Fo(Z° ~ all) B ( Z ° ~ ( + {) B(Z ° ~ ve + ~e) Rz

0.21

0.22

0.23

0.24

0.25

84.0 94.5 0.259 2.33 3.11 112 9 0.095 0.185 2.26 3.10 0.031 0.060 23.8

82.1 93.0 0.242 2.18 2.90 !12 9 0.089 0.176 2.11 2.91 0.031 0.060 23.7

80.3 91.6 0.226 2.03 2.71 ±12 9 0.084 0.168 1.99 2.74 0.031 0.061 23.5

78.6 90.2 0.212 1.91 2.54 & 12 9 0.080 0.160 1.87 2.59 0.031 0.062 23.3

77.0 88.9 0.200 1.80 2.40 12

9 0.077 0.154 1.77 2.46 0.031 0.063 23.0

All quoted decay rates are in units of GeV. * The quantity sin 2 0w given in (1.3) (i.e., the value measured in present day neutral current experiments) may actually be slightly larger than sin 2 0w = 1 - M w2 / M z2 which is appropriate for our analysis. However, we expect the difference to be <~0.01. ** In obtaining these results we u s e d gZ/4q'r ~ 0.0078/sin 2 0w. We note that because natural units are employed (h = c = 1), 1 GeV corresponds to -~1.52 x 1024 sec -1 in the decay rates.

D. Albert et al. / Decays of intermediate vector bosons

467

The validity of the WS model and basic features of QCD will be tested; furthermore sin 2 0w will be precisely determined. [In principle, the generalized Cabibbo mixing angles Oi in (2.7) can also be accurately determined by measuring the various hadronic decay modes of W ± boson.] An anticipated use of a precise measurement of the Z°'s total decay width, F(Z°-~aI1), is to determine the total number of neutrino species, That is, in addition to the three known species Ue, U~ and ~,, there may be other massless neutrinos which belong to fermion generations containing very massive quarks and charged leptons. If they enter the WS model in the same way as the three known generations (sequential left-handed doublets and right-handed singlets), then the decays Z °-~ ~'i + ~i, where u~ represents the new neutrino species, occur with the rate given in (2.10b). So, assuming that the additional charged leptons and quarks that accompany the ui are too massive to contribute to the Z°'s width, then by measuring the Z°'s total decay width, one can in principle determine the total number of neutrino species N, *. It is given by N~ x F(Z ° ~ u + ~) = F(Z ° ~ all) - F(Z ° ~ hadrons) - F(Z °--, charged leptons). (2.17) For the WS model the lowest-order rates in (2.10) and (2.11) imply (for 6 flavors) N~ = 10 sin 3 20wF(Z ° + all) - (21-48 sin 2 0w + 64 sin 4 0w).

(2.18)

As a numerical example, take sin 2 8w = 0.23, then (2.18) becomes N~ - 5.96F(Z ° ~ all) - 13.3,

(for sin 2 8w = 0.23).

(2.19)

Of course, to pinpoint N~ from (2.18), sin 2 0w must first be precisely determined; in addition, other effects must be corrected for. One such calculable effect is due to radiative corrections; their effects on partial decay widths and N~ is examined in sect. 3. Another interesting question is: Are there deviations from e-/x-r universality in the leptonic decays of the W ± and Z°? Up to insignificant phase-space effects of 2 2 order m e/MB, universality (as in the WS model) implies F(W-~e+~e)=F(W

~U+~,)=F(W

--,r+~),

F(Z° ~ e + ~) = F(Z°-,/x +t2) = F(Z ° ~ r + ~).

(2.20) (2.21)

However, it was previously demonstrated in the case of W ± decays [12] that for typical experimental detection setups, electromagnetic (QED) radiative corrections can cause what appear to be large deviations from the lowest-order equality in (2.20). In sect. 3 we review those results and exhibit similar but even larger QED corrections for the decays Z ° ~ f + ~. What about the qualitative features of hadronic decays of the W ~ and Z°? In lowest order they decay into quark + antiquark which subsequently hadronize into * Arguments based on big bang cosmology and the observed helium abundance in the universe imply an upper bound of 3 or 4 on the total number of neutrino species [11].

468

D. Albert et al. / Decays of intermediate vector bosons

two high-energy narrow jets of physical uncolored hadrons. Within the framework of QCD, one can make such jet predictions quantitative by analyzing the gluonic corrections to the lowest-order quark decay rates. We examine this possibility at the end of sect. 3.

3. Radiative corrections In this section we examine the effects of QED and QCD radiative corrections on the decay rates of the W :~ and Z ° given in sect. 2 and consider their implications for branching ratios, determination of the number of neutrino species, deviations from e - ~ - r universality and properties of hadronic jets produced by IVB decays. In general, electromagnetic (QED) radiative corrections to IVB decays are of order a/rr ~4~6 and hence should not be very important unless they are enhanced by large logarithmic factors. Such enhancements do occur if we severely restrict the final-state configurations, as we shall see when e-/~-r universality is discussed; however, for unconstrained inclusive decay rates the QED corrections turn out to be fairly insignificant. By unconstrained we mean inclusive rates defined in the following way (here we consider the Z °, the W ~ is discussed in ref. [12]): F ( Z ° ~ f + f + (7)) =- F ( Z ° ~ f+ f)* F(Z°~ f + f + ~) +F(Z° ~ f + f + 2"y) + F(Z ° ~ f + f + pair of massless fermions) + ....

(3.1)

where ... denotes rates for higher-order final-state configurations containing any number of photons and massless pairs of charged fermions * in addition to the zeorth-order ff pair. The rate F ( Z ° ~ f + f + ('g)) is inclusive in that it includes all final-state configurations consistent with energy-momentum conservation [e.g., soft and hard bremsstrahlung are integrated over in (3.1)]. Using this definition, F(Z°-~ f + f + (~)) is guaranteed to be free of infrared photonic divergences by the BlochNordsieck theorem [13] and fermion mass singularities (ln mf terms) by the Kinoshita-Sirlin-Lee-Nauenberg (KSLN) theorem [14] order by order in perturbation theory; hence it should not differ appreciably from Fo(Z°~ f + f'), the lowestorder contribution. We note, however, that the KSLN theorem holds only if we do not introduce mass singularities via our renormalization prescription. Perturbatively expanding in c~ = 1~7, the usual fine structure constant would introduce In ( m f / M z ) terms in order a 2 and higher. To avoid such logarithms we expand in a(Mz), the effective coupling renormalized at q2 = - M z2 rather than 0. For later numerical illustrations we will employ the estimate [2] 1 a (Mz) ~- a (Mw) ~ 128.-----5'

(3.2)

• of course, in reality there are no charged massless fermions; however, for 4m 2 <
D. Albert et al. / Decays of intermediate vector bosons

469

I

*z a

~z i

4g

(o)

(b)

(c)

I

i

I

I

I

~Z,

f&Z

(d)

(e)

Fig. 3. F e y n m a n diagrams contributing to the order a radiative corrections to Z ° ~ f + f.

To order a(Mz), we find from the one-loop virtual photonic corrections and one-photon bremsstrahlung diagrams in fig. 3 that (they are evaluated in appendix A) 3 _za(Mz) O(a2)] ' F(Z°+ f + f + (Y)) = F°(Z°-* f+ f)[ 1 +4 Of 7r + J

(3.3)

with Fo(Z°~ f + f) the lowest-order rate in (2.10)* and Of the fermion's electric charge. As anticipated, the electromagnetic corrections are indeed small = 0.2%. Additional weak radiative corrections (which are also free of mass singularities) can be expected from virtual graphs involving W ±, Z °, Higgs scalar and fermion loops; however, because of our definitions of the renormalized quantities g in (2.1b) and sin 8w in (1.1) the residual weak corrections should also be fairly insignificant. The result in (3.3) can also be obtained from the two-loop corrections to the Z ° propagator illustrated in fig. 4 and the relationship in (2.5). In the massless fermion limit which we are considering, this calculation becomes equivalent to the famous Jost-Luttinger two-loop correction to the photon propagator [15]. Indeed, taking the imaginary part of their result, one recovers the correction in (3.3)**. This provides an additional check on our result. In the case of W :~ decay, the QED corrections to inclusive decay rates were calculated in ref. [12] and also found to be very small. Turning now to the OCD corrections to decays involving quarks, Z°-~ q + q, we make the observation (discussed in appendix A) that to leading order, the QCD corrections come from the diagrams in fig. 3 with photons replaced by gluons. * Unless otherwise stated, we always take mf = 0. ** The correction ~(~/~r)O 2 is also obtainable from the ratio of the second and first terms in the Q E D beta function. The O(o~ 2) corrections to (3.3) can similarly be found from the analysis of the photon propagator given in ref. [16]. (Modifying terms to include m a n y charged fermions.)

470

D. Albert et al. / Decays of intermediate vector bosons

f

f

f f Fig. 4. Two-loop electromagnetic corrections to the Z°'s two-point function.

Hence the QCD radiative corrections are obtainable from the QED corrections in (3.3) with the replacement

O ~ ~ a4s , (3.4) 2 where as = gs/4Cr (the QCD coupling) and the 4 factor comes from the quadratic SU(3) color Casimir invariant of the quark field. Therefore, defining a QCD inclusive decay rate in analogy with (3.1) but with gluons (g) replacing photons (~), we find to order as* from virtual gluonic corrections and one gluon bremsstrahlung (summing over color) F(Z° ~ q + ~1+ (g))= Fo(Z°-~ q + q ) [ l + c~s(Mz)+ O ( c 2 ) ] , q = any quark flavor.

(3.5)

Although this result corresponds to the case of free quarks and gluons, it should also carry over to physical uncolored hadrons in the final state. Indeed, the correction in (3.5) is exactly the same as the QCD corrections to R -= ~(e+e - ~ hadrons)/o'(e÷e - ~ ~ + ~ - ) obtained by Appelquist and Georgi and Zee [18]. For Z ° decays, the correction factor in (3.5) should be quite reliable since as(Mz) is presumably small as a result of asymptotic freedom. Employing the effective asymptotic coupling which comes from using the first term in the QCD beta function, one finds 6~r c~s(Mz) (3.6) (33 - 2Nv) In (mz/A) ' where Nv = number of quark flavors and A is a phenomenologically determined mass scale ~-0.2-0.5 GeV; so for Nv = 6, we expect as(Mz) =0.15 - 0 . 1 7 .

(3.7)

This range of values for as(Mz) implies about a 5% correction in (3.5) from QCD effects. Because color interactions are flavor blind, the QCD corrections in (3.5) also carry over to hadronic decays of the W boson, W ~ q-1/3 + ft2/3 (i.e., the diagrams in fig. 3 with y ~ g are also applicable to W decays). Therefore, for hadronic inclusive rates F(W--* q_1/3 + ~t2/3 + (g))= F o ( W - ~ q_a/3 + ft2/3)[ 1+ c~s(Mw)+ O ( ~ : ) ] .

(3.8)

* The O(a 2) corrections to (3.5) are the same as the a~ corrections to e+e-~ hadrons which were recently calculated in ref. [17]. They are small for a conventional definition of as.

D. Albert et al. / Decays of intermediate vect.or bosons

471

Since Mw and M z aren't appreciably different, we also expect the corrections in (3.8) to enhance hadronic decays of the W by about 5%. Because of the 1 + a~/Tr QCD enhancement factor, we find the following corrected decay rates for the W ± and Z ° [compare with (2.9)-(2.11)]:

eM

F(W- ~ q-1/3 + q2/3 + (g)) = g ~ I K M I 2 ( I + IO7/"

s(Mw)),

(3.9)

\

2M

g z 8 • 20w+~sin4Ow)[l+C~s(Mz)] F(Z°~ qa/3 + qz/3 + (g)) 32~-----s cos- Ow ( 1 - ~ s m t 7r J ' (3.10a)

gZMz Uw +, ~8 sin 4 0w) 1/3+(g))_32~.cosZ0w( 1 - 54 sin • 2~

I-(ZO~q_l/3+q

[ as(Mz)] x 1+ , 77

.I

F(W- ~ hadrons) = 3g2Mw( 1 + 167r \ F(Z ° ~ hadrons)

(3.10b) (3.11 a)

3g 2Mz _ 2 sin2 0w+~Q sin4 0w)[ 1+ °~s(Mz)] 167r cos 2 0w (1

(3.11b) while leptonic decay rates are unaffected. (We are neglecting the small QED and weak radiative corrections to inclusive rates.) Including these QCD corrections, the ratios in (2.15) and (2.16) become R w = 9(1 + as(Mw)/~'), Rz =

1 8 - 3 6 sin e 0 w + 4 0 sin 4 0Wfl 1 - 4 sin 2 0 w + 8 sin 4 0w L +as(Mz)/Tr].

(3.12) (3.13)

Similarly (but somewhat more complicated), the branching ratios in (2.13) and (2.14) are decreased by QCD corrections in their denominators. To numerically estimate the effect of the QCD corrections, we take for definiteness a~(Mw)= a~(Mz)= 0.15 and give in table 2 the modified decay rates and branching ratios corresponding to sin 2 0w in the range 0.21 to 0.25 which follow from (3.8)-(3.13). Compare tables 1 and 2. The differences which are due to QCD corrections should be measureable when the W ± and Z bosons are discovered and very precise data becomes available. Note, however, that some quantities such as decay rates are much more sensitive to small changes in sin 2 0w than QCD corrections. So, to reliably estimate those quantities, sin 2 0w must first be precisely determined. We expect a very precise determination to occur when the W boson's mass Mw is measured, so that (1.1) can be used to obtain sin 0w = 38.5/Mw or when the ratio M w / M z = cos 0w is accurately measured.

D. Albert et al. / Decays of intermediate vector bosons

472

Table 2 Q C D corrected decay rates and branching ratios of the W

and Z ° which follow from the

formulas in (3.8)-(3.11) sin 2 0w F ( W - ~ ( + ~Te) F ( W - -* hadrons) F ( W - ~ all) B ( W - ~ 4 + ~Te) Rw F ( Z ° ~ 4 + ~) F(Z°~ve+~e) F(Z ° ~ hadrons) F(Z ° ~ all) B ( Z °--* 4 + t7) B ( Z ° ~ ve + re) Rz

0.21

0.22

0.23

0,24

0.25

0.259 2.44 3.22 0.080 9.43 0.095 0.185 2.36 3.20 0.030 0.058 24.9

0.242 2.29 3.02 0.080 9.43 0.089 0.176 2.21 3.01 0.030 0.058 24.8

0.226 2.13 2.81 0.080 9.43 0.084 0.168 2.09 2.85 0,030 0,059 24.6

0.212 2.00 2.64 0.080 9.43 0.080 0.160 1.96 2.68 0.030 0.060 24.4

0.200 1.88 2.48 0.081 9.43 0.077 0.154 1.85 2.54 0.030 0.061 24.1

In obtaining these estimates we used a~(Mw)= a s ( M z ) = 0.15.

3.1. Number of Neutrino Species The lowest-order predictions in (2.18) and (2.19) for the number of neutrino species, N~, as a function of the Z°'s total width are also modified by QCD corrections. From the results in (3.9)-(3.11) we find that the formula in (2.18) becomes N~ = 10 sin 3 20wF(Z °-~ all) - (21 - 48 sin 2 0w + 64 sin g 0w) as(Mz) (18 - 36 sin 2 0 w + 4 0 sin g 0w).

(3.14)

71"

For as(Mz)=0.15 and sin 2 0w = 0.23 this implies N~ = 5.96F(Z ° ~ all) - 13.9,

for sin 2 0w = 0.23.

(3.15)

Compare (3.15) with (2.19). Notice that the lowest-order and QCD corrected predictions differ by almost one neutrino species (actually about 0.6). So, if QCD corrections are not taken into account, the value of N~ obtained by measuring F ( Z ° + all) and using (2.19) could represent an overestimate by almost one unit. Of course, a better way of determining N~ would be to separately measure F ( Z ° o all), F ( Z ° ~ hadrons) and F ( Z ° ~ ~'+ 2) experimentally and then use (2.17). However, even going by that route, one will have to contend with radiative corrections as we shall now demonstrate.

3.2. e-#-r universality An attractive feature of the WS model is that it naturally allows e-/x-r universality in the weak interactions, i.e., the fact that these three leptons have the

D. Albert et al. / Decays o/intermediate vector bosons

473

same charged and neutral current couplings. Universality is one of the basic guiding principles of weak interactions, primarily because e-/~ universality in charged current weak interactions has been well verified experimentally*. However, for the ~-, charged current universality is still an unsettled issue and in the case of neutral current interactions, e-~-r universality is as yet totally unexplored experimentally. Therefore, tests of e - # - r universality in the leptonic decays of the W ~ and Z ° are very important. As mentioned earlier, for some experimental detection setups, which because of the cuts they apply actually measure exclusive leptonic decay rates, one may observe what appear to be deviations from e-~-7- universality. Such deviations are due to large radiative corrections which result from restrictions on finalstate configurations imposed by the experimentalist. Such effects can be substantial and therefore must be corrected for before inferences from the data can be made. For the leptonic decays of the W-, W - ~ ( + ~ + (y), it was previously pointed out in refs. [12, 20] that if one requires that the energy Ee of the charged lepton lie in the interval E ~ - A E ~
F(W--~(+~e+(y),AE)=Fo(W-~(+ie)

[o{( 1 ~

Mw ) In Mw

41n2--~-3

mg

Mw

where ( = e , # or r and F o ( W - ~ ( + i e ) is the zeroth-order rate in (2.9a)**. Note that the radiative corrections in (3.16a) are free of infrared divergences because soft photon bremsstrahlung (photon energy ko ~ AE) collinear to ( has not been included due to the energy cut imposed. For AE small, the correction in (3.16a) can be quite substantial; for example, if AE = 1 GeV and Mw = 80 GeV, then the decay rate for W e + ~ e + ( y ) is reduced by about 16% below the zeroth-order prediction in (2.9a). For AE very small the corrections in (3.16a) must be exponentiated (i.e., summed

* We note, however, that at present the best test of charged current e-~ universality is the ratio F(rr + ~ e + + ~,e)/F(~r+ ~ t~~ + ~', ) and the theoretical prediction for this quantity differs from the experimental value by almost 2 standard deviations [19]. Hopefully, new experimental measurements will clarify this situation. ** If me is retained in (2.2), the phase-space corrections to (2.9a) are of order m~/Mw z 2 ; an insignificant effect for m~ <
474

D. Albert et al. / Decays of intermediate vector bosons

to all orders) because perturbation theory then breaks down. In that way (3.16a) becomes F(W- ~ ( + ~Te+ (,/), hE) = Fo(W- -->( + ZTe)

exp[T{(4'n -3) a

Mw

\ 1 Mw

Mw

(3.16b)

These rather significant corrections can be very important if one is using the electron decay mode as a test of the WS model; clearly radiative corrections must be taken into account for any such comparison between theory and experiment. The magnitude of the correction is easily understood on physical grounds, i.e., light charged particles when accelerated become efficient radiators of hard bremsstrahlung (particularly in the forward direction); so the lighter the lepton the larger the correction in (3.16). In the case of branching ratios, for simplicity assuming a single experimental cut AE = ±Eel = AEev one finds from (3.16a) F(W-~&+6~+(Y)'AE)-I-~--~

F(W---' & + ~6 + {~,), ~E)

c~(4 In Mw_3"] I n -m& -. 2AE

]

me1

(3.17)

Since an experimental deviation from 1 for this ratio might be naively interpreted as a fundamental deviation from universality, it is important to be cognizant of the corrections in (3.17). As a numerical illustration of the deviations from unity that one can expect, we have given in table 3 values for the ratios of possible leptonic decay rates, predicted by (3.17) for a variety of AE values [for ±E very small, exponentiated formulas have been used, see (3.16b)]. We have calculated in appendix B the QED radiative corrections to exclusive leptonic decay modes of the Z °. From those results, we can perform a similar universality analysis for the leptonic decays* Z°~ ( + 2 + ( T ) where ( = e , / x or r. In this case we consider two different types of experimental constraints on the final state charged leptons. (I) The sum of the energies of ~ and ( a r e required to lie in the interval M z AE ~ Ee + E~ ~ Mz.

(II) Each lepton's energy separately satisfies the constraint ½(Mz-AE) Ee, E~ ~ ½Mz.

In both cases we assume AE <( Mz for simplicity. These two distinct restrictions represent typical experimental cuts that might be applied to the set of all possible leptonic decays. For experiments which detect each lepton separately, the second constraint is much more natural; we consider the first one only because it is closely connected with the Sterman-Weinberg definition of a QCD hadronic jet which we * The decays Z°-->v~+ fie are unaffected by photonic corrections in order a.

475

D. Albert et al. / Decays of intermediate vector bosons Table 3 Leptonic branching ratios predicted by eq. (3.17) for a range of b E values 2~E(GeV)

0.1

0.5

1.0

2.0

5.0

F(W +e+ ~+(y), AE) F(W -~ ~ + ,7,,+ (y), aE)

0.87

0.91

0.93

0.94

0.96

F(W ~e+~e+(y), AE) F(W -~ r + ~ +(y), AE)

0.81

0.86

0.89

0.91

0.95

F(W --, ~ + ~,~+ (y), bE) 0.93 F(W-~ ~+ G +(y), AE)

0.95

0.96

0.97

0.98

Numbers quoted were obtained using Mw = 80 GeV and o~(Mw) = 121.5.

will subsequently discuss. In the case of configuration I, we find (see appendix B)

Mz

mz 121]

- ~ { (4 In 2 - ~ - 3) In ~-e - 3~" ,1_1' (3.18) while for configuration II we find Fn(ZO

(+{+(y),AE)=Fo(ZO

(+2)[1

a Mz M z 1 2/1 - ~ { (4 In S - ~ - 3 ) l n - ~ - e - ~ r ~J, (3.19)

where Fo(Z°~ ( + Z) is the lowest-order decay in (2.10a). These results exhibit the following noteworthy features: the leading logs (mass singularities) in (3.18) and (3.19) are multiplied by 2 relative to the analogous terms in (3.16). This factor results because now there are two charged particles in the final state which can radiate hard bramsstrahlung and both of them are being experimentally restricted to have nearly maximal energy. The rate in (3.18) is larger than that in (3.19) (remember AE <
476

D. Albert et al. / Decays of intermediate vector bosons

therefore, one should sum up higher-order infrared effects in which case (3.18) and (3.19) exponentiate to*'**

[o( Mz ) ln--~ee+zl] ga

FI(Z°~ e+ ~+ (y), 2rE) = Fo(Z°~ g+,f) exp - ~ 4 In 2 - - ~ - 3

,

(3.20) FH(zO~g+~+(y),AE)=r,o(ZO

g+eV)exp[_~(41nM~_3) l n .~~M e +z lazr] . (3.21)

These rates vanish, as they should for AE-~ 0 due to radiation damping. Clearly the radiative corrections to Z ° decay, when experimental restrictions such as I or II are applied, are of such a large magnitude that they must be taken into account before useful information can be extracted from experimental measurements. Regarding e-~-~" universality in the leptonic decays of the Z °, we find from (3.18)-(3.21) for 41, dE=e, tz or r FI(Z°~gl+71+(Y),AE) FI(Z°~ ~2 + ~ + (y), AE)

1_~(41 n Mz_3~lnme2 2AE / me~ Mz m erJ

FII(Z°-~t~+g~+(3,),~E) , o~/,, M z FH(Z°-" (2 + gz + (Y), AE) = l - g~,* m ~ -

,,\ me2 J) In m~,x

(3.22a) (3.22b) (3.23a)

=exp I - a ( 4 In ~ - ~ - 3 ) I n me:] . melJ

(3.23b)

We have given unexponentiated and exponentiated forms. To illustrate the magnitude of the deviations from unity one can expect from the effects of radiative corrections, we list in table 4 the predicted branching ratios for a variety of AE values, using Mz = 92 GeV and a (Mz)= ~-~.5.1 Compare tables 3 and 4. Notice that because of the extra factor of 2 in the radiative corrections to Z ° decay relative to the corrections for W- decay, the values in table 4 are shifted about twice as much as those in table 3 from unity. Once again we find that one might misconstrue an experimental measurement of the leptonic branching ratios different from 1 as a fundamental breakdown of e-/z-~" universality if QED radiative corrections are not properly accounted for. Of course, if one does not experimentally constrain the * For a discussionof the exponentiationof infrared effectsin QED, see ref. [21]. ** The division of non-logarithmicterms into exponentiated and unexponentiatedfactors is somewhat arbitrary.

D. Albert et al. / Decays of intermediate vector bosons

477

Table 4 Leptonic branching ratios of the Z ° predicted by the WS model formulas in (3.22b) and (3.23b) for a range of AE values AE(GeV) FI(Z ° + e + E + (y), AE)

0.1

0.5

1.0

2.0

5.0

0.75

0.82

0.85

0.88

0.93

0.65

0.74

0.78

0.82

0.89

0.86

0.90

0.92

0.94

0.96

0.73

0.79

0.82

0.85

0.89

0.61

0.70

0.74

0.78

0.84

0.84

0.88

0.90

0.92

0.94

FffZ°-+/, +12 +(y), AE) FI(Z ° ~ e + 6 + (y), AE) Ft(Z ° ~ r + ~ + (y), AE) F I ( Z ° ~ # +/~ + ("/), AE) FI(Z ° ~ ¢ + 'r + (y), AE) Ih ( Z ° ~ e + e + (y), AE) F~,(Z° ~ tz +fi + ( y ) , AE) Fn(Z ° ~ e + E + (y), AE) Fu(Z~)-+ r +'Y + ( y ), AE) Fii(Z ° -~/z + 5 + (T), AE) FII(Z 0 -~ r + ~ + (y), AE)

Numbers quoted were obtained using M z = 92 GeV and a ( M z )

--

1

128.5.

final states but instead measures inclusive leptonic decay rates (i.e., include all hard bremsstrahlung configurations) then the QED corrections become just those in (3.3) and e-#-r universality is clearly manifested.

3.3. Hadronic jets Because of color confinement (a much anticipated but as yet unproven property of QCD), the W ± and Z ° will actually decay into colorless physical hadrons rather than free quarks and gluons. However, fortunately another property of QCD, asymptotic freedom, assures us that the free quark calculations and gluon corrections that we have employed to estimate the hadronic decay rates of the W ± and Z ° should be reliable. The scenario we envision is as follows: at very short distances (q2= M 2 or M 2) strong interaction effects should be fairly insignificant because the effective QCD coupling as(q 2) is small. Therefore, although nonperturbative hadronization effects will subsequently become large, they are unimportant when the decay is initiated and hence do not influence the total hadronic decay rate. Furthermore, because as is small, we trust the perturbative corrections in (3.5). Indeed, the use of free quark calculations in conjunction with perturbative QCD corrections has been somewhat validated by the excellent agreement between experimental measurements of R =- o'(e+e ~ hadrons)/o'(e+e ~ tz +tz-) and theoretical QCD predictions. In the case of W ± and Z ° decays, q2= M 2 and MZz,

478

D. Albert et al. / Decays of intermediate vector bosons

values considerably larger than the momentum transfers in present day e+e - annihilation experiments; so our use of QCD perturbation theory to study these decays should be even more justified. It has been pointed out by Sterman and Weinberg (SW) [22] that the appearance of hadronic jets (narrow energetic conical streams of hadrons) in high-energy processes can be easily understood within the framework of perturbative QCD. They suggest a scenario in which jetlike configurations of quarks and gluons are established at short distances and then materialize as final-state jets of physical hadrons. Perturbatively, this makes sense because the initial quarks and gluons do tend to emit other gluons and quark-antiquark pairs in their forward direction (when accelerated). However, this explanation of jet production makes the unproven (although plausible at high energies) assumption that the dynamics of quark-gluon hadronization does not disturb the initial jets' kinematic structure. We accept the SW picture of jet production in QCD and now examine its consequences for hadronic decays of the W ± and Z °. We begin by reviewing the Sterman-Weinberg criterion for a two-jet event as applied to IVB decays*. SWjets: An SW two-jet event for IVB ~ hadrons is one in which all but at most a fraction • <~1 of the total available energy, E = MB in the IVB's rest frame, is emitted within a pair of oppositely directed cones of opening half-angle S <<1, i.e., the total jet energy must satisfy the constraint M B - •MB ~
D. Albert et aL / Decays of intermediate vectorbosons

479

undergo the same QCD renormalization. We have already noted this feature for QCD corrections to inclusive hadronic decay rates [see (3.5)]; it is also true for the case we are now discussing, hadronic jet production. Therefore, we can use the results of Sterman and Weinberg [22] to discuss hadronic jet decays. For the SW definition of a jet event we find (see appendix C for details) that the decay rate of an intermediate vector boson B = W ± or Z ° into two SW jets is given by* 12 ~] , F(B ~ 2 SW jets) = Fo(B ~ hadrons) [ 1 4 as(MB){(41 n 2e + 3) In c~+ 57r - 5} 3 ~r (3.24)

where Fo(B ~ hadrons) is the zeroth-order decay rate in (2.11). The fraction of all hadronic decay events which should satisfy the SW criterion is given by f

F(B~2SWjets)[1 ---

=

4as(MB){(41n2e+3)ln6+lrr2-7}]

(3.25)

3

where the denominator includes the 1 +as/~" QCD correction factor. The results in (3.24) and (3.25) are identical to the formulas derived by Sterman and Weinberg for jet production in e+e - annihilation, as advertised. Now compare (3.24) with (3.18). Notice the similarity under the replacement 8--; mdMz. Indeed, if one performs a calorimetric leptonic detection experiment in which the total energy of the charged leptons and their accompanying photons deposited in a small solid angle is measured rather than just the energy of the charged leptons, then the corrections in (3.24) (with a~as~ a) would be appropriate rather than those in (3.18). If E is very small, then just as in the photonic case, we expect the logarithmic corrections in (3.24) and (3.25) to exponentiate because of radiation damping. So, for example (3.25) should be modified to [24, 25] +3)lnS+1rr2_7}]. f-~ exp [ 43 as(MB).{(41n2e 7r

(3.26)

This fraction vanishes in the limit e ~ 0 as it should. The SW jet predictions just given for IVB decays are much firmer than the corresponding results for present day e+e - experiments because oe~ is smaller and non-perturbative fragmentation effects which our analysis neglects should be less important. Experimental confirmation of the above jet predictions would provide strong evidence in support of QCD. A modified definition of a hadronic jet was recently given in ref. [24] which may he more appropriate for IVB decays (depending on the detection setup). Defining an equipartitioned (EP) 2-jet event to be one in which all but at most a fraction * Because we assume e, 8 <( 1, terms proportional to E and 8 have been dropped.

D. Albertet al. / Decaysof intermediatevectorbosons

480

• << 1 of the total available energy is emitted within a pair of oppositely directed cones of opening half-angle 6 << 1 and demanding that each jet's allowable energy individually satisfies the constraint

½(MB-EMB)<~Ei ~<~MB,

i = 1, 2

(3.27)

also leads to jet decay rates which are free of infrared divergences and quark mass singularities. The equipartitioned definition is, however, more restrictive and thus for a given (8, e) leads to smaller decay rates than the SW definition. (Equipartitioned jets automatically satisfy the SW criterion; but the converse is not true.) In addition, because the equipartitioned jet definition is more selective, it provides a cleaner experimental signature. Obviously, if each jet's energy is separately measured, then the equipartitioned jet definition is more appropriate while for calorimeter experiments which detect total energy deposited, the SW criterion is more sensible. Employing the equipartioned criterion, we find (see appendix C for technical details) that the decay rate for an intermediate vector boson B = W ± or Z ° into two EP jets is given by* F(B ~ 2 EP jets)= Fo(B ~ hadrons)[ 1 43 as(MB)~r{(4 In • + 3)ln 6 + ~rr 2 - ~}], (3.28) where Fo(B ~ hadrons) is given in (2.11). The fraction of all hadronic decays which satisfy the EP jet criterion is given by f,_F(B~2EPjets) [1 = F-~ ~ r o - - ~ s ) -

4 as(MB){(41ne+3)lnS+~.2_¼}] 3 7r

(3.29)

or, if we exponentiate, f' -~ exp

[ 4as(MB){(41ne+3)lnS+~Tr2-7}]. 3 7r

(3.30)

In the region where our calculations are valid, E << 1 and 8 <( 1, f' is less than f as required by definition. If both f and f are measured, then the quantity

f-f~_l-exp[~adMB)(41n21n6+~rr2)]

(3.31)

which represents the fraction of SW jets that do not satisfy the equipartitioned criterion provides a good quantitative check of QCD. In fact, since (3.31) is a ratio, some uncertainties common to the numerator and denominator should cancel; so it may be more reliable than the individual predictions for f and f'. Note, it is independent of e. * Notice the similarity between (3.28) and (3.19).

481

D. Albert et al. / Decays of interrnediate vector bosons Table 5 QCD predictions for f, f', and ( f - f ' ) / f from (3.26), (3.30) and (3.31) corresponding to a variety of E, 6 values (E, 6)

(0.05, 0.10)

(0.05, 0.15)

(0.05, 0.20)

(0.10, 0.15)

(0.10, 0.20)

(0.15, 0.20)

f f'

0.36 0.27 0.26

0.43 0.34 0.21

0.48 0.40 0.16

0.60 0.47 0.21

0.64 0.53 0.16

0.75 0.62 0.16

(f-f')/f

These estimates were obtained using as = 0.15.

To numerically illustrate the results of this subsection, we take as(MB)=0.15 and display in table 5 the QCD predictions for f, f ' and (f-f')/f corresponding to a variety of small e and 8 values. Notice, from the numbers in this table, that most hadronic decays of the W ± and Z ° are predicted to be very jet-like (highly energetic cones of hadrons with small opening angles). Indeed, for • = 0.05 about half of all hadronic decays satisfy the SW 2-jet criterion with S ~<0.2 = 12 °. For as(MB) = 0.15, (3.31) becomes

f-f'-

1 - 1.11~ 0 " 1 7 6

(3.32)

.

f This formula exhibits a sensitive dependence on 8 which we have plotted in fig. 5.

1.0 -

f_fl

f 0.5-

0.1 I 0.02

I O. I0

i 0,20

Fig. 5. Graph of the fraction ( f - f ' ) / f of SW jets which do not satisfy the equipartitioned 2-jet criterion, plotted as a function of 6 [see (3.32)]. For this estimate as = 0.15 was used.

482

D. Albert et al. / Decays of intermediate vector bosons

When precise experimental measurements become available, verification of the prediction in (3.32) can provide a nice check on QCD. 4. Discussion We have seen that both QED and QCD radiative corrections can be importar~t for accurately predicting the decay properties of the W :~ and Z °. In the case of inclusive decay rates, QED corrections [see (3.3)] were found to be fairly insignificant; however, for exclusive rates limited by experimental acceptance constraints [see (3.15) and (3.16)] the corrections were quite substantial because of the appearance of large logarithms. These large logs depended on the masses of the radiating fermions and therefore could fake a deviation from e-tz-~- universality if not separately accounted for in comparing theory with experiment. QCD corrections to inclusive decay rates were found to increase hadronic decay modes by about 5%. Although this effect is not very large, it can be important for specific applications. For example, in attempts to determine the total number of neutrino species from a measurement of the Z°'s total width, we found that unless QCD corrections were taken into account one would overestimate the actual number by almost one neutrino type. An important labor saving feature of QCD corrections to IVB decays that we noted was that they are often identical to corrections previously calculated for e+e annihilation into hadrons. So, for example, we were able to directly apply the theoretical QCD analysis of Sterman-Weinberg [22] and of equipartitioned jels [24] to IVB decays. Based on the results of those considerations, we expect rather spectacular highly collimated jet events (small opening angle) to dominate the hadronic decay modes of the Z ° and W :~. When the Z ° and W ± are finally discovered, precise measurements of their jet decay properties will provide important quantitative tests of QCD via the predictions in (3.24)-(3.32). In summary, discovery of the W ± and Z ° will mark an important step in the advancement of our understanding of the fundamental interactions. Their very existence will validate non-abelian gauge theories as legitimate descriptions of the weak interactions and measurements of their masses and decay properties will test the Weinberg-Salam model's specific predictions and provide a new way of determining sin2 0w and the KM matrix elements. Subsequent precise experimental measurements will probe the effects of radiative corrections that we have described in this paper; thus making the Z ° and W ± laboratories for exploring weak, electromagnetic and strong interactions. We wish to thank E. Derman, A. Sirlin and H.-S. Tsao for discussions related to this work. We also thank the Aspen Center for Physics and Stanford Linear Accelerator Center for their hospitality during the completion of part of this work. This research was supported in part by the US Department of Energy under Contract Grant No. EY-76-C-02-2232B.*000.

D. Albert et al. / Decays of intermediate vector bosons

483

Appendix A Radiative corrections to inclusive decay rates

In this appendix we calculate the order a electromagnetic corrections to the lowest order decays Z°-~ f + f. The result obtained also provides the order as QCD corrections to hadronic decays of the W ± and Z ° under the replacement aO~ -~ 4C~s as we will demonstrate. Our procedure is to set rnf = 0 from the start and use dimensional regularization [26] to simultaneously regularize both infrared divergences and fermion mass singularities which arise at intermediate steps of the computation [27]. In this way the singularities are manifested as poles at n (the dimension of space-time) equal to 4. At the end of our calculation, poles from virtual loop corrections cancel against bremsstrahlung contributions so that the limit n + 4 can be taken. Details of this regularization prescription along with all dimensionally generalized formulas that we need are given in ref. [27]. As a check we have also regularized this calculation by giving the proton a small mass ,~ (still setting mr-- 0) [22, 23]. For comparison we exhibit some of the results obtained by that method. In computing the decay rates for Z ° - ~ f + f and Z ° + f + f + 3 , in n dimensions, we employ the couplings and propagators illustrated in fig. 6. The factor (M27/4~r) 1 ,/4 has been introduced to keep the couplings dimensionless* and to reduce the f

_

f

T

)4

÷

~ Z I

I3

- Lg)(ij( ~ " ~ ) ( M~/4"rr) 14 f

i' Photon

K

LDp~(K)=-i. K z+ [E

Fermion

£P

L

[S(£)= +is

Fig. 6. Dimensionally generalized vertices and massless propagators used for the calculation of the order o~ radiative corrections.

* Introducing the unit of mass M z m e a n s that the expansion p a r a m e t e r will be a ( M z ) , the effective coupling at q Z = _ M 2.

484

D. Albert et al. / Decays of intermediate vector bosons

number of superfluous terms which could otherwise enter various parts of the calculation and eventually cancel among themselves [27, 29, 30]. We begin by computing the zeroth-order decay rate for Z°-~ f + f in n dimensions. ,From the diagram in fig. 1 with B ~ Z ° and f~ = f2 =-f, we find the lowestorder decay amplitude ~/[o = -"g( l M~/4~,~1_~/4 xz t~r.). u. .f .t .t ) y x - -a- ~+vby5 . . . . . e(q)

(A.1)

where e~(P) is the polarization vector of Z °. To obtain the decay rate Fo(Z°+ f + f), we work in the rest frame of the Z °, square the amplitude in (A.1), sum over fermion polarizations, average over initial polarizations of the Z ° and integrate over phase space, i.e.,

- ~1 f f d n 1l d R lq F°(Z°~ f+ f) = 2Mz J J 2/o(21r) ~-1 2qo(2rr) ~-1 (2~)n6n(P-l-q)½~po~ IM°12 ' (A.2) where

12 2 2-n/2 2 2 [ 2P.lP.q] 2 [Mol2 - 3g ( M z / 4 ~ ) (a + b ) l. q-~ -~/-~-z pol

'

(A.3)

The phase-space integrations were carried out in ref. [27]; the result is

2 Fo(Z°-+ f + f ) = g ~ ( a

1 ! 2 + b2)r F(gn ,,

(A.4)

where F is the gamma function (not to be confused with the decay rate). For n = 4, (F(2) = 1), this reduces to the result previously given in (2.3). To obtain the order a corrections to this lowest order rate, we next compute the one loop virtual corrections in figs. 3a,b,c and then add to it the photon bremsstrahlung rate corresponding to figs. 3d,e. Virtual Corrections: The three diagrams a,b and c in fig. 3 are separately gauge dependent and individually ultraviolet divergent (except in the Landau gauge rl = 0 [27]); however, their sum is gauge independent and free of ultraviolet divergences. From explicit calculation, we find

Ma+Mu+Mc=~o(-1)

22]

,/2 2aO~ F(3-½n)F2(½n-1)[ - 8 - ~ F(n-2) [(n-4) e n-4

(A.5) where a = e2/4~r and A/o is the zeroth-order amplitude in (A.1). Notice that (A.5) contains a double pole at n = 4 which corresponds to the product of an infrared

485

D. Albert et al. / Decays of intermediate vector bosons

divergence and mass singularity• The factor (-1) ~/2 2 when expanded about n = 4 gives (_l)n/2 2= l+i½~-(n -4)-~'trZ(n - 4 ) 2 + ....

(A.6)

The second term in this expansion gives rise to a relative imaginary pole while the third leads to a very important finite contribution. The imaginary pole terms can be summed to all orders in perturbation theory and thereby produce an infinite coulombic phase factor (they exponentiate) which cancels out in the amplitude squared ]dg]2; so they don't contribute to the rate. This extraneous infinite phase factor is a well-known classical phenomenon due to the long-range nature of the coulombic interaction between charged particles in the final state [31]. Adding the virtual corrections in (A.5) to (A.1), we obtain the order a corrected decay rate F(Z ° -+ f + f) = Fo(Z ° ~ f + f) a O 2 F(3 - ~1n ),.2,1 t ~n - 1)

x 1

2~

./2-2

8

2

F(n-2)

(A.7)

where Fo(Z ° ~ f +f) is given in (A.4). If one instead uses a photon mass regularization scheme, the corresponding result is F(Z°+f+f) = Fo(Z°+f+f) 1-

41n2(a/mz)+61n(a/mz)+~-~rr

I

(A.8) Bremsstrahlung: To complete this calculation, we need the dimensionally regularized rate for F(Z°+ f + f + y) that results from the diagrams in figs. 3d,e. To order a this decay rate is given by

1 II I d~ 11 d,-lq F(Z°+f+f+Y)=~-zz 2Io(2rr)" a 2qo(2rr),-i dn-lk

x 21- . . . . ~(2rr)'an(P-k-l-q)½ ~O~£~T)

y Iddd+ddel2'

pol

(A.9)

where 2 2 ,/2 , ~ f+l( a+bys ~ , dda=-igOfe(Mz/4Cr) ez(P)ev(k)&(1)Y~2-~, k y X - - - ~ v ~ t q ) ,

•

2

~[~ = t g O f e ( M z / 4 1 r )

2-n/2 ~

,, _ q+l( a+by5 ez(P)ev(k)uf(l)yx=----Ty~vr(q) Zq'K

Z

(A.10)

(A.11)

D. Albert et al. / Decays of intermediate eector bosons

486

correspond to figs. 3d,e, respectively. Squaring the amplitudes, performing, the polarization sums and using P = l + k + q, p2 = M~, l 2 = q2 = k 2 = 0 gives 2 I~g~+~gel ~ = gg ' 2 Uvlz/'c'rr) "~'~ ,4 ~,(a ~+b~)O~e~(,~_2) pol

[-4+

qMzi :+~ q : k

'

(1.12)

Integrating over the momentum of the photon, performing the angular integrations and changing to dimensionless variables x and y,

lo = ½xMz,

qo = ½yMz,

(1.13)

we find _ - =, aOf2(n -2) dF(Z°~f+f+Y)dx dy I'°(Z°-'t+ t)8-7~½ ~) (x+y - 1),,/2-2

,

x(1-x-y±xy)

2+2+

,~/2_212(x [

y ) (n-4)(2-x-y) ~

2

] -J.

(A.14)

If instead we first integrate over fermion momentum q and use 1

ko = ~zMz ,

(A. 15)

then we find d F ( Z ° ~ f + f + y)

dxdz

r ,,-,o f-~,~O2(n--2)

1)n/Z_ 2

--otZ, -+ + ~ 8rrr(--ln)- ( x + z -

×(1-X-Z+XZ) n/2 2[-4+ 4(1-z)+(n-2)z2]~~zz~ j .

(A.16)

To obtain the inclusive rate, we integrate over x and y in (A.14) using only energy conservation as our constraint (i.e., all possible decay configurations) 1

1

•~ ,~o - = . a O 2 ( n - 2 ) F ( Z ° - - > f + f + Y ) = l ° t L - + t + t ) 8-~F(--~ni f dx f 0

x(1-x-y*xy)

,./2_2[-2(x

2.}_

[

dy(x+y-

1)n/2_ 2

1 x

y ) ( n - ) (42 - x - y ) ~ 2+

2

-I ].

(A.17)

Making the change in variable y = 1 - vx, this becomes 1

1

aO2(n - 2 )

F(Z°-~f+f+'/)=Fo(z°-'f+f)

dt'x~-4

8rrr(ln)Idxl o

0

X (1 --x)n/2-3(1 -- V) n/2 2vn/2 312-4xv +2x2+2v2x2+ (n -4)(1 - x + vx)2]. (A.18a)

487

D. Albert et al. / Decays of intermediate vector bosons

Performing the integrations using 1

I d x x r l(l_x)m I

F(r)F(m) - F(r+m) '

(A.18b)

0

we find F(ZO f+~+y)=Fo(ZO

f+~)ceQZF2(an-1)[

8

+ 12 + +1] (A.19)

If one instead uses a photon mass regularization scheme, the corresponding result is 2

F(Z°--> f+ f + y) = Fo(Z°-~ f + f)-~-t [4 In2 (A/Mz)+ 6 In (A/Mz) +5

~ ~ ~ 2~

(A.20) Adding (A.19) to (A.7) [or (A.20) to (A.8)] and taking the limit n -->4 gives F(Z° ~ f+T) + F(Z°~ f + f + y) = Fo(Z° ~ f + f)[ 1 +~ - ~ +

O(c~2)] .

(1.21)

This is the correction to the inclusive decay rate given in (3.3). Our calculation carries over to the gluonic QCD corrections to hadronic decay rates when we sum over color. To order c~s the QCD diagrams are just those in fig. 3 with the photons replaced by gluons. Then the only real difference between these calculations is an extra factor of 1~ Lt" = 41 coming from the SU(3) matrices at the gluon-quark vertices. So, (A.21) is applicable to the QCD corrections with c~O~ 4~s. Therefore, the QCD corrections are

where we have used the effective coupling c~s(Mz) which is appropriate for this calculation (Mz is the only mass scale).

Appendix B Radiative corrections to exclusive decay rates

In this appendix we compute the order a electromagnetic corrections to the exclusive decay rates FI(Z°-> f + f + (y), AE) and FII(Z°--, f + f + (y), AE) which were defined and used in sect. 3 for our discussion of e-#-~- universality. To simplify things, we actually evaluate the rate for those events that do not satisfy the imposed phase-space constraint and subtract our result from the total

488

D. Albert et al. / Decays of intermediate vector bosons

rate F(Z°~f+f+(,/)) = Fo(Z°+ f+f)[1 +3(a/rr)O2]. In this way, infrared photonic divergences don't appear and mass singularities can be regularized in the limits of integration; therefore, n + 4 can be taken in (A.1.4) and (A.16) for this set of calculations. Those rates become, for n -+ 4, dF(Z°+ f + f + y) = Fo(ZO_,f +~)a02 x2+y 2 dx dy 27r ( 1 - x ) ( 1 - y ) '

(B.1)

dF(Z° ~ f + f + ~')= Fo(ZO__,f + h _ ~ [ 1--Z+Z2/2 (1 -x)(x +z - 1)

dx dz

1],

(B.2)

where x, y and z are the dimensionless energy variables of f, f and 3' respectively (see appendix A). For the inclusive rates we now consider, the allowed values of x, y and z are constrained by the particular restrictions imposed.

Configuration I: The sum of the energies of f and f are required to lie in the interval M z - AE < Ef + E~ < Mz (where AE <(Mz) or 2 - 22xE/Mz < x + y < 2. This constraint implies that the photon's energy must be less than AE (it doesn't allow hard bremsstrahlung) i.e., z <~2AE/Mz. For this case we employ (B.2). Using the complementary part of phase space (as explained above), we find FI(Z ° -+ f + f + (y), A E ) = F o ( Z ° ~ f + f')

(Mz--2mf)/(M z rnf)

x{ l+3aOr2 4

a02

I

Xmax

1-z+½z2

dz y dx[

(1--x--Tx-77-1)

7r

2AE/Mz

1]} '

Xmin

(B.3) where

Xmax=l_½z+½z(1 -

4m~ ~1/2 M~(1-z)] ' 4m~

Xmin=l--½Z--½Z( 1 M ~ " ~ 2 )

(B.4)

~1/2 }

•

Note that we keep the fermion mass in the limits of integration when it is required to regularize mass singularities. Making the useful change in variables x = 1-vz, carrying out the integration and dropping terms proportional to m f2/ M z2 and AE/Mz we find

---~--t

Mz

Mz

1 2 )']

(B.5) This is the result used in (3.18) for f = g (lepton) and O 2 = 1.

D. Albert et al. / Decays of intermediate vector bosons

489

Configuration II: Each fermion's energy must separately satisfy the constraint ½(Mz-AE)<~Ef, E?<~½Mz,(AE <
~O~ o~O~/

Fie(Z°-+ f + f + (Y)' AE) = F°(Z°~ f+ f)[ 1 +~ ~ --

-~t

f

dx

2mf/Mz Ymax

1-AE/M

z

Ymax

x~+y ~ x f

x~+y ~ ~]

dY(l_x)(l_y,+

Yrnin

f

dx f

1/2

x

dY(l_--~--y)lj,

(B.6)

where Y

max mln

[1 -Sx+5(x 1 1 2 - 4 m f2/ M z2)

1/2 2

2 2 ] +mf/Mz 1-½x + ~(x2- 4m2/M2z )l/2

(B.7)

The fermion mass has been retained only when necessary. Carrying out the phasespace integrations, we find F n ( Z ° + f + f + ( y ) , A E ) = F o ( Z ° ~ f + ~ )LI I - ~ O¢ ~r t/ ( 4 1 n M~--~--3 z ) In Mz ~ 2 }1 . mf

(B.8) This is the result given in (3.19) for f = g (lepton) and 02 = 1.

Appendix C

QCD /et calculations In this appendix we compute the QCD predicted decay rate of any intermediate vector boson B = W ± or Z ° into two hadronic jets. The cases considered are Sterman-Weinberg (SW) jets [22] and equipartitioned jets [24] (these two distinct definitions are described and discussed in sect. 3). As in appendix B, we avoid regularization by actually calculating the complementary phase-space integrals (i.e., the rate for events which do not satisfy the 2-jet criterion) and subtract the result from the total hadronic decay rate F(B + hadrons)= Fo(B + hadrons) [1 + a,(MB)/~r].

SWfets: For the SW definition of a 2-jet event (see sect. 3), we must include all phase-space configurations in which all but at most a fraction e <~1 of the total

490

D. Albert et al. / Decays of intermediate vector bosons

energy MB is emitted in a cone of opening half-angle 6 <<1. The complement of such configurations are those events in which more than the fraction • of the energy is emitted outside such a cone. To order a~ such complementary events can occur only if their is a gluon in the (perturbative) final state (B ~ q + fl + gluon). Therefore, we can use the differential decay rate in (B.2) with aO 2 gas 4 to describe them:

dF(B+q+cl+g)-Fo(B~q+q)~adM")( dx dz --~\il

1--Z+½Z2

1)

(C:I)

- x ] (-x--~z- 1)

(color has been summed over). Because we are neglecting terms proportional to • and 8, the phase space of the non-SW 2-jet events is quite simple in the z, x variables so that F(B ~ 2SW jets) = Fo(B -+ hadrons) 1

F(8)

X[~as(MB) f dz -~ 2~

f

dX(il 1 - z + ½ z 2 -x)(x+z-1)

1j,/]

(C.2)

l--F(8)

where COS2(~

F(6) = 1 - z sin26

(C.3)

is the rate for B ~ hadrons which do not satisfy the SW criterion. (Because we sum over color and flavor, Fo(B ~ q + Cl) and Fo(B ~ hadrons) are interchangeable.) Making the change in variables x = 1 + vz and integrating over v and z, we find 4 as(MB) e,1 2 73 F(B ~ 2SW jets) = Fo(B ~ hadrons) ;[(4 In 2e + 3) In o -- ~zr - aJ. ..3

"/7"

(C.4)

Therefore, the complementarity procedure gives F(B + 2SW jets) = Fo(B + hadrons)[1 + as(M~)/~']- F(B e; 2SW jets),

(C.5)

which from (C.4) becomes F(B+2SWjets)=Fo(B+hadrons)

4 as(MB){( 4 1 2 s~] 1- 5 ~r In2e+3)lnS+xTr-gtJ. (C.6)

This is the result given in (3.24). It of course agrees with the original formula derived for e+e - annihilation into 2 jets by Sterman and Weinberg [22].

Equipartitioned jets: For the more constrained equipartitioned 2-jet configurations (see sect. 3) it is convenient to use the differential decay rate in (B.1)

491

D. Albert et al. / Decays of intermediate vector bosons

(with

aO 2

--, ~%) to describe B - * q + q + g l u o n , i.e.,

dF(B~q+q+g) Fo(B _~q + _ c~s(MB) x 2 + y 2 dxdy q) ~ ( 1 - x ) ( 1 - y ) '

(C.7)

Then the decay rate of B not going into equipartitioned 2-jet configurations is given by (always neglecting contributions proportional to • and 3) 4 as(MB) F(B ~ 2EP jets) = Fo(B ~ hadrons)~ -7r 1/2

Vmax

1 ~= x2+y

x{ / O

dx I

2

dY(1-x)(1-y)~-I

Ymin

dx 1/2

Ymax

I

x

x2+y

2

,t},

(C.8)

where Ymin

I--X

1-x sin 2 6"

Ymax-

1 - 2 x sin 2 8 + x 2 sin 2 c~ 1 - x sin 2

(C.9)

Performing the integrations gives -, ! 2 7/ In • + 3)In ~ ,-6,]7" --41' F(B-~ 2EP jets) = Fo(B + hadrons)~4 a~(MB){(4 ~"

(C.10)

Now applying the complementarity procedure we find F(B + 2EP jets) = Fo(B 4 hadrons)[1 + ~s(MB)/rr] - F(B -~ 2EP jets),

(C.11)

which from (C.10) gives s] F(B ~ 2EP jets) = Fo(B ~ hadrons)[ 1 4 a~(MB){( 4 In e +3)In {~+ ~1- 2 -3} L 3 ¢r (C.12) This is the result given in (3.28). It was first derived in ref. [24] for the case of e+e - annihilation into jets. We end this appendix with a word of caution. The jet formulas that we have derived are purely perturbative results. To correct for non-perturbative hadronization (or fragmentation) effects, one must supply some phenomenological input, see for example ref. [32].

References

[1] s. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in Elementary particle physics: relativistic groups and analyticity (8th Nobel Syrup.) ed., N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) p. 367.

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D. Albert et al. / Decays of intermediate vector bosons

[2] W.J. Marciano, Phys. Rev. D20 (1979) 274. [3] C. Baltay, Proc. 19th Int. Conf. on High-energy physics, Tokyo, 1978, ed. S. Homma, M. Kawaguchi and H. Miyazawa (Phys. Soc. Japan, Tokyo, 1979) p. 882. [4] H. Hahn, M. Month and R. Rau, Rev. Mod. Phys. 49 (1977) 625. [5] L. Camilleri et al., CERN Yellow Report 76-18 (1976); Proc. LEP Summer Study (1978) CERN Yellow Report 79-01. [6] C. Quigg, Rev. Mod. Phys. 49 (1977) 297; R.F. Peierls, T.L. Trueman and L.-L. Wang, Phys. Rev. D16 (1977) 1397. [7] J. Ellis, in Weak interactions-present and future, Proc. SLAC Summer Inst. on Particle physics, 1978, ed., M. Zipf. [8] W.J. Marciano and H. Pagels. Phys. Reports 36 (1978) 137. [9] M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [10] S. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [11] J. Yang, D. Schramm, G. Steigman and R. Rood, Astrophys. J. 227 (1979) 697. [12] W. Marciano and A. Sirlin, Phys. Rev. D8 (1973) 3612. [13] F. Bloch and A. Nordsieck, Physi. Rev. 52 (1937) 54. [14] T. Kinoshita and A. Sirlin, Phys. Rev. 113 (1959) 1652; T. Kinoshita, J. Math. Phys. 3 (1962) 650; T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) B1549. [15] R. Jost and J. Luttinger, Helv. Phys. Acta 23 (1950) 201. [16] E. DeRafael and J. Rosner, Ann. of Phys. 82 (1974) 369. [17] M. Dine and J. Sapirstein, Phys. Rev. Lett. 43 (1979) 668. [18] T. Appelquist and H. Georgi, Phys. Rev. D8 (1973) 4000; A. Zee, Phys. Rev. D8 (1973) 4038. [19] D. Bryman and C. Picciotto, Phys. Rev. D l l (1975) 1337; W. Marciano and A. Sirlin, Phys. Rev. Lett. 36 (1976) 1425. [20] T. Appelquist, J. Primack and H. Quinn, Phys. Rev. D7 (1973) 2998. [21] D. Yennie, S. Frautschi and H. Surra, Ann. of Phys. 13 (1961) 379; K. Erickson, Nuovo Cim. 19 (1961) 1010; N. Papanicolaou, Phys. Reports 24 (1976) 229. [22] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436. [23] J. Bell and R. Jackiw, Nuovo Cim. 60A (1969) 47; S. Adler, Phys. Rev. 177 (1969) 2426. [24] W. Marciano, D. Wyler and Z. Parsa, Phys. Rev. Lett. 43 (1979) 22. [25] A. Smilga, Phys. Lett. 83B (1979) 357. [26] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189; C. Bollini and J. Giambiagi, Nuovo Cim. 12B (1972) 20. [27] W. Marciano, Phys. Rev. D12 (1975) 3861. [28] T. Appelquist and J. Primack, Phys. Rev. D4 (1971) 2454. [29] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [30] W. Marciano and A. Sirlin, Nucl. Phys. B88 (1975) 86. [31] N. Papanicolaou, Phys. Reports 24 (1976) 229. [32] G. Curci, M. Greco and Y. Srivastava, Phys. Rev. Lett. 43 (1979) 834. [33] J.D. Bjorken and S.D. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964).