- Email: [email protected]
Relations between the on-shell and MS frameworks and the mw-mz interdependence
Nuclear i Nort - oll d
331(1991)
h fd
efived. These expressions bridge the MS and ful to ctwnpute -Ir and sin' O w in the case of large inter, 'ence is expressed in two equ °alent ways: in ter o r a meter ® m ,. /saî~ ® ,( ). A li s;lins o the expressionss derived i illustrated nu ri l .
e relation 1 sin 20 W=
2
__
ntw(1 -`1r)
where sing ®w = 1 _M2 ,/mZ- s -', A = (za/ GA ) 1/2 _ (37.2803 ± 0.0003) Ge and ®r is a radiative correction, plays a significant role in current discussions of electroweak physics. For example, for given values of mz, m, and MH, eq. (1) allows us to compute sine ®w. This in turn can be used Lo predict mw, can be compared with other determinations of sine ®w to constraint the value of m, [2,31 and can also be used to predict asymmetries and other observables at the V-peak, as these are in many cases sensitive functions of this parameter. An analysis of the mass singularity structure of the radiative corrections to tL-decay has shown that insertion of Jr in the denominator of eq. (1) incorporates not only the leading logarithms of ®((a ln(m Z/m f))") There mf is a generic 0550-3213/91/$03 .50 ©1991 - Elsevier Science Publishers B.V. (North-Holland)
G. Degrassi et al. / ni «,.-mZ interdependence
50
fermion mass) but also, to a good approximation, the terms of O(a 2 1n(m Z /m f )) [4]. With increasing indications that the top quark is heavy, physicists have begun to study in greater detail and depth the effect of large m, on the radiative corrections . In particular, Consoli et al. have recently investigated the terms of O((afn2/1112)`) in the corrections to a-decay [5]. Describing the dominant contri(c 2 )Sp, where /s 2 butions to yr in terms of Ea = e2(I7yy(0) - H( O(tn 2 )) and function fermionic vacuum polarization one-loop unrenormalized IIYy(q 2) is the and 5p = 3afn2/16r~s -2 n1 2 , these authors found that the one-loop calculation of A r, when inserted in eq. (1), incorporates correctly the O(a 2 ) corrections of O(E.2) 2 and O(E.Sp) to ,u-decay but there is a mismatch in those involving 0((5j5) ). The first result is already contained in the conclusions of ref. [4] but the second one is novel and interesting for large int. In fact, the analysis of the terms of O((aü1i /i)1 2 )2 ) is unrelated to mass singularities and is therefore not covered by the methods of ref. [4]. In the present paper we present a formulation that automatically takes into account the reducible terms of O((am i/m w )" ), leading to simple relations between s' and the MS parameter sine âw(m z ) (henceforth abbreviated as s-' ) and suggesting useful novel ways of calculating Ar and other important quantities . We recall that use of the MS parameter s-' has recently led to the introduction of two new basic corrections of the electroweak theory, ®r [6] and vîw [7]. The relevant relations, suitable counterparts of (1), are ,. s-I =
,
rt122W (1 - ®rw)
A) A)
s c- =
mZ(1 - dî)
.
These equations summarize the meaning of ®î w and ®r : the former is the radiative correction that must be applied when s 2 is determined from a, G,, and mw , while the latter plays an analogous role when s 2 is extracted from a, G,, and mZ . The first step in the present analysis is to elucidate to what extent the one-loop corrections ®r and Arm, incorporate the leading higher-order contributions in eqs. .,nd (3). These arise from two distinct sources: (i) effects induced by resummation or iteration of one-loop effects and (ü) higher-order one-particle irreducible contributions . Throughout this section we will consider the first class which, for brevity, will be referred to as reducible higher-order contributions; in sect. 2 we will discuss how to incorporate the second class to O(a2 ) . The leading reducible contributions can be studied by a procedure suggested by the ingenious analysis of ref. [5]. One notices that these corrections are contained in the purely fermionic contributions to the /,-decay amplitude. In turn, the latter are given by _72=
8S()7 1 M 2M
+
(e021S20)~A") ww (0)]
+ ... ,
G. Degrassi et al. / in ,,- in z interdependence
51
where the 0 subscripts indicate unrenormalized parameters, (e02/s2 )A~~,(0) is the unrenormalized one-loop fermionic WW self-energy evaluated at q2 = 0, and the ellipses represent the bosonic contributions of O(a) as well as additional contributions of higher order. The parameter e2 in eq. (4) means 2 _
eo
e2 (1 -e 211(f)(0))
,
the familiar charge renormalization of QED. (The sign of IIYy follows the traditional conventions and is opposite to the one employed in ref. [5].) The authors of ref. [5] showed that the explicit term in eq. (4) is ultraviolet convergent to O(a2 )* and reached their conclusions concerning the leading higher-order reducible corrections by expressing this contribution in terms of the renormalized parameters e 2, s 2 and mom,. We now notice that an analogous analysis of eq. (4) in terms of e 2 , s2 and mw leads to e2/8^2M2 s
G~ [1 -e 2llyy(0)
- (e2/s2)Re((Aww(mw) -(0))/M2
where MS denotes both the MS renormalization [i.e . the subtraction of terms involving S = (n - 4) - ' + (y - ln(47))/2] and the choice tL = mz for the 't Hooft mass scale. The result (6) can readily be obtained by inserting eq. (5), s= ; s- - S 2 and in w(, = mw - (e /s~ ) Re A(f, w(m ~,~, ) into eq. (4). The explicit term in eq. (4) becomes e'/8s'mW
[1 -e - Ilyy(0) - (e`/s- ) Re((Aww(m 2~,) -Aww(~))/rnw) - (gs-/s-')(1 -e-IIYY~(0))]
We now notice that in the MS scheme 5-s2 involves only divergent terms proportional to powers of S. When only fermionic contributions are included, as the answer is convergent through O(a 2 ), (,5s2/s2)(1 - e 2 llyy (0)) must exactly cancel the terms proportional to a(S + ln(mz/A)) in the second and third terms and cannot contain contributions of O(a2 ). This result has been confirmed by an explicit calculation through O(a 2) and leads immediately to eq. (6). The last term in the denominator of eq. (6) does not contain mass singularities or terms of O(m2/m 2 ). Thus, the leading contributions are given by -e 2IIyy'(0)Nts and are logarithmic in character. But these are precisely the leading corrections contained * The mass counterterms associated with the virtual fermions are needed to cancel divergences in the irreducible two-loop corrections . Therefore, it is understood that they are included in the latter.
G. Degrassi et al. / m`l.-»:Z iwerdepeiideaace
'52
in Itw ! Further, e -2HYy (0)Ms and E « contain the same mass singularities and are of the same order of magnitude. Thus, we reach the conclusion that when the one-loop expression 1'îw [7] is inserted into eq. (2), there are no unaccounted terms of O(E 2 , E Sp, (Sii)2) [8]! We also note that the factor (1 - A w) in eq. (2) transforms the a present i A 2 into a/(1 - _41^ w), which may be interpreted as an effective a evaluated at inz. q. (6) contains i s compact form the one-loop fermionic corrections to j,-decay as well as all the higher-order reducible contributions that arise solely from them. In order to obtain the complete one-loop expression for ,Irw we must add the O(a) bosonic contributions to the selfenergies, as well as the vertex and box diagrams in the tL-decay amplitude. We recall that, when both one-loop fermionic and bosonic corrections are included, eq. (5) is replaced by 1
e"
e(1 + 23e/e)
(7a)
where (cf. eq. (26) of ref. [1]) 2 Se
7e 2,
e
8â-
(S +In(
rft
1
lu
21
Including the additional contributions we obtain (8a) îw=
e2
s
112
Re
Aww(mw) - Aww( 0 ) MW
e2 7 In c 2 6 + , . 16 2g2
5
s2
-
25e e
5
Ms _-S2
3 c2 2 c2
(8b)
The A ww self-energies include now the fermionic and bosonic contributions of 0(a). Their detailed exprcssions are given in ref. [9] (see also the appendix of the present paper). In the last term we have exhibited explicitly the dependence of the vertex and box contributions on c 2 and c 2 = m W/mZ . This dependence can be gleaned from sect . 4 of ref. [10] (cf. eq . (4.5) et seq.). Analogous dependences are present in Aww . If one replaces c'- , s 2 --> c 2, s 2 in the O(a) contributions, eq . (8b) * In this reference the coupling constant g -' = e 2/s' is included in the definitions of A WW (g 2 ), A z7 (g 2 ) and A,,z(q°) . In the present paper this coupling has been factored out . Similarly, e2 has been factored out in the definition of H.,,(q`) .
G. Degrassi et al. / in iv-mz interdependence
53
becomes identical to eq. (6) of ref. [7]. As it will be explained later, the difference between the two procedures is of O((a/2ws 2 ) 2 mi/mz) and represents subleading ®(a 2 ) contributions. An alternative expression is obtained by keeping the vertex and box diagrams contributions in the numerator,
Ge
-
e2
1 + (é-'/16îr2~2)( V+ B) (e2/s2) 8s 2m2w [1 + 28e/e Re((Aww(mw) - Aww( 0))/m 2w)]is
Here V + B is an abbreviation for the expression between curly brackets in eq. (8b) (i.e. the vertex and box diagram contributions) and e 2 is the renormalized version of eq. (7a) in the MS scheme*, i.e. -2e
e2
(1 + 28ele)Ms
The difference between (8a, b) and (8c) is < 0 .01 %. Turning our attention to eq. (3) and comparing with eq. (2), we note that
where c2
P - c2
mW m2c2 Z
.
(10)
To evaluate P we write s2 = s 2 - 8s 2 = s2-5s 2 and note that sô = 1 - Yt1 Wp/mzo. Thus, we have 8s 2 Y-
(5
= -C2y'
/mw - 8mz/mz) (1 - 8mz/mz)
2 MW
where Buzz and 8naw are the mass renormalization counterterms expressed in * This
definition does not follow exactly t"e usual conventions because eq . (8d) contains contributions from the top quark even when it is more massive than in z. Numerically, the differences are quite small even for in, = 250 CieV (see appendix A).
G. Degrassi et al. / in `t-ntz interdependence
>4
terms of unrenormalized coupling constants. Therefore, we find c-' =c -' - C 2 Y - 5S2 =C-'(1 - Ys),
(12a)
1
(12b)
YMs)
(1 -
Use of eq. (9) then yields Ar = A1`,`r
- PYMS( 1-
(12c)
A w) .
To evaluate YMs it is useful to note that eq. (1 lb) can be written as Y=
&nW «)
in W
bin2
-
,
in -Z
(13a)
(l-Y),
leading to Sm 2W
5171 z
inw
YMS
,. .. Ar = Ar W
-
In zP
MS
j5in 2
amW
P
(13b)
2
in W
mZ Ms
,. (1 - ArW) .
(13c)
To obtain expressions for p = c 2 /c'- and s 2 c 2 inZV G,,/7a that contain all the ®(a) terms and incorporate the leading reducible contributions, we insert the expressions for Sm z/m 2 and Bin w /m w into eqs. (13b, c) and obtain YMS
e2 ~2Pmz
-
e2
Re
~2
P
S2
®r =APW -
l
ê2 S
2
Aww(mw) 12
-A ZZ ( M22 m
(A y z( mz)/mz) Re 1 + e211yy(MZ) (1 - L1iW )
+(1 -®Pw )
2
rn z ^2 L'
s
,-
Re
2
e
MS
2
(14a)
-' MS
Aww(mw) ^2 C
2
-Azz(mz)
(A yz(M2 )lin2 )2 1
+ L' 2 ~j yy
(in2
)
MS
(14b ) _ MS
G. Degrassi et al. / mu,- mz interdependence
55
In these equations the A's and II,y,, stand for the one-loop expressions. They are given in ref. [9] (the bosonic parts of Ayz(mz) and HY ,y (m 2 ) are studied in greater detail in ref. [11]). It is worth noting that tadpole contributions cancel in eqs. (14a, b). The last terms in these expressions are O(a 2 , a 3 ) and represent the contributions of yZ-mixing to 8mz (cf. eq. (22d) of ref. [11]). As it was the case with eq. (8), eqs. (14a, b) contain all the higher-order reducible contributions that arise solely from the one-loop fermionic contributions. We note that A yz(m 2 )/mz does not contain mass singularities or terms proportional to M2/M /m z . Therefore, neglecting the very small O(a 2 ) terms involving this quantity, eqs. (14a, b) simplify to YCIS
ê2
Pm2
s
z
,. ,, Ar = ®rw -
Aww(mw) e2
Re ê2 s2
( 1 - ~rw) m2Z
- Azz(mz)
(15a)
Ims
Aww(mw) c2 - Azz(mï)
Re
(15b)
Ims
Use of eqs. (8b) and (15a, b) leads, without further approximations, to ®r =
e2 S2fn2z
+
Re Azz(mz) -
e2p 167r2s2
In C2 7
6+
s
2
A ww(0)
2-
C2 5
-S
2
28e
- e
- s2
5-
Ms
3
2
c2
c
It is instructive to consider the leading one-loop contributions for large can be gleaned from eqs. (B.1) and (B.2) of ref. [9], namely Re A ww = -
3 16 ;r2
3
Re Azz=-16
~r,
mt 1 mt 8 + In ( ~ ) - ) + . . . ,
4
m~ e2 0
8+1n
Mt
~
+
(15c)
2
Mt .
These
(16a)
(16b)
where the ellipses represent non-leading contributions of OW . Inserting these 2 (a + In(m,/ti)) expressions into (15a, b) we see that the terms proportional to 111
G. Degrassi et al. / m in,-mz interdependence
56
cancel exactly and we are left with YMS
3a in 2 ... , 167rs 2 fnw + 25e e
(16c)
3â(1 - ®r w ) in, 16-rrs -c - mZ
Ms
where â = ê-'/47,-. If non-leading contributions of O((a/27rs 2 ) Z mi/mZ) are neglected, we can set c -', s -' -~ c-, -2 (i.e. P --~ 1) and â(1 - Arw) --> a in the O(a) terms. In that case (15b, c) coincide with the results of ref. [6]. We reach the conclusion that when the one-loop expression for Jr is inserted into eq. (3), there are no unaccounted reducible contributions of O(E2 , Ea Sj5' (Sp ) 2 ) [8]! It is important to note that, in order this result to hold, the O(afnt/in Z) term must be written as in eq. (16d). For example, - 3anz t / 16 7-s- fn w, which coincides to O(a ), will not do because it does not incorporate certain contributions of O((ari1i /m z )2). It is also worthwhile to point out that the explicit term in eq. (16c) is the well-known Veltman correction [12". Again it differs from the more familiar form 3G. rn t/ C 87r` in subleading terms of Ma/2 7g2)2 in ë /jn z ). Eqs. 05a-c) contain all the O(a) terms as well as the leading higher-order reducible contributions . They are therefore very useful for practical calculations . 20
elations between
e on-shell and
S frameworks
Having shown the desirable properties of ®P w and d r explained in sect. 1, we can now apply them to obtain accurate expressions and relations for other important quantities . The parameter p = C2/e2 can be calculated from eq. (9). It can also be expressed as P where ®P =
1 (1 _ ® P)
( ®r w _'I r)
(17a)
= YMS .
(17b)
From here we find c2 = c 2 (1 - ®P), or s2 = s2 1 +
c2
Al . ®P s2
(18)
G. Degrassi et al. / m ",-mZ interdependence
Alternatively, we have c2
=c ep,
57
or
®r w - ® r 1 - Arw
c2 s 2 = e2 1 _ s2
(19)
Note that (®r w - ®r)/(1 - Ar"w ) = p Ap and that eq. (19) is obtained from eq. (18) by interchanging s2 H s2, r w H r. Comparing eqs. (1) and (2), we haves2(1 - r u, ) = s2 (1 - Ar ). Use of eq. (18) leads to 1 -Ar= I1
+ s2 ®p I[1 -AM, c2
(20)
or c2 Ai- = 4î'W - ., ®p(1 - ® r w ) s-
.
(21)
Recalling (9), (10), (17b) and (19), we can also write eq. (21) as dr =arw -
( A w -ar) - (C2/S2)(A1 . s2 [1 p)/( e2
)] .
(22)
Eqs. (19) and (22) are particularly useful for practical calculations . Because Jr w and AP depend very mildly on in, the iterative evaluation of s -' from m Z, based on [6] 1
s2 - -
2
1-
1-
1/2
4A2 mz(1
,
_Ar)
( 23)
converges quickly. Inserting fir, GPw and s 2 into eqs. (19) and (22), we obtain then determinations of s 2 and Ar that automatically take into account the leading reducible terms. The mw-m Z interdependence can be expressed in the usual manner, m 22u mz
1 =2 1+ 1-
1/2
4A2
mz(1
-
(24)
3r)
Alternatively, using eqs . (10) and (23), we have mw mz
11
2
-
4A2 m7P(1 - ~~ w)
1/2
58
G. Degrnssi
t sal,
f aaa
t~-aar Z interdependence
Some of these equations have a structure reminiscent of the approximate expressions of ref. [5]. For example, ft,-jr the leading terms, the authors of ref. [5] find 2 and an expression 1 - ;Ar = (1 - E.)(1 + (r'/s2)Sp), where Sp = 3nî2, G11/8 ,- -'~_ that looks like (25) with -Ai"w -* t*. . and P -~ p = 1 /(1 - Sp). An obvious advantage of eqs. (20) and (25) is that they- include the complete O(a) corrections in a very simple way (not just the leading asymptotic terms), and moreover they suggest that the natural parameter in discusskng the rn w-in z interdependence is P rather than p. The two quantities approach each other asymptotically for large ni t , but there are significant differences. This is also intuitively natural, as one expects that a relation between nz w and should significantly involve the on-shell selfenergies as in (15a) and (15b), r they than the self-energies evaluated at q' = 0. It is important to emphasize that, i P and ,r are given by the expressions of this section, then eq. (24) and eq. (25) are, in fact, identical! This observation may perhaps serve to clarify some ° ather confusing statements made in the recent literature concerning the validity- of eq. (24) [5]. /j't 2 )') can be included by simply The irreducible contributions of ()(C,)(111 2/111 replacing everywhere
--~ ,r -F
19) (,rw -,dr)2 (1-Ar) ' 3
(27r ` -
(26)
where :,r w and Jr in the r.h.s. stand for the one-loop expressions considered before . Indeed, when the secon term in (26) is added to the r.h .s. of eq. (15b) and the resulting expression is inserted into eq. (9), one obtains 1 Ms
(2 77.2 _ 19) 3
(
Ms)
2
(27)
where YMs is the one-loop cor ection given in eq. (15a). We now note that the leading irreducible corrections given in refs. [5,13] are expressed in terms of G~m2 and that, to leading order, one can do the same with YMs [cf. the discussion after eq. (16d)]. Therefore, eq. (27) can be directly compared with the results given in those papers and one then verifies that the last term in this equation correctly contains the irreducible corrections of 0((G~ m 2)2) or, equivalently, O(«2(m2/mw)`). If there are triplets in the theory, P is no longer calculable via eq. (9), but it becomes an additional paramete r that must be extracted from experiment [14]. Writing P5() = 1/( 1 -®P() )for the tree-level expression and assuming that the radiative corrections are given to good approximation by those of the minimal
G. Degrassï et al / aadj -m z interdepeadeetre
519
theory, we have
with AP given by eq. ( l
3. In order to evaluate the basic corrections r^w and .hr^, we employ eq. (8b) eq. (15b), respectively [alternatively, FF can calculated from eq. (15c)ß. e correction lr w is dominated the contributions of leptons an five quark flavors to - (2Se%) , which amounts to 0. (see a evaluating the self-energies and the vertex and box contributions in eq. (8b). well as in the second term of eq. (15b), two strategies present themselves : (1) ee explicit the distinction between r-' = m2 /In 2 and P (and re between s= and s -') or (2) replace in the a) corrections / z = c2 -j, c and similarly s 2 --q2, with other mass ratios expressed in to of z. as in 2t/ and n4I /mz. The difference between the two approaches are subleading corrections of 0((a/2-s-' j2)-'M2/1,1 z ). They are expected to quite small in -lrw and x -4 fir, roughly = 2 10 even for mt as large as 250 GeV. (It should remembered, however, that in calculating 1r, differences in lrw - A ; are enhanced by a factor of c -'/s2 [cf. eq. (22)] and therefore may be as large as (0.8-1 .5) x 10 - 3 for in, = 250 GeW The simpler strategy (2) was used in refs. [6. 7], where the domain in, < 2tî0 GeV was considered. In particular, we note that in the factor c2(1 - .1r ß,. ) in the second term of eq. (15b), -vrw and the contribution (2&e%)Ms- to 22 [cf. eq . (8d)] have the same mass singularities and are of the same order of magnitude. Thus, in the second terms of eqs. (15b) and (16d) we can approximate ê2(1 - .lrw ) --~ e 2 with a subleading error of O((a/2srs2 )2 m 2 /M 2 ) . Such an approximation was, in fact, used in ref. [6]. In the present, more detailed work, we adopt strategy (1), that is we keep the distinction between c2 and c'- (and similarly between s2 and s2 ) in evaluating the self-energy and vertex and box contributions and retain the factor ê-'(1 -,IrW ) in the second term of eq. (15b). The necessary analytic formulae are listed in the appendix. Two other refinements introduced, with respect to the calculations of refs. [6, 7], are : (i) inclusion of the small irreducible effects of O((ain/m2 )2 ) in the manner explained at the end of sect. 2 and (ii) small QCD corrections to the contributions of the two light isodoublets (u, d, c, s) in the Aww and AZZ self-energies appearing in dry, and ar, as well as in some of the contributions of the first five quark flavors to -(25e/e)Ms (see the appendix) . Starting with the expressions for dr and drw , the quantities discussed in this paper can readily be obtained. Use of eqs. (8b), (15b), (23) and (19) allows us to
. egr~issi et al / in,, .-in, imerdc ."ndedJce
o
r calculate iteratively - , w, 9' and s :!; we then evaluate shown in tables 1-4 as F nctions of in,, for from eq . 17b). These quantities are in, = 91 .17 GeV [15] and in,, = 2-5, 100. 500. 1000 GeV, res , .1 a 1 ed cted values of in w. I columns 2- we have kept re because rou off errors there may affect the last digit i the Ar determina1 1 se a columns re note that the values i t e e less elaborate calculations of refs . [6,71. t s o i tio s the (t-) i o let contributions t that in these cal es have e compute i the free fiel theory limit. n principle, f rections which necessarily involve a specific i te ret ti consider t e ey have e iscussed in the case of r [161, where t They have bee eV an = + 2. x 10 -1 or in, -estimated to be = 1 .5 X 1 ' for in , = 15 e 2 o nit > e ', relt a remaining uncertainty of 1 .5 x 10 -3 On,/25 For i i r leads to a change '. fixe n, a shift
an correspondingly C81ia
w=
- On z/ 2c)3S2 .
2
.,r = , so that a sift example, for n, = 240 eV, we have s 2 -- .21 a r = ' induces Ss 2 = 2.9 x 10' an Sin w = -15 MeV. Using tale 1 an eqs. (29a, b) we see that, if the QCD corrections of ref. [161 were included, we eV predicted in w values = 27 eV [43 would obtain for and = 5 eV 2 1eV] lower than indicated in tables 1-4, with an additional uncertainty o = ±9 MeV [ - 20 eV]. These C corrections have not been specifically studied in the case of and .1;w. However, in .,p they are expected to be smaller than in jr by roughly a factor of C2/S2, and greatly diminished in APw. An alternative procedure for large in, is to determine the additional terms of a 2 ) that must be added to the usual one-loop calculation of v r [11, carried out in the on-shell scheme, to take into account the leading higher-order reducible contributions. Neglecting subleading corrections of ®((a/27rs - ) 2 in /MZ), we can achieve this in the following manner. Replacing é 2 (1 - ®î w) ~ e'- , c 2 --~ c 2, we write eq. (15b) in the form s
(30a)
Aiw - ®r = where 2 ^2 2 ^2 F = Re[ Aww(mze) - Azz(mz)e ] Ms . m2z
(30b)
1 ~ sl ~ ,~ a
6.6~0 7~ 6.8~ 6. 6. 7 .9 6. 6.974 6. 5 7.015 7.34 7.051 7. 7. 7. 7.114 7.128
~
~,>
ff
r fe~
. 6.4 6 1 6.319
. . 0. 0.
317
6.177 6. 7 . 11 5.919 5.821 5.716 5. 5.~1 5.367 5®239 5.1 4. -1.816
. .~ 7 0®231 . 1~19 0.~ l l6 0.~ ? .~ 7 .~ 0.2.~ 7 0.2_~924 0.~~ 0.~°r 0.22I 0.2_~7
~ 2~~8
~ 91 .17~
6zl l Sv75 5.43 Sr l 4.45 4®1 3.72 3~3 2ß91 2.47 2. 1.,5 .42 -0.17 -0.79 -1 .46
~~~~ 2 1. ~n ~able ~®r rra~ = 1 10~~r 80.0 .0 1 .0 110.0 120 .0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0
6.761 6.841 6.896 6.939 6.976 7.007 7.036 7.061 7.084 7.105 7.125 7.144 7.162 7.179 7.195 7.210 7.225 7.240
6. 6.611 6.556 6.494 6.426 6.352 6.272 6.186 6.094 5 .996 5.892 5.781 5.665 5.543 5.415 5.280 5 .139 4.992
0.23380 0.Zî362 0.23342 0.23320 0.23296 0.23270 0.23241 0.23210 0.23178 0.23143 0.23107 0.23068 0.23028 0.22985 0.22941 0.22894 0.22846 0.22796
6.43 6: 5.76 5.45 5.13 4.79 4.44 4.07 3.68 3 .27 2.83 2.37 1 .87 1.35 0.80 0.22 - 0.40 -1 .05
0.~~95
0.43
0~2261 0.2~~ .~~ 0.~~~ 0.2197 .2182 0.2167 0~ l5 l 0?135 0.2l18 0~1 ' 0~ 1
1.5-1 1 .65 1. 1. 2.1E1 2~43
81 . 3 81 .1
~ie~ ~~
10~~~
0.2330 0.2317 0.23 0.2295 0.2284 0.2273 0.2261 0.2249 0.2236 0.2222 0.22~ 0.2194 0.2179 0 .2163 0.2146 0.2129 0.2111 0.2093
(1.11 0.25 0.3~ 0.48 0.59 0.70 0.81 0.93 1.05 1.18 1.31 1.45 1.59 1.73 1.88 2 .04 2.20 2.37
79.85 79 .91 79 .97 84 .03 .OS 84.14 84.20 84.27 84.33 80.40 80.48 80.55 80.63 80 .71 80 .80 80.88 80.98 81 .07
G. Degrassi et nl. / r'a `~~-rra Z ifaterdependerace
62
TABLE 3
As in table 1, for rra H
=
500 GeV
10`'ar~y
10`'ar"
s`
10`'ar
s`'
10`'ap
m~y (GeV)
80.0 90.0
6.851 6.93 i
6,901 6,852
0.23467
7,01
0.2351
- 0,05
79,74
130.0
7.096
6.590
4.70 4,31 3,
0.2270 0.2?i7 0.2243
0.78 0.90 1 .03
80.16 80 .23 80 .30 80 .37
rra~ (GeV)
i00,fl 110.0 120.0 1-10.0 i50,0 160.0 170,0 180.0 190,0
'_(10 .0 10.0 220 .0 ?30,0 2-10.0 2_~0.0
6,985 7.029 7.065 7.124 7,150 7 .173 7 .19-1 7 .21-1 7 .?°~3
7.?~ 1 7,?68 7,284 7,3
7.315 7.329
6,796 6.733 6.665 6.510 6.424 6 .331 6,?33 6 .125 ti,018
5,901 5,779 5,650 5,515 5.374 5.226
0.23449 0,23429 0.23406 0.23352 0,23355 0.23326 0.2_i295 0.23262 ®.?~? 27 0.?~ 1
6.67 6.36 6.05 5.74 5,41 5.06
i).23151 0.2_~110
3.47 3,02 2.53
0?2975 0?2927 0.22876
0.90 0.30 - 0.35
0.23067 0.23022
2,02 1,48
0.2338 0.2327 0.2316 0.2305 0.2294 0.2282
0.2229 0.2214 0.2 i 99 0,2153 0,2167
0.2149 0.2131 0.2113
0.09 0.20 U.32 0.43 0.54 0.66
i.16 1 .29 1 .43 1 .58 1 .73 1 .89 2.05 2.22
79.80 79,86 79,92 79,97 80,03 80.10
80.44 80.52 80.61 80.69 80.78 80.87 80.97
T :asLE 4 As in table 1, for rra a{ = 1 TeV
rra i tGeV)
10`aî"~~~
10 -'aî-
s`'
10`'ar~
s`
10`ap
rrt~,~ (GeV)
80.0 90.0 100.0 1 i 0.0 120.0 130.0 140.0 150.0 160.0 170 .0 180 .0 190 .0 200 .0 10 .0 220 .0 230 .0 240 .0 250 .0
6.882 6.961 7.016 7.059 7.095 7 .126 7 .154 7 .180 7 .203 7 .224 7 .244 7 .263 7 .281 7 .298 7.314 7 .330 7 .345 7 .359
7.014 6.964 6.908 6.845 6.776 6 .702 6.621 6 .534 6 .442 6 .343 6 .238 6 .128 6 .011 5 .888 5 .759 5 .624 5 .482 5 .334
0 .23508 0 .23490 0 .23470 OZ3447 0 .23422 0 .23395 0 .23366 0 .23335 0, 23302 0 .23266 0 .23229 0 .23190 0 .23149 0 .23105 0.23060 0.23013 0.22964 0.22913
7 .31 6 .97 6 .66 6 .36 6 .04 5 .71 5 .37 5 .01 4.63 4.22 3 .80 3 .34 2 .86 2 .36 1 .82 1 .25 0 .64 0 .01
0 .2362 0 .2349 0 .2338 0 .2327 0 .2316 0 .2304 0 .2293 0 .2280 0 .2267 0 .2254 0 .2240 0 .2225 0 .2210 0 .2194 0.2177 0.2160 0.2142 0.2123
- 0 .14 - 0.00 0.12 0.23 0.34 0 .46 0.57 0 .69 0 .81 0 .94 1 .07 1 .21 1 .35 1 .50 1 .65 1 .81 1 .97 2 .14
79 .68 79 .75 79 .80 79 .86 79 .92 79 .98 80 .04 8O .10 80 .17 80 .24 80 .31 80 .39 80 .47 80 .55 80.64~ 80 .73 80 .82 80 .92
G. Degrassi et al. / ni ll -mz interdependence
63
Combining eqs . (30a), (17b) and (21) and recalling sZc2(1 - r) = s2c'(1 obtain 2
1
® ~r = r w(c ) - Ç' F ~-' ( s2 r s 2c2 ( ) 1 2
e
r), we
r,~
where, for brevity, we have not indicated the dependence of (P) an , ,(c 2 ) on the mass ratios . Splitting (1 - Ar^w)/(1 - :1r) = 1 + ( :1r - ,Irw )/(1 - Ar ). e have c 2 e2 s-'c-' F(c2) s-'
-
c2 e2 ( r-A w ) F(c -') s 2 s-c2 (1 r
In the last term we replace :fir - Jr^w by the expression (31a) and employ e yielding ,r = Jîw(c') -
c
e7
s - s -c
c-' ) 2 F(
+
ç2 7
s-
8G,,miF
( c2 )
(1
- .I;w ) .
(31c)
Substituting AP~,,(c-) -~ ~;w(c2 ), F(c2 ) --+ F(c 2 ) (a replacement that does not affect the leading terms (- 2 be/e )K s and F = 3mt/ -'1t,z + . . .1, we obtain c-' e-' c-' BG,,ntZ ,r=®rw(c - ) - 2 s 2 C ,F(c`) + S, F(c- ) (1 - Arw) . (32a)
s
Recalling (8b) and (30b) one verifies that the first two terms on the r.h .s. coincide with the one-loop expression for ,r in the on-shell scheme [ 1 ]. us the last to represents the higher-order correction to that expression . To tv4"o-loop order it reduces asymptotically for large fn t to (c28p/s 2 )2, in agreement with the results of ref. [5]. We note that 8Gmm`z
F
- 8Gmn,W. Re Aww(mw) - Azz(mz) ( c) fnz r2 mw 2
Ms
(
32b )
This is a familiar combination of self-energies in which the coupling e -'/s2 has been replaced by 8G,,m~,/~ . The two-loop irreducible contribution for large mt is taken into account by appending a factor of [ 1 + (2 T,- - - 19)s'/3c 2 ] to the last term in eq. (32a). Values of Jr obtained from eq. (32a) are given in table 5. They include the two-loop irreducible contribution for large m t and small QCD corrections associ-
On Ir C%vlual
f Y. 432)
IM
51M MI
It a
MU) A) A 0
6.67
30 114 4S2 41 113 3.76
a' 13 ',
113 4,$! T47
vil
3.73 33-1 2-S119 141 IAS 1M R190 0.+,2 ,.2sS -o.92
I rAt ,t 4) 1~ 1 k10
MW it) Mu .0 i410
7-4)
2,
2'07 I q~S IA M _M -"7 - IX!
6w
3.73 5, 4.73 4_15% 3.93 .432 3.07 1M) 2.10 I_% 101 0.40 -0.2-3
à%
5.73 53 4. 4.26 ~S4 1,A) 2.91 143 1.90 1_34 0.75 0.12
- and on ji-maxis-es are exvre-.~scd in GeV The table slwm-, 100r.
with the light quarks (cf. the discussion at the beginning of this section and in the appendix). A detailed comparison shows that for in,, = 25 GeV, 100 GeV, 5 GeV. I TeV, the values of -Ir from eq. (32a) am layer than those obtained from 10-3 eq. by (fn, 3, - 2, - 4) X 10' for in, = 80 GeV and by (1 .4,13,12,12) X for in, = 250 GeV. This is compatible with our rough estimate of the subleading contributions of (vt/2., ) nq'/#nj) I z . We have also compared tables 1-5 with the values of -Ir obtained in the one-loop approximation within the on-shell scheme, i.e. the first two terms of eq. (32a). We find that, although the one-loop truncation of (32a) is a rather good approximation for m, < 180 GeV (the largest error in this range being - 2 X 10-3 for in = 180 GeV and m,, = 25 GeV), for precision studies with large top masses the more complete calculations are desirable. Because the two-loop irreducible contributions of O((a/2 z have not been computed, it seems impossible to prove rigorously at the present time which of the two calculations of Ar (from eq. (22) or eq. (32a)) is more accurate . 5p, (16P)2) On the other hand, in the case of the leading contributions of we know that the reducible effects are by far dominant. Assuming that the same is true for the subleading effects, we are inclined to believe that the calculation based on eq. (22) is more accurate, as it incorporates the reducible higher-order contributions in great detail . (22 1)
i2
2
t
7r S2)2 M2 t /M2
0(c2, fat a
erd /M
I
Mt
tt
le [ tee perha cease with i tween the tw, a ,ree ent quite
side ftom the QCD corrections associated with large m, (see the ion), current ° that the error in Ir discussion in this flavors 1 -' [ 18 analysis of the first five quark contribution is = + gh4, in arising from higher-order uncertainties is = + 1(l ®~ 1191 (this. is with our previous discussion of subleading corrections). Combining errors in quadrature, one obtains an uncertainty of = + 1 .3 10- ` in lr correct translates for s' - ®.225 into -6m w = ± 22 eV. Inclusion of the of ref. 16] would further shift the central values and the errors in the manner outlined after eq. (29b). Accurate determinations of m . are indeed eagerly awaited! While preparing this manuscript, we have received a preprint by J . R ner [21], that bears some relation to our work. Among other considerations and results. Rosner discusses two parameters similar to s? and gyp, and an approximate expression that resembles our accurate eq. (18). One of us (A.S.) would like to thank W.J. Marciano for a very interesting conversation. This work was supported in part by NSF grant PHY-8715995. Appendix A In this appendix we list the analytical formulae needed to calculate J ; w and ..1; when the dependence on c -' and c' is differentiated . The basic relations are eq. (8b) and eq. (15b) and all the explicit expressions are given in the 't HooftFeynman gauge. The starting point in the calculation of ( - 28e/e)MS is eq. (7b). The contribution of the first five quark flavors is analyzed by considering H( (0 -ez Re(IIY'y(mz) e 2 y) )MS = HYy ( 0)) +e2 Re M5y(mz) ms- (A.l)
G. Degrassi et al. /inil,-in z interdependence
66
The first term is evaluated using dispersion relations and detailed information on e + e - --* hadrons [ 1, 18, 20] while the second is studied perturbatively. Using the results of ref. [181 for the first term and in z = 91.17 GeV, we find 2 8e e
= 2«
In
37r >
MS
mz
( m,
1 + 3a + 0.02875 ± 0 .00090 47r )
SSa -(I+ SQCD) + 277r
-
8a
m )
97r In m z`
a
+ 2Tr (' In
c2 - ; ) ,
(A.2)
where the summation is over the charged leptons and the quoted uncertainty corresponds to the error in the first term of (A.1). In the evaluation of e -' Re 11(5)(1112 )I MS we have include . 1 a QCD correction factor of (1 + SQCD ),where [171 5QCD =
ajinz)
7r
+ 1 .405
as( mz)
(A-3)
7r
Numerically, BQCD = 4.02 x 10 -- for a S(mz) = 0.12. Similarly, we have appended a factor (1 + 3a/47r) to the leptonic contributions [4,171. Eq. (A.2) leads to 2 5e e
- = 0.06902 ± 0.00090 MS
8a
97r
In
in, )
`
mz
a (7 -, In c- - ~ ) . 27r
+
(A .4)
Eq . (A.4) gives the dominant contribution to 1Yw and can also be used to compute â = ê2 /47r via eq. (8d). As mentioned in sect . 1, it depends slightly on m,, varying from â -' = 127.8 ± 0 .1 for m, = in z to â -' = 128.0 ± 0.1 for in, = 250 GeV* . The fermionic contribution to the first term in eq. (8b) is given by e2 s-
Re
ww(mw) ,- A(ww( 0)
A(
n1 W
Ms
5
- + In 3
27rs`'
+
8
- (1+2~~)-
2
1)2
1 +
2
In
m, ~ In z
(A .5)
* For approximate calculations that clo nW take full cognizance of' thc radiative corrections, our suggestion is to employ thc value of o witl i the toi) yuark clccoupl`cl, . .e with m, = ny . i
G. Degrassi et a1. / m u, m z interdependence
67
where ~ 2 = m2 fm ZC2 . The corresponding bosonic contribution is e2 Aww(mw) - A`ww( 0) I
a - 3 (~In~-c21nc2) H +I2( -2 ,_ C2)) Q - c2) 4 47rS 2 c2 1
+ In
2~' c2 - C2 (4c2 + .1) -
-
+ 9 S2
1 (2-2+ .1)
-
2
S4
+ 4s2 +
+
s4
c2
c2
-
c2 s2 (
4c 2 + ~ -
(A.6)
,
where f =M2 lin z, H(x) is given analytically in ref. [9], and 4c 2 - 1
I,(c 2 , c - ) _
c`
In c 2
2 11 tan - '( 4c 2 - 1 )[-ê2(5 + , 3c` 3c 4 5
1
7
1
2
s4
1
s~
g4
c-
1
3c -
+
1
12c`
2e 2
3c + c2 2c 2 4 + 2 c + 4c 2 1 6c 2 101 6 2 1 1 ( 1 + c - 9 + 2- +-+ 1 (A , , 9 c 3c 4 2c6c~
s4
.7)
c`
Inserting eqs. (AA)-(A.7) into eq. (8b), we obtain a complete analytic formula for Are . In order to evaluate ®r we need to study the second term in eq. (15b). The fermionic contribution is e2 Aww(mw) _Ac f Re zz) (m 2z ) 112 2 s mz c^2 a
Ims
c2
(lnc 2 - 1) (1 + 8 Qcn ) + 2Trs2 e2 +
S2 8
(1+2~ 2 )-
In c2 5 2 = 3 + ln
m (MZ z
( 2_ 1)2 2 S S 2 1+ 2 In
~2
a m2 + Il( D) 1 4 1 - 2 7rs2c2 + 4-rrs'c2 M2z ~ 3a
I (
m` + 1654 21n +1 x 2a r - -42 .3S 9 I mz -'3 [(1 - 25 2
+ ~~
-~
(3 + D)(A(D) - 1)
.1 5 4 )(1 + SQco) + ( ~ - 3 S2 + 9S4) + -, (1 - 2s2 + 4S4)l
(A.8)
G. Degrassi et al. / rrr ei - -iii z iwerdepende®rce
68
where D --- 4(~n i /1,1 Z ) - 1 and A(D) = D' /-' tan - '(D - ' /-') for D > 0. The bosonic contribution is A"~(~rt 2v) _ c° s-nli e~
,-
,. ~ c -s.2- c - J
cti, zz ('~a z )
IMS
c_)) -H(~)
+ 1(e4 + _Lg4) _ 3 2
-
C
-
C 2( -L
LS-3 -2
9S )
In c-' 1 c 2 (-! + 3s-' - s4/c -' )
+
1 - 3s 4
+
where 21
s.,4 -~
2c -
x c -' 10C? 1
-5
1
i tan -
) 18
lOs 4
- ~ s-' JC' ? ~
s~ + 'c- + - -
J
1
t
4c -' - 1 s
6 C2 J
- 1
(A.10)
e note that in (A.5) and (A.8) we have also included the QCD correction factor (1 + aQCD) in the contribution of the two light quark isodoublets. Inserting eqs. (A.8)-(A.10) into eq. (15b) we obtain the complete analytic expression for vi. In the limit c'- --, c'- , eqs . (A .5)-(A .10) as well as the vertex and box diagrams contributions in eq. (8b) reduce to corresponding expressions of ref. [9] (the functions I,(c 2 , c - ) and 12( c -', c'- ) are called I,(c 2 ) and I,(c'- ) in that work). eferences [11 A . Sirlin, Phys . Rev. D22 (1980) 971 [2] U . Amaldi et al ., Phys . Rev . D36 (1987) 1385 ; G . Costa et al ., Nucl . Phys . B297 (1988) 244 [3] P . Langacker, Phys . Rev . Lett . 63 (1989) 1920 ; J . Ellis and G . L. Fogli, report CERN -TH .5511 /89, SARI-TH/89-60 [4] A . Sirlin, Phys . Rev . D29 (1984) 89 [5] M . Consoli, W . Hollik and F . Jegerlehner, Phys. Lett . B227 (1989) 167 [6] A . Sirlin, Phys . Lett . B232 (1989) 123 [7] S . Fanchiotti and A . Sirlin, Phys . Rev . D41 (1990) 319 [8] A . Sirlin, in Z °-physics, Proc . XXVth Rencontre de Morfond, ed . J . Tran Thanh Vân (Editions Frontières, Gif-sur-Yvette, 1990) p . 103 [9] W. J . Marciano and A . Sirlin, Phys . Rev. D22 (1980) 2695 [101 A . Sirlin, Rev . Mod . Phys. 50 (1978) 573 [ 11 ] A . Sirlin, Nucl . Phys . B332 (1990) 20 [121 M . Veltman, Nucl . Phys. B123 (1977) 89
G. Degrassi et al. / m jj - m z interdependence
69
[13] J.J. Van der Bij and F. Hoogeveen, Nucl . Phys. B283 (1987) 477 [14] B.W. Lee, in Proc. XVI Int . Conf. on High-energy physics, ed. J.D. Jackson and A. Roberts, Fermi National Laboratory, Batavia, 1L. 1972. Vol. IV, p. 266; D. Ross and M. Veltman, Nucl. Phys. B95 (1975) 135 ; P.Q. Hung and J.J. Sakurai, Nucl. Phys . B143 (1978) 81 [15] G. Rolandi, in ZED-physics, Pry. XXV1h Rencontre de Moriond, ed. J. bran Thanh Vin (Editions Frontières, Gif-sur-Yvette, 1990) p. 247 [16] B.A. Kniehl, J.H. Kiihn and R.G. Stuart . Phys. Lett. W14 (1988) 621 : in Polarization at LEP, ed. G. Alexander et al., CERN 88-06. Vol . 1, p. 158 ; B.A. Kniehl, Max Planck Institut preprint MPI-PAE/PTH 5/89 (1989) and references therein [17] G. Burgers and F. Jegerlehner, in Z"-Physics at LEP 1. Vol. 1 . ed. G. Altar`lli, R. aies and C. Verzegnassi, CERN 89-08, p. 55 [18] H. Burkhardt et al., in Polarization at LEP, ed. G. Alexander et al.. CERN 88 . Vol . 1 . P. 158 [19] F. Jegerlehner. Z. Phys. C32 (1986) 425 [20] F. JegeTlehner, Z. Phys. C32 (1986) 19-5 [21] J.L. Rosner . EFI preprint 90-18
331(1991)
h fd
efived. These expressions bridge the MS and ful to ctwnpute -Ir and sin' O w in the case of large inter, 'ence is expressed in two equ °alent ways: in ter o r a meter ® m ,. /saî~ ® ,( ). A li s;lins o the expressionss derived i illustrated nu ri l .
e relation 1 sin 20 W=
2
__
ntw(1 -`1r)
where sing ®w = 1 _M2 ,/mZ- s -', A = (za/ GA ) 1/2 _ (37.2803 ± 0.0003) Ge and ®r is a radiative correction, plays a significant role in current discussions of electroweak physics. For example, for given values of mz, m, and MH, eq. (1) allows us to compute sine ®w. This in turn can be used Lo predict mw, can be compared with other determinations of sine ®w to constraint the value of m, [2,31 and can also be used to predict asymmetries and other observables at the V-peak, as these are in many cases sensitive functions of this parameter. An analysis of the mass singularity structure of the radiative corrections to tL-decay has shown that insertion of Jr in the denominator of eq. (1) incorporates not only the leading logarithms of ®((a ln(m Z/m f))") There mf is a generic 0550-3213/91/$03 .50 ©1991 - Elsevier Science Publishers B.V. (North-Holland)
G. Degrassi et al. / ni «,.-mZ interdependence
50
fermion mass) but also, to a good approximation, the terms of O(a 2 1n(m Z /m f )) [4]. With increasing indications that the top quark is heavy, physicists have begun to study in greater detail and depth the effect of large m, on the radiative corrections . In particular, Consoli et al. have recently investigated the terms of O((afn2/1112)`) in the corrections to a-decay [5]. Describing the dominant contri(c 2 )Sp, where /s 2 butions to yr in terms of Ea = e2(I7yy(0) - H( O(tn 2 )) and function fermionic vacuum polarization one-loop unrenormalized IIYy(q 2) is the and 5p = 3afn2/16r~s -2 n1 2 , these authors found that the one-loop calculation of A r, when inserted in eq. (1), incorporates correctly the O(a 2 ) corrections of O(E.2) 2 and O(E.Sp) to ,u-decay but there is a mismatch in those involving 0((5j5) ). The first result is already contained in the conclusions of ref. [4] but the second one is novel and interesting for large int. In fact, the analysis of the terms of O((aü1i /i)1 2 )2 ) is unrelated to mass singularities and is therefore not covered by the methods of ref. [4]. In the present paper we present a formulation that automatically takes into account the reducible terms of O((am i/m w )" ), leading to simple relations between s' and the MS parameter sine âw(m z ) (henceforth abbreviated as s-' ) and suggesting useful novel ways of calculating Ar and other important quantities . We recall that use of the MS parameter s-' has recently led to the introduction of two new basic corrections of the electroweak theory, ®r [6] and vîw [7]. The relevant relations, suitable counterparts of (1), are ,. s-I =
,
rt122W (1 - ®rw)
A) A)
s c- =
mZ(1 - dî)
.
These equations summarize the meaning of ®î w and ®r : the former is the radiative correction that must be applied when s 2 is determined from a, G,, and mw , while the latter plays an analogous role when s 2 is extracted from a, G,, and mZ . The first step in the present analysis is to elucidate to what extent the one-loop corrections ®r and Arm, incorporate the leading higher-order contributions in eqs. .,nd (3). These arise from two distinct sources: (i) effects induced by resummation or iteration of one-loop effects and (ü) higher-order one-particle irreducible contributions . Throughout this section we will consider the first class which, for brevity, will be referred to as reducible higher-order contributions; in sect. 2 we will discuss how to incorporate the second class to O(a2 ) . The leading reducible contributions can be studied by a procedure suggested by the ingenious analysis of ref. [5]. One notices that these corrections are contained in the purely fermionic contributions to the /,-decay amplitude. In turn, the latter are given by _72=
8S()7 1 M 2M
+
(e021S20)~A") ww (0)]
+ ... ,
G. Degrassi et al. / in ,,- in z interdependence
51
where the 0 subscripts indicate unrenormalized parameters, (e02/s2 )A~~,(0) is the unrenormalized one-loop fermionic WW self-energy evaluated at q2 = 0, and the ellipses represent the bosonic contributions of O(a) as well as additional contributions of higher order. The parameter e2 in eq. (4) means 2 _
eo
e2 (1 -e 211(f)(0))
,
the familiar charge renormalization of QED. (The sign of IIYy follows the traditional conventions and is opposite to the one employed in ref. [5].) The authors of ref. [5] showed that the explicit term in eq. (4) is ultraviolet convergent to O(a2 )* and reached their conclusions concerning the leading higher-order reducible corrections by expressing this contribution in terms of the renormalized parameters e 2, s 2 and mom,. We now notice that an analogous analysis of eq. (4) in terms of e 2 , s2 and mw leads to e2/8^2M2 s
G~ [1 -e 2llyy(0)
- (e2/s2)Re((Aww(mw) -(0))/M2
where MS denotes both the MS renormalization [i.e . the subtraction of terms involving S = (n - 4) - ' + (y - ln(47))/2] and the choice tL = mz for the 't Hooft mass scale. The result (6) can readily be obtained by inserting eq. (5), s= ; s- - S 2 and in w(, = mw - (e /s~ ) Re A(f, w(m ~,~, ) into eq. (4). The explicit term in eq. (4) becomes e'/8s'mW
[1 -e - Ilyy(0) - (e`/s- ) Re((Aww(m 2~,) -Aww(~))/rnw) - (gs-/s-')(1 -e-IIYY~(0))]
We now notice that in the MS scheme 5-s2 involves only divergent terms proportional to powers of S. When only fermionic contributions are included, as the answer is convergent through O(a 2 ), (,5s2/s2)(1 - e 2 llyy (0)) must exactly cancel the terms proportional to a(S + ln(mz/A)) in the second and third terms and cannot contain contributions of O(a2 ). This result has been confirmed by an explicit calculation through O(a 2) and leads immediately to eq. (6). The last term in the denominator of eq. (6) does not contain mass singularities or terms of O(m2/m 2 ). Thus, the leading contributions are given by -e 2IIyy'(0)Nts and are logarithmic in character. But these are precisely the leading corrections contained * The mass counterterms associated with the virtual fermions are needed to cancel divergences in the irreducible two-loop corrections . Therefore, it is understood that they are included in the latter.
G. Degrassi et al. / m`l.-»:Z iwerdepeiideaace
'52
in Itw ! Further, e -2HYy (0)Ms and E « contain the same mass singularities and are of the same order of magnitude. Thus, we reach the conclusion that when the one-loop expression 1'îw [7] is inserted into eq. (2), there are no unaccounted terms of O(E 2 , E Sp, (Sii)2) [8]! We also note that the factor (1 - A w) in eq. (2) transforms the a present i A 2 into a/(1 - _41^ w), which may be interpreted as an effective a evaluated at inz. q. (6) contains i s compact form the one-loop fermionic corrections to j,-decay as well as all the higher-order reducible contributions that arise solely from them. In order to obtain the complete one-loop expression for ,Irw we must add the O(a) bosonic contributions to the selfenergies, as well as the vertex and box diagrams in the tL-decay amplitude. We recall that, when both one-loop fermionic and bosonic corrections are included, eq. (5) is replaced by 1
e"
e(1 + 23e/e)
(7a)
where (cf. eq. (26) of ref. [1]) 2 Se
7e 2,
e
8â-
(S +In(
rft
1
lu
21
Including the additional contributions we obtain (8a) îw=
e2
s
112
Re
Aww(mw) - Aww( 0 ) MW
e2 7 In c 2 6 + , . 16 2g2
5
s2
-
25e e
5
Ms _-S2
3 c2 2 c2
(8b)
The A ww self-energies include now the fermionic and bosonic contributions of 0(a). Their detailed exprcssions are given in ref. [9] (see also the appendix of the present paper). In the last term we have exhibited explicitly the dependence of the vertex and box contributions on c 2 and c 2 = m W/mZ . This dependence can be gleaned from sect . 4 of ref. [10] (cf. eq . (4.5) et seq.). Analogous dependences are present in Aww . If one replaces c'- , s 2 --> c 2, s 2 in the O(a) contributions, eq . (8b) * In this reference the coupling constant g -' = e 2/s' is included in the definitions of A WW (g 2 ), A z7 (g 2 ) and A,,z(q°) . In the present paper this coupling has been factored out . Similarly, e2 has been factored out in the definition of H.,,(q`) .
G. Degrassi et al. / in iv-mz interdependence
53
becomes identical to eq. (6) of ref. [7]. As it will be explained later, the difference between the two procedures is of O((a/2ws 2 ) 2 mi/mz) and represents subleading ®(a 2 ) contributions. An alternative expression is obtained by keeping the vertex and box diagrams contributions in the numerator,
Ge
-
e2
1 + (é-'/16îr2~2)( V+ B) (e2/s2) 8s 2m2w [1 + 28e/e Re((Aww(mw) - Aww( 0))/m 2w)]is
Here V + B is an abbreviation for the expression between curly brackets in eq. (8b) (i.e. the vertex and box diagram contributions) and e 2 is the renormalized version of eq. (7a) in the MS scheme*, i.e. -2e
e2
(1 + 28ele)Ms
The difference between (8a, b) and (8c) is < 0 .01 %. Turning our attention to eq. (3) and comparing with eq. (2), we note that
where c2
P - c2
mW m2c2 Z
.
(10)
To evaluate P we write s2 = s 2 - 8s 2 = s2-5s 2 and note that sô = 1 - Yt1 Wp/mzo. Thus, we have 8s 2 Y-
(5
= -C2y'
/mw - 8mz/mz) (1 - 8mz/mz)
2 MW
where Buzz and 8naw are the mass renormalization counterterms expressed in * This
definition does not follow exactly t"e usual conventions because eq . (8d) contains contributions from the top quark even when it is more massive than in z. Numerically, the differences are quite small even for in, = 250 CieV (see appendix A).
G. Degrassi et al. / in `t-ntz interdependence
>4
terms of unrenormalized coupling constants. Therefore, we find c-' =c -' - C 2 Y - 5S2 =C-'(1 - Ys),
(12a)
1
(12b)
YMs)
(1 -
Use of eq. (9) then yields Ar = A1`,`r
- PYMS( 1-
(12c)
A w) .
To evaluate YMs it is useful to note that eq. (1 lb) can be written as Y=
&nW «)
in W
bin2
-
,
in -Z
(13a)
(l-Y),
leading to Sm 2W
5171 z
inw
YMS
,. .. Ar = Ar W
-
In zP
MS
j5in 2
amW
P
(13b)
2
in W
mZ Ms
,. (1 - ArW) .
(13c)
To obtain expressions for p = c 2 /c'- and s 2 c 2 inZV G,,/7a that contain all the ®(a) terms and incorporate the leading reducible contributions, we insert the expressions for Sm z/m 2 and Bin w /m w into eqs. (13b, c) and obtain YMS
e2 ~2Pmz
-
e2
Re
~2
P
S2
®r =APW -
l
ê2 S
2
Aww(mw) 12
-A ZZ ( M22 m
(A y z( mz)/mz) Re 1 + e211yy(MZ) (1 - L1iW )
+(1 -®Pw )
2
rn z ^2 L'
s
,-
Re
2
e
MS
2
(14a)
-' MS
Aww(mw) ^2 C
2
-Azz(mz)
(A yz(M2 )lin2 )2 1
+ L' 2 ~j yy
(in2
)
MS
(14b ) _ MS
G. Degrassi et al. / mu,- mz interdependence
55
In these equations the A's and II,y,, stand for the one-loop expressions. They are given in ref. [9] (the bosonic parts of Ayz(mz) and HY ,y (m 2 ) are studied in greater detail in ref. [11]). It is worth noting that tadpole contributions cancel in eqs. (14a, b). The last terms in these expressions are O(a 2 , a 3 ) and represent the contributions of yZ-mixing to 8mz (cf. eq. (22d) of ref. [11]). As it was the case with eq. (8), eqs. (14a, b) contain all the higher-order reducible contributions that arise solely from the one-loop fermionic contributions. We note that A yz(m 2 )/mz does not contain mass singularities or terms proportional to M2/M /m z . Therefore, neglecting the very small O(a 2 ) terms involving this quantity, eqs. (14a, b) simplify to YCIS
ê2
Pm2
s
z
,. ,, Ar = ®rw -
Aww(mw) e2
Re ê2 s2
( 1 - ~rw) m2Z
- Azz(mz)
(15a)
Ims
Aww(mw) c2 - Azz(mï)
Re
(15b)
Ims
Use of eqs. (8b) and (15a, b) leads, without further approximations, to ®r =
e2 S2fn2z
+
Re Azz(mz) -
e2p 167r2s2
In C2 7
6+
s
2
A ww(0)
2-
C2 5
-S
2
28e
- e
- s2
5-
Ms
3
2
c2
c
It is instructive to consider the leading one-loop contributions for large can be gleaned from eqs. (B.1) and (B.2) of ref. [9], namely Re A ww = -
3 16 ;r2
3
Re Azz=-16
~r,
mt 1 mt 8 + In ( ~ ) - ) + . . . ,
4
m~ e2 0
8+1n
Mt
~
+
(15c)
2
Mt .
These
(16a)
(16b)
where the ellipses represent non-leading contributions of OW . Inserting these 2 (a + In(m,/ti)) expressions into (15a, b) we see that the terms proportional to 111
G. Degrassi et al. / m in,-mz interdependence
56
cancel exactly and we are left with YMS
3a in 2 ... , 167rs 2 fnw + 25e e
(16c)
3â(1 - ®r w ) in, 16-rrs -c - mZ
Ms
where â = ê-'/47,-. If non-leading contributions of O((a/27rs 2 ) Z mi/mZ) are neglected, we can set c -', s -' -~ c-, -2 (i.e. P --~ 1) and â(1 - Arw) --> a in the O(a) terms. In that case (15b, c) coincide with the results of ref. [6]. We reach the conclusion that when the one-loop expression for Jr is inserted into eq. (3), there are no unaccounted reducible contributions of O(E2 , Ea Sj5' (Sp ) 2 ) [8]! It is important to note that, in order this result to hold, the O(afnt/in Z) term must be written as in eq. (16d). For example, - 3anz t / 16 7-s- fn w, which coincides to O(a ), will not do because it does not incorporate certain contributions of O((ari1i /m z )2). It is also worthwhile to point out that the explicit term in eq. (16c) is the well-known Veltman correction [12". Again it differs from the more familiar form 3G. rn t/ C 87r` in subleading terms of Ma/2 7g2)2 in ë /jn z ). Eqs. 05a-c) contain all the O(a) terms as well as the leading higher-order reducible contributions . They are therefore very useful for practical calculations . 20
elations between
e on-shell and
S frameworks
Having shown the desirable properties of ®P w and d r explained in sect. 1, we can now apply them to obtain accurate expressions and relations for other important quantities . The parameter p = C2/e2 can be calculated from eq. (9). It can also be expressed as P where ®P =
1 (1 _ ® P)
( ®r w _'I r)
(17a)
= YMS .
(17b)
From here we find c2 = c 2 (1 - ®P), or s2 = s2 1 +
c2
Al . ®P s2
(18)
G. Degrassi et al. / m ",-mZ interdependence
Alternatively, we have c2
=c ep,
57
or
®r w - ® r 1 - Arw
c2 s 2 = e2 1 _ s2
(19)
Note that (®r w - ®r)/(1 - Ar"w ) = p Ap and that eq. (19) is obtained from eq. (18) by interchanging s2 H s2, r w H r. Comparing eqs. (1) and (2), we haves2(1 - r u, ) = s2 (1 - Ar ). Use of eq. (18) leads to 1 -Ar= I1
+ s2 ®p I[1 -AM, c2
(20)
or c2 Ai- = 4î'W - ., ®p(1 - ® r w ) s-
.
(21)
Recalling (9), (10), (17b) and (19), we can also write eq. (21) as dr =arw -
( A w -ar) - (C2/S2)(A1 . s2 [1 p)/( e2
)] .
(22)
Eqs. (19) and (22) are particularly useful for practical calculations . Because Jr w and AP depend very mildly on in, the iterative evaluation of s -' from m Z, based on [6] 1
s2 - -
2
1-
1-
1/2
4A2 mz(1
,
_Ar)
( 23)
converges quickly. Inserting fir, GPw and s 2 into eqs. (19) and (22), we obtain then determinations of s 2 and Ar that automatically take into account the leading reducible terms. The mw-m Z interdependence can be expressed in the usual manner, m 22u mz
1 =2 1+ 1-
1/2
4A2
mz(1
-
(24)
3r)
Alternatively, using eqs . (10) and (23), we have mw mz
11
2
-
4A2 m7P(1 - ~~ w)
1/2
58
G. Degrnssi
t sal,
f aaa
t~-aar Z interdependence
Some of these equations have a structure reminiscent of the approximate expressions of ref. [5]. For example, ft,-jr the leading terms, the authors of ref. [5] find 2 and an expression 1 - ;Ar = (1 - E.)(1 + (r'/s2)Sp), where Sp = 3nî2, G11/8 ,- -'~_ that looks like (25) with -Ai"w -* t*. . and P -~ p = 1 /(1 - Sp). An obvious advantage of eqs. (20) and (25) is that they- include the complete O(a) corrections in a very simple way (not just the leading asymptotic terms), and moreover they suggest that the natural parameter in discusskng the rn w-in z interdependence is P rather than p. The two quantities approach each other asymptotically for large ni t , but there are significant differences. This is also intuitively natural, as one expects that a relation between nz w and should significantly involve the on-shell selfenergies as in (15a) and (15b), r they than the self-energies evaluated at q' = 0. It is important to emphasize that, i P and ,r are given by the expressions of this section, then eq. (24) and eq. (25) are, in fact, identical! This observation may perhaps serve to clarify some ° ather confusing statements made in the recent literature concerning the validity- of eq. (24) [5]. /j't 2 )') can be included by simply The irreducible contributions of ()(C,)(111 2/111 replacing everywhere
--~ ,r -F
19) (,rw -,dr)2 (1-Ar) ' 3
(27r ` -
(26)
where :,r w and Jr in the r.h.s. stand for the one-loop expressions considered before . Indeed, when the secon term in (26) is added to the r.h .s. of eq. (15b) and the resulting expression is inserted into eq. (9), one obtains 1 Ms
(2 77.2 _ 19) 3
(
Ms)
2
(27)
where YMs is the one-loop cor ection given in eq. (15a). We now note that the leading irreducible corrections given in refs. [5,13] are expressed in terms of G~m2 and that, to leading order, one can do the same with YMs [cf. the discussion after eq. (16d)]. Therefore, eq. (27) can be directly compared with the results given in those papers and one then verifies that the last term in this equation correctly contains the irreducible corrections of 0((G~ m 2)2) or, equivalently, O(«2(m2/mw)`). If there are triplets in the theory, P is no longer calculable via eq. (9), but it becomes an additional paramete r that must be extracted from experiment [14]. Writing P5() = 1/( 1 -®P() )for the tree-level expression and assuming that the radiative corrections are given to good approximation by those of the minimal
G. Degrassï et al / aadj -m z interdepeadeetre
519
theory, we have
with AP given by eq. ( l
3. In order to evaluate the basic corrections r^w and .hr^, we employ eq. (8b) eq. (15b), respectively [alternatively, FF can calculated from eq. (15c)ß. e correction lr w is dominated the contributions of leptons an five quark flavors to - (2Se%) , which amounts to 0. (see a evaluating the self-energies and the vertex and box contributions in eq. (8b). well as in the second term of eq. (15b), two strategies present themselves : (1) ee explicit the distinction between r-' = m2 /In 2 and P (and re between s= and s -') or (2) replace in the a) corrections / z = c2 -j, c and similarly s 2 --q2, with other mass ratios expressed in to of z. as in 2t/ and n4I /mz. The difference between the two approaches are subleading corrections of 0((a/2-s-' j2)-'M2/1,1 z ). They are expected to quite small in -lrw and x -4 fir, roughly = 2 10 even for mt as large as 250 GeV. (It should remembered, however, that in calculating 1r, differences in lrw - A ; are enhanced by a factor of c -'/s2 [cf. eq. (22)] and therefore may be as large as (0.8-1 .5) x 10 - 3 for in, = 250 GeW The simpler strategy (2) was used in refs. [6. 7], where the domain in, < 2tî0 GeV was considered. In particular, we note that in the factor c2(1 - .1r ß,. ) in the second term of eq. (15b), -vrw and the contribution (2&e%)Ms- to 22 [cf. eq . (8d)] have the same mass singularities and are of the same order of magnitude. Thus, in the second terms of eqs. (15b) and (16d) we can approximate ê2(1 - .lrw ) --~ e 2 with a subleading error of O((a/2srs2 )2 m 2 /M 2 ) . Such an approximation was, in fact, used in ref. [6]. In the present, more detailed work, we adopt strategy (1), that is we keep the distinction between c2 and c'- (and similarly between s2 and s2 ) in evaluating the self-energy and vertex and box contributions and retain the factor ê-'(1 -,IrW ) in the second term of eq. (15b). The necessary analytic formulae are listed in the appendix. Two other refinements introduced, with respect to the calculations of refs. [6, 7], are : (i) inclusion of the small irreducible effects of O((ain/m2 )2 ) in the manner explained at the end of sect. 2 and (ii) small QCD corrections to the contributions of the two light isodoublets (u, d, c, s) in the Aww and AZZ self-energies appearing in dry, and ar, as well as in some of the contributions of the first five quark flavors to -(25e/e)Ms (see the appendix) . Starting with the expressions for dr and drw , the quantities discussed in this paper can readily be obtained. Use of eqs. (8b), (15b), (23) and (19) allows us to
. egr~issi et al / in,, .-in, imerdc ."ndedJce
o
r calculate iteratively - , w, 9' and s :!; we then evaluate shown in tables 1-4 as F nctions of in,, for from eq . 17b). These quantities are in, = 91 .17 GeV [15] and in,, = 2-5, 100. 500. 1000 GeV, res , .1 a 1 ed cted values of in w. I columns 2- we have kept re because rou off errors there may affect the last digit i the Ar determina1 1 se a columns re note that the values i t e e less elaborate calculations of refs . [6,71. t s o i tio s the (t-) i o let contributions t that in these cal es have e compute i the free fiel theory limit. n principle, f rections which necessarily involve a specific i te ret ti consider t e ey have e iscussed in the case of r [161, where t They have bee eV an = + 2. x 10 -1 or in, -estimated to be = 1 .5 X 1 ' for in , = 15 e 2 o nit > e ', relt a remaining uncertainty of 1 .5 x 10 -3 On,/25 For i i r leads to a change '. fixe n, a shift
an correspondingly C81ia
w=
- On z/ 2c)3S2 .
2
.,r = , so that a sift example, for n, = 240 eV, we have s 2 -- .21 a r = ' induces Ss 2 = 2.9 x 10' an Sin w = -15 MeV. Using tale 1 an eqs. (29a, b) we see that, if the QCD corrections of ref. [161 were included, we eV predicted in w values = 27 eV [43 would obtain for and = 5 eV 2 1eV] lower than indicated in tables 1-4, with an additional uncertainty o = ±9 MeV [ - 20 eV]. These C corrections have not been specifically studied in the case of and .1;w. However, in .,p they are expected to be smaller than in jr by roughly a factor of C2/S2, and greatly diminished in APw. An alternative procedure for large in, is to determine the additional terms of a 2 ) that must be added to the usual one-loop calculation of v r [11, carried out in the on-shell scheme, to take into account the leading higher-order reducible contributions. Neglecting subleading corrections of ®((a/27rs - ) 2 in /MZ), we can achieve this in the following manner. Replacing é 2 (1 - ®î w) ~ e'- , c 2 --~ c 2, we write eq. (15b) in the form s
(30a)
Aiw - ®r = where 2 ^2 2 ^2 F = Re[ Aww(mze) - Azz(mz)e ] Ms . m2z
(30b)
1 ~ sl ~ ,~ a
6.6~0 7~ 6.8~ 6. 6. 7 .9 6. 6.974 6. 5 7.015 7.34 7.051 7. 7. 7. 7.114 7.128
~
~,>
ff
r fe~
. 6.4 6 1 6.319
. . 0. 0.
317
6.177 6. 7 . 11 5.919 5.821 5.716 5. 5.~1 5.367 5®239 5.1 4. -1.816
. .~ 7 0®231 . 1~19 0.~ l l6 0.~ ? .~ 7 .~ 0.2.~ 7 0.2_~924 0.~~ 0.~°r 0.22I 0.2_~7
~ 2~~8
~ 91 .17~
6zl l Sv75 5.43 Sr l 4.45 4®1 3.72 3~3 2ß91 2.47 2. 1.,5 .42 -0.17 -0.79 -1 .46
~~~~ 2 1. ~n ~able ~®r rra~ = 1 10~~r 80.0 .0 1 .0 110.0 120 .0 130.0 140.0 150.0 160.0 170.0 180.0 190.0 200.0 210.0 220.0 230.0 240.0 250.0
6.761 6.841 6.896 6.939 6.976 7.007 7.036 7.061 7.084 7.105 7.125 7.144 7.162 7.179 7.195 7.210 7.225 7.240
6. 6.611 6.556 6.494 6.426 6.352 6.272 6.186 6.094 5 .996 5.892 5.781 5.665 5.543 5.415 5.280 5 .139 4.992
0.23380 0.Zî362 0.23342 0.23320 0.23296 0.23270 0.23241 0.23210 0.23178 0.23143 0.23107 0.23068 0.23028 0.22985 0.22941 0.22894 0.22846 0.22796
6.43 6: 5.76 5.45 5.13 4.79 4.44 4.07 3.68 3 .27 2.83 2.37 1 .87 1.35 0.80 0.22 - 0.40 -1 .05
0.~~95
0.43
0~2261 0.2~~ .~~ 0.~~~ 0.2197 .2182 0.2167 0~ l5 l 0?135 0.2l18 0~1 ' 0~ 1
1.5-1 1 .65 1. 1. 2.1E1 2~43
81 . 3 81 .1
~ie~ ~~
10~~~
0.2330 0.2317 0.23 0.2295 0.2284 0.2273 0.2261 0.2249 0.2236 0.2222 0.22~ 0.2194 0.2179 0 .2163 0.2146 0.2129 0.2111 0.2093
(1.11 0.25 0.3~ 0.48 0.59 0.70 0.81 0.93 1.05 1.18 1.31 1.45 1.59 1.73 1.88 2 .04 2.20 2.37
79.85 79 .91 79 .97 84 .03 .OS 84.14 84.20 84.27 84.33 80.40 80.48 80.55 80.63 80 .71 80 .80 80.88 80.98 81 .07
G. Degrassi et nl. / r'a `~~-rra Z ifaterdependerace
62
TABLE 3
As in table 1, for rra H
=
500 GeV
10`'ar~y
10`'ar"
s`
10`'ar
s`'
10`'ap
m~y (GeV)
80.0 90.0
6.851 6.93 i
6,901 6,852
0.23467
7,01
0.2351
- 0,05
79,74
130.0
7.096
6.590
4.70 4,31 3,
0.2270 0.2?i7 0.2243
0.78 0.90 1 .03
80.16 80 .23 80 .30 80 .37
rra~ (GeV)
i00,fl 110.0 120.0 1-10.0 i50,0 160.0 170,0 180.0 190,0
'_(10 .0 10.0 220 .0 ?30,0 2-10.0 2_~0.0
6,985 7.029 7.065 7.124 7,150 7 .173 7 .19-1 7 .21-1 7 .?°~3
7.?~ 1 7,?68 7,284 7,3
7.315 7.329
6,796 6.733 6.665 6.510 6.424 6 .331 6,?33 6 .125 ti,018
5,901 5,779 5,650 5,515 5.374 5.226
0.23449 0,23429 0.23406 0.23352 0,23355 0.23326 0.2_i295 0.23262 ®.?~? 27 0.?~ 1
6.67 6.36 6.05 5.74 5,41 5.06
i).23151 0.2_~110
3.47 3,02 2.53
0?2975 0?2927 0.22876
0.90 0.30 - 0.35
0.23067 0.23022
2,02 1,48
0.2338 0.2327 0.2316 0.2305 0.2294 0.2282
0.2229 0.2214 0.2 i 99 0,2153 0,2167
0.2149 0.2131 0.2113
0.09 0.20 U.32 0.43 0.54 0.66
i.16 1 .29 1 .43 1 .58 1 .73 1 .89 2.05 2.22
79.80 79,86 79,92 79,97 80,03 80.10
80.44 80.52 80.61 80.69 80.78 80.87 80.97
T :asLE 4 As in table 1, for rra a{ = 1 TeV
rra i tGeV)
10`aî"~~~
10 -'aî-
s`'
10`'ar~
s`
10`ap
rrt~,~ (GeV)
80.0 90.0 100.0 1 i 0.0 120.0 130.0 140.0 150.0 160.0 170 .0 180 .0 190 .0 200 .0 10 .0 220 .0 230 .0 240 .0 250 .0
6.882 6.961 7.016 7.059 7.095 7 .126 7 .154 7 .180 7 .203 7 .224 7 .244 7 .263 7 .281 7 .298 7.314 7 .330 7 .345 7 .359
7.014 6.964 6.908 6.845 6.776 6 .702 6.621 6 .534 6 .442 6 .343 6 .238 6 .128 6 .011 5 .888 5 .759 5 .624 5 .482 5 .334
0 .23508 0 .23490 0 .23470 OZ3447 0 .23422 0 .23395 0 .23366 0 .23335 0, 23302 0 .23266 0 .23229 0 .23190 0 .23149 0 .23105 0.23060 0.23013 0.22964 0.22913
7 .31 6 .97 6 .66 6 .36 6 .04 5 .71 5 .37 5 .01 4.63 4.22 3 .80 3 .34 2 .86 2 .36 1 .82 1 .25 0 .64 0 .01
0 .2362 0 .2349 0 .2338 0 .2327 0 .2316 0 .2304 0 .2293 0 .2280 0 .2267 0 .2254 0 .2240 0 .2225 0 .2210 0 .2194 0.2177 0.2160 0.2142 0.2123
- 0 .14 - 0.00 0.12 0.23 0.34 0 .46 0.57 0 .69 0 .81 0 .94 1 .07 1 .21 1 .35 1 .50 1 .65 1 .81 1 .97 2 .14
79 .68 79 .75 79 .80 79 .86 79 .92 79 .98 80 .04 8O .10 80 .17 80 .24 80 .31 80 .39 80 .47 80 .55 80.64~ 80 .73 80 .82 80 .92
G. Degrassi et al. / ni ll -mz interdependence
63
Combining eqs . (30a), (17b) and (21) and recalling sZc2(1 - r) = s2c'(1 obtain 2
1
® ~r = r w(c ) - Ç' F ~-' ( s2 r s 2c2 ( ) 1 2
e
r), we
r,~
where, for brevity, we have not indicated the dependence of (P) an , ,(c 2 ) on the mass ratios . Splitting (1 - Ar^w)/(1 - :1r) = 1 + ( :1r - ,Irw )/(1 - Ar ). e have c 2 e2 s-'c-' F(c2) s-'
-
c2 e2 ( r-A w ) F(c -') s 2 s-c2 (1 r
In the last term we replace :fir - Jr^w by the expression (31a) and employ e yielding ,r = Jîw(c') -
c
e7
s - s -c
c-' ) 2 F(
+
ç2 7
s-
8G,,miF
( c2 )
(1
- .I;w ) .
(31c)
Substituting AP~,,(c-) -~ ~;w(c2 ), F(c2 ) --+ F(c 2 ) (a replacement that does not affect the leading terms (- 2 be/e )K s and F = 3mt/ -'1t,z + . . .1, we obtain c-' e-' c-' BG,,ntZ ,r=®rw(c - ) - 2 s 2 C ,F(c`) + S, F(c- ) (1 - Arw) . (32a)
s
Recalling (8b) and (30b) one verifies that the first two terms on the r.h .s. coincide with the one-loop expression for ,r in the on-shell scheme [ 1 ]. us the last to represents the higher-order correction to that expression . To tv4"o-loop order it reduces asymptotically for large fn t to (c28p/s 2 )2, in agreement with the results of ref. [5]. We note that 8Gmm`z
F
- 8Gmn,W. Re Aww(mw) - Azz(mz) ( c) fnz r2 mw 2
Ms
(
32b )
This is a familiar combination of self-energies in which the coupling e -'/s2 has been replaced by 8G,,m~,/~ . The two-loop irreducible contribution for large mt is taken into account by appending a factor of [ 1 + (2 T,- - - 19)s'/3c 2 ] to the last term in eq. (32a). Values of Jr obtained from eq. (32a) are given in table 5. They include the two-loop irreducible contribution for large m t and small QCD corrections associ-
On Ir C%vlual
f Y. 432)
IM
51M MI
It a
MU) A) A 0
6.67
30 114 4S2 41 113 3.76
a' 13 ',
113 4,$! T47
vil
3.73 33-1 2-S119 141 IAS 1M R190 0.+,2 ,.2sS -o.92
I rAt ,t 4) 1~ 1 k10
MW it) Mu .0 i410
7-4)
2,
2'07 I q~S IA M _M -"7 - IX!
6w
3.73 5, 4.73 4_15% 3.93 .432 3.07 1M) 2.10 I_% 101 0.40 -0.2-3
à%
5.73 53 4. 4.26 ~S4 1,A) 2.91 143 1.90 1_34 0.75 0.12
- and on ji-maxis-es are exvre-.~scd in GeV The table slwm-, 100r.
with the light quarks (cf. the discussion at the beginning of this section and in the appendix). A detailed comparison shows that for in,, = 25 GeV, 100 GeV, 5 GeV. I TeV, the values of -Ir from eq. (32a) am layer than those obtained from 10-3 eq. by (fn, 3, - 2, - 4) X 10' for in, = 80 GeV and by (1 .4,13,12,12) X for in, = 250 GeV. This is compatible with our rough estimate of the subleading contributions of (vt/2., ) nq'/#nj) I z . We have also compared tables 1-5 with the values of -Ir obtained in the one-loop approximation within the on-shell scheme, i.e. the first two terms of eq. (32a). We find that, although the one-loop truncation of (32a) is a rather good approximation for m, < 180 GeV (the largest error in this range being - 2 X 10-3 for in = 180 GeV and m,, = 25 GeV), for precision studies with large top masses the more complete calculations are desirable. Because the two-loop irreducible contributions of O((a/2 z have not been computed, it seems impossible to prove rigorously at the present time which of the two calculations of Ar (from eq. (22) or eq. (32a)) is more accurate . 5p, (16P)2) On the other hand, in the case of the leading contributions of we know that the reducible effects are by far dominant. Assuming that the same is true for the subleading effects, we are inclined to believe that the calculation based on eq. (22) is more accurate, as it incorporates the reducible higher-order contributions in great detail . (22 1)
i2
2
t
7r S2)2 M2 t /M2
0(c2, fat a
erd /M
I
Mt
tt
le [ tee perha cease with i tween the tw, a ,ree ent quite
side ftom the QCD corrections associated with large m, (see the ion), current ° that the error in Ir discussion in this flavors 1 -' [ 18 analysis of the first five quark contribution is = + gh4, in arising from higher-order uncertainties is = + 1(l ®~ 1191 (this. is with our previous discussion of subleading corrections). Combining errors in quadrature, one obtains an uncertainty of = + 1 .3 10- ` in lr correct translates for s' - ®.225 into -6m w = ± 22 eV. Inclusion of the of ref. 16] would further shift the central values and the errors in the manner outlined after eq. (29b). Accurate determinations of m . are indeed eagerly awaited! While preparing this manuscript, we have received a preprint by J . R ner [21], that bears some relation to our work. Among other considerations and results. Rosner discusses two parameters similar to s? and gyp, and an approximate expression that resembles our accurate eq. (18). One of us (A.S.) would like to thank W.J. Marciano for a very interesting conversation. This work was supported in part by NSF grant PHY-8715995. Appendix A In this appendix we list the analytical formulae needed to calculate J ; w and ..1; when the dependence on c -' and c' is differentiated . The basic relations are eq. (8b) and eq. (15b) and all the explicit expressions are given in the 't HooftFeynman gauge. The starting point in the calculation of ( - 28e/e)MS is eq. (7b). The contribution of the first five quark flavors is analyzed by considering H( (0 -ez Re(IIY'y(mz) e 2 y) )MS = HYy ( 0)) +e2 Re M5y(mz) ms- (A.l)
G. Degrassi et al. /inil,-in z interdependence
66
The first term is evaluated using dispersion relations and detailed information on e + e - --* hadrons [ 1, 18, 20] while the second is studied perturbatively. Using the results of ref. [181 for the first term and in z = 91.17 GeV, we find 2 8e e
= 2«
In
37r >
MS
mz
( m,
1 + 3a + 0.02875 ± 0 .00090 47r )
SSa -(I+ SQCD) + 277r
-
8a
m )
97r In m z`
a
+ 2Tr (' In
c2 - ; ) ,
(A.2)
where the summation is over the charged leptons and the quoted uncertainty corresponds to the error in the first term of (A.1). In the evaluation of e -' Re 11(5)(1112 )I MS we have include . 1 a QCD correction factor of (1 + SQCD ),where [171 5QCD =
ajinz)
7r
+ 1 .405
as( mz)
(A-3)
7r
Numerically, BQCD = 4.02 x 10 -- for a S(mz) = 0.12. Similarly, we have appended a factor (1 + 3a/47r) to the leptonic contributions [4,171. Eq. (A.2) leads to 2 5e e
- = 0.06902 ± 0.00090 MS
8a
97r
In
in, )
`
mz
a (7 -, In c- - ~ ) . 27r
+
(A .4)
Eq . (A.4) gives the dominant contribution to 1Yw and can also be used to compute â = ê2 /47r via eq. (8d). As mentioned in sect . 1, it depends slightly on m,, varying from â -' = 127.8 ± 0 .1 for m, = in z to â -' = 128.0 ± 0.1 for in, = 250 GeV* . The fermionic contribution to the first term in eq. (8b) is given by e2 s-
Re
ww(mw) ,- A(ww( 0)
A(
n1 W
Ms
5
- + In 3
27rs`'
+
8
- (1+2~~)-
2
1)2
1 +
2
In
m, ~ In z
(A .5)
* For approximate calculations that clo nW take full cognizance of' thc radiative corrections, our suggestion is to employ thc value of o witl i the toi) yuark clccoupl`cl, . .e with m, = ny . i
G. Degrassi et a1. / m u, m z interdependence
67
where ~ 2 = m2 fm ZC2 . The corresponding bosonic contribution is e2 Aww(mw) - A`ww( 0) I
a - 3 (~In~-c21nc2) H +I2( -2 ,_ C2)) Q - c2) 4 47rS 2 c2 1
+ In
2~' c2 - C2 (4c2 + .1) -
-
+ 9 S2
1 (2-2+ .1)
-
2
S4
+ 4s2 +
+
s4
c2
c2
-
c2 s2 (
4c 2 + ~ -
(A.6)
,
where f =M2 lin z, H(x) is given analytically in ref. [9], and 4c 2 - 1
I,(c 2 , c - ) _
c`
In c 2
2 11 tan - '( 4c 2 - 1 )[-ê2(5 + , 3c` 3c 4 5
1
7
1
2
s4
1
s~
g4
c-
1
3c -
+
1
12c`
2e 2
3c + c2 2c 2 4 + 2 c + 4c 2 1 6c 2 101 6 2 1 1 ( 1 + c - 9 + 2- +-+ 1 (A , , 9 c 3c 4 2c6c~
s4
.7)
c`
Inserting eqs. (AA)-(A.7) into eq. (8b), we obtain a complete analytic formula for Are . In order to evaluate ®r we need to study the second term in eq. (15b). The fermionic contribution is e2 Aww(mw) _Ac f Re zz) (m 2z ) 112 2 s mz c^2 a
Ims
c2
(lnc 2 - 1) (1 + 8 Qcn ) + 2Trs2 e2 +
S2 8
(1+2~ 2 )-
In c2 5 2 = 3 + ln
m (MZ z
( 2_ 1)2 2 S S 2 1+ 2 In
~2
a m2 + Il( D) 1 4 1 - 2 7rs2c2 + 4-rrs'c2 M2z ~ 3a
I (
m` + 1654 21n +1 x 2a r - -42 .3S 9 I mz -'3 [(1 - 25 2
+ ~~
-~
(3 + D)(A(D) - 1)
.1 5 4 )(1 + SQco) + ( ~ - 3 S2 + 9S4) + -, (1 - 2s2 + 4S4)l
(A.8)
G. Degrassi et al. / rrr ei - -iii z iwerdepende®rce
68
where D --- 4(~n i /1,1 Z ) - 1 and A(D) = D' /-' tan - '(D - ' /-') for D > 0. The bosonic contribution is A"~(~rt 2v) _ c° s-nli e~
,-
,. ~ c -s.2- c - J
cti, zz ('~a z )
IMS
c_)) -H(~)
+ 1(e4 + _Lg4) _ 3 2
-
C
-
C 2( -L
LS-3 -2
9S )
In c-' 1 c 2 (-! + 3s-' - s4/c -' )
+
1 - 3s 4
+
where 21
s.,4 -~
2c -
x c -' 10C? 1
-5
1
i tan -
) 18
lOs 4
- ~ s-' JC' ? ~
s~ + 'c- + - -
J
1
t
4c -' - 1 s
6 C2 J
- 1
(A.10)
e note that in (A.5) and (A.8) we have also included the QCD correction factor (1 + aQCD) in the contribution of the two light quark isodoublets. Inserting eqs. (A.8)-(A.10) into eq. (15b) we obtain the complete analytic expression for vi. In the limit c'- --, c'- , eqs . (A .5)-(A .10) as well as the vertex and box diagrams contributions in eq. (8b) reduce to corresponding expressions of ref. [9] (the functions I,(c 2 , c - ) and 12( c -', c'- ) are called I,(c 2 ) and I,(c'- ) in that work). eferences [11 A . Sirlin, Phys . Rev. D22 (1980) 971 [2] U . Amaldi et al ., Phys . Rev . D36 (1987) 1385 ; G . Costa et al ., Nucl . Phys . B297 (1988) 244 [3] P . Langacker, Phys . Rev . Lett . 63 (1989) 1920 ; J . Ellis and G . L. Fogli, report CERN -TH .5511 /89, SARI-TH/89-60 [4] A . Sirlin, Phys . Rev . D29 (1984) 89 [5] M . Consoli, W . Hollik and F . Jegerlehner, Phys. Lett . B227 (1989) 167 [6] A . Sirlin, Phys . Lett . B232 (1989) 123 [7] S . Fanchiotti and A . Sirlin, Phys . Rev . D41 (1990) 319 [8] A . Sirlin, in Z °-physics, Proc . XXVth Rencontre de Morfond, ed . J . Tran Thanh Vân (Editions Frontières, Gif-sur-Yvette, 1990) p . 103 [9] W. J . Marciano and A . Sirlin, Phys . Rev. D22 (1980) 2695 [101 A . Sirlin, Rev . Mod . Phys. 50 (1978) 573 [ 11 ] A . Sirlin, Nucl . Phys . B332 (1990) 20 [121 M . Veltman, Nucl . Phys. B123 (1977) 89
G. Degrassi et al. / m jj - m z interdependence
69
[13] J.J. Van der Bij and F. Hoogeveen, Nucl . Phys. B283 (1987) 477 [14] B.W. Lee, in Proc. XVI Int . Conf. on High-energy physics, ed. J.D. Jackson and A. Roberts, Fermi National Laboratory, Batavia, 1L. 1972. Vol. IV, p. 266; D. Ross and M. Veltman, Nucl. Phys. B95 (1975) 135 ; P.Q. Hung and J.J. Sakurai, Nucl. Phys . B143 (1978) 81 [15] G. Rolandi, in ZED-physics, Pry. XXV1h Rencontre de Moriond, ed. J. bran Thanh Vin (Editions Frontières, Gif-sur-Yvette, 1990) p. 247 [16] B.A. Kniehl, J.H. Kiihn and R.G. Stuart . Phys. Lett. W14 (1988) 621 : in Polarization at LEP, ed. G. Alexander et al., CERN 88-06. Vol . 1, p. 158 ; B.A. Kniehl, Max Planck Institut preprint MPI-PAE/PTH 5/89 (1989) and references therein [17] G. Burgers and F. Jegerlehner, in Z"-Physics at LEP 1. Vol. 1 . ed. G. Altar`lli, R. aies and C. Verzegnassi, CERN 89-08, p. 55 [18] H. Burkhardt et al., in Polarization at LEP, ed. G. Alexander et al.. CERN 88 . Vol . 1 . P. 158 [19] F. Jegerlehner. Z. Phys. C32 (1986) 425 [20] F. JegeTlehner, Z. Phys. C32 (1986) 19-5 [21] J.L. Rosner . EFI preprint 90-18