Mass Relation of the Weak Gauge Bosons as a Clean Test of Electroweak Higher Order Effects: Analysis Including Leading-Log Corrections

- - Analysis Including Leading-Log Corrections-- 0 0 Zenro HIOKI Department of Physics, Kyoto University , Kyoto 606 (Received November 2, 1983) The relation between W and Z boson masses, which was previously derived at I-loop level, is improved by including the leading logarithmic corrections. The O(anlnn) contributions (n;;::2) are found to increase Mw-Mz difference slightly and to be comparable to the 0(11') non-leading effects. Some applications of this relation to heavy particle .search are also discussed. - The discovery of W and Z bosons at CERN PP colliderl),2) has convinced many particle physicists in their belief for validity of the 5U(2) x UO) electroweak theory.') We are now in a position to step towards the more detailed studies of the theory beyond tree approximation. Concerning this point, one elementary question is: How can we clearly see the electroweak higher order effects? The size of these effects to vari ous cross-sections and decay-widths which are normalized by GF, the Fermi coupling constant, were found generally small.*) On the contrary, they have been found non-negligible when we compute the W and Z boson masses from the data of low energy processes,*) i.e., GF and sin28w. In fact, the tree calculations receive ~ (2 ~ 3)% corrections at one-loop level, and these deviations seem to be detectable as the direct. experimental information on these masses will be obtained with the precision of 0.1 ~0.2 GeY (although the pres entpreliminary data are not so accurate yet). Unfortunately, however, this is not true because the present experimental data on sin28w include rather large uIlcertainties,S) at least ~S%, and thereby the one-loop effects become totally un clear. Instead of this, Maiani6) and F) have indepen dently pointed out that the Mw-Mz relation result ing from the one-loop-corrected f./ decay width and its experimental data is very important as a clean test of the higher order effects since such a relation does not suffer from the above uncer tainty on sin28w. Subsequently, this relation has been found also useful for heavy fermion *) A list (and summary) of the studies on higher order effects is given, e.g., in Ref. 4). search.B)-'O) In this short note, I will improve our previous works by taking the all leading logarithmic cor rections anlnn(m/M) into account. (M and m represent masses of weak boson and light fermion respectively.) I think these computations are significant due to the following reasons although the O(a2 In2) effeCts have already been studied by Maiani.6),B) i) It is important to examine how the leading-log terms affect the bound on t-quark mass, mt ::S250 GeV, derived in Ref. 10) by using the Mw-Mz relation and the UAI data. ii) This procedure in terms only of the directly measurable quantities, a and various masses, is very transparent. Before going into these studies, let us briefly summarize our previous calculations. The start ing point is the following simple equation: where the left-hand side is the f./-decay width calculated as a function of the physical param eters (mf and m, are fermion and Higgs-scalar masses), and the right-hand side is the exper imental data usually expressed by GF (see beloW). At first, the tree prediction for W mass, Mw(O), is obtained by using the width at the lowest order, r(O)(a, Mw, Mz), on the left-hand side ofEq. (1) as Here, 11'-'=137.036 and GF=(1.l66320.00002) x lO-sGey-2 which is determined from the muon lifetime, r", by Mw[a+ 2:anlnn] Mw[a+aln] Mz Then, by putting r 1)=rO)+Llr(rl}: one-Ioop corrected width) and solving Eq. (1) pertur batively, the Mw-Mz relation including the one loop effects (the O(a) plus the O(a In) effects) is obtained as Mw=Mw(OJ _[Llr(a, Mw, Mz, mf, m p)] a;;'w r(OJ(a, Mw, Mz) What I want to do here is to add the O(anlnn(m /M(nz.2) effects to LJr, and consequently to the Mw-Mz relation. This can be performed by the following operator analysis with the renor malization group technique, the basis of which was given by Kazama and Yao.ll) Generally, an effective Hamiltonian of a weak process renormalized at light fermion mass scale, m, is expressed as where Oi are composite operators relevant to the process under consideration, and coefficient func tions Ci obey the renormalization group equation. Fortunately, the situation is quite simple in the present case, since the relevant operators, 01 = il"yA1- Ys)lIe" eyap' , have both vanishing anomalous dimensions. Therefore the decay amplitude including leading log corrections becomes [A(p. ..... ellil )]1.1. -xa(M)Mz2 - (1 ) 2Mw2(Mi- Mw2) II"Ya - Ys lie (~f: the sum in all flavors and colors, Qf: .the corresponding electric charge in lei unit) In our previous works, only a and a2 terms in Eq. (7b) (plus other O(a) finite corrections) were taken into account. Explicit form of the mass relation is Mw=Mw(O)+LlMw-[ a(~~-a x Mw(Mz2-Mw2)] 2Mw2..:. Mi where LlMw represents the O(a) (non-leading) terms. In the Table, I show the numerical results for some typical values of Mz (As for the quark and Higgs masses, I have taken as mu = md = ms =0.1 GeV, mc=1.5 GeV, mb=4.7 GeV, m,=30 GeV and mp=10 Gev.) There Mw[a], Mw[a +aln] and Mw[a+~anlnn]express respectively the value of Mw with the O(a)(non-Ieading) correc tions only, with the full oneloop-corrections and with full one-loop-effects plus all leading-log cor rections. It is found that i) the O( anlnn) contributions (nz.2) increase slightly the devia tion from the tree prediction on M w, which is an interesting tendency for the test of the higher order effects, and ii) the size of them is compara ble to the O(a) contributions (but with the oppo site sign). Finally, let us discuss the heavy fermion search by the Mw-Mz relation. OJ The muon decay amplitude in the leading-log approximation was also derived by this approach in Ref. 12), where Mwand Mz were computed from GF and sin2 Ow, and the O(a2In2 ) effects were shown to be non-negligible. For a given Mz, Mwth("th" represents "theo retical value") is an increasing function of m, in the region m,<100 GeV.8).,.10) Concerning the data, 92.7 GeV~MzxP~97.7 GeV and 79.0 GeV ~MwxP~83.0 GeV (VAl data in Ref. 2)), I have found that an overlapping region of MwxP and Mwth (Mwth(Mz=92.7)~Mwth~Mwth(Mz=97.7)) disappears if m, is heavier than 250 GeV. (That is, the Mwth'm, curve for Mz=92.7 GeV crosses This is the I-quark mass bound mentioned in the preced ing paragraph. Since the O(<<nlnn) effects have been found to decrease the value of Mwth slightly for given Mz and mt, the above crossover point moves a little. I have estimated this shift as ~ +5 GeV, i.e., mt ~255 GeV. Since, there are still rather large ambiguities in the present data on Mw and Mz,' we should not take this bound too seriously. However, this analysis is a good. example of the fact that our approach for heavy fermion search is in fact effective. In conclusion, I have improved the theoretical relation between Mw and Mz by summing up the leading-log (...truncated)