

 24 Apr 95

EVIDENCE FOR ELECTROWEAK CORRECTIONS IN THE

STANDARD MODEL\Lambda

ALBERTO SIRLIN Department of Physics, New York University, 4 Washington Place

New York, NY 10003, USA E-mail: sirlin@mafalda.physics.nyu.edu

ABSTRACT The phenomenological evidence for electroweak corrections in the Standard Model, both at very low energies and the Z0 scale, is discussed. In particular, we review a simple but sharp argument for the presence of Electroweak Bosonic Corrections.

1. Universality of the Weak Interactions.

Historically, the first important application involving large radiative corrections to allowed weak-interaction processes is the analysis of Universality of the Weak Interactions.1 In modern language, the test of universality reduces to the question of whether or not the CKM matrix is unitary, a fundamental tenet of the Standard Model (SM). The most precise test involves the relation

jVudj2 + jVusj2 + jVubj2 = 1 : (1) The term jVudj2 is obtained from the ratio of the decay propabilities of the eight accurately measured Fermi transitions (a well-known example is O14 ! N14 + e+ + *) and _ decay, while Vus is extracted from K`3 and hyperon decays. (jVubj2 plays an essentially negligible role at present). If only the very large Fermi Coulomb corrections are included, the test does not work : the l.h.s. of Eq.(1) is found to be ss 1:04, with uncertainties of O(0:1%). Thus, the SM is not tenable under such a simplified analysis and it is necessary to evaluate the additional O(ff) corrections. There is however, a basic theoretical difficulty : in a rigorous analysis, one cannot simply use elementary Feynman diagrams because fi-decay involves complex hadronic systems at very small momentum transfers. Instead, it is possible to express the radiative corrections in terms of current correlation functions, i.e. Fourier trasforms of matrix elements of time-ordered products of current operators, and make use of their associated Ward identities and short distance expansions.2 Remarkably, the calculation can be carried out to good accuracy if one assumes that current conservation is softly broken, i.e. by mass terms, and that the strong interactions are asymptotically free

\Lambda To appear in the proceedings of the Ringberg Workshop on "Perspectives for electroweak interactions in e+e

\Gamma collisions" hosted by the Max Planck Institut, at the Ringberg Castle, M"unich,

February 5-8, 1995. Edited by B. Kniehl.

(as in QCD). In the local Fermi theory of weak interactions, the O(ff) corrections to the ratio is divergent, while the SM, being a renormalizable theory, provides a finite answer. Furthermore, for reasons that are not well understood, in versions of the SU(2)L \Theta U(1) theory where the Higgs scalars trasform as singlets and doublets, so that cos2 `W = M 2W =M 2Z is a natural relation, the answer is the same as in the local theory with the cutoff \Lambda replaced by MZ. (This assumes that the couplings of the Higgs scalars to leptons and light quarks are very small in analogy with the minimal version of the theory). Schematically, one obtains for the fi-decay probability2;3

P = P 0 ae1 + 3ff2ss ^ln ` MZ2E

m ' + 2

_Q ln `MZM ' + :::*oe ; (2)

where _Q = (2=3 \Gamma 1=3)=2 = 1=6 is the average charge of the underlying fundamental fields (in this case the u and d quarks), Em is the end-point energy of the positron, M is a hadronic mass of O(1GeV), and the ellipsis stand for significant but smaller contributions that have been studied in detail. In the O14 case, Em ss 2:3MeV and the first logarithmic term in Eq.(2) leads to a ss 3:45% correction. This contribution literally rescues the SM from obvious contradiction ! A recent analysis4 of the eight superallowed Fermi transitions leads to Vud = 0:9736 \Sigma 0:0007. Combining this result with Vus = 0:2205 \Sigma 0:0018, and Vub = 0:004 \Sigma 0:002, one obtains4

jVudj2 + jVusj2 + jVubj2 = 0:9965 \Sigma 0:0015 : (3) This falls short of unity by 2.3 times the estimated error, which is mainly theoretical. Part of this uncertainty is due to the nuclear overlap correction ffic, for which there exist at present somewhat different competing evaluations. A more recent approach4;5 attempts to take into account the contribution to ffic from core nucleons, introducing a phenomenological correction factor 1 + aZ (Z is the charge of the daughter nucleus), and determining a from the data. This leads4 to Vud = 0:9745 \Sigma 0:0007 and

jVudj2 + jVusj2 + jVubj2 = 0:9983 \Sigma 0:0015 ; (4) which is consistent with unity. The agreement with universality is even better in Wilkinson's recent analysis (second paper of Ref.5), which is also based on a Zdependent phenomenological correction factor ; he finds Vud = 0:97545 \Sigma 0:00082 andP

i=d;s;b jVuij2 = 1:0001 \Sigma 0:0018.5 Thus, at present it is not clear whether there is disagreement or not but, if so, it is at the 0:3% rather than the 4% level, which would

be devastating.

An interesting question is whether these are genuine electroweak corrections. To answer this query the following observations are relevant : i) One needs a renormalizable theory, such as the SM, to evaluate them. ii) It may be argued however, that the result can be reproduced with a local theory calculation involving only electromagnetic corrections, provided that one regularizes the result with a suitable cutoff

\Lambda . iii) Point ii) can be answered by noting that the cutoff is important and that one needs the complete theory to determine it accurately. For example, if the cutoff were \Lambda = v = (p2G_)(\Gamma 1=2) = 246GeV, a value which is rather reasonable and was in fact anticipated before the emergence of the SM, we would obtain

jVudj2 + jVusj2 + jVubj2 = ( 0:9919 \Sigma 0:00150:9937 \Sigma 0:0015 ; (5) where the upper and lower entries correspond to the treatments leading to Eq.(3) and Eq.(4), respectively. These results differ from unity by 5:4oe and 4:2oe, respectively, and are in clear disagreement with unitarity. Thus, an accurate determination of \Lambda is necessary and this can only be provided by the complete theory. iv) One may also inquire what diagrams are relevant in the SM calculation. The analysis shows that one must consider the corrections associated with the complete gauge sector, not just the photon, and this includes all the vertex and box diagrams involving virtual fl; Z0 and W \Sigma . For example, the fermionic couplings of fl and Z0 are not universal, being different for leptons and quarks, and one must study all these diagrams in order to obtain meaningful results. Furthermore, it is only after all the gauge-sector contributions are combined that the amplitudes become convergent (after renormalization) and can be analyzed with short distance expansions, necessary for the control of strong-interaction effects ! On the other hand, the Higgs sector has an indirect effect: Eq.(2) holds exactly only when cos2 `W = M 2W =M 2Z at the tree-level. We recall that this is the case in the presence of any number of Higgs doublets and singlets. When triplets and other representations are present so that cos2 `W = M 2W =aeM 2Z , there is an additional contribution [ff(ae \Gamma 1)=ss] ln(M 2W =M 2Z )[M 2W =M 2Z \Gamma 1]\Gamma 1 to the expression between curly brackets in Eq.(2).2 However, this contribution is practically negligible, as current phenomenology shows that ae is very close to unity.

2. Evidence at High Energies.

I follow the discussion of Ref.6. An alternative approach is developed in Ref.7. In the high-energy processes currently investigated the dominant electroweak corrections involve virtual fermions. Their effect is responsible for the large logarithms associated with the running of ff 8 and the contributions from the t \Gamma b isodoublet from which the Mt constraints are derived.

It is natural to ask whether there is evidence in high-energy phenomena for corrections not contained in the running of ff and, more specifically, in ff(MZ ). One way to quantify this question is to "measure" (\Delta r)res,6 the residual part of \Delta r 9 after extracting the effect of the running of ff. One has

ff 1 \Gamma \Delta r =

ff(MZ ) 1 \Gamma (\Delta r)res : (6)

It is worth noting that ff(MZ ) is scheme-dependent. Two frequently employed choices are : i) ff(MZ ) = ff=(1\Gamma \Delta ff), where \Delta ff = e2Re h\Pi (f)flfl (0) \Gamma \Pi (f)flfl (MZ)i is the fermionic contribution to the conventional QED vacuum-polarization function. A recent determination gives \Delta (5)Rad = 0:0280 \Sigma 0:0007 for the five-flavour component which, when combined with the leptonic and very small top contributions, leads to ff\Gamma 1(MZ ) = 128:899 \Sigma 0:090.10 Recent alternative evaluations are given in the papers of Ref.11. ii) The MS definition ^ff(MZ) = ff=[1 \Gamma e2\Pi flfl(0)MS]; where the MS subscript reminds us that the MS renormalization has been implemented and _ = MZ has been chosen. Updating the analysis of Ref.12 with the new result from Ref.10, one finds e2\Pi flfl(0)MS = 0:0666 \Sigma 0:0007 or ^ff\Gamma 1(MZ ) = 127:91 \Sigma 0:09:

Inserting the direct world-average determination MW = 80:23 \Sigma 0:18GeV 13 in the basic relation9

M 2W (1 \Gamma M 2W =M 2Z ) = (ssff=p2G_)=(1 \Gamma \Delta r) ; (7)

one obtains \Delta r = 0:0442 \Sigma 0:0104. Using ff\Gamma 1(MZ ) = 128:899; Eq.(6) gives (\Delta r)res =\Gamma

0:0161 \Sigma 0:0111; which differs from zero by only ss 1:5oe, not a very strong signal. The MW \Gamma MZ interdependence, in conjuction with the experimental value of MW , has also been extensively used by Z. Hioki14 to examine the effect of various components of \Delta r.

The constraint is much sharper if we interpret the data in the framework of the fully fledged SM, treated as a Quantum Field Theory with its plethora of radiative corrections and interlocking relations.6 The recent precision electroweak analysis, including all direct and indirect information, leads to MW = 80:32 \Sigma 0:06+0:01\Gamma 0:01 GeV,15 where the last error reflects the uncertainty in MH . Taking MW = 80:31 \Sigma 0:06GeV, the worst case for the analysis, one finds \Delta r = 0:0396 \Sigma 0:0035 which implies (\Delta r)res = \Gamma 0:0210 \Sigma 0:0037. This value differs from zero by 5:6oe. If we employ

^ff(MZ ) in Eq.(6) instead of ff(MZ ), the evidence is even sharper : (\Delta r)res becomes\Gamma

0:0289 \Sigma 0:0037, or 7:8oe away from a null result !

A scheme-independent argument can be obtained by considering two different definitions of sin2 `W which are physical observables6 : i) sin2 `lepteff ii) sin2 `W =

1 \Gamma M 2W =M 2Z . From the global fits one has : sin2 `lepteff = 0:2320 \Sigma 0:0003+0:0000\Gamma 0:0002 and sin2 `W = 0:2242 \Sigma 0:0012+0:0003\Gamma 0:0002, and we see that they differ by 6:3oe ! As the two definitions agree at the tree level (because in the SM Lagrangian there is a single mixing angle), the difference must be due to radiative corrections. In particular, there is no B.A. involving a single mixing angle, whether related to ff(MZ ) or not, that can accomodate all the information derived from the data using the full SM.

3. Evidence for Bosonic Electroweak Corrections in the SM.

I follow the discussion of Ref.16. There are also detailed studies based on an effective Lagrangian approach.17

By definition, at the one-loop level the electroweak bosonic corrections (E.B.C.)

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