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excitation of both degrees of freedom. The ground state of each flavour configuration cannot do anything but decay weakly, by disintegration of one of the heavy quarks, and sometimes by exchange of a W-boson between the constituents. A variety of fin
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arXiv:hep-ph/9702222v2 18 Apr 1997
The Weak-Magnetic Moment of Heavy Quarks
J. Bernab´ eu, J. Vidal Departament de F´ısica Te`orica, Universitat de Val`encia and IFIC, Centre Mixt Univ. Valencia-CSIC E-46100 Burjassot (Val`encia), Spain and G.A. Gonz´ alez-Sprinberg Instituto de F´ısica, Facultad de Ciencias, Universidad de la Rep´ ublica, CP 10773 11200 Montevideo, Uruguay
abstract With initial and final particles on-shell, the anomalous weak-magnetic dipole moments of b and c quarks are electroweak gauge invariant quantities of the effective couplings Zb¯b and Zc¯ c, respectively, and good candidates to test the Standard Model and/or new physics. Here we present a complete computation of these quantities within the Standard Model. We show that decoupling properties with respect to heavy particles do take place in the weak magnetic moment. The obtained values, ab (MZ2 ) = (2.98 − 1.56i) × 10−4 and ac (MZ2 ) = (−2.80 + 1.09i) × 10−5 are dominated by one-gluon exchange diagrams. The electroweak corrections are less than 1% of the total magnitude.
The neutral current sector of the Standard Model (SM) has been subjected to a detailed precision scrutiny in the past few years . This has led to establish definite quantum electroweak corrections to an impressive list of physical observables which see their treelevel values modified at the percent level. The agreement between the experiment and the theory proves the correctness of the SM and the machinery of renormalization in the quantum field theory. Although the issue is not still close, it seems  that even the Z-vertex to heavy quarks, which contains non-decoupling effects , is in agreement with the SM. An alternative to this methodology consists in isolating new observables in the quantum theory which were absent in the tree-level Lagrangian. In this paper we study the anomalous weak-magnetic moment (AWMM) of heavy quarks. The anomalous weak-magnetic moment of fermions carries important information about their interactions with other particles. It may be seen as the coefficient of a chirality-flipping term in the effective Lagrangian of the Z coupled to fermions. Therefore, at q 2 6= 0, it is expected to be proportional to the mass of the fermion, and only heavy fermions (leptons or quarks) are good candidates to have a measurable anomalous weakmagnetic moment. The already mentioned chirality properties indicate that some insight into the mechanism of mass generation may be obtained from it. These properties have also been considered in the context of extended models . In previous work  we have studied the case of the tau and shown that it is possible to construct polarization observables sensitives to the AWMM. In this paper we focus on quarks, in particular on the AWMM of the b- and c-quarks. In Ref.  different strategies to detect polarization effects for the b-quark are suggested and discussed, so that the observables may become feasible in the future.
Anomalous Weak-Magnetic Moment
As the AWMM is proportional to the mass of the particle, in principle, only heavy fermions might have a sizeable value for it. The heaviest quark, the top-quark , would
seem to be the perfect candidate. The problem arises there in the electroweak gauge invariant properties of the defined form factor. As it is already well known [5, 8] only the on-shell definition of the AWMM is electroweak gauge invariant and free of uncertainties. Nevertheless, recently some procedures to move off-shell the gauge invariant form factors have been proposed , but their invariant properties and physical significance are still under discussion . In this paper we concentrate ourselves in the study of the AWMM for the heavy quarks produced from on-shell Z’s, i.e. bottom b and charm c quarks. This is of order αs -strong or α-electroweak radiative correction to the Zq q¯ vertex. Using Lorentz covariance, the matrix element of the i-quark vector neutral current can be written in the form: vi (q 2 ) µ aw (q 2 ) µη u¯i (p) V (p, p¯) vi (¯ p) = e u¯i (p) γ +i i σ qη vi (¯ p) 2sw cw 2mi "
where q 2 = (p + p¯)2 is the 4-momentum squared in the center of mass frame, e is the proton charge and sw , cw are the weak mixing angle sine and cosine, respectively. The first term vi (q 2 ) is the Dirac vertex (or charge radius of the fermion i) form factor and it is present at tree level with a value vi (q 2 ) = Ti 3 − 2Qi s2w , whereas the second form factor, 2 aw i (q ), is the AWMM and it appears due to quantum corrections. As already mentioned,
at q 2 = MZ2 , it is a linearly independent and gauge invariant form factor of the Lorentz covariant matrix element, contributing to the physical Z −→ q q¯ decay amplitude.
Figure 1: Contributing Feynman diagrams to the anomalous weak magnetic moment. In the t’Hooft-Feynman gauge, there are 14 diagrams that contribute to aw i . In Fig. 1 we show the generic one-loop diagram contribution. From now on we denote by qi (qI ) the internal quark in the loop with the same (different) charge as the external quark; α, 3
β and δ are the particles circulating in the loop as shown in Fig. 1; χ and Φ are the neutral would-be Goldstone boson and physical Higgs, and σ ± are the charged would-be Goldstone bosons. Then, all the contributions may be written with the compact notation: [ai ]αβδ =
α m2i X cjk Ijk αβδ (i) 4π MZ2 jk
with (α β δ) standing for: (N q¯i qi ), (C ± q¯I qI ), (qI C + C − ) and (qi N N ′ ). N and N ′ are the neutral particles γ, Z, χ, Φ (with N 6= N ′ ), and C ± are the charged bosons W ± and σ ± . cjk are coefficients, and Ijk αβδ (i) ≡ Ijk (m2i , q 2 , m2i , m2α , m2δ , m2β )
are the on-shell (p2 = m2i , p¯ 2 = m2i and q 2 = (p + p¯)2 = MZ2 ) scalar, vector and tensor functions defined, from the one-loop 3-point functions I00;
We are only interested in the AWMM so that, for each diagram, we have to pick up only the σ µν qν coefficient shown in Eq.(1). Though the AWMM receives its leading ¯ µν ψ Z µν terms in the contribution from one loop diagrams (renormalizability excludes ψσ Lagrangian), it is finite, and can be extracted from them with no need of renormalization. Notice also that only vertex corrections may contribute to the AWMM, because the renormalization of the external legs does not change the (V-A) Lorentz structure of the vertex. As a test of our calculation we have verified the conservation of the vector current q µ u¯(p) V µ (p, p¯) v(¯ p) = 0
by explicitly checking that the coefficient of the q µ term of the matrix element (1) vanishes. This conservation does not occur on each diagram, but it can be confirmed by considering cancellations among some of them and, of course, in the overall sum. In the following, we list all contributions to the AWMM that are written, in a selfexplanatory notation, as:
Zbb [aw b ]
4vb Q2b MZ2 [I10 + I22 − I21 ]γbb sw cw
h vb 2 v 2 (I22 − I21 + I10 ) + M s3w c3w Z b
α =− 4π
γbb [aw =− b ]
a2b (3I22 − 3I21 + 11I10 − 4I00 ) χbb [aw b ]
m2b vb MZ2 [I22 − I21 ]χbb 2 3 3 MZ 2sw cw
m2b vb MZ2 [I22 − I21 + 2I10 ]Φbb 2 3 3 MZ 2sw cw
(vt + at ) |Vtb |2 MZ2 [I22 − I21 + 3I10 − I00 ]W tt s3w cw
with ai,I , VIi being the axial vector Zq q¯ coupling, and the Kobayashi-Maskawa qI qi mixing matrix element, respectively. Diagrams with the Higgs (Φ) and the neutral would-be Goldstone boson (χ) coupled to the Z only contribute to the axial form factor and not to the magnetic moment, so that one gets the result of Eq.(19). The natural scale of each diagram is
2 mb MZ,Φ
but those with an exchange of a Higgs
χbb Φbb (physical or not) between the two b’s (see [aw , [aw ) are suppressed by an extra b ] b ]
2 mb MZ,Φ
factor coming from the Higgs-b-b coupling. For similar reasons, due to the high
value of the top mass  one could then think that those diagrams with Higgs particles σtt w tσσ coupled to the t-quark ([aw ) would be the dominant ones. In fact, Eqs. (12) b ] , [ab ]
and (14) show the
expected factor, which should make sizeable the contribution
coming from these diagrams. Nevertheless, contrary to what happens in the charge radius (γ µ ) form factor , where non-decoupling effects take place, the behaviour with mt of the Ijk integrals given in Eqs. (12) and (14) prevents the product m2t Ijk to have a hard dependence with large mt . An expansion of the scalar functions Iijtσσ and Iijσtt –up to leading order– in terms of 1/t ≡ (MZ /mt )2 gives: ! ! 1 c2w c2w 1 2 tσσ MZ I00 = log + 2fw − 1 + O 2 log t t t t MZ2
1 = t
1 c2 1 1 c2 + O 2 log w log w + fw − 2 t 4 t t
1 = t
1 c2 2(1 − c2w ) 1 2c2 log w + fw − + w 3 t 3 9 3
1 = − t
(21) 1 c2 + O 2 log w t t
c2 1 + 2c2w 2c2 1 c2 1 1 + O 2 log w log w + fw − w + 6 t 3 3 36 t t !
c2 1 1 = − + O 2 log w t t t
c2 1 1 = − + O 2 log w 4t t t
1 c2 1 = − + O 2 log w 9t t t
c2w 1 1 + O 2 log = 18t t t
with fw =
4c2w − 1 arctan 1/ 4c2w − 1 . As can be seen from the previous expressions,
only a mild (MZ /mt )2 log (mt /MZ )2 dependence is got from the four diagrams that may give non-decoupling effects with the top-quark mass. The chirality flipping property of the magnetic moment makes the difference with respect to the charge radius, where nontW σ tσW decoupling effects are seen. In addition, for the [aw and [aw amplitudes, the b ] b ]
AWMM selects a product of left and right projectors that gives no linear contribution on m2t . Adding all these terms, we end up with the following result: (σtt)+(tσσ)+(tσW )+(tW σ) = [aw b ]lead.ord.in mt
α = 4π
1 2s3w c3w
23 11c2w MZ2 MZ2 log − +O 36 9 m2t m2t
and we conclude that the non-decoupling of a heavy top reduces to a constant term for the AWMM. The Ijk αβγ functions are analytically computed in terms of dilogarithm functions. As a check we confronted the result with a numerical integration in the mb → 0 limit. For mt = 174 GeV, MZ = 91.19 GeV, s2w = 0.232, α = 1/127.9 and mb = 4.5 GeV, the following numerical contributions for each diagram are found:
where the values between parenthesis in Eqs. (32) and (37) correspond to
(38) MΦ MZ
= 1, 2, 3.
The other values agreee with the result of Ref.  for the SM. We have taken the Kobayashi-Maskawa matrix being unity (VIi = diag(1, 1, 1)) for numerical results. Finally, the electroweak contribution to the b-AWMM is 2 −6 aw b (MZ ) = [ −(1.1; 2.0; 2.4) + 0.2i] × 10 ,
[MΦ = MZ , 2MZ , 3MZ ]
An immediate consequence of these results is that the AWMM contribution to the total electroweak width is very small. This is easily seen just by considering that the w ratio Γ(aw b )/ΓTree is given by the interference with the AWMM amplitude.
Γ(aw sw cw vb b ) ≈6 2 Re(aw b) w ΓTree vb + a2b
Then, Eq. (39) shows that only approximately 1 over 106 parts of the width is given by the electroweak contribution to the AWMM. Contrary to what happened for the tau weak magnetic moment, where the lepton vector neutral coupling was responsible for the suppression of the Higgs mass dependence, we observe here that the mass of the physical Higgs has a sizeable effect on the final electroweak magnetic moment (39) for the b-quark. For the selected range of MΦ , it changes the real part of the AWMM in more than 100%. This is so because, as can be seen from Eq. (37), the contribution of the bZΦ diagrams –Eqs. (15) and (16)– are almost of the same order as the leading ones. Unfortunately, these effects will not be observable because, as we will show in the following, the magnetic moment is dominated by the QCD contributions. In addition to the purely electroweak contributions to the AWMM of the b-quark given above, we now consider the QCD contributions to ab . To lowest order, there is only one relevant diagram of the type shown in Fig. 1: the one with α being now a gluon. The evaluation of that diagram only differs from the γb¯b diagram (Eq. (7)) in the couplings, so that it is straightforward to find the result h
1 − 4(mb /MZ )2 and αs = 0.117, which is in good agreement with the analyt-
ical expression found in Ref. , when expressed in terms of an AWMM. The final value we get for the weak magnetic moment of the b-quark is then QCD 2 ab (MZ2 ) = aw (MZ2 ) = (2.98 − 1.56i) × 10−4 b (MZ ) + ab
for MΦ = MZ . Eqs. (39) and (41) show that even though different values of the Higgs mass modify considerably the purely electroweak AWMM, this effect does not translate into an appreciable change of the total AWMM (for which only a 0.4% of variation is found if MΦ moves from MZ to 3 MZ ) because the electroweak contribution is less than 1% (for MΦ = 3MZ ) of the total one. 9
Due to the fact that there is no enhancement of the electroweak contributions coming from the presence of a heavy top-quark, all the electroweak diagrams (except those already mentioned with Higgs exchanged between two b’s) are of the same order, in particular the γb¯b diagram. Then, Eq. (41) leads to the conclusion that the QCD contribution is two orders of magnitude bigger than the electroweak one. In fact, the next to leading order contribution in perturbative QCD would be probably comparable to the computed leading order in the electroweak sector. For the c-quark, all the previous discussion holds, and one expects the electroweak contribution to be of the order 2 aw c (MZ )
2 aw b (MZ )
. That is
≈ O 10−7
The one-loop QCD contribution will also be dominant, and its magnitude can be easily computed from the analytic expression of Eq. (41), adapted to the c-quark. For mc = 1.6 GeV, we get the value: ac (MZ2 ) ≈ aQCD (MZ2 ) = c
αs 4 [aw ]γcc = (−2.80 + 1.09i) × 10−5 α 3Q2c c
We have calculated the electroweak contributions to the anomalous weak magnetic moment of the b-quark, within the Standard Model, and found that it is of the order 10−6 . One loop QCD contributions to the AWMM are dominant and increase its value to 10−4 . The result tells us that in the magnetic moment form factor: 1) the contributions from new physics to the electroweak sector are hidden by the dominant strong interaction contribution, 2) the Zb¯b width is rather insensitive to electroweak contributions in the AWMM sector, and 3) contrary to what happens for the charge radius form factor, nondecoupling effects do not take place in the AWMM. The value of the AWMM for the c-quark is also computed (up to first order in QCD) and it is, as expected, smaller than that for the b-quark by a factor (mc /mb )2 × vc /vb .
Acknowledgments We would like to thank Denis Comelli, Germ´an Rodrigo and Arcadi Santamaria for clarifying discussions. This work has been supported in part by CICYT, under Grant AEN 96-1718, by I.V.E.I., and by CONICYT under Grant 1039/94. G.A.G.S. thanks the Spanish Ministerio de Educaci´on y Ciencia for a postdoctoral grant at the University of Valencia at the beginning of this work and the hospitality received at the I.C.T.P. where this work was completed.
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