The definition of the physical properties of particles in perturbative gauge
theories must satisfy gauge invariance as a requisite. The Pinch Technique
provides a framework to define the electromagnetic form factors and the
electromagnetic static properties of fundamental particles in a consistent and
gauge-invariant form. We apply a simple prescription derived in this formalism
to check the calculation of the gauge-invariant one-loop bosonic electroweak
corrections to the muon anomalous magnetic moment.Comment: 6 pages and 1 eps figur

Cabral-Rosetti, L. G.

Castro, G. López

Pestieau, J.

English

arXiv

The definition of the physical properties of particles in perturbative gauge
theories must satisfy gauge invariance as a requisite. The Pinch Technique
provides a framework to define the electromagnetic form factors and the
electromagnetic static properties of fundamental particles in a consistent and
gauge-invariant form. We apply a simple prescription derived in this formalism
to check the calculation of the gauge-invariant one-loop bosonic electroweak
corrections to the muon anomalous magnetic moment.
 Comment: 6 pages and 1 eps figur

Cabral-Rosetti, L.G.

Lopez Castro, G.

Pestieau, Jean

DIAL UCLouvain

arXiv:hep-ph/0211437v1  27 Nov 2002hep-ph/0211437One-Loop Electroweak Corrections to the MuonAnomalous Magnetic Moment Using the PinchTechniqueL. G. Cabral-Rosettia,1 , G. Lo´pez Castrob,2 and J.Pestieauc,3aInstituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xico,Circuito Exterior C.U., A.P. 70-543, 04510 Me´xico D.F. Me´xicobDepartamento de F´ısica, Centro de Investigacio´n y de EstudiosAvanzados del IPN, Apartado Postal 14-740, 07000 Me´xico D.F. Me´xicoc Institut de Physique The´orique, Universite´ catholiquede Louvain, B-1348 Louvain-La-Neuve, BelgiumAbstractThe definition of the physical properties of particles in perturbative gauge theoriesmust satisfy gauge invariance as a requisite. The Pinch Technique provides a frameworkto define the electromagnetic form factors and the electromagnetic static properties offundamental particles in a consistent and gauge-invariant form. We apply a simpleprescription derived in this formalism to check the calculation of the gauge-invariantone-loop bosonic electroweak corrections to the muon anomalous magnetic moment.A definition of the neutrino charge radius that satisfies good physical requirements, i.e.it is a physical observable, has been provided recently [1] in the framework of the PinchTechnique (PT) formalism [2]. In the PT formalism, the construction of a gauge-independentand gauge-invariant one-loop vertex and, in particular, of an effective electromagnetic form1E-mail: luis@nuclecu.unam.mx2E-mail: Gabriel.Lopez@fis.cinvestav.mx3E-mail: pestieau@fyma.ucl.ac.be1factor for the neutrino amounts to compute [1] the one-loop vertex corrections using a simpleprescription in the linear RLξ gauge, where gauge-boson propagatorsP Vµν(q) =− iq2 −M 2V[gµν + (1− ξ) qµqνξq2 −M 2V], (1)are taken in the ’t Hooft-Feynman gauge ξ = 1, and the usual three-boson vertexΓαµν(q, k,−q − k) = (q − k)νgαµ + (2k + q)αgµν − (2q + k)µgαν , (2)is replaced by the truncated vertex [3]:ΓFαµν = (2k + q)αgµν + 2qνgαµ − 2qµgαν , (3)which satisfies [1] a simple Ward identity:qαΓFαµν = (k + q)2gµν − k2gµν .We emphasize that this prescription should be applicable not only for the case of theneutrino but also for the electromagnetic form factors of quarks and leptons [4]. In particular,it looks very appealing to compute in a simple form the electroweak contributions to thestatic properties of fermions. In this note we apply the PT prescription to give an alternativeevaluation of the one-loop W -boson contribution to the anomalous magnetic moment of themuon, aµ ≡ (g − 2)/2.The complete one-loop electroweak corrections to aµ were computed long time ago inrefs. [5] (the Higgs boson contribution and subleading muon mass terms are neglected):aweakµ =GFm2µ8pi2√2{103+13[(1− 4 sin2θW )2 − 5]}. (4)The first term in Eq. (4) accounts for the W -boson (plus unphysical scalars) contributionsand the second term for the Z0-boson correction to the vertex. Each one of these contribu-tions is independent of the ξ-gauge parameters (in the linear RLξ gauges) [5]. In the following,we are concerned first with the derivation of the first term in Eq. (4) using the PT pre-scription mentioned above. It is worth to mention that, in contradistinction with the PinchTechnique, the evaluation of the muon anomalous magnetic form factor (for a non-vanishingq2 value) is gauge-dependent with the methods used in refs. [5].Instead of performing an explicit evaluation of the W -boson corrections to the vertex,we can take advantage of a result derived, in another context, by Brodsky and Sullivan and2γBγ ,νµW W(a)µ µνµγBγ ,φ(b)µ µφνµφ Wγµ µ(c)νµW φγµ µ(d)Figure 1: W -boson contributions to aµ: in the BFM formalism only diagrams (a-b) arerequired, while the four diagrams are necessary in the RLξ gauge calculation.independently by Burnett and Levine in the late sixties [6]. Using the W -boson propagatorof Eq. (1) and the electromagnetic vertex of the W -boson as proposed by Lee and Yang [7](all particles are incoming, namely k1 + k2 + k3 = 0):Vµαβ = ie{gαβ(k1−k2)µ−gαµ(k1+κWk1+ξk2+κWk2)β+gβµ(k2+κWk2+ξk1+κWk1)α } , (5)it can be shown that the prescription of the PT formalism for the W -boson propagator andelectromagnetic vertex (see eqs. (1) and (3)) is obtained by choosing4:ξ = 1 and κW = 1 . (6)The W -boson contribution (Fig. 1a) to aweakµ obtained in refs. [6] using the Feynmanrules of Eqs. (1) and (5) is:aWWµ =GFm2µ8pi2√2{2(1− κW ) ln ξ + 103}. (7)As it can be easily checked by inserting the values given in eq. (6), the PT prescription forthis correction gives the correct result for the W -boson contributions to aµ (first term inEq. (4)). The contribution from the Z0-boson corresponding to the PT prescription (ξ = 1)computed in [5] must be added to Eq. (7). Therefore, we recover, in the leading muon massapproximation, the usual result for the electroweak corrections to aµ at the one-loop level.As a further check of the results obtained using the PT formalism, we have computed theone-loop electroweak corrections to aµ using the Background Field Method (BFM) [8]. It has4The usual electromagnetic vertex for the W -boson in gauge theories is recovered for the special choiceξ = 0 and κW = 1 in eq. (5).3been shown that when we fix the gauge parameter in the electroweak BFM to a particularvalue, namely ξQ = 1, one recovers the results of the PT formalism at the one-loop level [9].This equivalence of both formalisms has been shown to hold at the two-loop level in Ref.[10] and very recently it has been proved at all orders in [11]. Since the different diagramsthat contribute to the S-matrix amplitude in the framework of the PT have been rearrangedin a gauge-invariant form to produce an effective electromagnetic vertex, this becomes auseful test of the calculation. Using the BFM formalism, the calculation of the W -bosonand two-scalar contribution (The BFM only requires the contribution of Figs. 1a and 1b) tothe anomalous magnetic moment of the muon becomes (we keep some terms of higher orderin the muon mass):aWµ |BFMFig.1a =GFm2µ8pi2√2{103+56(mµmW)2+76(mµmW)4+ · · ·}, (8)aφφµ |BFMFig.1b =GFm2µ8pi2√2{−13(mµmW)2− 14(mµmW)4+ · · ·}. (9)The sum of Eqs. (8) and (9) give :aµ|BFMFig.1a+b =GFm2µ8pi2√2{103+12(mµmW)2+1112(mµmW)4+ · · ·}, (10)in good agreement with the results obtained using the PT recipe and with previous results [5].As a further check of the subleading terms inmµ, one can compute the different contributionsfrom Figs. (1.a-d) in the linear RLξ gauge. Choosing the gauge ξ = 1 we obtain:aWµ∣∣∣RLξFig.1a=GFm2µ8pi2√2{73+12(mµmW)2+56(mµmW)4+ · · ·}, (11)aφφµ∣∣∣RLξFig.1b=GFm2µ8pi2√2{−13(mµmW)2− 14(mµmW)4+ · · ·}, (12)aWµ∣∣∣RLξFig.1c= aWµ∣∣∣RLξFig.1d=GFm2µ8pi2√2{12+16(mµmW)2+16(mµmW)4+ · · ·}. (13)When we add Eqs. (11–13), we obtain the same result as in Eq. (10). Let us note that toobtain the above results we have made extensive use of the expressions for the two- (B0) andthree-point (C0) Passarino-Veltman functions derived in Ref. [12].In summary, the application of the prescription given in Eqs. (1) (with ξ = 1) and (3),shows the robustness and simplicity of the PT formalism. In particular, the PT could beuseful to verify the independence of the result with respect to the gauge-parameter in a given4gauge structure and to clarify the evaluation of the complete contributions to the two-loopelectroweak corrections to aµ, since it has been proved that gauge invariance is satisfied to allorders [10, 11] in this method. 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One-Loop Electroweak Corrections to the Muon Anomalous Magnetic Moment Using the Pinch Technique

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