# The neutrino charge radius is a physical observable

J. Bernabéu, J. Papavassiliou, and J. Vidal Departamento de Física Teórica and IFIC,
Universidad de Valencia-CSIC,
E-46100, Burjassot, Valencia, Spain
###### Abstract

We present a method which allows, at least in principle, the direct extraction of the gauge-invariant and process-independent neutrino charge radius (NCR) from experiments. Under special kinematic conditions, the judicious combination of neutrino and anti-neutrino forward differential cross-sections allows the exclusion of all target-dependent contributions, such as gauge-independent box-graphs, not related to the NCR. We show that the remaining contributions contain universal, renormalization group invariant combinations, such as the electroweak effective charge and the running mixing angle, which must be also separated out. By considering the appropriate number of independent experiments we show that one may systematically eliminate these universal terms, and finally express the NCR entirely in terms of physical cross-sections. Even though the kinematic conditions and the required precision may render the proposed experiments unfeasible, at the conceptual level the analysis presented here allows for the promotion of the NCR into a genuine physical observable.

PACS numbers: 11.10 Gh, 11.15Ex, 12.15.Lk, 14.80.Bn

###### pacs:
12.15.Lk, 13.15.+g, 13.40.Gp, 14.60.Lm
preprint: FTUV-02-1003, IFIC/02-47

## I Introduction

The diagrammatic definition of off-shell electromagnetic form-factors in the context of non-Abelian gauge theories is known to be different from the scalar or QED cases, in the sense that the “single-photon” approximation gives rise to gauge-dependent, and therefore unphysical results Fujikawa:1972fe ; Abers:1973qs . The neutrino electromagnetic form-factor has been a celebrated example of this general fact Papavassiliou:ex . Within the Standard Model the effective photon-neutrino interaction generated through one-loop radiative corrections is expected to give rise to a non-zero neutrino charge radius (NCR) Bernstein:jp ; Bardeen:1972vi ; Lee:1972fw ; Lee:1977ti ; Dolgov:1981hv , which, heuristically speaking, has been traditionally associated with the electromagnetic “size” of the neutrino SL . The extraction of this quantity from an off-shell one-loop photon-neutrino vertex has been carried out in various gauge-fixing schemes, leading to the general conclusion that, in the absence of a definite guiding principle, important physical requirements such as gauge-invariance, finiteness, and target-independence Lucio:1984mg ; Monyonko:1984gb ; Grau:1986cn ; Auriemma:ak ; Vogel:1989iv ; Degrassi:1989ip ; Musolf:1991sa ; Cabral-Rosetti:2000ad , could not be simultaneously satisfied. The non-trivial task in this context is to identify the correct subset of Feynman graphs, which would give rise to a gauge-invariant and finite result for the NCR, while, at the same time, retaining the process-independence of the definition. In particular, the most obvious manifestly gauge-invariant alternative of computing the entire process, and then forcing the answer to assume the form of a “single-photon” interaction is (by definition) process-dependent, because in the computation of the entire amplitude enter non-“single-photon” contributions. Therefore, adopting such a procedure precludes the conceptually appealing possibility of interpreting the resulting form-factor as an intrinsic property of the particle in question.

A definite solution to this problem has recently been presented in Bernabeu:2000hf . In that work the necessary guiding principle is provided by the pinch technique (PT) formalism Cornwall:1982zr ; Cornwall:1989gv ; Papavassiliou:1990zd ; Degrassi:1992ue , which implements the separation of a physical amplitude into electroweak gauge-invariant effective self-energy, vertex and box sub-amplitudes. The conceptual requirement that the effective electromagnetic vertex of a particle has to be process-independent, i. e., independent of the target used to probe the properties of the particle, is automatically implemented in this construction. In particular, the NCR is extracted from an effective one-loop proper vertex , which is independent of the gauge-fixing parameter, and satisfies a QED-like Ward identity. As has been demonstrated by means of detailed calculations in Bernabeu:2000hf , the PT construction of the vertex amounts to computing directly the corresponding proper vertex in the Feynman gauge of the Background Field Method Abbott:1980hw , using the Feynman rules derived in Denner:1994xt ; this fact is in accordance with the generally known correspondence between the PT and the Background Field Method, at one Denner:1994xt ; Hashimoto:1994ct ; Pilaftsis:1996fh and two loops Papavassiliou:2000az ; Papavassiliou:2000bb ; Binosi:2002bs .

The next important question in this context is whether the NCR so defined constitutes a genuine physical observable, and in particular how it can be extracted, even in principle, from experiment. The general strategy of how to accomplish this from and cross-sections has been addressed in a recent brief communication Bernabeu:2002nw . In this paper we present a detailed proof of the observable character of the NCR by its explicit separation from other renormalization group invariant (RGI) combinations in the physical cross-sections.

As has been explained in Bernabeu:2002nw , the main difficulty one needs to overcome in this context is the following: The PT rearrangement of the -matrix makes possible the definition of distinct sub-amplitudes, which are individually endowed with desirable theoretical properties; one of these sub-amplitudes, , is directly related to the NCR. However, the remaining sub-amplitudes, even though they do no enter into the definition of the NCR, still contribute numerically to the entire -matrix. Thus, in order to extract the NCR, one must conceive of an experiment, or combination of experiments, such that all contributions not related to the NCR will be eliminated.

In this paper we study in detail a set of such experiments involving neutrinos and anti-neutrinos. In particular, we elaborate on the “neutrino–anti-neutrino method”, which allows for the elimination of the box contributions. The general idea is to study appropriate combinations involving the one-loop cross-sections of (elastic) processes of the type and , and exploit the fact that the box diagrams behave differently than vertex or self-energy diagrams under the exchange , or equivalently, under charge conjugation Sarantakos:1983bp . It turns out that the sum of the total cross-sections of the two processes mentioned above is free of box contributions. This, together with the fact that the vertex corrections not related to the NCR, together with the Bremsstrahlung contributions vanish in the special kinematic limit of zero momentum transfer, where the NCR is actually defined, allows for the isolation of three distinct parts: the desired NCR, which depends explicitly on the flavour of the neutrino one is considering, together with two universal parts, i.e. contributions that are completely flavour- and target-independent, one consisting of the tree-level and one-loop -boson propagator, and the other of the one-loop mixing between and . There is an important theoretical difference however between the flavour-dependent NCR and the two universal pieces: The NCR is ultraviolet finite, whereas the universal parts are ultraviolet divergent, and they must therefore be renormalized. This fact raises an important issue which we will address in detail in this paper.

Specifically, in order to assign an observable character to individual sub-amplitudes, one needs in addition an explicit separation into RGI quantities. Otherwise the separation will depend on the way the renormalization subtraction is carried out, i.e. it would be scheme-dependent. In this paper we will employ the concepts of the electroweak effective charge, and of the effective (running) electroweak mixing angle, in order to cast the aforementioned universal contributions into manifestly RGI combinations Hagiwara:1994pw ; Papavassiliou:1996fn ; Papavassiliou:1997pb . An important consequence of this analysis is in fact related to the very definition of the NCR. Specifically, the universal PT one-loop self-energy is gauge-independent and couples electromagnetically to the target fermions. Therefore, it could be considered as a flavour-independent contribution to the NCR, to be added to the flavor-dependent one stemming from the proper vertex; in fact this point of view has been often advocated in the literature. However, the self-energy is a scheme-dependent and therefore unphysical contribution, which, as such cannot form part of the NCR definition. Instead, it must be appropriately combined with the tree-level contribution mediated by the -boson (which is certainly not part of the NCR), in order to form the universal RGI quantity known as the effective (running) electroweak mixing angle. Once the decomposition of the result into RGI quantities has been accomplished, one may proceed unambiguously into their experimental separation, by considering the appropriate number of different processes. Thus, the Standard Model prediction for the NCR, together with the two universal RGIs, can be finally expressed individually in terms of specific combinations of physical cross-sections.

The paper is organized as follows: In section II we review the PT rearrangement of the elastic scattering amplitude, focusing on the relevant kinematic limit of vanishing momentum transfer, where the NCR is defined. To simplify the analysis we choose the charged target fermions to be right-handedly polarized electrons, and we show that both the vertex corrections not related to the NCR as well as the Bremsstrahlung corrections vanish in the aforementioned kinematic limit. In section III we present the neutrino–anti-neutrino method in detail, using as target fermions both right-handedly polarized as well as unpolarized fermions. In section IV we review the renormalization properties of the relevant one-loop PT self-energies appearing in the neutral sector of the Standard Model, and we show how the resulting amplitudes may be written in terms of the ultraviolet finite NCR and two manifestly RGI building blocks. In section V we present two methods which allow for the individual extraction from an appropriate set of experiments of the NCR and the two RGI quantities introduced in the previous section. In addition, we present the theoretical Standard Model predictions for these three quantities. Finally, in section VI we present our conclusions.

## Ii The PT reorganized forward amplitude

In this section we will first review briefly some of the main results presented in Bernabeu:2000hf in an attempt to fix the notation and stress the relevant conceptual points. Then, we will show that in the special kinematic limit of zero momentum transfer, in which the NCR is in fact defined, the vertex corrections not related to the NCR, together with the Bremsstrahlung contributions, vanish.

For concreteness we will focus on the process , shown in Fig.1 The above process is chosen to be elastic with the Mandelstam variables defined as , , , and . The reason for considering instead of is because in this way one eliminates the charged channel mediated by a -boson (Fig.1j).

The two relevant tree-level photon () and -boson vertices and are given by

 ΓμAf¯f = −ieQfγμ=−ieQfγμ(PL+PR) ΓμZf¯f = −i(gwcw)γμ[(s2wQf−Tfz)PL+s2wQfPR] (1) = −i(gwcw)γμ[af+bfγ5]

with and . In the above formulas is the electric charge of the fermion , its -component of the weak iso-spin, and is the chirality projection operator, , and the electric charge is related to the gauge coupling by . Figure 1: The various classes of diagrams contributing to the one-loop amplitude (a–i). The charged channel (j) vanishes for neutrino flavours other than νe.

The one-loop contributions to the amplitude are shown in Fig.1b – Fig.1i . We will assume throughout that the PT rearrangement of the amplitudes has been carried out, exactly as described in Bernabeu:2000hf . In particular, after a series of crucial gauge-cancellations enforced by the elementary Ward identities of the theory Cornwall:1982zr ; Cornwall:1989gv ; Papavassiliou:1990zd , the amplitude has been split into individually gauge-invariant sub-amplitudes which correspond kinematically to self-energies, vertices, and boxes. As has been explained in detail in the literature Denner:1994xt ; Hashimoto:1994ct ; Pilaftsis:1996fh , these latter PT quantities coincide with the corresponding Green’s functions computed in the framework of the Background Field Method, at the special value of the (quantum) gauge fixing parameter . Notice that the gauge-invariant “pure” box contributions coincide with the conventional box contributions computed in the renormalizable Feynman gauge ( gauges, with ).

It is well-known Watson:1994tn that the PT rearrangement of the amplitude may be carried out regardless of the kinematical details, as for example the specific values of the Mandelstam variables, or the masses of the external particles. In what follows we will consider the above amplitude in the zero transfer limit, , , where the NCR is actually defined. In addition, we will assume that all external (on-shell) fermions are massless. As a result of this special kinematic situation we have the following relations:

 p21 = p22=k21=k22=p1⋅p2=k1⋅k2=0 p1⋅k1 = p1⋅k2=p2⋅k1=p2⋅k2=s/2. (2)

As we will see in a moment, in the aforementioned limit of , the one-loop PT vertex corrections to the and vertices vanish, and so do the Bremsstrahlung contributions. Moreover, the special kinematic relations given in Eq.(2) are crucial for the validity of the neutrino–anti-neutrino method which we will present in the next section.

In the center-of-mass system we have that , where and are the energies of the neutrino before and after the scattering, respectively, and , where is the scattering angle. Clearly, the condition corresponds to the exactly forward amplitude, with ,   . Equivalently, in the laboratory frame, where the (massive) target fermions are at rest, the condition of corresponds to the kinematically extreme case where the target fermion remains at rest after the scattering. Figure 2: The relevant vertex graphs, (a)–(b), and the fermion wave-function correction, (c).

The relevant quantities which will appear in our calculations are the and self-energies, to be denoted by and , respectively, and three one-loop vertices , , and , to be denoted by , , and , respectively. In the PT framework is transverse, for both the fermionic and the bosonic contributions, i.e. it may be written in terms of the dimension-less scalar function as

 ˆΣμνAZ(q2)=(q2gμν−qμqν)ˆΠAZ(q2). (3)

On the other hand, is of course not transverse. In what follows we will discard all longitudinal pieces, since they vanish between the conserved currents of the massless external fermions, and will keep only the part proportional to ; its dimension-full cofactor will be denoted by , i.e.

 ˆΣμνZZ(q2)=ˆΣZZ(q2)gμν (4)

The closed one-loop expressions for and can be found in various places in the literature Degrassi:1992ue ; Degrassi:1993kn ; Hagiwara:1994pw .

It is relatively easy to convince oneself that, if one were to relax the masslessness condition for external (target) fermions, all additional contributions due to their non-vanishing masses always appear proportional to positive powers of and/or , where stands for the mass of the or bosons. Clearly, the terms are naturally suppressed because of the heaviness of the gauge bosons. On the other hand, the terms can be made arbitrarily small, by letting the variable , which in principle can be controlled by adjusting the energies of the incoming particles, reach sufficiently high values.

Turning to the vertex corrections, the NCR, to be denoted by , will be defined from the vertex , which is given by the two graphs of Fig.2a and Fig.2b (with a photon and neutrinos entering into the vertex). As has been explained in detail in the literature, the construction of the gauge independent and gauge invariant one-loop vertex by means of the PT finally amounts to using the Feynman gauge for the all gauge-boson propagators, and replacing the usual three-boson vertex

 Γαμν(q,k,−k−q)=(q−k)νgαμ+(2k+q)αgμν−(2q+k)μgαν (5)

by the tree-level vertex

 ΓFαμν=(2k+q)αgμν+2qνgαμ−2qμgαν. (6)

The vertex satisfies the elementary Ward identity

 qαΓFαμν=(k+q)2gμν−k2gμν, (7)

Equivalently, one may use directly the Feynman rules of the Background Field Method Denner:1994xt , choosing for the gauge-fixing parameter of the (quantum) bosons the value .

It is straightforward to evaluate the two aforementioned vertex graphs; their sum gives a ultra-violet finite result, from which one can extract the dimension-less electromagnetic form-factor . In particular, since is proportional to , we may define the dimension-full form-factor as

 (8)

depends on the mass of the charged iso-doublet partner of the neutrino, which appears in the two relevant Feynman diagrams. In the limit of both , is infrared divergent, whereas it is infrared finite in the limit , . After canceling the against the photon propagator, we can take the limit , keeping non-zero. Defining as usual , we finally arrive at Bernabeu:2000hf

 ⟨r2νi⟩=GF4√2π2[3−2log(m2iM2W)],i=e,μ,τ (9)

where is the Fermi constant. Notice that the logarithmic term in the above expression originates entirely from the Abelian-like diagram of Fig.2b.

### ii.1 Vanishing of the Zee and Zνν vertex corrections

We will show that the sum of the one-loop vertex and wave-function corrections, which are collectively depicted in Fig.1h and Fig.1i vanishes in the limit . Since these diagrams are multiplied by a massive tree-level propagator , which is regular (non-divergent) in this limit, they do not contribute to the scattering amplitude we consider. This is to be contrasted with the , which is accompanied by a photon-propagator, thus giving rise to a contact interaction between the target-fermion and the neutrino, described by the NCR.

It is known Papavassiliou:1990zd that the one-loop PT vertex with or , shown in Fig.2a and Fig.2b, satisfies a QED-like Ward identity, relating it to the PT inverse fermion propagators , shown in Fig.2c , i.e

 qαˆΓαZff(q,p1,p2)=ˆΣf(p1)−ˆΣf(p2) (10)

Eq.(10) is a straightforward consequence of the tree-level Ward identity of Eq.(7). By virtue of Eq.(10), when the proper vertex graphs are combined with the renormalization of the external fermions, the net result is ultraviolet finite (because, as in QED, ). Below we give the results of the individual graphs contributing to vertex , corresponding to the vertex graphs of Fig.2, with a -boson entering into the vertex. These graphs are calculated in the limit where , the fermions appearing in the loop are considered strictly massless, and terms proportional to vanish, because they are contracted with a conserved current, since the external fermions are considered massless as well. The graph in Fig.2b containing the virtual photon is infrared divergent, even if the fermions are massive. We will regulate this divergence by introducing a fictitious photon mass , a procedure which is compatible with gauge-invariance. Equivalently one may use dimensional regularization to regularize both ultraviolet and infrared divergences Gastmans:uv ; Marciano:tv . In any case, all contributions will cancel algebraically, before the infrared cutoff is removed. The two quantities which naturally appear when calculating the diagrams using standard techniques, such as Feynman parametrization and dimensional integration, are the following:

 I1(M2) = 12(−2+ϵ)2∫10dx(1−x)[2ϵ−ln(xM2)] I2(M2) = (−2+ϵ)∫10dxx[2ϵ−ln(xM2)] (11)

where . It is elementary to verify that ; notice that this relation holds not only for the divergent parts, but also for the parts that are finite and non-vanishing as . In terms of and we have (we suppress a common factor ):

 [2b](e)A = s2wc2wI1(m2A)γα(ae+beγ5), [2c](e)A = s2wc2wI2(m2A)γα(ae+beγ5), (12) [2b](e)Z = I1(M2Z)γα[ae(a2e+3b2e)+be(b2e+3a2e)γ5], [2c](e)Z = I2(M2Z)γα[ae(a2e+3b2e)+be(b2e+3a2e)γ5], (13) [2b](ν)Z = I1(M2Z)γαPL, [2c](ν)Z = I2(M2Z)γαPL, (14) [2a](e)WW = −c4w2I2(M2W)γαPL, [2b](e)W = c2w4I1(M2W)γαPL, [2c](e)W = −c2w2(ae−be)I2(M2W)γαPL, (15) [2a](ν)WW = c4w2I2(M2W)γαPL, [2b](ν)W = c2w2(ae−be)I1(M2W)γαPL, [2c](ν)W = −c2w4I2(M2W)γαPL. (16)

where the subscripts on the left hand-side denote the virtual gauge boson(s) appearing inside the corresponding graphs, and the superscripts specify the type of incoming fermion. It is straightforward to see that . To prove the cancellations we have also used that .

### ii.2 Vanishing of the Bremsstrahlung

In this subsection we will show that the differential cross-section corresponding to the Bremsstrahlung diagrams vanishes in the kinematic limit of zero momentum transfer Passera:2000ug . The two diagrams contributing to the Bremsstrahlung process and are shown in Fig.3. The conservation of four-momentum assumes the form , and . A direct consequence of this special kinematic choice are the relations

 k1⋅k2+k1⋅k3−k2⋅k3=0,pi⋅(k2+k3−k1)=0,i=1,2 (17) Figure 3: The Bremsstrahlung diagrams

The -matrix element consists of the two parts, and , corresponding to the diagram (a) and diagram (b) of Fig.3, respectively, i.e.

 MBμ=MBaμ+MBbμ (18)

with

 MBaμ = [¯u(k2)γρ(ae+beγ5)Se(k1−k3)γμu(k1)][¯u(p2)γρPLu(p1)] MBbμ = [¯u(k2)γμSe(k2+k3)γρ(ae+beγ5)u(k1)][¯u(p2)γρPLu(p1)] (19)

where we have suppressed a common multiplicative factor originating from the coupling constants. The Bremsstrahlung differential cross-section, , is proportional to the square of the amplitude:

 κB = MB∗MB† (20) = (MBa+MBb)∗(MBa+MBb)† = MBa∗MB†a+MBb∗MB†b+2Re(MBa∗MB†b) ≡ κBaa+κBbb+2κBab

The sum over polarizations for the photon introduces the usual polarization tensor , with an arbitrary four-vector. Of course, gauge invariance furnishes the tree-level Ward identity , so that effectively only the piece of contributes.

Now for the electron propagators inside the above diagrams we use that

 Se(k1−k3)=−⧸k1−⧸k32k1⋅k3,Se(k2+k3)=⧸k2+⧸k32k2⋅k3 (21)

and so

 κBaa = Taa(2k1⋅k3)2 κBbb = Tbb(2k2⋅k3)2 κBab = −Tab(2k1⋅k3)(2k2⋅k3) (22)

with

 Taa = Tr(γμ(⧸k1−⧸k3)γρ(ae+beγ5)⧸k2γσ(ae+beγ5)(⧸k1−⧸k3)γμ⧸k1)% Tr(γρPL⧸p2γσPL⧸p1) Tbb = Tr(γρ(ae+beγ5)(⧸k2+⧸k3)γμ⧸k2γμ(⧸k2+⧸k3)γσ(ae+beγ5)⧸k1)% Tr(γρPL⧸p2γσPL⧸p1) Tab = Tr(γρ(ae+beγ5)(⧸k2+⧸k3)γμ⧸k2γσ(ae+beγ5)(⧸k1−⧸k3)γμ⧸k1)%Tr(γρPL⧸p2γσPL⧸p1) (23)

One can then show, employing the standard properties of the trace of matrices together with the special kinematic relations given above, that

 Taa = −32(2k1⋅k3)X1 Tbb = −32(2k2⋅k3)X2 Tab = −16[(2k1⋅k3)X2+(2k2⋅k3)X1] (24)
 X1 = (ae−be)2(p2⋅k2)(p1⋅k3)+(ae+be)2(p2⋅k3)(p1⋅k2) X2 = (ae+be)2(p2⋅k1)(p1⋅k3)+(ae−be)2(p2⋅k3)(p1⋅k1) (25)

Thus,

 κBaa = −16X1k1⋅k3 κBbb = −16X2k2⋅k3 κBab = 8X1k1⋅k3+8X2k2⋅k3 (26)

From these relations and Eq.(20) follows immediately that , as announced; evidently, in the kinematic limit considered, a completely destructive interference takes place, which forces the cross-section to vanish. Notice that in arriving at the above result nowhere have we actually assumed that the emitted photon is soft; in fact, the vanishing of the cross-section has been shown simply by evaluating the traces, without need to enter into the specifics of the (implicit) three-body phase-space integration.

## Iii The neutrino – anti-neutrino method

In this section we will present the neutrino – anti-neutrino method in detail. As we will see, this method isolates the process-dependent box contributions, at the expense of doubling the number of experiments needed.

### iii.1 Right-handed electrons

We begin our analysis by choosing the target electrons to be right-handedly polarized. This choice eliminates the one-loop box of Fig.1g, and makes the introduction to the neutrino–anti-neutrino method calculationally easier box . In addition, it will be used in section V as one of the processes that need be considered for the experimental extraction of the NCR. We emphasize that this particular choice of target-fermions constitutes no loss of generality; in fact, as we will see later in detail, all results derived using this particular process will be generalized after minor calculational adjustments, to the case of left-handed or unpolarized target fermions.

For the special case of right-handed fermions the two vertices of Eq.(1) assume the form

 ΓμAfR¯fR = −ieQfγμPR ΓμZfR¯fR = −ieQf(swcw)γμPR (27)

The differential cross-section in the center-of-mass system is given by

 dσdΩcm=164π2s|M|2 (28)

where and is the amplitude.

For the tree-level amplitude (Fig.1a), mediated only by an off-shell , we have

 M(0)νμeR=c(a)A(a)CL (29)

with

 CL=[¯uR(k2)γμuR(k1)][¯u(p2)γμPLu(p1)], (30)

where , is the right-handed electron spinor, and

 c(a)=ie22c2w ,          A(a)=−1M2Z=DZ(0), (31)

where is the scalar cofactor of the tree-level propagator .

At one-loop,

 M(1)νμeR=(∑ncnAn)CL+BZZ (32)

with , namely the corresponding diagrams relevant for the process, shown in Fig.1 . The coefficients are given by

 c(b)=ie22c2w ,c(c)=iegw2cw ,c(d)=2ie2, (33)

and

 A(b) = −iˆΣZZ(0)M4Z, A(c) = iˆΠAZ(0)M2Z, A(d) = ˆFνi(0). (34)

Notice that the factor of 2 appearing in is due to the fact that the usual definition of , given in Eq.(8), involves instead of the which appears in of Eq.(30)

Finally, denotes the contribution of the box graphs shown in Fig.1e and Fig.1f,

 BZZ = −KZZ[¯uR(k2)γρSe(ℓ−k1)γσuR(k1)][¯u(p2)γρPLSνμ(ℓ+p1)γσPLu(p1)] (35) +KZZ[¯uR(k2)γρSe(ℓ+k2)γσuR(k1)][¯u(p2)γσPLSνμ(ℓ+p1)γρPLu(p1)]

with

 KZZ≡e44c4w∫d4ℓ(2π)4DZ(ℓ)DZ(ℓ+p1−p2) (36)

where is the virtual four-momentum, and is the massless tree-level fermion propagator. The relative minus sign in the contribution of the direct and crossed boxes originates from the fact that in the former graph the direction of the fermion propagator is opposite to the flow of the four-momentum (see Fig.1e and Fig.1f).

Let us next consider the process , i.e. the same process as before with , and identical kinematics. For the tree-level contribution we have

 M(0)¯νμeR=A(a)¯CL (37)

with

 ¯CL=[¯uR(k2)γμuR(k1)][¯v(p1)γμPLv(p2)] (38)

Similarly, since the one-loop analysis can be repeated unaltered, we have for the one-loop amplitude

 M(1)¯νμeR=(∑ncnAn)¯CL+¯BZZ (39)

with

 ¯BZZ = +KZZ[¯uR(k2)γρSe(ℓ−k1)γσuR(k1)][¯v(p1)γρPLS¯νμ(ℓ+p1)γσPLv(p2)] (40) −KZZ[¯uR(k2)γρSe(ℓ+k2)γσuR(k1)][¯v(p1)γσPLS¯νμ(ℓ+p1)γρPLv(p2)]

Having set up the two amplitudes, we next turn to the specifics of the neutrino – anti-neutrino method. The basic observation is that the tree-level amplitudes as well as the part of the one-loop amplitude consisting of the propagator and vertex corrections, i.e. the first term in Eq.(39), which too is proportional to the tree-level amplitude, has different transformation properties under the replacement than the part of the amplitude originating from the box. In particular, the coupling of the boson to a pair of on-shell anti-neutrinos may be written in terms of on-shell neutrinos provided that one changes the chirality projector from to and supplying a relative minus sign Sarantakos:1983bp . Specifically, when sandwiched between external states the is given by

 (41)

whereas
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