Loop-induced Neutrino Non-Standard Interactions

Non-Standard Interactions (NSI) of neutrinos may originate from models in which new particles interact with neutrinos. In scalar extensions of the SM, the typical approach to obtain NSI requires Fierz transformations and charged Higgses, which suffer from strong constraints from collider searches or charged lepton flavor violation processes. We propose here an alternative approach to generate NSI, namely via loop processes. We show that such loop-induced NSI from secret neutrino interactions can reach sizes of ${\cal O}(0.1\sim1)$ compared to standard Fermi interaction. This approach can also give rise to neutrino-quark NSI.


I. INTRODUCTION
In the era of neutrino oscillation precision measurements, the standard three-neutrino oscillation framework is being tested with increasing precision [1][2][3]. Hence it is important to consider any new physics that could have significant effects on neutrino oscillations. One particularly interesting possibility is provided by so-called Non-Standard Interactions (NSI) of neutrinos, which has raised rather general interest in the literature (see, e.g., Refs. [4][5][6][7] for reviews on NSI). By introducing new flavor-changing neutral-current interactions (ν α γ µ ν β ψγ µ ψ) of neutrinos (ν α , ν β ) with other Standard Model fermions ψ, such NSI cause via coherent forward scattering flavor transitions in matter, disturbing the determination of the standard neutrino physics parameters. The effects of NSI in current and future long-baseline experiments (T2K, NOνA, DUNE, etc.), especially on the determination of δ CP , have been extensively studied [8][9][10][11][12][13][14].
From the theoretical point of view, NSI of neutrinos are well motivated. Generally speaking, neutrinos have long been considered as the portal of new physics, even more so after they were found to be massive. It is reasonable to speculate that the new physics related to neutrinos also brings new interactions to neutrinos. A well-known example is the type II seesaw model [15][16][17]. In this model, a scalar triplet is introduced to the SM and acquires a small vacuum exception value to generate neutrino masses 1 . Since the triplet couples both to electrons and neutrinos, NSI of neutrinos with electrons can be generated [19]. In addition to the type II seesaw model, other scalar extensions of the SM can also generate NSI in the same way, including scalar singlet models [20][21][22][23] or two-Higgs-doublet models [24,25], etc. In all these scalar extensions, the approach of generating NSI is to integrate out a charged scalar mediator to get scalar four-fermion interactions which are then converted by a Fierz transformation to vector form (containing γ µ ). The mediator must be charged due to the Fierz transformation rules (as we will demonstrate explicitly later), which is potentially a problem of obtaining sizable NSI because charged Higgses usually face stronger collider constraints than neutral ones.
In this paper, we propose a different way to generate NSI, namely loop-induced NSI. The approach is also based on scalar extensions of the SM 2 , but without using Fierz transformations. Instead, as the name implies, the loopinduced NSI are generated by loop diagrams. Although loop contributions are in general expected to be subdominant compared with tree level contributions, in some models this way can produce fairly sizable NSI which is absent at tree level. The advantage of loop-induced NSI compared to the usual one obtained by the Fierz transformation and charged Higgses is that the source of flavor violation can be confined to the neutrino sector with "hidden" scalar interactions. Hence, large NSI can be obtained without causing problems in other well-measured processes. Other scenarios can also give rise to neutrino-quark NSI, which are absent in the previous models.
The remainder of this paper is organized as follows. In Sec. II, we first briefly review how NSI can be generated in scalar models by Fierz transformations, and then introduce our concept of generating NSI by loop diagrams, with some general results presented while the detailed calculation is delegated to the appendices. Then we apply the results to several explicit models in Sec. III A to III C. Confronting these models with experimental constraints, we estimate the order of magnitude of the loop-induced NSI in these models in Sec. IV. Finally we conclude in Sec. V.

II. GENERAL ANALYSIS
In this section, we study the generation of NSI in a general framework which introduces a new scalar boson φ. It has Yukawa interactions with neutrinos and probably other SM fermions. Let us consider how the following NSI may be generated, Here P L ≡ (1 − γ 5 )/2 and ψ stands for electrons or quarks which can be chiral (e.g. ψ = e L , u R , d L , · · · ). Throughout the paper, we use α, β, · · · to denote the flavor indices.
In practice, NSI are usually expressed in terms of non-chiral neutrons (n), protons (p) and electrons (e): The NSI couplings in Eq. (2) can be connected to the chiral form in (1) by 3 Currently the experimental constraints on these NSI parameters, depending on the specific channels, range from O(10 −2 ) to O(1)-for a recent update, see Ref. [7].
To obtain the operator in Eq. (1), we need two essentials: one is flavor-sensitive interactions of the new scalar boson and the other is the conversion of the scalar form 4 to vector form. More technically, the NSI operators contain γ µ while the new scalar boson only introduces interactions which do not contain γ µ . Here we introduce two approaches to achieve the conversion, by the Fierz transformation and by loop corrections. We will refer to the corresponding NSI as Fierz-transformed NSI and loop-induced NSI respectively.

A. Fierz-transformed NSI
Applying the Fierz transformations in some scalar extensions of the SM (e.g. the type II seesaw model) to obtain NSI has been considered in the literature [19,21,24,27]. Generally, if a heavy scalar boson φ has Yukawa interactions ψ 1 ψ 2 φ and ψ 3 ψ 4 φ, integrating it out will lead to the four-fermion effective operator ψ 1 ψ 2 ψ 3 ψ 4 . The Fierz transformation (see, e.g., [28]) of this operator gives where the third term on the right-hand-side is a vector form interaction. Recall that only the vector form interaction leads to NSI effects in terrestrial matter [29]. In the SM and many extensions, the Yukawa interactions are based on chiral fermions. So it is also useful to provide the Fierz transformations of chiral fermions: which can be obtained by replacing It is noteworthy that Eq. (6) produces only scalar and tensor form interactions while Eq. (7) produces only vector form interactions. Therefore, in a chiral theory only when the effective operator has a chirality structure as in Eq. (7), the vector form NSI can be obtained. If Eq. (5) or Eq. (7) is used to generate NSI, we should identify ψ 2 and ψ 3 with neutrinos, and ψ 4 and ψ 1 with electrons or quarks. Note that ψ 1 and ψ 4 need to be identical to generate NSI terms from coherent forward scattering in matter. This is clear from comparing Eqs. (5), (7) with Eq. (1). In addition, limits from flavor physics strongly contrains cases with ψ 1 = ψ 4 . Hence we can infer that the effective operator before the Fierz transformation should be ψν L ν L ψ, where ψ stands for charged fermions. Since ψν L and ν L ψ have nonzero electric charges, the new scalar boson must be charged. If ψ is a quark, then the scalar boson has to be colored. Such leptoquarks are severely constrained.
In conclusion, the Fierz transformation approach requires a charged scalar boson to generate NSI. If the boson is a singlet under SU (3) c , then neutrino-quark NSI can not be generated. Note further that since strong limits on additional charged scalars exist, the particle responsible for the Fierz-transformed NSI can not be light (MeV-scale), which is often discussed (see e.g. [30,31]) in the context of matter-induced NSI by coherent forward scattering.

B. Loop-induced NSI
We will demonstrate now that if neutrinos have Yukawa interactions with a new scalar boson, then NSI can be generated at the loop level 5 . Both neutral and charged Higgses can generate such terms. Here we discuss two possible diagrams for loop-induced NSI, as shown in Fig. 1. The first one is based on loop corrections to the neutrino-Z vertex (left panel) which we will refer to as the triangle diagram. The other is a box diagram, which consists of pure Yukawa interactions and does not involve any gauge interactions. The external fermion lines are two neutrinos of different flavor, and two charged fermions, which can be either electrons or quarks. The internal fermion lines can be charged or neutral fermions and do not need to be identical, depending on the models. As discussed above, the external lines should be identical fermions.
As we have mentioned, the flavor violation is introduced by the scalar-neutrino interactions and needs to be converted to vector form interactions. In the triangle diagram, this is achieved by the fact that the triangle loop generates an effective flavor-changing vertex Z µ ν Lα γ µ ν Lβ . In the box diagram, the effective four-fermion operator also has γ µ 's between the fermion fields because of the internal fermion propagator.
In computing the loop-induced NSI, we need to consider the UV divergences. By simple power counting, one can see that the triangle diagram contains a logarithmic UV divergence Λ d 4 k 1 k 4 ∼ log Λ while the box diagram is not divergent because d 4 k 1 k 6 is finite. In a renormalizable model, the UV divergence in any physical process should be canceled by adding all relevant diagrams and counterterms together. For the triangle diagram considered here, because at tree level the neutral current interactions are flavor conserving, there is no corresponding counter term. Therefore in a renormalizable and complete model, one simply needs to sum over the relevant diagrams to obtain a finite result.

Triangle diagrams:
In Appendix A, we compute the triangle diagram and the result is presented as follows. If the UV divergences cancel out, the effective flavor changing Z − ν vertex in Fig. 1 is The notation is as follows: • ψ int is the internal fermion appearing in the triangle loop. The Yukawa vertices are formulated as which defines the Yukawa couplings y α and y β . In the triangle diagram, ψ int can be any SM fermion that couples to the Z boson. In the loop calculation, we assume the fermion masses are all negligibly small compared to the boson masses (both φ and Z). This is fine as long as only leptons are coupling to the scalar, but one could also accomodate more exotic models where quarks including the top couple to scalars and neutrinos.
• g/c W is the gauge coupling attached to the Z boson, and Q Z is the corresponding Z charge of a fermion. Both are defined by the covariant derivative For convenience we list the Z charges of the SM fermions in Tab. I.
• The scalar boson φ in the triangle loop has mass m φ . Depending on models, it may also have a Z-charge Q (φ) Z . In Appendix A, we show that if the Z charges are conserved in the model, then the UV divergences cancel. In Eq. (9), the Z charge conservation requires Any renormalizable model satisfies Eq. (11), as we demonstrate explicitely in In Sec. III A.
• f (r) and h(r) are two finite functions of the mass ratio The explicit forms of f (r) and h(r) are rather complicated and can be found in Appendix A. These functions have simple limits for r 1 and r 1: For general values of r, we numerically evaluate them and show the results in Fig. 2.
Given the effective Zνν vertex in Eq. (8), the corresponding low-energy four-fermion interaction is where

Box diagrams:
The box diagram is always finite. After computing the loop integral (see Appendix B), we obtain the effective Lagrangian generated by the box NSI: Here we adopt the same definition of y α and y β as in Eq. (8). In addition, y ψ is the Yukawa vertex marked in the box diagram in Fig. 1. The corresponding Yukawa interaction is where ψ and ψ are the external and internal fermion lines (left part of the box diagram).
To summarize, we combine the above loop-induced NSI as with the individual contributions Here αβ and αβ denote the contributions of the triangle and box diagrams respectively; Q (ψ) Z is the Z charge of ψ (electrons/quarks) as listed in Tab. I, g (1) αβ is given by Eq. (8). Note that the fermions considered here are chiral. The usually considered vector NSI, cf. Eqs. (3,4) can be obtained by summing for the triangle diagram their corresponding Q Z charges from Tab. I. The box diagram needs to be multiplied by 2. We stress here that since Yukawa couplings can be complex, the various can also be complex. This is in contrast to typical models in which integrating out a gauge boson generates NSI.

III. APPLICATION TO MODELS
We will apply the above general results now to explicit models.
A. Model A: the minimal charged Higgs model The first model we consider is a very simple extension of the SM by adding only a scalar singlet φ with hypercharge Y φ = 1 to the SM. After electroweak symmetry breaking, φ will eventually obtain one unit of electric charge. For this reason, we will refer to the model as the minimal charged Higgs model. The model has been studied in, e.g., Ref. [20][21][22] (the latter two discuss tree-level NSI effects), and has also been considered as a part of larger SM extensions such as the Zee model [32].
Because the hypercharge is Y φ = 1, the only new Yukawa interaction allowed by symmetry is L c iσ 2 Lφ, where L = (ν L , e L ) T is a SM lepton doublet with hypercharge Y L = −1/2; L c is the charge conjugate of L so L c has the same hypercharge as L; iσ 2 is necessary to form an SU (2) L invariant. Note that for any two Dirac spinors ψ 1 and ψ 2 , the combination ψ c 1 ψ 2 = ψ c 2 ψ 1 is symmetric under the interchange of 1 ↔ 2 (similar to the well-known fact that a Majorana mass matrix is always symmetric). On the other hand, the SU (2) L product with iσ 2 is anti-symmetric. As a result, L c iσ 2 Lφ vanishes if the two lepton doublets are of the same flavor: the Yukawa interactions of φ can be non-vanishing only when there are at least two different flavors. Adding the new Yukawa interactions to the SM, the Lagrangian of this model is where the Yukawa matrix y αβ is anti-symmetric. The SM Higgs doublet is denoted as H and V (φ, H) denotes all quartic terms involving φ and H together or φ only. The scalar mass m 2 φ is assumed to be larger than the electroweak scale to avoid direct constraints from collider searches. For convenience of later use, we explicitly expand the new Yukawa terms: The covariant derivative is where B µ is the U (1) Y gauge boson. After the Weinberg rotation, we obtain with the Z-and electric charges Here Q (φ) A = 1 implies that φ has the same electric charge as the proton, Q The Z charges of the SM fermions have already been defined in Eq. (10) and listed in Tab. I. It is important to notice that the Z charges in the Yukawa term (22) are conserved which is crucial for the UV divergences in the relevant loops to cancel. Eq. (27) is not an accidental result because the model here is renormalizable and UV divergences should not appear in any physical processes. Next, we shall discuss the neutrino NSI in this model, using the general results which have been obtained in Sec. II.

Fierz-transformed NSI:
We first integrate out φ, which generates the scalar form effective operator Because y αβ is anti-symmetric, β and α can only be µ or τ , but not e. So Eq. (28) can not generate NSI between ν e and e. According to Eq. (7), with the replacement (ψ 1R , ψ 2L , ψ 3L , ψ 4R ) → (ν c βL , e L , e L , ν c αL ), we obtain where in the second step we have used the identity (C10). Eq. (29) is the Fierz-transformed NSI in this model, which is only possible for coupling to electrons. We stress the known fact that only µτ , µµ and τ τ can be generated in this model via Fierz transformation, and that its magnitude is constrained to be rather small [21,22], typically around O(10 −3 ). We will show next that loop-induced NSI terms can generate all flavor terms, though later it turns out that those terms are also constrained to be small. Nevertheless, the analysis illustrates the potential importance of loop effects.

Loop-induced NSI:
Without loss of generality, let us first focus on how g µe can be generated according to the results in Sec. II B and Eq. (22). The relevant terms in Eq. (22) are By comparing this expression to Eq. (9), we have the mapping Using Eq. (8) and assuming For other flavors, one can straightforward derive similar results accordingly. The general result is Eq. (33) combined with Eq. (19) gives the triangle NSI in this model: The box NSI in this model also exists, but only for electron-neutrino NSI because ψ in the right panel of Fig. 1 can only be an electron. The box NSI parameter αβ can be directly obtained from Eq. (19) with the Yukawa couplings replaced by which leads to Recall that the usually considered vector form for is twice the value of Eq. (36). It is noteworthy that all flavor terms αβ can be generated, while the Fierz-transformed NSI was only possible for the µτ case.

B. Model B: Secret neutrino interactions
Secret neutrino interactions are a type of interactions that only exist among neutrinos. They are generally difficult to be tested in terrestrial experiments because electrons and quarks are not involved in such interactions. However, secret neutrino interactions could have interesting cosmological and astrophysical effects, in supernova dynamics, cosmic neutrino propagation, Big Bang Nucleosynthesis (BBN), etc. Therefore it has been considered in many references [33][34][35][36][37][38]. The simplest secret neutrino interaction is a scalar boson interacting with the left-handed neutrinos φν L ν L where ν L is in the Weyl spinor notation 6 . In the Dirac notation, and including the flavor indices, the interaction should be formulated as We demonstrate now that the secret neutrino interaction in Eq. (37) to find [cf. Eq. (8)]: Then the corresponding triangle NSI parameter in Eq. (19) is: Note that the internal fermions in the triangle diagram are left-handed neutrinos, and recall that r = m 2 Z /m 2 φ . However, one should note that Eq. (37) is not a complete model so the UV divergences cannot be fully canceled without introducing new particles or new interaction terms. Consequently there is a UV divergence, explicitly shown in Eq. (A14) and not given here. In a complete and renormalizable model containing the secret neutrino interaction (37), this UV divergence will be canceled by additional diagrams, potentially modifying the result (39). Since this depends on the details of the complete model, we refrain from going further into detail and keep Eq. (39), which should be order-of-magnitude wise correct. Regarding Eq. (37) there is not necessarily lepton number violation because φ could carry two units of lepton number. However, if the lepton number is violated by, e.g., non-zero φ , then such a secret interaction can also be responsible for a Majorana neutrino mass. If this term is the only term responsible for neutrino mass, it is interesting to note that αβ ∝ (m ν m † ν ) αβ , which would result in eµ eτ µτ , where the proportionality factor between eµ and µτ is about ∆m 2 21 /|∆m 2 31 | [39]. Due to a lack of very stringent terrestrial constraints on the secret neutrino interactions, the loop-induced NSI in this model can be in principle much larger than in the previous model. We will discuss possible sizes of the NSI later in Sec. IV.

C. Model C: Neutral scalar boson
Neutrinos could also have new scalar interactions with the charged fermions mediated by a neutral scalar, which can be expressed by the following Lagrangian: L ⊃ y ν αβ φν α ν β + y ψ φψψ + h.c. (ψ = e, u, d).
In this model, because the scalar boson is neutral, there is no Fierz-transformed NSI. In the triangle diagram (Fig. 1), since the external neutrino lines are left-handed neutrinos, the internal fermion lines can only be righthanded neutrinos 7 because φν α ν β = φν αR ν βL + φν αL ν βR . Since right-handed neutrinos do not couple to the Z boson there is no triangle NSI in this model, i.e.
However, this model has loop-induced NSI from the box diagram. By comparing Eq. (40) to Eq. (9) and (17), we have the mapping which gives the box NSI parameter: Although Eq. (40) is not a complete model, in contrast to model B, there is no UV divergence in computing the loop-induced NSI because the box diagram is always finite.

IV. HOW LARGE CAN LOOP-INDUCED NSI BE?
Now that we have derived loop-induced NSI both in the general framework and in several specific models, a natural question to ask is how large they can be. The answer of course depends on the models as well as the experimental constraints. In this section, we summarize some experimental constraints on the three models and estimate the allowed magnitude of loop-induced vector NSI for couplings to electrons, protons and neutrons, whose definition is given in Eqs. (3,4). We selectively consider three most relevant experimental constraints, namely the invisible Z decay width, elastic neutrino scattering and charged lepton flavor violation. When all these constraints are taken into consideration, we find that loop-induced NSI in the three models can reach the magnitude listed in Tab. II.
Invisible Z decay width Since in the triangle diagram the Zνν vertices are modified in models A and B, it is necessary to consider the effect on the invisible Z decay width which has been measured precisely [50]: Adding Eq. (8) to the SM Zνν terms, we have the following Zνν interactions For one generation of neutrinos, the decay width Γ Z→νν is proportional to the absolute square of the vertex coupling. Generalizing to three generations, it holds that Γ Z,inv ∝ tr[λλ † ], from which we can infer Therefore, the invisible Z decay width should give a strong constraint on tr[λλ † ]. However, one should note that even when tr[λλ † ] is fixed at 3, large values of g (1) αβ are still allowed due to cancellations in the matrix product. The constraint from invisible Z decay is only useful when it is combined with the elastic neutrino scattering constraints to be introduced next.

Elastic neutrino scattering
New neutrino interactions can be directly constrained by elastic neutrino scattering experiments [6,48,49]. Some neutrino-electron scattering experiments (e.g. CHARM II [42,43], LSND [44], TEXONO [45]) already have precision measurement of the SM process and most recently coherent elastic neutrino-nucleus scattering has been successfully observed and will also be precisely measured in the near future [40,41].
In general when there are new neutrino interactions, elastic neutrino scattering is sensitive to the ratio between the new and SM cross sections (ignoring spectral effects): where the final neutrino state in the numerator can be of any flavor and the cross section σ new is a sum over all possible flavors. The target particle ψ can be either an electron or a nucleus. Considering the specific models in this paper, the Zνν vertices are modified in model A and model B, while in model A and model C the scalar bosons make tree-level contributions to neutrino-electron/nucleus scattering.
Given the modified Zνν vertices in Eq. (45), it is straightforward to derive 8 It is interesting to note that Eq. (46) can be expressed in term of the ratios R α : According to the ν µ and ν e elastic scattering data [42,44,45], R e and R µ cannot have large deviations from 1: This combined with the Z decay observation |N ν − 3| 1 implies that R τ should also be close to 1: Using Eq. (45), we can convert the constraints on R α to constraints on g (1) αβ : Thus, the corresponding constraints on αβ are where for ψ = e or p the result is suppressed by their small Z charges |Q (e) The tree-level contribution of the scalar boson in model C is roughly Assuming y ν αβ y ψ is O(1), the box NSI in model C could reach Similar constraints also exist for model A, which should be approximately of the same magnitude. However, as we will see, the constraints from charged lepton decay are much more stringent than those from elastic neutrino scattering in model A.

Charged lepton flavor violation
Charged lepton flavor violation (CLFV) could cause rare lepton decays such as µ → eγ, µ → 3e, τ → µγ, etc. Currently all lepton flavor violating decays have not been observed, which yields very strong constraints on models containing CLFV. In this paper, we only need to consider CLFV in model A because the other two models only have flavor violations limited to the neutrino sector. Here we would like to refer to Ref. [21] which has studied these decay processes in model A. We present the results in Ref. [21] with the experimental bounds updated.
Combining all the constraints discussed above, the strongest constraints on the loop-induced NSI parameters come from CLFV for model A, and elastic neutrino scattering for modela B and C. The results are summarized in Tab. II.

V. CONCLUSION
In scalar extensions of the SM, complex NSI can be generated at the loop level, denoted here loop-induced NSI. There are two types of loop diagrams that are responsible for loop-induced NSI, triangle diagrams and box diagrams shown in Fig. 1. We computed the loop diagrams and derived general formulae for loop-induced NSI, given by Eqs. (8) and (19).
To be more concrete, we applied our results to three specific and frequently discussed models, which contain charged or neutral scalar bosons. With the experimental constraints on these models taken into consideration, we estimated how large the loop-induced NSI can be, which is summarized in Tab. II. Testable NSI are possible.
Our calculations were performed in the limit of heavy scalars (heavier than the fermion masses in loops), though a similar analysis could also be performed for light particles. Loop-induced NSI are not neccesarily obtainable by scalar particles only, but also by vector bosons, leptoquarks etc. The relevant phenomenology will differ and deserves future study.

ACKNOWLEDGMENTS
IB is supported by the IMPRS-PTFS and enrolled at Heidelberg University. WR is supported by the DFG with grant RO 2516/6-1 in the Heisenberg program.

Appendix A: The triangle diagrams
In this Appendix, we compute the triangle diagrams in a general U (1) model. The result can be directly applied to more complicated models such as the SM extended by various scalar particles. Various useful identities and relations necessary for our calculations can be found in Appendix C.
The U (1) model being considered here contains a massive scalar φ and three massless fermions ψ 1 , ψ 2 and ψ 3 . They are all charged under the U (1) gauge symmetry so the Lagrangian is where In Eq. (A1) we have included only two Yukawa couplings y 12 and y 23 because this is the minimal requirement to obtain the effective flavor-changing operator A µ ψ 1 γ µ ψ 3 at the 1-loop level, as indicated in Fig. 3. Including other Yukawa terms (ψ 1 φψ 3 or ψ 1 φ * ψ 3 ) would only complicate the scenario and may not allowed by the U (1) charges 9 . Due to the U (1) charge conservation, the Yukawa interactions in Eq. (A1) are allowed when which further implies There are two potential problems when applying the above U (1) model to SM extensions. First, the Z boson in the SM is massive while here the U (1) gauge boson is massless if the gauge symmetry is unbroken. To make the result in this appendix applicable to massive gauge bosons, we manually introduce a mass m A for the gauge boson. The gauge boson mass could be generated by introduce another scalar field with spontaneous symmetry breaking but we would rather refrain from involving such details.
Another problem is about the chiral fermions. The SM, as a chiral theory of fermions, has different gauge interactions for left-handed and right-handed fermions. For example, e L and e R have different Z-vertices. The U (1) model considered here is parity conserving, i.e. both the left-handed and right-handed components of ψ i are equally coupled to the gauge and scalar bosons. This problem can be easily solved if the Dirac spinors are decomposed into Weyl spinors. In the U (1) model, the two sets of Weyl components (ψ L1 , ψ R2 , ψ L3 ) and (ψ R1 , ψ L2 , ψ R3 ) do not couple to each other directly because Therefore, in the diagrams in Fig. 3, all the fermion lines can also be regarded as either (ψ L1 , ψ R2 , ψ L3 ) or (ψ R1 , ψ L2 , ψ R3 ). At the end of this section, we will also present the result for a chiral U (1).
It is important to notice that the sum of all the four diagrams in Fig. 3 is finite, as pointed out in Ref. [20]. The UV divergences necessarily cancel out if the model is renormalizable, otherwise there is no corresponding counter term to cancel the infinity. We will show the cancellation explicitly in the following calculation. Now let us compute the four diagrams (a)-(d) in Fig. 3. The relevant interactions are Then it is straightforward to write down the amplitudes Here we assume that the fermions ψ 1, 3 have very small masses m 1, 3 so that the standard technique of extracting form factors can be applied. After that, we will take the limit m 1,3 → 0. For simplicity, we evaluate the amplitudes with all external momenta on shell so that amplitudes can be organized by three form factors F 1 , F 2 and F 3 as follows Since / q = / p 3 − / p 1 and / p 1 u(p 1 ) = m 1 , u(p 3 ) / p 3 = m 3 , the F 1 term actually reduces to γ µ F 1 (q 2 ) in the zero mass limit (m 1,3 → 0). We use the computer program Package-X [52] to compute the loop integrals in dimensional regularization. The form factors can be directly extracted by using the corresponding projectors in Package-X. In this paper, we are only interested in the F 1 form factor. Let us first check the UV divergence in F 1 . Since we are using dimensional regularization, the UV divergent part is proportional to 1/ ≡ 2/(d − 4): where the terms proportional to Q 2 , Q φ , Q 3 , and Q 1 correspond to the contributions of diagrams (a), (b), (c), and (d), respectively (note that each of these diagrams has a distinct gauge interaction vertex and a characteristic U (1) charge). Eq. (A13) can also be written as which, according to Eq. (A3), implies that the UV divergence vanishes if the U (1) charges are conserved.
Note that the result is applicable only when the U (1) charges are conserved-see Eq. (A3). For a chiral U (1) theory, we can still use the above result by simply replacing the Dirac spinors with the chiral spinors. For example, if only (ψ L1 , ψ R2 , ψ L3 ) are present in the model, then we have where g 31 is the same as Eq. (A24), and (Q 1 , Q 2 , Q 3 ) should be the U (1) charges of (ψ L1 , ψ R2 , ψ L3 ) respectively.
where we define the integral (B4) It can be evaluated straightforwardly: . (B5) In the equal mass limit (m 2 a = m 2 b = m 2 φ ), it is In summary, the box diagram generates the following four-fermion effect operator In this work, we need to frequently transform Dirac matrices and spinor products from one to another. Besides, in the loop calculation, we also need some useful identities about the dilogarithm functions. Therefore, we compile them in this appendix.
The left-and right-handed components of a Dirac spinor ψ are defined as ψ L ≡ P L ψ, ψ R ≡ P R ψ.
The charge conjugate of ψ is defined as The left-and right-handed projections and the charge conjugation are related by