An alternative formulation of left-right symmetry with $B-L$ conservation and purely Dirac neutrinos

We propose an alternative formulation of a Left-Right Symmetric Model (LRSM) where the difference between baryon number ($B$) and lepton number ($L$) remains an unbroken symmetry. This is unlike the conventional formulation, where $B-L$ is promoted to a local symmetry and is broken explicitly in order to generate Majorana neutrino masses. In our case $B-L$ remains a global symmetry after the left-right symmetry breaking, allowing only Dirac mass terms for neutrinos. In addition to parity restoration at some high scale, this formulation provides a natural framework to explain $B-L$ as an anomaly-free global symmetry of the Standard Model and the non-observation of $(B-L)$-violating processes. Neutrino masses are purely Dirac type and are generated either through a two loop radiative mechanism or by implementing a Dirac seesaw mechanism.


I. INTRODUCTION
With the discovery of the Higgs boson the last missing piece of evidence confirming the Standard Model (SM) of particle physics has been obtained. However, the observation of neutrino oscillations has established non-vanishing neutrino masses, which is undeniable evidence of physics beyond the SM. In the SM the left-handed fermions transform as electroweak doublets while the right-handed fermions transform as singlets due to parity violation. Thus, it is natural to seek for a left-right symmetric theory at a high energy scale, where both the left-handed and the right-handed fermions transform on an equal footing under the gauge group and parity is restored. At some high energy the left-right symmetric gauge group and parity is broken spontaneously, which explains the observed parity violation at low energies.
This motivates the left-right symmetric model (LRSM), in which the SM gauge group [1]. Several versions of the LRSM exist in literature (see for example [2] for a recent review) and in almost all of these models one identifies the generator of the group U (1) X with the B − L symmetry, where B is the baryon number and L is the lepton number 1 . For the SM particles this identification follows simply from the charge equation relating the SM gauge group to the LRSM gauge group which is broken by the conventional choice of a triplet Higgs scalar. If this choice is generalised for the right-handed neutrinos one can then generate small Majorana neutrino masses for the neutrinos through the seesaw mechanism [5]. However, in general this choice is not unique for new fermions added to the SM spectrum or for alternative Higgs sectors.
In the conventional LRSM, at some high energy scale compared to the electroweak symmetry breaking scale the left-right symmetric gauge symmetry group can be written as which breaks down to the SM gauge group SU (3) c × SU (2) L × U (1) Y . The electric charge is related to the generators of the gauge groups by the relation In the conventional case the quantum number X is identified with the B − L symmetry, so that B − L becomes a local gauge symmetry of the model. Consequently, the left- 1 For some exceptions see e.g. [3,4].
Left-right symmetry naturally includes the right-handed neutrinos ν R . The symmetry breaking pattern is given by where M R corresponds to the SU (2) R breaking scale. The relevant scalar sector is given by : [1,1,3,2] .
In the absence of any additional symmetry, the gauge symmetry allows the interactions of the Higgs triplets with the fermions which determine the B − L quantum number of ∆ L,R uniquely, allowing the identification When this conventional model is embedded in grand unified theories like SO(10) GUT, the theory contains diquarks (∆ qq that couple to two quarks) or leptoquarks (∆ lq that couple to a quark and a lepton or an anti-lepton) [6]. All these scalar fields belong to one 126-dimensional representation of SO (10) and their quantum numbers are determined by the quantum numbers of the fermions, which dictates X = B −L. In this conventional formalism there are many sources of B − L violation, all of which could affect the lepton asymmetry of the universe, and hence, the baryon asymmetry of the universe. Usually one considers mainly the interactions of the right-handed neutrinos N R for studying leptogenesis [7,8]  A complete study should thus address all the following interactions: (i) Interactions of the gauge boson W R with the right-handed leptons, and also with the Higgs triplet ∆ R which violate B − L quantum numbers [9][10][11][12][13][14][15][16]18]. In some models, these interactions can also generate a lepton asymmetry. (ii) Interactions of the diquark Higgs scalars ∆ qq with themselves and with the dilepton Higgs scalars [19][20][21][22][23]. When a model predicts neutronantineutron oscillation, light diquark Higgs scalar are predicted. These models may wash out the lepton asymmetry generated by the right-handed neutrino decays. (iii) The interactions of the right-handed triplet Higgs scalars ∆ R [24][25][26][27] can also affect the lepton asymmetry generated by other mechanisms. (iv) The left-handed triplet Higgs scalars ∆ L can generate a lepton asymmetry and also a neutrino mass, after the right-handed neutrinos decay [28][29][30][31][32].
Even when M ∆ > M N , the Higgs decay can generate an asymmetry, which is not affected by the slow lepton number violating decays of the right-handed neutrinos.
In what follows, we will construct a formulation of an LRSM with an unbroken B − L symmetry, where all these interactions are absent because B −L is not spontaneously broken and consequently one ends up with very different phenomenology and signatures. Firstly, we note that in general, one can define a new quantum number ζ, such that in Eq. (1) we 2 For a recent review with some relevant discussion see for example [17]. have, Thus, if ζ = 0, then B − L can as well become a global unbroken symmetry, independent of the left-right symmetry. In this work we point out an alternative scheme of left-right symmetry breaking, where B − L is no longer considered to be a local gauge symmetry, but remains an unbroken symmetry. Consequently, all fermions including the neutrinos are Dirac particles. Interestingly, in this model the neutrinos can have tiny Dirac masses generated through either a two loop radiative correction or a Dirac seesaw mechanism [33] depending on the Higgs sector of the model. This formulation also provides a natural framework to explain B − L as a global symmetry of the SM and can explain the non-observation of any (B − L)-violating processes. The baryon asymmetry of the universe can be explained in this formulation through (B − L)-conserving neutrinogenesis mechanism [34][35][36].
The plan for rest of the paper is as follows. In Section II, we present an LRSM with an unbroken B −L symmetry where no Higgs bi-doublet is present and the Higgs sector consists of a right-handed doublet, a left-handed doublet and a parity-odd singlet. In this scenario the quark masses and the charged lepton masses are generated through a seesaw mechanism introducing new vector-like states, while the neutrino masses are generated radiatively at the two loop level. In Section III, we study the two loop radiative contribution in the con- Section V, we outline the observable phenomenology of this formulation and discuss constraints from various considerations. Finally in Section VI we conclude and comment on the possible implementation of a dark matter candidate and leptogenesis mechanisms to generate the observed baryon asymmetry of the universe in this scenario.
II. LEFT-RIGHT SYMMETRIC MODEL WITH AN UNBROKEN B − L SYMME-

TRY
The fermion content of this model is the same as given in Eq. (3). In addition we will add vector-like fermions. For the left-right symmetry breaking, we now use a doublet Higgs scalars χ R ≡ [1, 1, 2, 1], whose vacuum expectation value (VEV) breaks the left-right [37,38]. It is crucial to note that this field does not have any exclusive interaction with the SM fermions, and hence the B − L quantum number is no longer uniquely determined as compared to the conventional LRSM.
Therefore for χ R , we can choose B − L = 0, and hence ζ = 1 in Eq. (7). The left-right symmetry ensures that we have a second doublet Higgs scalar χ L ≡ [1, 2, 1, 1], with the same assignment of B − L = 0 and ζ = 1. Interestingly, these assignments do not require any additional global symmetries, but will allow B − L to remain as a global unbroken symmetry after the electroweak symmetry breaking.
A priori we have two choices of Higgs sector for breaking the electroweak symmetry.
The first choice is that we keep the Higgs bi-doublet from the conventional model; after electroweak symmetry breaking it will then generate Dirac masses for all the fermions. Such a scenario is the subject of discussion in Section IV. In this section we will be primarily interested in the alternative possibility, with no Higgs bi-doublet and where the left-handed Higgs doublet χ L ≡ [1, 2, 1, 1] breaks the electroweak symmetry. In such a scenario the quark masses and the charged lepton masses are generated through a seesaw mechanism introducing new vector-like states [33]. Interestingly, in this scenario the neutrino masses can be generated radiatively at the two loop level induced by W L − W R mixing at the one loop level [39]. The field content of this model is summarised in Table I.
The necessity of the scalar field ρ in the model is justifiable from an examination of the relevant scalar potential [40]. In the absence of the Higgs bi-doublet the general scalar potential of this model can be written as Redefining λ 1 and λ 2 in terms of λ + = (λ 1 + λ 2 /2)/2 and λ − = (λ 1 − λ 2 /2)/2 and using the parametrisation χ 0 L = r sin β, χ 0 R = r cos β and ρ = s, we can recast the scalar Minimising the scalar potential with respect to r, β and s we obtain µ ρχ r 2 s sin 2β − 2λ − r 4 cos 2β sin 2β = 0 , From Eq. (11) it is evident that for µ ρχ = 0, i.e. if the ρ field is decoupled from the model then β = π/4, π/2, · · ·. Here β = π/4 corresponds to the unbroken parity symmetry case χ 0 L = χ 0 R and β = π/2 corresponds to the case where χ 0 L = 0, χ 0 R = 0, leading to massless quarks and charged leptons. Therefore we conclude that it is crucial for the model to have a ρ field with µ ρχ = 0 thus giving cos 2β = µ ρχ s/2λ − r 2 , leading to a realistic mass spectrum for quarks and charged leptons of the model. Thus, we will consider the symmetry breaking pattern In this scheme, the usual Dirac mass terms for the SM fermions are not allowed due to the absence of a Higgs bi-doublet scalar. However, under the presence of vector-like copies of quark and charged lepton gauge isosinglets, the charged fermion mass matrices can assume a seesaw structure. The relevant Yukawa interaction Lagrangian in this model is given by where we suppress the flavour and colour indices on the fields and couplings for brevity.
where τ 2 is the usual second Pauli matrix. Note that, in general, if the parity symmetry is broken by the VEV of a singlet scalar at some high scale as compared to the left-right symmetry breaking scale then the Yukawa couplings corresponding to the right-type and left-type Yukawa terms may run differently under the renormalisation group below the parity breaking scale. This approach where the left-right parity symmetry and SU (2) R breaking scales are decoupled from each other was first proposed in [41]. Therefore, while writing the Yukawa terms above we distinguish the left-and right-handed couplings explicitly with the subscripts L and R.
After electroweak symmetry breaking we can write the mass matrices for the quarks as [42][43][44][45][46] where χ L,R = u L,R . Up to leading order in h uL u L , the SM and heavy vector partner up-quark masses are given by The down-quark masses and mixing are obtained in an analogous manner. Note that in writing the above equations we have dropped the flavour indices of the Yukawa couplings h uL,uR which determine the observed quark and charged lepton mixings. The hierarchy of the quark masses can be explained by assuming either a hierarchical structure of the Yukawa couplings or a hierarchical structure of more than one generation of the vector-like quark masses.
Similarly, the charged lepton masses are generated through a Dirac seesaw mechanism.
However, we explicitly assume more than one generation of vector-like charged lepton and work in a basis where the vector-like charged lepton masses are diagonal. In such a basis the SM charged lepton masses are given by The charged lepton mass matrix given in Eq. (17) can be diagonalised by the bi-unitary where l m L(R) = U L(R) l f L(R) with the superscripts m and f correspond to the mass and flavour bases, respectively. Light Dirac neutrino masses are generated through a two loop contribution employing the mixing of W L and W R occurring at a one loop level [39]. The relevant Feynman diagram is shown in Fig. 1. The computation of this diagram leads to the following neutrino mass term where I k = m E k I k corresponds to a diagonal matrix, with Here p and k denote the momenta of the W L and b in the loops, respectively. Note that to simplify the analysis we have made the assumption that the top and the bottom quarks (20) the neutrino mass matrix turns out to be directly proportional to the charged lepton mass matrix and consequently, the PMNS matrix turns out to be diagonal which is ruled out by the current neutrino oscillation data 3 . However, we would like to emphasise that the above argument is no longer true in the case where more than one generation of vector-like charged lepton with a hierarchical mass spectrum is considered. In Section III, we shall focus on this scenario and show that it is indeed possible to accommodate non-trivial mixings in the PMNS mixing matrix using only the two loop radiative contribution if more than one generation of heavy vector-like charged lepton is present.
In Appendices A and B we sketch two alternative methods of evaluating the two loop integral given in Eq. (20). Note that even though such integrals have been evaluated in the literature for one heavy vector-like charged lepton state under some simplifying assumptions [39], it is crucial to evaluate them more generally to understand the dependence of the integral on the vector-like quark masses, which generate a non-trivial mixing for the neutrinos in addition to nonzero masses. Following the approach outlined in Appendix A, the final neutrino mass is given by where The neutrino mass matrix given in Eq. (21) can be diagonalised by the bi-unitary transformation where U ν L and U ν R are the left-and right-handed unitary matrices corresponding to the bi-unitary transformation diagonalising the neutrino mass matrix.

III. A LEFT-RIGHT SYMMETRIC PARAMETRISATION OF THE RADIATIVE NEUTRINO MASSES AND MIXING
To analyse the two loop radiative neutrino masses and mixings phenomenologically, it is convenient to parametrise the charged lepton and neutrino masses. From Eqs. (17) and (18) the diagonal charged lepton matrix is given by where the matrices have been made bold to distinguish them from numbers andM −1 Similarly, from Eqs. (21) and (23) the diagonal neutrino mass matrix is given by where is a diagonal matrix with J being the diagonal matrix corresponding to the integral Eq. (22). If J is not proportional to m −1 E then one can have a non-trivial PMNS mixing matrix U L = U l † L U ν L by solving Eqs. (24) and (25) simultaneously, for h L and h R to fit the neutrino oscillation data. A comprehensive numerical analysis of the 3 × 3 general left-right asymmetric mixing case is highly non-trivial and involves a large number of parameters. This is beyond the scope of the current work and will be addressed in a future work. Here we will focus on a particularly interesting limiting case where the left-and right-handed unitary rotation matrices and the Yukawa couplings are identical i.e. U l,ν L ≡ U l,ν R ≡ U l,ν and h L ≡ h R ≡ h. This helps us in constructing a new parametrisationà la Casas-Ibarra [47] which immensely simplifies the underlying numerical analysis of simultaneously solving Eqs. (24) and (25). Even though such a simplifying assumption need not to be true in general, it enables us to explore the qualitative dependence of the PMNS mixing angle on different model parameters by using a phenomenological approach. As noted before, for a diagonal m E and J,M −1 E and M Eν are diagonal matrices in generation space, which allows us to write the identities which can then be substituted in Eq. (28) to solve for the PMNS mixing matrix up to an In order to understand the dependence of the PMNS mixing angle on different model parameters (in particular, the left-right symmetry breaking scale and mass scale of the heavy vector-like fermions) and the arbitrary unitary rotation matrices qualitatively, we explore the discussed parametrisation to solve Eq. (29) and (30) simultaneously for a 2 × 2 case.
Furthermore we restrict ourselves to the case where all the Yukawa matrices and rotation matrices are real. With these simplifying assumptions, the arbitrary rotation matrices R l,ν and the PMNS matrix U can now be parametrised in terms of one rotation angle each   It is evident from these plots that one requires a SU (2) R breaking VEV of u R ∼ O(10 2 ) TeV to generate the correct neutrino mass splitting and a maximal PMNS mixing angle. Although, a priori, it appears to be relatively high as compared to the currently accessible mass scales at the LHC, it is interesting to note that such mass scales are in agreement with the strong cosmological bounds (discussed in Section V) on the SU ( In Fig. 3, we present the numerical solutions for the case of inverted ordering using the best-fit values for the atmospheric and solar neutrino mass squared differences from the global oscillation analysis of [52], setting the lightest neutrino mass to be 10 −2 eV as a benchmark choice and using the second and third generation neutrino masses as the diagonal entries of m diag ν . We note that in this case one also requires a SU (2) R breaking VEV of u R ∼ O(10 2 ) TeV to generate the correct neutrino mass splitting and a maximal PMNS mixing angle.

IV. LEFT-RIGHT SYMMETRIC MODEL WITH A GLOBAL B − L SYMMETRY IN THE PRESENCE OF A BI-DOUBLET HIGGS
In this alternative scenario a Higgs bi-doublet breaks the electroweak symmetry. The field content and their transformations are summarised in Table. III. The quarks acquire their masses through the vacuum expectation value of the bi-doublet while the Yukawa couplings giving rise to lepton masses are forbidden by some symmetry 4 . Both the charged and the neutral leptons would then acquire Dirac seesaw masses in this scenario [33] 5 . For this purpose, we introduce four singlet vector-like fermions, which are the charged and neutral heavy leptons:  whereΦ = τ 2 Φ * τ 2 . Using the notation χ L = u L , χ R = u R , Φ = diag(v 1 , v 2 ) and ρ = s, we minimise the scalar potential to obtain where µ hφ = (µ hΦ1 v 2 + µ hΦ2 v 1 )/(v 1 + v 2 ). The effective mass terms µ 2 L and µ 2 R are given by From Eqs. (34), (35) one gets One can derive the seesaw relation from the above equation as Assuming the hierarchy u L u R s, M yields priately.
The Yukawa term for the quarks involving the Higgs bi-doublet is given by whereΦ = τ 2 Φτ 2 and τ 2 is the second Pauli matrix. After the electroweak symmetry is broken via the VEV of the Higgs scalar bi-doublet, one can obtain the Dirac mass terms for the SM quarks. Thus, the case of quark masses is similar to the conventional LRSM case and we will not repeat the details here 6 . On the other hand, for the charged and neutral leptons there is no Dirac mass term due to the Higgs bi-doublet as mentioned earlier 7 . The Yukawa interactions giving mass to the leptons are given by The charged lepton masses are generated through a Dirac seesaw mechanism (similar to Section II) and the mass matrix is given by For convenience of analysing the neutrino sector we shall work with the CP conjugates of the right-handed fields so that the neutrino mass matrix can be written in the basis ( ν L N L σ L Σ L ) as Here a = f L u L ; b = f R u R ; and c = m σ . This gives six Dirac neutrinos, three very heavy ones with mass ∼ c, and three light ones with mass ∼ ab/c [57][58][59]. The heavy Dirac neutrinos are 6 Note that, in the presence of vector-like quarks in the model there can be a seesaw type contribution as well [42]. 7 To ensure this we assume that under a discrete Z 2 symmetry the right-handed fields L R , σ R , and E R are odd, while all other fields are even. made of σ L and Σ L , while the light Dirac neutrinos are the usual neutrinos, a combination of ν L and N L or ν R . Note that one can a priori draw a two loop diagram similar to Fig. 1, without the vector-like fields in a scenario where the charged lepton and quark masses are generated by the bi-doublet Higgs and only neutrino masses are vanishing at the tree level.
However, in such a diagram the external neutrino lines can be folded to generate a tadpole correction to the VEV of the neutral component of bi-doublet Higgs which diverges and therefore must be cancelled by adding a counterterm [60]. Therefore, one must ensure that the bi-doublet VEV satisfies the constraint 0|Tr[Φτ 2 Φ * τ 2 ]|0 = 0 at the tree level, implying that there is no mixing between W L − W R at the tree level. Over the mass range 1 eV < M Z < 1 TeV, BBN puts a varying upper bound on the coupling g from the condition that the thermalisation of ν R does not contribute considerably to N eff . For masses M Z < 1 MeV, for example, ν R must thermalise after the photon temperature ∼ 1 MeV, giving g < 3 × 10 −7 keV/M Z . Natural couplings of order unity are similarly excluded for 1 MeV < M Z < 10 GeV, but both of these regimes are clearly not of interest in our scenario. For M Z > 10 GeV thef f ↔ν R ν R process can be treated at T BBN as a four-fermion contact interaction and constraints are thus made on the ratio M Z /g . This is analogous to a constraint on the ratio M W R /g R , which also leads to thermalisation viā f f ↔ν R ν R . The constraint presented in Ref. [61] is M Z /g > 6.7 TeV. For the benchmark couplings considered in the radiative and Higgs bi-doublet cases in this paper, this puts a lower bound on M W R in the range 1 − 6 TeV. Ref. [62] similarly investigates the regime M Z T BBN , but instead studies the relationship between N eff and the temperature T dec ν R at which ν R decouples. Enforcing the interaction rate Γ(T dec ν R ) to be equal to the Hubble rate H(T dec ν R ) at this temperature, a bound of similar size can be placed on M W R /g R . Direct searches for additional gauge bosons have also been performed at colliders, with analyses probing large values of M Z . The study of LEP 2 data [63] in Ref. as those of the SM [69]. For the benchmark value of g R considered in this paper and the subsequent lower bound of M W R 5 TeV, we can safely expect the additional gauge boson W R to be out of reach at the high-luminosity LHC 8 .

VI. CONCLUSION
The question of how neutrinos acquire their masses, needed to understand the observed oscillation phenomena, remains one of the main outstanding issues in particle physics. The overwhelming majority of explanations work by generating ∆L = 2 Majorana masses for neutrinos, with the type-I seesaw mechanism as the most prominent example. While this approach clearly has theoretical and phenomenological advantages, it is also important to pursue other potential solutions.
In this paper, we have proposed an alternative formulation of a Left-Right Symmetric 8 For a relevant discussion of W R multi-leptonic decay modes see [70,71].
Instead, B − L remains a global symmetry after the left-right symmetry breaking, allowing only Dirac mass terms for neutrinos. While parity is restored at a high scale, this formulation provides a natural framework to explain B − L as an anomaly-free global symmetry of the SM. In this model, a bi-doublet Higgs is not present and the charged SM fermion masses fundamentally originate from a Dirac seesaw mechanism connected to heavy vector-like fermion partners. The lightness of neutrinos in the instance is explained as neutrino Dirac mass terms are induced at the two-loop level, cf. Fig. 1. Alternatively, a Dirac seesaw mechanism can be invoked for the neutrinos as well if the corresponding heavy vector-like neutrino partners exist. We show that for an appropriate spectrum of heavy states, not only the lightness of neutrinos relative to the charged fermions can be explained, but also a large two flavour mixing in the leptonic sector. An analysis of the full three-flavour framework will be reported elsewhere.
Our models may be enhanced in several directions. For example, while neutrinoless double beta decay is strictly forbidden, one can add a light charge-neutral scalar particle φ with quantum numbers [1, 1, 2, 1] under our model gauge group. This particle can potentially be a Dark Matter candidate [72][73][74] with a Yukawa coupling to the heavy N of the form g φ N N φ.
In this case, 0νββφ decay with emission of the light neutral scalar φ via a single effective dimension-7 operator of the form Λ −3 NP (ūOd)(ēOν)φ is possible. This provides a working example of a scenario where purely Dirac neutrinos can mimic the conventional 0νββ decay associated with violation of lepton number by two units and thus the Majorana nature of neutrinos. This supports the necessity of searches for extra particles in double beta decay in order to fully understand the nature of neutrinos [75].
Finally, we would also like to make some remarks on the possibility of realising leptogenesis in this formalism. In our scenarios, leptogenesis may occur through neutrinogenesis [34,35]. To give an example, the scalar field χ R can decay as χ R → R +E R and χ R → Φ † +Φ because of the coupling χ † R χ R Φ † Φ when χ R acquires a VEV. Through self-energy diagrams there can then be an interference and these decays can generate an asymmetry in the ζ quantum number which means that there will be more E R compared to E L , since R and φ has ζ = 0. However, since B − L is conserved, the asymmetry in E R will be same as the asymmetry in R . Since B − L is conserved, the out-of-equilibrium three-body decays of E R and E L will produce different amounts of ν L and ν R . Since the Yukawa couplings responsible for ν R + φ → ν L + W L are not allowed, the amount of lepton asymmetry stored in ν R will not be converted into ν L . Thus although there is no B − L asymmetry, there is an asymmetry in ν L and an equal and opposite amount of asymmetry in ν R . Since the ν R asymmetry will not get converted to a baryon asymmetry in the presence of sphalerons, the ν L asymmetry will generate a baryon asymmetry of the universe. Since B − L is an unbroken symmetry in this model, there are no other washout interactions that can affect the baryon asymmetry of the universe. Alternatively, one can also add an additional heavy doublet scalar field to implement a neutrinogenesis mechanism similar to [36]. In this section we outline the evaluation of the two loop integral given in Eq. (20) using the Passarino-Veltman integral reduction. Note that the first term in the numerator of Eq. (20) is suppressed by m 2 W L /m 2 W R with respect to the second term and therefore can be neglected to obtain Next using Partial-fraction decomposition and Passarino-Veltman reduction formula the integral can be simplified to obtain where B 0 is the Passarino-Veltman function defined as [76] B 0 k 2 , m 2 1 , m 2 Next performing a Wick rotation and defining the dimensionless quantities α k = m 2 B /m 2 E k and β k = m 2 T /m 2 E k the integral given in Eq. (A2) can be further simplified to obtain (A4) Appendix B: Evaluation of the two loop integral using master integral reduction In this section we outline another alternative approach using master integral reduction for the evaluation of the two loop integral given in Eq. (A1). Using Feynman parametrisation the two loop integral can be written as where the master integral is given by [77] G(m 1 , 2; m 2 , 1; m 3 , 1; 0) = π 4 2 2 + 1 [−1 + 2γ + 2 log(π m 2 1 )] + where Sp(z) corresponds to the Spence function and the following notations are used (B6)