Witten's loop in the minimal flipped $SU(5)$ unification revisited

In the simplest potentially realistic renormalizable variants of the flipped $SU(5)$ unified model the right-handed neutrino masses are conveniently generated by means of the Witten's two-loop mechanism. As a consequence, the compactness of the underlying scalar sector provides strong correlations between the low-energy flavor observables such as neutrino masses and mixing and the flavor structure of the fermionic currents governing the baryon and lepton number violating nucleon decays. In this study, the associated two-loop Feynman integrals are fully evaluated and, subsequently, are used to draw quantitative conclusions about the central observables of interest such as the proton decay branching ratios and the absolute neutrino mass scale.


I. INTRODUCTION
Though not a genuine grand unified theory (GUT), the flipped SU (5) gauge theory [1][2][3] still attracts significant attention [4][5][6][7] due to several rather unique features it exhibits. In particular, one-stage symmetry breaking down to the Standard Model (SM) can be achieved regardless of whether or not a TeV-scale supersymmetry is assumed. The corresponding Higgs sector can also be very small, as it is sufficient to employ just a single 10-dimensional representation to accomplish the necessary symmetry breaking. This is to be compared to the 24 of the Georgi-Glashow SU (5) [8] and/or 45 ⊕ 16 (or even 45 ⊕ 126) of the minimal SO (10) GUTs (see, e.g., Refs. [9,10] and references therein).
Flipped SU (5) models also share several other nice features with their truly unified cousins. From the point of view of phenomenology, two such features stand out as being particularly relevant due to their immediate experimental consequences. Firstly, as in the SO(10) GUTs, 3 right-handed (RH) neutrinos are enforced in the spectrum, allowing for the use of a type-I seesaw mechanism to generate the light neutrino masses. Additionally, as in SU (5) there is only one heavy gauge boson, which typically yields somewhat stronger correlations between the flavor structure of the baryon and lepton number violating (BLNV) currents and the low-energy flavor observables, and hence one can often say quite a bit about, e.g., the proton lifetime.
However, upon closer inspection one finds a certain level of tension between the practical implications of these two points. For example, in order to implement the standard type-I seesaw with the RH neutrinos at hand, a 50-dimensional four-index scalar 50 S of SU (5) is typically added [11] together with a 3 × 3 complex symmetric Yukawa matrix Y 50 in order to generate the * Electronic address: harries@ipnp.troja.mff.cuni.cz † Electronic address: malinsky@ipnp.troja.mff.cuni.cz ‡ Electronic address: zdrahal@ipnp.troja.mff.cuni.cz desired RH Majorana mass term via a renormalizable coupling such as Y IJ 50 10 T F I C −1 10 F J 50 S . Besides enlarging the scalar sector enormously (and, hence, disposing of the uniquely small size of the "minimal" Higgs sector noted above as one of the most attractive structural features of the framework), the extra scalar and the associated Yukawa at play reduces the value of the lowenergy neutrino masses and the lepton mixing data as constraints for the proton lifetime estimates as it essentially leaves the neutrino sector on its own.
Remarkably enough, this dichotomy may be overcome by noticing [12,13] that the RH neutrino masses in flipped SU (5) models may be generated even without the unpleasant extra 50 S at the two-loop level by means of a variant of the mechanism first identified by Witten in the SO (10) context [14]. The two main features [13] of this scenario are, first, a simple relation among the seesaw and the GUT scales where the former is, roughly speaking, given by the latter times an extra two-loop suppression and, second, a rigid correlation between the flavor structures of the neutrino and charged sectors, which in most cases may be transformed into a set of strong constraints for, e.g., the proton decay partial widths and branching ratios.
To this end, the Witten's-loop-equipped flipped SU (5) may even be viewed as the most economical renormalizable theory of the BLNV nucleon decays, much simpler than, e.g., the potentially realistic variants of the SO (10) and even the SU (5) GUTs.
From this perspective, it is interesting that in Ref. [13] most of the basic features of this framework may have been identified even without an explicit calculation of the graphs involved in Witten's mechanism. In this work we intend to overcome this drawback by a careful inspection of the Feynman graphs and their evaluation which, as we shall see, will clarify several other points left unaddressed in the preceding studies. In particular, the calculation will make it very clear that the minimal potentially realistic and renormalizable incarnation of the scheme under consideration is the variant featuring a pair of 5-dimensional scalars in the Higgs sector (besides a single copy of the "obligatory" 10-dimensional 10 H scalar). Second, it will be shown that, in this framework, the light neutrino spectrum is always forced to be on the heavy side (actually, within the reach of the KATRIN experiment [15]), which, among other things, provides a clear smoking gun signal of the scheme.
In Section II we first provide a brief review of the flipped SU (5) gauge theory context, identify the Feynman graphs underpinning the radiative RH neutrino mass generation in the minimal and next-to-minimal models, and exploit the seesaw formula in order to get strong constraints on their parameter space. Section III is devoted to a detailed analysis of the relevant two-loop graphs in the scenario with one copy of the 5-dimensional scalar in the Higgs sector; this setting is simple enough to allow for a complete analytic understanding of the results. In Section IV these findings are used for the identification of the minimal potentially realistic model of this kind, which is subsequently shown to be strongly constrained and potentially highly predictive. Most of the technical details of the lengthy calculations are deferred to a set of appendices.

II. FLIPPED SU (5) À LA WITTEN
The defining feature of the flipped SU (5) unifications is the "non-standard" embedding of the SM hypercharge operator within its SU (5) ⊗ U (1) X gauge symmetry algebra, namely where T 24 stands for the usual hypercharge-like generator of the standard SU (5) (normalized in such a way that the electric charge obeys Q = T 3 L + T 24 ) and X is the unique non-trivial anomaly-free generator of the additional U (1) normalized in such a way that it receives integer values over the three basic irreps accommodating each generation of the SM matter, (2) where the first number in brackets gives the SU (5) representation and the second the charge under U (1) X . Compared to the standard SU (5) case, the SM matter fields u c L and d c L are swapped with respect to their usual assignments, i.e., the former is a member of 5 M while the latter resides in 10 M . Similarly, e c L is found in the SU (5) singlet and the compulsory RH neutrino ν c L replaces it in the 10-plet.
The minimal Higgs sector sufficient for breaking the SU (5) ⊗ U (1) symmetry down to the SM and, subsequently, to the SU (3) ⊗ U (1) of QCD+QED consists of 10 H = (10, +1) 1 , in which the SM singlet occupies the same position as the RH neutrino does in 10 M , and 5 H = (5, −2) containing the SM Higgs doublet. The breakdown of SU (5) ⊗ U (1) X to the SM gauge symmetry takes place after the SM singlet present in 10 H develops a non-zero vacuum expectation value (VEV), V G , generating masses for the gauge bosons X µ , where g 5 is the SU (5) gauge coupling. The color triplet, SU (2) L singlet components of 10 H and 5 H also mix at this stage to form a pair of massive color triplets ∆ 1,2 transforming under the SM gauge symmetry as (3, 1, − 1 3 ), with masses m ∆1,2 . Further details regarding the tree-level scalar spectrum in this minimal flipped SU (5) model are given in Appendix B.
For the above embedding of the SM matter content and minimal set of Higgs scalars, one can readily write the most general renormalizable 2 Yukawa Lagrangian (suppressing all flavor indices) with Y 10 , Y 5 and Y 1 denoting the relevant 3 × 3 complex Yukawa coupling matrices; note that the first of these, unlike the latter two, is required to be symmetric in its flavor indices, i.e., Y 10 = Y T 10 . In the broken phase, the second term in Eq. (4) yields a strong correlation among the Dirac neutrino mass matrix M D ν and the up-type quark mass matrix M u , namely, at the GUT scale. The flavor symmetric nature of Y 10 also means that the down-type quark mass matrix satisfies M d = M T d , while the couplings in Eq. (4) say nothing specific about the mass matrix M e of the charged leptons. 1 It may be worth pointing out here that, due to the non-zero U (1) X charge of 10 H inherent to the flipped SU (5) models, there is no way to build a non-renormalizable d = 5 operator (presumably Planck-scale suppressed) that might, in the broken phase, affect the gauge-kinetic form and hence introduce significant theoretical uncertainties in the high-scale gauge-matching conditions and the determination of the GUT scale. As a result, one of the primary sources of irreducible uncertainties hindering the predictive power of the "standard" GUTs (such as the Georgi-Glashow SU (5) or the non-minimal SO(10) models with either 54 or 210 breaking the unified symmetry) is absent from this class of models. 2 Note that in non-renormalizable settings the benefits of the scheme may be lost as the Witten's loop contribution may be swamped by the effects of, e.g., the d = 5 non-renormalizable operators of the 10 M 10 M 10 H 10 H type.
As we shall see, these correlations will turn out to be central for the high degree of predictivity of this framework 3 entertained in the following sections.

A. The RH neutrino masses and type-I seesaw
So far, we have left aside any discussion of the physical light neutrino masses in the current scenario. Obviously, Eq. (5) cannot be the whole story here and, thus, one has to employ a variant of the seesaw mechanism in some way; since the type-II and/or type-III options cannot be realized with the minimal scalar and fermionic sectors at hand one is left with the type-I seesaw implemented through the Majorana mass term for the RH neutrinos.
This may be most easily devised by employing a 50-dimensional scalar [11] that can couple to the 10 T M C −1 10 M fermionic bilinear; the VEV of a singlet therein then gives rise to the desired mass term. As was noted in Section I, however, the associated single-purpose extra Yukawa matrix does not bring any additional insight into the flavor structure of the model, and limits the extent to which low-energy data can be used in constraining proton decay observables. Therefore we do not adopt this option here and, instead, consider the effects emerging at the quantum level in the minimal model.

The Witten's loop structure
The simultaneous presence of the diquark-type of interactions, mediated by the X µ and ∆ 1,2 bosons, together with their leptoquark counterparts (involving the same set of fields) in the model implies that even ∆L = 2 Feynman diagrams corresponding to the Majorana type of neutrino masses may be constructed at some higher order level. This, indeed, is the central point behind every radiative (Majorana) neutrino mass generation mechanism; in the flipped SU (5) framework, it finds its incarnation in a pair of two-loop topologies depicted in FIG. 1, which can be viewed as "reduced" versions of Witten's original SO(10) graph(s) [14].
Note that in our analysis we shall work in the broken phase perturbation theory with masses in the free Hamiltonian 4 and in the unitary gauge so that there are 3 To this end, it is worth noting that these relations remain intact even in models with more than a single copy of 5 H in the scalar sector; as we shall see, this (especially the symmetry of M d ) will be crucial for the construction of the minimal potentially realistic scenario identified in Section IV B. 4 Hence, we are avoiding the need to sum over an infinite tower of graphs (like the one drawn in Witten's original work [14]) with increasing numbers of VEV insertions. On the other hand, the explicit proportionality to the µ parameter governing the mixing between the 10 H and 5 H multiplets (see Appendix B), which is obvious in the massless perturbation theory, becomes more involved in the massive case where µ emerges at the level of the no Goldstone modes around. This reduces the number of relevant graphs considerably, albeit at the cost of making the Feynman integrals somewhat more complicated compared to other cases.
Based on the graphs in FIG. 1 that remain in this approach, it is immediately possible to make several comments on both the flavor structure and overall scale of the generated Majorana mass matrix M M ν . The flavor structure in particular plays a central role in what follows, and is governed by the Yukawa couplings appearing in each of the contributing graphs. In each of the two topologies there is only a single Yukawa coupling present, associated with the couplings of ∆ i to the fermions. These couplings involve only the 5 H components of ∆ i , since it is only these components that couple to the fermions through the Yukawa interactions in Eq. (4). Moreover, since all of the fermions appearing in the two graphs in FIG. 1 reside in 10 M , the single relevant Yukawa coupling matrix is the symmetric Y 10 . Hence, in the minimal model there is a tight correlation between the radiatively generated RH neutrino Majorana mass matrix and the mass matrix of the down-type quarks, making the scheme rather predictive.
The overall scale of M M ν , on the other hand, depends on both the Yukawa couplings in Y 10 as well as the gauge couplings and the sizes of the mass parameters entering into each of the graphs. One can initially estimate it to be proportional to the dominant mass entry in the relevant graphs suppressed by the appropriate two-loop factor and the combination of gauge (entering raised to the fourth power) and Yukawa couplings.
Of the various mass parameters appearing in the evaluation of the graphs, the fermionic masses m f should play no role in the integrals as the singlet Majorana mass generation does not rely on the electroweak symmetry relevant mixing matrix in the scalar sector, Eq. (B8).
breaking. Hence, in dealing with the Feynman integration we shall work in the chiral limit with all SM fermions massless. This, in principle, may lead to spurious IR divergences in the form of, e.g., log(m f /Q) arising in individual partial fractions of the integrands, where Q is the renormalization scale, but as a whole M M ν should be stable in the m f → 0 limit.
Similarly, it is natural to expect that in the other extreme case corresponding to one of the scalars ∆ i becoming significantly lighter with respect to the X µ boson masses (and, hence, bringing about another practically massless propagator) M M ν should also remain regular; hence, the only mass that can make it to the denominators in the final result is m X . This also suggests that, barring the couplings, each individual graph should be governed by powers of the m ∆i /m X ratio which, in turn, makes it merely a function of a single 5 parameter.
be such that the unification pattern is consistent with the low-energy data and compatible with the non-observation of proton decay with at least 10 34 years of lifetime [16].
Hence, demanding consistency of Eq. (7) with the data one can derive constraints on m max ν and, in particular, on U ν , which is central to the BLNV phenomenology of the model. Indeed, U ν drives all the proton decay branching ratios into neutral mesons including the "golden channel" p → π 0 e + final state: where the C i 's are various low-energy factors calculable using chiral Lagrangian techniques (see, e.g., Ref. [17] and references therein) and V CKM and V P M N S are the Cabibbo-Kobayashi-Maskawa and the Pontecorvo-Maki-Nakagawa-Sakata mixing matrices, respectively. In this sense, the minimal flipped SU (5) unification equipped with the Witten's loop mechanism can be viewed as a particularly simple (if not the most minimal of all) theory of the absolute neutrino mass scale and, at the same time, the two-body BLNV nucleon decays.

B. Consistency constraints and implications
Let us now work out the aforementioned consistency constraints in more detail and give some basic examples of their possible implications. Firstly, it should be noted that there is a lower limit on the largest entry of W ν that depends on m max ν and the shape of U ν . Taking into account the typical 50% reduction of the running top quark Yukawa between M Z and the unification scale (at around 10 16 GeV) and taking, for example, m max ν = 1 eV and U ν = 1 one finds that the (3, 3) entry of W ν is as large as about The same magnitude, however, may not so easily be achieved for the (3, 3) entry of M M ν as required by Eq. (7) due to the generic 10 −3 geometrical suppression in the relevant two-loop graphs and a possible further suppression associated with the Yukawa coupling Y 10 ; the latter may be especially problematic in the minimal scenario (4) because then Y 10 is fixed by the down-type quark masses and, thus, brings about another suppression of some 10 −2 to (M M ν ) 33 . However, this correlation is loosened if there is more than a single copy of 5 H in the scalar sector. As was already indicated in Ref. [13], the additional Y 10 associated to an extra 5 H can conspire with the original Y 10 to do two things at once: they may partially cancel in the down-type quark mass formula to account for the moderate suppression of M d /M Z yet their other combination governing M M ν (weighted by the appropriate scalar mixings) may still remain large, thus avoiding the problematic additional 10 −2 suppression. In what follows, we shall model this situation by imposing a humble |y| 4π perturbativity criterion on all the Y 10 and Y 10 entries.
However, even in such a case the ∼ 10 13 GeV lower limit on the largest entry (W ν ) 33 , may still be problematic because, for U ν = 1, it may be further enhanced by the admixture of the yet larger (2, 2) and, in particular, the (1, 1) entry of m diag ν −1 ; as a matter of fact the latter is not constrained at all given that the lightest neutrino mass eigenstate may still be extremely light. Thus, the lower bound on the magnitude of the largest element of W ν gets further boosted over the naïve estimate of 10 13 GeV whenever U ν departs significantly from unity, which in turn constrains all of the partial widths, Eqs. (8).
Hence, a thorough evaluation of the graphs in FIG. 1 will decide several important questions, namely: 1. Can the elements of M M ν ever be big enough to be consistent (at least in the most optimistic scenario with U ν ∼ 1) with W ν , as required by Eq. (7) This is what we turn our attention to in the remainder of this article.

III. WITTEN'S LOOP CALCULATION
The leading contribution to the radiatively generated RH neutrino mass in the current scheme may be computed by considering the graphs in FIG. 1 evaluated at zero external momentum, see Appendix C, with the relevant interaction terms given in Appendix A. In the minimal renormalizable model containing only a single 10 H and one or more 5 H representations, no one-loop contribution to the RH neutrino mass matrix can be generated, nor do there exist any one-loop counterterm graphs. The resulting expression for the RH Majorana neutrino mass matrix in the case of a single 5 H multiplet reads where the scalar mixing matrix elements (U ∆ ) ij are given in Appendix B, and I 3 (s i ) is the sum of the corresponding loop integrals evaluated at zero external incoming momentum, regarded as a function of s i = m 2 ∆i /m 2 X . Recall that there is an overall extra factor of 2 included in Eq. (10) related to the permutation of the two external neutral field lines (for I = J) or to the symmetry of Y 10 (for I = J). The integrals Σ P 1 (0) and Σ P 2 (0), corresponding to topology 1 and 2 respectively, are given by The integrals in Eq. (12) and Eq. (13) are evaluated by reducing them to expressions involving (variants of) the brackets by Veltman and van der Bij [18], which may be evaluated directly [18][19][20][21][22][23]. The details of this reduction, and the resulting analytic expressions for the two-loop integrals, are given in Appendix D. In particular, using the results given in Ref. [18] and appropriate generalizations thereof, it is found that the contributing brackets are free of potential IR divergences in the limit of massless internal fermions, such that the fermion masses may safely be allowed to vanish as in Eq. (12) and Eq. (13).
On the other hand, each graph is individually UV divergent. Setting = 2 − D 2 , where D is the spacetime dimensionality, the divergences are found to be −(4π) 4 Σ P,div and −(4π) 4 Σ P,div It follows from Eq. (11) that the total contribution I 3 (s i ) to the RH neutrino mass matrix is UV finite, as must be the case here due to the absence of the necessary counterterms.

IV. RESULTS
The behavior of the result for the purely kinematic piece of the RH neutrino mass matrix, I 3 (s), is shown in FIG. 2. Notably, the magnitude of I 3 (s) is bounded for all s ≥ 0. Indeed, from the analytic result given in Eq. (D31), one has that for s → 0, while in the opposite limit with s → ∞,

A. RH neutrino masses in the minimal model
With I 3 (s) determined, we may proceed to evaluate the size of M M ν in Eq. (10). Substituting in the explicit forms of the mixing matrix elements in Eq. (B8) one obtains and ν = µ/V G . We note thatĨ → 0 as µ → 0, reflecting the fact that the graphs rely on the 10 H − 5 H mixing. It is also clear from Eq. (19) that, since I 3 (s) is bounded,Ĩ cannot be made arbitrarily large to compensate for the suppression factors noted in Section II. To develop some sense of the allowed size ofĨ, it is useful to substitute for s i from Eq. (B7) and inspectĨ as a function of ν, λ 2 , λ 5 , and g 5 , neglecting all terms that are of the order of v 2 /V 2 G , where v is the electroweak VEV, see Eq. (B2). Requiring that the tree-level vacuum be locally stable implies [13] λ 2,5 < 0 and |ν| ≤ λ 2 λ 5 .
Qualitatively different behavior results in the more general case that ν does not saturate the bound given in Eq. (20). This is demonstrated in FIG. 4, in which the value ofĨ is plotted as a function of λ 2 = λ 5 = λ with for several values of α. AlthoughĨ remains invariant under λ 2 ↔ λ 5 , with the maximum value of |Ĩ| still occurring for λ 2 = λ 5 , for values of |α| < 1, |Ĩ| now tends to zero for large values of the scalar couplings λ 2 , λ 5 . This is due to the fact that, for |α| = 1, both s 1 , s 2 grow with increasing |λ| such that I 3 (s 1 ), I 3 (s 2 ) → −3, while the coefficients of each in Eq. (19) are equal in magnitude but of opposite sign, resulting in the two terms cancelling. Physically, this corresponds to the expected dynamical decoupling of the heavy scalar states in the m ∆1,2 → ∞ limit. For α = 1, at least one color triplet scalar is massless at tree-level for all values of λ 2 and λ 5 . Consequently, this state never decouples andĨ therefore does not vanish. Technically, this arises because I 3 (s 1 ) = 3 while I 3 (s 2 ) → −3, with the two contributions still enteringĨ with coefficients of equal magnitude but opposite sign.
However, even in the most optimistic case with |Ĩ| → 3, the above results make it clear that there is little hope for a viable prediction of the light neutrino spectrum in the minimal scenario under consideration. For acceptable values of m X ∼ 10 17 GeV, and taking g 5 ≈ 0.5, the elements of M M ν are found to be 10 12 GeV after taking into account the ∼ 10 −2 suppression associated with presence of Y 10 . This is to be compared with the (optimistic) lower bound of ∼ 10 13 GeV for the elements of the left-hand side of Eq. (7). Evidently, in the case when only a single 5 H is present in the spectrum the answer to whether Eq. (7) can be satisfied is negative. In fact, in this minimal model the problem is exacerbated by the fact that Y 10 ∝ M d , which implies a far too hierarchical pattern of light neutrino masses irrespective of their absolute size, as was previously noted in Ref. [13]. Thus we are immediately led to consider the remaining questions raised in Section II concerning the viability of the model with an additional 5 H representation instead.

B. Minimal potentially realistic model
As noted above, the addition of a second 5 H multiplet in principle allows both the Y 10 suppression and the overly hierarchical flavor structure to be avoided. At the same time, the overall predictive power of the theory is not significantly harmed by this addition; in particular, doing so does not spoil the key Yukawa relations used in obtaining Eq. (7). With a second 5 H multiplet, the Yukawa sector of the model reads L Y 10 10 M 10 M 5 H + Y 10 where Y 10 is of course also flavor symmetric. In this scenario, the Dirac neutrino mass matrix still remains tightly correlated with the up-type quark masses, with the GUT scale relation holding at tree-level, where v is the VEV associated with the electrically neutral component of 5 H , see Appendix B 2. By contrast, the analogous relationship between the down-type quark masses and the generated RH neutrino Majorana masses, GeV. (26) Note that in both cases we have used the (numerical) upper limit which is completely analogous to the limit discussed in Section IV A for the single-5 H case.
Remarkably, for the typical flipped SU (5) value of m X = 10 17 GeV (see, e.g., Ref. [13]) the case i) limit, Eq. (25), is just on the borderline of compatibility with the optimistic lower limit in Eq. (9) on |W ν |, while the latter case ii) in principle admits lower 6 values of m X .
This, in turn, implies that there is generally not much room for any significant admixture of the second neutrino (inverse) mass within the element (W ν ) 33 , hence, the only allowed U ν 's in Eq. (7) are those for which (U ν ) 13 and (U ν ) 23 are small.
To this end, the model clearly calls for a dedicated numerical analysis including a detailed calculation of the heavy spectrum that conforms to, among other things, the requirement of a significant spread of the scalar triplets in order to maximize |Ĩ|. This, however, is beyond the scope of the current study and will be elaborated on elsewhere.
At this point, let us just illustrate the typical situation by evaluating the most significant proton-decay two-body branching ratios (neglecting the kinematically suppressed vector-meson channels for simplicity) in the (U ν ) 13 = (U ν ) 23 = 0 limit with the 1-2 mixing angle θ 12 therein chosen in such a way that Γ(p → π 0 µ + ) is maximized (see Ref. [13] for further details): Needless to say, for non-extremal values of θ 12 these branching ratios may vary; in particular, Br(p → π 0 e + )/ Br(p → π 0 µ + ) should increase.
Finally, let us say a few words about the lower limits on the mass of the heaviest SM neutrino in the two cases (25) 6 These, however, may not be that simple to get within potentially realistic unification chains, see Appendix C of Ref. [13]. and (26). As for the former, one obtains 7 while for the latter one has m 3 0.08 10 17 GeV m X eV (30) which, actually, turns out to be independent on the specific form of the U ν matrix as long as the 1-3 and 2-3 mixings therein are small (see the discussion above). With this at hand, any specific experimental upper limit on the absolute neutrino mass scale may be readily translated into a lower limit on m X and, subsequently, the proton lifetime.

V. CONCLUSIONS AND OUTLOOK
The two-loop radiative RH neutrino mass generation mechanism originally identified by Witten in 1980s in the SO(10) context finds a beautiful incarnation in the class of renormalizable flipped SU (5) unified theories where, among other effects, it avoids the need for the 50-dimensional scalar representation. This, in turn, renders the simplest potentially realistic scenarios perhaps the most minimal (partially) unified gauge theories on the market, with strong implications for some of the key beyond-Standard-Model observables such as the absolute neutrino mass scale and proton decay.
In this work we have focused on a thorough evaluation of the relevant Feynman graphs in these scenarios paying particular attention to their analytic properties and the absolute size of the effect which turns out to be the key to the consistency of the scenario as a whole. It has been shown that there is no way to be consistent with the data with only one 5-dimensional scalar multiplet at play and, hence, the minimal potentially realistic setup must include two such irreps in the scalar sector (along with the 10-dimensional tensor).
As it turns out, such a minimal flipped SU (5) model is subject to strong constraints on its allowed parameter space that lead to rather stringent limits on the absolute light neutrino mass scale as well as the BLNV two-body nucleon decays. A thorough numerical analysis of the corresponding correlations is deferred to a future study. contract no. 17-04902S. We would like to thank Helena Kolešová, Jiří Novotný, Catarina Simões and Diego Aristizabal Sierra for illuminating discussions.

Appendix A: The interaction Lagrangian
The radiative generation of the RH neutrino masses involves only a small subset of the interactions associated with the full flipped SU (5) Lagrangian. Working in the SU (5) ⊗ U (1) X broken phase, we extract the required interactions from the kinetic terms and general Yukawa Lagrangian, Eq. (4), making use of FeynRules [24,25] and FeynArts [26,27] to verify that all terms and contributing diagrams are accounted for. As discussed in Section II, when the model contains only a single 5 H representation the relevant diagrams are found to be those in FIG. 1, arising from the interaction Lagrangian where i, j, k and α, β denote the SU ( . Following the breakdown of the SU (5) ⊗ U (1) X symmetry due to the non-zero VEV V G , the scalar states D and T mix to form the SU (3) C ⊗ SU (2) L ⊗ U (1) Y eigenstates ∆ 1,2 , as described in Appendix B.
Let us note that in deriving the central formula Eq. (10), especially the overall factor of 3 therein, the color and isospin factors in Eq. (A1) play a crucial role. It is also worth noting that the exact cancellation of the UV divergences discussed in Section III, which relies on the extra factor of 2 in Eq. (11), emerges from the difference of the overall numerical factors in the last two terms in Eq. (A1).
After including an additional 5 H to arrive at the minimal realistic model discussed in Section IV B, the interaction Lagrangian remains rather similar. The addition of Yukawa couplings involving 5 H leads to the set of interaction terms (with color indices suppressed for simplicity) where T denotes the additional (3, 1, − 1 3 ) multiplet contained in 5 H , which mixes with the states D and T to yield a set of SU (3) C ⊗SU (2) L ⊗U (1) Y eigenstates ∆ 1,2,3 .
For the sake of completeness and matching to the SM Yukawa couplings we also present the terms involving the doublet Higgs interactions here: The scalar basis is chosen such that the spontaneous breaking of SU (5) ⊗ U (1) X and the subsequent electroweak symmetry breaking takes place via the non-zero VEVs Requiring that this corresponds to a stationary point of the scalar potential yields the conditions which permit the parameters m 2 5 , m 2 10 to be eliminated in favor of the VEVs.
After the breakdown of SU (5) ⊗ U (1) X to SU (3) C ⊗ SU (2) L ⊗ U (1) Y , the charged vector bosons X µ associated with the broken generators acquire masses m X given by Eq. (3). The scalar states T and D of relevance to the generation of the RH neutrino masses mix, with the mass matrix (in the basis (D † , T )) where Eq. (B3) with v = 0 has been used to eliminate m 2 10 . This is diagonalized by a unitary matrix U ∆ according to which, in the electroweak vacuum, simplifies into (B7) The elements of the mixing matrix U ∆ read (B8)

Model with two 5H representations
In the minimal realistic model with two 5 H representations, we take the tree-level scalar potential to be given by The field basis is again chosen such that the fields 10 H and 5 H acquire non-zero VEVs given by Eq. (B2), while The corresponding conditions that must hold for this to be a stationary point of the potential are where In deriving the above, and in all expressions below, we restrict our attention to the case where all couplings are real.
In the SU (3) C ⊗SU (2) L ⊗U (1) Y symmetric phase, i.e., for V G = 0, v = v = 0, the set of scalar color triplets that mix is extended to include the color triplet T associated with 5 H . The 3 × 3 mass matrix, in the basis (D † , T, T ), where Eq. (B14) with v = v = 0 has been used to eliminate the dependence on m 2 10 . The resulting mass eigenstates (∆ 1 , ∆ 2 , ∆ 3 ) are obtained through the rotation where the unitary matrix U ∆ diagonalizes M 2 ∆ according to

Appendix C: Radiative fermion mass generation
In general, the physical mass of a single spin-1/2 fermion is obtained as the value of m for which (/ k + m)Γ (2) where Γ (2) (k) is the renormalized two-point 1PI Green's function, In this expression, Z(k) corresponds to the wavefunction renormalization and Σ(0) is the zero incoming momentum contribution to the appropriate sum of Feynman diagrams. Taken together, Eq. (C1) and Eq. (C2) imply that which generally amounts to a transcendental equation to be solved for the physical mass m. An expression for m may be obtained perturbatively by writing Z(m 2 ) = 1 + ∆Z(m 2 ), Σ(0) = m 0 + ∆m 0 , where the first and second term in each expression correspond to the treelevel and loop corrections to each quantity, respectively. One finds the result where we show only the leading part of the higher-order contribution. Therefore, in the general case with m 0 = 0, a calculation of the leading higher-order contribution to the physical mass would require the evaluation of the loop corrections to both Σ(0) and Z(k 2 ).
However, for the case studied in this article in which the RH neutrinos are massless at tree-level, Eq. (C4) reads simply m = ∆m 0 = Σ(0) at leading order.

Veltman-Van der Bij brackets
Remarkably enough, there is an entire industry concerning the evaluation methods for the zero-externalmomentum two-point 1PI graphs, see, e.g., Ref. [18] or Ref. [23] and references therein.
The principal object in these methods are the so-called Veltman-Van der Bij brackets. As the original paper uses an Euclidean metric and a different choice of dimensional regularization parameter , we give here all of the relevant expressions in our particular convention, i.e., in Minkowski metric g = diag(1, −1, −1, −1) and with the number of spacetime dimensions equal to D = 4 − 2 .
We introduce the brackets in the following way With the last expression we have introduced a shorthand notation that simplifies the form of this appendix.
Note that the brackets are invariant under the exchange of positions of the individual groups of components, which can be obtained by the change of variables (p ↔ q) and (p + q → p, −q → q).
By a partial cancellation of fractions we can derive various reduction formulae of the type which are dimensionless (cf. Ref. [18]). The operation transcribing simple brackets into double brackets is 't Hooft's p-operation [28]. In our notation it reads (D9)

Topology 1
Topology 1 of FIG. 1 leads to the kinematic form (i.e., neglecting the specific form of the vertices) of the integral given in Eq. (12). By using D-dimensional gamma matrix gymnastics, it can be simplified into The slashed product can be rewritten into / p / q = p · q − ip µ σ µν q ν . After performing the p integration the second term would have to be of the form iq µ σ µν q ν and, due to the antisymmetry of σ µν , such a term will not contribute. After the operations given above, we obtain This may be rewritten in terms of the simple brackets using relations similar to those in Eq. (D6).

Topology 2
Neglecting the specific form of the vertices, Topology 2 of FIG. 1 leads to the second integral in Eq. (12). It can be simplified into (again making use of the antisymmetry of σ µν ) The result after simplification reads (D13)

Integrals
For the reader's convenience, we list here the results of the integrals appearing in the expressions in our convention. As integrals A 0 (M 2 A ) appear in the results in the second power, we need to evaluate also the term linear in . This gives where with Q being the renormalization scale and γ the Euler-Mascheroni constant.
As was already stated, all of the simple brackets can be obtained from the double brackets using Eq. (D9). Therefore, we give here the result only for them. It reads and the function f (a, b) is given by