A note on dimension-5 operators in GUTs and their impact

Quantum gravity or string compactification can lead to effective dimension-5 operators in Grand Unified Theories which modify the gauge kinetic terms. We exhaustively discuss the group-theoretic nature of such operators for the popular SU(5), SO(10), and E(6) models. In particular, for SU(5) only a Higgs in the 200 representation can help bring the couplings to unification below the Planck scale and in consistency with proton decay limits while for a supersymmetric version 24, 75, or 200 representations are all acceptable. The results also have a direct application in non-universality of gaugino masses in a class of supersymmetric models where identical group-theoretic features obtain.


I Introduction
The remarkable success of electroweak unification has been a motivation to seek a Grand Unified Theory (GUT) linking together the strong and electroweak interactions in a framework with quarklepton unification [1]. The merits of this programme need no underscoring and rightfully it has been attracting continual attention over several decades. It has all along been also realised that this is but the penultimate step, unification of all interactions with gravity being the final objective.
The Standard Model (SM) is based on the gauge group G SM ≡ SU(3) c ⊗ SU(2) L ⊗ U(1) Y which has three independent couplings g 3 , g 2 , and g 1 . The minimal scheme of grand unification envisions the placement of quarks and leptons in a common multiplet of the GUT group and the unification of the three SM couplings into one unified coupling g GU T at high energies. The couplings evolve logarithmically with energy and so the unification, if achieved, is at a high scale of O (10 15 ) GeV or more. The current low energy measured values of the couplings, in fact, are not consistent with unification in the minimal model. TeV-scale Supersymmetry (SUSY) is one much-discussed remedy for this. Another consequence of grand unification is the instability of the proton. Proton decay has so far not been experimentally observed, implying a scale of grand unification at least an order of magnitude higher than 10 15 GeV.
A full quantum-theoretic treatment of gravity is not available currently. Nonetheless, it has been found useful to attempt to mimic some of its implications on grand unification through higher dimension effective contributions, suppressed by powers of the Planck mass, M P l . In a string theory setting, similar effective operators may also originate from string compactification, M P l being then replaced by the compactification scale M c .
In this work we focus on the corrections to the gauge kinetic term: where F µν = Σ i λ i .F µν i is the gauge field strength tensor with λ i being the matrix representations of the generators normalised to T r(λ i λ j ) = c δ ij . Conventionally, for SU(n) groups the λ i are chosen in the fundamental representation with c = 1/2. In the following, we will often find it convenient to utilise other representations.
The lowest order contribution from quantum gravitational (or string compactification) effects, which is what we wish to consider here, is of dimension five and has the form: where Φ D denotes the D-component Higgs multiplet which breaks the GUT symmetry and η parametrises the strength of this interaction. In order for it to be possible to form a gauge invariant of the form in eq. 2, Φ D can be in any representation included in the symmetric product of two adjoint representations of the group. For example, in SU (5), (24 ⊗ 24) sym = 1 ⊕ 24 ⊕ 75 ⊕ 200 and Φ D may be in these representations only.
When Φ D develops a vacuum expectation value (vev) v D , which sets the scale of grand unification M X , an effective gauge kinetic term is generated from eq. 2. Depending on the structure of the vev, this additional contribution usually will not be the same for the different subgroups in G SM , leading, after an appropriate scaling of the gauge fields, to an alteration of the gauge coupling unification condition to: wherein the δ i , i = 1, 2, 3, and ǫ = ηv D /2M P l ∼ O(M X /M P l ) arise from eq. 2. Thus, the presence of the dimension-5 terms in the Lagrangian modifies the usual boundary conditions on gauge couplings where they are expected to unify at M X . It is indeed possible that this change will be just enough to entail the unification programme to succeed with the current low energy values of the coupling constants as a boundary condition. Here, we show that this is the case for SU(5) GUT.
In this work we work out the consequences of such dimension-5 operators for the unified theories based on SU(5), SO (10), and E (6). For ordinary SU(5) as well as its SUSY variant, using one-and two-loop renormalisation group evolution of the gauge couplings we examine the consistency of the low energy measurements with grand unification while remaining within the proton decay restrictions. We also make a brief remark about the applicability of these results to non-universality of gaugino masses in a class of SUSY models. Table 1: Effective contributions to gauge kinetic terms from different Higgs representations in eq. 2 for SU (5). (see eq. 3.)
III sin 2 θ W , gauge couplings The shift in the gauge couplings as dictated by eq. 3 leaves its mark at low energies through the Renormalization Group (RG) equations. Moreover, besides the low energy value of the weak mixing angle sin 2 θ W , even its GUT-level prediction is affected.
The weak mixing angle sin 2 θ W = g ′2 /(g 2 + g ′2 ) is expressed in terms of the SM gauge couplings g ′ and g ≡ g 2 . Of these, the U(1) Y coupling g ′ is related to the coupling g 1 arising in a unified theory through g 2 1 = c 2 g ′2 where c 2 = 5 3 . In the limit of unification of all couplings at a GUT-scale, M X , this leads to the prediction sin 2 θ W (M X ) = 3/8. Now, due to the modified GUT relationship of eq. 3 one has for the weak mixing angleθ W : The experimentally determined value of sin 2 θ W at low energies receives further RG-dependent corrections to which we now turn.
The RG equations governing gauge coupling evolution are: where at two-loop order [3] β i, j = 1,2,3 for U(1) Y , SU(2) L , and SU(3) c , respectively. The coefficients b i and b ij are: and n H and n G are respectively the number of Higgs doublets and the number of fermion generations in the theory. The RG equations must satisfy the boundary conditions set by eq. 3 on the g 2 i (M X ).
In our numerical analyses below we show the full two-loop RG equation results. For ease of discussion if only the lowest order contributions are retained, then in the absence of dimension-5 operators α i = g 2 i /4π. These equations can be combined to yield: SU (5) ǫ ǫ M X (GeV) Representations (from eq. 17) (using eq .
where α -the fine structure constant -and α 3 are the couplings at the scale M Z .
Inclusion of the boundary condition, eq. 3, dictated by the dimension-5 interactions alters eqs. 13 and 14 to: which reduces to eq. 14 in the appropriate limit. In fact, The first term on the r.h.s. of eq. 17 is fixed from eq. 14. From the measured value of sin 2 θ W [4] one can extract the value of ǫ. These are presented for the different Φ representations in Table 2.
These O(ǫ) one-loop analytic results can be cross-checked using the full RG equations in eq. 8 with n G = 3 and n H = 1. Using the low energy (∼ M Z ) measured values [4], sin 2 θ W = 0.231 19 (14) and α 3 = 0.11 76 (20), the RG equations can be numerically integrated. The scale M X is fixed through the requirement that the modified unification condition, eq. 3, is satisfied there. From this analysis one can determine ǫ and M X . The conclusions from one-loop RG running are shown in Table 2 and Fig. 1.
The two-loop results, shown as insets in Fig. 1, incorporate the proper matching conditions [5] as well as eq. 3 at M X , namely, where C i is the quadratic Casimir for the i-th subgroup. It is noteworthy that the results are not significantly affected and the coupling constants still unify. The unification scales, M X , are found to be 5.01 ×10  In [6], noting that the dimension-5 operator in eq. 2 with Φ 24 cannot by itself provide satisfactory gauge unification, it has been proposed that including gravitational contributions in the beta functions can help ameliorate this problem. Alternatively, within SUSY SU (5) it has been argued in [7] that one-loop (as well as two-loop) RG evolution with Φ 24 -driven boundary conditions in eq. 3 can yield satisfactory unification solutions provided the possible modification of the Planck scale itself due to the large number of GUT fields is given consideration.  IV SO(10) and E (6) SU (5) is admittedly the GUT model which has been examined most in the literature by far. Nonetheless, there are also strong motivations for going to larger GUT groups like SO(10) and E (6). The minimal SU(5) model, as noted earlier, is in conflict with proton decay measurements. Further, the larger gauge groups offer the possibility of left-right symmetry. Here, we briefly discuss the effect of dimension-5 interactions, as in eq. 2, on SO(10) and E(6) models. The detailed analysis of RG evolution of the gauge couplings for these cases is not presented in this paper [8].

IV.1 SO(10) GUT
SO(10) [9] is now the widely preferred model for grand unification, offering the option of descending to G SM through a left-right symmetric route [10] -the intermediate Pati-Salam G P S ≡ SU(4) c ⊗ SU(2) L ⊗ SU(2) R group -or via an SU(5) ⊗ U(1) theory. The latter option will not give any novel feature beyond what has already been discussed in the context of SU(5) GUTs. So, here we consider the breaking chain SO(10) → G P S → G SM . Table 3: Effective contributions to gauge kinetic terms from different Higgs representations in eq. 2 for SO (10). (see eq. 3.)

SO(10) Representations
In SO (10), one usually requires g 4c = g 2L = g 2R at the unification scale. The presence of any dimension-5 effective interactions, of the form of eq. 2, will affect this relation introducing corrections such that The adjoint representation 45 of SO(10) satisfies (45 ⊗ 45) sym = 1 ⊕ 54 ⊕ 210 ⊕ 770 implying that Φ D can be chosen only in the 54, 210, and 770-dimensional representations. The vev < Φ D > must ensure that G P S is unbroken.

IV.2 E(6) GUT
The exceptional group E(6) has been proposed as a viable GUT alternative [13]. It offers SU (3) (3) R . Needless to say, any one of these fields or linear combinations thereof may be chosen to break the symmetry. In particular, two linear combinations may be identified which respect δ 3L = ±δ 3R . These are: One can readily read off δ 3c = δ 3L = δ 3R = − 3 √ 26 . In Table 4 we collect the findings for the different representations of E(6).  In a supersymmetric theory, the fermions, Higgs scalars, and gauge bosons of the GUT model are endowed with superpartners constituting chiral and vector supermultiplets. It is well known that gauge coupling unification is possible if SUSY is manifested at the TeV scale [14]. If dimension-5 interactions are also present then that will further affect this unification. In fact, it was shown within SUSY SU (5) that if the δ i (i = 1, 2, 3) in eq. 3 are fixed as determined (see Table 1) by the 24dimensional representation [15] or permitted to vary arbitrarily [16] then unification, at the one-loop level, is always possible.
Here we perform a one-loop as well as a two-loop analysis. Above the SUSY scale (chosen as 1 TeV) this entails the replacement of eqs. 10 and 11 by (for n G = 3, n H = 2) We find that unification is allowed for all three choices of Φ -namely, 24, 75, and 200 -when the δ i (i = 1, 2, 3) are appropriately identified. The results are presented in Table 5. Unlike the non-SUSY alternative in Table 2, now for every case one gets M X ∼ 10 16 GeV. In line with expectation, the size of ǫ is reduced in this SUSY case as the couplings tend to unify even without these interactions. The trend of agreement between the one-loop and two-loop results is gratifying.
The degenerate fermionic SUSY partner of Φ might play a role in the generation of realistic neutrino masses through the see-saw mechanism. Continuing with the SU(5) model for the purpose of illustration, if Φ is in the 24 representation then its fermion partner multiplet contains fields transforming as (1,1,0) and (1,3,0) under SU(3) c ⊗SU(2) L ⊗U(1) Y . Through a Yukawa coupling of the form 5 F 24 F 5 H they can serve as the heavy exchanged fermionic mode of type I [17] and type III [18] see-saw models, respectively 1 . These fermions reside at the M X scale and will lead to neutrino masses typically at the 10 −2 − 10 −3 eV level. In fact, the above Yukawa coupling will result in simultaneous type I and type III see-saw contributions which together will lead to an enhancement. A similar situation obtains if Φ is chosen in the 200 representation. In the 75 of SU(5) the SU(3) c ⊗SU(2) L ⊗U(1) Y (1,3,0) field is not present. So only type I see-saw is feasible for this alternative. For SO (10) and E(6) SUSY GUTs, over and above these options, there is even room for the type II see-saw [19] through heavy scalars transforming as (1,3,0).

V.2 Non-universality of gaugino masses
The results presented in the earlier sections have a direct application -specifically in the generation of non-universal gaugino masses -in SUSY models which emerge from a GUT where supersymmetry breaking and GUT symmetry breaking are tied together. These arise from the gauge kinetic term which can be schematically written as where f αβ is a function of the chiral superfield Φ whose scalar component is responsible for the GUT symmetry breaking. f αβ is symmetric in the GUT gauge indices α, β and W α are the GUT gauge superfields. When the F-component of the chiral superfield, F Φ , gets a non-zero vev at the GUT scale, the gauginos λ α develop an effective mass term where M is the mass scale of the GUT symmetry breaking. This scenario, including detailed phenomenological implications, has been widely discussed in the context of SUSY SU(5) [15,16]. Some cases have also been examined for SUSY SO(10) [20].
It is obvious that the group-theoretic structures of eqs. 2 and 29 are identical. Hence the results discussed in the previous sections can be taken over mutatis mutandis. For example, for SU (5) This is to be compared with eq. 3. The δ 1,2,3 can be found in Table 1. The non-universality of gaugino masses has far-reaching implications for SUSY phenomenology [21].
The deviation from universality for different Higgs representations is controlled by the δ 3c,3L,3R which are listed in Table 4.

VI Conclusions
Non-perturbative effects arising from quantum gravity or string compactification can be mimicked through higher dimensional non-renormalisable interactions. We have considered one class of such dimension-5 interactions which modify the unification condition of gauge coupling constants in grand unified theories. For SU (5), SO (10), and E(6) GUTs we have exhaustively worked out their implications for gauge coupling unification conditions. For the case of SU (5), we have shown how low energy physics can constrain these interaction strengths and the manner in which this would affect the unification scale. We also briefly examine the status of gauge unification in an SU(5) SUSY GUT model with the dimension-5 interactions and discuss a possible route to generate realistic neutrino masses through the see-saw mechanism. A corollary of the exercise is an application to SUSY models where non-universal gaugino mass relations obtain from a similar group-theoretic structure.