due to singlet VEVs (or instantons) some additional suppression is expected when the couplings arise from the Kähler potential. This fact is particularly appealing for the generation of theµ-term as it can be sufficiently small, if induced from the Kähler potential along the lines of the so-called Giudice-Masiero (GM) mechanism [114].
In order toavoid fast proton decay, the U(1) symmetries must also forbid the following superpoten-tial and Kähler potensuperpoten-tial couplings:
βi 5i5Hu ⊃βiLiHu,
λi jk5i5j10k ⊃λ0i jkLiLje¯k+λ1i jkd¯iLjQk+λ2i jkd¯id¯ju¯k, δi jk10i10j10k5Hd ⊃δ1i jkQiQjQkHd+δ1i jkQiu¯je¯kHd, γi 5i5Hd5Hu5Hu ⊃γiLiHdHuHu,
κi jk10i10j5k ⊃κi jk1 Qiu¯jL¯k+κ2i jke¯iu¯jdk+κi jk3 QiQjdk, κi 5Hu5Hd10i ⊃κ1iH∗uHde¯i.
(7.23)
For a consistent model we need to require that upon breakdown of the U(1) symmetries these operators are not generated. This is for example achieved by demanding the presence of an effective matter parity symmetry.
At next we have to decompose the dimension five proton decay inducing operators and check for their absence individually:
ωi jkl10i10j10k5l ⊃ω1i jklQiQjQkLl+ω2i jklu¯iu¯je¯kd¯l+ω3i jklQiu¯je¯kLl. (7.24) We demand these operators to be forbidden by the U(1) charges, keeping in mind that they can be generated in a similar fashion as theµ-term. We also expect that while the operators in (7.23) remain absent, it is possible to generatefull rank Yukawa matrices3
W ⊂Yi,ujQiujHu+Yi,jduidjHd+Yi,jLeiLjHd. (7.25) This necessarily implies that the charges of the desired operators must differ in comparison to the un-desired ones. As a consequence, one observes that the fieldHd has to come from a different curve than all the other leptons and triplets to guarantee that dimension four operators (such asλ0andλ1in (7.23)) are not introduced together with the Yukawa entries.
7.4 Results of the scan
We split up the general results into the models within the spectral cover and those obtained from thetops.
Within the spectral cover models we find that the flux restrictions of having only the MSSM spectrum an no other leptons coming from theHdcurve restricts the overall amount of models to only six models, all descending from the 2+2+1 split. All these models have phenomenological unappealing features. We give one example model with the flux restrictions on the curves given in Table7.1. This model suffers of the following structure of operator charges:
q(Q2u¯2Hu)=q(HuHd)=q(Q1d¯1Li)=q( ¯u1u¯1d¯k)=(−5,−5). (7.26)
3Recall that, as the matter fields need not to arise from complete SU(5) multiplets, so that the down and lepton Yukawas do not necessarily coincide.
Curve q1 q2 M N Matter 101 1 5 2 1 Q1,2+u¯1+e¯1,2,3 105 −4 0 1 -1 Q3+u¯2,3
511 2 10 0 -1 Hcu
515 −3 5 0 1 Hd
535 −3 −5 3 0 (L+d)¯1,2,3
Table 7.1: Spectral cover model and its corresponding flux quanta along the different matter curves in the 2+2+1 splitting. The lower indices of the matter representations are family indices.
It is a notorious feature of these models that operators of competing relevance have the same charge. As we do not know of any mechanism at this point to generate a hierarchy among the above operators, we have to assume that the Yukawa coupling is generated at the same strength a the dimension four proton decay inducing operator. Even worse, we find that theµterm is generated as well.
At next we focus on the models obtained from the SU(5)topsgiven in table5.6 and repeat the same strategy. Before we present our findings we focus on the particular feature that we have only one10 curve in these models. It follows from (7.11) and (7.12) that there can be no flux put along these curves and the whole SU(5) structure is maintained for these curves. Furthermore, all three families come from the same10curve and hence have the same U(1) charge.
This structure allows us to relate the charges of all relevant operators presented above: First of all, the presence of a tree level top Yukawa fixes theHucharge to be
q(Hu)=−2q(10). (7.27)
For the subsequent discussion, we introduce the following notation for the charges of the operators:
µ : q(HdHu) =q(Hd)+q(Hu) :=qµ, YL : q(¯eHdLi) :=qYiL,
Yd : q( ¯uHdd¯i) :=qYid,
βi : q(LiHu) =q(Li)+q(Hu) :=qβj.
(7.28)
Among these operators, all but theβiterms should be induced upon breakdown of the U(1) symmetries.
Now we can express the charges of all unwanted operators in terms of the charges defined above. The dangerous dimension four proton decay operators are:
λ0i j : q(Qd¯iLj)=qYid +q(Hd)−q(Lj)=qYid +qµ−qβj λ1i j : q(¯eLiLj) =qYiL+q(Hd)−q(Lj)=qYiL+qµ−qβj
λ2i j : q( ¯ud¯id¯j) =qYid+q(Hd)−q( ¯dj) =qYid +qµ−q(Hu)−q( ¯dj).
(7.29)
Since we want to generate the down-type Yukawa matrices, we see that the previous couplings are only forbidden due to the charge difference between the Hd- andLj-curves in the case of the λ0i j and λ1i j couplings, and due to the charge difference betweenHd- and ¯dj-curves in the case of the λ2i j. Thus, as already pointed out, it is necessary that the5Hd-curve contains only the down-type Higgs, since any lepton or down-type quark with identical charge will automatically induce a dangerous operator. As we
7.4 Results of the scan
want to induce theµ-term as well we observe that no ¯difield can arise from theHu-curve either.4 In general we can write the charges in terms of theβi operators that clarifies the overall structure even more. Thus, if we find a configuration such that the Yukawa couplings and theµ-term is induced but the βi-terms stay forbidden, the dimension four operators stay forbidden as well. Furthermore, we observe that the dimension five operators in the superpotential
ω1i , ω3i : q(QQQLi)=q(Q¯u¯eLi)=−qµ+qYiL :=q(10 10 10Li),
ω2i : q(QQu¯d¯i) = q( ¯uu¯¯ed¯i) =−qµ+qYid :=q(10 10 10d¯i), (7.30) will unavoidably be induced together with the Yukawa couplings and theµ-term. It should be noted that theµ-term charge enters with a minus sign in the previous equations. This implies that the mechanism (such as a singlet VEV) which induces the ωi-terms in the superpotential will not induce the µ-term directly in the superpotential but can generate it from the Kähler potential after SUSY breakdown. Note also that it is possible to induce a Weinberg operator
Wi j: q(LiLjHuHu)=qβi +qβj (7.31) without inducing the βi-terms by using, for example, singlet VEVs with charge q(si) = −2qβi which implies at least a remnantZ2N symmetry in the VEV configuration.
In a similar fashion, we observe that the operators
δ1, δ2: q(QQQHd) =q(Q¯u¯eHd)=−qβi +qYiL,
γi: q(LiHdHuHu)=qµ+qβi, (7.32) will remain absent as long as the βi-terms are not induced. The same holds for the Kähler potential terms
κ1i : q(Q¯uL∗i) =−qβi,
κ: q(¯eHu∗Hd)=qµ+qβi, (7.33) with the exception of
κ2i , κi3: q(QQd¯∗i)=q( ¯ue¯d¯∗i)=−q(Hu)−q( ¯dj)=−qµ+q(Hd)−q( ¯di) :=q(10 10d¯i∗), (7.34) for which one has to ensure that no triplets emerge from the Higgs curves as a necessary (but not sufficient) condition. Note that the above observations are independent of the number of 5-curves and U(1) symmetries. However, there remains a crucial interplay between the Higgs charges compared to those of the down-type quarks and those of the singlet fields which have to be checked on a case by case analysis. Within the four tops and for all flux configurations, we find only four models that are based on τ5,1 andτ5,2 that meet all our phenomenological criteria. We summarize the details of one model that we call Benchmark model A in Table 7.2. There we summarize spectrum as well as the charge of all relevant operators. We indeed see, that many operators are forbidden as long as the βi terms are not induced. We find, that the singlet s1 induces theµ-term from the Kähler potential and induces all Yukawa couplings but also dimension five proton decay inducing operators, while dimension four
4Note that ifHuand ¯djcome from the same curve their U(1) charges carry opposite signs.
1. Spectrum 2. Singlet VEVs: s1,a
q(s1)=(0,−5), q(a)=(10,0) .
Curve q1 q2 M N Matter
10 −1 2 3 0 (Q+u¯+e)¯ 1,2,3
51 3 −1 1 −1 d¯1 3.µ- andβi-terms
52 −2 4 0 −1 Hu
q(HuL¯i)=(5,0), q(HuHd)=(0,−5).
54 3 4 2 1 L1,2,3+d¯2,3
55 −2 −1 0 1 Hd
4. Yukawa couplings
q(Qiu¯jHu)=(0,0), q(Qid¯jHd)=
(0,0) (0,5) (0,5)
j
, q(¯eiLjHd)=(0,5).
5. Allowed dimension five proton decay and Weinberg operators
q(10 10 10Li)=(0,10), q(10 10 10d¯i)=
(0,5) (0,10) (0,10)
j
, q(LiLjHuHu)=(10,0).
6. Forbidden operators
q( ¯ud¯id¯j)=
(5,0) (5,0) (5,0) (5,0) (5,10) (5,10) (5,0) (5,10) (5,10)
i,j
, q(10 10d¯i∗)=
(−5,5) (−5,0) (−5,0)
.
Table 7.2: Details of benchmark model A. We give the charges of all relevant operators and VEV fields.