aa r X i v : . [ h e p - ph ] D ec September 13, 2011
New Flavor U (1) F Symmetry for SUSY SU (5) Zurab Tavartkiladze Center for Elementary Particle Physics, ITP, Ilia State University, 0162 Tbilisi, Georgia
Abstract
Within supersymmetric SU (5) Grand Unified Theory, we present several new scenarioswith anomaly free flavor symmetry U (1) F . Within each scenario, a variety of cases offer manypossibilities for phenomenologically interesting model building. We present three concrete andeconomical models with anomaly free U (1) F leading to natural understanding of observedhierarchies between charged fermion masses and CKM mixing angles. Noticeable hierarchies between charged fermion masses and mixings remain unexplained within theStandard Model and its minimal supersymmetric (SUSY) extension. Grand Unification (GUT) [1]gives some interesting asymptotic mass relations (like m b = m τ within SU (5) GUT), but problemof flavor still remains unresolved. It may well be that the resolution of this puzzle has somephysical origin, and a nice idea is existence of the flavor symmetry acting between different flavorsof quarks and leptons. The simplest possibility is the Abelian U (1) F flavor symmetry [2], which hasbeen extensively investigated with U (1) F being an anomalous symmetry [3] of a stringy origin [4].Some attempts to find anomaly free setup with U (1) F symmetry, for explanation of fermion masshierarchies, also exist in a literature [5, 6]. With anomaly free U (1) F , without relying on somespecific string construction, one can investigate a given scenario (within MSSM [5] or GUT [6])based on conventional field theoretical arguments.In this Letter within SUSY SU (5) GUT, we suggest new way of finding non-anomalous U (1) F flavor symmetries. We present several scenarios with anomaly free U (1) F symmetries, which providenatural explanation of hierarchies between charged fermion masses and mixings.The Letter is organized as follows. In the next section we pursue the way of finding non-anomalous U (1) F symmetries by possible embedding of SU (5) × U (1) F into the anomaly freenon-Abelian symmetry, and present our findings. In section 3, after listing requirements which we E-mail: [email protected] member of Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia. θ µτ turns out to be naturally large, giving good background for building promising scenarios forneutrino masses and mixings. Our conclusions are given in Sect. 4, while in Appendix A we discussthe possibility of consistent U (1) F symmetry breaking, needed for realistic model building. SU (5) and Non-Anomalous Flavor U (1) F As already noted, we are working within the framework of SUSY SU (5) GUT and looking fornon-anomalous flavor U (1) F symmetry. Minimal chiral content for the fermion sector consiststo 10 + ¯5 multiplets per generation, whose SM matter composition and quantum numbers under SU (3) C × SU (2) L × U (1) Y ≡ G gauge group is10 = q (3 , , − √
60 ) + u c (¯3 , , √
60 ) + e c (1 , , − √
60 ) , ¯5 = d c (¯3 , , − √
60 ) + l (1 , , √
60 ) . (1)Last entries in the brackets represent corresponding hypercharge Y with SU (5) normalization (beinggenerator of SU (5), the Y has the form Y = √ Diag(2 , , , − , − U (1) F charge assignments which are anomaly free. Clearly, for one of the families,out of three, the simplest assignment 10 + ¯5 (subscripts indicate U (1) F charges) is anomaly free.The non-zero charge assignment would require addition of new states which can be SU (5) singletscharged under U (1) F . We will look for extensions with minimal possible content. By minimalcontent we mean that non-trivial SU (5) representations, which we introduce, will be just those ofminimal SUSY SU (5) GUT. These are three families of matter (10 + ¯5)-supermultiplets, one pair ofHiggs superfields H (5) + ¯ H (¯5) (including MSSM Higgs doublet superfields h u and h d respectively),and an adjoint Σ(24) of SU (5) (needed for symmetry breaking SU (5) → G ). We assume thatΣ has no U (1) F charge, Q (Σ) = 0, and thus does not contribute to anomalies. Thus SU (5) stateswhich may contribute to anomalies are three 10 i -plets ( i = 1 , , k -plets ( k = 1 , , ,
4) andone 5-plet. With this set, the SU (5) anomaly A = 3 A (10) + 4 A (¯5) + A (5) = 0 vanishes because A (10) = − A (¯5) = A (5). As far as the anomalies SU (5) · U (1) F , ( U (1) F ) and (Gravity) · U (1) F are concerned, they should satisfy the following conditions: SU (5) · U (1) F : A = 32 X i =1 Q (10 i ) + 12 X k =1 Q (¯5 k ) + Q (5) ! = 0 , (2)( U (1) F ) : A = 10 X i =1 Q (10 i ) + 5 X k =1 Q (¯5 k ) + Q (5) ! + X s Q s = 0 , (3)(Gravity) · U (1) F : A GG = Tr Q = 10 X i =1 Q (10 i ) + 5 X k =1 Q (¯5 k ) + Q (5) ! + X s Q s = 0 , (4)where Q s denotes U (1) F charges of SU (5) singlet states. Upon finding the anomaly free assignmentswe will limit ourself with scenarios involving small number of singlets. All other mixed anomaliesvanish due to properties of SU (5) generators. Unless U (1) F charge assignments are such that anomalies coming from different families cancel each other. U (1) F is Abelian symmetry, there is no overall normalization for the charges. However,in order to have realistic phenomenology, all charge ratios should be rational numbers; i.e. in theunit of one field’s charge, all remaining states’ charges should be rational numbers. To find suchanomaly free charge assignment, leading to desirable phenomenology, one way is to find solution(s)of system of Eqs. (2)-(4) in a straightforward way [5], [6]. Another way might be to extract U (1) F (as a subgroup) from anomaly free non-Abelian flavor symmetries [7] (which are compatible with SU (5) GUT). Different, and unexplored yet, way of finding is to embed SU (5) × U (1) F (as asubgroup) in higher non-Abelian symmetries with anomaly free content. In this work, we followthe latter way in order to find anomaly free flavor U (1) F symmetries within SUSY SU (5) GUT.For this purpose, consider higher gauge symmetries containing SU (5) as their subgroups plus U (1) factors. Clearly, the rank of such non-Abelian groups should be ≥
5. Since all states willbelong to non-Abelian groups, the condition Tr Q = 0 of Eq. (4) will be automatically satisfied.However, vanishing of other anomalies will require specific selection of the field content [8]. Onesimple possibility emerges via SO (10) group which has a maximal subgroup is SU (5) × U (1) ′ . SO (10)’s spinorial representation - the 16-plet - decomposes under the SU (5) × U (1) ′ as [9]16 = 10 + ¯5 − + 1 , (5)where subscripts are U (1) ′ charges which can be identified with U (1) F charges. In this way, U (1) F is anomaly free since all anomalies [ SU (5) , SU (5) · U (1) F , etc.] vanish. The SU (5)’s singlet 1 ,charged under U (1) F , plays important role for anomaly cancellation.For finding another assignment let us consider 27-plet of E group (the rank six exceptionalgroup). With E → SO (10) × U (1) ′′ → SU (5) × U (1) ′′ decomposition we have [9]27 = 16 + 10 − + 1 ′ = (10 + ¯5 + 1) + (5 + ¯5 ′ ) − + 1 ′ , (6)where subscripts denote U (1) ′′ charges. In this case U (1) ′′ can be identified with U (1) F . We seethat, in this case anomaly cansellation requires two SU (5) singlets (charged under U (1) F ) and extracharged 5 , ¯5 plets of SU (5).Each anomaly free content (5) and (6), we presented so far, includes one 10-plet of SU (5). Thishappened because of simple and single anomaly free SO (10) and E representations 16 and 27-pletsrespectively. Another (higher) representations might give more 10-plets. Since those higher stateswould also involve extra exotic states, we do not consider such possibilities here.As far as the unitary groups with rank greater than five, we start discussion with SU (7). Lowergroup SU (6) is subgroup of E which was already considered above (detailed comment about this isgiven at the end of this section). As it will turn out, the SU (7) group can give an interesting anomalyfree field content. Consider SU (7)’s one particular set of chiral representations 35 + 2 × ¯7, which isanomaly free. Here 35 is three index antisymmetric representation and ¯7 is an anti-fundamentalof SU (7). Their decomposition via the chain SU (7) → SU (6) × U (1) → SU (5) × U (1) × U (1) is35 = 20 + 15 − = (10 − + 10 ) + (10 + 5 − ) − , ¯7 = ¯6 − + 1 = (¯5 − + 1 ) − + (1 ) , (7) We omit normalization factor, which is not essential here. Other SU (7)’s anomaly free chiral sets like 21 + 3 × ¯7 and 21 + 35 + ¯7 etc., involve either too many SU (5)singlets, or unwanted SU (5) states and thus will not be considered here. U (1) and U (1) charges respectively are indicated assubscripts. Note that U (1) and U (1) are coming from SU (6) and SU (7) respectively. Their corre-sponding generators are Y U (1) = √ Diag (1 , , , , , −
5) and Y U (1) = √ Diag (1 , , , , , , − √ and √ are omitted in Eq. (7). Note that set ofEq. (7) includes SU (5)’s 10-plet, which we did not intend to introduce. However, there is oneloophole which helps in this situation. Since consideration of SU (7) symmetry was just the wayof finding the anomaly free U (1) F , we will consider SU (5) × U (1) F gauge symmetry, important isthat SU (5) and anomalies of Eqs. (2)-(4) vanish. So, if the pair of (10 + 5)-plets is replaced by(10 + ¯5) then SU (5) anomaly will not be changed (i.e. will still vanish). With this substitution10 →
10, 5 → ¯5, without changing the U (1)-charges, the mixed and cubic anomalies (2)-(4) willremain intact. Therefore, we will consider the following content(10 − + 10 ) + (10 + ¯5 − ) − + 2 × [(¯5 − + 1 ) − + (1 ) ] , (8)which involves three pairs (three families!) of (10 + ¯5)-plets.Higher rank gauge groups SU ( N > , SO ( N > , E , E etc. with corresponding anomalyfree representations will give extra (unwanted) non-trivial representations of SU (5) and we do notconsider them. Therefore, we will use the sets (5), (6) and (8) in our further studies for modelbuilding.With Abelian symmetries U (1) ′ , U (1) ′′ , U (1) and U (1) various linear superpositions can beconstructed. Starting with U (1) ′ and U (1) ′′ , which respectively transform the sets given in Eqs.(5) and (6), let us consider the superposition Q sup = aQ U (1) ′′ + bQ U (1) ′ . (9)In order to construct such a superposition, the content of (5) should be extended with extra singlet1 ′ and 5 , ¯5 ′ plets, with U (1) ′ charges Q U (1) ′ (1 ′ ) = 0, Q U (1) ′ (5) = 2 q , Q U (1) ′ (¯5 ′ ) = − q , where q issome number. Q sup can be identified with U (1) F (= U (1) sup ). In order that U (1) F = U (1) sup beanomaly free some constraints on a, b and q should be imposed. Simplest possibility, leading torealistic models, is to require cancellation of mixed anomalies U (1) ′ · [ U (1) ′′ ] and [ U (1) ′ ] · U (1) ′′ .If these mixed anomalies will vanish, then U (1) sup also will be anomaly free (because separately U (1) ′ and U (1) ′′ are anomaly free). One can easily make sure that with q = ± U (1) sup is anomalyfree for arbitrary values of a and b . Note that q = − SO (10) normalization, i.e. SU (5)’s multiplets 5 and ¯5 coming from the SO (10)’s fundamental 10-plet, should have charges − q = −
1. Thus, anomaly free field content is:10 a + b + ¯5 a − b + 1 a +5 b + 5 − a − b + ¯5 ′− a +2 b + 1 ′ a . (10)Similarly, from U (1) and U (1) charges of the fields given in Eq. (8) we can build superposition¯ Q sup = ¯ aQ U (1) + ¯ bQ U (1) . (11)Note that ¯ Q sup is automatically anomaly free for arbitrary ¯ a and ¯ b , because the orthogonal gener-ators Y U (1) and Y U (1) originate from single SU (7). Thus, using (8) we can write the anomaly freeset 10 − a +3¯ b + 10 a +3¯ b + 10 a − b + ¯5 − a − b + 2 × (cid:0) ¯5 − ¯ a − ¯ b + 1 a − ¯ b + 1 ′ b (cid:1) , (12)4here subscripts denote ¯ Q sup charges. These charges could be identified with charges of flavor U (1) F .Summarizing all possibilities discussed above, we can have the following options for flavor U (1) F charge assignments: A : 10 + ¯5 , (13) B : 10 α + ¯5 − α + 1 α , ( α = 0) , (14) C : 10 a + b + ¯5 a − b + 1 a +5 b + 5 − a − b + ¯5 ′− a +2 b + 1 ′ a , ( a = 0 , a = − b ) , (15) D : 10 − a +3¯ b + 10 a +3¯ b + 10 a − b + ¯5 − a − b + 2 × (cid:0) ¯5 − ¯ a − ¯ b + 1 a − ¯ b + 1 ′ b (cid:1) , (¯ b = 0) . (16)The conditions in brackets are imposed in order to avoid repetition of identical cases. For example,in case B , with α = 0 we recover case A with extra neutral SU (5) singlet. Likewise, in case C , with a = 0 or a = − b we obtain case B augmented with extra vector-like states with opposite U (1) F charges. Also, condition ¯ b = 0 for case D guarantees that we will not deal with case obtainedfrom embedding of SU (5) × U (1) F in SU (6) group. Indeed, with ¯ b = 0 together with states10 − a + 10 a + 2 × ′ (which do not contribute in SU (5) · U (1) F , ( U (1) F ) and Tr Q anomalies) weget set 10 a + ¯5 − a + 2 × (¯5 − ¯ a + 1 a ). The latter field content can be obtained via SU (6) embeddingas follows. Consider SU (6) field content 15 + 2 × ¯6 which is anomaly free. Decomposition of15 and ¯6 under SU (6) → SU (5) × U (1) is 15 = 10 + 5 − and ¯6 = ¯5 − + 1 , where for U (1) charges the normalization factor 1 / √
60 is neglected. Now making replacement 5 − → ¯5 − andadding the pair 10 − + 10 the field content will remain anomaly free. Adding to these two neutralsinglets (1 ′ taken two times) we will get the field content of D with ¯ b = 0. Note that discussingembedding of SU (5) × U (1) F in unitary groups, we skipped the SU (6) group. The reason wasthat the case C , obtained from E embedding, includes the case of SU (6) embedding. This is notsurprising since one of E ’s maximal subgroup is SU (6) × SU (2) and 27 (of E ) decomposition E → SU (6) × SU (2) is 27 = (15 ,
1) + (¯6 , C a = 5 / b = 3 / + 5 − + 2 × (¯5 − + 1 ). These are obtained by SU (6) → SU (5) × U (1) decompositionof 15 + 2 × ¯6. That’s why consideration of unitary groups has been started from SU (7).Before closing this section, let us mention that for case D , in constructing the ¯ Q sup charges,besides Q U (1) and Q U (1) , one can also use another U (1)s - either charge of U (1) ′ or U (1) ′′ , or bothtogether. However, one should make sure that superposition is such that all anomalies are zero. Forexample, use U (1) ′′ symmetry. Then instead of Eq. (11) we will have ¯ Q sup = ¯ aQ U (1) + ¯ bQ U (1) +¯ cQ U (1) ′′ . To do this, we should pick up from set D
10 + ¯5 + 1 + ¯5 + ¯5 + 1 and (according to lastequation in (6)) assign U (1) ′′ charges 1 , , , − , − , U (1) ′′ charges p and − p , while U (1) ′′ charges of remaining two singlets are k and − k . Thus, the set with (one simple possible) ¯ Q sup charge assignment will look:10 − a +3¯ b + p ¯ c + 10 a +3¯ b − p ¯ c + 10 a − b +¯ c + ¯5 − a − b − c + ¯5 − ¯ a − ¯ b +¯ c + ¯5 − ¯ a − ¯ b − c +1 a − ¯ b +¯ c + 1 a − ¯ b +4¯ c + 1 ′ b + k ¯ c + 1 ′ b − k ¯ c , with 30¯ a (3 + 2 p ) = ¯ c (2 k + 10 p − . (17)Relations between ¯ a, ¯ c, k and p (imposed for ¯ c = 0) given in (17) insures that all anomalies vanish.Clearly, with rational selection of ¯ a, k and p the value of ¯ c also will be rational. The set given inEq. (17) is one simple selection among several options and opens up many possibilities for modelbuilding with realistic phenomenology. 5 SU (5) × U (1) F Models
For model building with U (1) F symmetry we list and discuss requirements which should be satisfiedin order to obtain phenomenologically viable and economical setups. (a) In total, we should have three 10-plets of SU (5), four ¯5-plets and one 5-plet. Out of thesemultiplets three pairs of (10 + ¯5) are matter superfields (containing quark and lepton superfields,as given in Eq. (1)). The 5-plet and one remaining ¯5-plet are scalar superfields which will bedenoted by H and ¯ H respectively. (b) In order to have top quark Yukawa coupling λ t ∼
1, the U (1) F symmetry should allowcoupling 10 H at renormalizable level. At the same time, all 10-plets should have different U (1) F charges in order to generate adequately suppressed hierarchies of λ u /λ t and λ c /λ t .Since for U (1) F charge assignments we have options given in (13)-(16), for building three gen-eration models with U (1) F flavor symmetry we can consider different combinations of these assign-ments. For example, one pair of 10 , ¯5-plets can have U (1) F assignment A (of Eq. (13)), anotherpair of 10 , ¯5-plets can have assignment B and third pair of 10 , ¯5-plets can come from selection C .This collection can be refereed as ABC model. This model involves three 10-plets, four ¯5-pletsand one 5-plet (satisfying requirement (a) ). Other collections, such as
ABB , BBB , etc., are alsopossible. However, selections like
ACC , CCC , etc. are not allowed since they would involve extra5-plet(s) (not satisfying requirement (a) ). Note, considering, say,
ABB model, for two sets of 10 , ¯5coming with B charge assignments should be taken α and α ′ = α (for satisfying requirement (b) ).At the same time, for this selection extra pair of 5 , ¯5-plets should be introduced with opposite U (1) F charges. (c) Upon model building, one should make sure that only one U (1) (identified with U (1) F )emerges. For instance, if ABC model is considered, the parameters α, a, b should not be indepen-dent. They should be fixed as α = m n β , a = m n β , b = m n β ( m i , n i are integers). This would avoidextra global U (1) symmetries.Summarizing, satisfying all this requirements, we will group models in following five classes: ABB BBB DABC BBC (18)Each of these includes several possibilities. Clarification of varieties of these possibilities is in order. • Model ABB
In this case we combine sets given by Eqs. (13) and (14), and take: 10 + ¯5 , 10 α + ¯5 − α + 1 α and 10 α ′ +¯5 − α ′ +1 α ′ . In addition, we introduce the pair 5 q +¯5 − q . Thus, for this class, the completefield content is: 10 + ¯5 , α + ¯5 − α + 1 α α ′ + ¯5 − α ′ + 1 α ′ , q + ¯5 − q . (19)This selection is not unique. We can exchange 5-plet’s U (1) F charge with one of the ¯5-plets’ charge.With this, anomaly cancellation conditions are not changed. Thus, for U (1) F charge of the 5-plet,identified with Higgs superfield H (5), we have three (qualitatively different) options Q H = 0 , − α The scalar superfield Σ(24) (neutral under U (1) F ), needed for the symmetry breaking SU (5) → G , is alsoassumed. q . In counting these options, we took into account that the charge selection Q H = − α ′ doesnot differ from selection Q H = − α (former is obtained from the latter by substitution α → α ′ ).Also, the case with Q H = − q is obtained from case Q H = q by substitution q → − q . From the(remaining) four ¯5-plets one should be identified with the Higgs superfield ¯ H (¯5). For each given Q H , one should count how many qualitatively different charge assignments is possible for ¯ H . Onecan make sure that for the pair ( Q H , Q ¯ H ) eight different possibilities are allowed:( Q H , Q ¯ H ) ( i ) = { ( q, − q ) , ( q, , ( q, − α ) , (0 , q ) , (0 , − α ) , ( − α, q ) , ( − α, , ( − α, − α ′ ) } , (20)where i = 1 , , · · · , H and ¯ H .Thus, the content ABB of Eq. (19) forms class with these different charge assignments. Tomake clear which particular U (1) F charge assignment for H, ¯ H is considered, it is instructive to usenotation ABB ( i ) . For instance, ABB ( i =3) would mean that we are taking ( Q H , Q ¯ H ) ( i =3) = ( q, − α )(see Eq. (20)). • Model ABC
In this case, we collect together sets of Eqs. (13), (14) and (15). Thus, the field content is:10 + ¯5 , α + ¯5 − α + 1 α , a + b + ¯5 a − b + 1 a +5 b + 5 − a − b + ¯5 ′− a +2 b + 1 ′ a . (21)Since (21) includes three 10-plets, four ¯5’s and one 5-plet, we do not need to introduce any additionalvector-like states. Also in this case, we can exchange U (1) F charge of 5-plet with one of the ¯5’scharge. It turns out that here we will have the following 20 possibilities for ( Q H , Q ¯ H ) pair selection:( Q H , Q ¯ H ) ( i ) = { ( − a − b, b − a ) , ( − a − b, , ( − a − b, − α ) , ( − a − b, a − b ) , (0 , − α ) , (0 , a − b ) , (0 , − a − b ) , (0 , b − a ) , ( − α, , ( − α, a − b ) , ( − α, − a − b ) , ( − α, b − a ) , ( a − b, , ( a − b, − α ) , ( a − b, − a − b ) , ( a − b, b − a ) , (2 b − a, , (2 b − a, − α ) , (2 b − a, a − b ) , (2 b − a, − a − b ) } . (22)Thus, this ABC ( i ) ( i = 1 , , · · · ,
20) class unifies twenty possible charge assignments for the pair( H, ¯ H ). • Model BBB
For constructing this case, we pick up the set of Eq. (14) three times (with corresponding chargeassignments) and add the pair 5 q + ¯5 − q . Thus, the complete content is:10 α + ¯5 − α + 1 α , α ′ + ¯5 − α ′ + 1 α ′ , α ′′ +¯5 − α ′′ +1 α ′′ , q + ¯5 − q . (23)Here for ( Q H , Q ¯ H ) pair selection we have four qualitatively different cases:( Q H , Q ¯ H ) ( i ) = { ( q, − q ) , ( q, − α ) , ( − α, − α ′ ) , ( − α, q ) } . (24)Therefore, this BBB ( i ) ( i = 1 , · · · ,
4) class unifies four options for the pair ( Q H , Q ¯ H ). • Model BBC B -type charge assignments,in combination of set (15). This gives the field content:10 α + ¯5 − α + 1 α , α ′ + ¯5 − α ′ + 1 α ′ , a + b + ¯5 a − b + 1 a +5 b + 5 − a − b + ¯5 ′− a +2 b + 1 ′ a . (25)The list of possible ( Q H , Q ¯ H ) pairs is:( Q H , Q ¯ H ) ( i ) = { ( − a − b, − α ) , ( − a − b, a − b ) , ( − a − b, b − a ) , (2 b − a, − α ) , (2 b − a, a − b ) , (2 b − a, − a − b ) , ( − α, − α ′ ) , ( − α, a − b ) , ( − α, − a − b ) , ( − α, b − a ) , ( a − b, − α ) , ( a − b, − a − b ) , ( a − b, b − a ) } , (26)giving thirteen possibilities unified in this BBC ( i ) ( i = 1 , , · · · ,
13) class. • Model D
The field content of this model is given in (16). It includes three 10 and three ¯5-plets. So, wedo not need to combine this content with other ones, but must add to it the pair 5 q + ¯5 − q . If the U (1) F charge assignments are just those given in (16), then for the pairs ( Q H , Q ¯ H ) we will haveeight options. However, as already discussed, it is possible to build charge assignments utilizingadditional U (1)-charges, as was done in the example given in Eq. (17). The latter case offers 13distinct options for the pairs ( Q H , Q ¯ H ). These, open up varieties for the model building. Oneexample from this D -class of models is presented in Sect. 3.4. In order to proceed with model building, first we give all acceptable up-type Yukawa texturesobtained by U (1) F symmetry. In our approach, among up-type quarks only top quark has renor-malizable Yukawa coupling. Yukawa couplings λ u and λ c emerge after U (1) F flavor symmetrybreaking. The breaking of U (1) F should be achieved by flavon superfields. Here we consider simpleset of flavon pair X + ¯ X with U (1) F charges Q ( X ) = − β , Q ( ¯ X ) = β . (27)In general, scalar components of X and ¯ X have different VEVs h X i and h ¯ X i respectively. Detaileddiscussion of possibility for U (1) F symmetry breaking, giving fixed VEVs for X and ¯ X , is presentedin Appendix A. We introduce the notations | X | M Pl = ǫ , | ¯ X | M Pl = ¯ ǫ , (28)where M Pl ≃ . · GeV is reduced Planck scale, which will be treated as natural cut off for allhigher-dimensional non-renormalizable operators. Thus, the hierarchies between Yukawa couplingsand CKM mixing angles will be expressed by powers of small parameters ǫ, ¯ ǫ ≪ H (5) ⊃ h u ,the up-type quark masses emerge through the Yukawa couplings of the form 10 · · H , wherefamily and SU (5) indices are suppressed. As it turns out, within this setup, three acceptableYukawa textures emerge for up-type quarks. These textures will be referred as U1 , U2 and U3 .8 i) Up Quark Yukawa Texture U1 The U (1) F charges of three 10-plets and the Higgs superfield H are: Q (10 ) = nβ ( nβ + β ) , Q (10 ) = nβ − β , Q (10 ) = nβ − β , Q ( H ) = 6 β − nβ . (29)This selection provides the following Yukawa texture10 U1 : 10 ǫ ( ǫ ) ǫ ( ǫ ) ǫ ( ǫ ) ǫ ( ǫ ) ǫ ǫ ǫ ( ǫ ) ǫ H , (30)where dimensionless couplings (whose magnitudes are assumed to be ∼ / −
3) are not displayed.With ǫ = 1 / − /
5, the matrix (30) gives right hierarchies between up-type quark Yukawas. (ii) Up Quark Yukawa Texture U2
In this case we use the following assignment Q (10 ) = nβ + 3 β , Q (10 ) = nβ , Q (10 ) = nβ + β , Q ( H ) = − nβ − β , (31)which gives the texture: 10 U2 : 10 ǫ ǫ ǫ ǫ ¯ ǫ ¯ ǫǫ ¯ ǫ H . (32)With selection ¯ ǫ = 1 / − / ǫ ∼ (1 / − / · ¯ ǫ , the needed hierarchies for the ratios λ u /λ c , λ c /λ t are generated. (iii) Up Quark Yukawa Texture U3 Finally, with U (1) F charge selections Q (10 ) = nβ − β , Q (10 ) = nβ , Q (10 ) = nβ + β , Q ( H ) = − nβ − β , (33)the up-type quark Yukawa couplings will be 10 U3 : 10 ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ H , (34)which for ¯ ǫ ∼ / − /
10 gives successful explanation of hierarchies λ u /λ c ∼ ¯ ǫ and λ c /λ t ∼ ¯ ǫ .This classification of up-type Yukawa textures helps to build models emerging from classesof Eq. (18) (for each class, see discussion after Eq. (18)). As one can see, there are manypossibilities to be considered in order to see which one gives phenomenologically viable model.Detailed investigation and complete list of acceptable scenarios will be presented in a longer paper[10]. Below we present three models with successful explanation of hierarchies between chargedfermion masses and mixings. 9able 1: U (1) F charge assignment for ABC ( i =4) -U1 ( n =1) model.10 ¯5 ¯5 ¯5 H (5) ¯ H (¯5) 1 Q U (1) F β − β − β − β β β − β β − β ( i =4) -U1 ( n =1) Model
In this model, the content of Eq. (21) is considered and charges are matched in such a way as toobtain with up-type Yukawa texture U1 of Eq. (30). Here, selection n = 1 is made. Thus, accordingto Eq. (29), the charge of H (5)-plet is Q H = 4 β , while charges of 10-plets are Q (10 i ) = { β, , − β } .From the set (21) we will identify 10 α , 10 and 10 a + b with 1 st , 2 nd and 3 rd families respectively, and5 − a − b with H . Making the charge matching α = β , a + b = − β and selection a = − β/
4, we willhave { α, a, b } = { β, − β/ , − β/ } . (35)Furthermore, since we are dealing with ABC ( i =4) model, using Eqs. (22), (35) we have ( Q H , Q ¯ H ) ( i =4) = ( − a − b, a − b ) = (4 β, β ). The charges of remaining ¯5-plets: ¯5 , ¯5 − α and ¯5 − a +2 b , which weidentify with 1 st , 2 nd and 3 rd families of matter ¯5-plets respectively, will be Q (¯5 i ) = { , − β, − β } .The U (1) F charge assignment of all states of content (21) is summarized in Table 1. With thisassignment, the Yukawa coupling matrices are determined as follows:10 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ H , ¯5 ¯5 ¯5 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ¯ H . (36)Taking into account Eq. (1) and H ⊃ h u , ¯ H ⊃ h d , Eq. (36) yield: λ u : λ c : λ t ∼ ǫ : ǫ : 1 , λ t ∼ ,λ e : λ µ : λ τ ∼ ǫ : ǫ : 1 , λ d : λ s : λ b ∼ ǫ : ǫ : 1 . (37)Assuming that in (36) there are dimensionless Yukawa couplings with natural values - in a range ∼ / −
3, with selection ǫ ≃ .
2, the hierarchies in (37) can fit well with the experimental data.Notice that λ b,τ ∼
1, which means that in this scenario tan β ≈ −
60. As far as the CKM mixingangles are concerned, from (36) one can obtain: | V us | ∼ ǫ , | V cb | ∼ ǫ , | V ub | ∼ ǫ . (38)These are also of right magnitudes (with ǫ ≃ . Q (¯5 ) = Q (¯5 ),corresponding entries in 2 nd and 3 rd columns of the second matrix of Eq. (36) have comparablesizes. Taking into account that ¯5 ⊃ l , this leads to the naturally large mixing between l and l lepton flavors: tan θ µτ ∼ , (39)10roviding good explanation for large ν µ − ν τ neutrino oscillations. To demonstrate this, we alsodiscuss neutrino sector in some extent. Let us work in a basis where the matrix responsible forthe charged lepton masses (2 nd matrix in Eq. (36)) is diagonal. Thus, the mixing matrix emergingfrom the neutrino sector will coincide with lepton mixing matrix. We will apply the singlet state1 (see Tab. 1) as a right-handed neutrino. The relevant couplings are λ ν (¯5 + t ¯5 )1 H + ˆ M ǫ ,with λ ν , t being dimensionless couplings and ˆ M some scale. Moreover, we also include higher orderoperators λ ǫ ¯5 ¯5 HH/M ′ and λ ǫ ¯5 ¯5 HH/M ′′ . Integration out of the state 1 , together withlatter operators, give the neutrino mass matrix: M ν = t t t m + δ
00 0 0 m , (40)with m = λ ν v u ˆ Mǫ , m = λ ǫ M ′ v u and δ = λ M ′ λ M ′′ ǫ . The first matrix at r.h.s. of (40) (emerged by integratingout the 1 state) is mostly responsible for the mass m ν and leptonic θ mixing. Indeed, in the limit m →
0, we get tan θ = | t | . This, for | t | ∼ θ ≈ o . Inclusion of the m terms are responsible for mixing angles θ , θ and masses m ν , . With a selection m = 0 .
029 eV, m = 0 . t = 0 . δ = 0 . m = m ν − m ν ≃ . · − eV , ∆ m = m ν − m ν ≃ . · − eV , θ = 34 o , θ = 45 . o , θ = 9 o . These agree well with a recent data [11].In this considered case neutrinos are hierarchical in mass: m ν i = (0 . , . , . λ ν, , ∼ ǫ ≃ .
25, ˆ M ∼ GeV, M ′ ∼ GeV, M ′′ ∼ GeV. Although the values of these scales remain unexplained withinthis scenario, we have showed that the model can be compatible with neutrino sector. More detailedstudy of this and related issues will be presented in [10].As in minimal SU (5) GUT, some care is needed to cure the problem of M D − M E mass degen-eracy. For fixing this problem one can use either an extension by scalar 45-supermultiplets [12],or include powers of adjoint 24-plet in the Yukawa couplings [13], or utilize extra heavy mattersupermultiplets [14]. Study of this problem is beyond the scope of this Letter.Before closing this subsection, let us mention that within this scenario the splitting betweenmasses of doublets and triplets (coming from H, ¯ H ) should be obtained via fine tuning (as in minimalSUSY SU (5)) of the model parameters. However, one should make sure that this is possible toachieve. Due to the U (1) F symmetry, renormalizable superpotential couplings ( M H + λ H Σ) H ¯ H areforbidden. However, in this scenario we have extra SU (5) singlet states charged under U (1) F (seeTable 1). For instance, picking up the states 1 and 1 and announcing them as scalar superfields(with positive matter R -parity), the relevant lowest superpotential couplings (including them) willbe M ǫ + M ¯ ǫ + M Pl ¯ ǫ , where dimensionless couplings have been neglected (assumingthat they are of the order of unity). One can check that vanishing of the F -terms F = F = 0lead to the induced VEVs h i ∼ M Pl ¯ ǫ and h i ∼ M Pl ǫ / ¯ ǫ . With selection ¯ ǫ ∼ .
25 we willhave h i ∼ − M Pl , h i ∼ . M Pl without affecting anything in the discussion above. However,the couplings 1 ( λ H + λ ′ H Σ M Pl ) H ¯ H with h Σ i = V · Diag(2 , , , − , −
3) and tuning condition λ H =3 λ ′ H V /M Pl (satisfied with λ H ∼ . λ ′ H ∼ −
8, rendering theory self consistent) lead to themassless doublets ( M H = 0) and colored triplets with masses M H = λ H h i ∼ few · M GUT .11able 2: U (1) F charge assignment for BBB ( i =2) -U3 ( n = − / model.10 ¯5 ¯5 ¯5 H (5) ¯ H (¯5) 1 Q U (1) F − β − β β β β β − β − β − β − β β ( i =2) -U3 ( n = − / Model
Within
BBB model with content (23), one successful scenario is obtained with i = 2 (in Eq. (24))and with up-type Yukawa texture U3 with n = − / Q (10 i ) = { nβ − β, nβ, nβ + β } = { α ′′ , α ′ , α } . With this, usingEqs. (24), (33) we will have ( Q H , Q ¯ H ) ( i =2) = ( q, − α ) = ( − nβ − β, − nβ − β ). Remaining¯5-plets, ¯5 − α ′′ , ¯5 − α ′ and ¯5 − q will be identified as 1 st , 2 nd and 3 rd families respectively of the matter¯5 states. Therefore, Q (¯5 i ) = { β − nβ, − nβ, β + 2 nβ } . With selection n = − /
5, the U (1) F charges of all states from the content (23) are given in Table 2. With these assignments, couplingsresponsible for up-type quark Yukawas are given in Eq. (34), while couplings generating chargedlepton and down quark masses are: ¯5 ¯5 ¯5 ǫ ¯ ǫ ¯ ǫ ǫ ¯ ǫ ¯ ǫǫ ¯ H . (41)These (with ǫ < ¯ ǫ ) give λ u : λ c : λ t ∼ ¯ ǫ : ¯ ǫ : 1 , λ t ∼ ,λ e : λ µ : λ τ ∼ ǫ : ¯ ǫ : 1 , λ d : λ s : λ b ∼ ǫ : ¯ ǫ : 1 . (42)Taking ¯ ǫ ∼ / − /
10 and ǫ ∼ · − , the pattern (42) describe well hierarchies betweencharged fermion Yukawa couplings. Also, the CKM matrix elements are properly suppressed: | V us | ∼ ¯ ǫ, | V ub | ∼ ¯ ǫ , | V cb | ∼ ¯ ǫ . Because of the large mixing between ¯5 and ¯5 states, also in thiscase for leptonic mixing we expect tan θ µτ ∼
1, providing large ν µ − ν τ oscillations. Demonstrationof this can be done in a same way as for the model presented in Sect. 3.2.The doublet-triplet splitting within this scenario can be achieved in the same manner as wasdiscussed at the end of the Sect. 3.2 for ABC ( i =4) -U1 ( n =1) model. Without going in this discussion,let us proceed to consider another scenario. ( n =1) Model: Content of Eq. (17)
The field content of this model is given in Eq. (17) augmented with Higgs superfields H (5) and ¯ H (¯5)of U (1) F charges q and − q respectively. We will match charges of the 10-plets with assignmentsof U3 texture (see Eq. (33)) as follows Q (10 i ) = { nβ − β, nβ, nβ + β } = { a − b + ¯ c, a + 3¯ b − p ¯ c, − a + 3¯ b + p ¯ c } . Therefore, q = − nβ − β . A phenomenologically viable model is obtainedwith the selection p = k = − /
3. This, with the matching given above and condition in Eq. (17),give (¯ a, ¯ b, ¯ c, n ) = (cid:0) β, β, − β, (cid:1) . Furthermore, we make the identification of flavors of ¯5-plets12able 3: U (1) F charge assignment for D-U3 ( n =1) model with content of Eq. (17). For parametersthe following selection is made (¯ a, ¯ b, ¯ c ) = (cid:0) β, β, − β (cid:1) , p = k = − / ¯5 ¯5 ¯5 H (5) ¯ H (¯5) 1 Q U (1) F β β β − β − β − β β β β − β as: (¯5 , ¯5 , ¯5 ) = (¯5 − ¯ a − ¯ b − c , ¯5 − a − b − c , ¯5 − ¯ a − ¯ b +¯ c ). With these selections and parameters determinedabove, all U (1) F charges get fixed (in the unit of β ). In Table 3 we summarize the charges of allstates. With these assignments, the Yukawa couplings are:10 ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ ¯ ǫ H , ¯5 ¯5 ¯5 ǫ ¯ ǫ ¯ ǫ ǫ ¯ ǫ ¯ ǫǫ ¯ H . (43)These textures lead to: λ u : λ c : λ t ∼ ¯ ǫ : ¯ ǫ : 1 , λ t ∼ ,λ e : λ µ : λ τ ∼ ǫ : ¯ ǫ : 1 , λ d : λ s : λ b ∼ ǫ : ¯ ǫ : 1 . (44)With ¯ ǫ ∼ / − /
10 and ǫ ≈ .
3, the ratios in Eq. (44) describe well observed hierarchiesbetween charged fermion masses. Also, the CKM mixing angles have adequately suppressed values: | V us | ∼ ¯ ǫ, | V ub | ∼ ¯ ǫ , | V cb | ∼ ¯ ǫ , while for the leptonic mixing angle θ µτ one expects tan θ µτ ∼ H and ¯ H have opposite U (1) F charges, the doublet-tripletsplitting can be obtained in the same way (by fine tuning) as within minimal SUSY SU (5). Thus,no additional effort is needed, unlike the scenarios considered in Sections 3.2 and 3.3.Finally, let us note that by proper shift of U (1) F charges of the states of Table 3, one canobtain the charge assignments of model BBB ( i =2) -U3 ( n = − / given in Tab. 2. However, thelatter’s assignment leads to different phenomenology (such as the different couplings required forthe doublet-triplet splitting etc.). That’s why, as a different model, this scenario has been presentedseparately.Since within considered scenarios matter superfields ( f i ) have family dependent U (1) F charges Q f i , there is potentially new source for sfermion mass non-universality. In particular, as given at theend of Appendix, after SUSY breaking D U (1) F -term becomes 2( m X − m X ) / ˜ g , where ˜ g is U (1) F ’scoupling constant and m X and m X are soft mass ’s of the scalar components of the flavon superfields X and ¯ X respectively. Non-zero D U (1) F -term give non-universal contribution to the sfermion massesof the form ∆ m f i = Q f i ( m X − m X ) / N = 1SUGRA [15], due to m X = m X universality, this contribution vanish and we have no additionalsource for flavor violation. Note that the relation m X = m X is quite stable against radiative Since with this assignment 1 ’s U (1) F charge is zero and it does not contribute to anomalies, there is no needfor introducing state 1 . However, presence of four singlets (including 1 ) is required with more general assignmentof Eq. (17). X and ¯ X states with matter. However, this kind of couplings, appearing at high-dimensionaloperator level, are strongly suppressed. This insures stability of the relation m X = m X . Note alsothat below the U (1) F symmetry breaking scale the D U (1) F -term is not renormalized. Therefore, weconclude that in order to avoid new contributions to the FCNC (which is common problem withingeneric SUGRA) one should work within framework (such as minimal SUGRA) giving universalityof soft masses. In this Letter we have presented new examples of non-anomalous flavor U (1) F symmetries withinSUSY SU (5) GUT. Our way of finding of such U (1) F s was to embed the SU (5) × U (1) F in non-Abelian group with anomaly free content. Our selection was based on the requirement that non-trivial SU (5) states should be just those of minimal SUSY SU (5), while the number of additionalsinglet states should not be large. The latter, within concrete scenario, can be exploited for modelbuilding with realistic phenomenology. For demonstrative purposes we have presented three mod-els which nicely explain hierarchies between charged fermion masses and mixings. We have notaddressed the problem of wrong asymptotic mass relations M D = M TE , common also for minimal SU (5) GUT. Solution of this problem can be achieved either by inclusion of scalar 45 super-multiplets [12], or appropriate powers of the Higgs supermultiplet of 24 (adjoint) in the Yukawainteractions [13], or specific extension of the matter sector [14] can be considered. Within themodels, we have found, many varieties of possibilities emerge which require detailed investigation.Complete study of these, together with neutrino sector (some of the singlets, involved in the con-sidered models, can serve as right-handed neutrinos) and other phenomenological issues will bepresented in forthcoming publication [10]. Acknowledgement
I thank K.S. Babu and M.C. Chen for useful comments, and interesting and stimulating discussions.I wish to thank the Center for Theoretical Underground Physics and Related Areas (CETUP*)in Lead, South Dakota and the High Energy Physics Group at Oklahoma State University forwarm hospitality. Partial support from Shota Rustaveli National Science Foundation is kindlyacknowledged.
A Breaking of U (1) F In this appendix we discuss the breaking of U (1) F gauge symmetry and show that desired VEVsfor the flavon fields can be generated. As was mentioned in the text of the paper, the minimalsetup of the charged flavon superfields, which we consider is X and ¯ X with U (1) F charges given inEq. (27). Since we are dealing with Abelian flavor symmetry, in general the Fayet-Iliopoulos (FI)term is allowed and we will include it in our consideration. It has the form ξ R d θV U (1) F , where ξ isparameter with dimension of mass squire. This FI term together with standard D -term Lagrangian14ouplings, for V U (1) F ’s auxiliary component give: D U (1) F = ξ − β | X | + β | ¯ X | . (A.1)Moreover, in order to fix all VEVs we need to have some superpotential couplings. For thispurpose we introduce the superfield S which is neutral ( Q ( S ) = 0) under U (1) F . The most generalrenormalizable superpotential involving X , ¯ X and S will have the form W = λS ( X ¯ X − µ ) + 12 m S S + 13 σS , (A.2)where µ and m S are some mass parameters, while λ and σ are dimensionless couplings. From (A.2),for F -components we derive − F ∗ S = λ ( X ¯ X − µ ) + m S S + σS , F ∗ X = − λS ¯ X , F ∗ ¯ X = − λSX . (A.3)In the unbroken SUSY limit D and F -terms should satisfy F S = F X = F ¯ X = D U (1) F = 0, whichusing (A.1) and (A.3) gives | X | − | ¯ X | = ξ/β , X ¯ X = µ , S = 0 . (A.4)These give non-zero VEVs for X and ¯ X fields: | X | = 1 √ ξβ + s ξ β + 4 | µ | ! / , | ¯ X | = √ | µ | ξβ + s ξ β + 4 | µ | ! − / . (A.5)From (A.5) we see that | X | and | ¯ X | have different values. It is interesting to consider two limitingcases: a ) : ξ/β < , | µ | ≪ − ξ/β , | X | ≃ | µ | p − ξ/β , | ¯ X | ≃ p − ξ/β , | X | ≪ | ¯ X | , b ) : ξ/β > , | µ | ≪ ξ/β , | X | ≃ p ξ/β , | ¯ X | ≃ | µ | p ξ/β , | X | ≫ | ¯ X | . (A.6)Thus, with notations of Eq. (28), case a) gives ǫ ≪ ¯ ǫ , while in case b) we have ǫ ≫ ¯ ǫ . When thescales satisfy relation ξβ ∼ | µ | , Eq. (A.5) gives ǫ ∼ ¯ ǫ . Note that with solution (A.5) and h S i = 0,all states coming from the superfields X , ¯ X and S get masses.Including soft SUSY breaking terms in the potential, VEVs of the fields will be slightly shifted.In particular, with soft mass squires m X and m X for the fields X and ¯ X respectively, one can readilycheck that their VEVs are shifted in such a way that D U (1) F ≃ m X − m X ) / ˜ g (˜ g is U (1) F ’s couplingconstant). As discussed in the end of Sect. 3, this would have impact on flavor violating processes.On the other hand, within minimal SUGRA scenario, the universality m X = m X insures that D U (1) F = 0. 15 eferences [1] J. C. Pati and A. Salam, Phys. Rev. D (1974) 275;H. Georgi and S. L. Glashow, Phys. Rev. Lett. (1974) 438;H. Georgi, H. R. Quinn and S. Weinberg, Phys. Rev. Lett. (1974) 451.[2] C. D. Froggatt, H. B. Nielsen, Nucl. Phys. B147 (1979) 277.[3] L. E. Ibanez and G. G. Ross, Phys. Lett. B (1994) 100;P. Binetruy and P. Ramond, Phys. Lett. B (1995) 49;V. Jain and R. Shrock, Phys. Lett. B (1995) 83.[4] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B (1987) 589;J. J. Atick, L. J. Dixon and A. Sen, Nucl. Phys. B (1987) 109;M. Dine, I. Ichinose and N. Seiberg, Nucl. Phys. B (1987) 253. .[5] E. Dudas, S. Pokorski and C. A. Savoy, Phys. Lett. B (1995) 45.[6] M. C. Chen, D. R. T. Jones, A. Rajaraman and H. B. Yu, Phys. Rev. D (2008) 015019.[7] K. S. Babu, M. Frank and S. K. Rai, Phys. Rev. Lett. (2011) 061802.[8] For anomaly calculus within various groups see:J. Banks and H. Georgi, Phys. Rev. D (1976) 1159;S. Okubo, Phys. Rev. D (1977) 3528.[9] R. Slansky, Phys. Rept. (1981) 1.[10] Z. Tavartkiladze, in preparation.[11] T. Schwetz, M. Tortola and J. W. F. Valle, New J. Phys. (2011) 109401.[12] H. Georgi and C. Jarlskog, Phys. Lett. B (1979) 297;J. A. Harvey, P. Ramond and D. B. Reiss, Phys. Lett. B (1980) 309;S. Dimopoulos, L. J. Hall and S. Raby, Phys. Rev. Lett. (1992) 1984.[13] J. R. Ellis and M. K. Gaillard, Phys. Lett. B (1979) 315.[14] Q. Shafi and Z. Tavartkiladze, Phys. Lett. B (1999) 129; Phys. Lett. B (1999) 563;Z. Tavartkiladze, Phys. Rev. D76 (2007) 055012.[15] A. H. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. (1982) 970;R. Barbieri, S. Ferrara and C. A. Savoy, Phys. Lett. B119