Baryon Non-Invariant Couplings in Higgs Effective Field Theory
aa r X i v : . [ h e p - ph ] M a r FTUAM-16-45IFT-UAM/CSIC-16-135
Baryon Non-Invariant Couplings in Higgs Effective Field Theory
Luca Merlo, Sara Saa, and Mario Sacrist´an-Barbero Departamento de F´ısica Te´orica and Instituto de F´sica Te´orica, IFT-UAM/CSIC,Universidad Aut´onoma de Madrid, Cantoblanco, 28049, Madrid, Spain (Dated: October 2, 2018)The basis of leading operators which are not invariant under baryon number is constructed withinthe Higgs Effective Field Theory. This list contains 12 dimension six operators, which preserve thecombination B − L , to be compared to only 6 operators for the Standard Model Effective FieldTheory. The discussion on the independent flavour contractions is presented in detail for a genericnumber of fermion families adopting the Hilbert series technique. I. INTRODUCTION
The Standard Model (SM) cannot explain the presentmatter-antimatter asymmetry in our universe [1–3]. Apossibility to tackle this problem is to consider addi-tional sources of baryon number violation, as predicted inseveral Beyond the SM (BSM) contexts, such as GrandUnified Theories [4]. On the other side, no baryon num-ber ( B ) violating (BNV) process has been observed sofar, despite the numerous experimental searches on BNVdecays of nucleons – which provide the most stringentconstraints – hadrons, heavy quarks and leptons, and Z boson [5].Without assuming any specific model, an effective fieldtheory (EFT) approach can be adopted to describe BNVprocesses. The first attempt in this direction goes backto the late 1970’s [6–9], followed by a few more recentstudies [10–12]. All these analyses are performed in theso-called SM Effective Field Theory (SMEFT) context,characterised by the construction of non-renormalisableoperators, invariant under the SM gauge symmetries, andbuilt up in terms of SM fermions, gauge bosons and the SU (2) L -doublet scalar boson (“Higgs” for short) [13–15].The cut-off of the theory suppressing these operators willbe referred to as Λ B . At the lowest order in the expansionin 1 / Λ B , four BNV independent structures of canonicaldimension d = 6 were identified [6–9], O = ¯ d CRα u Rβ ¯ Q CLγi L Lj ǫ ij ǫ αβγ , O = ¯ Q CLiα Q Ljβ ¯ u CRγ e R ǫ ij ǫ αβγ , O = ¯ Q CLiα Q Ljβ ¯ Q CLγk L Ll ǫ il ǫ kj ǫ αβγ , O = ¯ d CRα u Rβ ¯ u CRγ e R ǫ αβγ , (1)where Q L ≡ ( u L , d L ) T , u R , d R , L L ≡ ( ν L , e L ) T , and e R are the SM fermions, ǫ αβγ and ǫ ij are the antisymmetrictensors for the colour and electroweak (EW) contractions.If right-handed (RH) neutrinos, N R , are considered inaddition, this set is extended by two operators: O = ¯ Q CLiα Q Ljβ ¯ d CRγ N R ǫ ij ǫ αβγ , O = ¯ u CRα d Rβ ¯ d CRγ N R ǫ αβγ , (2)The operators listed in the previous equations refer onlyto one generation of fermions. Moving to the three gener-ation case does not require the introduction of additional structures, but only to insert explicitly flavour indices onthe fermion fields.The operators in Eqs. (1) and (2) preserve B − L with∆ B = +1 = ∆ L , and then a baryon can only decay intoan anti-lepton and a meson. The constraints on the pro-ton lifetime [16–18] translate into a lower bound on thecut-off Λ B of about 10 GeV, independently of the spe-cific flavour contraction that can be considered for eachoperator. On the contrary, when a flavour symmetry isconsidered, such as the so-called Minimal Flavour Vio-lation ansatz in its global [19–28] or gauged [24, 29–34]versions, the scale Λ B can be lowered, but still it will bemuch larger than the electroweak scale v ≈
246 GeV.The basic ingredient of the SMEFT construction is thetreatment of the Higgs field as an exact EW doublet. Al-though this hypothesis is currently supported by collidersearches (see for example Ref. [35]), the present uncer-tainties leave open the possibility for alternative descrip-tions of the EW symmetry breaking (EWSB) mechanism,potentially free from the Hierarchy problem. Still in thecontext of effective approaches, a description that allowsfor deviations from the exact EW doublet representa-tion for the Higgs field is the so-called Higgs EffectiveField Theory (HEFT) Lagrangian that generalises theSMEFT one. The HEFT Lagrangian is the most generaldescription of gauge and Higgs couplings, respecting theparadigm of Lorentz and SU (3) c × SU (2) L × U (1) Y gaugeinvariance: it is a very useful tool to describe an extendedclass of “Higgs” models, from the SM and the SMEFTscenarios, to Goldstone Boson Higgs models [36–42] anddilaton-like constructions [43–47].The aim of this paper is to construct the BNV oper-ator basis in the HEFT context, completing in this wayprevious studies on the HEFT framework.In the next section, the HEFT setup is summarisedand the BNV basis is presented. The comparison be-tween the HEFT basis and the corresponding one in theSMEFT setup is discussed in Sect. III. The counting ofthe distinct flavour contractions, considering a genericnumber of fermions, is performed in Sect. IV, based onthe Hilbert series technique. The latter is a mathemati-cal method from Invariant Theory to count the numberof independent structures invariant under a certain sym-metry group (for recent phenomenology applications seeRefs. [48–53]). II. THE BNV HEFT LAGRANGIAN
The crucial difference between the SMEFT and theHEFT is the relationship between the physical Higgs field h ( x ) and the SM Goldstone bosons (GBs) −→ π ( x ): in theSMEFT, the four fields belong to the SU (2) L doubletΦ( x ), Φ( x ) = U ( x ) v + h ( x ) √ ! , (3)where U ( x ) ≡ e i −→ σ ·−→ π ( x ) /v (4)is the GB matrix. In the HEFT, instead, the physicalHiggs and the GB matrix are treated as independent ob-jects [54–62]. This fact, together with the adimension-ality of the GB matrix, leads to a much larger numberof operators in the HEFT with respect to the SMEFT,at the same order in the expansion. As a consequence,HEFT exhibits the following distinguishing features [62–68]: - several correlations typical of the SMEFT, such asthose between triple and quartic gauge couplings,are lost in the HEFT;- Higgs couplings are completely free in the HEFT,while they can be correlated to pure gauge cou-plings in the SMEFT;- some couplings that are expected to be stronglysuppressed in the SMEFT, are instead predictedwith higher strength in the HEFT and are poten-tially visible in the present LHC run.In Ref. [62], the complete HEFT Lagrangian, invariantunder baryon and lepton numbers, has been presented atfirst order in the expansion on the new physics scale [69],making explicit the custodial symmetry nature of the op-erators. The building blocks used for these structures arethe SM gauge bosons, the GB matrix U ( x ), the physi-cal Higgs field h ( x ) and the SM fermions arranged indoublets of the global SU (2) L or SU (2) R symmetries.Arranging the RH fermions in doublets of SU (2) R , Q R = (cid:18) u R d R (cid:19) L R = (cid:18) N R e R (cid:19) , (5)allows to distinguish the custodial symmetry preservingoperators from those that instead violate it. Further-more, this notation is consistent with the HEFT for-malism, where the GBs matrix U ( x ) transforms as a bi-doublet of the global SU (2) L × SU (2) R symmetry, U ( x ) → L U ( x ) R † , (6) RH neutrinos are considered as part of the SU (2) R lepton dou-blet, but the origin of their masses will not be discussed here. being L , R the unitary transformation associated to SU (2) L,R , respectively. Indeed, the Yukawa couplingsare given by¯ Q L U ( x ) Y Q Q R , ¯ L L U ( x ) Y L L R , (7)where the Yukawa matrices are written in a compact no-tation as 6 × Y Q = diag( Y u , Y d ) and Y L = diag( Y ν , Y e ). Fur-thermore, it is useful to introduce the scalar chiral field T ( x ), T ( x ) ≡ U ( x ) σ U ( x ) † , T ( x ) → L T ( x ) L † , (8)that breaks SU (2) R , while preserving SU (2) L , and there-fore behaves as a spurion for the custodial symmetry.In the HEFT Lagrangian, the dependence on thephysical Higgs is conventionally described throughadimensional generic functions F ( h/v ) [54, 70], being v the EW vacuum expectation value. These functions arecommonly written as a polynomial expansion in h/v , F ( h/v ) = 1 + α ( h/v ) + β ( h/v ) + . . . , which followsfrom the fact that the physical Higgs is an isosingletscalar of the EW symmetry. The study of the scalarfield manifold, depending on the specific F ( h ), canindeed lead to phenomenological consequences, allowingto disentangle between different frameworks. This hasbeen analysed in Refs. [71–73].One could expect that the basis of BNV operatorsintroduced in Eqs. (1) and (2) will not be modified inthe HEFT framework, as they are purely fermionic. In-deed, these six operators are simply rewritten in terms of SU (2) L and SU (2) R fermion doublets. However, the factthat the GB matrix U and the chiral scalar field T areadimensional allows to construct additional independentstructures with the same canonical dimensions.The set of operators that constitutes the BNV HEFTbasis, at the first order in the expansion on Λ B , consistsof 12 independent structures: R = ¯ Q CLiα Q Ljβ ¯ Q CLkγ L Ll ǫ il ǫ kj ǫ αβγ F ( h ) R = ¯ Q CLiα Q Ljβ ¯ Q CLkγ ( T L L ) l ǫ il ǫ kj ǫ αβγ F ( h ) R = ¯ Q CRiα Q Rjβ ¯ Q CRkγ L Rl ǫ il ǫ kj ǫ αβγ F ( h ) R = ¯ Q CRiα Q Rjβ ¯ Q CRkγ ( U † TU L R ) l ǫ il ǫ kj ǫ αβγ F ( h ) R = ¯ Q CRiα Q Rjβ ¯ Q CLkγ L Ll ǫ ij ǫ kl ǫ αβγ F ( h ) R = ¯ Q CRiα Q Rjβ ¯ Q CLkγ ( T L L ) l ǫ ij ǫ kl ǫ αβγ F ( h ) R = ( ¯ Q CRα U t ) i ( TU Q Rβ ) j ¯ Q CLkγ L Ll ǫ il ǫ kj ǫ αβγ F ( h ) R = ( ¯ Q CRα U t ) i ( TU Q Rβ ) j ¯ Q CLkγ ( T L L ) l ǫ il ǫ kj ǫ αβγ F ( h ) R = ¯ Q CLiα Q Ljβ ¯ Q CRkγ L Rl ǫ ij ǫ kl ǫ αβγ F ( h ) R = ¯ Q CLiα Q Ljβ ¯ Q CRkγ ( U † TU L R ) l ǫ ij ǫ kl ǫ αβγ F ( h ) R = ( ¯ Q CLα U ∗ ) i ( U † T Q Lβ ) j ¯ Q CRkγ L Rl ǫ il ǫ kj ǫ αβγ F ( h ) R = ( ¯ Q CLα U ∗ ) i ( U † T Q Lβ ) j ¯ Q CRkγ ( U † TU L R ) l ǫ il ǫ kj ǫ αβγ F ( h ) . (9)Other BNV operators can be constructed, but are re-dundant with respect to the structures in this list. Forexample, one could consider an operator similar to R ,with T contracted to the second quark doublet insteadthan to the lepton doublet: however, R , R and this al-ternative operator are not independent among each otherand one should choose only two of them. Other exampleswill be discussed in Sect. IV.All the operators in this list have canonical mass di-mension 6 and therefore are suppressed by Λ B . Indeed,the insertion of the scalar chiral field T or of the GBmatrix does not lead to any additional mass suppression.Among these 12 operators, only 4 of them are custodialsymmetry preserving, R , R , R and R , and thus donot contain the custodial spurion T .When ignoring RH neutrinos, the number of indepen-dent operators reduces to 9: in particular, R , R and R turn out to be vanishing or redundant with respectto the other structures. III. COMPARISON WITH THE SMEFT
The BNV SMEFT operators in Eqs. (1) and (2) andthe ones in Eq. (9) present a series of similarities:- all the operators can be written in terms of scalarcurrents, being the other type of contractions van-ishing or redundant by Fierz identity;- both bases contain operators classified intofour distinct classes: schematically, Q L Q L Q L L L , Q R Q R Q R L R , Q L Q L Q R L R and Q R Q R Q L L L ;- the operators in both bases preserve B − L .On the other side, there is not a one-to-one relation be-tween the two sets of operators, as indeed:- the d = 6 SMEFT basis consists of only 6 indepen-dent operators, while the HEFT one presents 12structures;- only two combinations of SMEFT operators, O −O and O + O , in Eqs. (1) and (2), contain sourcesof custodial symmetry breaking; on the other hand,all the operators in Eq. (9) are custodial symmetrybreaking, except for R , R , R and R ;- B − L non-invariant operators can be found in theSMEFT Lagrangian at dimensions different fromsix [74], while this is not the case in the HEFT,where indeed B − L invariance is guaranteed by hy-percharge invariance. This follows from two facts:first, hypercharge can be identified with B − L in theories invariant under the SU (2) L × SU (2) R symmetry, such as in left-right symmetric mod-els [75, 76]. In these frameworks, as the RHfermions also belong to an SU (2) doublet represen-tation, and they have the same electric charge astheir left-handed (LH) counterparts, both LH and RH fields must have the same hypercharge, − / B − L : ψ L → e i ( B − L ) θ ( x ) ψ L ψ R → e i ( B − L ) θ ( x ) e iθ ( x ) σ ψ R , (10)where θ ( x ) is the transformation parameter. Thesecond fact which guarantees the identification ofhypercharge and B − L is that the only spurionbreaking SU (2) R , in the HEFT context is the scalarchiral field T . As it does not carry hypercharge, itsinsertion in an operator cannot lead to hyperchargeviolation, neither of B − L .In the SMEFT, where hypercharge and B − L areindependent, SM gauge invariant operators can vi-olate B − L , and the lowest dimensional example isthe so-called Weinberg operator ( ¯ L cL ˜Φ ∗ )( ˜Φ † L L ). InHEFT, this operator cannot be constructed, unlessother sources of SU (2) R violation are considered.As a title of example, one could consider the Paulimatrix σ + = ( σ + iσ ) /
2, that allows to write theequivalent to the Weinberg operator in HEFT [77]:( ¯ L cL U ∗ ) σ + ( U † L L ) . (11)This operator preserves hypercharge, but violates SU (2) R and lepton number by two units, as it canbe seen by writing explicitly the transformation un-der hypercharge of the GB matrix: U ( x ) → U ( x ) e − iθ ( x ) σ . (12)Notice that this is a three dimensional operatorand therefore provides a direct mass term for thelight active neutrinos. In contrast, the Weinbergoperator in the SMEFT is of d = 5 and thus sup-pressed by a power of the mass scale at which lep-ton number is broken. This is an example of thestrong impact of the adimensionality of the GB ma-trix U with respect to the SU (2) L doublet Higgsof the SMEFT. In the rest of the paper, no othersources of SU (2) R violation will be considered be-side T , consistently with previous studies in theHEFT context.It is interesting to determine the connection betweenthe operators in Eq. (9) and those in Eqs. (1) and (2), asit will help to identify possible ways to distinguish the twodescriptions. The connection for the HEFT operatorswhich do not contain GBs is straightforward: R → O R → O + O R → −O + O R → −O R → O − O R → −O − O . (13)Notice, indeed, that the combination U † TU appearing in R , R and R simplifies to σ once using the definitionof T in Eq. (8). This list shows that there is a linearcorrespondence between 6 operators of the HEFT basisand the 6 operators of the d = 6 SMEFT one. Theother HEFT operators contain interactions that can bedescribed by SMEFT operators with dimension 8. Anexample is the following: R → ¯ Q CLiα Q Ljβ ¯ Q CLkγ h(cid:16) ˜Φ ˜Φ † − ΦΦ † (cid:17) L L i l ǫ il ǫ kj ǫ αβγ (14)where the h -independent couplings of the combination˜Φ ˜Φ † − ΦΦ † in the unitary gauge play the same role asthe scalar chiral field T in R .The study of the connections between the HEFT andSMEFT operators leads to the conclusion that severalcorrelations typical of the SMEFT are lost in the HEFTand that some couplings that are expected to be stronglysuppressed in the SMEFT are instead predicted to be rel-evant in HEFT. This fact has already been pointed outin Refs. [60, 62, 64] for the B and L invariant couplingsand is confirmed here for the B and L non-invariant ones.An example is the comparison between the decay ratesof the proton and of the neutron: Γ( p → π e + ) andΓ( n → π ¯ ν e ). In the d = 6 SMEFT framework, the val-ues of these two observables are predicted to be exactlythe same, while this correlation can be broken considering d = 8 operators. On the other side, in the HEFT context,the operators R , R , R , R , R , R contribute dif-ferently to the two decay rates, and no correlation arisesat any order. An experimental discrepancy among thesetwo observables could then be explained either in termsof the SMEFT, but advocating d = 8 contributions, orin terms of the HEFT Lagrangian. The magnitude ofthe discrepancy is what could tell which is the correctdescription: a relative difference between the two decayrates larger than about (cid:0) v / Λ B (cid:1) cannot be compatiblewith the d = 8 SMEFT Lagrangian, and instead couldwell be accounted for in the HEFT context.At present, the non-observation of the proton decayputs a lower bound on the ratio Λ B /c i of about 10 GeV,where c i represents the combination of the operator co-efficients entering the proton decay rate. As a result,this strategy to disentangle the two frameworks is an in-teresting feature from the theoretical side, although ex-perimentally is not viable yet. Moreover, it allows toestimate the order of magnitude of the contributions tothese decay rates from the d = 8 SMEFT operators ofabout 10 − , with respect to those from the d = 6 ones. IV. FLAVOUR CONTRACTION COUNTING
The number of independent flavour contractions can becounted directly considering the symmetries of the opera-tors in Eq. (9). Alternatively, one can adopt the Hilbertseries technique, which provides a polynomial functionwhose terms can be matched with the operators in Eq. (9) and the corresponding coefficients count the number ofindependent flavour contractions. Although the match-ing is straightforward in the absence of scalar fields, as forthe BNV HEFT operators considered here, one shouldbe careful when dealing with structures containing thefields T and U , in order to remove the redundancies dueto T = and U † U = .The discussion on the number of flavour contractionsadopting the Hilbert series technique is presented below,considering in all generality N f fermion families.The counting for R is N f (2 N f + 1) / R is the same as R , as T only adds a flipof sign in the second component of the lepton doublet. Afew cases with T insertions in the Q L Q L Q L L L ( LLLL for brevity) operators are redundant and have been sub-tracted from the total counting.For the Q R Q R Q R L R ( RRRR ) operators, R and R ,which are written exclusively in terms of SU (2) R dou-blets, the counting simply mirrors that of the LLLL onesand each operator presents N f (2 N f + 1) / RRRR structures, O and O , is 2 N f . This apparent contradiction is easily solvednoticing that the SU (2) R symmetry is still partially pre-served in the operators R and R and prevents part ofthe possible flavour contractions among four RH singletfermions. Indeed, rewriting explicitly the flavour indices a, b, c, d , one gets R abcd = O { bc } ad + O { bc } ad , R abcd = −O { bc } ad + O { bc } ad , (15)where the brackets should be read as O { ab } cdi ≡ O abcdi + O bacdi . This shows that R and R only contain theflavour symmetric contractions in b and c of the SMEFToperators. The flavour antisymmetric contractions areinstead described by two additional structures: R ′ = ¯ Q CRiα ( U † TU Q Rβ ) j ¯ Q CRkγ L Rl ǫ il ǫ kj ǫ αβγ , R ′ = ¯ Q CRiα ( U † TU Q Rβ ) j ¯ Q CRkγ ( U † TU L R ) l ǫ il ǫ kj ǫ αβγ . (16)These two operators are redundant with respect to R and R for N f = 1, but they should be added to the listin Eq. (9) for N f > RRRR operators sums up to 2 N f matchingthe result for the SMEFT case.The operators R – R exhibit a Q R Q R Q L L L ( RRLL )structure. Among these, only R can be directly relatedto a d = 6 operator of the SMEFT Lagrangian. Rewrit-ing the expression for R in Eq. (13), making explicit theflavour indices, one can see that R only contains part ofthe interactions described by O : R abcd = −O { ab } cd . (17)Similarly, the operator R abcd contains only the flavourcontractions symmetric in a and b . It is therefore nec-essary to introduce two additional operators that com-pletely break the SU (2) R structure between the first two SU (2) R quark doublets in R and R : R ′ = ¯ Q CRiα (cid:0) U † TU Q Rβ (cid:1) j ¯ Q CLkγ L Ll ǫ ij ǫ kl ǫ αβγ F ( h ) . R ′ = ¯ Q CRiα (cid:0) U † TU Q Rβ (cid:1) j ¯ Q CLkγ ( T L L ) l ǫ ij ǫ kl ǫ αβγ F ( h ) . (18)As for the previous case, these two structures are redun-dant with R and R for N f = 1, otherwise they shouldbe added to the basis. R ′ and R ′ contain the interac-tions with the combinations antisymmetric in a and b .Therefore R and R ′ provide altogether the flavour con-tractions of the SMEFT operator O . On the other hand,the interactions of R and R ′ are described by a d = 8operator of the SMEFT Lagrangian.The independent structures contained in the two re-maining RRLL operators, R and R , read in the uni-tary gauge¯ u CRαa u Rβb ¯ d CLγc e Ld ε αβγ , ¯ d CRαa d Rβb ¯ u CLγc ν Ld ε αβγ , (19)and are non-vanishing only for the combinations anti-symmetric in a and b . As a result, the number of inde-pendent flavour contractions for each of these operatorsis N f ( N f − / Q L Q L Q R L R ( LLRR ) operatorsis not fully analogous to that of the
RRLL ones. Theinteractions in R and R are described by linear com-binations of the operators O and O of the SMEFT La-grangian, as in Eq. (13). The number of their flavour con-tractions is N f ( N f + 1) / R and R is analogous to the one fortheir RRLL counterparts, R and R : N f ( N f − / R → N f (2 N f + 1) / R → N f (2 N f + 1) / R , R ′ → N f R , R ′ → N f R , R ′ → N f R , R ′ → N f R → N f ( N f − / R → N f ( N f − / R → N f ( N f + 1) / R → N f ( N f + 1) / R → N f ( N f − / R → N f ( N f − / . (20) This analysis completes previous studies on the HEFTLagrangian, which received much attention in the lastyears for its relevance in collider searches. This paperprovides, for the first time, the complete set of leadingoperators which are not invariant under baryon and lep-ton numbers, but do preserve B − L combination.A detailed comparison with the SMEFT Lagrangianis also presented, pointing out a strategy to distinguishbetween the two approaches. Finally, the Hilbert seriestechnique, which has recently undergone a revival of in-terest, has been adopted to discuss the number of flavourindependent contractions for a generic number of fermionfamilies. ACKNOWLEDGMENTS
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