Baryon noninvariant couplings in Higgs effective field theory
Abstract
The basis of leading operators which are not invariant under baryon number is constructed within the Higgs effective field theory. This list contains 12 dimension six operators, which preserve the combination \(BL\), to be compared to only 6 operators for the standard model effective field theory. The discussion of the independent flavour contractions is presented in detail for a generic number of fermion families adopting the Hilbert series technique.
Keywords
Lepton Number Effective Field Theory Standard Model Gauge Standard Model Fermion Minimal Flavour Violation1 Introduction
The standard model (SM) cannot explain the present matter–antimatter asymmetry in our universe [1, 2, 3]. A possibility to tackle this problem is to consider additional sources of baryon number violation, as predicted in several beyond the SM (BSM) contexts, such as Grand Unified Theories [4]. On the other side, no baryon number (B) violating (BNV) process has been observed so far, despite the numerous experimental searches on BNV decays of nucleons—which provide the most stringent constraints—hadrons, heavy quarks and leptons, and Z boson [5].
The operators in Eqs. (1) and (2) preserve \(BL\) with \(\Delta B=+1=\Delta L\), and then a baryon can only decay into an antilepton and a meson. The constraints on the proton lifetime [16, 17, 18] translate into a lower bound on the cutoff \(\Lambda _{B}\) of about \(10^{15}\mathrm {\,GeV}\), independently of the specific flavour contraction that can be considered for each operator. On the contrary, when a flavour symmetry is considered, such as the socalled Minimal Flavour Violation ansatz in its global [19, 20, 21, 22, 23, 24, 25, 26, 27, 28] or gauged [24, 29, 30, 31, 32, 33, 34] versions, the scale \(\Lambda _{B}\) can be lowered, but still it will be much larger than the electroweak scale \(v\approx 246\mathrm {\,GeV}\).
The basic ingredient of the SMEFT construction is the treatment of the Higgs field as an exact EW doublet. Although this hypothesis is currently supported by collider searches (see for example Ref. [35]), the present uncertainties leave open the possibility for alternative descriptions of the EW symmetry breaking (EWSB) mechanism, potentially free from the Hierarchy problem. Still in the context of effective approaches, a description that allows for deviations from the exact EW doublet representation for the Higgs field is the socalled Higgs effective field theory (HEFT) Lagrangian that generalises the SMEFT one. The HEFT Lagrangian is the most general description of gauge and Higgs couplings, respecting the paradigm of Lorentz and \(SU(3)_c\times SU(2)_L\times U(1)_Y\) gauge invariance: it is a very useful tool to describe an extended class of “Higgs” models, from the SM and the SMEFT scenarios, to Goldstone Boson Higgs models [36, 37, 38, 39, 40, 41, 42] and dilatonlike constructions [43, 44, 45, 46, 47].
The aim of this paper is to construct the BNV operator basis in the HEFT context, completing in this way previous studies on the HEFT framework.
In the next section, the HEFT setup is summarised and the BNV basis is presented. The comparison between the HEFT basis and the corresponding one in the SMEFT setup is discussed in Sect. 3. The counting of the distinct flavour contractions, considering a generic number of fermions, is performed in Sect. 4, based on the Hilbert series technique. The latter is a mathematical method from invariant theory to count the number of independent structures invariant under a certain symmetry group (for recent phenomenology applications see Refs. [48, 49, 50, 51, 52, 53]).
2 The BNV HEFT Lagrangian

several correlations typical of the SMEFT, such as those 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 couplings in the SMEFT;

some couplings that are expected to be strongly suppressed in the SMEFT, are instead predicted with higher strength in the HEFT and are potentially visible in the present LHC run.
In the HEFT Lagrangian, the dependence on the physical Higgs is conventionally described through adimensional generic functions \(\mathcal {F}(h/v)\) [54, 70], being v the EW vacuum expectation value. These functions are commonly written as a polynomial expansion in h / v, \(\mathcal {F}(h/v)=1+\alpha (h/v)+\beta (h/v)^2+\cdots \), which follows from the fact that the physical Higgs is an isosinglet scalar of the EW symmetry. The study of the scalar field manifold, depending on the specific \(\mathcal {F}(h)\), can indeed lead to phenomenological consequences, allowing to disentangle between different frameworks. This has been analysed in Refs. [71, 72, 73].
One could expect that the basis of BNV operators introduced in Eqs. (1) and (2) will not be modified in the HEFT framework, as they are purely fermionic. Indeed, these six operators are simply rewritten in terms of \(SU(2)_L\) and \(SU(2)_R\) fermion doublets. However, the fact that the GB matrix \(\mathbf {U}\) and the chiral scalar field \(\mathbf {T}\) are adimensional allows one to construct additional independent structures with the same canonical dimensions.
All the operators in this list have canonical mass dimension 6 and therefore are suppressed by \(\Lambda ^2_{B}\). Indeed, the insertion of the scalar chiral field \(\mathbf {T}\) or of the GB matrix does not lead to any additional mass suppression. Among these 12 operators, only four of them are custodial symmetry preserving, \(\mathcal {R}_{1}\), \(\mathcal {R}_{3}\), \(\mathcal {R}_{5}\) and \(\mathcal {R}_{9}\), and thus do not contain the custodial spurion \(\mathbf {T}\).
When ignoring RH neutrinos, the number of independent operators reduces to nine: in particular, \(\mathcal {R}_4\), \(\mathcal {R}_{10}\) and \(\mathcal {R}_{12}\) turn out to be vanishing or redundant with respect to the other structures.
3 Comparison with the SMEFT

all the operators can be written in terms of scalar currents, being the other type of contractions vanishing or redundant by the Fierz identity;

both bases contain operators classified into four 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 \(BL\).

the \(d=6\) SMEFT basis consists of only six independent operators, while the HEFT one presents 12 structures;

only two combinations of SMEFT operators, \(\mathcal {O}_{4}\mathcal {O}_{6}\) and \(\mathcal {O}_{2}+\mathcal {O}_{5}\), in Eqs. (1) and (2), contain sources of custodial symmetry breaking; on the other hand, all the operators in Eq. (9) are custodial symmetry breaking, except for \(\mathcal {R}_{1}\), \(\mathcal {R}_{3}\), \(\mathcal {R}_{5}\) and \(\mathcal {R}_{9}\);
 \(BL\) noninvariant operators can be found in the SMEFT Lagrangian at dimensions different from six [74], while this is not the case in the HEFT, where indeed \(BL\) invariance is guaranteed by hypercharge invariance. This follows from two facts: first, hypercharge can be identified with \(BL\) in theories invariant under the \(SU(2)_L\times SU(2)_R\) symmetry, such as in left–right symmetric models [75, 76]. In these frameworks, as the RH fermions also belong to an SU(2) doublet representation, and they have the same electric charge as their lefthanded (LH) counterparts, both LH and RH fields must have the same hypercharge, \(1\) for leptons and 1 / 3 for quarks, in a given convention. In a compact notation, then the hypercharge can be written as \(BL\):where \(\theta (x)\) is the transformation parameter. The second fact which guarantees the identification of hypercharge and \(BL\) is that the only spurion breaking \(SU(2)_R\), in the HEFT context is the scalar chiral field \(\mathbf {T}\). As it does not carry hypercharge, its insertion in an operator cannot lead to hypercharge violation, neither of \(BL\). In the SMEFT, where hypercharge and \(BL\) are independent, SM gauge invariant operators can violate \(BL\), and the lowest dimensional example is the socalled Weinberg operator \((\bar{L}^c_L\tilde{\Phi }^*)(\tilde{\Phi }^\dag L_L)\). In HEFT, this operator cannot be constructed, unless other sources of \(SU(2)_R\) violation are considered. As a title of example, one could consider the Pauli matrix \(\sigma _+=(\sigma _1+i\sigma _2)/2\), which allows one to write the equivalent to the Weinberg operator in HEFT [77]:$$\begin{aligned} \psi _L&\rightarrow e^{i(BL)\theta (x)}\psi _L\nonumber \\ \psi _R&\rightarrow e^{i(BL)\theta (x)}e^{i\theta (x)\sigma _3}\psi _R,\nonumber \end{aligned}$$(10)This operator preserves hypercharge, but violates \(SU(2)_R\) and lepton number by two units, as can be seen by writing explicitly the transformation under hypercharge of the GB matrix:$$\begin{aligned} (\bar{L}^c_L\mathbf {U}^*)\sigma _+(\mathbf {U}^\dag L_L). \end{aligned}$$(11)Notice that this is a three dimensional operator and therefore provides a direct mass term for the light active neutrinos. In contrast, the Weinberg operator in the SMEFT is of \(d=5\) and thus suppressed by a power of the mass scale at which lepton number is broken. This is an example of the strong impact of the adimensionality of the GB matrix \(\mathbf {U}\) with respect to the \(SU(2)_L\) doublet Higgs of the SMEFT. In the rest of the paper, no other sources of \(SU(2)_R\) violation will be considered beside \(\mathbf {T}\), consistently with previous studies in the HEFT context.$$\begin{aligned} \mathbf {U}(x)\rightarrow \mathbf {U}(x) e^{i\theta (x)\sigma _3}. \end{aligned}$$(12)
The study of the connections between the HEFT and SMEFT operators leads to the conclusion that several correlations typical of the SMEFT are lost in the HEFT and that some couplings that are expected to be strongly suppressed in the SMEFT are instead predicted to be relevant in HEFT. This fact has already been pointed out in Refs. [60, 62, 64] for the B and L invariant couplings and is confirmed here for the B and L noninvariant ones. An example is the comparison between the decay rates of the proton and of the neutron: \(\Gamma (p\rightarrow \pi ^0e^+)\) and \(\Gamma (n\rightarrow \pi ^0\bar{\nu }_e)\). In the \(d=6\) SMEFT framework, the values of these two observables are predicted to be exactly the same, while this correlation can be broken considering \(d=8\) operators. On the other side, in the HEFT context, the operators \(R_2\), \(R_6\), \(R_7\), \(R_8\), \(R_{11}\), \(R_{12}\) contribute differently to the two decay rates, and no correlation arises at any order. An experimental discrepancy among these two observables could then be explained either in terms of the SMEFT, but advocating \(d=8\) contributions, or in terms of the HEFT Lagrangian. The magnitude of the discrepancy is what could tell which is the correct description: a relative difference between the two decay rates larger than about \((v^2/\Lambda _{B}^2)^2\) cannot be compatible with the \(d=8\) SMEFT Lagrangian and, instead, could well be accounted for in the HEFT context.
At present, the nonobservation of the proton decay puts a lower bound on the ratio \(\Lambda _{B}/c_i\) of about \(10^{15}\,\mathrm {GeV}\), where \(c_i\) represents the combination of the operator coefficients entering the proton decay rate. As a result, this strategy to disentangle the two frameworks is an interesting feature from the theoretical side, although experimentally is not viable yet. Moreover, it allows one to estimate the order of magnitude of the contributions to these decay rates from the \(d=8\) SMEFT operators of about \(10^{51}\), with respect to those from the \(d=6\) ones.
4 Flavour contraction counting
The number of independent flavour contractions can be counted directly considering the symmetries of the operators in Eq. (9). Alternatively, one can adopt the Hilbert series technique, which provides a polynomial function whose terms can be matched with the operators in Eq. (9) and the corresponding coefficients count the number of independent flavour contractions. Although the matching is straightforward in the absence of scalar fields, as for the BNV HEFT operators considered here, one should be careful when dealing with structures containing the fields \(\mathbf {T}\) and \(\mathbf {U}\), in order to remove the redundancies due to \(\mathbf {T}^2 =\mathbbm {1}\) and \(\mathbf {U}^\dag \mathbf {U}=\mathbbm {1}\).
The discussion of the number of flavour contractions adopting the Hilbert series technique is presented below, considering in all generality \(N_f\) fermion families.
The counting for \(\mathcal {R}_1\) is \(N_f^2(2N_f^2+1)/3\) and coincides with the one in Ref. [12], where it is discussed in terms of flavour representations by using Young tableaux. The counting of \(\mathcal {R}_2\) is the same as \(\mathcal {R}_1\), as \(\mathbf {T}\) only adds a flip of sign in the second component of the lepton doublet. A few cases with \(\mathbf {T}\) insertions in the \(Q_{L}Q_{L}Q_{L}L_{L}\) (LLLL for brevity) operators are redundant and have been subtracted from the total counting.
The counting for the \(Q_{L}Q_{L}Q_{R}L_{R}\) (LLRR) operators is not fully analogous to that of the RRLL ones. The interactions in \(\mathcal {R}^9\) and \(\mathcal {R}^{10}\) are described by linear combinations of the operators \(\mathcal {O}_2\) and \(\mathcal {O}_5\) of the SMEFT Lagrangian, as in Eq. (13). The number of their flavour contractions is \(N_f^3(N_f+1)/2\) for each of them, in agreement with Refs. [12, 78]. Finally, the counting of the flavour contractions of \(\mathcal {R}_{11}\) and \(\mathcal {R}_{12}\) is analogous to the one for their RRLL counterparts, \(\mathcal {R}_7\) and \(\mathcal {R}_8\): \(N_f^3(N_f1)/2\).
A detailed comparison with the SMEFT Lagrangian is also presented, pointing out a strategy to distinguish between the two approaches. Finally, the Hilbert series technique, which has recently undergone a revival of interest, has been adopted to discuss the number of flavour independent contractions for a generic number of fermion families.
Footnotes
 1.
RH neutrinos are considered as part of the \(SU(2)_R\) lepton doublet, but the origin of their masses will not be discussed here.
Notes
Acknowledgements
We specially acknowledge discussions with Ilaria Brivio, Belén Gavela, Elizabeth Jenkins and Aneesh Manohar. We also acknowledge Enrique Fernández Martínez for pointing out a typo in the text. S.S. is grateful to the Physics Department of the University of California, Berkeley, for hospitality during the completion of this work. L.M. and S.S. acknowledge partial financial support by the European Union through the FP7 ITN INVISIBLES (PITNGA2011289442), by the Horizon2020 RISE InvisiblesPlus 690575, by CiCYT through the projects FPA201231880 and FPA201678645, and by the SpanishMINECO through the Centro de excelencia Severo Ochoa Program under grant SEV20120249. The work of L.M. is supported by the Spanish MINECO through the “Ramón y Cajal” programme (RYC201517173). The work of S.S. was supported through the grant BES2013066480 of the Spanish MICINN.
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