On the non-minimal character of the SMEFT

When integrating out unknown new physics sectors, what is the minimal character of the Standard Model Effective Field Theory (SMEFT) that can result? In this paper we focus on a particular aspect of this question:"How can one obtain only one dimension six operator in the SMEFT from a consistent tree level matching onto an unknown new physics sector?"We show why this requires conditions on the ultraviolet field content that do not indicate a stand alone ultraviolet complete scenario. Further, we demonstrate how a dynamical origin of the ultraviolet scales assumed to exist in order to generate the masses of the heavy states integrated out generically induces more operators. Therefore, our analysis indicates that the infrared limit captured from a new sector in consistent matchings induces multiple operators in the SMEFT quite generically. Global data analyses in the SMEFT can and should accommodate this fact.

When integrating out unknown new physics sectors, what is the minimal character of the Standard Model Effective Field Theory (SMEFT) that can result?In this paper we focus on a particular aspect of this question: "How can one obtain only one dimension six operator in the SMEFT from a consistent tree level matching onto an unknown new physics sector?"We show why this requires conditions on the ultraviolet field content that do not indicate a stand alone ultraviolet complete scenario.Further, we demonstrate how a dynamical origin of the ultraviolet scales assumed to exist in order to generate the masses of the heavy states integrated out generically induces more operators.Therefore, our analysis indicates that the infrared limit captured from a new sector in consistent matchings induces multiple operators in the SMEFT quite generically.Global data analyses in the SMEFT can and should accommodate this fact.

I. INTRODUCTION.
Despite the null results on beyond the Standard Model (SM) resonance searches from Run I and II at LHC, the arguments in favor of physics beyond the SM are very strong.It is reasonable to expect that the low energy effects of a new physics sector, which has a mass gap in its typical mass scale(s) 1 compared to the electroweak scale v ≃ 246 GeV, could be resolved in the future.This is particularly the case if Λ 4 π v, which is consistent with expectations of ultraviolet (UV) physics motivated by naturalness concerns for the Higgs mass.Broad classes of new physics scenarios consistent with this minimal decoupling assumption can be constrained efficiently using effective field theory methods to analyse scattering data limited to energies √ s ∼ v ≪ Λ.This formalism has come to be known as the Standard Model Effective Field Theory (SMEFT) recently [1][2][3][4][5][6][7][8][9][10] where the SM is supplemented with a series of higher dimensional operators2 In this approach, the null results of lower energy tests for physics beyond the SM can be consistent with beyond-the-SM UV physics in the ∼ TeV mass scale range.However, a large set of experimental measurements that test the symmetry breaking patterns of the SM must be accommodated.This can be accomplished in a manner that avoids fine tuning.In this work, we use a symmetry assumption that new physics in the TeV mass range leads to a L 6 correction to the SM that (approximately) respects the global symmetry group G = U(1) B ⊗U(1) L ⊗SU(3) 5 , and in addition a discrete CP symmetry. 3Some of these symmetries cannot be exact, as the SM defines a minimal symmetry breaking in the SMEFT.However, the G and CP symmetry breaking pattern of the SM can allow the effect of the UV physics sectors in the ∼ TeV scales of experimental interest because of two reasons: it follows a Minimal Flavor Violating (MFV) pattern for flavor changing measurements [12][13][14]; and it is proportional to the Jarlskog invariant [15,16] in the case of SM CP violation.
In this paper, we consider matching patterns of operators that result from integrating out new physics sectors.We study the effect of simultaneously integrating out not only the heavy fields, but also a UV sector that generates the required heavy mass scale(s) Λ > v.We initially focus on the question of when, if ever, only one operator can be obtained in such a tree level matching in our chosen basis for L 6 [2].We examine this question in the SMEFT in terms of the matching effects of spin-{1,1/2,0} fields that can couple to the SM through (d ≤ 4) mass dimension interactions.Higher spin composite fields (and spin towers) are possible and even required in the presence of UV confining strong interactions.Similarly, when a UV sector with a strong interaction is present, there is no particular reason in general for tree level matchings, as opposed to non-perturbative matchings, to be the largest contribution to the Wilson coefficients.Such extra matching contributions are difficult to characterize (other than by using the full SMEFT formalism) and only reinforce our main point on the non-minimal character of the SMEFT, so we do not focus on these contributions.
The number of operators induced in matching is operator basis dependent.However, the conditions uncovered on the UV field content to reduce the operator profile (i.e. the number of independent SU(3) × SU(2) × U(1) operators) are still meaningful.The conditions can be framed in terms of symmetries and several simple observations on new physics spectra and dynamics that can generate a scale Λ, as we show.Practically speaking, most global analyses are being constructed using the well defined Warsaw basis [2], so we focus on this basis when examining the one operator question.We use the notation Q i to denote an operator defined in the Warsaw basis in this work, and refer the reader to Ref. [2] for the explicit operator definitions.Note also that we refer to one operator with the understanding that, consistent with our assumptions of G symmetry, flavor indices are not used to distinguish oper-ators.
The structure of this paper is as follows.Following a brief comment on the dimension-5 operator and Fermi theory in Section II, we provide in Section III.A a comprehensive discussion on the SMEFT matching at tree level onto L 6 when a massive spin-1 state present in a UV physics sector is integrated out.We focus this discussion on the "one operator induced at tree level" question consistent with the assumed (approximate) G symmetry.We demonstrate why such a simple UV sector cannot be a complete scenario if a mechanism to generate the heavy state's mass is demanded.We then discuss the spin-1/2 case, drawing a similar conclusions in Section III.B.In Section III.C we examine the case of integrating out a scalar field focused on the "one operator" question.We show how the scalar case is more subtle, but still argues for more operators when UV complete scenarios are demanded.Section IV contains our conclusions.

II. TWO EXCEPTIONAL EFT CASES
When considering the one operator question, we note that a few historical accidents in EFTs can be misleading.First of all, L 5 and L d with d ≥ 6 are distinct when considering this question.Due to the charges of the SM field content, only one operator (with flavor indices) can be constructed in L 5 .The operator that results [17,18], is the well known example where one operator at a particular mass dimension does result when integrating out UV physics. 4The interplay of global U(1) L number violation and the constraints of the SM field's representations leading to one operator in L 5 is an exception that is not repeated at higher orders in the SMEFT operator expansion [6][7][8][9][10].Historically, Fermi theory has frequently been used as a prototypical EFT to build intuition.This can be unfortunate, as Fermi theory is atypical and has a number of non-trivial accidental features that are not generic.In Fermi theory, the four-fermion operator is generated when the W boson is integrated out.This effective operator is used in the process µ − → e − + νe +ν µ to infer the Fermi constant, G F .The UV sector in the case of Fermi theory is the SM which does induce a series of other operators at tree level, in addition to the operator Q ℓℓ .These four-fermion operators are due to the Higgs field and the Z boson.However, the highly suppressed Yukawa couplings of the SM Higgs to light fermions leads to an exceptional situation numerically in terms of the operator profiles.The small Yukawa couplings are not formally the consequence of a fine tuning, as they are protected by the full chiral symmetry of the SM.More discussion on the accidents in Fermi theory, and how it is commonly misunderstood, can be found in Ref. [19].Arguably, there is some theoretical evidence based on the structure and particle content of the SM in the direction of embedding this model into SU (5) or SU (10), see for example the arguments in Ref. [20].This could be interpreted as a hint to an underlying theory, similar to the chiral structure of the SM being a low energy hint of its UV structure.However, the problems of TeV scale grand unified theories are very well known.In this work we make a more phenomenologically motivated choice and assume approximate G symmetry (and CP symmetry).

III. G SYMMETRIC TREE LEVEL MATCHINGS A. Spin 1 states
Spin-1 fields that couple to the SM quark bi-linears in the manner assumed are given by Table I [21][22][23].The requirement of linear couplings of mass dimension less than four, together with Lorentz symmetry and invariance under the full SM gauge group constrains the possible quantum numbers of UV field content.Fields with other representations that give SMEFT matchings respecting G are possible, if these conditions are relaxed.Our notation is that Q c , L c are the right handed conjugate doublet fields of the SM fermions.The global flavor symmetry in the  quark and lepton sectors are defined as It is also possible to have a vector field couple to lepton bi-linears, to quark-lepton bi-linears or have an interaction with the SM Higgs field.We list the corresponding fields in Table II and Table III.Cases V XII , V XIII have fields that carry a global lepton number and V XIV − V XX carry both lepton and baryon numbers.Although counterintuitive, UV fields that carry flavor quantum numbers do not necessarily lead to lower energy signatures of flavor violation -outside of the MFV pattern.Similarly, fields carrying lepton number do not necessarily lead to lower energy signatures of lepton flavor violation at tree level. 5ields that carry both lepton and baryon number are potentially more problematic in inducing proton decay, but

Case
Qi generated at tree level Examples of the sets of L6 operators in the SMEFT obtained by integrating out various massive vectors.
such phenomenological constraints are not the focus of this paper.

Dimension-6 operator matching
Solving the classical equations of motion (EOM) for the heavy vector fields and substituting the classical solution into the Lagrangian results in a direct tree level matching in terms of a product of currents.We define the currents as and the tree level matching is given by Here Φ represents H or H = i σ 2 H ⋆ and ⊗ indicates a group product characterized by the SU(2) L representation of vector fields. 6For a vector field of the form considered in Tables I, II and III, the current product falls into one of three types: • four-fermion: (J µ ψ ) † J ψ,µ , • scalar derivative: • mixed scalar-fermion: We have systematically examined the profile in terms of operators obtained in tree level matchings to the Warsaw basis from the fields listed in Tables I,II and III, finding the following rule: Flavour singlet vector fields that do not break G Q × G L induce more than one operator at tree level when matching onto the SMEFT Warsaw basis.
Operators induced at tree level when the massive vector case is integrated out.The cases are grouped in the table into the chiral (J µ ψ ) † J ψ,µ operator classes induced.The top section refers to LLLL operators.The middle section of the table refers to LLRR operators.The bottom section of the table refers to RRRR operators induced at tree level.
This result is easy to demonstrate.Fields that are SU(3) C and SU(2) L singlets couple to (quark and lepton) fermion fields and also the scalar currents, inducing a large number of operators at tree level.A vector field can be made to couple to the left-handed doublets by assigning the field to a 3 of SU(2) L .The scalar and leptonic couplings can be removed by assigning the field to a 8 of SU(3) C .In this case the operator profile is reduced to at least two (J µ ψ ) † J ψ,µ operators via the relation [2] ( Qp here p, r, s, t are flavor indices.We show some examples of the multiple operators induced when integrating out vector fields at tree level in Table IV.Introducing G Q × G L symmetry, vector fields can be reduced in their infrared (IR) SMEFT operator profile to one operator in the Warsaw basis in the limit of vanishing Yukawa matrices; see Table V.Note that with the exception of case V VII which has a bi-linear flavor breaking spurion in Y † d and Y u , the presence of a U(1) Y charge is also associated with the lack of Higgs scalar currents induced.This has an important consequence when the self interactions of the vector are studied for unitarity violation, as will be discussed shortly.
A spurion analysis allows the corrections due to the nonzero Yukawa matrices of the SM (that break the flavor symmetry in a phenomenologically safe MFV pattern) to be systematically studied.We define the SM Yukawa G Q ×G L symmetry is restored if we endow the Yukawa matrices with the transformation properties under , Introducing G Q × G L symmetry breaking when the Y u , Y d , Y e matrices take on their SM values gives more operators at tree level for fields with flavor quantum numbers.On general grounds, the (J µ H ) † J H,µ current products are induced proportional to two spurions breaking the flavor symmetry, and the (J µ ψ ) † J H,µ , (J µ H ) † J ψ,µ current products are induced proportional to one flavor breaking spurion insertion.Here we refer to the spurions listed in Table V that are bi-linear in Yukawa matrices.As a specific example consider VIII that is a 8 under SU(3) QL .The Lagrangian 7 is given by Note that the largest spurion that restores the flavor symmetry for the second line is T A Y † u Y u and some indices are suppressed in Eqn.(11).The additional spurion breaking proportional to Y † d Y d is neglected in what follows.Integrating by parts and the EOM for the vector field are used to manipulate the derivative to appear as shown on the second line in Eqn.(11).Integrating out the field V (1) VIII using the classical EOM gives Hq pr , 7 Recall the flavor adjoint 8 representation is real.
where the dummy labels a and b are summed over {u, d} and {e, d}, respectively.A similar pattern of matchings onto the class 3 (D 2 H 4 ), 5 (H 3 ψψ) and 7 (H 2 D ψψ) operators of the Warsaw basis is present for almost all color singlet fields with flavor quantum numbers listed in Tables II and III.The exceptional case is the field V XII whose non-trivial SU(2) L representation and U(1) Y charge forbids a scalar current from being induced at tree level in this manner.
The pattern of tree level matchings is strongly dictated by the charges and representations of the UV fields under SU(3) C × SU(2) L × U(1) Y , G Q and G L .We emphasize, data fits to subsets of operators in the SMEFT formalism can be justified by appealing to UV field content with U(1) Y charges and non-trivial representations under SM groups when only retaining tree level matching contributions.See Table V for details on cases that generate only one operator at a time.
This conclusion is subject to the following qualifications.First, the single operators obtained in tree level matchings to the vectors in Tables II, III are limited to (J µ ψ ) † J ψ,µ operator forms.Such operators at LHC contribute to continuum parton production in a fashion dictated by the power counting of the theory.Conversely, the precise measurements made on a scattering through a SM resonance (with mass M and width Γ) parametrically has a Γ/M suppression, compared to the leading resonant behavior, when considering the interference with (J µ ψ ) † J ψ,µ operators.Second, as y t ≃ 1, a flavor symmetry spurion breaking proportional to only powers of Y u can induce operators of class 3, 5 and 7 without significant numerical suppression.This makes it difficult to justify "one at a time" data fits to (J µ ψ ) † J ψ,µ SMEFT operators with up quark field content (consistent with our assumptions).On the other hand, one at a time data fits to (J µ ψ ) † J ψ,µ operators that only have leptonic or down quark field content can be potentially justified.In these cases the induced scalar currents proportional to MFV like flavor breaking spurious are numerically suppressed compared to pure up quark spurions by at least y b /y t ∼ 10 −2 .
Finally, we also note that we never obtain only one operator in such a tree level matching that involves the Higgs field, in the cases of massive vector UV field content considered.
Arguments against orphaned vectors.
The vector fields listed in Tables I, II and III inducing a single L 6 SMEFT operator at tree level, carry at least one non-trivial representation under the SM gauge symmetry and flavor symmetries. 8Non-trivial representations and U(1) Y charges reduce the interactions for SM particles with the new sector, which consequently minimizes the IR + TABLE VI.One loop renormalization results.Here O indicates the matrix element of the vector-fermion bilinear interaction term and δZ 3 corresponds to the divergence present in this three point interaction at one loop from the vector-fermion coupling.The notation is such that We have labeled several of the numerical factors in the table with the group space (SU(3)C, SU(3) Fl , SU(2)L) that generates them, with the subscript F l indicating a SU(3) flavour group.SMEFT operator profile.However, such fields in general do not indicate a stand alone UV complete scenario (where the vector could be an "orphan") for the following reasons.
(1) Landau poles and triviality.The β function of the coupling of the vector fields to the fermion bi-linears (denoted g V in Eqn.(11)) is determined by renormalizing the fermion fields and vector field two point functions, and subsequently extracting the β function for g V .We relate the bare (0) and renormalized (r) fields and couplings as where Z x = 1 + δ Zx for x = {V, g V , ψ, ψ}.We use a renormalization scheme employing MS subtraction and d = 4 − 2ǫ dimensions using standard methods.The relevant diagrams are shown in Figure 1.The β-function for the running of the coupling g V is given by where the renormalization factors δ Z 's for the various vector field cases are presented in Table VI.The general expectation is that g V will have a positive β function -indicating Landau poles [24], quantum triviality [25] and a UV incompletion.This is indeed the case for all vector fields inducing one (J µ ψ ) † J ψ,µ operator, with the exception of color 3 vectors coupling to quark bi-linears; i.e. cases V 3 X,XI .In this exceptional case, the SU(3) C vector-fermion coupling mimics the effect of a non-abelian interaction.
An oversimplified UV scenario afflicted with an internal inconsistency indicated by the presence of Landau poles cannot formally generate a consistent IR limit.This indicates that further new physics must be present below the Landau pole scale Λ L approximated by However, numerically corrections suppressed by Λ L are smaller than one loop matching effects.
(2) Unitarity and vector self-interactions.A more intractable problem is generated by O(1) self interactions of orphan vector fields.The four point vector self interaction is not forbidden by any symmetry.Conversely the three point interaction can be forbidden by the presence of a U(1) Y charge in the composite field.Consider the 2 → 2 longitudinal vector scattering displayed in Fig. 2 that is dictated by such a four-point and three-point interactions at tree level.The relevant Lagrangian involving a general vector field with self-interactions is The amplitudes at leading order with the high-energy approximation for the vector polarization ǫ µ L ≃ p µ /M V through a s-, t-and u-channel vector exchange and a four-point contact interaction, respectively, read Here abstract group structure constants F s,t,u for three channels have been introduced.For example, in the model V IX : F s = f ABE f CDE f ijn f kln where A, B, C, D, E refer to the flavor index and i, j, k, l, n denote the iso-spin index.
If λ = (g ′ ) 2 is accomplished by a global symmetry then the amplitudes M L 3 will cancel with three terms in M L 4 with an identical F factor respectively, through the Mandelstam relation s + t + u = 4 M 2 V .As a result, the leading scaling in ∼ (p 2 ) 2 /M 4 V disappears.The full amplitudes then grow as ∼ p 2 /M 2 V .However, if the three-point interaction is forbidden -for example due to the field carrying a U(1) Y charge -then the amplitude cannot be so moderated in its growth at high energies, and scales as ∼ (p 2 ) 2 /M 4 V .In this manner, the presence of a U(1) Y charge forbidding the scalar current simultaneously turns off the three-point interaction that is required to moderate the high energy scattering behavior of an orphaned vector field.
Standard partial wave unitarity arguments [26][27][28] give that the unitarity violation scale associated with the vector field without a three point interaction is where F t , an F u are determined by a particular scattering cross section.A quick onset of unitarity violation follows from a sizable four-point interaction that is expected to emerge from a strongly interacting composite sector on general grounds.Even introducing a loop suppression to the vector self interaction, that is λ ∼ (16π) −1 , is of little help -one still finds Λ ∼ M V due to the presence of a fourth root in Eqn.(22).Hence, the UV strong sector should be simultaneously considered to define a consistent matching onto the SMEFT.This would increase the low energy operator profile of such a scenario in the SMEFT beyond one operator generically due to non perturbative matchings, and a "one at a time" analysis invoking a tree level matching would be logically incoherent.
(3) Siblings of massive vectors with non-trivial representations.A massive vector field with non-trivial representations under subgroups of G is also generically accompanied by more "sibling" fields.If the massive vector gains a mass by a UV Higgs mechanism, the corresponding sibling field includes at least a scalar (S) obtaining a vacuum expectation value (vev).Define this expectation value as S † S = v ′2 /2.We require dim(V) + 1 ≤ dim(S) so that all of the components of the vector become massive in the presence of a scalar field obtaining a vev, through eaten Goldstone components of S. 9  One can use the global symmetry rotations on S to rotate the new vev to a (uneaten) component of S, denoted s.The interaction of s with H † H cannot be forbidden by an explicit G breaking without violating our assumptions.This would introduce highly constrained low energy effects into the SMEFT through the vev v ′ leading to the vector mass matrix.A vacuum misalignment [32] is assumed to 9 An additional non goldstone incomplete scalar multiplet is famously required when introducing a vev in this manner [29][30][31].
make the vector mass matrix symmetric under G in this work.This results in the Higgs portal coupling not being suppressed by a G Q × G L breaking spurion.Concretely consider the Lagrangian Here the covariant derivative is D µ = ∂ µ + ig V V a h a with h a an abstract group generator that defines the non-trivial representations that the V multiplet carries.S is expanded as a where ρ a corresponds to the goldstone components of the S multiplet that are eaten to generate the vector mass, and the • • • fill out the full dimension of S. The vev v ′ must be arranged to break the dim(V) h ′ a generators.Simultaneously v ′ must not break the G subgroup, so g a S = 0, where the generators of G are denoted g a . 10Integrating out s after UV symmetry breaking gives in addition to the operators induced by integrating out the vector field.Here the scalar mass is m 2 s = λ ′ v ′2 /2.In addition, L 4 terms are induced that require a finite redefinition of λ and v in the SM to rearrange L SM back into standard from.Here we have neglected many higher order effects including subdominant mass splitting terms.Note the sizable vector four point interaction, that is enhanced in the λ ′ → 0 limit, indicating unitarity violation when the UV Higgs is integrated out of the spectrum.In this limit it is of interest to not neglect mass splitting effects proportional to λ SH .
In order to avoid assuming a UV Higgs mechanism, we can consider a massive vector generated by a hypothetical UV strong sector, with spin-1/2 constituents Ψ, so that the vector fields are V µ ∼ Ψγ µ Ψ condensates.This composite field carries at least one non-trivial representation under one of the groups G Q , G L , SU(3) C or SU(2) L to reduce the SMEFT operator profile to one operator.Denote this non-trivial representation as N, and the corresponding group as G ′ .The Ψ are charged under G ′ or a larger group H with H ⊃ G ′ .
We can consider G ′ or the proper subgroup case where G ′ ⊂ H without loss of generality with the following arguments.The Ψ belongs to SU(3), and N ∈ {3, 3, 6, 8}, or SU (2) with N = {2, 3} for the vector fields of interest.The non-trivial representations in N can be generated from tensor products of the Ψ irreducible representations.In the case where the Ψ belongs to SU(3) we denote the irreducible representations as P, R, which need not have the same dimension.When N is generated by P ⊗ P the singlet representation is also generated in the tensor product.A color singlet sibling under SU(3) C is expected with a mass proximate to a color octet vector, which induces a number of operators in L 6 when integrated out.Similarly, a flavor singlet sibling under a flavor group is also expected for flavor octets.Interestingly, the flavor 8 vector fields we considered all have zero U(1) Y charges, so their flavor singlet siblings with the same U(1) Y charge are not forbidden by the flavor symmetry to have the coupling with the corresponding quark bi-linear and also with the J H,µ of vanishing U(1) Y charge, inducing more than one operator in L 6 when integrated out.When N ∈ P ⊗ P multiple representations result, for example in the case of P = 3, the 3 and 6 fields are simultaneously present.Such fields (V X,XI ) can induce the same operator when integrated out.On the other hand, these fields necessarily carry U(1) Y , and thus have a cut off scale proximate to the massive vectors mass scale for the cases consistent with our assumptions. 11Next we consider the cases when the non-trivial representation is generated by bi-linears of Ψ carrying representations of unequal dimension N ∈ P ⊗ R. By inspection of the tensor products of SU(3) with triality 0 and 1 [37] it is possible to generate each N ∈ {3, 3, 6, 8} for SU(3) in such a manner.However, for each P and R one can also form a condensate Ψγ µ Ψ with zero U(1) Y charge from the product P ⊗ P, R ⊗ R. Two more pure singlet spin one bound states proximate in mass to M V are expected in the spectrum, unless forbidden by another symmetry. 12Restricting the discussion for non-trivial SU(2) representations to the vector cases that do not carry U(1) Y and induce one operator at tree level, we are left with the field V 1 IX .Further, V 1 IX has a large flavor breaking spurion proportional to the top Yukawa generating more operators at tree level when integrated out, see Table V.
For all of these reasons, orphaned vector fields with nontrivial representations of the SM symmetry groups demand siblings and a "good UV home".

B. Spin 1/2 states
If heavy spin-1/2 states are integrated out, the mass 13 of the massive fermion(s) (denoted M with M ≫ v) must be introduced in some manner.As discussed in the previous section, a chiral fermion with a UV Higgs mechanism induces more operators at tree level when integrating out the UV scalar field.In this section, we confine the discussion to general vector like fermions.The general Lagrangian associated with a pair of heavy vector-like quark (VLQ) denoted by Q L , Q R that are flavor singlets includes where for the SU(2) L singlet and doublets and for the 3 of SU(2) L .The interaction term L int Q for the VLQs to the SM fermions through the Higgs doublet is defined as The requirement that the action be stationary under variations of the heavy VLQ fields QL , QR results in two coupled EOMs: Mathematically, the coupled Eqns.( 29) and ( 30) can be solved iteratively.Taking the limit of large M , one can expand the classical solutions schematically as When substituted back into Eqn.(25) the effect of the leading term in these solutions vanishes due to chirality.We generically find that multiple operators are induced at tree level when integrating out a vector like fermion.The cases where the vector like quark do not carry flavor quantum numbers are shown in Table VII.In the cases that the VLQs carry flavor quantum numbers, previously discussed in Refs.[34][35][36], multiple operators are again obtained.We show some sample cases of this type in Table VIII.Multiple operators at tree level are also obtained in the case of integrating out vector like leptons, see Table IX.
(1,3,1) dRH † √ √ TABLE VIII.Tree level L6 operators induced in the SMEFT with massive quarks integrated out in some sample cases with flavor quantum numbers, see Refs.[34][35][36] for more discussion on the phenomenology of these fields.

C. Spin 0 states
Unlike the cases of massive vectors and spin-1/2 fields, a massive scalar can couple into the SM through a number of interactions and naively generate many operators in the IR SMEFT matching limit.However, the examples (S A , S B and S C in Table X) discussed in Refs.[8,38] show that only one operator Q H , can be obtained if an explicit scale is introduced without a dynamical origin to give the scalar a mass.For instance, S A couples through linear and bilinear interactions in the full multi-scalar potential, denoted V (H, S A ) in Table X.To reduce the operator profile of S A to one operator, it is assumed that S A has a discrete or additional U(1) symmetry.Such a symmetry forbids a large number of four-fermion operators at tree level, and also a number of linear S interactions in the scalar potential that otherwise generate Q H✷ .Similarly, S B,C also have minimal one operator profiles containing only Q H .However, this again follows from the UV scale being introduced without a dynamical origin.In all these cases, a hierarchy problem in the UV sector is also introduced.
Table X also lists the cases of flavor singlet scalar fields that couple to through the S 2 H † H interaction and in addition have an independent SH † H interaction via a dimensionfull coupling.In these cases, the operators Q H and Q H✷ are simultaneously produced in tree level matchings.
As in the case of massive vectors and fermions, scalars can carry non-trivial representations under G Q or G L to isolate the coupling to a single fermion bi-linear.These states have been studied previously in Refs.[39][40][41][42][43].To avoid an explicit breaking of G Q or G L in this coupling, all of these states carry at least two non-trivial representations under the flavor (G Q or G L ) or gauge (SU(3) C or SU(2) L ) groups.For instance, consider integrating out "di-quark" states of this form discussed in Ref. [41] at tree level.A scalar current operator of the form ψ1L ψ 2R ψ2R ψ 1L is directly obtained.This operator can be projected into the Warsaw basis via Fierz transformation, As the "di-quark" scalars are in non-trivial representations under SU(2) L and/or SU(3) C groups, the index associated with these symmetries are not contracted between the fermions in each vector current, c.f. the right hand side of Eqn.(33).When reducing to the Warsaw basis one uses the SU(3) and SU(2) relations Concretely, performing this mapping for the "di-quark" scalars that couple to ūR Q L and dR Q L induce the operators Q (1,8) qu and Q (1,8) qd respectively.Similarly, the "diquark" scalars coupling to qq .On the other hand, exceptional cases that can generate only one operator do exist in "di-quark" scalars that couple to right handed SU(2) L bi-linears of the same fermion field i.e. to pairs of u R , d R and e R .These scalars can induce the single operator Q uu , Q ee , Q dd that are defined in the Warsaw basis, see the examples in Table XI.
In spite of only Q H being induced at tree level in cases S A,B,C and only one of the operators Q uu , Q dd and Q ee obtained at tree level in the cases S III -S VII , the arguments based on the mass scale generation from a UV Higgs mechanism with an associated extra scalar degree of freedom still hold.The heavy scalar (S) can be embedded in a larger scalar multiplet S ′ that develops a vev, or not so embedded, when a UV Higgs mechanism is invoked to introduce a new scale Λ ≫ v. Due to the fact that any field obtaining a vev with its self conjugate forms a singlet under G this leads to Q H✷ (as shown in Eqn. ( 24)) in either case, in addition to any matchings of S integrated out at tree level.
Alternatively, if a strong sector is present and the "diquark" scalar is composite, then the arguments in favor of "sibling" fields imply an extended spectrum that generically contains singlet composite states.Additionally, in the presence of a confining strong sector, both spin-0 and spin-1 composite states are expected to be embedded in a spin tower [45].Finally, some form of dimensional transmutation can be used to generate a scale.This can take place in the context of weaker couplings using the Coleman-Weinberg (CW) mechanism [44], or in the case with stronger couplings with a mechanism similar to the generation of Λ QCD .Of course, the (weak coupling version) of CW requires multiple couplings and generically a non-minimal UV particle spectrum.
It is also important to notice that there are scalar four-fermion current operators with the chiral structure ( LR)( LR) and ( LR)( RL) defined in the Warsaw basis.However, these operators are not constructed out of a pair of bi-linears with the same SM field content.As a result, additional vector current operators are induced when the ( LR)( LR) and ( LR)( RL) operators are obtained with a tree level exchange.Nevertheless, the presence of multiple four-fermion operators induced at tree level from the projection of some four-fermion scalar currents into the form of the Warsaw basis is clearly a more basis dependent conclusion than other arguments made in this paper.

IV. CONCLUSIONS
Driven by the question: "Can one obtain only one dimension six operator in the SMEFT from a consistent tree level matching onto an unknown new physics sector?", in this paper we have examined the non-minimal character of the SMEFT.
We addressed this question using a (G and CP) symmetry assumption to accommodate the large set of lower energy measurements that probe the symmetry breaking pattern of the SM into the TeV mass scale range and beyond.We have focused on the tree level matchings capturing the consistent IR limit of a new physics sector.Due to the extensive mixing of the operators in L 6 under renormalization [3][4][5], the SMEFT clearly has a non-minimal character once loop induced effects are considered, requiring many operators for consistent lower energy data analysis. 14  14 Such studies can also require mapping the SMEFT to a lower We have uncovered some cases where only one operator (neglecting flavor indices) is naively induced at tree level.These operators in the Warsaw basis are (H † H) 3 , or of the four-fermion form, and come about due to a massive scalar or vector field having non-trivial representations under the symmetry groups of the SM.We have found that vector fields can carry non-zero U(1) Y charge to reduce the operator profile by avoiding Higgs scalar currents being induced, but these fields have severe unitarity problems due to the lack of a three-point vector self-interaction.This indicates the presence of large non-perturbative matching corrections in addition to tree level matching effects.On the other hand, when the massive vector fields are not charged under U(1) Y , flavor symmetries can be introduced to reduce the IR SMEFT operator profile.In this case, a spurion symmetry breaking analysis shows scalar currents are still induced, leading to more operators at tree level.In practice, fitting to one pure lepton or down quark fourfermion operator is not as poorly motivated as fitting to up quark four-fermion operators due to the relative magnitudes of the flavor breaking spurions in each case.
In contrast to the vector fields, the presence of scalar fields in a UV sector do not directly cause severe unitarity problems.Integrating them out could have a relatively minimal operator profile, i.e.only Q uu , Q dd or Q ee is induced at tree level in the cases shown in Table XI, and only Q H in some cases in Table X.However, requiring a mass generation mechanism for these fields would lead to more matching contributions to the SMEFT operators.The scenario of a UV Higgs mechanism, if present, generically induces more operators that are constructed out of the SM Higgs field at tree level.This occurs without a suppression by a G Q × G L symmetry breaking spurion.When a UV Higgs mechanism is avoided by assuming compositeness and a new strong interaction, we have argued that the requirement of non-trivial representations for the vector and scalar fields to reduce the operator profile would indicate the presence of an extended spectrum of the composite states -including singlet fields -that couple through many SM portals.This would lead to more operators with tree level matchings when the extended spectrum is integrated out.
The SMEFT is a complicated field theory.It is natural and reasonable to seek a reduction of this complexity to use in data analyses in the SMEFT framework.Using symmetry assumptions is widely accepted.We have examined in this work if an alternative ad-hoc approach of using "one operator at a time" in data analyses can be representative of a consistent tree level matching to an unknown new energy Lagrangian, as in studies of B decays.Using the mapping of the SMEFT to C 9 and C 10 as reported in [46] our results support simultaneous fits to C 9 and C 10 .We do not find examples where the combination of Wilson coefficients in the SMEFT at tree level naturally cancel out in these lower energy parameters.This is largely due to the chirality of the relevant SMEFT operators.We thank a reviewer for a suggestive inquiry on this point in the review process.
physics sector.Our results show that the SMEFT has a non-minimal character quite generically and thus this approach should be avoided, if possible.To ensure the right conclusions are being drawn on the degree of constraint on unknown UV physics sectors, multiple operators should be retained in data analyses.Fortunately global data analyses in the SMEFT can already be performed with multiple operators, by using symmetries to simplify the number of parameters present.Further the growing LHC data set makes such global analyses even more feasible to execute in practice.In some cases, the resulting constraints can be relaxed by orders of magnitude [47,48] compared to a "one operator at a time" analysis.Nevertheless, our analysis shows that retaining multiple operators is preferred, and a relaxation of constraints can be required to obtain a consistent IR limit of an underlying UV physics sector, when a dynamical origin of the UV scales introduced is demanded.

TABLE VII .
Tree level L6 operators induced in the SMEFT with massive quarks integrated out.

TABLE IX .
Tree level L6 operators induced in the SMEFT with massive leptons integrated out.

TABLE XI .
[41]cases where a single L6 operator is generated at tree level for different scalar representations that are not singlets under the flavor group, without the insertion of spurion Yukawa fields, from Ref.[41].