On the non-minimal character of the SMEFT
aa r X i v : . [ h e p - ph ] A p r On the non-minimal character of the SMEFT
Yun Jiang and Michael Trott
Niels Bohr International Academy, University of Copenhagen,Blegdamsvej 17, DK-2100 Copenhagen, Denmark
When integrating out unknown new physics sectors, what is the minimal character of the StandardModel Effective Field Theory (SMEFT) that can result? In this paper we focus on a particularaspect of this question: “How can one obtain only one dimension six operator in the SMEFT froma consistent tree level matching onto an unknown new physics sector?” We show why this requiresconditions on the ultraviolet field content that do not indicate a stand alone ultraviolet completescenario. Further, we demonstrate how a dynamical origin of the ultraviolet scales assumed to existin 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 consistentmatchings induces multiple operators in the SMEFT quite generically. Global data analyses in theSMEFT 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, thearguments in favor of physics beyond the SM are verystrong. It is reasonable to expect that the low energyeffects of a new physics sector, which has a mass gap in itstypical mass scale(s) compared to the electroweak scale v ≃
246 GeV, could be resolved in the future. This isparticularly the case if Λ . π v , which is consistent withexpectations of ultraviolet (UV) physics motivated by nat-uralness concerns for the Higgs mass. Broad classes of newphysics scenarios consistent with this minimal decouplingassumption can be constrained efficiently using effectivefield theory methods to analyse scattering data limitedto energies √ s ∼ v ≪ Λ. This formalism has come tobe known as the Standard Model Effective Field Theory(SMEFT) recently [1–10] where the SM is supplementedwith a series of higher dimensional operators L SMEFT = L SM + L + L + L + L + · · · (1)In this approach, the null results of lower energy tests forphysics 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 thesymmetry breaking patterns of the SM must be accommo-dated. This can be accomplished in a manner that avoidsfine tuning. In this work, we use a symmetry assumptionthat new physics in the TeV mass range leads to a L cor-rection to the SM that (approximately) respects the globalsymmetry group G = U(1) B ⊗ U(1) L ⊗ SU(3) , and in addi-tion a discrete CP symmetry. Some of these symmetriescannot be exact, as the SM defines a minimal symmetrybreaking in the SMEFT. However, the G and CP sym-metry breaking pattern of the SM can allow the effect of Schematically denoted as Λ and assumed to be in the ∼ TeV range. The complete sum of all non-redundant operators at each massdimension order d defines L d here. This hypothesis is not the only way to accommodate TeV fieldcontent, see for example the discussion in Ref. [11]. the UV physics sectors in the ∼ TeV scales of experimen-tal interest because of two reasons: it follows a MinimalFlavor Violating (MFV) pattern for flavor changing mea-surements [12–14]; and it is proportional to the Jarlskoginvariant [15, 16] in the case of SM CP violation.In this paper, we consider matching patterns of opera-tors that result from integrating out new physics sectors.We study the effect of simultaneously integrating out notonly the heavy fields, but also a UV sector that generatesthe required heavy mass scale(s) Λ > v . We initially focuson the question of when, if ever, only one operator can beobtained in such a tree level matching in our chosen basisfor L [2]. We examine this question in the SMEFT interms of the matching effects of spin- { } fields thatcan couple to the SM through ( d ≤
4) mass dimensioninteractions. Higher spin composite fields (and spin tow-ers) are possible and even required in the presence of UVconfining strong interactions. Similarly, when a UV sectorwith a strong interaction is present, there is no particularreason in general for tree level matchings, as opposed tonon-perturbative matchings, to be the largest contributionto the Wilson coefficients. Such extra matching contribu-tions are difficult to characterize (other than by using thefull SMEFT formalism) and only reinforce our main pointon the non-minimal character of the SMEFT, so we do notfocus on these contributions.The number of operators induced in matching is oper-ator basis dependent. However, the conditions uncoveredon the UV field content to reduce the operator profile (i.e.the number of independent SU(3) × SU(2) × U(1) opera-tors) are still meaningful. The conditions can be framedin terms of symmetries and several simple observationson new physics spectra and dynamics that can generatea scale Λ, as we show. Practically speaking, most globalanalyses are being constructed using the well defined War-saw basis [2], so we focus on this basis when examining theone operator question. We use the notation Q i to denotean operator defined in the Warsaw basis in this work, andrefer the reader to Ref. [2] for the explicit operator defi-nitions. Note also that we refer to one operator with theunderstanding that, consistent with our assumptions of Gsymmetry, flavor indices are not used to distinguish oper-ators.The structure of this paper is as follows. Following abrief comment on the dimension-5 operator and Fermi the-ory in Section II, we provide in Section III.A a comprehen-sive discussion on the SMEFT matching at tree level onto L when a massive spin-1 state present in a UV physicssector is integrated out. We focus this discussion on the“one operator induced at tree level” question consistentwith the assumed (approximate) G symmetry. We demon-strate why such a simple UV sector cannot be a completescenario if a mechanism to generate the heavy state’s massis demanded. We then discuss the spin-1/2 case, drawinga similar conclusions in Section III.B. In Section III.C weexamine the case of integrating out a scalar field focusedon the “one operator” question. We show how the scalarcase is more subtle, but still argues for more operatorswhen UV complete scenarios are demanded. Section IVcontains our conclusions. II. TWO EXCEPTIONAL EFT CASES
When considering the one operator question, we notethat a few historical accidents in EFTs can be mislead-ing. First of all, L and L d with d ≥ L . The operator that results [17, 18], L = c ij (cid:16) L cL,i ˜ H ⋆ (cid:17) (cid:16) ˜ H † L L,j (cid:17) + h . c . (2)is the well known example where one operator at a par-ticular mass dimension does result when integrating outUV physics. The interplay of global U(1) L number viola-tion and the constraints of the SM field’s representationsleading to one operator in L is an exception that is not re-peated at higher orders in the SMEFT operator expansion[6–10].Historically, Fermi theory has frequently been used asa prototypical EFT to build intuition. This can be un-fortunate, as Fermi theory is atypical and has a numberof non-trivial accidental features that are not generic. InFermi theory, the four-fermion operator Q ℓℓ = (cid:0) L L γ µ L L (cid:1) (cid:0) L L γ µ L L (cid:1) , (3)is generated when the W boson is integrated out. This ef-fective operator is used in the process µ − → e − + ¯ ν e + ν µ toinfer the Fermi constant, G F . The UV sector in the case ofFermi theory is the SM which does induce a series of otheroperators at tree level, in addition to the operator Q ℓℓ .These four-fermion operators are due to the Higgs fieldand the Z boson. However, the highly suppressed Yukawacouplings of the SM Higgs to light fermions leads to anexceptional situation numerically in terms of the operator Here and below our notation with a c superscript indicates a chargeconjugate representation of a SM field. Case SU(3) C SU(2) L U(1) Y G Q G L Couples to V (1 , d R γ µ d R V (1 , u R γ µ u R V (1 , Q L γ µ Q L V (1 , Q L σ I γ µ Q L V (1 , d R γ µ d R V (1 , u R γ µ u R V (1 , d R γ µ u R V (1 , Q L γ µ Q L V (1 , Q L σ I γ µ Q L V (¯3 , ¯3,6 2 -1/6 (1,3,3) (1,1) ¯ d R γ µ Q cL V (¯3 , ¯3,6 2 5/6 (3,1,3) (1,1) ¯ u R γ µ Q cL TABLE I. Vector representations [21, 22] consistent with ourassumptions. The first three rows are the same field sub-classified. Superscripts on the field label indicate the repre-sentation under color. The Gell-Mann matrix T A (for bothcolor and flavor 8’s) is present but suppressed in the couplingto some fermion bi-linears. σ I is the Pauli matrix. The tablelargely follows from the SU(3) group relations ⊗ ¯3 = ⊕ and ⊗ = ⊕ ¯3 . profiles. The small Yukawa couplings are not formally theconsequence of a fine tuning, as they are protected by thefull chiral symmetry of the SM. More discussion on theaccidents in Fermi theory, and how it is commonly misun-derstood, can be found in Ref. [19].Arguably, there is some theoretical evidence based onthe structure and particle content of the SM in the direc-tion of embedding this model into SU(5) or SU(10), seefor example the arguments in Ref. [20]. This could be in-terpreted as a hint to an underlying theory, similar to thechiral structure of the SM being a low energy hint of itsUV structure. However, the problems of TeV scale grandunified theories are very well known. In this work we makea more phenomenologically motivated choice and assumeapproximate G symmetry (and CP symmetry). III. G SYMMETRIC TREE LEVEL MATCHINGSA. Spin states Spin-1 fields that couple to the SM quark bi-linears inthe manner assumed are given by Table I [21–23]. The re-quirement of linear couplings of mass dimension less thanfour, together with Lorentz symmetry and invariance un-der the full SM gauge group constrains the possible quan-tum numbers of UV field content. Fields with other repre-sentations that give SMEFT matchings respecting G arepossible, if these conditions are relaxed. Our notation isthat Q c , L c are the right handed conjugate doublet fieldsof the SM fermions. The global flavor symmetry in the Case SU(3) C SU(2) L U(1) Y G Q G L Couples to V (1)I e R γ µ e R V (1)I L L γ µ L L V (1)IV L L σ I γ µ L L V XII L cL γ µ e R V XIII e R γ µ e R V XIV ¯3 2 -1/6 (¯3,1,1) (¯3,1) ¯ L cL γ µ u R V XV ¯3 2 5/6 (1,¯3,1) (¯3,1) ¯ L cL γ µ d R V XVI ¯3 1 -2/3 (1,¯3,1) (1,3) ¯ e R γ µ d R V XVII ¯3 1 -5/3 (¯3,1,1) (1,3) ¯ e R γ µ u R V XVIII e R γ µ Q cL V XIX ¯3 1 -2/3 (1,1,¯3) (3,1) ¯ L L γ µ Q L V XX ¯3 3 -2/3 (1,1,¯3) (3,1) ¯ L L σ I γ µ Q L V XXI L L γ µ ¯ L L V XXII L L σ I γ µ ¯ L L TABLE II. Different vector representations that couple tofermion bi-linears respecting G, without the insertion of aYukawa matrix.Case SU(3) C SU(2) L U(1) Y G Q G L Couples to V (1)I , V (1)II , V (1)III H † iD µ H V (1)IV H † σ I iD µ H V (1)XXIII H T iD µ H V (1)XXIV H T iσ I D µ H TABLE III. Vector representations coupling to currents con-structed from Higgs fields. quark and lepton sectors are defined asG Q = SU(3) u R × SU(3) d R × SU(3) Q L , (4)G L = SU(3) L L × SU(3) e R . (5)It is also possible to have a vector field couple to leptonbi-linears, to quark-lepton bi-linears or have an interac-tion with the SM Higgs field. We list the correspondingfields in Table II and Table III. Cases V XII , V XIII havefields that carry a global lepton number and V XIV − V XX carry both lepton and baryon numbers. Although coun-terintuitive, UV fields that carry flavor quantum numbersdo not necessarily lead to lower energy signatures of flavorviolation – outside of the MFV pattern. Similarly, fieldscarrying lepton number do not necessarily lead to lowerenergy signatures of lepton flavor violation at tree level. Fields that carry both lepton and baryon number are po-tentially more problematic in inducing proton decay, but This was previously noted in Ref. [23] in the lepton number casefor flavor singlet fields.
Case Q i generated at tree level V (1)IV Q ll , Q (3) qq,lq , Q (3) Hq,Hl , Q H , Q H D , Q H ✷ , Q eH , Q uH , Q dH V (8)IV Q (1) qq , Q (3) qq V (8)IX Q (1) qq , Q (3) qq V XX Q (1) lq , Q (3) lq V (1)XXIII Q H , Q H D , Q H ✷ , Q eH , Q uH , Q dH V (1)XXIV Q H , Q H D , Q H ✷ , Q eH , Q uH , Q dH TABLE IV. Examples of the sets of L operators in the SMEFTobtained by integrating out various massive vectors. such phenomenological constraints are not the focus of thispaper. Dimension-6 operator matching
Solving the classical equations of motion (EOM) for theheavy vector fields and substituting the classical solutioninto the Lagrangian results in a direct tree level matchingin terms of a product of currents. We define the currentsas J a = { J µψ , J µH } = { ¯ ψ γ µ ⊗ ψ, ( D µ H ) † ⊗ Φ } , (6)and the tree level matching is given by∆ L ⊃ − M V ( J µa ) † J µb . (7)Here Φ represents H or ˜ H = i σ H ⋆ and ⊗ indicates agroup product characterized by the SU(2) L representationof vector fields. For a vector field of the form consideredin Tables I, II and III, the current product falls into oneof three types: • four-fermion: ( J µψ ) † J ψ,µ , • scalar derivative: ( J µH ) † J H,µ , • mixed scalar-fermion: ( J µψ ) † J H,µ , ( J µH ) † J ψ,µ .We have systematically examined the profile in terms ofoperators obtained in tree level matchings to the Warsawbasis from the fields listed in Tables I,II and III, findingthe following rule: Flavour singlet vector fields that do not break G Q × G L induce more than one operator at tree level when matchingonto the SMEFT Warsaw basis. This notation is consistent with Ref. [23]. Note also that a furthercurrent of the form D µ F µν with F = { B, W, G } is redundant [23]. Case Op U(1) Y G Q , G L Spurion V (1)VIII Q (1) qq T A Y † u Y u , T A Y † d Y d V (1)IX Q (1) qq T A Y † u Y u , T A Y † d Y d V XIX Q (1) lq -2/3 / V (¯3 , Q (1) qd -1/6 / V (¯3 , Q (1) qu V XVIII Q qe -5/6 / V XII Q le V XIV Q lu -1/6 / V XV Q ld V (1)V Q dd T A Y † d Y d V (1)VI Q uu T A Y † u Y u V (1)VII Q (1) ud -1 Y † d Y u V XIII Q ee T A Y † e Y e V XVI Q ed -2/3 / V XVII Q eu -5/3 /TABLE V. Operators induced at tree level when the massivevector case is integrated out. The cases are grouped in thetable into the chiral ( J µψ ) † J ψ,µ operator classes induced. Thetop section refers to LLLL operators. The middle section ofthe table refers to LLRR operators. The bottom section of thetable 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) fermionfields and also the scalar currents, inducing a large numberof operators at tree level. A vector field can be made tocouple to the left-handed doublets by assigning the fieldto a of SU(2) L . The scalar and leptonic couplings canbe removed by assigning the field to a of SU(3) C . Inthis case the operator profile is reduced to at least two( J µψ ) † J ψ,µ operators via the relation [2]( ¯ Q pL σ I T A γ µ Q rL )( ¯ Q sL σ I T A γ µ Q tL ) = − Q (3) qqptsr + 34 Q (1) qqptsr − Q (3) qqprst , (8)here p, r, s, t are flavor indices. We show some examples ofthe multiple operators induced when integrating out vec-tor 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 War-saw basis in the limit of vanishing Yukawa matrices; seeTable V. Note that with the exception of case V (1)VII whichhas a bi-linear flavor breaking spurion in Y † d and Y u , thepresence of a U(1) Y charge is also associated with the lackof Higgs scalar currents induced. This has an importantconsequence when the self interactions of the vector arestudied for unitarity violation, as will be discussed shortly.A spurion analysis allows the corrections due to thenonzero Yukawa matrices of the SM (that break the fla-vor symmetry in a phenomenologically safe MFV pattern)to be systematically studied. We define the SM Yukawa matrices Y u , Y d , Y e as L Y = − ( Y u ) pr ¯ u R,p Q rL ˜ H † − ( Y d ) pr ¯ d R,p Q rL H † − ( Y e ) pr ¯ e R,p L rL H † + h . c . (9)G Q × G L symmetry is restored if we endow the Yukawa ma-trices with the transformation properties under { G Q , G L } Y u ∼ (3 , , ¯3 , , , Y d ∼ (1 , , ¯3 , , ,Y e ∼ (1 , , , ¯3 , . (10)Introducing G Q × G L symmetry breaking when the Y u , Y d , Y e matrices take on their SM values gives more op-erators at tree level for fields with flavor quantum numbers.On general grounds, the ( J µH ) † J H,µ current products areinduced proportional to two spurions breaking the flavorsymmetry, and the ( J µψ ) † J H,µ , ( J µH ) † J ψ,µ current productsare induced proportional to one flavor breaking spurioninsertion. Here we refer to the spurions listed in TableV that are bi-linear in Yukawa matrices. As a specificexample consider V (1)VIII that is a under SU(3) Q L . TheLagrangian is given by L SM + L V (1)VIII where L V (1)VIII = −
12 ( D µ V ν D µ V ν − D µ V ν D ν V µ ) − M V V ν V ν + (cid:0) λ V V µ,A T A Y † u Y u ( D µ H ) † H + h . c . (cid:1) , (11)+ g V V µ,A ( ¯ Q L T A γ µ Q L ) + · · · . Note that the largest spurion that restores the flavor sym-metry for the second line is T A Y † u Y u and some indices aresuppressed in Eqn. (11). The additional spurion breakingproportional to Y † d Y d is neglected in what follows. In-tegrating by parts and the EOM for the vector field areused to manipulate the derivative to appear as shown onthe second line in Eqn. (11). Integrating out the field V (1)VIII using the classical EOM gives∆ L ⊃ g V M V (cid:20) Q (1) qqrssr − Q (1) qqrrss (cid:21) + 14 M V h ((Im λ V ) − (Re λ V ) ) Q H ✷ + 4(Im λ V ) Q HD +2 i (Re λ V )(Im λ V )( Y † b Q bH − Y b Q † bH ) − i (Re λ V )(Im λ V )( Y † u Q uH − Y u Q † uH ) i × (cid:20) Tr[( Y † u Y u )( Y † u Y u )] − (diag( Y † u Y u )) (cid:21) (12) − g V Im[ λ V ]2 M V Q (1) Hqpr (cid:20) ( Y † u Y u ) pr − diag( Y † u Y u )3 δ pr (cid:21) + i g V Re[ λ V ]2 M V (cid:20) (( Y † u Y u ) Y † a ) mi Q aHim − ( Y a ( Y † u Y u )) im Q † aHmi (cid:21) − i g V Re[ λ V ]6 M V Tr[ Y † u Y u ] (cid:20) ( Y † a ) mi Q aHim − ( Y a ) im Q † aHmi (cid:21) , Recall the flavor adjoint representation is real. where the dummy labels a and b are summed over { u, d } and { e, d } , respectively. A similar pattern of matchingsonto the class 3 ( D H ), 5 ( H ¯ ψψ ) and 7 ( H D ¯ ψψ ) op-erators of the Warsaw basis is present for almost all colorsinglet fields with flavor quantum numbers listed in Ta-bles II and III. The exceptional case is the field V XII whosenon-trivial SU(2) L representation and U(1) Y charge for-bids a scalar current from being induced at tree level inthis manner.The pattern of tree level matchings is strongly dictatedby the charges and representations of the UV fields underSU(3) C × SU(2) L × U(1) Y , G Q and G L . We emphasize,data fits to subsets of operators in the SMEFT formal-ism can be justified by appealing to UV field content withU(1) Y charges and non-trivial representations under SMgroups when only retaining tree level matching contribu-tions. See Table V for details on cases that generate onlyone operator at a time.This conclusion is subject to the following qualifications.First, the single operators obtained in tree level matchingsto the vectors in Tables II, III are limited to ( J µψ ) † J ψ,µ op-erator forms. Such operators at LHC contribute to contin-uum parton production in a fashion dictated by the powercounting of the theory. Conversely, the precise measure-ments made on a scattering through a SM resonance (withmass M and width Γ) parametrically has a Γ /M suppres-sion, compared to the leading resonant behavior, whenconsidering the interference with ( J µψ ) † J ψ,µ operators.Second, as y t ≃
1, a flavor symmetry spurion breakingproportional to only powers of Y u can induce operators ofclass 3, 5 and 7 without significant numerical suppression.This makes it difficult to justify “one at a time” data fits to( J µψ ) † 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 µψ ) † J ψ,µ operators that onlyhave leptonic or down quark field content can be poten-tially justified. In these cases the induced scalar currentsproportional to MFV like flavor breaking spurious are nu-merically suppressed compared to pure up quark spurionsby at least y b /y t ∼ − .Finally, we also note that we never obtain only one op-erator in such a tree level matching that involves the Higgsfield, in the cases of massive vector UV field content con-sidered. Arguments against orphaned vectors.
The vector fields listed in Tables I, II and III inducinga single L SMEFT operator at tree level, carry at leastone non-trivial representation under the SM gauge sym-metry and flavor symmetries. Non-trivial representationsand U(1) Y charges reduce the interactions for SM particleswith the new sector, which consequently minimizes the IR In all cases but one, multiple non-trivial representations arepresent. The one exceptional case is V XIII which is only an under SU(3) e R . Case π ǫ δ Z g V hOi − π ǫ δ Z ¯ ψ g V − π ǫ δ Zψ g V − π ǫ δ ZV g V β y V (1)VIII F (3 Fl ) C (3 Fl ) F C (3 Fl ) F (cid:0) (cid:1) Fl · C + V (1)IX F (2) F (3 Fl ) C (2) F C (3 Fl ) F C (2) F C (3 Fl ) F · C (cid:0) (cid:1) Fl (cid:0) (cid:1) L + V XIX Fl · C Fl 23 · V (¯3)X , XI − C Fl ( − C · Fl ( − C 23 · ( − C −V (6)X , XI Fl · C · Fl · C 23 · C + V XVIII Fl · C · Fl 23 + V XII Fl Fl · + V XIV Fl · C Fl · + V XV Fl · C Fl · + V (1)V F (3 Fl ) C (3 Fl ) F C (3 Fl ) F (cid:0) (cid:1) Fl · C + V (1)VI F (3 Fl ) C (3 Fl ) F C (3 Fl ) F (cid:0) (cid:1) Fl · C + V (1)VII Fl Fl 23 · C + V XIII F (3 Fl ) C (3 Fl ) F C (3 Fl ) F (cid:0) (cid:1) Fl + V XVI Fl · C Fl 23 + V XVII Fl · C Fl 23 +TABLE VI. One loop renormalization results. Here h O i indi-cates the matrix element of the vector-fermion bilinear inter-action term and δ Z corresponds to the divergence present inthis three point interaction at one loop from the vector-fermioncoupling. The notation is such that F ( N ) ≡ C ( N ) F − N with C ( N ) F = N − N . We have labeled several of the numerical factorsin 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.
V V ψ ¯ ψψ ¯ ψ V FIG. 1. Diagrams relevant for the renormalization of g V . SMEFT operator profile. However, such fields in generaldo not indicate a stand alone UV complete scenario (wherethe vector could be an “orphan”) for the following reasons. (1) Landau poles and triviality.
The β function ofthe coupling of the vector fields to the fermion bi-linears(denoted g V in Eqn. (11)) is determined by renormalizingthe fermion fields and vector field two point functions, andsubsequently extracting the β function for g V . We relatethe bare (0) and renormalized ( r ) fields and couplings as V (0) µ = p Z V V ( r ) µ , g (0) V = Z g V g ( r ) V µ ǫ , (13) ψ (0) i = p Z ψ i ψ ( r ) i , (14)where Z x = 1 + δ Z x for x = { V, g V , ¯ ψ, ψ } . We use arenormalization scheme employing MS subtraction and d = 4 − ǫ dimensions using standard methods. The rele-vant diagrams are shown in Figure 1. The β -function forthe running of the coupling g V is given by β g V = 2 g V ǫ (cid:18) − δ Z hOi − δ Z ¯ ψ − δ Z ψ − δ Z V (cid:19) , (15)where the renormalization factors δ Z ’s for the various vec-tor field cases are presented in Table VI. The general ex-pectation is that g V will have a positive β function – in-dicating Landau poles [24], quantum triviality [25] anda UV incompletion. This is indeed the case for all vectorfields inducing one ( J µψ ) † J ψ,µ operator, with the exceptionof color ¯3 vectors coupling to quark bi-linears; i.e. cases V ¯3X , XI . In this exceptional case, the SU(3) C vector-fermioncoupling mimics the effect of a non-abelian interaction.An oversimplified UV scenario afflicted with an internalinconsistency indicated by the presence of Landau polescannot formally generate a consistent IR limit. This indi-cates that further new physics must be present below theLandau pole scale Λ L approximated byΛ L ∼ M V exp [ g V /β g V ] . (16)However, numerically corrections suppressed by Λ L aresmaller than one loop matching effects. (2) Unitarity and vector self-interactions. A moreintractable problem is generated by O (1) self interactionsof orphan vector fields. The four point vector self inter-action is not forbidden by any symmetry. Conversely thethree point interaction can be forbidden by the presence ofa U(1) Y charge in the composite field. Consider the 2 → L V = λ V † µ V µ V † ν V ν + g ′ ∂ µ V µ V † ν V ν + . . . (17)The amplitudes at leading order with the high-energyapproximation for the vector polarization ǫ µL ≃ p µ /M V through a s-, t- and u- channel vector exchange and afour-point contact interaction, respectively, read M L , s = ( g ′ ) F s st − su M V , (18) M L , t = ( g ′ ) F t st − ut M V , (19) M L , u = ( g ′ ) F u us − ut M V , (20) M L = λ (cid:18) F s t − u M V + F t s − u M V + F u s − t M V (cid:19) . (21)Here abstract group structure constants F s,t,u for threechannels have been introduced. For example, in the model V (1)IX : F s = f ABE f CDE f ijn f kln where A, B, C, D, E referto the flavor index and i, j, k, l, n denote the iso-spin index.If λ = ( g ′ ) is accomplished by a global symmetry thenthe amplitudes M L will cancel with three terms in M L with an identical F factor respectively, through the Man-delstam relation s + t + u = 4 M V . As a result, the leading V A,aµi V B,bνj V C,cρ,k V D,dσ,l
VV VV VV VV
FIG. 2. 2 → scaling in ∼ ( p ) /M V disappears. The full amplitudesthen grow as ∼ p /M V . However, if the three-point inter-action is forbidden - for example due to the field carrying aU(1) Y charge - then the amplitude cannot be so moderatedin its growth at high energies, and scales as ∼ ( p ) /M V .In this manner, the presence of a U(1) Y charge forbiddingthe scalar current simultaneously turns off the three-pointinteraction that is required to moderate the high energyscattering behavior of an orphaned vector field.Standard partial wave unitarity arguments [26–28] givethat the unitarity violation scale associated with the vectorfield without a three point interaction isΛ . . M V ( F t + F u ) − / λ − / , (22)where F t , an F u are determined by a particular scatteringcross section. A quick onset of unitarity violation followsfrom a sizable four-point interaction that is expected toemerge from a strongly interacting composite sector ongeneral grounds. Even introducing a loop suppression tothe vector self interaction, that is λ ∼ (16 π ) − , is of littlehelp - one still finds Λ ∼ M V due to the presence of afourth root in Eqn. (22). Hence, the UV strong sectorshould be simultaneously considered to define a consistentmatching onto the SMEFT. This would increase the lowenergy operator profile of such a scenario in the SMEFTbeyond one operator generically due to non perturbativematchings, and a “one at a time” analysis invoking a treelevel matching would be logically incoherent. (3) Siblings of massive vectors with non-trivialrepresentations. A massive vector field with non-trivialrepresentations under subgroups of G is also generically ac-companied by more “sibling” fields. If the massive vectorgains a mass by a UV Higgs mechanism, the correspond-ing sibling field includes at least a scalar ( S ) obtaining avacuum expectation value (vev). Define this expectationvalue as h S † S i = v ′ /
2. We require dim( V ) + 1 ≤ dim(S)so that all of the components of the vector become massivein the presence of a scalar field obtaining a vev, througheaten Goldstone components of S . One can use the global symmetry rotations on S to ro-tate the new vev to a (uneaten) component of S , denoted s . The interaction of s with H † H cannot be forbidden byan explicit G breaking without violating our assumptions.This would introduce highly constrained low energy effectsinto the SMEFT through the vev v ′ leading to the vectormass matrix. A vacuum misalignment [32] is assumed to An additional non goldstone incomplete scalar multiplet is fa-mously required when introducing a vev in this manner [29–31]. make the vector mass matrix symmetric under G in thiswork. This results in the Higgs portal coupling not beingsuppressed by a G Q × G L breaking spurion. Concretelyconsider the Lagrangian L SH = ( D µ S ) † ( D µ S ) − λ ′ S † S − v ′ + λ SH S † S H † H. (23)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-trivialrepresentations that the V multiplet carries. S is expandedas S = ( · · · , v ′ + s + · · · ) / √ h ′ a ρ a where ρ a corre-sponds to the goldstone components of the S multipletthat are eaten to generate the vector mass, and the · · · fill out the full dimension of S . The vev v ′ must be ar-ranged to break the dim( V ) h ′ a generators. Simultaneously v ′ must not break the G subgroup, so g a h S i = 0, wherethe generators of G are denoted g a . Integrating out s after UV symmetry breaking gives∆ L = − λ SH λ ′ m s Q H ✷ − g V λ ′ ( V µ V µ ) + · · · (24)in addition to the operators induced by integrating outthe vector field. Here the scalar mass is m s = λ ′ v ′ /
2. Inaddition, L terms are induced that require a finite redef-inition of λ and v in the SM to rearrange L SM back intostandard from. Here we have neglected many higher ordereffects including subdominant mass splitting terms. Notethe sizable vector four point interaction, that is enhancedin the λ ′ → λ SH .In order to avoid assuming a UV Higgs mechanism, wecan consider a composite massive vector generated by ahypothetical UV strong sector, with spin-1/2 constituentsΨ, so that the vector fields are V µ ∼ h ¯Ψ γ µ Ψ i conden-sates. This composite field carries at least one non-trivialrepresentation under one of the groups G Q , G L , SU(3) C orSU(2) L to reduce the SMEFT operator profile to one op-erator. Denote this non-trivial representation as N , andthe corresponding group as G ′ . The Ψ are charged underG ′ or a larger group H with H ⊃ G ′ .We can consider G ′ or the proper subgroup case whereG ′ ⊂ H without loss of generality with the following ar-guments. The Ψ belongs to SU(3), and N ∈ { , ¯3 , , } ,or SU(2) with N = { , } for the vector fields of inter-est. The non-trivial representations in N can be generatedfrom tensor products of the Ψ irreducible representations.In the case where the Ψ belongs to SU(3) we denote theirreducible representations as P , R , which need not havethe same dimension. When N is generated by P ⊗ ¯P the In general one expects the symmetry breaking pattern to be suchthat there will be uneaten goldstone bosons, or additional massivevectors in the spectrum. Here we are considering an exceptionalminimal spectrum when examining the one operator question. singlet representation is also generated in the tensor prod-uct. A color singlet sibling under SU(3) C is expected witha mass proximate to a color octet vector, which induces anumber of operators in L when integrated out. Similarly,a flavor singlet sibling under a flavor group is also expectedfor flavor octets. Interestingly, the flavor vector fields weconsidered all have zero U(1) Y charges, so their flavor sin-glet siblings with the same U(1) Y charge are not forbiddenby the flavor symmetry to have the coupling with the cor-responding quark bi-linear and also with the J H,µ of van-ishing U(1) Y charge, inducing more than one operator in L when integrated out. When N ∈ P ⊗ P multiple rep-resentations result, for example in the case of P = , the ¯3 and fields are simultaneously present. Such fields ( V X , XI )can induce the same operator when integrated out. On theother hand, these fields necessarily carry U(1) Y , and thushave a cut off scale proximate to the massive vectors massscale for the cases consistent with our assumptions. Nextwe consider the cases when the non-trivial representationis generated by bi-linears of Ψ carrying representationsof unequal dimension N ∈ P ⊗ R . By inspection of thetensor products of SU(3) with triality 0 and 1 [37] it is pos-sible to generate each N ∈ { , ¯3 , , } for SU(3) in such amanner. However, for each P and R one can also form acondensate h ¯Ψ γ µ Ψ i with zero U(1) Y charge from the prod-uct P ⊗ ¯P , R ⊗ ¯R . Two more pure singlet spin one boundstates proximate in mass to M V are expected in the spec-trum, unless forbidden by another symmetry. Restrict-ing the discussion for non-trivial SU(2) representations tothe vector cases that do not carry U(1) Y and induce oneoperator at tree level, we are left with the field V . Fur-ther, V has a large flavor breaking spurion proportionalto the top Yukawa generating more operators at tree levelwhen integrated out, see Table V.For all of these reasons, orphaned vector fields with non-trivial representations of the SM symmetry groups demandsiblings and a “good UV home”. B. Spin / states If heavy spin-1/2 states are integrated out, the mass ofthe massive fermion(s) (denoted M with M ≫ v ) must beintroduced in some manner. As discussed in the previoussection, a chiral fermion with a UV Higgs mechanism in-duces more operators at tree level when integrating out theUV scalar field. In this section, we confine the discussionto general vector like fermions. The general Lagrangianassociated with a pair of heavy vector-like quark (VLQ) This is a generic expectation if the Ψ are charged under U(1) Y . For example, some of the expected spectra degeneracy can be liftedin analogy to the η ′ [33]. Here we are referring to the dominant component of the mass ofthe fermion from the new sector. In addition, there will be masscontributions and splitting proportional to ∼ v . As such effectsdo not act to reduce the IR SMEFT operator profile, we neglectthese contributions. Case SU(2) L U(1) Y J Q L Q uH Q dH Q (1) Hq Q (3) Hq Q (1)I − ¯ Q L H √ √ √Q (3)I − σ I ¯ Q L H √ √ √ √Q (1)II ¯ Q L H ∗ √ √ √Q (3)II σ I ¯ Q L H ∗ √ √ √ √ Case SU(2) L U(1) Y J Q R Q uH Q dH Q Hu Q Hd Q Hud Q III ¯ u R H T √ √ √ √ √Q IV ¯ d R H † √ √ √ √ √Q V ¯ u R H † √ √Q VI − ¯ d R H T √ √ TABLE VII. Tree level L operators induced in the SMEFTwith massive quarks integrated out. denoted by Q L , Q R that are flavor singlets includes L Q = L Q + L int Q , (25)where L Q = ¯ Q L i /D Q L + ¯ Q R i /D Q R − M (cid:0) ¯ Q L Q R + ¯ Q R Q L (cid:1) (26)for the SU(2) L singlet and doublets and L Q = Tr (cid:2) ¯ Q L i /D Q L + ¯ Q R i /D Q R (cid:3) − M Tr (cid:2) ¯ Q L Q R + ¯ Q R Q L (cid:3) (27)for the of SU(2) L . The interaction term L int Q for theVLQs to the SM fermions through the Higgs doublet isdefined as L int Q = J Q L Q R + J Q R Q L + h . c . (28)The requirement that the action be stationary under vari-ations of the heavy VLQ fields ¯ Q L , ¯ Q R results in two cou-pled EOMs: i /D Q L − M Q R + ( J Q R γ ) † = 0 , (29) i /D Q R − M Q L + ( J Q L γ ) † = 0 . (30)Mathematically, the coupled Eqns. (29) and (30) can besolved iteratively. Taking the limit of large M , one canexpand the classical solutions schematically as Q R = ( J Q R γ ) † M + i /DM ( J Q L γ ) † + · · · , (31) Q L = ( J Q L γ ) † M + i /DM ( J Q R γ ) † + · · · . (32)When substituted back into Eqn. (25) the effect of theleading term in these solutions vanishes due to chirality.We generically find that multiple operators are inducedat tree level when integrating out a vector like fermion.The cases where the vector like quark do not carry flavorquantum numbers are shown in Table VII. In the casesthat the VLQs carry flavor quantum numbers, previouslydiscussed in Refs. [34–36], multiple operators are againobtained. We show some sample cases of this type in Ta-ble VIII. Multiple operators at tree level are also obtainedin the case of integrating out vector like leptons, see Ta-ble IX. Case SU(2) L U(1) Y G Q J Q R Q uH Q dH Q Hu Q Hd Q VII (3,1,1) ¯ u R H T √ √Q VIII (1,3,1) ¯ d R H † √ √ TABLE VIII. Tree level L operators induced in the SMEFTwith massive quarks integrated out in some sample cases withflavor quantum numbers, see Refs. [34–36] for more discussionon the phenomenology of these fields.Case SU(2) L U(1) Y J L L Q (1) Hl Q (3) Hl Q eH Q (1) He L (1)I − L L H √ √ √L (3)I − σ I ¯ L L H √ √ √ Case SU(2) L U(1) Y J L R Q (1) Hl Q (3) Hl Q eH Q (1) He L III − ¯ e R H † √ √L IV − ¯ e R H T √ √ TABLE IX. Tree level L operators induced in the SMEFTwith massive leptons integrated out. 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 numberof interactions and naively generate many operators in theIR SMEFT matching limit. However, the examples ( S A , S B and S C in Table X) discussed in Refs. [8, 38] show thatonly one operator Q H , can be obtained if an explicit scaleis introduced without a dynamical origin to give the scalara mass. For instance, S A couples through linear and bilin-ear 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 ad-ditional U(1) symmetry. Such a symmetry forbids a largenumber of four-fermion operators at tree level, and alsoa number of linear S interactions in the scalar potentialthat otherwise generate Q H ✷ . Similarly, S B,C also haveminimal one operator profiles containing only Q H . How-ever, this again follows from the UV scale being introducedwithout a dynamical origin. In all these cases, a hierarchyproblem in the UV sector is also introduced.Table X also lists the cases of flavor singlet scalar fieldsthat couple to through the S H † H interaction and in addi-tion have an independent S H † H interaction via a dimen-sionfull 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, scalarscan carry non-trivial representations under G Q or G L toisolate the coupling to a single fermion bi-linear. Thesestates have been studied previously in Refs. [39–43]. Toavoid an explicit breaking of G Q or G L in this coupling,all of these states carry at least two non-trivial represen-tations under the flavor (G Q or G L ) or gauge (SU(3) C or SU(2) L ) groups. For instance, consider integratingout “di-quark” states of this form discussed in Ref. [41]at tree level. A scalar current operator of the form Case SU(2) L U(1) Y Couplings Q H Q H ✷ S A / V ( H, S A ) √S B / H ) † S B + h . c . √S C / H † S C H † H + h . c . √S S S I + ( S I ) † S I ) H † H √ √S S S I σH † H , ( S I ) † S I H † H √ √S − S S II H T H , ( S II ) † S II H † H √ √S − S S II σH T H , ( S II ) † S II H † H √ √ TABLE X. L operators obtained at tree level when flavor andcolour singlet scalars are integrated out. Λ S indicates a dimen-sionfull coupling.Case SU(3) C SU(2) L U(1) Y G Q Couples to Op S III u R u R Q uu S IV ¯6 1 -4/3 (¯6,1,1) u R u R Q uu S V d R d R Q dd S VI ¯6 1 2/3 (1,¯6,1) d R d R Q dd Case SU(3) C SU(2) L U(1) Y G L Couples to Op S VII e R e R Q ee TABLE XI. The cases where a single L operator is generatedat tree level for different scalar representations that are notsinglets under the flavor group, without the insertion of spurionYukawa fields, from Ref. [41]. ¯ ψ L ψ R ¯ ψ R ψ L is directly obtained. This operator can beprojected into the Warsaw basis via Fierz transformation,( ¯ ψ L ψ R )( ¯ ψ R ψ L ) = −
12 ( ¯ ψ L γ µ ψ L )( ¯ ψ R γ µ ψ R ) . (33)As the “di-quark” scalars are in non-trivial representationsunder SU(2) L and/or SU(3) C groups, the index associ-ated with these symmetries are not contracted betweenthe fermions in each vector current, c.f. the right handside of Eqn. (33). When reducing to the Warsaw basis oneuses the SU(3) and SU(2) relations T Aij T Akl = 12 δ il δ jk − δ ij δ kl , (34) σ Ijk σ Imn = 2 δ jn δ mk − δ jk δ mn . (35)Concretely, performing this mapping for the “di-quark”scalars that couple to ¯ u R Q L and ¯ d R Q L induce the op-erators Q (1 , qu and Q (1 , qd respectively. Similarly, the “di-quark” scalars coupling to Q L Q L generate Q (1 , qq . On theother hand, exceptional cases that can generate only oneoperator do exist in “di-quark” scalars that couple to righthanded SU(2) L bi-linears of the same fermion field i.e. topairs of u R , d R and e R . These scalars can induce the sin-gle operator Q uu , Q ee , Q dd that are defined in the Warsawbasis, 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 argu-ments based on the mass scale generation from a UV Higgs mechanism with an associated extra scalar degree of free-dom still hold. The heavy scalar ( S ) can be embeddedin a larger scalar multiplet S ′ that develops a vev, or notso embedded, when a UV Higgs mechanism is invoked tointroduce a new scale Λ ≫ v . Due to the fact that anyfield obtaining a vev with its self conjugate forms a sin-glet under G this leads to Q H ✷ (as shown in Eqn. (24)) ineither case, in addition to any matchings of S integratedout at tree level.Alternatively, if a strong sector is present and the “di-quark” scalar is composite, then the arguments in favorof “sibling” fields imply an extended spectrum that gener-ically contains singlet composite states. Additionally, inthe presence of a confining strong sector, both spin-0 andspin-1 composite states are expected to be embedded ina spin tower [45]. Finally, some form of dimensionaltransmutation can be used to generate a scale. This cantake place in the context of weaker couplings using theColeman-Weinberg (CW) mechanism [44], or in the casewith stronger couplings with a mechanism similar to thegeneration of Λ QCD . Of course, the (weak coupling ver-sion) of CW requires multiple couplings and generically anon-minimal UV particle spectrum.It is also important to notice that there are scalarfour-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 pairof 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 atree level exchange. Nevertheless, the presence of multiplefour-fermion operators induced at tree level from the pro-jection of some four-fermion scalar currents into the formof the Warsaw basis is clearly a more basis dependent con-clusion than other arguments made in this paper. IV. CONCLUSIONS
Driven by the question: “Can one obtain only one di-mension six operator in the SMEFT from a consistent treelevel matching onto an unknown new physics sector?”, inthis paper we have examined the non-minimal characterof the SMEFT.We addressed this question using a (G and CP) sym-metry assumption to accommodate the large set of lowerenergy measurements that probe the symmetry breakingpattern of the SM into the TeV mass scale range and be-yond. We have focused on the tree level matchings captur-ing the consistent IR limit of a new physics sector. Due tothe extensive mixing of the operators in L under renor-malization [3–5], the SMEFT clearly has a non-minimalcharacter once loop induced effects are considered, requir-ing many operators for consistent lower energy data anal-ysis. Such studies can also require mapping the SMEFT to a lower H † H ) , or ofthe four-fermion form, and come about due to a massivescalar or vector field having non-trivial representations un-der the symmetry groups of the SM. We have found thatvector fields can carry non-zero U(1) Y charge to reduce theoperator profile by avoiding Higgs scalar currents being in-duced, but these fields have severe unitarity problems dueto the lack of a three-point vector self-interaction. Thisindicates the presence of large non-perturbative match-ing corrections in addition to tree level matching effects.On the other hand, when the massive vector fields are notcharged under U(1) Y , flavor symmetries can be introducedto reduce the IR SMEFT operator profile. In this case, aspurion symmetry breaking analysis shows scalar currentsare still induced, leading to more operators at tree level.In practice, fitting to one pure lepton or down quark four-fermion operator is not as poorly motivated as fitting toup quark four-fermion operators due to the relative mag-nitudes of the flavor breaking spurions in each case.In contrast to the vector fields, the presence of scalarfields in a UV sector do not directly cause severe unitar-ity problems. Integrating them out could have a relativelyminimal operator profile, i.e. only Q uu , Q dd or Q ee is in-duced at tree level in the cases shown in Table XI, andonly Q H in some cases in Table X. However, requiringa mass generation mechanism for these fields would leadto more matching contributions to the SMEFT operators.The scenario of a UV Higgs mechanism, if present, gener-ically induces more operators that are constructed out ofthe SM Higgs field at tree level. This occurs without a sup-pression by a G Q × G L symmetry breaking spurion. Whena UV Higgs mechanism is avoided by assuming compos-iteness and a new strong interaction, we have argued thatthe requirement of non-trivial representations for the vec-tor and scalar fields to reduce the operator profile wouldindicate the presence of an extended spectrum of the com-posite states - including singlet fields - that couple throughmany SM portals. This would lead to more operators withtree level matchings when the extended spectrum is inte-grated out.The SMEFT is a complicated field theory. It is naturaland reasonable to seek a reduction of this complexity to usein data analyses in the SMEFT framework. Using symme-try assumptions is widely accepted. We have examined inthis work if an alternative ad-hoc approach of using “oneoperator at a time” in data analyses can be representativeof a consistent tree level matching to an unknown new energy Lagrangian, as in studies of B decays. Using the mappingof the SMEFT to C and C as reported in [46] our results supportsimultaneous fits to C and C . We do not find examples wherethe combination of Wilson coefficients in the SMEFT at tree levelnaturally cancel out in these lower energy parameters. This islargely due to the chirality of the relevant SMEFT operators. Wethank a reviewer for a suggestive inquiry on this point in the reviewprocess. physics sector. Our results show that the SMEFT has anon-minimal character quite generically and thus this ap-proach should be avoided, if possible. To ensure the rightconclusions are being drawn on the degree of constraint onunknown UV physics sectors, multiple operators should beretained in data analyses. Fortunately global data analy-ses in the SMEFT can already be performed with multipleoperators, by using symmetries to simplify the number ofparameters present. Further the growing LHC data setmakes such global analyses even more feasible to executein practice. In some cases, the resulting constraints canbe relaxed by orders of magnitude [47, 48] compared to a“one operator at a time” analysis. Nevertheless, our anal-ysis shows that retaining multiple operators is preferred,and a relaxation of constraints can be required to obtaina consistent IR limit of an underlying UV physics sector,when a dynamical origin of the UV scales introduced isdemanded. Acknowledgements
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