On the Non-Minimal Character of the SMEFT

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On the non-minimal character of the SMEFT

Jiang, Yun; Trott, Michael Robert

Published in: Physics Letters B

DOI: 10.1016/j.physletb.2017.04.053

Publication date: 2017

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Citation for published version (APA): Jiang, Y., & Trott, M. R. (2017). On the non-minimal character of the SMEFT. Physics Letters B, 770, 108-116. https://doi.org/10.1016/j.physletb.2017.04.053

Download date: 09. apr.. 2020 Physics Letters B 770 (2017) 108–116

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Physics Letters B

www.elsevier.com/locate/physletb

On the non-minimal character of the SMEFT ∗ Yun Jiang, Michael Trott

Niels Bohr International Academy, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark a r t i c l e i n f o a b s t r a c t

Article history: When integrating out unknown new physics sectors, what is the minimal character of the Standard Model Received 23 December 2016 Effective Field Theory (SMEFT) that can result? In this paper we focus on a particular aspect of this Received in revised form 20 April 2017 question: “How can one obtain only one dimension six operator in the SMEFT from a consistent tree level Accepted 22 April 2017 matching onto an unknown new physics sector?” We show why this requires conditions on the ultraviolet Available online 26 April 2017 field content that do not indicate a stand alone ultraviolet complete scenario. Further, we demonstrate Editor: B. Grinstein 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. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction the SM must be accommodated. This can be accomplished in a manner that avoids fine tuning. In this work, we use a symmetry Despite the null results on beyond the Standard Model (SM) assumption that new physics in the TeV mass range leads to a L6 resonance searches from Run I and II at LHC, the arguments in correction to the SM that (approximately) respects the global sym- 5 favor of physics beyond the SM are very strong. It is reasonable metry group G = U(1)B ⊗ U(1)L ⊗ SU(3) , and in addition a discrete to expect that the low energy effects of a new physics sector, CP symmetry.3 Some of these symmetries cannot be exact, as the 1 which has a mass gap in its typical mass scale(s) compared to SM defines a minimal symmetry breaking in the SMEFT. However, the electroweak scale v  246 GeV, could be resolved in the future. the G and CP symmetry breaking pattern of the SM can allow the This is particularly the case if  4 π v, which is consistent with effect of the UV physics sectors in the ∼ TeV scales of experimental expectations of ultraviolet (UV) physics motivated by naturalness interest because of two reasons: it follows a Minimal Flavor Violat- concerns for the Higgs mass. Broad classes of new physics sce- ing (MFV) pattern for flavor changing measurements [12–14]; and narios consistent with this minimal decoupling assumption can be it is proportional to the Jarlskog invariant [15,16] in the case of SM constrained efficiently using effective field√ theory methods to anal- CP violation. ∼  yse scattering data limited to energies s v . This formalism In this paper, we consider matching patterns of operators that has come to be known as the Standard Model Effective Field The- result from integrating out new physics sectors. We study the ef- ory (SMEFT) recently [1–10] where the SM is supplemented with fect of simultaneously integrating out not only the heavy fields, but 2 a series of higher dimensional operators also a UV sector that generates the required heavy mass scale(s)

LSMEFT = LSM + L5 + L6 + L7 + L8 +··· (1) > 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 In this approach, the null results of lower energy tests for physics chosen basis for L6 [2]. We examine this question in the SMEFT beyond the SM can be consistent with beyond-the-SM UV physics in terms of the matching effects of spin-{1,1/2,0} fields that can in the ∼ TeV mass scale range. However, a large set of experi- couple to the SM through (d ≤ 4) mass dimension interactions. mental measurements that test the symmetry breaking patterns of 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, * Corresponding author. E-mail address: [email protected] (M. Trott). 1 Schematically denoted as and assumed to be in the ∼ TeV range. 2 The complete sum of all non-redundant operators at each mass dimension order 3 This hypothesis is not the only way to accommodate TeV field content, see for d defines Ld here. example the discussion in Ref. [11]. http://dx.doi.org/10.1016/j.physletb.2017.04.053 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116 109 there is no particular reason in general for tree level matchings, as Table 1 opposed to non-perturbative matchings, to be the largest contribu- Vector representations [21,22] consistent with our assumptions. The first three rows tion to the Wilson coefficients. Such extra matching contributions are the same field sub-classified. Superscripts on the field label indicate the repre- sentation under color. The Gell-Mann matrix T A (for both color and flavor 8’s) is are difficult to characterize (other than by using the full SMEFT present but suppressed in the coupling to some fermion bi-linears. σI is the Pauli ¯ formalism) and only reinforce our main point on the non-minimal matrix. The table largely follows from the SU(3) group relations 3 ⊗ 3 = 1 ⊕ 8 and ¯ character of the SMEFT, so we do not focus on these contributions. 3 ⊗ 3 = 6 ⊕ 3.

The number of operators induced in matching is operator ba- Case SU(3)C SU(2)L U(1)Y GQ GL Couples to sis dependent. However, the conditions uncovered on the UV field V(1,8) ¯ μ I 1,8 1 0 (1,1,1) (1,1) dR γ dR content to reduce the operator profile (i.e. the number of inde- V(1,8) ¯ μ II 1,8 1 0 (1,1,1) (1,1) uR γ uR pendent SU(3) × SU(2) × U(1) operators) are still meaningful. The V(1,8) ¯ μ III 1,8 1 0 (1,1,1) (1,1) Q L γ Q L conditions can be framed in terms of symmetries and several sim- V(1,8) ¯ I μ IV 1,8 3 0 (1,1,1) (1,1) Q L σ γ Q L ple observations on new physics spectra and dynamics that can (1,8) ¯ V 1,8 1 0 (1,8,1) (1,1) d γ μ d generate a scale , as we show. Practically speaking, most global V R R (1,8) V 1,8 1 0 (8,1,1) (1,1) u¯ γ μ u analyses are being constructed using the well defined Warsaw ba- VI R R (1,8) ¯ V 1,8 1 −1(3,3,1)¯ (1,1) d γ μ u sis [2], so we focus on this basis when examining the one operator VII R R V(1,8) ¯ μ VIII 1,8 1 0 (1,1,8) (1,1) Q L γ Q L question. We use the notation Q i to denote an operator defined (1,8) ¯ I μ V 1,8 3 0 (1,1,8) (1,1) Q L σ γ Q L in the Warsaw basis in this work, and refer the reader to Ref. [2] IX¯ V(3,6) ¯ − ¯ μ c X 3,6 2 1/6 (1,3,3) (1,1) dR γ Q L for the explicit operator definitions. Note also that we refer to one (3¯,6) V 3,6¯ 2 5/6 (3,1,3) (1,1) u¯ γ μ Q c operator with the understanding that, consistent with our assump- XI R L tions of G symmetry, flavor indices are not used to distinguish operators. is generated when the W boson is integrated out. This effective − − The structure of this paper is as follows. Following a brief operator is used in the process μ → e + ν¯e + νμ to infer the comment on the dimension-5 operator and Fermi theory in Sec- Fermi constant, G F . The UV sector in the case of Fermi theory is tion 2, we provide in Section 3.1 a comprehensive discussion on the SM which does induce a series of other operators at tree level, L the SMEFT matching at tree level onto 6 when a massive spin-1 in addition to the operator Q . These four-fermion operators are state present in a UV physics sector is integrated out. We focus due to the Higgs field and the Z boson. However, the highly sup- this discussion on the “one operator induced at tree level” ques- pressed Yukawa couplings of the SM Higgs to light fermions leads tion consistent with the assumed (approximate) G symmetry. We to an exceptional situation numerically in terms of the operator demonstrate why such a simple UV sector cannot be a complete profiles. The small Yukawa couplings are not formally the conse- scenario if a mechanism to generate the heavy state’s mass is quence of a fine tuning, as they are protected by the full chiral demanded. We then discuss the spin-1/2 case, drawing a similar symmetry of the SM. More discussion on the accidents in Fermi conclusions in Section 3.2. In Section 3.3 we examine the case of theory, and how it is commonly misunderstood, can be found in integrating out a scalar field focused on the “one operator” ques- Ref. [19]. tion. We show how the scalar case is more subtle, but still argues Arguably, there is some theoretical evidence based on the struc- for more operators when UV complete scenarios are demanded. ture and particle content of the SM in the direction of embedding Section 4 contains our conclusions. 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 2. Two exceptional EFT cases theory, similar to the chiral structure of the SM being a low en- ergy hint of its UV structure. However, the problems of TeV scale When considering the one operator question, we note that a grand unified theories are very well known. In this work we make few historical accidents in EFTs can be misleading. First of all, L5 a more phenomenologically motivated choice and assume approx- and Ld with d ≥ 6are distinct when considering this question. Due imate G symmetry (and CP symmetry). to the charges of the SM field content, only one operator (with flavor indices) can be constructed in L5. The operator that results 3. G symmetric tree level matchings [17,18],    3.1. Spin 1 states c L = ij c ˜  ˜ † + 5 LL,i H H LL, j h.c. (2) 2 Spin-1 fields that couple to the SM quark bi-linears in the is the well known example where one operator at a particular manner assumed are given by Table 1 [21–23]. The requirement mass dimension does result when integrating out UV physics.4 The of linear couplings of mass dimension less than four, together interplay of global U(1)L number violation and the constraints of with Lorentz symmetry and invariance under the full SM gauge the SM field’s representations leading to one operator in L5 is an group constrains the possible quantum numbers of UV field con- exception that is not repeated at higher orders in the SMEFT oper- tent. Fields with other representations that give SMEFT matchings ator expansion [6–10]. respecting G are possible, if these conditions are relaxed. Our no- c c Historically, Fermi theory has frequently been used as a pro- tation is that Q , L are the right handed conjugate doublet fields totypical EFT to build intuition. This can be unfortunate, as Fermi of the SM fermions. The global flavor symmetry in the quark and theory is atypical and has a number of non-trivial accidental fea- lepton sectors are defined as tures that are not generic. In Fermi theory, the four-fermion oper- = × × ator GQ SU(3)uR SU(3)dR SU(3)QL , (4)    G = SU(3) × SU(3) . (5) μ L LL eR Q  = LL γ LL LL γμLL , (3) 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 4 Here and below our notation with a c superscript indicates a charge conjugate field. We list the corresponding fields in Table 2 and Table 3. representation of a SM field. Cases VXII, VXIII have fields that carry a global lepton number and 110 Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116

Table 2 Table 4 Different vector representations that couple to fermion bi-linears respecting G, Examples of the sets of L6 operators in the SMEFT obtained by integrating out without the insertion of a Yukawa matrix. various massive vectors.

Case SU(3)C SU(2)L U(1)Y GQ GL Couples to Case Q i generated at tree level (1) (1) (3) (3) V ¯ μ V 2 I 1 1 0 (1,1,1) (1,1) eR γ eR IV Q ll , Q qq,lq, Q Hq,Hl, Q H , Q HD, Q H , Q eH, Q uH, Q dH V(1) ¯ μ I 1 1 0 (1,1,1) (1,1) LL γ LL (8) (1) (3) (1) V Q , Q V 1 3 0 (1,1,1) (1,1) L¯ σ I γ μ L IV qq qq IV L L V(8) (1) (3) IX Q qq , Q qq V ¯ ¯ ¯c μ (1) (3) XII 1 2 3/2 (1,1,1) (3,3) LL γ eR VXX Q , Q μ lq lq VXIII 1 1 0 (1,1,1) (1,8) e¯ R γ eR (1) V ¯ − ¯ ¯ ¯c μ V Q , Q , Q 2, Q , Q , Q XIV 32 1/6 (3,1,1) (3,1) LL γ uR XXIII H HD H eH uH dH V ¯ ¯ ¯ ¯c μ V(1) XV 325/6(1,3,1) (3,1) LL γ dR XXIV Q H , Q HD, Q H2, Q eH, Q uH, Q dH ¯ ¯ μ VXVI 31−2/3 (1,3,1) (1,3) e¯ R γ dR ¯ ¯ μ VXVII 31−5/3 (3,1,1) (1,3) e¯ R γ uR V − ¯ μ c XVIII 32 5/6 (1,1,3) (1,3) eR γ Q L Table 5 V ¯ − ¯ ¯ μ XIX 31 2/3 (1,1,3) (3,1) LL γ Q L Operators induced at tree level when the massive vector case is integrated out. The V ¯ − ¯ ¯ I μ μ † XX 33 2/3 (1,1,3) (3,1) LL σ γ Q L cases are grouped in the table into the chiral ( Jψ ) Jψ,μ operator classes induced. ¯ μ ¯ VXXI 1 1 0 (1,1,1) (8,1) LL γ LL The top section refers to LLLL operators. The middle section of the table refers to ¯ I μ ¯ VXXII 1 3 0 (1,1,1) (8,1) LL σ γ LL LLRR operators. The bottom section of the table refers to RRRR operators induced at tree level.

Case Op U(1)Y G Q , G L Spurion Table 3 V(1) (1) A † A † Vector representations coupling to currents constructed from Higgs fields. VIII Q qq 0 T Yu Yu , T Yd Yd (1) (1) † † V Q 0 T A Y Y , T A Y Y Case SU(3) SU(2) U(1) G G Couples to IX qq u u d d C L Y Q L (1) VXIX Q −2/3 / V(1) V(1) V(1) † μ lq I , II , III 1 1 0 (1,1,1) (1,1) H iD H (1) ¯ V 1 3 0 (1,1,1) (1,1) H† I iDμ H V(3,6) (1) − IV σ X Q qd 1/6 / ¯ V(3,6) (1) (1) T Q qu 5/6 / V 11−1 (1,1,1) (1,1) H iDμ H XI XXIII V Q −5/6 / V(1) − T XVIII qe XXIV 13 1 (1,1,1) (1,1) H iσI Dμ H VXII Q le 3/2 / VXIV Q lu −1/6 / VXV Q ld 5/6 / VXIV − VXX carry both lepton and baryon numbers. Although coun- V(1) A † terintuitive, UV fields that carry flavor quantum numbers do not V Q dd 0 T Yd Yd V(1) A † necessarily lead to lower energy signatures of flavor violation – VI Q uu 0 T Yu Yu V(1) (1) − † outside of the MFV pattern. Similarly, fields carrying lepton num- VII Q ud 1 Yd Yu V A † ber do not necessarily lead to lower energy signatures of lepton XIII Q ee 0 T Ye Ye V − flavor violation at tree level.5 Fields that carry both lepton and XVI Q ed 2/3 / VXVII Q eu −5/3 / baryon number are potentially more problematic in inducing pro- ton decay, but such phenomenological constraints are not the focus of this paper. We have systematically examined the profile in terms of opera- tors obtained in tree level matchings to the Warsaw basis from the 3.1.1. Dimension-6 operator matching fields listed in Tables 1, 2 and 3, finding the following rule: Solving the classical equations of motion (EOM) for the heavy Flavor singlet vector fields that do not break GQ × GL induce more vector fields and substituting the classical solution into the La- than one operator at tree level when matching onto the SMEFT Warsaw grangian results in a direct tree level matching in terms of a prod- basis. uct of currents. We define the currents as This result is easy to demonstrate. Fields that are SU(3)C and ={ μ μ}={¯ ⊗ † ⊗ } SU(2) singlets couple to (quark and lepton) fermion fields and Ja Jψ , J H ψ γμ ψ,(Dμ H) , (6) L also the scalar currents, inducing a large number of operators at and the tree level matching is given by 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 1 L ⊃− μ † μ leptonic couplings can be removed by assigning the field to a 8 of 6 ( Ja ) Jb . (7) M2 SU(3)C. In this case the operator profile is reduced to at least two V μ † ( Jψ ) Jψ,μ operators via the relation [2] ˜ =  ⊗ Here represents H or H i σ2 H and indicates a group prod- ¯ p I A μ r ¯ s I A μ t 6 (Q σ T γ Q )(Q σ T γ Q ) = uct characterized by the SU(2)L representation of vector fields. For L L L L a vector field of the form considered in Tables 1, 2 and 3, the cur- 1 (3) 3 (1) 1 (3) − Q + Q − Q , (8) rent product falls into one of three types: qq qq qq 4 ptsr 4 ptsr 6 prst • μ † here p, r, s, t are flavor indices. We show some examples of the four-fermion: ( Jψ ) Jψ,μ, • μ † multiple operators induced when integrating out vector fields at scalar derivative: ( J H ) J H,μ, × μ μ tree level in Table 4. Introducing GQ GL symmetry, vector fields • mixed scalar-fermion: ( J )† J , ( J )† J . ψ H,μ H ψ,μ can be reduced in their infrared (IR) SMEFT operator profile to one operator in the Warsaw basis in the limit of vanishing Yukawa ma- (1) trices; see Table 5. Note that with the exception of case V which 5 VII This was previously noted in Ref. [23] in the lepton number case for flavor sin- † has a bi-linear flavor breaking spurion in Y and Y , the presence glet fields. d u 6 This notation is consistent with Ref. [23]. Note also that a further current of the of a U(1)Y charge is also associated with the lack of Higgs scalar μ form D Fμν with F ={B, W , G} is redundant [23]. currents induced. This has an important consequence when the Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116 111 self interactions of the vector are studied for unitarity violation, where the dummy labels a and b are summed over {u, d} and as will be discussed shortly. {e, d}, respectively. A similar pattern of matchings onto the class A spurion analysis allows the corrections due to the nonzero 3 (D2 H4), 5 (H3ψψ¯ ) and 7 (H2 Dψψ¯ ) operators of the Warsaw Yukawa matrices of the SM (that break the flavor symmetry in a basis is present for almost all color singlet fields with flavor quan- phenomenologically safe MFV pattern) to be systematically stud- tum numbers listed in Tables 2 and 3. The exceptional case is ied. We define the SM Yukawa matrices Yu , Yd, Ye as the field VXII whose non-trivial SU(2)L representation and U(1)Y charge forbids a scalar current from being induced at tree level in L =− p ¯ r ˜ † − p ¯ r † Y (Yu)r uR,p Q L H (Yd)r dR,p Q L H this manner. − p ¯ r † + The pattern of tree level matchings is strongly dictated by (Ye)r eR,p LL H h.c. (9) the charges and representations of the UV fields under SU(3)C × × GQ GL symmetry is restored if we endow the Yukawa matrices SU(2)L × U(1)Y, GQ and GL. We emphasize, data fits to subsets of with the transformation properties under {GQ, GL} operators in the SMEFT formalism can be justified by appealing to UV field content with U(1) charges and non-trivial represen- ∼ ¯ ∼ ¯ Y Yu (3, 1, 3, 1, 1), Yd (1, 3, 3, 1, 1), tations under SM groups when only retaining tree level matching ¯ contributions. See Table 5 for details on cases that generate only Ye ∼ (1, 1, 1, 3, 3). (10) one operator at a time. Introducing GQ × GL symmetry breaking when the Yu, Yd, Ye ma- This conclusion is subject to the following qualifications. First, trices take on their SM values gives more operators at tree level the single operators obtained in tree level matchings to the vectors μ † for fields with flavor quantum numbers. On general grounds, the in Tables 2, 3 are limited to ( Jψ ) Jψ,μ operator forms. Such opera- μ † ( J H ) J H,μ current products are induced proportional to two spu- tors at LHC contribute to continuum parton production in a fashion μ † μ † dictated by the power counting of the theory. Conversely, the pre- rions breaking the flavor symmetry, and the ( Jψ ) J H,μ, ( J H ) Jψ,μ current products are induced proportional to one flavor breaking cise measurements made on a scattering through a SM resonance spurion insertion. Here we refer to the spurions listed in Table 5 (with mass M and width ) parametrically has a /M suppression, that are bi-linear in Yukawa matrices. As a specific example con- compared to the leading resonant behavior, when considering the (1) μ † V 7 interference with ( J ) Jψ,μ operators. sider VIII that is a 8 under SU(3)QL . The Lagrangian is given by ψ L + L Second, as y  1, a flavor symmetry spurion breaking propor- SM V(1) where t VIII tional to only powers of Yu can induce operators of class 3, 5 and   2 1 MV 7 without significant numerical suppression. This makes it difficult L =− V μVν − V νVμ − V Vν μ † V(1) Dμ ν D Dμ ν D ν to justify “one at a time” data fits to ( J ) Jψ,μ SMEFT operators VIII 2 2 ψ   with up quark field content (consistent with our assumptions). On A † μ † μ + V V + † λ μ,A T Yu Yu (D H) H h.c. , (11) the other hand, one at a time data fits to ( Jψ ) Jψ,μ operators that only have leptonic or down quark field content can be potentially + V ¯ A μ +··· gV μ,A (Q L T γ Q L ) . justified. In these cases the induced scalar currents proportional Note that the largest spurion that restores the flavor symmetry for to MFV like flavor breaking spurious are numerically suppressed −2 A † compared to pure up quark spurions by at least yb/yt ∼ 10 . the second line is T Yu Yu and some indices are suppressed in † Finally, we also note that we never obtain only one operator Eqn. (11). The additional spurion breaking proportional to Y Y is d d in such a tree level matching that involves the Higgs field, in the neglected in what follows. Integrating by parts and the EOM for cases of massive vector UV field content considered. the vector field are used to manipulate the derivative to appear as shown on the second line in Eqn. (11). Integrating out the field (1) 3.1.2. Arguments against orphaned vectors V using the classical EOM gives VIII The vector fields listed in Tables 1, 2and 3 inducing a single L6   2 SMEFT operator at tree level, carry at least one non-trivial repre- gV (1) 1 (1) 8 L ⊃ Q − Q sentation under the SM gauge symmetry and flavor symmetries. 6 2 qq qq 4 MV rssr 3 rrss Non-trivial representations and U(1) charges reduce the interac-  Y 1 tions for SM particles with the new sector, which consequently + 2 − 2 + 2 ((ImλV ) (ReλV ) )Q H2 4(ImλV ) Q HD minimizes the IR SMEFT operator profile. However, such fields in 4 M2 V general do not indicate a stand alone UV complete scenario (where † † + 2i(ReλV )(ImλV )(Y Q − Y Q ) the vector could be an “orphan”) for the following reasons. b bH b bH − † − † (1) Landau poles and triviality. The β function of the cou- 2i(ReλV )(ImλV )(Yu Q uH Yu Q uH) pling of the vector fields to the fermion bi-linears (denoted gV in † 2 Eqn. (11)) is determined by renormalizing the fermion fields and × [ † † ]−(diag(Yu Yu)) Tr (Yu Yu)(Yu Yu) (12) vector field two point functions, and subsequently extracting the β 3 function for gV . We relate the bare (0) and renormalized (r) fields [ ] † and couplings as gV Im λV (1) † p diag(Yu Yu) − Q (Y Y ) − δ 2 Hq u u r pr 2MV pr 3 (0) = (r) (0) = (r)    Vμ Z V Vμ , gV Z gV gV μ , (13) [ ] gV Re λV † † m † i † (0) (r) + i ((Y Y )Y ) Q − (Y (Y Y )) Q ψ = Zψ ψ , (14) 2 u u a i aH a u u m aH i i i 2MV im mi   ¯ where Zx = 1 + δZ for x ={V , gV , ψ, ψ}. We use a renormaliza- gV Re[λV ] † † † x − i Tr[Y Y ] (Y )m Q − (Y )i Q , tion scheme employing MS subtraction and d = 4 − 2 dimensions 2 u u a i aH a m aH 6MV im mi

8 In all cases but one, multiple non-trivial representations are present. The one 7 V Recall the flavor adjoint 8 representation is real. exceptional case is XIII which is only an 8 under SU(3)eR . 112 Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116

Fig. 2. 2 → 2 vector scattering diagrams.

symmetry. Conversely the three point interaction can be forbidden Fig. 1. Diagrams relevant for the renormalization of gV . 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 Table 6 dictated by such a four-point and three-point interactions at tree One loop renormalization results. Here O  indicates the matrix element of the level. The relevant Lagrangian involving a general vector field with vector-fermion bilinear interaction term and δZ3 corresponds to the divergence present in this three point interaction at one loop from the vector-fermion coupling. self-interactions is (N) (N) 2− The notation is such that F(N) ≡ C − 1 N with C = N 1 . We have labeled F 2 F 2N λ  several of the numerical factors in the table with the group space (SU(3) , SU(3) , † μ † ν μ † ν C Fl LV = VμV VνV + g ∂μV VνV + ... (17) SU(2)L) that generates them, with the subscript Fl indicating a SU(3) flavor group. 4

2 − 2 2 2 16 16π  δZ ¯ −16π  δZ − π  δZ3 ψ ψ 16π  δZ V The amplitudes at leading order with the high-energy approxima- Case 2 O 2 2 2 βy μ gV gV gV gV  μ   tion for the vector polarization L p /MV through a s-, t- and V(1) F (3Fl) (3Fl ) 4 1 · + VIII (3Fl) C F C F 3C u-channel vector exchange and a four-point contact interaction, re- 3 2 Fl   · spectively, read V(1) F F (2) (3Fl) (2) (3Fl ) 2 3C 1 1 + IX (2) (3Fl) C F C F C F C F 3 2 Fl 2 L V · 2 · + st − su XIX 13Fl 3C 3Fl 3 2 L  2 M = (g ) F s , (18) ¯ 3,s 4 V(3) − − · − 2 · − − 4 MV X,XI 1C 3Fl( 2)C 23Fl( 2)C 3 ( 1)C V(6) · · · 2 · +  st − ut X,XI 13Fl 1C 23Fl 1C 3 1C ML = 2 3,t (g ) Ft , (19) V · · 2 + 4 XVIII 13Fl 3C 23Fl 3 4 MV V · 2 + XII 13Fl 3Fl 2 3 us − ut ML =  2 V 13· 3 3 · 2 2 + 3,u (g ) Fu , (20) XIV Fl C Fl 3 4 M4 V · · 2 + V XV 13Fl 3C 3Fl 2 3     t2 − u2 s2 − u2 s2 − t2 V(1) F (3Fl) (3Fl ) 2 1 · + L V (3Fl) C F C F 3C M = λ F + F + F . (21) 3  2 Fl 4 s 4 t 4 u 4 4 MV 4 MV 4 MV V(1) F (3Fl) (3Fl ) 2 1 · + VI (3Fl) C F C F 3C 3 2 Fl V(1) 2 · + Here abstract group structure constants F for three channels VII 13Fl 3Fl 3 3C s,t,u   (1) V F (3Fl) (3Fl ) 2 1 + have been introduced. For example, in the model V : F s = XIII (3Fl) C F C F IX 3 2 Fl V · 2 + f ABE f CDE f ijn f kln where A, B, C, D, E refer to the flavor index and XVI 13Fl 3C 3Fl 3 V 13· 3 3 2 + i, j,k,l,n denote the iso-spin index. XVII Fl C Fl 3  If λ = (g )2 is accomplished by a global symmetry then the ML ML amplitudes 3 will cancel with three terms in 4 with an using standard methods. The relevant diagrams are shown in Fig. 1. identical F factor respectively, through the Mandelstam relation The β-function for the running of the coupling gV is given by 2 2 2 4 s + t + u = 4 MV . As a result, the leading scaling in ∼ (p ) /MV  2 2 disappears. The full amplitudes then grow as ∼ p /MV . However, δZ3 1 1 1 βgV = 2 gV  − − δZ ¯ − δZ − δZ , (15) if the three-point interaction is forbidden – for example due to O 2 ψ 2 ψ 2 V the field carrying a U(1)Y charge – then the amplitude cannot where the renormalization factors δZ ’s for the various vector field be so moderated in its growth at high energies, and scales as 2 2 4 cases are presented in Table 6. The general expectation is that gV ∼ (p ) /MV . In this manner, the presence of a U(1)Y charge for- will have a positive β function – indicating Landau poles [24], bidding the scalar current simultaneously turns off the three-point quantum triviality [25] and a UV incompletion. This is indeed the interaction that is required to moderate the high energy scattering μ † behavior of an orphaned vector field. case for all vector fields inducing one ( Jψ ) Jψ,μ operator, with the exception of color 3¯ vectors coupling to quark bi-linears; i.e. cases Standard partial wave unitarity arguments [26–28] give that the ¯ unitarity violation scale associated with the vector field without a V3 . In this exceptional case, the SU(3) vector-fermion coupling X,XI C three point interaction is mimics the effect of a non-abelian interaction. An oversimplified UV scenario afflicted with an internal incon- −1/4 −1/4  0.2 MV (F + F ) λ , (22) sistency indicated by the presence of Landau poles cannot for- t u mally generate a consistent IR limit. This indicates that further new where Ft , an Fu are determined by a particular scattering cross physics must be present below the Landau pole scale L approxi- section. A quick onset of unitarity violation follows from a siz- mated by able four-point interaction that is expected to emerge from a   strongly interacting composite sector on general grounds. Even in- ∼ L MV exp gV /βgV . (16) troducing a loop suppression to the vector self interaction, that is − λ ∼ (16π) 1, is of little help – one still finds ∼ MV due to the However, numerically corrections suppressed by L are smaller than one loop matching effects. presence of a fourth root in Eqn. (22). Hence, the UV strong sector should be simultaneously considered to define a consistent match- (2) Unitarity and vector self-interactions. A more intractable ing onto the SMEFT. This would increase the low energy operator problem is generated by O(1) self interactions of orphan vector profile of such a scenario in the SMEFT beyond one operator gener- fields. The four point vector self interaction is not forbidden by any ically due to non-perturbative matchings, and a “one at a time” Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116 113

  analysis invoking a tree level matching would be logically incoher- We can consider G or the proper subgroup case where G ⊂ H ent. without loss of generality with the following arguments. The  belongs to SU(3), and N ∈{3, 3¯, 6, 8}, or SU(2) with N ={2, 3} for (3) Siblings of massive vectors with non-trivial representa- the vector fields of interest. The non-trivial representations in N tions. A massive vector field with non-trivial representations under can be generated from tensor products of the  irreducible repre- subgroups of G is also generically accompanied by more “sibling” sentations. In the case where the  belongs to SU(3) we denote fields. If the massive vector gains a mass by a UV Higgs mecha- the irreducible representations as P, R, which need not have the nism, the corresponding sibling field includes at least a scalar (S) same dimension. When N is generated by P ⊗ P¯ the singlet repre- obtaining a vacuum expectation value (vev). Define this expecta-  sentation is also generated in the tensor product. A color singlet tion value as S† S = v 2/2. We require dim(V) + 1 ≤ dim(S) so sibling under SU(3) is expected with a mass proximate to a color that all of the components of the vector become massive in the C octet vector, which induces a number of operators in L when in- presence of a scalar field obtaining a vev, through eaten Goldstone 6 tegrated out. Similarly, a flavor singlet sibling under a flavor group components of S.9 is also expected for flavor octets. Interestingly, the flavor 8 vector One can use the global symmetry rotations on S to rotate the fields we considered all have zero U(1) charges, so their flavor new vev to a (uneaten) component of S, denoted s. The interac- Y singlet siblings with the same U(1) charge are not forbidden by tion of s with H† H cannot be forbidden by an explicit G breaking Y the flavor symmetry to have the coupling with the corresponding without violating our assumptions. This would introduce highly  quark bi-linear and also with the J of vanishing U(1) charge, constrained low energy effects into the SMEFT through the vev v H,μ Y inducing more than one operator in L when integrated out. When leading to the vector mass matrix. A vacuum misalignment [32] 6 N ∈ P ⊗ P multiple representations result, for example in the case is assumed to make the vector mass matrix symmetric under G of P = 3, the 3¯ and 6 fields are simultaneously present. Such fields in this work. This results in the Higgs portal coupling not being (VX,XI) can induce the same operator when integrated out. On the suppressed by a GQ × GL breaking spurion. Concretely consider the other hand, these fields necessarily carry U(1) , and thus have a cut off Lagrangian Y scale proximate to the massive vectors mass scale for the cases consis-   2 11 μ † λ † v 2 † † tent with our assumptions. Next we consider the cases when the LSH = (D S) (Dμ S) − (S S − ) + λSHS SH H. 4 2 non-trivial representation is generated by bi-linears of  carry- ∈ ⊗ (23) ing representations of unequal dimension N P R. By inspection of the tensor products of SU(3) with triality 0 and 1 [37] it is μ = μ + V a ¯ Here the covariant derivative is D ∂ igV ah with ha possible to generate each N ∈{3, 3, 6, 8} for SU(3) in such a man- an abstract group generator that defines the non-trivial repre- ner. However, for each P and R one can also form a condensate sentations that the V multiplet carries. S is expanded as S = ¯ ¯ ¯   √ γμ with zero U(1)Y charge from the product P ⊗ P, R ⊗ R. ···  + +··· +  , v s / 2 ha ρa where ρa corresponds to the gold- Two more pure singlet spin one bound states proximate in mass stone components of the S multiplet that are eaten to generate the to MV are expected in the spectrum, unless forbidden by another ··· vector mass, and the fill out the full dimension of S. The vev symmetry.12 Restricting the discussion for non-trivial SU(2) repre-  V  v must be arranged to break the dim( ) ha generators. Simulta- sentations to the vector cases that do not carry U(1) and induce   = Y neously v must not break the Gsubgroup, so ga S 0, where one operator at tree level, we are left with the field V1 . Further, 10 IX the generators of G are denoted ga. Integrating out s after UV V1 IX has a large flavor breaking spurion proportional to the top symmetry breaking gives Yukawa generating more operators at tree level when integrated 2 2 out, see Table 5. 2 λSH 4 gV μ 2 L6 =− Q H2 − (Vμ V ) +··· (24) For all of these reasons, orphaned vector fields with non-trivial  2 λ λ ms representations of the SM symmetry groups demand siblings and in addition to the operators induced by integrating out the vec- a “good UV home”. 2 =   2 L tor field. Here the scalar mass is ms λ v /2. In addition, 4 terms are induced that require a finite redefinition of λ and v in 3.2. Spin 1/2 states the SM to rearrange LSM back into standard from. Here we have neglected many higher order effects including subdominant mass If heavy spin-1/2 states are integrated out, the mass13 of the splitting terms. Note the sizable vector four point interaction, that massive fermion(s) (denoted M with M  v) must be introduced  is enhanced in the λ → 0 limit, indicating unitarity violation when in some manner. As discussed in the previous section, a chiral the UV Higgs is integrated out of the spectrum. In this limit it is of fermion with a UV Higgs mechanism induces more operators at interest to not neglect mass splitting effects proportional to λSH. tree level when integrating out the UV scalar field. In this section, In order to avoid assuming a UV Higgs mechanism, we can we confine the discussion to general vector like fermions. The gen- consider a composite massive vector generated by a hypothetical eral Lagrangian associated with a pair of heavy vector-like quark UV strong sector, with spin-1/2 constituents , so that the vector (VLQ) denoted by QL , QR that are flavor singlets includes fields are Vμ ∼ ¯ γ μ  condensates. This composite field carries L = L0 + Lint at least one non-trivial representation under one of the groups Q Q Q , (25) GQ, GL, SU(3)C or SU(2)L to reduce the SMEFT operator profile to where one operator. Denote this non-trivial representation as N, and the     0 ¯ ¯ ¯ ¯ corresponding group as G . The  are charged under G or a larger LQ = QL iD/QL + QR iD/QR − M QL QR + QR QL (26)  group Hwith H ⊃ G .

11 This is a generic expectation if the  are charged under U(1)Y. 9 An additional non-goldstone incomplete scalar multiplet is famously required 12 For example, some of the expected spectra degeneracy can be lifted in analogy  when introducing a vev in this manner [29–31]. to the η [33]. 10 In general one expects the symmetry breaking pattern to be such that there 13 Here we are referring to the dominant component of the mass of the fermion will be uneaten goldstone bosons, or additional massive vectors in the spectrum. from the new sector. In addition, there will be mass contributions and splitting Here we are considering an exceptional minimal spectrum when examining the one proportional to ∼ v. As such effects do not act to reduce the IR SMEFT operator operator question. profile, we neglect these contributions. 114 Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116

Table 7 Table 9 Tree level L6 operators induced in the SMEFT with massive quarks integrated out. Tree level L6 operators induced in the SMEFT with massive leptons integrated out.

Q (1) (3) L (1) (3) (1) Case SU(2)L U(1)Y J L Q uH Q dH Q Hq Q Hq Case SU(2)L U(1)Y J L Q Hl Q Hl Q eH Q He √√√ √√√ (1) 1 ¯ (1) ¯ Q 1 − Q H L 1 −1 LL H I 3 L I √√√ √√√√ (3) I ¯ (3) 1 I ¯ L 3 −1 σ LL H Q 3 − σ Q L H I I 3 √√√ (1) ∗ Q 2 ¯ L (1) (3) (1) II 1 3 Q L H √√√√ Case SU(2)L U(1)Y J R Q Hl Q Hl Q eH Q He (3) ∗ Q 3 2 σ I Q¯ H √√ II 3 L L 2 − 1 e¯ H† III 2 R √√ Q L 2 − 3 e¯ H T Case SU(2)L U(1)Y J Q uH Q dH Q Hu Q Hd Q Hud IV 2 R R √√√√√ Q 2 1 u¯ H T III 6 R √√√√√ Q 2 1 d¯ H† IV 6 R √√ Table 10 7 † Q 2 u¯ H L6 operators obtained at tree level when flavor and color singlet scalars are inte- V 6 R √√ Q − 5 ¯ T grated out. S indicates a dimensionfull coupling. VI 2 6 dR H Case SU(2)L U(1)Y Couplings Q H Q H2 √ S S A 21/2 V (H, A ) √ Table 8 3 † SB 43/2 (H ) SB + h.c. Tree level L operators induced in the SMEFT with massive quarks integrated out √ 6 S 41/2 H†S H† H + h.c. in some sample cases with flavor quantum numbers, see Refs. [34–36] for more C C √√ S1 10( S + (S )†S )H† H discussion on the phenomenology of these fields. I S I I I √√ S3 S † S †S † I 30S Iσ H H, ( I) I H H √√ Q S1 − S T S †S † Case SU(2)L U(1)Y GQ J R Q uH Q dH Q Hu Q Hd II 1 1 S II H H, ( II) II H H √√ √√ S3 3 −1 S σ H T H, (S )†S H† H Q 2 1 (3,1,1) u¯ H T II S II II II VII 6 R √√ Q 1 ¯ † VIII 2 6 (1,3,1) dR H obtained if an explicit scale is introduced without a dynamical ori- gin to give the scalar a mass. For instance, SA couples through for the SU(2)L singlet and doublets and     linear and bilinear interactions in the full multi-scalar potential, 0 ¯ ¯ ¯ ¯ S LQ = Tr QL iD/QL + QR iD/QR − MTr QL QR + QR QL (27) denoted V (H, A ) in Table 10. To reduce the operator profile of SA to one operator, it is assumed that SA has a discrete or addi- int for the 3 of SU(2)L. The interaction term LQ for the VLQs to the tional U(1) symmetry. Such a symmetry forbids a large number of SM fermions through the Higgs doublet is defined as four-fermion operators at tree level, and also a number of linear S Q Q interactions in the scalar potential that otherwise generate Q H2. Lint = J Q + J Q + h c (28) Q L R R L . . Similarly, SB,C also have minimal one operator profiles containing only Q . However, this again follows from the UV scale being in- The requirement that the action be stationary under variations of H ¯ ¯ troduced without a dynamical origin. In all these cases, a hierarchy the heavy VLQ fields QL , QR results in two coupled EOMs: problem in the UV sector is also introduced. Q iD/Q − MQ + ( J γ 0)† = 0, (29) Table 10 also lists the cases of flavor singlet scalar fields that L R R S2 † Q couple to through the H H interaction and in addition have /Q − Q + 0 † = † iD R M L ( J L γ ) 0. (30) an independent S H H interaction via a dimensionfull coupling. In these cases, the operators Q and Q 2 are simultaneously pro- Mathematically, the coupled Eqns. (29) and (30) can be solved it- H H duced in tree level matchings. eratively. Taking the limit of large M, one can expand the classical As in the case of massive vectors and fermions, scalars can carry solutions schematically as non-trivial representations under GQ or GL to isolate the coupling Q 0 † to a single fermion bi-linear. These states have been studied pre- ( J R γ ) iD/ Q 0 † QR = + ( J γ ) +··· , (31) viously in Refs. [39–43]. To avoid an explicit breaking of G or G M M2 L Q L Q in this coupling, all of these states carry at least two non-trivial ( J γ 0)† iD/ Q = L + Q 0 † +··· representations under the flavor (GQ or GL) or gauge (SU(3)C or L ( J R γ ) . (32) M M2 SU(2)L) groups. For instance, consider integrating out “di-quark” states of this form discussed in Ref. [41] at tree level. A scalar cur- When substituted back into Eqn. (25) the effect of the leading term rent operator of the form ψ¯ ψ ψ¯ ψ is directly obtained. This in these solutions vanishes due to chirality. 1L 2R 2R 1L operator can be projected into the Warsaw basis via Fierz transfor- We generically find that multiple operators are induced at tree mation, level when integrating out a vector like fermion. The cases where the vector like quark do not carry flavor quantum numbers are ¯ ¯ 1 ¯ ¯ μ (ψ1L ψ4R )(ψ3R ψ2L ) =− (ψ1L γμψ2L)(ψ3R γ ψ4R ). (33) shown in Table 7. In the cases that the VLQs carry flavor quantum 2 numbers, previously discussed in Refs. [34–36], multiple operators As the “di-quark” scalars are in non-trivial representations under are again obtained. We show some sample cases of this type in SU(2)L and/or SU(3)C groups, the index associated with these sym- Table 8. Multiple operators at tree level are also obtained in the metries are not contracted between the fermions in each vector case of integrating out vector like leptons, see Table 9. current, cf. the right hand side of Eqn. (33). When reducing to the Warsaw basis one uses the SU(3) and SU(2) relations 3.3. Spin 0 states A A 1 1 T T = δilδ jk − δijδkl, (34) Unlike the cases of massive vectors and spin-1/2 fields, a mas- ij kl 2 6 sive scalar can couple into the SM through a number of interac- I I = − σ jk σmn 2δ jn δmk δ jk δmn. (35) tions and naively generate many operators in the IR SMEFT match- ing limit. However, the examples (SA , SB and SC in Table 10) Concretely, performing this mapping for the “di-quark” scalars that ¯ ¯ (1,8) (1,8) discussed in Refs. [8,38] show that only one operator Q H , can be couple to uR Q L and dR Q L induce the operators Q qu and Q qd Y. Jiang, M. Trott / Physics Letters B 770 (2017) 108–116 115

Table 11 tree level matchings capturing the consistent IR limit of a new The cases where a single L operator is generated at tree level for different scalar 6 physics sector. Due to the extensive mixing of the operators in L6 representations that are not singlets under the flavor group, without the insertion under renormalization [3–5], the SMEFT clearly has a non-minimal of spurion Yukawa fields, from Ref. [41]. character once loop induced effects are considered, requiring many Case SU(3)C SU(2)L U(1)Y GQ Couples to Op operators for consistent lower energy data analysis.14 S − III 31 4/3 (3,1,1) uR uR Q uu We have uncovered some cases where only one operator (ne- S ¯ − ¯ IV 61 4/3 (6,1,1) uR uR Q uu glecting flavor indices) is naively induced at tree level. These oper- S V 3 1 2/3 (1,3,1) dR dR Q dd † 3 ¯ ¯ ators in the Warsaw basis are (H H) , or of the four-fermion form, SVI 612/3(1,6,1) dR dR Q dd and come about due to a massive scalar or vector field having non- Case SU(3)C SU(2)L U(1)Y GL Couples to Op trivial representations under the symmetry groups of the SM. We ¯ SVII 112(1,6) eR eR Q ee 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 respectively. Similarly, the “di-quark” scalars coupling to Q L Q L lack of a three-point vector self-interaction. This indicates the pres- (1,3) generate Q qq . On the other hand, exceptional cases that can gen- ence of large non-perturbative matching corrections in addition to erate only one operator do exist in “di-quark” scalars that couple tree level matching effects. On the other hand, when the massive to right handed SU(2) bi-linears of the same fermion field i.e. to L vector fields are not charged under U(1)Y, flavor symmetries can pairs of uR , dR and eR . These scalars can induce the single oper- be introduced to reduce the IR SMEFT operator profile. In this case, ator Q uu, Q ee, Q dd that are defined in the Warsaw basis, see the a spurion symmetry breaking analysis shows scalar currents are examples in Table 11. still induced, leading to more operators at tree level. In practice, S In spite of only Q H being induced at tree level in cases A,B,C fitting to one pure lepton or down quark four-fermion operator is and only one of the operators Q uu, Q dd and Q ee obtained at tree not as poorly motivated as fitting to up quark four-fermion opera- S S level in the cases III– VII, the arguments based on the mass scale tors due to the relative magnitudes of the flavor breaking spurions generation from a UV Higgs mechanism with an associated extra in each case. S scalar degree of freedom still hold. The heavy scalar ( ) can be In contrast to the vector fields, the presence of scalar fields in a S embedded in a larger scalar multiplet that develops a vev, or UV sector do not directly cause severe unitarity problems. Integrat- not so embedded, when a UV Higgs mechanism is invoked to in- ing them out could have a relatively minimal operator profile, i.e. troduce a new scale  v. Due to the fact that any field obtaining only Q uu, Q dd or Q ee is induced at tree level in the cases shown in a vev with its self conjugate forms a singlet under G this leads to Table 11, and only Q H in some cases in Table 10. However, requir- Q H2 (as shown in Eqn. (24)) in either case, in addition to any ing a mass generation mechanism for these fields would lead to S matchings of integrated out at tree level. more matching contributions to the SMEFT operators. The scenario Alternatively, if a strong sector is present and the “di-quark” of a UV Higgs mechanism, if present, generically induces more op- scalar is composite, then the arguments in favor of “sibling” fields erators that are constructed out of the SM Higgs field at tree level. imply an extended spectrum that generically contains singlet com- This occurs without a suppression by a GQ × GL symmetry break- posite states. Additionally, in the presence of a confining strong ing spurion. When a UV Higgs mechanism is avoided by assuming sector, both spin-0 and spin-1 composite states are expected to be compositeness and a new strong interaction, we have argued that embedded in a spin tower [45]. Finally, some form of dimensional the requirement of non-trivial representations for the vector and transmutation can be used to generate a scale. This can take place scalar fields to reduce the operator profile would indicate the pres- in the context of weaker couplings using the Coleman–Weinberg ence of an extended spectrum of the composite states – including (CW) mechanism [44], or in the case with stronger couplings with singlet fields – that couple through many SM portals. This would a mechanism similar to the generation of QCD. Of course, the lead to more operators with tree level matchings when the ex- (weak coupling version) of CW requires multiple couplings and tended spectrum is integrated out. generically a non-minimal UV particle spectrum. The SMEFT is a complicated field theory. It is natural and rea- It is also important to notice that there are scalar four-fermion ¯ ¯ ¯ ¯ sonable to seek a reduction of this complexity to use in data current operators with the chiral structure (LR)(LR) and (LR)(RL) analyses in the SMEFT framework. Using symmetry assumptions defined in the Warsaw basis. However, these operators are not con- is widely accepted. We have examined in this work if an alter- structed out of a pair of bi-linears with the same SM field content. native ad-hoc approach of using “one operator at a time” in data As a result, additional vector current operators are induced when ¯ ¯ ¯ ¯ analyses can be representative of a consistent tree level match- the (LR)(LR) and (LR)(RL) operators are obtained with a tree ing to an unknown new physics sector. Our results show that the level exchange. Nevertheless, the presence of multiple four-fermion SMEFT has a non-minimal character quite generically and thus this operators induced at tree level from the projection of some four- approach should be avoided, if possible. To ensure the right conclu- fermion scalar currents into the form of the Warsaw basis is clearly sions are being drawn on the degree of constraint on unknown UV a more basis dependent conclusion than other arguments made in physics sectors, multiple operators should be retained in data anal- this paper. yses. Fortunately global data analyses in the SMEFT can already be performed with multiple operators, by using symmetries to sim- 4. Conclusions plify the number of parameters present. Further the growing LHC data set makes such global analyses even more feasible to execute 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 14 Such studies can also require mapping the SMEFT to a lower energy Lagrangian, the non-minimal character of the SMEFT. as in studies of B decays. Using the mapping of the SMEFT to C9 and C10 as re- We addressed this question using a (G and CP) symmetry as- ported in [46] our results support simultaneous fits to C9 and C10. We do not find examples where the combination of Wilson coefficients in the SMEFT at tree level sumption to accommodate the large set of lower energy measure- naturally cancel out in these lower energy parameters. This is largely due to the chi- ments that probe the symmetry breaking pattern of the SM into rality of the relevant SMEFT operators. 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