Interpreting top-quark LHC measurements in the standard-model effective field theory
J. A. Aguilar Saavedra, C. Degrande, G. Durieux, F. Maltoni, E. Vryonidou, C. Zhang, D. Barducci, I. Brivio, V. Cirigliano, W. Dekens, J. de Vries, C. Englert, M. Fabbrichesi, C. Grojean, U. Haisch, Y. Jiang, J. Kamenik, M. Mangano, D. Marzocca, E. Mereghetti, K. Mimasu, L. Moore, G. Perez, T. Plehn, F. Riva, M. Russell, J. Santiago, M. Schulze, Y. Soreq, A. Tonero, M. Trott, S. Westhoff, C. White, A. Wulzer, J. Zupan
CCERN-LPCC-2018-01
Interpreting top-quark LHC measurementsin the standard-model effective field theory
J. A. Aguilar Saavedra, C. Degrande, G. Durieux, F. Maltoni, E. Vryonidou, C. Zhang (editors),D. Barducci, I. Brivio, V. Cirigliano, W. Dekens, , J. de Vries, C. Englert, M. Fabbrichesi, C. Grojean, , U. Haisch, , Y. Jiang, J. Kamenik, , M. Mangano, D. Marzocca, E. Mereghetti, K. Mimasu, L. Moore, G. Perez, T. Plehn, F. Riva, M. Russell, J. Santiago, M. Schulze, Y. Soreq, A. Tonero, M. Trott, S. Westhoff, C. White, A. Wulzer, , , J. Zupan. Departamento de Física Teórica y del Cosmos, U. de Granada, E-18071 Granada, Spain CERN, Theoretical Physics Department, Geneva 23 CH-1211, Switzerland DESY, Notkestraße 85, D-22607 Hamburg, Germany Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université catholique de Louvain,B-1348 Louvain-la-Neuve, Belgium Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China SISSA and INFN, Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University ofCopenhagen, DK-2100 Copenhagen, Denmark Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA New Mexico Consortium, Los Alamos Research Park, Los Alamos, NM 87544, USA Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany Rudolf Peierls Centre for Theoretical Physics, University of Oxford, OX1 3NP Oxford, UK Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Department of Particle Physics and Astrophysics, Weizmann Institute of Science,Rehovot 7610001, Israel Institut für Theoretische Physik, Universität Heidelberg, Germany CAFPE and Departamento de Física Teórica y del Cosmos, U. de Granada, E-18071 Granada, Spain Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA UNIFAL-MG, Rodovia José Aurélio Vilela 11999, 37715-400 Poços de Caldas, MG, Brazil Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University ofLondon, 327 Mile End Road, London E1 4NS, UK Institut de Théorie des Phénomènes Physiques, EPFL, Lausanne, Switzerland Dipartimento di Fisica e Astronomia, Universitá di Padova and INFN Padova, Italy Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA
Abstract
This note proposes common standards and prescriptions for the effective-field-theory in-terpretation of top-quark measurements at the LHC. a r X i v : . [ h e p - ph ] F e b ontents B - and L -conserving degrees of freedom 25D Less restrictive flavour symmetry 28E FCNC degrees of freedom 29 Summarising efforts undertaken under the auspices of the LHC TOP Working Group, this theorynote aims at establishing basic common standards for the interpretation of top-quark measure-ments at the LHC in the standard-model effective field theory (SMEFT).Guiding principles are first stated (Section 2). In a nutshell, we rely on the Warsaw basis ofgauge-invariant dimension-six operators, focus on the ones which give rise to top-quark interactions(Section 3) and limit ourselves to the tree level. Three different assumptions about beyond-the-standard-model (BSM) flavour structures are considered to prioritize studies among the otherwiseoverwhelming number of four-fermion operators (Section 4). Top-quark flavour-changing neutralcurrents (FCNCs) are examined separately (Appendix E). For convenience, degrees of freedomare defined from combinations of Warsaw-basis operator coefficients (Appendices C and D). Theyare aligned with the directions of the effective-field-theory (EFT) parameter space which appearin given processes, in interferences with standard-model (SM) amplitudes, and in top-quark inter-actions with some of the gauge boson mass eigenstates. Naming and normalization conventionsare fixed. Indicative constraints on these degrees of freedom are compiled in Appendix A. Modelimplementations are provided for tree-level Monte Carlo simulation (Appendix B). Finally, asimple example of analysis strategy is outlined to illustrate how the challenges of a global EFTtreatment could be addressed and to identify the experimental outputs that would be useful inthis case (Section 5). It rests on the common knowledge established over the years through manyEFT studies in the top-quark, Higgs, electroweak or flavour sectors. The literature quoted in thisnote is by no means comprehensive or even representative of the fields of top-quark physics andEFTs.
1. The so-called Warsaw basis of dimension-six operators [1] is adopted. See also Ref. [2, 3] forearly discussions of top-quark related operators.2. Our discussion exclusively concerns processes involving at least a top quark. Only oper-ators involving such a particle are considered. Other operators affecting the considered2rocesses are assumed to be well constrained by measurements in processes that do not in-volve top quarks. This assumption may not always be justified and explicit checks shouldbe performed. It was for instance shown that jet production sufficiently tightly constrainmodifications to the triple gluon vertex [4].3. Three different assumptions about BSM flavour structures are considered —mostly basedon symmetries— to effectively reduce the overwhelming number of four-quark operators [5].A baseline scenario is defined. Less and more restrictive assumptions are also considered.Top-quark FCNC operators are treated separately.Minimal flavour violation in the quark sector is used in the baseline scenario, under theassumption of a unit Cabibbo-Kobayashi-Maskawa (CKM) matrix and finite Yukawa cou-plings only for the top and bottom quarks, i.e. we impose a U (2) q × U (2) u × U (2) d flavoursymmetry among the first two generations. In the lepton sector, flavour diagonality, i.e.[ U (1) l + e ] , is chosen as baseline.4. For convenience, we identify the linear combinations of Warsaw-basis operators that appearin interferences with SM amplitudes and in interactions with physical fields after electroweaksymmetry breaking. This may reduce the number of relevant parameters and unconstrainedcombinations in a given measurement. Independent combinations are defined and referred toas degrees of freedom from now on. Normalizations and notations are fixed. We recommendto provide experimental and theoretical results in terms of those parameters in the future.We refrain from stating which degrees of freedom are relevant for which process as such astatement is observable dependent. Different factorization schemes (four- or five-flavour) andapproximations (e.g., about the CKM matrix, light fermion masses and Yukawa couplings,or subleading electroweak contributions) would impact numerical results. We recommend todetermine systematically the dependence of each observable of interest on the listed degreesof freedom. Some illustrative dependences are provided in Appendix B.3 for total rates.More sophisticated observables may be sensitive to additional degrees of freedom or haveenhanced dependences on particular combinations.5. As a starting point, we rely on a tree-level description, working at zeroth order in the loopexpansion. All tree-level contributions are considered on an equal footing. Hierarchies be-tween them are only derived from the available experimental constraints. The dependenceof a specific observable on a given EFT parameter is possibly omitted only if other measure-ments constrain that parameter much below the level of sensitivity of this observable. As aresult, the inclusion of its dependence should affect neither the pre-existing constraints onthat parameter, nor the resulting constraints on others, in a combination of this new mea-surement with existing ones. This approach is very phenomenological and agnostic aboutspecific theories. In practice, as further and further constraints are collected and combined,a picture will progressively emerge of what specific measurement is particularly relevant toconstrain a given direction in parameter space.The higher-order dependences of observables on SM couplings will be considered in a secondstep. They could either induce corrections to existing tree-level contributions, or generate adependence on new EFT parameters. A discussion of next-to-leading order QCD effects onEFT predictions is planned. A variety of results is already available in the literature [6–13]. The definitions of the operators that will be relevant to our discussion —those which containa top quark for a suitable flavour assignment— are collected here. The associated degrees offreedom, following the flavour assumptions detailed in Section 4, will be defined in Appendix Cand used in the rest of this note. The notation employed in this section is that of Ref. [1] withflavour indices labelled by ijkl ; left-handed fermion doublets denoted by q , l ; right-handed fermionsinglets by u , d , e ; the Higgs doublet by ϕ ; the antisymmetric SU (2) tensor by ε ≡ iτ ; ˜ ϕ = εϕ ∗ ;3 ϕ † i ←→ D µ ϕ ) ≡ ϕ † ( iD µ ϕ ) − ( iD µ ϕ † ) ϕ ; ( ϕ † i ←→ D Iµ ϕ ) ≡ ϕ † τ I ( iD µ ϕ ) − ( iD µ ϕ † ) τ I ϕ where τ I are the Paulimatrices; T A ≡ λ A / λ A are Gell-Mann matrices.Four-quark operators: O ijkl ) qq = (¯ q i γ µ q j )(¯ q k γ µ q l ) , (1) O ijkl ) qq = (¯ q i γ µ τ I q j )(¯ q k γ µ τ I q l ) , (2) O ijkl ) qu = (¯ q i γ µ q j )(¯ u k γ µ u l ) , (3) O ijkl ) qu = (¯ q i γ µ T A q j )(¯ u k γ µ T A u l ) , (4) O ijkl ) qd = (¯ q i γ µ q j )( ¯ d k γ µ d l ) , (5) O ijkl ) qd = (¯ q i γ µ T A q j )( ¯ d k γ µ T A d l ) , (6) O ( ijkl ) uu = (¯ u i γ µ u j )(¯ u k γ µ u l ) , (7) O ijkl ) ud = (¯ u i γ µ u j )( ¯ d k γ µ d l ) , (8) O ijkl ) ud = (¯ u i γ µ T A u j )( ¯ d k γ µ T A d l ) , (9) ‡ O ijkl ) quqd = (¯ q i u j ) ε (¯ q k d l ) , (10) ‡ O ijkl ) quqd = (¯ q i T A u j ) ε (¯ q k T A d l ) , (11)Two-quark operators: ‡ O ( ij ) uϕ = ¯ q i u j ˜ ϕ ( ϕ † ϕ ) , (12) O ij ) ϕq = ( ϕ † i ←→ D µ ϕ )(¯ q i γ µ q j ) , (13) O ij ) ϕq = ( ϕ † i ←→ D Iµ ϕ )(¯ q i γ µ τ I q j ) , (14) O ( ij ) ϕu = ( ϕ † i ←→ D µ ϕ )(¯ u i γ µ u j ) , (15) ‡ O ( ij ) ϕud = ( ˜ ϕ † iD µ ϕ )(¯ u i γ µ d j ) , (16) ‡ O ( ij ) uW = (¯ q i σ µν τ I u j ) ˜ ϕW Iµν , (17) ‡ O ( ij ) dW = (¯ q i σ µν τ I d j ) ϕW Iµν , (18) ‡ O ( ij ) uB = (¯ q i σ µν u j ) ˜ ϕB µν , (19) ‡ O ( ij ) uG = (¯ q i σ µν T A u j ) ˜ ϕG Aµν , (20)Two-quark-two-lepton operators: O ijkl ) lq = (¯ l i γ µ l j )(¯ q k γ µ q l ) , (21) O ijkl ) lq = (¯ l i γ µ τ I l j )(¯ q k γ µ τ I q l ) , (22) O ( ijkl ) lu = (¯ l i γ µ l j )(¯ u k γ µ u l ) , (23) O ( ijkl ) eq = (¯ e i γ µ e j )(¯ q k γ µ q l ) , (24) O ( ijkl ) eu = (¯ e i γ µ e j )(¯ u k γ µ u l ) , (25) ‡ O ijkl ) lequ = (¯ l i e j ) ε (¯ q k u l ) , (26) ‡ O ijkl ) lequ = (¯ l i σ µν e j ) ε (¯ q k σ µν u l ) , (27) ‡ O ( ijkl ) ledq = (¯ l i e j )( ¯ d k q l ) , (28)Baryon- and lepton-number-violating operators: ‡ O ( ijkl ) duq = ( d ciα u jβ )( q ckγ εl l ) (cid:15) αβγ , (29) ‡ O ( ijkl ) qqu = ( q ciα εq jβ )( u ckγ e l ) (cid:15) αβγ , (30) In the latest version of Ref. [1], O , qqq are merged into one single operator with SU (2) L indices mixed betweenthe two fermion bilinears. The two conventions are technically speaking equivalent [14]. O ijkl ) qqq = ( q ciα εq jβ )( q ckγ εl l ) (cid:15) αβγ , (31) ‡ O ijkl ) qqq = ( q ciα τ I εq jβ )( q ckγ τ I εl l ) (cid:15) αβγ , (32) ‡ O ( ijkl ) duu = ( d ciα u jβ )( u ckγ e l ) (cid:15) αβγ , (33)Non-Hermitian operators are marked with a double dagger (only in the above list): ‡ O . ForHermitian operators which involve vector Lorentz bilinears, complex conjugation is equivalent tothe transposition of generation indices: O ( ij ) ∗ = O ( ji ) and by extension, for four-fermion operators, O ( ijkl ) ∗ = O ( jilk ) . The corresponding effective Lagrangian then takes the form: L = X a (cid:18) C a Λ ‡ O a + h.c. (cid:19) + X b C b Λ O b , (34)where the conjugates of the Hermitian operators, labelled here with an index b , are not added.Conventionally and unless otherwise specified, the arbitrary scale Λ will be set to 1 TeV. Equiva-lently, one could consider that numerical values quoted are in units of TeV − for the dimensionfulcoefficients ˜ C i ≡ C i / Λ . It is understood that the implicit sum over flavour indices only includesindependent combinations, i.e., the symmetry in flavour space of the operator coefficients is takeninto account. As mentioned in the introduction, prioritizing the study of certain flavour structures is requiredamong the otherwise overwhelming number of four-quark operators. In the lepton sector, itseems manageable to only assume flavour diagonality, i.e. [ U (1) l + e ] which includes a separate U (1) l × U (1) e diagonal subgroup for each of the three lepton generations. This leaves independentparameters for each lepton-antilepton pair of a given generation. This assumption adopted asbaseline could, for instance, easily be further restricted to [ U (1) l × U (1) e ] , U (3) l + e or even U (3) l × U (3) e . As in the quark sector (see below), the third generation of leptons could also be putaside and U (2) symmetries assumed among the first two generations. We discuss different flavourassumptions for the quark sector in the next three sections, starting with a baseline symmetry(Section 4.1) subsequently made less and more restrictive (Sections 4.2 and 4.3). U (2) q × U (2) u × U (2) d scenario As a baseline flavour scenario in the quark sector and motivated —as detailed below— by theminimal flavour violation (MFV) ansatz [15–17], we impose a U (2) q × U (2) u × U (2) d symmetryamong the first two generations. The MFV expansion of quark bilinear coefficients is the following:¯ q i q j : a I + a Y u Y † u + a Y d Y † d + · · · (35)¯ u i u j : b I + b Y † u Y u + . . . (36)¯ d i d j : c I + c Y † d Y d + . . . (37)¯ u i d j : d Y † u Y d + . . . (38)¯ q i u j : e Y u + e Y u Y † u Y u + e Y d Y † d Y u + . . . (39)¯ q i d j : f Y d + f Y d Y † d Y d + f Y u Y † u Y d + . . . (40)where a , b , etc. are order-one coefficients (see Ref. [18] for the resummation of large contributionsin such series). As a first approximation, we assume a unit CKM matrix, and retain only thetop and bottom Yukawa couplings, so that Y u = diag(0 , , y t ) and Y d = diag(0 , , y b ). Whenconsidering low-energy constraints and restoring the full CKM matrix, one would need to worryabout whether UV flavour structures are aligned with the up-, down-quark sectors, or in between5hose limits [19]. Denoting the left-handed quark doublet and right-handed quark singlets of thethird generation as Q , t , and b ,¯ q i q i , ¯ u i u i , ¯ d i d i bilinears are allowed in the first two generations,¯ QQ, ¯ tt, ¯ bb, ¯ tb, ¯ Qt, ¯ Qb bilinears are allowed in the third generation,under the above assumptions. The coefficients of the first-generation bilinears do not depend onthe i ∈ { , } index which is thus implicitly summed over. Fierz transformations may be requiredon four-fermion operators to bring such quark-antiquark pairs in the same Lorentz bilinear. Equiv-alently, a U (2) q × U (2) u × U (2) d symmetry is assumed between the first two quark generations andno restriction is imposed on the third-generation bilinears. This assumption simplifies four-fermionoperators but does not affect third-generation two-fermion ones. Compared to flavour diagonality,i.e. [ U (1) q + u + d ] , which would just force quarks and antiquarks to appear in same-flavour pairs, U (2) q × U (2) u × U (2) d effectively imposes the following additional requirements:1. the right-handed charged currents of the first generations (¯ ud , ¯ du ) are forbidden,2. the chirality-flipping bilinears of the first generations (¯ qu , ¯ qd ) are forbidden,3. the coefficients of the bilinears of the first and second generations are forced to be identical.The U (2) q × U (2) u × U (2) d flavour symmetry assumption is used by default in this note where nototherwise specified. The following numbers of degrees of freedom are produced for the operatorsof each category of field content:four heavy quarks 11 + 2 CPVtwo light and two heavy quarks 14two heavy quarks and bosons 9 + 6 CPVtwo heavy quarks and two leptons (8 + 3 CPV) × U (2) q × U (2) u × U (2) d scenario would retain only thefour-fermion operators and exclude the operators with two heavy quarks and bosons. This wouldbe justified when heavy bosons only couple to the SM fermions, so that the low-energy effects aredominated by the tree-level exchanges of heavy mediators between fermionic currents. U (2) q + u + d scenario In order to allow for the light-quark bilinears listed in item 1 and item 2 above, one can limit theflavour symmetry imposed to U (2) q + u + d only, the diagonal subgroup of U (2) q × U (2) u × U (2) d .The additional 10 + 10 CPV degrees of freedom that then appear for operators containing twolight and two heavy quarks are discussed in Appendix D. top-philic scenario A more restrictive top-philic scenario is not obtained by imposing a specific flavour symmetry butrather by assuming that new physics couples dominantly to the left-handed doublet and right-handed up-type quark singlet of the third generation as well as to bosons. All possible operatorswith this field content are thus constructed. Purely bosonic operators which lead to flavour-universal effects are discarded. A projection onto the Warsaw basis is subsequently performed,notably by employing the equations of motion to trade operators with more derivatives for oper-ators with more fields. In this process, the CKM matrix is again approximated by a unit matrixand all Yukawa couplings but the top and bottom ones are neglected. Only a limited number of6ndependent Lorentz and colour structures are generated. In terms of the degrees of freedom ofthe baseline scenario, the ones generated in this case are: c [ I ] tϕ , c − ϕq , c ϕq , c ϕt , c [ I ] tW , c [ I ] tB , c [ I ] tG , (41) c [ I ] ϕtb and c [ I ] bW appear proportional to y b (42) c QQ , c QQ , c Qt , c Qt , c tt , (43) c QDW = c , Qq = c ‘ ) Ql , (44) c QDB = 6 c , Qq = 32 c Qu = − c Qd = − c Qb = − c ‘ ) Ql = − c ( ‘ ) Qe , (45) c tDB = 6 c tq = 32 c tu = − c td = − c tb = − c ( ‘ ) tl = − c ( ‘ ) te , (46) c QDG = c Qq = c Qu = c Qd = c Qb , (47) c tDG = c tq = c tu = c td = c tb . (48)All other degrees of freedom then vanish. Counting the degrees of freedom in this scenario,distinguishing operator categories by their field content, one obtains:two heavy quarks and bosons 9 + 6 CPVfour heavy quarks 5four fermions 5. An example of analysis strategy is sketched in this section to illustrate how the challenges ofglobal EFT interpretations could be addressed and to highlight what would, in that case, be usefulexperimental outputs. It relies on fiducial observables defined at, and unfolded to, the particlelevel. We stress that other strategies could be equally suitable, more practical to implement insome cases or lead to better sensitivities, by avoiding unfolding. As our aim is illustrative only andas it is in any case difficult to make prescriptions applicable to any type of experimental analysis,we refrain from considering other possibilities in detail. We expect, however, many of the pointsraised and recommendations made to apply more generally to other analysis strategies. A wealthof EFT analyses has for instance been carried out in the Higgs and electroweak sectors at the LHCand treats issues related to those discussed here (see e.g. Refs. [20–25]).The approach presented here is meant to be simultaneously practical and useful on a long-termbasis. Importantly, it allows for relatively precise reinterpretations without full detector simulation.It could thus easily benefit from future improvements in the accuracy of our EFT predictions andadapt to the evolving picture of global EFT constraints or to changes in underlying assumptions.It should allow to derive robust global constraints and be applicable in a wide variety of situationsbut could however not lead to the tightest constraints on individual operators. Our rationale is thatglobal constraints have more value than individual ones. A global analysis covers systematicallythe theory space in direct vicinity of the SM and is able to identify new physics through correlateddeviations in precise measurements. A tentative recipe could be the following:1. Define observables at the particle level in a fiducial phase-space volume close enough to thedetector level so as to make the unfolding model independent.(a) Several bins of a differential distribution would qualify as examples of observables. Atotal rate would qualify too, as well as ratios [26, 27], asymmetries, etc. A higher level, like the parton one, could be envisioned but would enhance the model-dependence of theunfolding. If advantageous, the forward folding of particle-level predictions to the detector level could be consideredinstead.
C++ code taking asinput a particle-level event sample or kinematic variables.(c) Statistically optimal observables [28–30], similar to matrix element methods (see alsoRef. [31, 32]), could be very useful from both theoretical and experimental points ofview. Their definitions rely on firm theoretical bases, and encode our physical under-standing instead of requiring a resource-intensive and opaque training. They moreoverconstitute a discrete set exactly sufficient to maximally exploit the available kinematicinformation on which the effect of higher-order corrections (see e.g. Ref. [33]) or system-atic uncertainties can be transparently studied. Applications in experimental analysesinclude anomalous triple gauge coupling studies at LEP [34–37] and, recently, CP prop-erties of the Higgs boson in di-tau final state [38].2. Unfold the measurement of these observables, as well as the estimates for the various SMcontributions, to the particle level.(a) For a suitably defined fiducial region, the particle and detector levels should be suffi-ciently close to each other for the unfolding to be performed under the SM hypothesisonly. Full simulation at various EFT parameter points would then be avoided alto-gether. The validity of this approximation should however be checked explicitly.(b) The SM contributions would include both the signal and backgrounds of a SM mea-surement. It is important to detail the background composition as these processescould have some EFT dependence which may be neglected at first but which could bedesirable to account for at some later point in time.3. For the various measurements, provide the statistical and systematics likelihoods, errorsource breakdown, their correlations, and whether they follow a flat or Gaussian distribution.Notice that the information above should be sufficient for anybody able to generate an EFT(or any NP) sample at the particle level to set constraints. In order to obtain global EFTconstraints, one could further proceed as follows.4. Compute, numerically or analytically, for each observable O k , the linear S i and quadratic S ij contributions of dimension-six operators, in addition to the SM contributions B l alreadymentioned in item 2b. Schematically, the expansion in powers of dimension-six operatorcoefficients can the be written as: O k = B kl + C i Λ S ki + C i C j Λ S kij + · · · (49)Quadratic and higher powers of dimension-six operator coefficients can be generated at treelevel when several operator insertions are possible in one amplitude. The expansion ofnormalized distributions, ratios, asymmetries, or of the total width appearing in top-quarkpropagators also generates such higher powers. In some cases, the interference betweenSM amplitudes and EFT ones could be suppressed (see for instance helicity selection rulesdiscussed in Ref. [39] and references therein) or even vanishingly small (in the case of FCNCsfor instance). The dominant contribution could then arise at the quadratic level. Operatorsof dimension larger than six, in particular that of dimension-eight which were not consideredin this note, also start contributing at order 1 / Λ in the EFT expansion.(a) At tree level, with MG5_aMC@NLO , if a different coupling order
EFT i is given to eachof the relevant N operator coefficients C i , the computation of S ki and S kij can mosteasily be done by generating at most N + N ( N + 1) / DIM6_iˆ2==1 syntax of
MG5_aMC@NLO [40] in particular). If8he approximation in item 2a holds, these samples do not need to be passed throughdetector simulation.(b) As explained in Section 2, it is in principle desirable to include all operators thatcontribute at some given loop order, say first at the tree level. Exceptions could bemade for operators that are much better constrained than the others, at the time atwhich the limits are extracted. This comparison of the relative strength of differentmeasurements could however evolve with time and the goodness of this approximationbe re-evaluated.5. Within some statistical framework, use the measured O k , the estimated B kl with their sta-tistical and systematic uncertainties, and the S ki , S kij to derive global constraints on the C i operator coefficients.(a) It is instructive to also quote individual constraints, set by considering one operator ata time. A comparison between global and individual constraints gives some indicationabout the magnitude of approximate degeneracies between EFT parameters.(b) For easy combination with other measurements, provide the full covariance matrices ofuncertainties on the C i determination in the Gaussian approximation. The eigenvec-tors of its inverse corresponding to vanishing eigenvalues indicate which directions ofthe EFT parameter space remain unconstrained by the measured observables. Othereigenvectors and eigenvalues provide the constraints in other directions.(c) Repeat this procedure twice, with and without including the S kij quadratic EFT con-tributions. The comparison between those two sets of results can explicitly establishwhere quadratic dimension-six EFT contributions are subleading compared to linearones. Remarkably, when the linear dependence dominates, the constraints obtainedcan easily be translated from one basis of dimension-six operators to the other and aretherefore of greater generality.6. For the purpose of this note, let us define our EFT description to be valid in the regimewhere it is dominated by dimension-six operator contributions, at the linear level or at somefinite higher order. Determining the relative magnitude of the coefficients of operators ofvarious dimensions requires a specific model or power counting. Moreover, the magnitudes ofoperator-coefficient contributions to a specific observable also depend on E , the characteristicenergy scale of the process examined. In a production process at the LHC, the reconstructedpartonic centre-of-mass energy is often, theoretically, such a suitable quantity. Other proxiesmay also be considered like H T , the scalar sum of all transverse energies. Their correlationwith the physically relevant variables should then be studied, as done for instance in Ref. [23](see Fig. 2 and related discussion). In decays, this characteristic scale E is the mass of thedecaying particle.When it is practically feasible, displaying the variation of the limit as a function of an uppercut on E at E cut (see Fig. 1) allows for the valid interpretation of EFT results in a largerclass of models [41, 42]: featuring a new-physics scale lower than the nominal E , or leadingto significant contributions from operators of dimension larger than six when no E cut isapplied.Interpreting EFT results in specific models proceeds by expressing each C i / Λ in terms ofthe model couplings and mass scales, through a so-called matching procedure. The EFTdescription of a resonance of mass M would be dominated by the operators of lowest di-mension when M (cid:29) E cut . Stricter constraints may apply for instance if the resonance hasa large width.7. Besides validity, the quantum perturbativity of the dimension-six EFT can be examinedusing the same picture. This perturbativity is a necessary condition for the validity (definedas in the previous item) and can be established without reference to a specific model orpower counting. 9 i (cid:14) Λ E C i ( E cut / π Λ) = cst C i ( E cut / Λ) = cstlinear limitquadratic limit Figure 1: Illustration of the limit set on an EFTparameter as function of a cut on the character-istic energy scale of the process considered (seeitem 6). Qualitatively, one expects the limitsto be progressively degraded as E cut is pushedtowards lower and lower values. At high cut val-ues, beyond the energy directly accessible in theprocess considered, a plateau should be reached.The regions excluded when the dimension-sixEFT is truncated to linear and quadratic ordersare delimited by solid lines (see item 5c). Thehatched regions indicate where the dimension-six EFT loses perturbativity (see item 7). Inpractice, curves will not be symmetric with re-spect to C i / Λ = 0.(a) This quantum perturbativity of the dimension-six EFT is examined through the sum ofcontributions involving more and more operator insertions in diagrams with higher andhigher numbers of loops. The convergence of such a series (involving one additional op-erator insertion per loop) requires that C i ( E cut / π Λ) be smaller than a constant whichwould roughly be of order one if the normalization of C i is natural to the observableconsidered. The exact condition would need to be determined on a case-by-case basis.It defines a hyperbola in the ( C i / Λ , E ) plane. At tree level, all allowed powers of C i / Λ can moreover become relevant as soon as C i ( E cut / Λ) approaches a constant,possibly of order one. If contributions from higher dimensional operators at the sameorder in 1 / Λ had similar magnitudes, they would also become relevant at that point.(b) If it exists, the point at which the perturbativity hyperbola crosses the limit curvesestablishes a minimal E cut that has to be imposed for the limit computed perturbativelyto make sense. For a sufficiently tight constraint, the limit curves would only cross theperturbativity hyperbola for a cut value beyond the maximal energy directly accessiblein the process considered.(c) If it exists, the point at which the C i ( E cut / Λ) = constant hyperbola crosses the limitcurve provides a sense of where higher-order terms in the tree-level expansion in powersof dimension-six operator coefficients could become relevant. In particular, one mayexpect the linear and quadratic limits to start diverging around that point.(d) When discussing the relevance of higher-order contributions, definite statements aredifficult to make, as is always the case when trying to estimate the terms not computedin a truncated series. As mentioned in item 4, it is worth bearing in mind the firstterm could for instance have an accidental suppression as it generally arises from theinterference of a SM amplitude with an EFT one. In this note, we established the first bases of a framework for the EFT interpretation of top-quarkmeasurements at the LHC. Experimental collaborations and theorists are encouraged to use thiscommon language in future publications to facilitate comparisons and combinations. Having men-tioned the caveats associated with those choices, we limited our scope to dimension-six operatorsof the Warsaw basis which contain a top quark, prioritized the study of possible flavour structureswith consistent flavour scenarios, defined the degrees of freedom relevant in each case, compiledexisting indicative constraints on these parameters, discussed the use of simulation tools to ex-tract their dependence, and tabulated several benchmark results. An example of analysis strategy10as also sketched to illustrate one possible way in which the challenges of a global EFT analysiscould be addressed. The experimental outputs that are in that case desirable were highlighted.It is expected that the statements made in the discussion of this example could be transposed toconcrete analyses and adapted to different strategies.Having laid those bases, we did not address more involved issues like next-to-leading ordercorrections (especially in QCD), theoretical uncertainties, or the treatment of unstable tops. Theseimportant topics may deserve further discussion in the LHC TOP Working Group. It is left to thecommunities of both theorists and experimentalists to study the observables that are best suitedto constrain particular direction of the EFT parameter space and to examine the complementaritybetween them as well as between different processes. Combining the constraints from varioussources or determining which dependences are relevant in each observable are tasks that shouldalso be addressed to make progress in the program outlined here. Ultimately, it will be importantto combine top-quark, Higgs and electroweak measurements within the SMEFT. Studies targetingthese two other sectors are discussed in the corresponding LHC HXS and EW Working Groups.Several methodological points raised in this note, such as those discussed in Section 5, have alsobeen addressed in these contexts, see for example Ref. [43].Finally, as stressed several times in this note, the global EFT picture will necessarily evolvewith time as new measurements and more accurate predictions become available. Some of theassumptions made here may therefore need to be revised in the future as a finer picture is obtained.
Acknowledgements
We would like to warmly thank the experimentalists of the ATLAS and CMS Collaborationshaving provided us with extensive feedback on the content of this note, the participants to theLHC TOP Working Group meetings in which it was discussed, and the conveners of the WorkingGroup for their support.
A Indicative constraints
As a reference, we collect here various limits set by theoretical studies on the relevant degrees offreedom. We note that there is presently no study considering all the operators listed above andtherefore no marginalised constraints are available. Typically, the fits considering one operator at atime (or marginalising over a smaller subset of operators) provide more stringent constraints. Thelimits listed should therefore only serve as a guidance for potential sensitivity studies or fits. Directlimits arising from the top-quark measurements are given in Table 1, where available. Indirectlimits from low-energy observables also exist, some of which can be very strong yet more modeldependent. Limits from B decays, dilepton production, electroweak precision observables, as wellas electric dipole moments and CP asymmetries are discussed in the following three subsections. A.1 Constraints from low-energy flavour physics
The CKM matrix can be approximated as the identity matrix when focusing on measurementsinvolving resonant top quarks. However, when confronting new physics models with flavourobservables, suppressed in the SM by small CKM matrix elements, it is of course crucial toconsistently keep all the CKM factors. By working in the down-quark mass basis, the mis-alignment between the up- and down- quark masses can be described by considering the quarkdoublet as q i = ( V ∗ ji u L,j , d
L,i ) T . Another basis often used is the one where up quarks havediagonal mass matrix. This can be obtained by simply rotating q i with the CKM matrix:˜ q i = ( V q ) i = ( u L,i , V ij d L,j ) T . This choice changes the flavour structure of the operators bythe same CKM rotations, for example the term C ( ij ) ¯ q i γ µ q j → ˜ C ( ij ) ¯˜ q i γ µ ˜ q j , where ˜ C = V CV † . Inthe spirit of this note, we assume that the (33) element of such flavour matrices is the largest one.The U (2) q flavour symmetry relates the off-diagonal components to the CKM matrix [48, 49]. Inthis well motivated framework, in general the new physics contributions are not necessarily aligned11our-heavy (11 + 2 CPV d.o.f.) Indicative direct limits c QQ ≡ C qq − C qq c QQ ≡ C qq ! c + QQ ≡ C qq + C qq [ − . , .
80] ( E cut = 3 TeV) [44] c Qt ≡ C qu [ − . , .
90] ( E cut = 3 TeV) [44] c Qt ≡ C qu [ − . , .
33] ( E cut = 3 TeV) [44] c Qb ≡ C qd c Qb ≡ C qd c tt ≡ C (3333) uu [ − . , .
80] ( E cut = 3 TeV) [44] c tb ≡ C ud c tb ≡ C ud c I ] QtQb ≡ [Im]Re { C quqd } c I ] QtQb ≡ [Im]Re { C quqd } Two-light-two-heavy (14 d.o.f.) c , Qq ≡ C ii qq + ( C i i ) qq − C i i ) qq ) [ − . , .
24] [45], [ − . , .
10] [44] c , Qq ≡ C i i ) qq − C i i ) qq [ − . , .
73] [44] c , Qq ≡ C ii qq + C i i ) qq + C i i ) qq [ − . , .
15] [44] c , Qq ≡ C i i ) qq + 3 C i i ) qq [ − . , .
93] [44] c Qu ≡ C ii ) qu [ − . , .
44] [44] c Qu ≡ C ii ) qu [ − . , .
05] [44] c Qd ≡ C ii ) qd [ − . , .
02] [44] c Qd ≡ C ii ) qd [ − . , .
39] [44] c tq ≡ C ii qu [ − . , .
84] [44] c tq ≡ C ii qu [ − . , .
49] [44] c tu ≡ C ( ii uu + C ( i i ) uu [ − . , .
57] [44] c tu ≡ C ( i i ) uu [ − . , .
75] [44] c td ≡ C ii ) ud [ − . , .
04] [44] c td ≡ C ii ) ud [ − . , .
31] [44]Two-heavy (9 + 6 CPV d.o.f.) c [ I ] tϕ ≡ [Im]Re { C (33) uϕ } c − ϕq ≡ C ϕq − C ϕq c ϕq [ − . , .
1] [45], [ − . , .
6] [46] c ϕQ ≡ C ϕq [ − . , .
0] [45], [ − . , .
3] [46] c ϕt ≡ C (33) ϕu [ − . , .
3] [45], [ − . , .
1] [46] c [ I ] ϕtb ≡ [Im]Re { C (33) ϕud } c [ I ] tW ≡ [Im]Re { C (33) uW } c tW [ − . , .
5] [45], [ − . , .
1] [46] c [ I ] tZ ≡ [Im]Re {− s W C (33) uB + c W C (33) uW } c tB [ − . , .
6] [45], [ − . , .
6] [46] c [ I ] bW ≡ [Im]Re { C (33) dW } c [ I ] tG ≡ [Im]Re { C (33) uG } c tG [ − . , .
24] [45]Two-heavy-two-lepton (8 + 3 CPV d.o.f. × c ‘ ) Ql ≡ C ‘‘ lq c − ( ‘ ) Ql ≡ C ‘‘ lq − C ‘‘ lq c ( ‘ ) Qe ≡ C ( ‘‘ eq c ( ‘ ) tl ≡ C ( ‘‘ lu c ( ‘ ) te ≡ C ( ‘‘ eu c S [ I ]( ‘ ) t ≡ [Im]Re { C ‘‘ lequ } c T [ I ]( ‘ ) t ≡ [Im]Re { C ‘‘ lequ } c S [ I ]( ‘ ) b ≡ [Im]Re { C ( ‘‘ ledq } Table 1: Indicative limits on top-quark operator coefficients for Λ = 1 TeV. For details on thefit procedure, information on the input data and set of operators over which the results aremarginalised please consult the corresponding references (see also Ref. [47]). Coefficients markedwith a ‘!’ are not independent of the ones previously defined. 12ith the up or down quarks, but will have some misalignment of the order of the relevant CKMelements. For definiteness, in the following we mostly work in the down-quark mass basis.Let us consider the charged-current transition b → c e i ¯ ν j and study the experimental limitswe can extract on the top-quark operators. From the general effective Hamiltonian at the B -meson mass scale, we can consider only the operators involving the left-handed c L quark, since c R necessarily arises from a second-generation family index. We are left with H b → ce ¯ ν eff = − v V cb (cid:16) ( δ ij + c ijV L )(¯ c L γ µ b L )(¯ e iL γ µ ν jL ) + c ijS R (¯ c L b R )(¯ e iR ν jL ) + h.c. (cid:17) , (50)where v ≈
246 GeV. The tree-level matching to the SMEFT is given by [50, 51] c ijV L = − v Λ P k V ck C ijk lq V cb ! + v Λ P k V ck C k ϕq V cb ! δ ij ,c ijS R = − v Λ P k V ck C ( ji k ) ledq V cb , (51)where k = 1 , , U (2) q symmetry, the contribution from the (33) element and from the (23) one are of the same order since,for example C ij lq /C ij lq ∼ O ( V cb ). In realistic fits to these observables it is thus importantto keep track of all contributions, which can lead to important phenomenological consequences(see e.g. Ref. [52]). Nevertheless, in order to extract the indicative numerical constraints, in thefollowing we assume down-alignment and keep as non vanishing only the (33) element of theseoperators.Consistently with the rest of the note, we neglect lepton-flavour-violating terms, while weallow possible deviations from lepton-flavour universality. In the case of operators with electronand muon one can derive the constraints from B → D ( ∗ ) ‘ν decays [53], which experimentallyagree with the SM prediction. In the case of the τ lepton, instead, the corresponding decays B → D ( ∗ ) τ ν show an interesting deviation from the SM prediction with a statistical significanceof more than 4 σ [54–58]. In particular, the relevant observables are ratios between the decaymodes to tau and light leptons in which hadronic uncertainties largely cancel. Combining thedifferent measurements one can express the result as (see e.g. Ref. [52]) R D ( ∗ ) ≡ B ( B → D ( ∗ ) τ ν ) B ( B → D ( ∗ ) ‘ν ) (cid:18) B ( B → D ( ∗ ) τ ν ) SM B ( B → D ( ∗ ) ‘ν ) SM (cid:19) − = | c ττV L | = 1 . ± . , (52)where we showed explicitly only the contribution from the vector operator. This corresponds to c ττV L = 0 . ± . c ττS R contributes to the observables above,with a different weight in R D compared to R D ∗ , a stronger constraint can be derived from the B c lifetime [59], due to the chiral enhancement of the scalar contribution. This gives an approximatebound | c ττS R | (cid:46) .
39, which should be compared to the value c S R ≈ . R D ∗ . In the case of down-alignment, a non-vanishing contribution to D − ¯ D mixing is generatedby the CKM rotation from the O qq and O qq operators: L eff ⊃ C qq + C qq Λ ( V ub V ∗ cb ) (¯ u L γ µ c L ) + h.c. . (53)This can be used to cast a limit on the combination above [60]. The limits expressed in terms ofthe degrees of freedom defined in this note are reported in Table 2.With the choice of down-alignment, contributions to b → s transitions, such as in B → K ( ∗ ) ‘ + ‘ − decays or in B s − ¯ B s mixing, arise proportionally to the off-diagonal (32) element of thequark flavour matrix of the various operators. While, as already mentioned, the SU (2) q flavoursymmetry predicts that this element should be of O ( V cb ), in the simplified discussion above this wasput to zero by hand, thus forbidding tree-level contributions to these processes. It is worthwhile to13our-heavy c + QQ ≡ C qq + C qq [ − ,
21] [60] (from D − ¯ D , with down-alignment)˜ c + QQ ≡ ˜ C qq + ˜ C qq [ − . , .
03] [60] (from B s − ¯ B s , with up-alignment)Two-heavy-two-lepton ‘ = e ‘ = µ ‘ = τc ‘ ) Ql ≡ C ‘‘ lq [ − . , .
22] [53] [ − . , .
57] [53] − . ± .
40 [52] c S ( ‘ ) b ≡ Re { C ( ‘‘ ledq } [ − ,
10] [53] [ − ,
13] [53] [ − ,
13] [59]Table 2: Indicative limits on top-quark operators arising from semileptonic B decays and heavymeson oscillations and for Λ = 1 TeV.Two-heavy-two-lepton ‘ = e ‘ = µ ‘ = τc − ( ‘ ) Ql + 2 c ‘ ) Ql ≡ C ‘‘ lq + C ‘‘ lq [ − . , .
20] [64] [ − . , .
34] [64] [ − . , .
6] [65] c ( ‘ ) Qe ≡ C ( ‘‘ eq [ − . , .
28] [64] [ − . , .
40] [64] – c S ( ‘ ) b ≡ Re { C ( ‘‘ ledq } – – [ − . , .
9] [65]Table 3: Indicative limits on top-quark operators arising from dilepton pair production at theLHC and with Λ = 1 TeV.briefly discuss what would happen in the opposite scenario of up-alignment , where the operatorsare written in terms of ˜ q i = ( u L,i , V ij d L,j ) T and only the (33) component of the flavour matrix isleft non-vanishing. In this case the b → d i transition would arise proportionally to V ib , while therewould be no b → c charged-current transition (which, in this case, would be strictly proportionalto the (32) element of the flavour matrices). ∆ F = 2 processes in the down sector, therefore, putstrong limits at tree level on four-quark operators: L eff ⊃ ˜ C qq + ˜ C qq Λ (cid:2) ( V ts V ∗ tb ) (¯ b L γ µ s L ) + ( V td V ∗ tb ) (¯ b L γ µ d L ) + ( V td V ∗ ts ) (¯ s L γ µ d L ) (cid:3) + h.c. . (54)In particular, the strongest constraint on the overall coefficient is from B s − ¯ B s mixing [60]. Themeasurement of various b → s‘ + ‘ − transitions would instead constrain the two-quark-two-leptonoperators ˜ O ij lq , ˜ O ij lq , and ˜ O ( ij eq (see e.g. Refs. [61, 62]). Loop-level contributions to B s mixing and electroweak precision measurements can instead be used to put constraints on the˜ O ϕq , ˜ O ϕq , and ˜ O (33) ϕu operators [63]. It is clear that both up- and down-alignment are veryparticular cases. Experimentally, since limits from ∆ F = 2 processes are stronger in the downsector, the down-alignment case might be preferred. Generic models of flavour are expected tointerpolate between the two scenarios, therefore a more general analysis, which is well beyond thepurpose of the present note, would be advisable. A.2 Constraints from high- p T di-lepton searches It is well known that the high-energy tail of 2 → q ¯ q → q ¯ q and q ¯ q → ‘ + ‘ − . These have been shown to provide verystrong constraints on four-fermion operators [50, 64–68]. Since the sensitivity comes from thehigh energy tail, the issue of the validity of the EFT expansion is a crucial one to be addressedin this case. In particular, the underlying assumption for these bounds to be valid is that themaximal centre of mass energy from which the sensitivity is gained, E max , has to be much smallerthan the mass scale of the heavy states which have been integrated out, E max (cid:28) M NP . Thisissue has been studied in detail in the references above and it has been shown that there existmodels in which the new states are heavy enough for the EFT approach to be valid, and for14hich the constraints obtained are relevant. For example, in Ref. [64], it has been shown that q ¯ q → e + e − , µ + µ − processes can be used to put significant constraints on some models addressingthe neutral-current B -physics anomalies. Instead, the EFT interpretation of the limits in the τ τ final state [65] should be taken as indicative only, since the mass scale of new physics cannot betoo high in that case. Nevertheless, comparing with the limits on explicit models [65] shows thatthe EFT ones still provide a good first-order indication. In Table 3 we report the limits from[64, 65] on the two-quark-two-lepton operators involving third generation quarks, in particular the b quark since it is the only one accessible in the initial state. A.3 Indirect constraints from electroweak precision observables
Electroweak precision observables (EWPOs) provide additional constraints on the top-quark op-erators. These measurements include the Z -pole data from LEP and SLC, the fermion pairproduction and W pair production from LEP2, W mass and width measurements from LEP andTevatron, and other low-energy measurements such as DIS and atomic parity violation. Thesemeasurements do not directly involve resonant top quarks. The constraints come only from thetop-quark loop-induced contributions in the two-point functions of the electroweak gauge bosons,which are modified by top-quark operators. Unlike in the SM, these contributions are in generalUV divergent, and therefore they need to be interpreted with care. On the one hand, we aim toget as much information as possible, while on the other hand, we would like to base our approachonly on minimum set of assumptions, so that the resulting constraints are meaningful for a widerange of BSM scenarios. The approach adopted in Refs. [69, 70] is an example of a reasonablebalance between these two aspects.Consider the following subset of the two-heavy degrees of freedom c − ϕQ , c ϕQ , c ϕt , c tW , c tB , c bW (55)that are relevant in the EWPOs. One specific contribution, c − ϕQ + 2 c ϕQ , (56)corresponds to Z → b ¯ b and can be constrained at the tree-level. All the other degrees of freedomenter only through the modification of the self-energies of W , Z and γ at the loop level. Thesecontributions consist of two parts: (a) the UV poles and the corresponding logarithmic terms, (b) the remaining finite terms.The UV poles in the first part need to be cancelled by the renormalization of the following twonon-top operators, to get physical results: O ϕW B = (cid:0) ϕ † τ I ϕ (cid:1) W Iµν B µν (57) O ϕD = ( ϕ † D µ ϕ ) ∗ ( ϕ † D µ ϕ ) (58)In other words, the top-quark operators mix into these two operators. After the renormalization,contribution (a) gives a linear q -dependent part in all the self-energy functions Π V V ( q ). Thesecontributions can be identified as a change of the S and T parameters. It is well known that theEWPOs places stringent bounds on the S and T parameters, and so naively one could apply thesebounds to constrain the top-quark operators. However, given that O ϕW B and O ϕD are included This statement is basis dependent. In this study we use the “HISZ” basis [71], which is convenient for obliquephysics effects, but we keep the definitions of O ϕWB and O ϕD to be consistent with the Warsaw basis. The physicsresult is basis-independent. In the Warsaw basis, some of the counter terms for the oblique parameters will beprovided by other operators involving fermion fields. c − ϕq + 2 c ϕQ ≡ C ϕq + C ϕq . ± . c − ϕq ≡ C ϕq − C ϕq . ± . c ϕt ≡ C (33) ϕu . ± . c tW ≡ Re { C (33) uW } − . ± . c tB ≡ Re { C (33) uB } . ± . c bW ≡ Re { C (33) dW } ± S and T . One finds,for example for c tW , ˆ S = c ϕW B ( µ ) v Λ c W s W − N c gc tW π √ vm t
53 ln m t µ (59)which prevents us from directly setting bounds on c tW , as the value of c ϕW B ( µ ) is unknown. Here µ is the renormalization scale. Indicative bounds may be obtained by assuming “no accidentalcancellation” between the two terms, allowing us to set c ϕW B ( µ ) = 0. However, since the mixingeffect exits, the results strongly depend on the scale µ at which this assumption is made. Nevertheless, useful information can be obtained from contribution (b). By definition the S and T parameters assume a linear q dependence for all two-point functions Π V V ( q ). This is notthe case when loop-contributions are the dominant ones, in particular the finite terms in (b) arein general not linear q functions. On the other hand, the EWPOs contain more information thanthe S and T parameters. To extract this information, one needs to abandon the oblique parameterformalism and perform a global fit for all measurements, including the top-loop contributions inall theory predictions. c ϕW B ( µ ) and c ϕD ( µ ) should be included, to obtain physical predictions. Inthe resulting χ , one then marginalizes over the c ϕW B ( µ ) and c ϕD ( µ ) coefficients. The remainingconstraints become weaker, but they are more reliable as they do not depend on any specificassumptions on these two coefficients. One can also check that the constraints obtained in thisway are independent of the renormalization scale µ , to confirm that these results are physical.The details of this analysis can be found in Ref. [70]. We summarize the results on therelevant degrees of freedom, as indicative constraints in Table 4, assuming only one top operatoris considered at a time. As a final remark, apart from O ϕW B and O ϕD , one could certainly include more non-topoperators and perform a more global EW fit. The results will be more model-independent, atthe cost of weakening the limits one would get after marginalizing over the non-top operators.The reason for only incorporating the O ϕW B and O ϕD operators here, is that they are the onlyones to which the top-quark operators mix, and therefore the minimum set of operators one hasto incorporate in order to extract exactly the part (b) contributions, which are µ independentand therefore more physical. This however implies that the resulting limits are based on theassumption that the dominant BSM effects are captured by top-quark operators. The most constraining bounds obtained in this way corresponds to setting µ = Λ in Eq. (59) and assuming c ϕWB (Λ) = 0. This is often called the renormalization-group-induced bounds in the literature. They are howeverstrongly model-dependent as assumptions about the matching scale are required, and are not consistent with theglobal EFT picture as a bottom-up approach. Therefore we do not discuss these bounds here and refer the interestedreaders to Refs. [63, 72–74]. Ref. [75] also constrained both c − ϕq + 2 c ϕQ and c − ϕq by combining e + e − → Z → b ¯ b measurements with singletop-quark t -channel production at hadron colliders. .4 Indirect constraints from low-energy probes of CP violation In addition to limits from direct observables, complementary constraints can be derived from low-energy measurements. Such indirect observables do not involve resonant top quarks, but theyare affected, in some cases very significantly, by virtual top quarks and as such can be used toprobe SMEFT top-quark operators. Here we focus on a subset of SMEFT operators, namely theoperators defined in Appendix C that violate CP, which are mainly constrained by electric-dipole-moment (EDM) experiments and asymmetry measurements in B → X s γ . Although we do notchange the flavour structure of the operators themselves, we will deviate slightly from the baselineflavour assumptions of Section 4. In particular, we restore the off-diagonal CKM matrix elementsand the light Yukawa couplings. Although these small SM parameters can be neglected in directprobes, they give rise to loop diagrams that, in some cases, induce the dominant contributions tolow-energy probes. Indirect constraints on the CP-even parts, coming from electroweak precisiondata, are discussed in the next subsection.As not all readers might be familiar with EDM phenomenology we give a brief introductionhere. EDMs of leptons, nucleons, atoms, and molecules are probes of flavour-diagonal CPV thatsuffer from essentially no SM background. While the CKM mechanism predicts nonzero EDMs,they are orders of magnitude below the current experimental limits. At present, the strongestEDM constraints arise from measurements on three different systems: the neutron, the Hgatom, and the polar molecule ThO. The limit on the latter can be interpreted (with care) as alimit on the electron EDM. In order to interpret measurements on these complicated systems interms of the SMEFT Wilson coefficients, several steps need to be taken. First of all, the SMEFToperators must be evolved to lower energies, typically up to a scale of a few GeV where QCD isstill perturbative. At that point, the SMEFT operators are matched to an effective Lagrangiandescribing the dynamics of the relevant low-energy degrees of freedom such as nucleons, pions,photons, and electrons. This effective Lagrangian is then used to calculate the EDMs of nucleons,nuclei, atoms, and molecules. A detailed discussion is beyond the scope of this note, and belowwe briefly describe how to connect the limit on the neutron and electron EDM to the SMEFTWilson coefficients.We start by discussing the limits from the neutron EDM. This observable obtains contributionsfrom several CPV operators, namely four of the top-Higgs couplings, c tϕ , c tG , c bW , and c ϕtb , as wellas the four-quark operator coefficients, c , QtQb . To estimate the limits, we evolve these operators,by using renormalization-group equations (RGEs), from the scale of new physics, Λ, to the scaleof the top-quark mass. At this scale we integrate out the top quark. Here both c tϕ and c tG inducea threshold contribution to a purely gluonic CPV operator without top quarks, the so-calledWeinberg operator L W = g S d W Λ f abc (cid:15) µναβ G aαβ G bµρ G c ρν , (60)with [76–81] d W ( m t ) = g S π v √ m t h ( m t , m h ) c Itϕ + g S (4 π ) v √ m t c ItG , (61)where h ( m t , m h ) ’ .
05 is a finite two-loop integral. Note that c Itϕ also contributes to the EDMs oflight quarks, proportional to their Yukawa couplings, through two-loop Barr-Zee diagrams [82–85].We include this effect in the limits discussed below.In addition, c ϕtb , c bW , and c , QtQb contribute to the Weinberg operator by first inducing thebottom chromo-EDM, O (33) dG . For the four-quark operators and O (33) dW the bottom chromo-EDM isinduced through renormalization-group evolution [86–88] between µ = Λ and µ = m t , while O ϕtb only provides a matching contribution d Im C (33) dG d ln µ = √ π ) m t v (cid:18) c IQtQb − N C c IQtQb (cid:19) + 2(4 π ) (cid:20) g W + 13 g Y (cid:21) c IbW . . . , C (33) dG ( m − t ) = Im C (33) dG ( m + t ) + 1 √ π ) m t v f W c Iϕtb , (62)where f W ’ . O (33) dG then induces a contribution to the Weinberg operator d W ( m − b ) = d W ( m + b ) + g S (4 π ) v √ m b Im( C (33) dG ( m + b )) . (63)The Weinberg operator can now be evolved to lower energies and, around the QCD scale, bematched to hadronic operators. In particular, it induces a contribution to the neutron EDM | d n | ’ (50 MeV) eg s (1 GeV) d W (1 GeV) / Λ . (64)The required hadronic matrix element suffers from large uncertainties and here we have taken theaverage of various estimates [77, 90, 91]. The impact of hadronic and nuclear uncertainties on low-energy precision constraints on the SMEFT operators can be significant and has been discussed indetail in Refs. [88, 92]. The constraints that result from employing Eq. (64) and the experimentallimit, d n < . · − e fm [93, 94], are collected in Table 5.Moving on to the electron EDM, there are again several operators that contribute, namely,three top-Higgs couplings c ItA,tW,tϕ , as well as the semi-leptonic operators c S ( e ) t,b and c T ( e ) t . Of thesemi-leptonic operators the tensor operator induces the electron EDM through a single top loop,while the scalar ones require additional loops. The relevant RGEs are given by [87, 88], dd ln µ d e c SI ( e ) t / Λ c T I ( e ) t / Λ c SI ( e ) b / Λ = 1(4 π ) − N C Q t m t − C F g S − N c ) y t y b g W + g Y C F g S
00 0 0 − C F g S · d e c SI ( e ) t / Λ c T I ( e ) t / Λ c SI ( e ) b / Λ , (65)where C F = ( N C − / N C , y b,t = m b,t √ /v , Q f stands for the electric charge, and we onlykept the electroweak terms that are relevant for the mixing of c SI ( e ) t,b into c T I ( e ) t . The solution ofthe above equations provides the leading logarithmic contributions to the electron EDM. These,combined with d e ≤ . · − e fm [95], give stringent constraints, which are again collected inTable 5. The same loops that induce the electron EDM also give a contribution to the electronanomalous magnetic moment, proportional to the real part of the semileptonic couplings. Al-though the resulting limits are weaker than the EDM limits they are still significant for two of thecouplings, we obtain | c S ( e ) t | (cid:46) · − and | c T ( e ) t | (cid:46) · − .Of the top-Higgs couplings, c Itϕ generates the electron EDM through two-loop Barr-Zee dia-grams [82–85], giving a stronger limit than the neutron EDM. In addition, when we evolve the c ItA,tW couplings from the scale of new physics, Λ, to lower energies they first mix into CPVHiggs-gauge couplings of the form ( ϕ † ϕ ) ˜ X µν X µν [88, 96, 97], where X denotes an SU (2) or U (1)gauge-field strength. In a second step these gauge-Higgs couplings mix into the electron and light-quark EDMs. This last step is proportional to the Yukawa couplings of the light fields leading to astrong suppression. Nevertheless, the experimental limit on the electron EDM is sufficiently strongto overwhelm the other probes of the CPV components of these two top-quark dipoles [88, 96].Finally, we briefly discuss limits from rare B decays. At the one-loop level, the couplings c Iϕtb,bW,tA,tW give contributions to flavour-changing dipole operators that mediate b → s transi-tions proportional to the CKM element V ts ’ .
04. The contributions of these flavour-changingoperators to the CP asymmetry A CP ( B → X s γ ) [98], together with the experimental measurement[99], can be used set the limits in Table 5. It should be noted that measurements of the branchingratio can constrain the real parts of these couplings as well. This leads to limits which are typicallya factor of a few stronger than those on the imaginary parts, see, for example, Refs. [80, 100–102].18our-heavy c IQtQb ≡ Im { C quqd } [ − . , . · − ( d n ) c IQtQb ≡ Im { C quqd } [ − . , . · − ( d n )Two-heavy c Itϕ ≡ Im { C (33) uϕ } [ − . , .
7] ( d n ) [ − . , .
18] ( d e ) c Iϕtb ≡ Im { C (33) ϕud } [ − . , . d n ) [ − . , . B → X s γ ) c ItW ≡ Im { C (33) uW } [ − . , . · − ( d e ) [ − . , .
5] ( B → X s γ ) c ItA ≡ Im { c W C (33) uB + s W C (33) uW } [ − . , . · − ( d e ) [ − . , .
0] ( B → X s γ ) c IbW ≡ Im { C (33) dW } [ − . , . · − ( d n ) [ − . , . · − ( B → X s γ ) c ItG ≡ Im { C (33) uG } [ − . , . · − ( d n )Two-heavy-two-lepton c SI ( e ) t ≡ Im { C lequ } [ − . , . · − ( d e ) c T I ( e ) t ≡ Im { C lequ } [ − . , . · − ( d e ) c SI ( e ) b ≡ Im { C (1133) ledq } [ − . , . · − ( d e )Table 5: Constraints from the electron and neutron EDMs as well as A CP ( B → X s γ ). Here weturn on one coupling at a time and assume Λ = 1 TeV. The source of the constraints are indicatedin brackets. Summary
All the above discussed constraints are collected in Table 5. With the exception of c Itϕ , the CPVcoefficients are constrained at the percent level or stronger by EDM experiments. The semi-leptonic operators are more stringently constrained, which is mainly due to the fact that theircontribution to d e is proportional to m t (where one would naively expect m e ). Instead, theconstraints from B → X s γ are particularly strong for the c IbW and c Iϕtb couplings because of an m t /m b enhancement. In most cases, the constraints on the imaginary parts are stronger than thecorresponding limits on the real parts of the top-quark couplings. As a result, it will be difficultto reach a similar sensitivity by studying CPV observables at the LHC.The interpretation of these low-energy constraints requires some care however. In derivingthese limits we have assumed one dimension-six operator to be present at the scale Λ at a time.This assumption is no longer valid if multiple top operators are important at the scale Λ, orif one makes less restrictive assumptions about the flavor structure, such as a non-linear flavorsymmetry [18]. A global analysis involving all top-quark operators, for example, would leave somecombinations of operator coefficients unconstrained [88]. The difference between individual and global constraints is typically large for the CPV components as only a handful of sensitive low-energy measurements exist, in contrast to a much larger range of high-energy measurements ofthe real components. In a global setting, collider constraints on the CPV Wilson coefficients aretherefore necessary to bound unconstrained directions in the parameter space. An example is therecent ATLAS measurement [103] of a CPV phase in t → bW decays which significantly impactsthe global fit of CPV top-Higgs interactions [88, 89].In addition, the low-energy observables get contributions from SMEFT operators that do notinvolve top quarks. For example, if an electron EDM is generated at the scale Λ with exactlythe right size, it could weaken the limits on the semi-leptonic operators significantly. In fact, theleading logarithmic contributions we considered here result from divergent loops and require non-top operators, such as d e , to absorb the divergences. In these cases one might expect d e (Λ) = 0,which could in principle lead to cancellations and weakened limits. These cancellations have tobe very severe in order to avoid the strong low-energy constraints. In any case, in order to evadethe low-energy limits strong correlations between SMEFT operators are required, and this would19ose highly non-trivial constraints on models of beyond-the-SM physics. B UFO models
The dim6top implementation of the degrees of freedom introduced in this note, as well the
SMEFTsim implementation of Warsaw-basis operators will be described in this appendix. SuchUFO [104] models can be used for Monte Carlo simulation.
B.1 The dim6top implementation
The flavour-, B - and L -conserving parameters implemented in the dim6top UFO model (avail-able at https://feynrules.irmp.ucl.ac.be/wiki/dim6top ) are listed in Table 6. The FCNCdegrees of freedom defined in Appendix E and implemented in this model are listed in Table 7.
General comments – The implementation is a tree-level one.– Λ is conventionally fixed to 1 TeV. Equivalently, EFT input parameters can be thought ofas being the dimensionful ˜ c i ≡ c i / Λ expressed in units of TeV − .– The CKM matrix is assumed to be a unit matrix.– The masses of u, d, s, c, e, µ fermions are set to zero by default.– The unitary gauge is used and Goldstone bosons are removed.– Are only added to the Lagrangian, the Hermitian conjugates of the non-Hermitian operators,and independent flavour assignments.– Two versions of the model are made available. In the first one, dim6top_LO_UFO , all theflavour-, B - and L -conserving parameters are assigned the same coupling order : DIM6=1 .Similarly, all the flavour-changing parameters are assigned one single coupling order : FCNC=1 .In a second version, dim6top_LO_UFO_each_coupling_order , an individual coupling order is additionally assigned to each EFT parameter: the cQQ1 and cqq11x3331 parameters arefor instance assigned
DIM6_cQQ1 and
FCNC_cqq11x3331 coupling orders .This allows for the selection of individual degrees-of-freedom interferences in
MG5_aMC@NLO ,using a syntax such as > generate p p > t t˜ FCNC=0 DIM6ˆ2==1 DIM6_ctZˆ2==1> generate p p > t t˜ FCNC=0 DIM6ˆ2==2 DIM6_ctZˆ2==1 DIM6_ctWˆ2==1 which would for instance respectively yield the interference between SM amplitudes and thatin which ctZ is inserted once, and between amplitudes in which ctZ and ctW are insertedonce. Specifying the order of the squared amplitude is however not supported yet whendecay chains are specified.A positive
QED=n coupling order has also been assigned to EFT parameters correspond-ing to operators involving n Higgs doublet fields in the unbroken phase. Given that theHiggs vacuum expectation value has
QED = − , this avoids technical problems related to theappearance of interactions with net negative QED coupling order . Syntax • In MG5_aMC@NLO , import the model by > import model dim6top_LO_UFO , or > import model dim6top_LO_UFO_each_coupling_order • By default, the bottom quark is massive and the four-flavour scheme is used. Loading themodel with a restriction card where
MB=0 would automatically switch to the five flavourscheme. One can otherwise redefine manually: 20 our-heavy c QQ cQQ1 c QQ cQQ8 c Qt cQt1 c Qt cQt8 c Qb cQb1 c Qb cQb8 c tt ctt1 c tb ctb1 c tb ctb8 c I ] QtQb cQtQb1[I] ( I stands for imaginary part) c I ] QtQb cQtQb8[I]
Two-heavy-two-light c , Qq cQq13 U (2) q × U (2) u × U (2) d assumed c , Qq cQq83 c , Qq cQq11 c , Qq cQq81 c Qu cQu1 c Qu cQu8 c Qd cQd1 c Qd cQd8 c tq ctq1 c tq ctq8 c tu ctu1 c tu ctu8 c td ctd1 c td ctd8 Two-heavy c [ I ] tϕ ctp,ctpI c − ϕQ cpQM c ϕQ cpQ3 c ϕt cpt c ϕb cpb (implemented but involves no top) c [ I ] ϕtb cptb,cptbI c [ I ] tW ctW,ctWI c [ I ] tZ ctZ,ctZI c [ I ] bW cbW,cbWI c [ I ] tG ctG,ctGI Two-heavy-two-lepton c ‘ ) Ql cQl3(l) assuming lepton flavour diagonality: c − ( ‘ ) Ql cQlM(l) (l) ∈ { , , } c ( ‘ ) Qe cQe c ( ‘ ) tl ctl(l) c ( ‘ ) te cte(l) c S [ I ]( ‘ ) t ctlS[I](l) c T [ I ]( ‘ ) t ctlT[I](l) c S [ I ]( ‘ ) b cblS[I](l) Two-heavy-two-light, preserving only U (2) q + u + d c I ] tQqu ctQqu1[I] c I ] tQqu ctQqu8[I] c I ] bQqd cbQqd1[I] c I ] bQqd cbQqd8[I] c I ] Qtqd cQtqd1[I] c I ] Qtqd cQtqd8[I] c I ] Qbqu cQbqu1[I] c I ] Qbqu cQbqu8[I] c I ] btud cbtud1[I] c I ] btud cbtud8[I] Table 6: dim6top
UFO model parameter namesfor the flavour-, B - and L -conserving degrees offreedom defined in Appendices C and D. One-light-three-heavy c I ](333 a ) qq cqq11[I]x333(a) (a) ∈ { , } : light quark gen. c I ](333 a ) qq cqq13[I]x333(a) I : imaginary part c [ I ](333 a ) uu cuu1[I]x333(a) c I ](333 a ) qu cqu1[I]x333(a) c I ](333 a ) qu cqu8[I]x333(a) c I ](3 a qu cqu1[I]x3(a)33 c I ](3 a qu cqu8[I]x3(a)33 c I ](333 a ) qd cqd1[I]x333(a) c I ](333 a ) qd cqu8[I]x333(a) c I ](3 a qd cqd1[I]x3(a)33 c I ](3 a qd cqd8[I]x3(a)33 c I ](333 a ) ud cud1[I]x333(a) c I ](333 a ) ud cud8[I]x333(a) c I ](3 a ud cud1[I]x3(a)33 c I ](3 a ud cud8[I]x3(a)33 c I ](333 a ) quqd cquqd1[I]x333(a) c I ](33 a quqd cquqd1[I]x33(a)3 c I ](3 a quqd cquqd1[I]x3(a)33 c I ]( a quqd cquqd1[I]x(a)333 c I ](333 a ) quqd cquqd8[I]x333(a) c I ](33 a quqd cquqd8[I]x33(a)3 c I ](3 a quqd cquqd8[I]x3(a)33 c I ]( a quqd cquqd8[I]x(a)333 Three-light-one-heavy c , I ](3 a ) qq cqq11[I]x3(a)ii c , I ](3 a ) qq cqq13[I]x3(a)ii c , I ](3 a ) qq cqq81[I]x3(a)ii c , I ](3 a ) qq cqq83[I]x3(a)ii c I ](3 a ) uu cuu1[I]x3(a)ii c I ](3 a ) uu cuu8[I]x3(a)ii c I ](3 a ) ud cud1[I]x3(a)ii c I ](3 a ) ud cud8[I]x3(a)ii c I ](3 a ) qu cqu1[I]x3(a)ii c I ](3 a ) qu cqu8[I]x3(a)ii c I ]( a qu cqu1[I]xii3(a) c I ]( a qu cqu8[I]xii3(a) c I ](3 a ) qd cqd1[I]x3(a)ii c I ](3 a ) qd cqd8[I]x3(a)ii One-light-one-heavy c [ I ](3 a ) tϕ ctp[I]x3(a) c [ I ]( a tϕ ctp[I]x(a)3 c − [ I ](3+ a ) ϕq cpQM[I]x3(a) c I ](3+ a ) ϕq cpQ3[I]x3(a) c [ I ](3+ a ) ϕu cpt[I]x3(a) c [ I ](3 a ) ϕud cptb[I]x3(a) c [ I ]( a ϕud cptb[I]x(a)3 c [ I ](3 a ) uW ctW[I]x3(a) c [ I ]( a uW ctW[I]x(a)3 c [ I ](3 a ) uZ ctZ[I]x3(a) c [ I ]( a uZ ctZ[I]x(a)3 c [ I ](3 a ) dW cdW[I]x3(a) c [ I ]( a dW cdW[I]x(a)3 c [ I ](3 a ) uG cdG[I]x3(a) c [ I ]( a uG cdG[I]x(a)3 One-light-one-heavy-two-lepton c ‘, a ) lq cQl3[I]x(l)x3(a) (l) ∈ { , , } lepton gen. c − ( ‘, a ) lq cQlM[I]x(l)x3(a) c ( ‘, a ) eq cQe[I]x(l)x3(a) c ( ‘, a ) lu ctl[I]x(l)x3(a) c ( ‘, a ) eu cte[I]x(l)x3(a) c S ( ‘, a ) lequ ctlS[I]x(l)x3(a) c S ( ‘,a lequ ctlS[I]x(l)x(a)3 c T ( ‘, a ) lequ ctlT[I]x(l)x3(a) c T ( ‘,a lequ ctlT[I]x(l)x(a)3 c S ( ‘, a ) ledq cblS[I]x(l)x3(a) (implemented but involve c S ( ‘,a ledq cblS[I]x(l)x(a)3 no top FCNC) Table 7: dim6top
UFO model parameter namesfor the FCNC degrees of freedom introduced inAppendix E. 21 define p = p b b˜> define j = p and set
MB=0 in the param_card . For consistency, one may then also set ymb=0 . • Processes can be generated through commands like > generate p p > t t˜ FCNC=0 DIM6=1 which allows for one (or no) insertion of a param-eter of
DIM6 coupling order per amplitude. • To focus on the leading QCD amplitudes, one can also restrict their maximal allowed
QED order using for instance: > generate p p > t t˜ FCNC=0 DIM6=1 QED=0> generate p p > t t˜ Z FCNC=0 DIM6=1 QED=1
B.2 The
SMEFTsim implementation
The
SMEFTsim package [105] (available at http://feynrules.irmp.ucl.ac.be/wiki/SMEFT ) pro-vides a complete
FeynRules [106] implementation of the B -conserving dimension-six Lagrangianin the Warsaw basis [1], that automatically performs the field and parameter redefinitions re-quired to have canonically normalized kinetic terms and following from the choice of an inputparameters set. The package contains FeynRules models as well as pre-exported UFO models forsix different Lagrangians, corresponding to two possible input parameters sets ( { ˆ α ew , ˆ m Z , ˆ G F } or { ˆ m W , ˆ m Z , ˆ G F } ) and to three possible assumptions on the flavour structure of the theory: • a flavour general case, in which all the flavour indices are explicitly kept and the Wilsoncoefficients of the fermionic operators are defined as tensorial parameters. • a U (3) flavour symmetric case in which flavour contractions are fixed by the symmetry.The Yukawa couplings are treated as flavour-breaking spurions and consistently inserted inoperators with chirality-flipping fermion currents. • a linear MFV case [17] that, unlike the U (3) symmetric setup, does not contain CPviolating parameters beyond the CKM phase while it includes insertions of the flavour-violating spurions Y f Y † f , Y † f Y f up to linear order.Two versions of SMEFTsim are available, denominated “set A” and “set B”: these implementationsare independent but completely equivalent, and their simultaneous availability is meant to help across-check of the results.
General comments – Analogously to dim6top , SMEFTsim is a leading order, unitary gauge implementation.– The Wilson coefficients are dimensionless quantities and the EFT cutoff Λ is an externalparameter of the UFO, set by default to 1 TeV (
LambdaSMEFT = 1.e+03 ).– The definitions of the relevant parameters (in the Warsaw basis) are given in Table 8. Thecorrespondence with the parameters used in dim6top can be deduced from the definitionssummarized in Table 1.A
QED=-1 interaction order has been assigned to
LambdaSMEFT , in order to avoid couplingswith negative
QED order. – Complex Wilson coefficients are parametrized by their absolute value and complex phase ciAbs Exp[I ciPh] rather than by real and imaginary parts. When testing the imaginarypart of a given operator it is therefore necessary to set the corresponding phase to π/ . One coupling with negative
QED order is still present nonetheless, namely the C ϕ correction to the h vertex,which plays no role here. (33) uϕ cuHAbs33, cuHPh33 C (33) ϕu cHuAbs33, cHuPh33 C ϕq cHq1Abs33 C ϕq cHq3Abs33 C (33) ϕud cHudAbs33, cHudPh33 C (33) uW cuWAbs33, cuWPh33 C (33) dW cdWAbs33, cdWPh33 C (33) uB cuBAbs33, cuBPh33 C (33) uG cuGAbs33, cuGPh33 C qq cqq1Abs3333 C qq cqq3Abs3333 C aa qq cqq1Abs(aa)33 C aa qq cqq3Abs(aa)33 C a a ) qq cqq1Abs(a)33(a) C aa qq cqq3Abs(a)33(a) C qu cqu1Abs3333 C qu cqu8Abs3333 C aa qu cqu1Abs(aa)33 C aa qu cqu8Abs(aa)33 C aa ) qu cqu1Abs33(aa) C aa ) qu cqu8Abs33(aa) C qd cqd1Abs3333 C qd cqdAbs3333 C aa ) qd cqd1Abs33(aa) C aa )) qd cqdAbs33(aa) C (3333) uu cuuAbs3333 C ( aa uu cuuAbs(aa)33 C ( a a ) uu cuuAbs(a)33(a) C ud cud1Abs3333 C ud cud8Abs3333 C aa ) ud cud1Abs33(aa) C aa ) ud cud8Abs33(aa) C quqd cquqd1Abs3x3x3x3, cquqd1Ph3x3x3x3 C quqd cquqd8Abs3x3x3x3, cquqd8Ph3x3x3x3 C ll lq clq1Abs(ll)33 C ll lq clq3Abs(ll)33 C ( ll lu cluAbs(ll)33 C ( ll eq cqeAbs33(ll) C ( ll eu ceuAbs(ll)33 C ll lequ clequ1Abs(l)x(l)x3x3,clequ1Ph(l)x(l)x3x3 C ll lequ clequ3Abs(l)x(l)x3x3, clequ3Ph(l)x(l)x3x3 C ( ll ledq cledqAbs(l)x(l)x3x3, cledqPh(l)x(l)x3x3 Table 8: Flavour conserving parameters in the
SMEFTsim-A
UFO implementation, that are indirect correspondence with the degrees of freedom defined in Appendix C. The index a = { , } runs over the light quark generations, while l = { , , } indicates lepton flavours. flavour case relevant parameters total numbergeneral [ C uH , C uB , C uW , C uG ] i , j , [ C dW , C Hud ] j [ C Hu , C (1) , (3) Hq ] i ,
618 abs. values[ C (1) , (3) qq , C uu , C (1) , (8) qu ] jkl,i kl,ij l,ijk , [ C (1) , (8) ud , C qe , C (1) , (8) qd ] jkl,i kl ,
557 phases[ C (1) , (3) lq , C eu , C lu , C (1) , (3) lequ ] ij l,ijk , [ C ledq ] ijk , [ C (1) , (8) quqd ] jkl,i kl,ij l C H (cid:3) , C HD , C HWB , [ C ll ] , [ C (3) Hl ] , U (3) C uH , C Hu , C (1) , (3) Hq , C Hud , C uW , C uB , C uG , C dW , C (1) , (3) qq , C (1) , (3) qq ,
34 abs. values C (1) , (3) lq , C uu , C uu , C (1) , (8) ud , C eu , C lu , C qe , C (1) , (8) qu , C (1) , (8) qd , C (1) , (8) quqd , C H (cid:3) , C HD , C HWB , C ll , C ll , C (3) Hl linear MFV C uH , C ∗ Hu , C (1) , (3) ∗∗ Hq , C Hud , C uW , C uB , C uG , C dW , C (1) , (3) ∗∗∗∗ qq ,
83 abs. values C (1) , (3) qq , C (1) , (3) ∗∗ lq , C ∗∗ uu , C uu , C (1) , (8) ∗∗ ud , C ∗ eu , C ∗ lu , C ∗∗ qe ,C (1) , (8) ∗∗∗ qu , C (1) , (8) ∗∗∗ qd , C (1) , (8) quqd ,C H (cid:3) , C HD , C HWB , C ll , C ll , C (3) Hl Table 9: Detailed list of the parameters that are relevant for top-quark physics for each flavourassumption in
SMEFTsim . These are all set to 1 when importing the restrict_TopEFT restrictionin any given model. The indices i, j, k, l take values in {1,2,3}. The terms in the last rows areincluded as they contribute to top couplings via parameter redefinitions. In the MFV case all theparameters are real and coefficients with n asterisks admit n independent insertions of flavour-violating spurions. 23 The SM couplings of the Higgs boson that first arise at loop level (i.e. hgg , hγγ , hZγ ) areincluded and parametrized as effective vertices with a coefficient that reproduces the SMloop function. These vertices are assigned an interaction order SMHLOOP = 1 .– All the flavour contractions are automatically implemented in the model, with a parametriza-tion that allows to fix only the independent ones. This means that, for instance, setting C aa qq = 1 introduces both the O aa qq and O aa ) qq terms in the Lagrangian.– An interaction order NP=1 has been defined for the Wilson coefficients, equivalent to
DIM6 in dim6top . There is no analogue of FCNC nor of the individual
DIM6_ci . Syntax • In MG5_aMC@NLO import the model, restricted by the file restrict_XXX.dat by > import model SMEFTsim_A_general_alphaScheme-XXX All the restrictions available leave MB and ymb non-vanishing. It is possible to switch to thefive flavour scheme either modifying manually the restriction card or setting > define p = p b b˜> define j = p after importing the model and then modifying the param_card after generating the processoutput. • Processes can be generated with > generate p p > t t˜ NP==1 SMHLOOP=0 to allow the insertion of one Wilson coefficient in the amplitude and excluding SM hgg , hγγ ,and hZγ vertices or, for instance, with > generate p p > t t˜ NPˆ2==1 SMHLOOP=0 to select for the interference term. QED and
QCD interaction orders can also be specified toselect for specific diagrams. To ensure the inclusion of all the tree-level SM contributions, itis advisable to specify
QED<=8 QCD<=8 . B.3 Benchmark results
The linear and quadratic EFT dependences of some total rates are displayed in Tables 10-21. Theseare meant to provide benchmark results against which simulation procedures could be checked.They are also indicative of which degree of freedom is relevant for which process.These numbers are obtained using
MG5_aMC@NLO (v2.6.0 or 2.6.1) [40] for the 13 TeV LHC, with nn23lo1 as PDF set. All fermion masses and Yukawa couplings but the top ones (
MT=172 ) aretaken vanishing. The five-flavour scheme, default running renormalization and factorization scalesare used, and simple p T >
20 GeV, | η | < .
5, ∆
R > . b -)jets,charged leptons and photons.All tree-level contributions are included irrespectively of their QED order. Additional depen-dences may also be generated: with finite fermion masses and Yukawas (especially the bottomones), when considering other observables (e.g., sensitive to CPV contributions), beyond the treelevel (e.g., at NLO in QCD or when accounting for loop-induced Higgs couplings), etc. Cuts mayalso affect the hierarchies of sensitivities.Obtained with dim6top , these benchmark results have been cross checked with
SMEFTsim_A version 2.0, in the flavour general setup and with the { ˆ α ew , ˆ m Z , ˆ G F } input scheme (equally validresults can be obtained with set B and/or switching to a { ˆ m W , ˆ m Z , ˆ G F } input scheme):– A set of dedicated restriction cards is provided with the SMEFTsim_A_general_alphaScheme model to reproduce these numerical results. All of them set to zero the masses and Yukawacouplings of all fermions except the bottom and top quarks. They also approximate theCKM matrix by a unit matrix and fix the input parameters to values consistent with thoseadopted in dim6top . In particular they modify 24
B = ymb = 4.7 , MT = 172.0 , MH = 125.0 , MW = 79.824360aEW = 7.818608e-03 , aS = 0.1184 compared to the default settings. In addition: restrict_SMlimit_top: sets to zero all the Wilson coefficients. This restriction is availablefor all the flavour setups. restrict_TopEFT: sets to 1 the absolute values of all the Wilson coefficients that are rel-evant for top physics and to 0 the remaining ones. This restriction is available for allthe flavour setups, and the list of coefficients retained in each case is given in Table 9. restrict_(ci)_top: with (ci) one of the degrees of freedom of Table 6 in the notation of dim6top : turns on the combination of Wilson coefficients equivalent to setting ci = 1 ,while vanishing the others.– As dim6top does not include loop-level interactions of the Higgs ( hgg , hγγ , or hZγ ), ameaningful comparison with SMEFTsim would require setting
SMHLOOP=0 .– SMEFTsim utilizes the opposite convention, compared to dim6top , for the sign of covariantderivatives (schematically: D µ = ∂ µ + iA µ ). At the linear level, some interferences with SMamplitudes are affected, in particular those proportional to ctW[I] and ctZ[I] .– Due to different choices concerning the inclusion of non-independent flavour contractions(only independent flavour assignments are included in dim6top while all are in SMEFTsim ),factor-of-two differences arise when considering the coefficients cQq83, cQq81, cQq13, cQq11,ctu1, ctu8 , that stem from operators with four identical fields: Q , qq , O uu .– Due to the different parametrization of complex coefficients (in (Abs, Arg) rather than (Re,Im)), the evaluation of S ki for purely imaginary coefficients can be less accurate in SMEFTsim .– In
SMEFTsim , due to the internal implementation of Warsaw basis operators instead of thedegrees of freedom specific to top-quark physics, some small interferences may need to beobtained from cancellations between large contributions and thus suffer from larger numericaluncertainties. This concerns for instance the evaluation of the vanishing linear dependenceon the colour octet cQq83 in single top production. Phenomenological consequences may belimited.
C Flavour-, B - and L -conserving degrees of freedom Let us define the EFT degrees of freedom, operator category by operator category. We recommendto quote results in terms of these degrees of freedom. Their definitions, in terms of Warsaw-basisoperator coefficients are also summarized in Table 1.
C.1 Four-heavy operators
We consider first four-quark operators. All vector ( ¯ LL ¯ LL , ¯ LL ¯ RR , ¯ RR ¯ RR ) operators have real C (3333) = C (3333) ∗ coefficients, unlike the scalar ( ¯ LR ¯ LR ) ones. For ¯ LL ¯ LL operators, one candefine the colour singlet and octets that would or not interfere with the QCD amplitudes: O qq O qq ! = (cid:18) − /
30 4 (cid:19) T (cid:18) ( ¯ Qγ µ Q ) ( ¯ Qγ µ Q )( ¯ Qγ µ T A Q ) ( ¯ Qγ µ T A Q ) (cid:19) , (66)where Q represents the third-generation left-handed quark doublet, while t and b will representthe right-handed quark singlets in the unbroken electroweak phase. In the electroweak broken Note that we wrote matrix equations involving operators as O = M T o . The Lagrangian C T O terms are thenequated to c T o to establish the definition of the coefficients of the o operators as c ≡ M C . O qq O qq ! = − /
30 81 1 T (¯ tγ µ P L t )(¯ tγ µ P L t )(¯ tγ µ P L t )(¯ bγ µ P L b )(¯ tγ µ T A P L t )(¯ bγ µ T A P L b )(¯ bγ µ P L b )(¯ bγ µ P L b ) (67)which is useful to identify the colour singlet and octet combinations of two top and two b quarks,four tops and four b ’s. We use the combinations leading to ¯ tt ¯ bb interactions as independent degreesof freedom. No further manipulation is performed on other operators featuring four heavy quarks.One thus defines: c QQ ≡ C qq − C qq ,c QQ ≡ C qq , c Qt ≡ C qu ,c Qt ≡ C qu ,c Qb ≡ C qd ,c Qb ≡ C qd , c tt ≡ C uu , c tb ≡ C ud ,c tb ≡ C ud , (68)and c I ] QtQb ≡ [Im]Re { C quqd } , c I ] QtQb ≡ [Im]Re { C quqd } . (69)In total, there are thus 11 + 2 CPV degrees of freedom for four-heavy operators. One can alsodefine the combination c + QQ ≡ C qq + C qq = (3 c QQ + c QQ ) / C.2 Two-light-two-heavy operators
Scalar operators involving a light-quark current are not allowed by our baseline flavour assumption(see Appendix D where this assumption is relaxed). The coefficients of the vector ¯ LL ¯ LL operators O , qq have the following symmetries under permutations of their generation indices: C ( ijkl ) qq = C ( klij ) qq , C ( jilk ) qq = C ( ijkl ) qq ∗ , (and thus C ( lkji ) qq = C ( ijkl ) qq ∗ ) , which namely implies that C ( iijj ) qq and C ( ijji ) qq elements are real. From the C ( ijkl ) qq and C ( klij ) qq combinations, only one is thus retained in the sum over flavour indices in the EFT Lagrangian ofEq. (34). For each of these two operators, there are thus two independent assignments of third-generation indices to a quark-antiquark pairs that are compatible with our U (2) q × U (2) u × U (2) d baseline flavour symmetry: C ( ii qq = C ( ii qq ∗ = C (33 ii ) qq = C (33 ii ) qq ∗ , C ( i i ) qq = C ( i i ) qq ∗ = C (3 ii qq = C (3 ii qq ∗ . (70)To understand better the structure of their interferences with SM amplitudes it is useful to decom-pose them, using Fierz identities, onto quadrilinears featuring a heavy- and a light-quark bilinear.One obtains: O aii ) qq O iia ) qq O aii ) qq O iia ) qq = / /
20 1 / − /
60 1 0 30 1 0 − T ( ¯ Qγ µ Q ) (¯ q i γ µ q i )( ¯ Qγ µ τ I Q ) (¯ q i γ µ τ I q i )( ¯ Qγ µ T A Q ) (¯ q i γ µ T A q i )( ¯ Qγ µ T A τ I Q ) (¯ q i γ µ T A τ I q i ) , (71)and similarly, for the ¯ RR ¯ RR operator O uu : O ( ii uu O ( i i ) uu ! = (cid:18) /
30 2 (cid:19) T (cid:18) (¯ tγ µ t ) (¯ u i γ µ u i )(¯ tγ µ T A t ) (¯ u i γ µ T A u i ) (cid:19) . (72)26here are four four-quark operators of ¯ LL ¯ RR type in the Warsaw basis: O ijkl ) qu , O ijkl ) qu , O ijkl ) qd , O ijkl ) qd which form six degrees of freedom when accounting for the two C ( ii qu = C ( ii qu ∗ and C (33 ii ) qu = C (33 ii ) qu ∗ flavour assignments. Altogether, one thus defines the following degrees of free-dom for two-light-two-heavy operators: c , Qq ≡ C ii qq + 16 C i i ) qq + 12 C i i ) qq ,c , Qq ≡ C ii qq + 16 ( C i i ) qq − C i i ) qq ) ,c , Qq ≡ C i i ) qq + 3 C i i ) qq ,c , Qq ≡ C i i ) qq − C i i ) qq , c tu ≡ C ( ii uu + 13 C ( i i ) uu ,c tu ≡ C ( i i ) uu ,c td ≡ C ii ) ud ,c td ≡ C ii ) ud , c tq ≡ C ii qu ,c Qu ≡ C ii ) qu ,c Qd ≡ C ii ) qd ,c tq ≡ C ii qu ,c Qu ≡ C ii ) qu ,c Qd ≡ C ii ) qd , (73)where i is either 1 or 2. In total, there are thus 14 degrees of freedom for two-light-two-heavyoperators. C.3 Two-heavy operators
The operator involving two quarks and boson which possibly contain a top quark where listedin Section 3. There are a few choices to be made in the definitions of the associated degrees offreedom, in view of the electroweak broken phase decomposition of O ϕq O ϕq O ϕq ! = − T + e s W c W (¯ tγ µ P L t ) Z µ ( v + h ) − e s W c W (¯ bγ µ P L b ) Z µ ( v + h ) es W √ (¯ tγ µ P L b ) W + µ ( v + h ) es W √ (¯ bγ µ P L t ) W − µ ( v + h ) , (74)and of electroweak dipole operators: O (33) uB O (33) uW ! = c W s W − s W c W T (¯ tσ µν P R t ) A µν ( v + h )(¯ tσ µν P R t ) Z µν ( v + h )(¯ bσ µν P R t ) W − µν ( v + h ) , (75)where s W and c W are the sine and cosine of the weak mixing angle (in the unitary gauge). Amongthe possible definitions, we choose to use as degrees of freedom the combinations that involvecharged W , Z bosons and tops in the broken phase. In this sector, one thus defines: c [ I ] tϕ ≡ [Im]Re { C (33) uϕ } , c − ϕQ ≡ C ϕq − C ϕq ,c ϕQ ≡ C ϕq ,c ϕt ≡ C (33) ϕu ,c [ I ] ϕtb ≡ [Im]Re { C (33) ϕud } , c [ I ] tW ≡ [Im]Re { C (33) uW } ,c [ I ] tZ ≡ [Im]Re {− s W C (33) uB + c W C (33) uW } ,c [ I ] bW ≡ [Im]Re { C (33) dW } ,c [ I ] tG ≡ [Im]Re { C (33) uG } . (76)On the other hand, the combination of O , ϕq operators that modifies the SM coupling of the b quark to the Z is c + ϕQ ≡ C ϕq + C ϕq = c − ϕQ + 2 c ϕQ and the combination appearing electromag-netic dipole of the top is c [ I ] tA ≡ [Im]Re { c W C (33) uB + s W C (33) uW } = ( c [ I ] tW − c W c [ I ] tZ ) /s W . In total, there arethus 9 + 6 CPV degrees of freedom for two-heavy operators. 27 .4 Two-heavy-two-lepton operators Let us know address the definition of the degrees of freedom associated to the operators involvingtwo quarks and two leptons. We decompose the O , lq operators in the broken phase, O ‘‘ lq O ‘‘ lq ! = − −
11 10 20 2 T (¯ ν ‘ γ µ P L ν ‘ )(¯ tγ µ P L t )(¯ ν ‘ γ µ P L ν ‘ )(¯ bγ µ P L b )(¯ ‘γ µ P L ‘ )(¯ tγ µ P L t )(¯ ‘γ µ P L ‘ )(¯ bγ µ P L b )(¯ ν ‘ γ µ P L ‘ )(¯ bγ µ P L t )(¯ ‘γ µ P L ν ‘ )(¯ tγ µ P L b ) , (77)and select as degrees of freedom the combinations that give rise to a top-quark interaction with apair of charged leptons, and to charged currents. One therefore defines: c − ( ‘ ) Ql ≡ C ‘‘ lq − C ‘‘ lq ,c ‘ ) Ql ≡ C ‘‘ lq , c ( ‘ ) tl ≡ C ( ‘‘ lu ,c ( ‘ ) Qe ≡ C ( ‘‘ eq , c ( ‘ ) te ≡ C ( ‘‘ eu , c S [ I ]( ‘ ) t ≡ [Im]Re { C ‘‘ lequ } ,c T [ I ]( ‘ ) t ≡ [Im]Re { C ‘‘ lequ } ,c S [ I ]( ‘ ) b ≡ [Im]Re { C ‘‘ ledq } . (78)These constitute 8 + 3 CPV degrees of freedom, for each of the three generations of leptons. D Less restrictive flavour symmetry
We examine here the consequences of relaxing the flavour symmetry among the first two genera-tions of quarks from U (2) q × U (2) u × U (2) d to U (2) q + u + d . New contributions are only generatedin the two-light–two-heavy category of four-quark operators as flavour-diagonal chirality-flippinglight quark-antiquark pairs as well as right-handed charged currents within the first generationsare now allowed. D.1 Four-quark ¯ LL ¯ RR operators For this category of operators, the ( i i ) and (3 ii
3) flavour assignments become allowed: C i i ) qu = C ii qu ∗ , C i i ) qu = C ii qu ∗ , (79) C i i ) qd = C ii qd ∗ , C i i ) qd = C ii qd ∗ . (80)Using Fierz identities, one then obtains: O i i ) qu O i i ) qu ! = (cid:18) − / − / − / (cid:19) T (cid:18) (¯ t Q ) (¯ q i u i )(¯ tT A Q ) (¯ q i T A u i ) (cid:19) (81)and similarly for O qd operators. The corresponding degrees of freedom c I ] tQqu = [Im]Re {− C i i ) qu − C i i ) qu } ,c I ] bQqd = [Im]Re {− C i i ) qd − C i i ) qd } , c I ] tQqu = [Im]Re {− C i i ) qu + 23 C i i ) qu } ,c I ] bQqd = [Im]Re {− C i i ) qd + 23 C i i ) qd } , (82)are complex. 28 .2 Four-quark ¯ LR ¯ LR operators The O quqd operators belong to that category. Proceeding to the Fierz decomposition: O i i ) quqd O ii quqd O i i ) quqd O ii quqd = − / − / − / − /
24 0 − /
18 0 − / /
24 00 − / − / − / − /
24 0 − / − / / T ( ¯ Q t ) ε (¯ q i d i )( ¯ Q T A t ) ε (¯ q i T A d i )( ¯ Qσ µν t ) ε (¯ q i σ µν d i )( ¯ Qσ µν T A t ) ε (¯ q i σ µν T A d i )( ¯ Q b ) ε (¯ q i u i )( ¯ Q T A b ) ε (¯ q i T A u i )( ¯ Qσ µν b ) ε (¯ q i σ µν u i )( ¯ Qσ µν T A b ) ε (¯ q i σ µν T A u i ) , (83)one can for instance select the scalar operator coefficients as independent degrees of freedom: c I ] Qtqd = [Im]Re {− C i i ) quqd − C i i ) quqd } ,c I ] Qbqu = [Im]Re {− C ii quqd − C ii quqd } , c I ] Qtqd = [Im]Re {− C i i ) quqd + 16 C i i ) quqd } ,c I ] Qbqu = [Im]Re {− C ii quqd + 16 C ii quqd } , (84)which are complex. D.3 Four-quark ¯ RR ¯ RR operators In this category, new structures only arise for O ud operators. The Fierz decomposition gives: O i i ) ud O i i ) ud ! = (cid:18) / / − / (cid:19) T (cid:18) (¯ bγ µ t ) (¯ u i γ µ d i )(¯ bγ µ T A t ) (¯ u i γ µ T A d i ) (cid:19) , (85)and two additional complex degrees of freedom can be defined: c I ] btud = [Im]Re { C i i ) ud + 49 C i i ) ud } , c I ] btud = [Im]Re { C i i ) ud − C i i ) ud } . (86) E FCNC degrees of freedom
We now somewhat relax the benchmark flavour symmetry imposed to consider the important caseof top-quark FCNC interactions. We therefore assume the U (2) q × U (2) u × U (2) d is broken by asmall parameter connecting generations 3 and a ∈ { , } . In each operator, a quark-antiquark pairmixing these two generations arises at the linear order in this breaking parameter. We restrictourselves to this order and require the other quark bilinear in a four-quark operator to still satisfythe U (2) q × U (2) u × U (2) d symmetry, possibly after a Fierz transformation has been applied. Wenow construct the degrees of freedom satisfying this condition, operator type by operator type. E.1 One-light-one-heavy operators
The FCNC operators in this category are trivially obtained from the flavour conserving ones. NoteHermitian couplings C ( ji ) ∗ = C ij have complex off-diagonal components. The degrees of freedom29re the following: c [ I ](3 a ) tϕ ≡ [Im]Re { C (3 a ) uϕ } ,c [ I ]( a tϕ ≡ [Im]Re { C ( a uϕ } , c − [ I ](3+ a ) ϕq ≡ [Im]Re { C a ) ϕq − C a ) ϕq } ,c I ](3+ a ) ϕq ≡ [Im]Re { C a ) ϕq } ,c [ I ](3+ a ) ϕu ≡ [Im]Re { C (3 a ) ϕu } ,c [ I ](3 a ) ϕud ≡ [Im]Re { C (3 a ) ϕud } ,c [ I ]( a ϕud ≡ [Im]Re { C ( a ϕud } , c [ I ](3 a ) uW ≡ [Im]Re { C (3 a ) uW } ,c [ I ]( a uW ≡ [Im]Re { C ( a uW } ,c [ I ](3 a ) uZ ≡ [Im]Re {− s W C (3 a ) uB + c W C (3 a ) uW } ,c [ I ]( a uZ ≡ [Im]Re {− s W C ( a uB + c W C ( a uW } ,c [ I ](3 a ) dW ≡ [Im]Re { C (3 a ) dW } ,c [ I ]( a dW ≡ [Im]Re { C ( a dW } ,c [ I ](3 a ) uG ≡ [Im]Re { C (3 a ) uG } ,c [ I ]( a uG ≡ [Im]Re { C ( a uG } . (87)Altogether, these are 15 CP-conserving degrees of freedom and 15 CP-violating ones. E.2 One-light-one-heavy-two-lepton operators
There is no difficulty either to list the FCNC degrees of freedom for operators containing twoquarks and two leptons: c ‘, a ) lq ≡ [Im]Re { C ‘‘ a ) lq } ,c − ( ‘, a ) lq ≡ [Im]Re { C − ( ‘‘ a ) lq } ,c ( ‘, a ) eq ≡ [Im]Re { C ( ‘‘ a ) eq } ,c ( ‘, a ) lu ≡ [Im]Re { C ( ‘‘ a ) lu } ,c ( ‘, a ) eu ≡ [Im]Re { C ( ‘‘ a ) eu } , c S ( ‘, a ) lequ ≡ [Im]Re { C ‘‘ a ) lequ } ,c S ( ‘,a lequ ≡ [Im]Re { C ‘‘a lequ } ,c T ( ‘, a ) lequ ≡ [Im]Re { C ‘‘ a ) lequ } ,c T ( ‘,a lequ ≡ [Im]Re { C ‘‘a lequ } , (88)These form 9 + 9 CPV degrees of freedom. E.3 One-light-three-heavy operators
We consider first operators involving three third-generation and one first- or second-generationquarks. The ¯ LL ¯ LL operators featuring one light field and three heavy ones satisfy O , a ) qq = O , a qq = O , a qq ∗ = O , a qq ∗ . So we impose the same equality on their coefficients andretain only one of them as independent complex parameter. In the broken electroweak phase,these operators are decomposed as follows: O a ) qq O a ) qq ! = / / − / / −
21 1 T (¯ tγ µ P L t ) (¯ tγ µ P L u a )(¯ tγ µ P L b ) (¯ bγ µ P L u a )(¯ tγ µ T A P L b ) (¯ bγ µ T A P L u a )(¯ bγ µ P L t ) (¯ tγ µ P L d a )(¯ bγ µ T A P L t ) (¯ tγ µ T A P L d a )(¯ bγ µ P L b ) (¯ bγ µ P L d a ) . (89)Note here that only the ¯ tt ¯ tu and ¯ bb ¯ bd field combinations unambiguously arise from flavour-changingneutral current interactions. Both combinations arise proportional to C a ) qq + C a ) qq . Theother field combinations could have been generated by flavour off-diagonal charged currents. Thecolour singlet ¯ tb ¯ bu and ¯ bt ¯ td operators for instance interfere with CKM-suppressed SM charged cur-rents, while the octets do not. The single ¯ RR ¯ RR operator contributing to three top production alsosatisfies O (333 a ) uu = O (3 a uu = O (33 a uu ∗ = O ( a uu ∗ while O (333 a ) ud = O (33 a ud ∗ and O (3 a ud = O ( a ud ∗ .30imilarly the ¯ LL ¯ RR operators satisfies O , a ) qu = O , a qu ∗ and O , a qu = O , a qu ∗ . The¯ LR ¯ LR operators have no symmetry of this kind. So, the relevant degrees of freedom are c I ](333 a ) qq = [Im]Re { C a ) qq } ,c I ](333 a ) qq = [Im]Re { C a ) qq } ,c [ I ](333 a ) uu = [Im]Re { C (333 a ) uu } , c I ](333 a ) qu = [Im]Re { C a ) qu } ,c I ](333 a ) qu = [Im]Re { C a ) qu } ,c I ](3 a qu = [Im]Re { C a qu } ,c I ](3 a qu = [Im]Re { C a qu } , c I ](333 a ) qd = [Im]Re { C a ) qd } ,c I ](333 a ) qd = [Im]Re { C a ) qd } ,c I ](3 a qd = [Im]Re { C a qd } ,c I ](3 a qd = [Im]Re { C a qd } , (90) c I ](333 a ) ud = [Im]Re { C a ) ud } ,c I ](333 a ) ud = [Im]Re { C a ) ud } ,c I ](3 a ud = [Im]Re { C a ud } ,c I ](3 a ud = [Im]Re { C a ud } , c I ](333 a ) quqd = [Im]Re { C a ) quqd } ,c I ](33 a quqd = [Im]Re { C a quqd } ,c I ](3 a quqd = [Im]Re { C a quqd } ,c I ]( a quqd = [Im]Re { C a quqd } , c I ](333 a ) quqd = [Im]Re { C a ) quqd } ,c I ](33 a quqd = [Im]Re { C a quqd } ,c I ](3 a quqd = [Im]Re { C a quqd } ,c I ]( a quqd = [Im]Re { C a quqd } . (91) E.4 Three-light-one-heavy operators
Again, we impose here the presence of a quark-antiquark pair satisfying the U (2) q × U (2) u × U (2) d symmetry. The suitable ¯ LL ¯ LL operators satisfy the O , aii ) qq = O , ii a ) qq = O , a ii ) qq ∗ = O , iia qq ∗ , O , iia ) qq = O , ia i ) qq = O , i ai ) qq ∗ = O , aii qq ∗ relations. Fierz transformation thenlead to O aii ) qq O iia ) qq O aii ) qq O iia ) qq = / /
20 1 / − /
60 1 0 30 1 0 − T ( ¯ Qγ µ q a ) (¯ q i γ µ q i )( ¯ Qγ µ τ I q a ) (¯ q i γ µ τ I q i )( ¯ Qγ µ T A q a ) (¯ q i γ µ T A q i )( ¯ Qγ µ T A τ I q a ) (¯ q i γ µ T A τ I q i ) (92)and one can thus define the following degrees of freedom: c , I ](3 a ) qq ≡ [Im]Re { C aii ) qq + 16 C iia ) qq + 12 C iia ) qq } , (93) c , I ](3 a ) qq ≡ [Im]Re { C aii ) qq + 16 ( C iia ) qq − C iia ) qq ) } , (94) c , I ](3 a ) qq ≡ [Im]Re { C iia ) qq + 3 C iia ) qq } , (95) c , I ](3 a ) qq ≡ [Im]Re { C iia ) qq − C iia ) qq } , (96)in a way similar to the flavour conserving case. For ¯ RR ¯ RR operators, one has first: O (3 aii ) uu O (3 iia ) uu ! = (cid:18) /
30 2 (cid:19) T (cid:18) (¯ tγ µ u a ) (¯ u i γ µ u i )(¯ tγ µ T A u a ) (¯ u i γ µ T A u i ) (cid:19) (97)in the case of the O uu operator whose coefficient with different flavour assignations satisfies thesame relations as that of O qq . One thus defines c I ](3 a ) uu = [Im]Re { C (3 aii ) uu + 13 C (3 iia ) uu } ,c I ](3 a ) uu = [Im]Re { C (3 iia ) uu } , (98)31hile no Fierzing is required for the definition of the remaining ¯ RR ¯ RR and ¯ LL ¯ RR degrees offreedom: c I ](3 a ) ud = [Im]Re { C aii ) ud } ,c I ](3 a ) ud = [Im]Re { C aii ) ud } , c I ](3 a ) qu = [Im]Re { C aii ) qu } ,c I ](3 a ) qu = [Im]Re { C aii ) qu } ,c I ]( a qu = [Im]Re { C ii a ) qu } ,c I ]( a qu = [Im]Re { C ii a ) qu } , c I ](3 a ) qd = [Im]Re { C aii ) qd } ,c I ](3 a ) qd = [Im]Re { C aii ) qd } . (99) E.5 Counting
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02 pb 0 .
016 pb 1 . . c QQ cQQ1 − . − . − × − . − . − . c QQ cQQ8 − . − . − − . − . − . c Qt cQt1 − . − . × − . − . − . c Qt cQt8 − . − . − − . − . − . c Qb cQb1 − . . − . − . − . − . c Qb cQb8 .
14 3 . .
12 0 .
35 0 .
16 0 . c tt ctt1 − . × c tb ctb1 − . . − . − . − . − . c tb ctb8 .
13 3 . .
11 0 .
26 0 .
31 0 . c QtQb cQtQb1 c QtQb cQtQb8 c IQtQb cQtQb1I c IQtQb cQtQb8I c , Qq cQq83 . − .
11 4 . − −
20 8 . c , Qq cQq81
12 7 . . ×
71 40 75 c tq ctq8
13 8 . . ×
62 51 74 c Qu cQu8 . . c tu ctu8 . c Qd cQd8 . c td ctd8 . c , Qq cQq13 . . . ×
22 11 18 c , Qq cQq11 . − . − . − . − . c tq ctq1 .
65 2 . − . . .
84 3 . . c Qu cQu1 .
57 1 . − . . . . c tu ctu1 . − . − . . . . c Qd cQd1 − .
19 0 . − − . − . − . c td ctd1 − . − . − − . − . − . c tϕ ctp − . − . − . − . − . × c − ϕQ cpQM − .
063 1 − − . − × − . − . c ϕQ cpQ3 .
68 22 0 .
065 0 .
46 3 . . . c ϕt cpt − .
024 2 . − .
36 68 − . − . c ϕtb cptb c tW ctW .
98 1 −
34 13 1 . . c tZ ctZ − .
54 0 .
028 27 − . − . − − . c bW cbW c tG ctG . × . × . × . × . × . × . × c Itϕ ctpI − . × − . − . − . . c Iϕtb cptbI c ItW ctWI . × − . − . − .
19 0 .
29 0 .
91 0 . c ItZ ctZI − . × − . − . . . − . − . c IbW cbWI c ItG ctGI − . .
48 0 .
66 0 . − . . − . c Ql cQl31 .
011 0 . c − (1) Ql cQlM1 − . − . c (1) Qe cQe1 − . c (1) tl ctl1 − . − . c (1) te cte1 − . c S (1) tl ctlS1 c T (1) tl ctlT1 c S (1) bl cblS1 c SI (1) tl ctlSI1 c TI (1) tl ctlTI1 c SI (1) bl cblSI1 c tQqu ctQqu1 c tQqu ctQqu8 c bQqd cbQqd1 c bQqd cbQqd8 c Qtqd cQtqd1 c Qtqd cQtqd8 c Qbqu cQbqu1 c Qbqu cQbqu8 c btud cbtud1 c btud cbtud8 c ItQqu ctQqu1I c ItQqu ctQqu8I c IbQqd cbQqd1I c IbQqd cbQqd8I c IQtqd cQtqd1I c IQtqd cQtqd8I c IQbqu cQbqu1I c IQbqu cQbqu8I c Ibtud cbtud1I c Ibtud cbtud8I
Table 10: Linear dependences on the various degrees of freedom of total top pair production ratesat the 13 TeV LHC. For convenience, numerical values for the S kj / P l B kl ratios are provided inpermil. Absolute SM rates (i.e. P l B kl ) are also quoted in picobarns. LO simulation at the partonlevel, the five-flavour scheme, NNPDF2.3LO1, running renormalization and factorization scalesare used. Simple p T >
20 GeV, | η | < .
5, ∆
R > . b -)jets, charged leptonsand photons. As input parameters, m t = 172 GeV (all other fermion masses and Yukawa couplingsare set to zero) and m h = 125 GeV are notably employed. Monte Carlo uncertainties are of theorder of 15%, but outliers are expected given the number of quantity evaluated. 39 p → tj pp → t e − ¯ ν pp → tj e + e − pp → tj γ pp → tj h SM sm
55 pb 2 . . .
39 pb 0 .
016 pb c QQ cQQ1 c QQ cQQ8 c Qt cQt1 c Qt cQt8 c Qb cQb1 c Qb cQb8 c tt ctt1 c tb ctb1 c tb ctb8 c QtQb cQtQb1 c QtQb cQtQb8 c IQtQb cQtQb1I c IQtQb cQtQb8I c , Qq cQq83 − . × − − . × − − . × − − . × − c , Qq cQq81 c tq ctq8 c Qu cQu8 c tu ctu8 c Qd cQd8 c td ctd8 c , Qq cQq13 − . × − . × − . × − . × c , Qq cQq11 c tq ctq1 c Qu cQu1 c tu ctu1 c Qd cQd1 c td ctd1 c tϕ ctp − c − ϕQ cpQM c ϕQ cpQ3 . × . × . × . × . × c ϕt cpt . c ϕtb cptb c tW ctW −
76 45 50 9 . × c tZ ctZ − − c bW cbW c tG ctG c Itϕ ctpI − . c Iϕtb cptbI c ItW ctWI . × − − . .
47 0 . − . c ItZ ctZI − .
87 0 . c IbW cbWI c ItG ctGI . c Ql cQl31 . c − (1) Ql cQlM1 . c (1) Qe cQe1 − . c (1) tl ctl1 − . c (1) te cte1 . c S (1) tl ctlS1 c T (1) tl ctlT1 c S (1) bl cblS1 c SI (1) tl ctlSI1 c TI (1) tl ctlTI1 c SI (1) bl cblSI1 c tQqu ctQqu1 c tQqu ctQqu8 c bQqd cbQqd1 c bQqd cbQqd8 c Qtqd cQtqd1 c Qtqd cQtqd8 c Qbqu cQbqu1 c Qbqu cQbqu8 c btud cbtud1 c btud cbtud8 c ItQqu ctQqu1I c ItQqu ctQqu8I c IbQqd cbQqd1I c IbQqd cbQqd8I c IQtqd cQtqd1I c IQtqd cQtqd8I c IQbqu cQbqu1I c IQbqu cQbqu8I c Ibtud cbtud1I c Ibtud cbtud8I
Table 11: Same as Table 10, for single top production processes. 40 QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ . · . ························· . − . . · − . × − − . ······································· c QQ · . · . ························· − . ·· . ·· . ····································· c Q t ·· . ························· . − . . · . − . ······································· c Q t ··· . ························· − . ·· . ·· . ····································· c Q b ···· . ·· . ···················· − . · − . · − . . ······································· c Q b ····· . ·· . ·························· . ····································· c tt ········································································· c t b ······· . ···················· − . · − . · − . . ······································· c t b ········ . ·························· . ····································· c Q t Q b ········· . . ···················· . ·· . ······································ c Q t Q b ·········· . ···················· . ·· − . ······································ c I Q t Q b ··········· . . ························ . ·· . ································ c I Q t Q b ············ . ························ . ·· − . ································ c , Q q ············· . . . ··················· . ····································· c , Q q ·············· . . ··················· . ····································· c t q ··············· . ··················· . ····································· c Q u ················ . . ················· . ····································· c t u ················· . ················· . ····································· c Q d ·················· . . ··············· . ····································· c t d ··················· . ··············· . ····································· c , Q q ···················· . . ····· − . · − . · − . ······································· c , Q q ····················· ····· − . · − . · − . ······································· c t q ······················ ····· − . · − . · − . ······································· c Q u ······················· . . ··· . · . · . − . ······································· c t u ························ . ··· . · . · . − . ······································· c Q d ························· . . · − . · − . · − . . ······································· c t d ·························· . · − . · − . · − . . ······································· c t ϕ ········································································· c − ϕ Q ···························· . − . . · − . − . ······································· c ϕ Q ····························· . − . · − . . · − . ····································· c ϕ t ······························ . · − . − . ······································· c ϕ t b ······························· . ·· − . ······································ c t W ································ − . · . ··· − × − ································· c t Z ································· . ···· . × − ·································· c b W ·································· . ······································ c t G ··································· ····· . ······························· c I t ϕ ········································································· c I ϕ t b ····································· . ·· − . ································ c I t W ······································ . − . · . ······························· c I t Z ······································· . ································· c I b W ········································ . ································ c I t G ········································· ······························· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· . ··················· c t Q q u ······················································ . ·················· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ························································· . ··············· c Q t q d ·························································· . ·············· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· . ········· c I t Q q u ································································ . ········ c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ··································································· . ····· c I Q t q d ···································································· . ···· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t ¯ t r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. N u m e r i c a l v a l u e ss m a ll e r t h a n − a r e o m i tt e d . F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ . · . ························ . − . − . − . · − . − . ···· . − . ·· − . − . . − . . ·························· c QQ · . · . ························ . − . − . · . . · . ·· . . · × − − . − . × − − . − . . ·························· c Q t ·· . ························ − . . − . − . · − . . ·· . × − · − . . ·· − . − . − . × − − . . ·························· c Q t ··· . ························ . − . − . · − . . · . ·· − . . · − . − . − . × − − . − . . ·························· c Q b ···· . ·· . ··················· − . × − . · − . · − . . ···· . . ··· . − . × − · . × − ·························· c Q b ····· . ·· . ··················· − . · . · . − . · . ·· . − . · . · − . . . − . ·························· c tt ········································································· c t b ······· . ··················· . × − . · − . · − . . ···· . . ···· − . × − . × − . ·························· c t b ········ . ··················· − . · . · . . · . ·· . . · − . · − . × − . − . − . ·························· c Q t Q b ········· . . ···················· . ·· . ·· . × − ·· − . ································ c Q t Q b ·········· . ···················· . ·· − . ·· − . ·· . ································ c I Q t Q b ··········· . . ·················· − . × − ·· . ·· . ·· . ································ c I Q t Q b ············ . ·················· . ·· − . ·· . ·· − . ································ c , Q q ············· − − . ············ . · − . · . . · − ·· − . − . · − . · . . − . − . ·························· c , Q q ·············· . ············ − . · . · . − . · ·· − . . · − . · − . . . − . ·························· c t q ··············· ············ − . · . · . · ·· . . · . · − . . − . − . ·························· c Q u ················ ·········· − . · . · . − . · . ·· − . . · . · − . − . . . ·························· c t u ················· . ·········· − . · . · . . · . ·· . − . · − . · − . . − . − . ·························· c Q d ·················· . . ········ − . · . · . − . · . ·· . − . · − . · − . . . − . ·························· c t d ··················· . ········ − . · . · . . · . ·· . − . · . · − . . − . − . ·························· c , Q q ···················· . × − − . ···· − . − · . · . ·· − . · . − . ··· − . − . . − . ·························· c , Q q ····················· . × ···· − . . · . · . − . ·· − . · . − . ··· . − . − . − . ·························· c t q ······················ × ···· . . · . · . − . ·· − . · − . − . ··· . . . − . ·························· c Q u ······················· ·· . − . · . · . − ·· − . · − . . ··· − . . . − . ·························· c t u ························ ·· − . − . · . · . − . ·· − . · − . . ··· − . . . × − − . ·························· c Q d ························· − . . · − . · − . . ·· . · . . ··· . − . − . . ·························· c t d ·························· . . · − . · − . ·· . · . . ··· . − . . . ·························· c t ϕ ··························· . − . − . − . · . . ···· − . × − . ································· c − ϕ Q ···························· . . − . · − . . · − − . · . . · − . . . − . . − . ·························· c ϕ Q ····························· . . · − . − . · − . − . · . − . · . . . . . − . ·························· c ϕ t ······························ · . . · ·· . − . · . − . . − . . − . ·························· c ϕ t b ······························· . ·· . ····· . ································ c t W ································ − · . · − . . · − . − . . . . . ·························· c t Z ································· . × · . − . · − . . · . − . − . − . − . − . ·························· c b W ·································· . ·· − . ··································· c t G ··································· × ·· − . − . · − . − . − . . − . − . ·························· c I t ϕ ···································· . · . . ································· c I ϕ t b ····································· . ·· . ································ c I t W ······································ − · − . − . − . . − . ·························· c I t Z ······································· . × · − . . − . . . ·························· c I b W ········································ . ································ c I t G ········································· . × . × − . . . − . ·························· c ( ) Q l ·········································· . . · − . ··························· c − ( ) Q l ··········································· . · − . ··························· c ( ) Q e ············································ . · − . ·························· c ( ) t l ············································· . ··························· c ( ) t e ·············································· . ·························· c S ( ) t l ··············································· . − . ·· . ····················· c T ( ) t l ················································ . × · − . ······················ c S ( ) b l ········································································· c S I ( ) t l ·················································· . − . ····················· c T I ( ) t l ··················································· . × ····················· c S I ( ) b l ········································································· c t Q q u ····················································· ··················· c t Q q u ······················································ ·················· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ························································· ··············· c Q t q d ·························································· . ·············· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· ········· c I t Q q u ································································ ········ c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ··································································· ····· c I Q t q d ···································································· . ···· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t ¯ t e + e − r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ············· . × − − . ············· − . ·· ·· − ·· − . ·· − . − . ······························ c , Q q ·············· . × ··················· ····· − . ······························· c t q ··············· ··················· ····· − . ······························· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ···················· × − . × − ···· − . − . . − . · . × . ·· . · − . . ·· . − . · − . ··························· c , Q q ····················· . × ···· . . − . . · − ·· − . · . − . ·· . . · . ··························· c t q ······················ . × ···· − . . − . . · − ·· − . · . . ·· − . . · . ··························· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ··························· . · − . . × − · . − . ···· − . . × − ·· − × − ······························ c − ϕ Q ···························· . . . · − . − . ·· . × − · − . . ·· . . × − · − . × − ··························· c ϕ Q ····························· . . · − . . ·· − . × − · . . ·· . − . · − . ··························· c ϕ t ······························ . · − . − . ·· − . × − · − . . ·· . ·· − . × − ··························· c ϕ t b ·································· . ····· − . ································ c t W ································ − . ·· − . · − . − . ·· . × − − . · − . ··························· c t Z ································· . ·· − . · . − . ·· . − . × − · . ··························· c b W ····································· . ··································· c t G ··································· ····· − . ······························· c I t ϕ ···································· . · . . × − ·· − . ······························ c I ϕ t b ········································ . ································ c I t W ······································ − . ·· − . − . · − . ··························· c I t Z ······································· . ·· . ·· . × − ··························· c I b W ········································································· c I t G ········································· . ······························· c ( ) Q l ·········································· . . · . ··························· c − ( ) Q l ··········································· . · . ··························· c ( ) Q e ········································································· c ( ) t l ············································· . ··························· c ( ) t e ········································································· c S ( ) t l ··············································· . − . ·· . ····················· c T ( ) t l ················································ . · − . ······················ c S ( ) b l ········································································· c S I ( ) t l ·················································· . − . ····················· c T I ( ) t l ··················································· . ····················· c S I ( ) b l ········································································· c t Q q u ····················································· . × ··················· c t Q q u ······················································ ·················· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ························································· . × ··············· c Q t q d ·························································· ·············· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· . × ········· c I t Q q u ································································ ········ c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ··································································· . × ····· c I Q t q d ···································································· ···· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t ¯ t e + ν r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ . · . ························· . − . . · − . − . ···· . × − . ································· c QQ · . · . ························· − . ·· . − . · . ·· − . . · . × − ······························· c Q t ·· . ························· . − . . · − . . ···· − . . ································· c Q t ··· . ························· − . ·· . . · . ·· . . × − · . × − ······························· c Q b ···· . ·· . ···················· − . · − . · − . . ···· − . × − . × − ································· c Q b ····· . ·· . ······················· . − . · . ·· − . . · − . × − ······························· c tt ········································································· c t b ······· . ···················· − . · − . · − . . ···· . × − . × − ································· c t b ········ . ······················· . − . · . ·· . − . · − . × − ······························· c Q t Q b ········· . . ···················· . ·· . ·· . × − ·· . ································ c Q t Q b ·········· . ···················· . ·· . ·· . × − ·· . ································ c I Q t Q b ··········· . . ·················· − . × − ·· − . ·· . ·· . ································ c I Q t Q b ············ . ·················· − . × − ·· − . ·· . ·· . ································ c , Q q ············· . . ················ . − . · . ·· . − . · . ······························· c , Q q ·············· . ················ . − . · . ·· . − . · . ······························· c t q ··············· ················ . − . · . ·· − . . · − . ······························· c Q u ················ . ·············· . − . · . ·· . − . · . ······························· c t u ················· . ·············· . − . · . ·· . − . · − . ······························· c Q d ·················· . . ············ . − . · . ·· . − . · − . ······························· c t d ··················· . ············ . − . · . ·· . − . · − . ······························· c , Q q ···················· . ····· − . · − . · . − ···· . . ································· c , Q q ····················· ····· − . · − . · . − . ···· − . − . ································· c t q ······················ ····· − . · − . · . − . ···· . − . ································· c Q u ······················· . ··· . · . · . − ···· − . − . ································· c t u ························ ··· . · . · . − . ···· . . ································· c Q d ························· . · − . · − . · − . ···· . . ································· c t d ·························· · − . · − . · − . . ···· . . ································· c t ϕ ········································································· c − ϕ Q ···························· . − . . · − . − . ···· . × − . ································· c ϕ Q ····························· . − . · − . . · − . ·· . − . · − . × − ······························· c ϕ t ······························ . · − . − . ···· − . ·································· c ϕ t b ······························· . ·· . ······································ c t W ································ . × − . × · ·· − . − . · − . ······························· c t Z ································· . × · − ·· − . . · . ······························· c b W ·································· . ·· . × − ··································· c t G ··································· ·· . . · − . ······························· c I t ϕ ········································································· c I ϕ t b ····································· . ·· . ································ c I t W ······································ . × − . × · ······························· c I t Z ······································· . × · − ······························· c I b W ········································ . ································ c I t G ········································· ······························· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· ··················· c t Q q u ······················································ ·················· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ························································· ··············· c Q t q d ·························································· . ·············· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· ········· c I t Q q u ································································ ········ c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ··································································· ····· c I Q t q d ···································································· . ···· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t ¯ t γ r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ . · . ························ . − . − . . · − . − . ·· − . × − · − . − . ································· c QQ · . · . ······················· . · − . ·· . ·· . − × − · − . ·· . ······························· c Q t ·· . ························ . . − . . · − . − . ·· . × − · − . − . ································· c Q t ··· . ······················· . · − . ·· . ·· . . × − · − . ·· − . ······························· c Q b ···· . ·· . ··················· . − . · − . · − . . ···· − . − . ································· c Q b ····· . ·· . ·················· − . ······· . ····· − . ······························· c tt ········································································· c t b ······· . ··················· . − . · − . · − . . ···· − . − . ································· c t b ········ . ·················· − . ······· . . × − ···· . ······························· c Q t Q b ········· . . ···················· . ·· − . ·· . ·· . ································ c Q t Q b ·········· . ···················· . ·· − . ·· . ·· . ································ c I Q t Q b ··········· . . ·················· − . ·· − . ·· . ·· − . ································ c I Q t Q b ············ . ·················· − . ·· − . ·· . ·· − . ································ c , Q q ············· . ··········· − . ······· . − . ···· − . ······························· c , Q q ·············· ··········· − . ······· − . ···· − . ······························· c t q ··············· ··········· − . ······· − . ···· − . ······························· c Q u ················ ········· − . ······· − . ···· . ······························· c t u ················· ········· − . ······· . ···· . ······························· c Q d ·················· . ······· − . ······· . ···· . ······························· c t d ··················· ······· − . ······· . ···· . ······························· c , Q q ···················· . × ···· − . − · − . · − . ·· − . · − . . ································· c , Q q ····················· . × ···· − . − . · − . · − ·· − . · − . − . ································· c t q ······················ . × ···· − . − . · − . · − ·· . · − . . ································· c Q u ······················· ·· − . . · . · − ·· − . · . . ································· c t u ························ ·· − . . · . · − ·· − . · . . ································· c Q d ························· . − . · − . · − ·· . · − . − . ································· c t d ·························· . − . · − . · − ·· . · − . − . ································· c t ϕ ··························· . . − . . · − . . · − − . · − . − . · . ······························· c − ϕ Q ···························· . . . · − . − . ·· . × − · − . . ································· c ϕ Q ····························· . − . · − . . · − . ·· . . · − . ······························· c ϕ t ······························ . · − . − . ·· × − · − . . ································· c ϕ t b ······························· . ·· − . ····· . ································ c t W ································ − · . − . · − . . · . ······························· c t Z ································· ·· . · . − . ································· c b W ·································· . ·· − . ··································· c t G ··································· × − . · − . ·· − . ······························· c I t ϕ ···································· . · − . . · ······························· c I ϕ t b ····································· . ·· − . ································ c I t W ······································ − · . ······························· c I t Z ······································· . ································· c I b W ········································ . ································ c I t G ········································· . × ······························· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· . × ··················· c t Q q u ······················································ ·················· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ························································· ··············· c Q t q d ·························································· ·············· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· . × ········· c I t Q q u ································································ ········ c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ··································································· ····· c I Q t q d ···································································· ···· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t ¯ t h r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ············· . × ··························································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ···················· . × ········ − ·· ····· − . ·································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ········································································· c − ϕ Q ········································································· c ϕ Q ····························· . ·· . ········································ c ϕ t ········································································· c ϕ t b ······························· . ·· − . ······································ c t W ································ ········································ c t Z ········································································· c b W ·································· ······································ c t G ········································································· c I t ϕ ········································································· c I ϕ t b ····································· . ·· − . ································ c I t W ······································ ·································· c I t Z ········································································· c I b W ········································ ································ c I t G ········································································· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· ··················· c t Q q u ······················································ ·················· c b Q q d ······················································· ················· c b Q q d ························································ ················ c Q t q d ························································· ··············· c Q t q d ·························································· ·············· c Q b q u ··························································· ············· c Q b q u ···························································· ············ c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· ········· c I t Q q u ································································ ········ c I b Q q d ································································· ······· c I b Q q d ·································································· ······ c I Q t q d ··································································· ····· c I Q t q d ···································································· ···· c I Q b q u ····································································· ··· c I Q b q u ······································································ ·· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t j r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ········································································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ········································································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ········································································· c − ϕ Q ········································································· c ϕ Q ····························· . ·· − . ·· ·· . ·· − . . ······························ c ϕ t ········································································· c ϕ t b ······························· . ·· . ····· − . ································ c t W ································ ·· − ····· . − . ······························ c t Z ········································································· c b W ·································· ·· . ··································· c t G ··································· ·· − . ··· . ······························ c I t ϕ ········································································· c I ϕ t b ····································· . ·· . ································ c I t W ······································ ·· − . ······························ c I t Z ········································································· c I b W ········································ ································ c I t G ········································· . ······························ c ( ) Q l ·········································· . ······························ c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ··············································· . − . ·· . ····················· c T ( ) t l ················································ . · − . ······················ c S ( ) b l ········································································· c S I ( ) t l ·················································· . − . ····················· c T I ( ) t l ··················································· . ····················· c S I ( ) b l ········································································· c t Q q u ········································································· c t Q q u ········································································· c b Q q d ········································································· c b Q q d ········································································· c Q t q d ········································································· c Q t q d ········································································· c Q b q u ········································································· c Q b q u ········································································· c b t u d ········································································· c b t u d ········································································· c I t Q q u ········································································· c I t Q q u ········································································· c I b Q q d ········································································· c I b Q q d ········································································· c I Q t q d ········································································· c I Q t q d ········································································· c I Q b q u ········································································· c I Q b q u ········································································· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t e − ν r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ············· . × ··························································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ···················· . × ······· − . − . × − . · . ···· . − . ·· − . − . . − . − . ·························· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ········································································· c − ϕ Q ···························· . . − . · . . ···· . − . ·· . . − . − . . ·························· c ϕ Q ····························· − . · . . ···· − . . ·· . . − . . . ·························· c ϕ t ······························ . · . − . ···· − . − . ·· − . − . . . − . ·························· c ϕ t b ······························· . ·· − . ····· − . ································ c t W ································ . × − ···· − . − . ·· . . . − . − . ·························· c t Z ································· ···· . . ·· − . − . − . . . ·························· c b W ·································· ·· . ··································· c t G ········································································· c I t ϕ ········································································· c I ϕ t b ····································· . ·· − . ································ c I t W ······································ . × − ·· . . − . − . . ·························· c I t Z ······································· ·· − . − . . . − . ·························· c I b W ········································ ································ c I t G ········································································· c ( ) Q l ·········································· · − . ··························· c − ( ) Q l ··········································· . · − . ··························· c ( ) Q e ············································ . · − . ·························· c ( ) t l ············································· . ··························· c ( ) t e ·············································· . ·························· c S ( ) t l ··············································· . − . ·· . ····················· c T ( ) t l ················································ · . ······················ c S ( ) b l ········································································· c S I ( ) t l ·················································· . − . ····················· c T I ( ) t l ··················································· ····················· c S I ( ) b l ········································································· c t Q q u ····················································· . × ··················· c t Q q u ······················································ ·················· c b Q q d ······················································· × ················· c b Q q d ························································ ················ c Q t q d ························································· . × ··············· c Q t q d ·························································· ·············· c Q b q u ··························································· . × ············· c Q b q u ···························································· ············ c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· . × ········· c I t Q q u ································································ ········ c I b Q q d ································································· × ······· c I b Q q d ·································································· ······ c I Q t q d ··································································· . × ····· c I Q t q d ···································································· ···· c I Q b q u ····································································· . × ··· c I Q b q u ······································································ ·· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t j e + e − r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ············· . × ··························································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ···················· . × ········ − ·· − ···· . . ································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ········································································· c − ϕ Q ········································································· c ϕ Q ····························· . ·· . − . ···· − . − . ································· c ϕ t ········································································· c ϕ t b ······························· . ·· − . ····· . ································ c t W ································ − ···· − . . ································· c t Z ································· ···· − . ·································· c b W ·································· ·· . ··································· c t G ········································································· c I t ϕ ········································································· c I ϕ t b ····································· . ·· − . ································ c I t W ······································ − ································· c I t Z ······································· ································· c I b W ········································ ································ c I t G ········································································· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· ··················· c t Q q u ······················································ ·················· c b Q q d ······················································· ················· c b Q q d ························································ ················ c Q t q d ························································· ··············· c Q t q d ·························································· ·············· c Q b q u ··························································· ············· c Q b q u ···························································· ············ c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· ········· c I t Q q u ································································ ········ c I b Q q d ································································· ······· c I b Q q d ·································································· ······ c I Q t q d ··································································· ····· c I Q t q d ···································································· ···· c I Q b q u ····································································· ··· c I Q b q u ······································································ ·· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t j γ r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e . QQ c QQ c Q t c Q t c Q b c Q b c tt c t b c t b c Q t Q b c Q t Q b c I Q t Q b c I Q t Q b c , Q q c , Q q c t q c Q u c t u c Q d c t d c , Q q c , Q q c t q c Q u c t u c Q d c t d c t ϕ c − ϕ Q c ϕ Q c ϕ t c ϕ t b c t W c t Z c b W c t G c I t ϕ c I ϕ t b c I t W c I t Z c I b W c I t G c ( ) Q l c − ( ) Q l c ( ) Q e c ( ) t l c ( ) t e c S ( ) t l c T ( ) t l c S ( ) b l c S I ( ) t l c T I ( ) t l c S I ( ) b l c t Q q u c t Q q u c b Q q d c b Q q d c Q t q d c Q t q d c Q b q u c Q b q u c b t u d c b t u d c I t Q q u c I t Q q u c I b Q q d c I b Q q d c I Q t q d c I Q t q d c I Q b q u c I Q b q u c I b t u d c I b t u d c QQ ········································································· c QQ ········································································· c Q t ········································································· c Q t ········································································· c Q b ········································································· c Q b ········································································· c tt ········································································· c t b ········································································· c t b ········································································· c Q t Q b ········································································· c Q t Q b ········································································· c I Q t Q b ········································································· c I Q t Q b ········································································· c , Q q ············· . × ··························································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c , Q q ···················· . × ······ . × · − . × ·· − ··· . · − . ·································· c , Q q ········································································· c t q ········································································· c Q u ········································································· c t u ········································································· c Q d ········································································· c t d ········································································· c t ϕ ··························· · − ·· − ··· . · . ·································· c − ϕ Q ········································································· c ϕ Q ····························· . × ·· . × ··· . · . ·································· c ϕ t ········································································· c ϕ t b ······························· ·· − ····· . ································ c t W ································ . × ··· ···································· c t Z ········································································· c b W ·································· × ·· − . ··································· c t G ········································································· c I t ϕ ···································· . · − ·································· c I ϕ t b ····································· ·· − ································ c I t W ······································ . × ·································· c I t Z ········································································· c I b W ········································ × ································ c I t G ········································································· c ( ) Q l ········································································· c − ( ) Q l ········································································· c ( ) Q e ········································································· c ( ) t l ········································································· c ( ) t e ········································································· c S ( ) t l ········································································· c T ( ) t l ········································································· c S ( ) b l ········································································· c S I ( ) t l ········································································· c T I ( ) t l ········································································· c S I ( ) b l ········································································· c t Q q u ····················································· . × ··················· c t Q q u ······················································ . × ·················· c b Q q d ······················································· . × ················· c b Q q d ························································ . × ················ c Q t q d ························································· . × ··············· c Q t q d ·························································· . × ·············· c Q b q u ··························································· . × ············· c Q b q u ···························································· . × ············ c b t u d ········································································· c b t u d ········································································· c I t Q q u ······························································· . × ········· c I t Q q u ································································ . × ········ c I b Q q d ································································· . × ······· c I b Q q d ·································································· . × ······ c I Q t q d ··································································· . × ····· c I Q t q d ···································································· . × ···· c I Q b q u ····································································· . × ··· c I Q b q u ······································································ . × ·· c I b t u d ········································································· c I b t u d ········································································· T a b l e : Q u a d r a t i c d e p e nd e n ce o n t h e v a r i o u s d e g r ee s o ff r ee d o m o f t h e t o t a l pp → t j h r a t e . F o r c o n v e n i e n ce s , nu m e r i c a l v a l u e s f o r S k i j / P l B k l a r e q u o t e d i np e r m il. F o r m o r e d e t a il ss ee t h e l e g e nd o f T a b l e ..