Dimension-seven operator contribution to the top quark anomalous interactions
DDimension-seven operator contribution to the top quark anomalous interactions
V.V. Denisov , S.R. Slabospitskii Moscow Institute of Physics and Technology,Dolgoprudny, Moscow Region, Russia NRC “Kurchatov Institute” - IHEP,Protvino, Moscow Region, RussiaE–mail: [email protected], [email protected]
Abstract
The contribution of dimension-seven operators to anomalous FCNC-interactions ofthe t -quarks with a photon and a gluon is considered. The phenomenological Lagrangianand Feynman rules are derived. There are evaluated the expressions for the widthsFCNC-decays of the t -quark into light quark and two photons, two gluons, a photon anda gluon, and three quarks. a r X i v : . [ h e p - ph ] M a r Introduction
At present, it is not known what type of New Physics (NP) beyond the Standard Modelframework will be responsible for possible deviations from predictions of the Standard Model(SM). Multiple scenarios of SM extensions lead to different predictions in the t -quark sectorwith their own a specific set of types of interactions and parameters (coupling constants, themasses of new objects). At the same time, different scenarios often predict the same or verysimilar effects, leading to processes with identical final states.To describe the various anomalous interactions of t -quarks it is widely used a universalapproach based on the formalism of an effective field theory [1]. In this approach, the anoma-lous interactions of t -quarks are described by a model-independent manner through the useof an effective (phenomenological) Lagrangian [2, 3, 4, 5]. This Lagrangian must be gauge-invariant with respect to the SM gauge group (otherwise, the input anomalous interactionsimmediately would lead to contradictions with modern precision measurements) and consistsof the terms with an increasing dimension, suppressed by increasingly higher degrees of theNP-scale. Such a Lagrangian of the anomalous t -quark interactions can be presented in thefollowing form [1, 3, 4]: L QF T = L SM + κ ¯ ψ q ˆ O (4) ψ t + κ Λ ¯ ψ q ˆ O (5) ψ t + κ Λ ¯ ψ q ˆ O (6) ψ t + · · · (1)where L SM is the SM Lagrangian (see, e.g. [9]), Λ is new physics mass scale, κ i are theanomalous couplings.The promising directions of searching for NP in the t -quark sector are processes with theFlavour changing neutral currents (FCNC) t → γ ( g, Z ) + c ( u )Within the SM framework such processes are highly suppressed (due to loop diagrams) [2]: B ( t → q γ/g/Z ) < O (10 − ÷ − ), which makes them practically impossible to observein the experiment. Thus, the experimental observation of the t -quark FCNC interactions t -quarks will unambiguously indicate the existence of NP beyond the SM.The FCNC processes with the dimension-5 and 6 operators were analyzed in detail ear-lier and there were built expressions for all possible FCNC interactions of t -quarks (see, forexample, [3, 4]).In this article we consider the role of the dimension-seven operator contributions to anoma-lous FCNC-interaction of t -quarks with a photon and gluon ( t γ q and t g q ): L (7) F CNC = κ Λ ¯ ψ q ˆ O (7) ψ t , q = u, c In this article, the upper q -quark in the ( tV q ) interaction will be denoted by the symbol u .Therefore, all the results will remain the same for u - or c -quarks. In this article, when constructing the operators ˆ O ( n ) there are considered the interactionswith only one external massless gauge boson V ( V = γ, g ). Due to a gauge invariance, theinteracting boson enters into operators in the form of the field strength tensor F µν (and the1ual tensor (cid:101) F µν ): L F CNC = 1Λ ¯ ψ u κ ˆ W (7) µν ψ t F µν + 1Λ ¯ ψ u κ ˆ W (7) µν ψ t (cid:101) F µν (2) (cid:101) F µν ≡ ε µναβ F αβ ; κ i = ξ i + ζ i γ F ( qed ) µν = F µν = ∂ µ A ν − ∂ ν A µ F ( qcd ) µν = G aµν = ∂ µ B aν − ∂ ν B aµ + if abc t c B bµ B cν where A µ , B aµ are the photon and gluon fields, ξ i ζ i - are the anomalous couplings (in thegeneral case, the complex numbers).In the general case, the operator ˆ W ( n ) is constructed from the Dirac, Gell-Man and covari-ant derivatives D µ , D ∗ µ [9]: −→ D µ = D µ = −→ ∂ µ − ie q A µ − ig s t a G µa ←− D µ = D ∗ µ = ←− ∂ µ + ie q A µ + ig s t a G µa where e q is the electric charge of the quark, g s is the constant of the strong interactions, and t a - Gell-Mann matrices. The covariant derivative D µ acts on the spinor ψ t , and D ∗ µ - tothe antispinor ¯ ψ u . We note that, because of covariant derivatives, in Lagrangians there areterms, describing the interaction with one, two and three bosons. In this article we presentan explicit form of expressions only for the interaction with one and two bosons. Second restriction on the form of the operators ˆ W ( n ) consists in the following. When con-structing such operators, it is assumed that covariant derivatives act only on quarks (spinors).In this case, they must have convolution by indices with a strength tensor. This assumptionrestricts the type of the operators ˆ W ( n ) . Indeed, let us consider the contribution of thedimension-seven operator: ¯ ψ u ˆ D ∗ γ µ D ν ψ t F µν , ˆ D ∗ = ←− D α γ α In this case, the action of the derivative ˆ D ∗ (which is not related to interacting boson), canbe treated, for example, as a form-factor: κ (7) Λ ¯ ψ u ˆ D ∗ γ µ D ν ψ t F µν = κ (7) Λ ¯ ψ u (cid:32) ←− D α γ α Λ (cid:33) γ µ D ν ψ t F µν ; κ (7) Λ ¯ ψ u (cid:32) ←− D α γ α Λ (cid:33) = ⇒ ˜ κ ( q )Λ ¯ ψ u = ⇒ κ (7) Λ ¯ ψ u ˆ D ∗ γ µ D ν ψ t F µν (cid:39) ˜ κ ( q )Λ ¯ ψ u γ µ D ν ψ t F µν The last expression is, in fact, represented by the contribution of an operator of dimension-six! Thus, taking into account these two assumptions (on the interaction with only one bosonand second, described above) the phenomenological Lagrangian of FCNC interactions can beconstructed only from the operators of dimensions-five, six and seven.
Dimension-seven operator for anomalous t -quark FCNC-interaction with a photon com-prises four gauge-invariant terms:ˆ W (7) µνγ : D µ D ν , D ∗ µ D ∗ ν , D ∗ µ D ν , D ν D ∗ µ F µν the contributions of the first and secondterms are equal to each other, and the last two are related by the relation: D µ D ν = D ∗ µ D ∗ ν = −
12 ∆ µν D ν D ∗ µ = D ∗ µ D ν − ∆ µν ∆ µν = ie q F µν + ig s t a G µνa Therefore, the operator ˆ W (7) µνγ comprises only two independent structures ( D ∗ µ D ν and ∆ µν ).The Lagrangian of t -quark FCNC-interaction with a photon has the form: L (7) F CNC ( t γq ) = e q Λ ¯ ψ u ˆ W µν ψ t F µν + e q Λ ¯ ψ u ˆ W µνD ψ t (cid:101) F µν (3)ˆ W µν = κ D ∗ µ D ν − κ ∆ µν W µνD = κ D ∗ µ D ν − κ ∆ µν κ i = ξ γi + ζ γi γ where ξ γi , ζ γi are the anomalous couplings.In what follows, the common numerical factors (of the type ±
1, 1/2, i, ... ) are included inthe anomalous couplings. We note that in the Lagrangian the operators D ∗ µ D ν × (cid:101) F µν and∆ µν × (cid:101) F µν result in the same expressions for terms describing the interaction with one or twobosons. Therefore, in this article we rely on κ = κ Omitting the trivial computations and using the the momentum representation, for each termsfrom ˆ W (7) and ˆ W (7) D we have¯ ψ u D ∗ µ D ν ψ t F µν → ¯ u u κ [ ˆ w + ˆ X + ˆ U + ˆ V (3)] u t ¯ ψ u ∆ µν ψ t F µν → ¯ u u κ [2 X + U + V (3)] u t ¯ ψ u D ∗ µ D ν ψ t (cid:101) F µν = ¯ ψ u ∆ µν ψ t (cid:101) F µν → ¯ u u κ [2 ˆ X + ˆ U + ˆ V (3)] u t where ¯ u u and u t are the spinors of the light u and t -quarks with momenta p and p , respec-tively. The expressions ˆ V i (3) , i = 1 , − ): w = e q p µ p ν ( q µ g να − q ν g µα ) A α X = e q (cid:104) ( q q ) g αβ − q β q α (cid:105) A α A β X = e q (cid:104) ( q + q ) g αβ − q α ( q + q ) β − q β ( q + q ) α (cid:105) A α A β X = e q ε µναβ q µ q ν A α A β U = e q g s t a (cid:104) ( q q ) g αβ − q β q α (cid:105) A α ( q ) B βa ( q ) U = e q g s t a ( q + q ) λ ( q λ g αβ − q β g λα ) A α ( q ) B βa ( q ) U = e q g s t a ε µναβ q µ q ν A α ( q ) B βa ( q ) (4)here q q are the momenta of the bosons. The corresponding Feynman rules are given in theAppendix. 3 .2 The anomalous interaction with a gluon Dimension-seven operator for anomalous t -quark FCNC-interaction with a gluon comprisesthree gauge-invariant terms:ˆ W a µν : D ∗ µ t a D ν , t a D µ D ν , D ∗ µ D ∗ ν t a Also, as in the case of interaction with a photon, the contributions of operators t a D µ D ν and D ∗ µ D ∗ ν t a are equal to each other. Thus, the Lagrangian of dimension-seven, describing theinteraction with the gluon, has the form: L (7) F CNC ( t g q ) = g s Λ ¯ ψ u ˆ W a µν ψ t G aµν + g s Λ ¯ ψ u ˆ W a µνD ψ t (cid:101) G aµν (5)ˆ W a µν = λ D ∗ µ t a D ν − λ t a ∆ µν W a µνD = λ D ∗ µ t a D ν − λ t a ∆ µν λ i = ξ gi + ζ gi γ where ξ gi , ζ gi are anomalous couplings, ¯ ψ u and ψ t are spinors describing the light u and t -quarks, the strength tensor G µνa is defined above (2).After transition into momentum space for each term in the Lagrangian (5) we get:¯ ψ u D ∗ µ t a D ν ψ t G aµν → ¯ u u λ [ w g + U + Y + V , ] u t ¯ ψ u t a D µ D ν ψ t G aµν → ¯ u u λ [ U + Y + V , ] u t ¯ ψ u D ∗ µ t a D ν ψ t (cid:101) G aµ ν → u u λ [ U + Y + V , ] u t ¯ ψ u t a D µ D ν ψ t (cid:101) G aµν → − u u λ [ U + Y + V , ] u t where ¯ u u and u t are spinors, describing a light u and t -quarks with momenta p and p ,respectively. Values of V , describe interactions with 3 and 4 bosons. The explicit form ofwhich is not given in this article. For the rest, we have (below, each expression contains thetotal coefficient Λ − ): w g = g s t a p µ p ν ( q µ g να − q ν g µα ) B αa U = e q g s t a (cid:2) (( q + q ) q ) g αβ − q α ( q + q ) β (cid:3) A α B βa U = e q g s t a (cid:104) ( q q ) g αβ − q α q β (cid:105) A α ( q ) B βa ( q ) U = e q g s t a ε µναβ q µ q ν A α B βa Y = g s (cid:110) t a t b (cid:104) (( p q ) − ( p q )) g αβ − p α p β + p α p β − q α p β + p α q β (cid:105) + t b t a (cid:104) (( p q ) − ( p q )) g αβ + p α p β − p α p β + q α p β − p α q β (cid:105)(cid:111) B α a B β b Y = g s ( δ ab + d abc t c )[( q q ) g αβ − q α q β ] B α a B β b Y = g s (cid:2)(cid:0) δ ab + d abc t c (cid:1) q µ q ν + 2 it k f kab p µ p ν (cid:3) ε µναβ B α a B β b Y = g s (cid:0) δ ab + d abc t c (cid:1) ε µναβ q µ q ν B α a B β b (6)here q q are the momenta of the bosons. The corresponding Feynman rules are given in theAppendix. t -quark decay widths We note that the amplitudes ( T ) containing the anomalous interaction vertex with onereal boson (photon or gluon), are always equal to zero. Indeed, taking into account the law of4onservation of momentum and choosing a calibration, which ensures the Lorentz condition(( qV ) = 0), for such vertices we obtain: T ∝ w = p µ p ν ( q µ g να − q ν g µα ) V α ; V α = A α , B αa , p = p + q, ( qV ) = 0 , q = 0 → w = [( p q ) p α − ( p q ) p α ] V α = (cid:2) ( p q ) p α − q p α − ( p q ) q α − ( p q ) p α (cid:3) V α = 0Therefore, in contrast to operators of dimension-5 and 6, for the considered interaction due todimension-seven operators in the lowest order perturbation theory the t -quark can decay intothe following three-body channels: t → u γ γ, t → u γ g, t → u g gt → u q ¯ q ; q (cid:54) = u, t → u ¯ u u (7)The diagrams describing these processes are presented below. t ( p ) u ( p ) q q ( a ) t ( p ) u ( p ) q q q ( a ) t ( p ) u ( p ) q q q ( a ) t ( p ) u ( q ) r p q ( a ) Figure 1: The diagrams describing the t -quark decays. Here p and p are the momenta of t and u -quarks, respectively, q and q are the momenta of the gauge bosons or ¯ qq pair.In all calculations we set: m is the mass of the t -quark, the masses of light quarks areassumed to be zero. We use axial gauge [8]: (cid:88) pol V µ V ν = ρ µν ( q ) = − g µν + q µ n ν + n µ q ν ( qn ) − n q µ q ν ( qn ) ; ρ µν n ν = 0 (8)where q is the gauge boson momentum, n is gauge fixing 4-vector. In what follows we take ita sum of q and q . Then, we get: n = q + q → ρ µν ( q ) = ρ µν ( q ) = ρ µν = − g µν + q µ q ν + q µ q ν ( q q ) ρ µν ρ αν = ρ µα ; ρ µν g µν = 2As is well known, the 1 → q = ( q + q ) ; x = q m , y = ( p + q ) m , x + y ≤ , ≤ { x, y } ≤ , (9)Then, we get the following form for the width for each decay channels: d Γ( t → uab ) = m π | T ( t → uab ) | dx dy (10)5here | T | is the square of the amplitude (with averaging on the spin and color of the initial t -quark is included in expression (10)), a and b are the corresponding final states (photons,gluons, and quarks). t → uγγ decay The amplitude of the decay of the t -quark into two photons is described by a single Feynmandiagram ( a in Fig. 1) and equals (see Appendix for the vertex t → qγγ ): T ( t → uγγ ) = e q Λ ¯ u ( p )( T + T + T ) u ( p ) T = κ (cid:104) ( q + q ) g αβ − q α ( q + q ) β − ( q + q ) α q β (cid:105) A α A β T = 2 κ (cid:104) ( q q ) g αβ − q β q α (cid:105) A α A β T = 2 κ ε µναβ q µ q ν A α A β , κ i = ξ γi + ζ γi γ where q q are the momenta of the photons. Using the axial gauge (8) we get: q = q = 0; ( q A ) = ( q A ) = ( q A ) = ( q A ) = 0 T + T = 2( κ + κ ) B ; T = 2 κ B B = ( q q )( A A ) = ( q / A A ); B = ε µναβ q µ q ν A α | B | = q A µ A µ A α A α = q ρ µα ρ µα = 12 q | B | = ε µναβ ε µ (cid:48) ν ; α (cid:48) β (cid:48) q µ q ν q µ (cid:48) q ν (cid:48) A α A β A α (cid:48) A β (cid:48) = 2( q q ) = 12 q | ¯ u ( p ) κ i u ( p ) | = Tr( ˆ p + m )( ξ ∗ − ζ ∗ γ ) ˆ p ( ξ + ζγ ) = 2( | ξ | + | ζ | )( m − q )As a result, the amplitude squared equals: | T | = 4 (cid:18) (cid:19) e q Λ K γγ ( m − q ) q ; K γγ = (cid:2) | ξ γ + ξ γ | + | ξ γ | + | ζ γ + ζ γ | + | ζ γ | (cid:3) where 3 is color coefficient, and 2 in the denominator takes into account the identity of thefinal photons. Below we present the expressions for the t -quark decay width: d Γ( t → uγγ ) /dx dy = e t α e µ ( m/ π ) K γγ x (1 − x )Γ( t → uγγ ) = e t α e µ ( m/ π ) K γγ (cid:41) (11)where α e is the fine structure constant and we use the notation µ = (cid:16) m Λ (cid:17) t → u γ g decay The decay of the t -quark into u -quark, the photon and the gluon occurs due to FCNC inter-actions induced by a photon and a gluon. The amplitude is described by a single Feynmandiagram ( a in Fig. 1) and equals: T ( t → qγ g ) = e q g s Λ ¯ u ( p )( T γg + T γg + T γg ) u ( p )6here T γg = κ t a (cid:104) (( q + q ) q )) g αβ − ( q + q ) α q β (cid:105) A α B β b + λ t a (cid:2) (( q + q ) q )) g αβ − q α ( q + q ) β (cid:3) A α B β b T γg = ( κ + λ ) (cid:104) ( q q ) g αβ − q α q β (cid:105) A α B β b T γg = ( κ + λ + λ ) t a ε µ ναβ q µ q ν A α B β b , κ i = ξ γi + ζ γi γ , λ i = ξ gi + ζ gi γ Expressions for the decay widths are equal to the corresponding expressions from (11) withthe replacement (cid:8) e t α e K γγ (cid:9) γγ → (cid:8) e t α e α s K gγ (cid:9) g γ (12)where α s is the QCD coupling. Here we have K g γ = | ξ γ + ξ γ + ξ g + ξ g | + | ξ γ + ξ g + ξ g | + | ζ γ + ζ γ + ζ g + ζ g | + | ζ γ + ζ g + ζ g | t → u g g decay The amplitude of the t -quark decay into u -quark and two gluons is described by two Feynmandiagrams ( a and a in Fig. 1) and equals: T ( t → ug g ) = g s Λ ¯ u ( p )( T gg + T gg + T gg + T gg + T gg ) u ( p )where T gg = λ t a p µ k ν ( q µ g να − q ν g µα ) − ρ αα (cid:48) ( q ) q F α (cid:48) βδabc B βb B δc T gg = λ t a t b (cid:104) (( p q ) − ( p q )) g αβ − p α p β + p α p β − q α p β + p α q β (cid:105) × B α a B β b + λ t b t a (cid:104) (( p q ) − ( p q )) g αβ + p α p β − p α p β + q α p β − p α q β (cid:105) × B α a B β b T gg = λ (cid:18) δ ab + t k d kab (cid:19) (cid:104) ( q q ) g αβ − q α q β (cid:105) B α a B β b T gg = λ (cid:20)(cid:18) δ ab + t k d kab (cid:19) q µ q ν + 2 it k f kab k µ p ν (cid:21) ε µ ναβ B α a B β b T gg = λ (cid:18) δ ab + t k d kab (cid:19) q µ q ν ε µ ναβ B α a B β b , λ i = ξ gi + ζ gi γ , here q q are the photon and gluon momenta. F α (cid:48) βδabc = ig s f abc (cid:104) ( − q − q ) δ g α (cid:48) β + ( − q + q ) α (cid:48) g δβ + ( q + q ) β g α (cid:48) δ (cid:105) ; ρ αα (cid:48) ( q ) = − g αα (cid:48) + q α q α (cid:48) q Then we get: | T ( t → u g g ) | = 4 g s ( m − q ) (cid:2) χ q + 3 χ ( m − q ) (cid:3) χ = | ξ g − ξ g | + | ξ g + ξ g | + | ζ g − ζ g | + | ζ g + ζ g | χ = | ξ g | + | ζ g | t -quark decay width: d Γ( t → u g g ) /dx dy = α s µ ( m/ π ) [7 χ x (1 − x ) + 3 χ (1 − x ) ]Γ( t → u g g ) = α s µ ( m/ π ) [7 χ + 18 χ ] (cid:41) (13) t → u ¯ qq and t → u ¯ uu decays The decay of the t -quark into a light u -quark and a quark-antiquark pair can proceed throughtwo channels t → u ¯ q q, q (cid:54) = u ; t → u ¯ u u The first decay is described by a single diagram a , and the second decay - by two diagrams a and a (see Fig. 1). The corresponding amplitudes are: T ( t → u ¯ qq ) q (cid:54) = u = ( g s / Λ ) W ; T ( t → u ¯ uu ) = ( g s / Λ ) ( W − W ) W = { ¯ u ( p ) λ t a u ( p ) } p µ p ν ( q µ g να − q ν g µα ) − ρ α (cid:48) α ( q ) q (cid:110) ¯ u ( q ) t a γ α (cid:48) v ( q ) (cid:111) W = { ¯ u ( q ) λ t a u ( p ) } p µ q ν ( r µ g να − r ν g µα ) − ρ αα (cid:48) ( r ) r (cid:110) ¯ u ( p ) t a γ α (cid:48) v ( q ) (cid:111) q = q + q ; r = p + q ρ αα (cid:48) ( q ) = − g αα (cid:48) + q α q α (cid:48) q ; ρ αα (cid:48) ( r ) = − g αα (cid:48) + q α r α (cid:48) + r α q α (cid:48) ( qr ) − q r α r α (cid:48) ( qr ) The expressions for the decay widths of the t -quark are equal to: d Γ( t → u ¯ q q ) /dxdy = α s µ ( m/ π ) ( | ξ g | + | ζ g | )(1 − x − y )(1 − x ) y Γ( t → u ¯ q q ) = α s µ ( m/ π ) ( | ξ g | + | ζ g | ) d Γ( t → u ¯ uu ) dxdy = α s µ ( m/ π ) ( | ξ g | + | ζ g | )(1 − x − y )( x + y − xy )Γ( t → u ¯ uu ) = α s µ (23 m/ π ) ( | ξ g | + | ζ g | ) (14) Estimation of the probability the t -quark “two-body” decays We note that during hadronization the pair of quarks (or quark and gluon) with a smallinvariant mass can form a hadronic jet ( j ). In this case, the decays of the t -quark in theexperiment can lead to the observed two-body final states: t → uγγ → j ( uγ ) + γ ; uγg → j ( ug ) + γ ; uγg → j ( uγ ) + j ( g ); ugg → j ( ug ) + j ( g ) , ... To estimate the probability of these two-body decays we require that the invariant mass of apair of final particles from the decays (7) should be less than 40 GeV (naturally, the realisticestimates can be obtained after a detailed modeling of processes): m min ≤
40 GeV → δ = (cid:16) m min m (cid:17) (cid:39) . β [ t → jj ] = Γ( t → jj ) / Γ( t → uab ) (15)8here a and b are two photons, gluons or light quarks. Then, for decays (7) we get: t → uγγ : β [ t → j ( uγ ) + γ ] = (5 / δ (cid:39) . t → uγg : β [ t → j ( uγ ) + j ] = (5 / δ + (5 / δ (4 − δ ) (cid:39) . t → ugg : β [ t → j + j ] = (5 / δ + (5 / − (1 − δ ) ) (cid:39) . t → u ¯ qq : β [ t → j + j ] = 5 δ (1 − δ ) (cid:39) . t → u ¯ uu : β [ t → j + j ] = (20 / δ (6 − δ + 2 δ ) (cid:39) . (16)Thus, it follows from the estimates (16) that approximately 25% the case of decay of t -quarks (7) due to considered FCNC interaction can lead to observable two-body final states. The contribution of dimension-seven operators to anomalous FCNC interactions of the t -quarks with a photon and a gluon is considered. A phenomenological Lagrangian of suchan interaction and the corresponding Feynman rules are derived. There are evaluated theexpressions for the widths of the FCNC decays of the t -quark to light u or c quarks and γγ , γ g , ¯ qq . It is shown that in a notable number of cases such decays of the t -quark due todimension-seven operators can lead to observable two-body (a jet and a photon or two hadronjets) to final states.In conclusion, the authors are sincerely grateful V. Kabachenko, V. Kachanov, M. Mangano,P. Mandrik, A. Razumov and R. Rogalyov for useful discussions. References [1] W. Buchmuller and D. Wyler, “Effective Lagrangian Analysis of New Interactions andFlavor Conservation,” Nucl. Phys. B (1986) 621. doi:10.1016/0550-3213(86)90262-2[2] M. Beneke et al. , ,,Top quark physics“, arXiv:hep-ph/0003033, in “Standart model physics(and more) at the LHC“, G. Altarelli and M. L. Mangano eds., Geneva, Switzerland:CERN (2000) 529 p.[3] J.A. Aguilar-Saavedra, “A Minimal set of top anomalous couplings,” Nucl. Phys. B (2009) 181 doi:10.1016/j.nuclphysb.2008.12.012 [arXiv:0811.3842 [hep-ph]].[4] B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, “Dimension-Six Terms inthe Standard Model Lagrangian,” JHEP (2010) 085 doi:10.1007/JHEP10(2010)085[arXiv:1008.4884 [hep-ph]].[5] C. Zhang, S. Willenbrock, “Effective-Field-Theory Approach to Top-Quark Productionand Decay“, Phys. Rev. D (2011) 034006, arXiv:1008.3869.[6] E.E. Boos, O. Brandt, D. Denisov, D.P. Denisov, P. Grannis, The top quark (20 yearsafter the discovery), Phys. Usp. et al. (Particle Data Group), Chin. Phys. C, , 100001 (2016)[8] Itzykson C. and Zuber J.-B., Quantum Field Theory (New York: McGraw-Hill, 1985).[9] V.I. Borodulin, R.N. Rogalyov and S.R. Slabospitskii, “CORE 3.1 (COmpendium of RE-lations, Version 3.1),” arXiv:1702.08246 [hep-ph].9 ppendix
Here we present the Feynman rules for the anomalous FCNC interactions of t -quarks due todimension-seven operators. All vertices contain the common factor Λ − and the notation foranomalous couplings is used: κ i = ξ γ + ζ γ γ ; λ i = ξ g + ζ g γ The interaction with a photon u t p ¯ u q p Aq e q κ p µ p ν [ q µ g να − q ν g µα ] A α u t p ¯ u q p A A e q κ (cid:104) ( q + q ) g αβ − q α ( q + q ) β − ( q + q ) α q β (cid:105) A α A β e q κ (cid:104) ( q q ) g αβ − q β q α (cid:105) A α A β e q κ ε µναβ q µ q ν A α A β u t p ¯ u q p AB a e q g s κ t a (cid:104) (( q + q ) q ) g αβ − ( q + q ) α q β (cid:105) A α B β a e q g s κ t a (cid:104) ( q q ) g αβ − q β q α (cid:105) A α B β a e q g s κ t a ε µναβ q µ q ν A α B β a The interaction with a gluon u t p ¯ u q p B a q g s λ p µ p ν [ q µ g να − q ν g µα ] B αa u t p ¯ u q p B b B a g s W abgg B αa B βb ; W abgg = λ t a t a (cid:104) (( p q ) − ( p q )) g αβ − p α p β + p α p β − q α p β + p α q β (cid:105) λ t b t a (cid:104) (( p q ) − ( p q )) g αβ + p α p β − p α p β + q α p β − p α q β (cid:105) λ (cid:0) δ ab / t k d kab (cid:1) (cid:104) ( q q ) g αβ − q α q β (cid:105) λ (cid:2)(cid:0) δ ab / t k d kab (cid:1) q µ q ν + 2 it k f kab p µ p ν (cid:3) ε µ ναβ λ (cid:0) δ ab / t k d kab (cid:1) ε µ ναβ q µ q ν u t p ¯ u q p AB a e q g s λ t a (cid:2) (( q + q ) q ) g αβ q α − q α ( q + q ) β (cid:3) A α B β a e q g s λ t a (cid:16) ( q q ) g αβ − q α q β (cid:17) A α B β b e q g s ( λ + λ ) t a ε µ ναβ q µ q ν A α B β aa