Lectures notes on Flavour physics

Transcription

1 Lectures notes on Flavour physics UZH, Spring Semester 016 Contents 1 The flavour sector of the SM 1.1 Properties of the CKM matrix Present status of CKM fits Theoretical tools for weak interactions at low energies 8.1 Low-energy effective Larangians Effective Lagrangian for F = amplitudes The gaugeless limit of F = amplitudes Systematic approaches to QCD corrections The matching procedure Matching at one-loop order RG-Improved Perturbation Theory Anomalous dimensions Effective Lagrangians for rare decays Non-leptonic processes Flavor-changing neutral-current processes

2 1 The flavour sector of the SM The Standard Model (SM) Lagrangian can be divided into two main parts, the gauge and the Higgs (or symmetry breaking) sector. The gauge sector is extremely simple and highly symmetric: it is completely specified by the local symmetry Glocal SM = SU(3) C SU() L U(1) Y and by the fermion content, L SM gauge = i=1...3 ψ=q i L...Ei R ψid/ ψ 1 G a 4 µνg a µν 1 a= a=1...3 W a µνw a µν 1 4 B µνb µν. (1) The fermion content consist of five fields with different quantum numbers under the gauge group. 1 Q i L(3, ) +1/6, U i R(3, 1) +/3, D i R(3, 1) 1/3, L i L(1, ) 1/, E i R(1, 1) 1, () each of them appearing in three different replica or flavours (i = 1,, 3). This structure give rise to a large global flavour symmetry of L SM gauge. Both the local and the global symmetries of L SM gauge are broken by the introduction of a SU() L scalar doublet H (the Higgs field). The local symmetry is spontaneously broken by the vacuum expectation value (vev) of H, ( ) φ + H =, H = 1 ( ) 0, (3) v where v is determined by the W boson mass: m W = g v 4 φ 0, v = ( G F ) 1/ 46 GeV. (4) The global flavour symmetry is explicitly broken by the Yukawa interaction of H with the fermion fields: L SM Yukawa = Y ij Q d i LDRH j + Yu ij Q i LURH j c + Ye ij L i LERH j + h.c. (H c = iσ H ). (5) The large global flavour symmetry of L SM gauge, corresponding to the independent unitary rotations in flavour space of the five fermion fields in Eq. (), is a U(3) 5 group. This can be decomposed as follows: G flavour = U(3) 5 = U(1) 5 G q G l, (6) where G q = SU(3) QL SU(3) UR SU(3) DR, G l = SU(3) LL SU(3) ER. (7) 1 The notation used to indicate each field is ψ(a, B) Y, where A and B denote the representation under the SU(3) C and SU() L groups, respectively, and Y is the U(1) Y charge.

3 Three of the five U(1) subgroups can be identified with the total barion and lepton number, which are not broken by L Yukawa, and the weak hypercharge, which is gauged and broken only spontaneously by H = 0. The subgroups controlling flavour-changing dynamics and flavour non-universality are the non-abelian groups G q and G l, which are explicitly broken by Y d,u,e not being proportional to the identity matrix. The diagonalization of each Yukawa coupling requires, in general, two independent unitary matrices, V L Y V R = diag(y 1, y, y 3 ). In the lepton sector the invariance of L SM gauge under G l allows us to freely choose the two matrices necessary to diagonalize Y e without breaking gauge invariance, or without observable consequences. This is not the case in the quark sector, where we can freely choose only three of the four unitary matrices necessary to diagonalize both Y d and Y u. Choosing the basis where Y d is diagonal (and eliminating the right-handed diagonalization matrix of Y u ) we can write Y d = λ d, Y u = V λ u, (8) where λ d = diag(y d, y s, y b ), λ u = diag(y u, y c, y t ), y q = m q v. (9) Alternatively we could choose a gauge-invariant basis where Y d = V λ d and Y u = λ u. Since the flavour symmetry do not allow the diagonalization from the left of both Y d and Y u, in both cases we are left with a non-trivial unitary mixing matrix, V, which is nothing but the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [1, ]. A generic unitary 3 3 [N N] complex unitary matrix depends on three [N(N 1)/] real rotational angles and six [N(N + 1)/] complex phases. Having chosen a quark basis where the Y d and Y u have the form in (8) leaves us with a residual invariance under the flavour group which allows us to eliminate five of the six complex phases in V (the relative phases of the various quark fields). As a result, the physical parameters in V are four: three real angles and one complex CP-violating phase. The full set of parameters controlling the breaking of the quark flavour symmetry in the SM is composed by the six quark masses in λ u,d and the four parameters of V. For practical purposes it is often convenient to work in the mass eigenstate basis of both up- and and down-type quarks. This can be achieved rotating independently the up and down components of the quark doublet Q L, or moving the CKM matrix from the Yukawa sector to the charged weak current in L SM gauge: J µ W quarks = ū i Lγ µ d i L u,d mass basis ū i LV ij γ µ d j L. (10) However, it must be stressed that V originates from the Yukawa sector (in particular by the miss-alignment of Y u and Y d in the SU(3) QL subgroup of G q ): in absence of Yukawa couplings we can always set V ij = δ ij. To summarize, quark flavour physics within the SM is characterized by a large flavour symmetry, G q, defined by the gauge sector, whose only breaking sources are the two Yukawa couplings Y d and Y u. The CKM matrix arises by the miss-alignment of Y u and Y d in flavour space. 3

4 1.1 Properties of the CKM matrix The standard parametrization of the CKM matrix [3] in terms of three rotational angles (θ ij ) and one complex phase (δ) is V = = V ud V us V ub V cd V cs V cb V td V ts V tb = R(s 1 ) R(s 13 ; e iδ ) R(s 3 ) c 1 c 13 s 1 c 13 s 13 e iδ s 1 c 3 c 1 s 3 s 13 e iδ c 1 c 3 s 1 s 3 s 13 e iδ s 3 c 13 s 1 s 3 c 1 c 3 s 13 e iδ s 3 c 1 s 1 c 3 s 13 e iδ c 3 c 13, (11) where c ij = cos θ ij, s ij = sin θ ij, R(θ 1 ) and R(θ 3 ) are real rotational matrices (among the 1 and 3 families, respectively), and R(s 13 ; e iδ ) is R(s 13 ; e iδ ) c 13 0 s 13 e iδ s 13 e iδ 0 c 13. (1) Under the phase field redefintions u i L e iαu i u i L and d i L e iαd i d i L, the CKM elements are transformed as V ij e i(αd j αu i ) V ij. (13) This implies that the moduli of the elements ( V ij ) and the combinations V ai V aj V bi V bj, (14) are phase-convention independent quantities. The off-diagonal elements of the CKM matrix show a strongly hierarchical pattern: V us and V cd are close to 0., the elements V cb and V ts are of order 4 10 whereas V ub and V td are of order The Wolfenstein parametrization, namely the expansion of the CKM matrix elements in powers of the small parameter λ. = V us 0., is a convenient way to exhibit this hierarchy in a more explicit way [4]: 1 λ λ Aλ 3 (ϱ iη) V = λ 1 λ Aλ + O(λ 4 ), (15) Aλ 3 (1 ϱ iη) Aλ 1 where A, ϱ, and η are free parameters of order 1. Because of the smallness of λ and the fact that for each element the expansion parameter is actually λ, this is a rapidly converging expansion. The Wolfenstein parametrization is certainly more transparent than the standard parametrization. However, if one requires sufficient level of accuracy, the terms of O(λ 4 ) and 4

5 Figure 1: The CKM unitarity triangle. O(λ 5 ) have to be included in phenomenological applications. This can be achieved in many different ways, according to the convention adopted. The simplest (and nowadays commonly adopted) choice is obtained defining the parameters {λ, A, ϱ, η} in terms of the angles of the exact parametrization in Eq. (11) as follows [5]: λ. = s 1, Aλ. = s3, Aλ 3 (ϱ iη). = s 13 e iδ. (16) The change of variables {s ij, δ} {λ, A, ϱ, η} in Eq. (11) leads to an exact parametrization of the CKM matrix in terms of the Wolfenstein parameters. Expanding this expression up to O(λ 5 ) leads to where 1 1 λ 1 8 λ4 λ + O(λ 7 ) Aλ 3 (ϱ iη) λ + 1 A λ 5 [1 (ϱ + iη)] 1 1 λ 1 8 λ4 (1 + 4A ) Aλ + O(λ 8 ) Aλ 3 (1 ϱ i η) Aλ + 1 Aλ4 [1 (ϱ + iη)] 1 1 A λ 4 (17) ϱ = ϱ(1 λ ) + O(λ4 ), η = η(1 λ ) + O(λ4 ). (18) The advantage of this generalization of the Wolfenstein parametrization is the absence of relevant corrections to V us, V cd, V ub and V cb, and a simple change in V td, which facilitate the implementation of experimental constraints. The unitarity of the CKM matrix implies the following relations between its elements: I) VikV ki = 1, II) VikV kj i. (19) k=1...3 k=1...3 These relations are a distinctive feature of the SM, where the CKM matrix is the only source of quark flavour mixing. Their experimental verification is therefore a useful tool to set bounds, or possibly reveal, new sources of flavour symmetry breaking. Among the relations of type II, the one obtained for i = 1 and j = 3, namely V ud V ub + V cd V cb + V td V tb = 0 (0) 5

6 Figure : Allowed region in the ϱ, η plane, from [6] (see also [7]). Superimposed are the individual constraints from charmless semileptonic B decays ( V ub ), mass differences in the B d ( m d ) and B s ( m s ) systems, CP violation in the neutral kaon system (ε K ) and in the B d systems (sin β), the combined constrains on α and γ from various B decays. or V ud V ub V cd V cb + VtdV tb + 1 = 0 [ ϱ + i η] + [(1 ϱ) i η] + 1 = 0, (1) V cd Vcb is particularly interesting since it involves the sum of three terms all of the same order in λ and is usually represented as a unitarity triangle in the complex plane, as shown in Fig. 1. It is worth to stress that Eq. (0) is invariant under any phase transformation of the quark fields. Under such transformations the triangle in Fig. 1 is rotated in the complex plane, but its angles and the sides remain unchanged. Both angles and sides of the unitary triangle are indeed observable quantities which can be measured in suitable experiments. 6

7 1. Present status of CKM fits The values of V us and V cb, or λ and A in the parametrization (17), are determined with good accuracy from K πlν and B X c lν decays, respectively. According to recent phenomenological analyses [6, 7] their numerical values are λ = 0.5 ± 0.001, A = 0.8 ± () Using these results, all the other constraints on the elements of the CKM matrix can be expressed as constraints on ϱ and η (or constraints on the CKM unitarity triangle in Fig. 1). The resulting constraints are shown in Fig.. As can be seen, they are all consistent with a unique value of ϱ and η: ρ = 0.13 ± 0.0, η = 0.35 ± (3) The consistency of different constraints on the CKM unitarity triangle is a powerful consistency test of the SM in describing flavour-changing phenomena. From the plot in Fig. it is quite clear, at least in a qualitative way, that there is little room for non-sm contributions in flavour changing transitions. 7

8 Theoretical tools for weak interactions at low energies.1 Low-energy effective Larangians The decays of B, D, and K mesons are processes which involve at least two different energy scales: the electroweak scale, characterized by the W boson mass, which determines the flavor-changing transition at the quark level, and the scale of strong interactions Λ QCD, related to the hadron formation. The presence of these two widely separated scales makes the calculation of the decay amplitudes starting from the full SM Lagrangian quite complicated: large logarithms of the type log(m W /Λ QCD ) may appear, leading to a breakdown of ordinary perturbation theory. This problem can be substantially simplified by integrating out the heavy SM fields (W and Z bosons, as well as the top quark) at the electroweak scale, and constructing an appropriate low-energy effective field theory (EFT) where only the light SM fields appear. The weak effective Lagrangians thus obtained contains local operators of dimension six (and higher), written in terms of light SM fermions, photon and gluon fields, suppressed by inverse powers of the W mass. To be concrete, let s consider the example of charged-current semileptonic weak interactions. The basic building block in the full SM Lagrangian is where full SM LW = g J µ W (x)w µ + (x) + h.c., (4) J µ W (x) = V ij ū i L(x)γ µ d j L(x) + ē j L(x)γ µ ν j L(x) (5) is the weak charged current already introduced in Eq. (10). Integrating out the W field at the tree level we contract two vertexes of this type generating the non-local transition amplitude it = i g which involves only light fields. Here D µν (x, m W ) is the W propagator in coordinate space: expanding it in inverse powers of m W, D µν (x, m W ) = d 4 xd µν (x, m W ) T [ J µ W (x), J ν W (0) ], (6) d 4 q e iq x ig µν + O(q µ, q ν ) (π) 4 q m W + iε = δ(x) ig µν m W +..., (7) the leading contribution to T can be interpreted as the tree-level contribution of the following effective local Lagrangian L (0) eff = 4G F g µν J µ W (x)j ν W (x), (8) 8

9 where G F / = g /(8m W ) is the Fermi coupling. If we select in the product of the two currents one quark and one leptonic current, L semi lept eff = 4G F V ij ū i L(x)γ µ d j L(x) ν L (x)γ µ e L (x) + h.c., (9) we obtain an effective Lagrangian which provides an excellent description of semileptonic weak decays. For B decays the neglected terms in the expansion (7) correspond to corrections of O(m B/m W ) to the decay amplitudes. In principle, these corrections could be taken into account by adding appropriate dimension-eight operators in the effective Lagrangian. However, in most cases they are safely negligible. The case of charged semileptonic decays is particularly simple since we can ignore QCD effects: the operator (9) is not renormalized by strong interactions (a more detailed explanation of why this happens will be presented in the next lecture). This is the main reason we can determine with high precision a few moduli of CKM matrix elements (in particular V cb and V us ) from semileptonic decays of B and K mesons. The situation is more complicated in four-quark interactions and flavor-changing neutralcurrent processes, where QCD corrections and higher-order weak interactions cannot be neglected, but the basic strategy is the same. First of all we need to identify a complete basis of local operators, that includes also those generated beyond the tree level. In general, given a fixed order in the 1/m W expansion of the amplitudes, we need to consider all operators of corresponding dimension (e.g. dimension six at the first order in the 1/m W expansion) compatible with the symmetries of the system. Then we must introduce an artificial scale in the problem, the renormalization scale µ, which is needed to regularize QCD (or QED) corrections in the EFT. The effective Lagrangian for generic F = 1 processes assumes the form L F =1 = 4 G F C i (µ)q i (30) where the sum runs over the complete basis of operators. As explicitly indicated, the effective couplings C i (µ) (known as Wilson coefficients) depend, in general, on the renormalization scale, similarly to the scale dependence of the QCD coupling α s (µ). The dependence from this scale cancels when evaluating the matrix elements of the effective Lagrangian for physical processes, that we can generically indicate as i M(i f) = 4 G F C i (µ) f Q i (µ) i. (31) The scale µ acts as a separator of short- and long-distance virtual corrections: short-distance effects are included in the C i (µ), whereas long-distance effects are left as explicit degrees of freedom in the EFT. We denote by F = 1 processes, the transitions with change of flavor by one unit, such as e.g. b sūu. Here we the initial state has b-flavor=1 and s-flavor=0, whereas the final state has b-flavor=0 and s-flavor=1, hence F b = F s = 1. i 9

10 In practice, the problem reduces to the following three well-defined and independent steps: 1. the evaluation of the initial conditions of the C i (µ) at the electroweak scale (µ m W ), that is done by an approrpiate matching procedure between the effective theory and the full theory;. the evaluation of the renormalization-group equations (RGE) which determine the evolution of the C i (µ) from the electroweak scale down to the energy scale of the physical process (µ m B ); 3. the evaluation of the matrix elements of the effective Lagrangian for the physical hadronic processes (which involve energy scales from the meson masses down to Λ QCD ). The first step is the one where New Physics (NP), namely physics beyond the SM, may contribute: if we assume NP is heavy, it may modify the initial conditions of the Wilson coefficients at the high scale, while it cannot affect the following two steps. While the RGE evolution and the hadronic matrix elements are not directly related to NP, they may influence the sensitivity to NP of physical observables. In particular, the evaluation of hadronic matrix elements is potentially affected by non-perturbative QCD effects: these are often a large source of theoretical uncertainty which can obscure NP effects. RGE effects do not induce sizable uncertainties since they can be fully handled within perturbative QCD; however, the sizable logs generated by the RGE running may dilute the interesting shortdistance information encoded in the value of the Wilson coefficients at the high scale. As we will discuss in the following, only in specific observables these two effects are small and under good theoretical control. A deeper discussion about the construction of low-energy effective Lagrangians, with a more detailed discussions of the first two steps mentioned above will be presented in the lecture n.3.. Effective Lagrangian for F = amplitudes Among four-quark interactions, a particularly interesting case is those of F = transitions, namely amplitudes with a double change of flavor (e.g. F b = F s = ). The amplitudes control the mixing of neutral mesons (e.g. Bs 0 B s 0 mixing). The effective Lagrangian relevant for Bd 0 B d 0 and Bs 0 B s 0 mixing can be conventionally written as L SM B= = q=d,s F = Cq ( b L γ µ q L bl γ µ q L ), (3) F = where the leading contribution to the Wilson coefficient Cq can be determined by computing the box diagrams in Fig. 3 (in the limit of small external momenta). The explicit 10

11 Figure 3: Box diagrams contributing to B d - B d mixing in the unitary gauge. Figure 4: Box diagrams contributing to B d - B d in the gaugeless limit. F = calculation for the coefficient Cd yields F = C = 1 loop d q =u,c,t q=u,c,t (Vq bv q d)(vqbv qd )F (x q, x q ) (33) (V tbv td ) [F (x t, x t ) + F (0, 0) F (0, x t )] (34) = (V tbv tq ) G F 4π m W S 0 (x t ) (35) where x q = m q/m W. The intermediate result in (33) follows from neglecting all quark masses but for m t, and using the unitarily of the CM matrix (that implies VubV ud + VcbV cd + VtbV td = 0). The latter implies in particular that the mass-independent contribution to the amplitude of up, charm, and top-quarks cancel (GIM mechanism [8]). The explicit expression of the loop function is and, as expected, vanishes in the limit x t 0. S 0 (x t ) = 4x t 11x t + x 3 t 4(1 x t ) 3x3 t ln x t (1 x t ) 3 (36).3 The gaugeless limit of F = amplitudes An interesting aspect of the loop function in Eq. (36) is the fact that it diverges in the limit m t /m W. This behavior is apparently strange: it contradicts the expectation that contributions of heavy particles decouple, at low energies, in the limit where their masses increase. The origin of this effect can be understand by noting that the leading contribution to the F = amplitude is generated only by the Yukawa interaction. This leading contribution can be better isolated in the gaugeless limit of the SM, i.e. if we send to 11

12 zero the gauge couplings. In this limit m W 0 and the derivation of the effective Lagrangian discussed in Sect..1 does not make sense. However, the leading contribution to the effective Lagrangians for F = processes remains unaffected. Indeed, the leading contribution to these processes is generated by Yukawa interactions of the type in Fig. 4, where the scalar fields are the Goldstone-bosons components of the Higgs field (which are not eaten up by the W in the limit g 0). Since the top is still heavy, we can integrate it out (i.e. we can compute the amplitude in the limit of a heavy top), obtaining the following result for L B= : L SM B= gi 0 = [(Y uyu ) bq ] ( b 18π m L γ µ q L ) t = G F m t 16π (V tbv tq ) ( b L γ µ q L ). (37) Taking into account that S 0 (x) x/4 for x, it is easy to verify that this result is equivalent to the one in Eq. (36) in the large m t limit. The last expression in Eq. (37), which holds in the limit where we neglect the charm Yukawa coupling, shows that the decoupling of the amplitude with the mass of the top is compensated by four powers of the top Yukawa coupling at the numerator. The divergence for m t can thus be understood as the divergence of one of the fundamental couplings of the theory. Note also that in the gaugeless limit there is no GIM mechanism. The contributions of the various up-type quarks inside the loops do not cancel each other: they are directly weighted by the corresponding Yukawa couplings, and this is why the top-quark contribution is the dominant one. This exercise illustrates the key role of the Yukawa coupling in determining the main properties flavour physics within the SM, as advertised at the beginning of this lecture. It also illustrates the interplay of flavour and electroweak symmetry breaking in determining the structure of short-distance dominated flavour-changing processes in the SM. 1

13 3 Systematic approaches to QCD corrections 3.1 The matching procedure The leading effective Lagrangian in (8) can be derived integrating-out the W field in the path integral formulation. However, the same method cannot be used to systematically get rid of the high-frequency modes of the quark and gluon fields. Two problems arise: first, the path integral is no longer Gaussian once QCD effects are taken into account; second, the strong interactions are not perturbative at low energy due to the confinement of colored particles into hadrons. One deals with these difficulties using a general procedure called matching, which consists of the following steps: 1. List all possible gauge-invariant operators of a given dimension allowed by the symmetries and quantum numbers associated with a given problem. The dimension of the operators is determined by the accuracy goal of the calculation, generally d = 6 is sufficient for most calculations in flavor physics.. Write down the effective Lagrangian with undetermined couplings C i, in our case L eff weak = 4 G F i C i Q i. (38) Note that the Wilson coefficients C i are process independent, namely the same coefficients arise in the calculation of many different weak-interaction amplitudes. 3. Determine the values of the coefficients C i (µ) such that A n = f n L SM i n = i C i (µ) f n Q i i n + higher power corrections (39) to a given order in perturbation theory (we need to study as many matrix elements as necessary to determine all the independent C i ). The crucial point is that, despite non-perturbative effects at low energies, we can determine the C i perturbatively at high energies thanks to asymptotic freedom. By construction, the effective theory has the same infrared (IR) strucutre of the full theory. Differences arise only at high energy, where the effective theory misses the high-frequency modes of the full theory. The corresponding contributions are absorbed into the Wilson coefficients. As along as the theory is perturbative (weakly coupled) at and above µ, the Wilson coefficients are calculable in perturbation theory. It follows that the Wilson coefficients in the effective Lagrangian are insensitive to any infrared physics unlike the amplitudes themselves. As a result, matching calculations can be done using arbitrary IR regulators and working with free quark and gluon states. Obviously, this is a great advantage for actual calculations in QCD. 13

14 In the specific case of the four-fermion weak Lagrangian, it is useful to note the following facts: i) four fermion fields already make dimension d = 6, so no derivatives, or extra fields, or external mass terms are allowed at this order; ii) weak interactions only involve left-handed fermion fields and chirality is preserved in strong-interaction processes once we set m q = 0; iii) for the quark bilinears ψ L Γψ L with Γ an element of the Dirac basis, only the possibility Γ = γ µ remains; iv) operators must be gauge invariant (in particular, color singlets) and Lorentz invariant. According to these rules, it is easy to realise that the semileptonic Lagrangian in (9) contains a single effective operator at d = 6. The situation is more involved for hadronic decays. For simplicity, let s consider hadronic processes without penguin topologies, and in particulare the quark transition b u c s. In this case the symmetry arguments listed above allow two operators differing in their color structure: L eff = 4G [ F Vcs V ub C1 (µ) s j Lγ µ c j L ū i Lγ µ b i L + C (µ) s i Lγ µ c j L ū j Lγ µ bl] i, (40) The tree-level matching condition implies C 1 = 1 + O(α s ) and C = O(α s ). Using a Fierz rearrangement, the second operator above can also be written as ū j Lγ µ c j L s i Lγ µ b i L. Note also that s L γ µ t a c L ū L γ µ t a b L = 1 si Lγ µ c j L ū j Lγ µ b i L 1 N c s L γ µ c L ū L γ µ b L (41) gives nothing new. Here t a are the generators of color SU(3). 3. Matching at one-loop order The one-loop QCD corrections to a generic semileptonic transition in both the full theory and the effective theory are shown by the first diagram, in each row in in Figure 5. In semileptonic processes diagrams b and c are absent and one finds that the radiative corrections in the two theories are identical. As a result, the matching procedure is trivial and the semileptonic Lagrangian (9) is not renormalised. In the b u c s case all six diagrams contribute. For diagrams a the result is the same in the two teories; however, this is not the case for diagrams b and c, where the expansion of the W propagator and integration over loop momenta do not commute. The reason is that in the last two diagrams in the top row the loop momentum flows through the W -boson propagator. Rather than going through the full calculation in detail, we just note that d D p 1 MW p f(p) 1 MW d D p ( 1 + p M W +... ) f(p). (4) While the left-hand side is non-analytic in M W, the right-hand side is obviously analytic. Differences between the two integrals arise from the region of large loop momenta where p M W. But for such large momenta QCD is weakly coupled. Perturbation theory can thus be trusted to compute the differences between the matrix elements in the two theories, 14

15 g W g W W g (a) g (b) (c) g g (a) (b) (c) Figure 5: One-loop QCD corrections to four-fermion weak-interaction amplitudes, both in the full theory (top row) and in the low-energy effective theory (bottom row). For semileptonic processes only the diagrams a are possible, whereas for b u c s all six diagrams contribute. which are accounted for by the Wilson coefficient functions. diagrams in Figure 5 gives (in the MS subtraction scheme) [9] Explicit calculation of the C 1 (µ) = ( α s (µ) ln M W 11 ) + O(α N c 4π µ 6 s), C (µ) = 3 α ( s(µ) ln M W 11 ) + O(α 4π µ 6 s). (43) Some important comments are in order: IR regulators (such as quark and gluon masses, external momenta, etc.) present in intermediate steps of the calculation in both theories cancel in the results for the Wilson coefficients C i. Matrix elements in the effective theory are often more singular than those in the full theory (which are ultraviolet (UV) finite in the present case) and require additional UV subtractions (operator renormalization). This gives rise to the renormalization-scale and -scheme dependence of the Wilson coefficients. The physical reason for this is that the mass M W acts as an UV regulator in the box diagrams of the full theory. Roughly speaking, the logarithms in the results for the 15

16 Wilson coefficients arise as follows: 1 + α s ln M W ( = 1 + α p }{{ s ln M ) ( ) W 1 + α µ } s ln µ +..., (44) p }{{}}{{} full theory C(µ) Q(µ) where the expression on the left is the matrix element in the full theory (which is UV finite and regularized in the IR by an off-shell momentum p ), while the expression on the right is the product of a Wilson coefficient and a matrix element in the effective theory. The EFT matrix element has the same dependence on the IR cutoff as the matrix element in the full theory, while all reference to the fundamental scale M W resides in the Wilson coefficient. Another way of representing this result is in the form of logarithmic integrations: M W dk M p k = W dk µ k + µ dk p k. (45) In general, the Wilson coefficients absorb the high-frequency contributions of the loop integrals, while the low-frequency contributions reside in the EFT matrix elements. 3.3 RG-Improved Perturbation Theory There are some important technical aspects which we have ignored in the discussion of the previous lecture. Recall the one-loop matching results for the Wilson coefficients C 1 and C from (43): C 1 (µ) = ( α s (µ) ln M W 11 ) + O(α N c 4π µ 6 s), C (µ) = 3 α ( s(µ) ln M W 11 ) + O(α 4π µ 6 s). (46) Ideally, we would like to integrate out all high-frequency modes perturbatively and then evaluate the remaining EFT matrix elements Q i (µ) at some low scale µ few GeV, below which perturbation theory becomes untrustworthy. The computation of these matrix elements must use a non-perturbative approach such as lattice QCD, heavy-quark expansions, or chiral perturbation theory. A glance at the above equations shows a potential problem: the expansion parameter is not αs αs 0.1, but ln M W π π µ 0.8. The problem is indeed generic: in the presence of widely separated scales M µ, perturbation theory often involves powers of α s ln M rather than powers of α µ s. Such large logarithmic terms must be resummed to all orders. The general solution to the problem of large logarithms is called renormalization-group (RG) improved perturbation theory. It provides a reorganization of perturbation theory 16

17 in which α s ln M µ is treated as an O(1) parameter, while α s 1. Large logarithms are resummed to all orders in perturbation theory by solving RG equations. The nomenclature of RG-improved perturbation theory is as follows: At leading order (LO) all terms of the form (α s ln M µ )n with n = 0,..., are resummed. The result is an O(1) contribution to the Wilson coefficient functions. At next-to-leading order (NLO), one also resums terms of the form α s (α s ln M µ )n, all of which count as O(α s ), and so on. Note that in cases where the term with n = 0 is absent (such as for C ), there may be O(1) effects after resummation that not seen at tree level in perturbation theory. In order to perform such resummations, it is useful to discuss in more detail the renormalization of the composite operators in the effective Lagrangian. 3.4 Anomalous dimensions Consider a complete set (a basis) {Q i (µ)} ; i = 1,..., n. (47) of operators of dimension δ allowed by the symmetries (quantum numbers, etc.) of a problem. Recall that by changing the scale µ one reshuffles terms from the matrix elements Q i into the coefficients C i, leaving the result for any observable unchanged, i.e. n n A = C i (µ) Q i (µ) = C i (µ δµ) Q i (µ δµ). (48) i=1 i=1 The fact that physical observables are scale independent implies that d d ln µ n C i (µ) Q i (µ) = 0. (49) i=1 Since the operator basis is complete, we can expand the logarithmic derivative of the operator matrix elements in terms of the same basis operators. We write d n d ln µ Q i(µ) γ ij (µ) Q j (µ). (50) j=1 If there is more than one operator present, we say that the operators mix under scale variation. The dimensionless coefficients γ ij measure the incremental change under scale variation and are free of large logarithms. They are called anomalous dimensions. Using this definition, it follows from (49) that n j=1 [ d d ln µ C j(µ) ] n C i (µ) γ ij (µ) Q j (µ) = 0. (51) i=1 17

18 Since by assumption the operators Q i are linearly independent, we conclude that d d ln µ C j(µ) n C i (µ) γ ij (µ) = 0. (5) i=1 This is the RG equation obeyed by the Wilson coefficient functions. In matrix notation, we can rewrite this equation as d d ln µ C(µ) = ˆγ T (µ) C(µ). (53) The dimensionless anomalous-dimension matrix ˆγ depends on the scale µ only through the running coupling α s (µ). Changing variables from ln µ to α s (µ), we find d C(µ) dα s (µ) = ˆγT (α s (µ)) C(µ) β(α s (µ)), (54) where β = dα s (µ)/d ln µ is the QCD β-function. The initial condition for the solution of the RG equation is set by the values C(M W ) of the Wilson coefficients at the weak scale. Equation (54) has the same structure as the Heisenberg equation for the time dependence of the Hamiltonian in quantum field theory. The unique solution to this equation is C(µ) = T α exp [ α s(µ) α s(m W ) dα ˆγT (α) β(α) ] C(M W ). (55) The matrix exponential is defined through its Taylor expansion, and the symbol T α means an ordering such that ˆγ T (α) with larger α stands to the left of those with smaller α. Such an ordering prescription is necessary because, in general, the matrices ˆγ T (α) at different α values do not commute. We now perform a (controlled) perturbative expansion of the quantities C(M W ), ˆγ(α), and β(α) entering the general solution, all of which are free of large logarithms. Consider, for simplicity, the case of a single Wilson coefficient (n = 1, no mixing). Writing γ(α s ) = γ 0 α s 4π + O(α s), we find the leading-order solution β(α s ) = α s C(µ) = To see that this sums the leading logarithms, note that ( ) γ 0 ( αs β (µ) 0 α s 1 + β 0 α s (M W ) 4π ln M W ) γ 0 β 0 γ 0 = 1 µ [ ] α s β 0 4π + O(α s), C(M W ) = 1 + O(α s ), (56) ( ) γ 0 αs β (µ) 0 [ 1 + O(αs ) ]. (57) α s (M W ) α s 4π ln M W µ + O(α s ln M W µ ). (58) 18

19 It is straightforward to go to higher orders in the expansion in α s. For the case of a single operator, the NLO solution reads C(µ) = ( ) γ 0 αs β (µ) 0 [1 + α ] s(µ) α s (M W ) α s (M W ) S + c 1 + O(α α s (M W ) 4π 4π s), (59) where and we have expanded S = γ ( 0 γ1 β ) 1, (60) β 0 γ 0 β 0 ( ) αs n+1 ( ) αs n+1 γ(α s ) = γ n, β(αs ) = α s β n, n=0 4π n=0 4π ( ) n αs (M W ) C(M W ) = 1 + c n. (61) n=1 4π The generalization to the case of operator mixing is discussed in great detail in the comprehesive review by Buchalla, Buras, and Lautenbacher [9]. The systematics of RG-improved perturbation theory is summarized in the following table: Order γ, β C(M W ) LO 1-loop tree-level NLO -loop 1-loop NNLO 3-loop -loop The LO approximation is really only good for illustration purposes. At NLO we achieve the same accuracy as in the case of a conventional one-loop calculation for a single-scale problem. Note, however, that two-loop anomalous dimensions are required at this order. The NNLO approximation is the state of the art for many applications, where high precision is of concern. 19

20 4 Effective Lagrangians for rare decays In the previous lecture we have discussed in some detail the derivation of the effective Lagrangian relevant to describe the b u c s decay. The latter is the simplest case where we encounter the phenomenon of operator mixing. In rare processes of the type b s qq or b s l + l the operator basis is more complicated due to the occurrence of a new class of diagrams, the so-called penguin-diagrams. The conceptual steps necessary to derive the corresponding effective Lagrangians are the same as those discussed in the previous lecture, but the procedure is more lengthy (and the operator list is longer) given the appearance of those new diagrams. In the following we will briefly review the structure of these effective Hamiltonians without discussing their detailed derivation. 4.1 Non-leptonic processes Let s start from non-leptonic processes where the underlying partonic transition is is b s + qq. In this case the relevant effective Lagrangian can be written as = 4 G F λ s 10 q C i (µ)q q i (µ) λ s t C i (µ)q i (µ), (6) L non lept b s q=u,c i=1, where λ s q = V qbv qs, and the operator basis is i=3 Q q 1 = b α Lγ µ ql α q Lγ β µ s β L, Q q = b α Lγ µ q β L q Lγ β µ s α L, Q 3 = b α Lγ µ s α L q q Lγ β µ q β L, Q 4 = b α Lγ µ s β L q q Lγ β µ ql α, Q 5 = b α Lγ µ s α L q q Rγ β µ q β R, Q 6 = b α Lγ µ s β L q q Rγ β µ qr α, Q 7 = b 3 α Lγ µ s α L q e q q Rγ β µ q β R, Q 8 = 3 b α Lγ µ s β L q e q q Rγ β µ qr α, Q 9 = b 3 α Lγ µ s α L q e q q Lγ β µ q β L, Q 10 = b 3 α Lγ µ s β L q e q q Lγ β µ ql α, (63) with {α, β} and e q denoting color indexes the electric charge of the quark q, respectively. Out of these operators, only Q c 1 and Q u 1 are generated at the tree-level by the W exchange. Indeed, comparing with the tree-level structure in (8), we find C u,c 1 (m W ) = 1 + O(α s, α), C u,c 10(m W ) = 0 + O(α s, α). (64) However, after including RGE effects and running down to µ m b, both C u,c 1 and C u,c become O(1) and running further down to µ 1 GeV also C 3 6 become O(1). The operators Q 3 10, often called penguin operators, are present only in processes of the type b s(d)+ qq or s d+ qq, where one-loop topologies of the type in Fig. 6 are allowed. Such operators are not present in processes of the type b cūd, that involve four different quark species. The operators Q 3 6 are generated at the one-loop level by the QCD penguins (Fig. 6 left), while Q 7 10 are generated by electroweak (EW) diagrams of the type in Fig. 6 0

21 Figure 6: One-loop penguins diagrams: QCD penguins (left) and EW penguins (right). right and by related box diagrams. These digrams involve all three types of up-type quarks inside the loops. However, since q=u,c,t VqbV qs = 0, we can always eliminate one CKM combination, as we have already seen in the case of the F = amplitudes. Moreover, if the amplitude is regular in the limit of vanishing up-quark mass, the mass-independent part of the loop amplitude cancels leaving a sizable contribution only in the part of the amplitude proportional to VtbV ts (GIM mechanism [8]): A peng = q=u,c,t ( m VqbV qs A q [ ( m = VcbV cs A c ( VtbV mt ts F m W ) m W ) ( )] [ ( ) ( )] m A u m + V m W m tbv ts A t m A u W m W m W ). (65) This is why in Eq. (6) the coefficients C 3 10 are multiplied only by the CKM combination λ s t. The coefficients of the penguin operators at the electroweak scale are potentially more sensitive to NP with respect to the initial values of the other four-quark operators (being free from tree-level SM contributions). However, it is hard to distinguish their contribution from those of the other four-quark operators in non-leptonic processes. Moreover, the relative contribution from long-distance physics (running down from m W to m b ) is sizable and dilute the interesting short-distance information at low energies. 4. Flavor-changing neutral-current processes For b s transitions with a photon or a lepton pair in the final state, denoted Flavorchanging neutral-current processes (or FCNC), additional dimension-six operators must be 1

22 included in the basis, L rare b s = L non lept b s + 4 G F λ s t (C 7γ Q 7γ + C 8g Q 8g + C 9V Q 9V + C 10A Q 10A ), (66) where Q 7γ = e 16π m b b α Rσ µν F µν s α L, Q 8g = g s 16π m b b α Rσ µν G A µνt A s α L, Q 9V = 1 b α Lγ µ s α L lγ µ l, Q 10A = 1 b α Lγ µ s α L lγ µ γ 5 l, (67) and G A µν (F µν ) is the gluon (photon) field strength tensor. The initial conditions of these operators are particularly sensitive to NP and, contrary to non-leptonic processes, in this case is easier to isolate their contribution in low-energy observables. The cleanest case is C 10A, which do not mix with any of the four-quark operators listed above and is not renormalised by QCD corrections: 3 [ C10A(m SM W ) = g x t 4 xt + 3x ] t 8π 8 1 x t (1 x t ) ln x t, x t = m t m W. (68) NP effects at the TeV scale could easily modify this result, and this deviation would directly show up in low-energy observables sensitive to C 10A, such as the branching ratio of the rare decays B s,d l + l. We finally note that while the operators in Eqs. (63) and (67) form a complete basis within the SM, this is not necessarily the case beyond the SM. In particular, within specific scenarios also right-handed current operators (e.g. those obtained from (67) for q L(R) q R(L) ) may appear. n. 3). 3 This happens because the corresponding operator has a vanishing anomalous dimension (see lecture

23 References [1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). [] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 65 (1973). [3] L. L. Chau and W. Y. Keung, Phys. Rev. Lett. 53 (1984) 180. [4] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). [5] A. J. Buras, M. E. Lautenbacher, and G. Ostermaier, Phys. Rev. D 50, 3433 (1994) [arxiv:hepph/ ]. [6] J. Charles et al. [CKMfitter Collaboration], [7] M. Bona et al. [UTfit Collaboration], [8] S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D, 185 (1970). [9] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68 (1996) 115 [arxiv:hepph/951380]. 3