Renormalization and Effective Field Theory

out of fields appearing in the theory. Λ is the characteristic energy scale of theory and R is the cutoff scale chosen just below Λ. Since the mass dimension of Lagrangian is 4, operators having mass dimension greater than 4 are numerically suppressed by powers of fundamental scale (i.e., Λ) of the theory. The higher the dimension of the operator, the more it will be suppressed. Thus, the lower dimensional operators are the important ones and the higher dimensional ones can be neglected. Physics from distances smaller thanR⁻¹ (i.e., contributions from scales higher thanR or the high energy interactions) is contained in the Wilson coefficientsCi and that from distances greater than that (i.e., the low energy interactions) are accounted for by the operators. This approach of expressing the Lagrangian in this form is called operator product expansion (OPE) [6, 17–23]. The Wilson coefficients and the local operators, both depend on the renormalization scale R and renormalization scheme. The scale and scheme dependence of Ci’s must be cancelled by the similar dependence of operators making the physical observables independent of them. The cutoff (R) is generally chosen to be of the order of the mass of the decaying hadron (i.e., O(m_b), O(m_c) andO(1−2 GeV) forB,D and K mesons). Depending on the precision goal one can truncate the series and thus, only a finite number of operators and couplings need to be retained.

The similarity of FCNC processes, in decays and mixing, allows us to describe almost all of them in terms of four fermion local operators. At tree level, the leading order effective Lagrangian for the four fermion process can be written as:

L^{ef f}_4f = G√F

2J_Lµ⁻J_L^µ+ ≡ G√F

2

∑

i

CiQi (1.16)

where, G_F

√2 ≡ g₂²

8M_W² ,GF being the Fermi constant.

For four-fermion interaction, since each fermion line has a mass dimension of 3/2, so the operators involved in this case should have at least dimension 6. So, a M_W⁻² scaling law for Wilson coefficients at leading order is needed. Thus, the non-local product of two charged current interactions can be expanded into a series of local operators, whose contributions are weighted by the effective coupling constants. In the SM, the dimension-6 operators can be classified in 6 categories. In the notation of [17], the ones which play a dominant role in the phenomenology of weak decays of B mesons are given as follows:

1.3. Renormalization and Effective Field Theory 13 Current-Current Operators (Figure 1.2(a))

Q1 = (¯s_αLγ_µc_βL) (¯c_βLγ^µb_αL) (1.17) Q2 = (¯s_Lγ_µc_L) (¯c_Lγ^µb_L) (1.18)

QCD-Penguin Operators (Figure 1.2(b)) Q3 = (¯sLγµbL) ∑

q=u,d,s,c,b

(¯qLγ^µqL) (1.19)

Q4 = (¯sαLγµb_βL) ∑

q=u,d,s,c,b

(¯q_βLγ^µqαL) (1.20) Q5 = (¯s_Lγµb_L) ∑

q=u,d,s,c,b

(¯q_Rγ^µq_R) (1.21)

Q6 = (¯s_αLγ_µb_βL) ∑

q=u,d,s,c,b

(¯q_βRγ^µq_αR) (1.22)

Electroweak-Penguin Operators (Figure 1.2(c)) Q3Q = 3

2 (¯s_Lγ_µb_L) ∑

q=u,d,s,c,b

e_q (¯q_Rγ^µq_R) (1.23) Q4Q = 3

2 (¯sLαγµbLβ) ∑

q=u,d,s,c,b

eq (¯qRβγ^µqRα) (1.24) Q5Q = 3

2 (¯s_Lγ_µb_L) ∑

q=u,d,s,c,b

e_q (¯q_Lγ^µq_L) (1.25) Q6Q = 3

2 (¯sLαγµbLβ) ∑

q=u,d,s,c,b

eq (¯qLβγ^µqLα) (1.26)

Magnetic-Penguin Operators (Figure 1.2(d)) Q7 = e

16π²mb¯sαLσ^µνbαRFµν (1.27) Q8 = g

16π²m_b¯s_αLσ^µνT_αβ^a b_βR G_µν^a (1.28)

∆F = 2 (∆S = 2 or ∆B = 2) Operators (Figure 1.2(e))

Q(∆S= 2) = (¯sLγµdL) (¯sLγ^µdL) (1.29) Q(∆B= 2) = (¯bLγµdL) (¯bLγ^µdL) (1.30)

b

W

c

c

s b

g W

c

c

s b

u,c,t u,c,t

g q

s

q W

(a) (b)

b

u,c,t u,c,t

γ,Z q

s

q W

b

W W

γ,Z q

s

q u,c,t

(c)

b

u,c,t u,c,t

g

s W

b

u,c,t u,c,t

γ,Z

s W

b

W W

γ,Z

s u,c,t

(d)

d b,s

u,c,t u,c,t

W W

b,s d

d b,s

W W

u,c,t u,c,t

b,s d

b

u,c,t u,c,t

γ,Z l

s

l W

(e) (f)

Figure 1.2: Feynman diagram representations of various operators.

(a) Current-Current (b) QCD Penguins (c) Electroweak Penguins (d) Magnetic Penguins (e) ∆F = 2processes (box processes) and (f ) Semi Leptonic Penguin processes.

1.3. Renormalization and Effective Field Theory 15 Semi-Leptonic Operators (Figure 1.2(f ))

Q9 = e²

16π²(¯s_Lγ_µb_L) (¯lγ^µl) (1.31) Q10 = e²

16π²(¯s_Lγµb_L) (¯lγ^µγ5l) (1.32) Qν¯ν = α_em

4π (¯s_Lγ_µb_L) (¯ν_Lγ^µν_L) (1.33) Qµ¯µ = αem

4π (¯sLγµbL) (¯µLγ^µµL) (1.34) where, the subscripts L and R refer to the left and right-handed components of the fermion field given by qL,R = (1∓γ5/2)q. e and g represent the electromagnetic and strong coupling constants (g₁ and g₃),e_q is the electric charge of the relevant quark and T_αβ^a ≡ λ^a_αβ (a=1,. . . 8) are the SU(3) color generators, α and β being the color indices.

The above set of operators is characteristic for any consideration of the interplay of QCD and electroweak effects. Since the ratio α/α_s (≡ 10⁻²) of the QED and QCD couplings is very small, electroweak penguins are expected to play a minor role in comparison with QCD penguins.

The effective Hamiltonian arising from this set of operators is usually written as:

Hef f = G_F

√2

∑

i

V_CKMⁱ Ci(R)Qi (1.35)

where the sum runs over all the operators andV_CKMⁱ is the suitable CKM factor (for e.g., V_ts^∗V_tb for theb→s transitions). The decay amplitude of a process can be written as:

A(M →F) =⟨F|Hef f|M⟩= G_F

√2

∑

i

V_CKMⁱ Ci(R)⟨F|Qi(R)|M⟩ (1.36)

where, ⟨F|Qi(R)|M⟩ is the hadronic matrix element of Qi between M and F evaluated at the renormalization scale R.

The Wilson coefficients can be computed by calculating the amplitude for simple processes in both full and effective theories and then comparing them. Since the Wilson coefficients are process independent, the values obtained by matching a specific process can then be used for the calculation of amplitude of other processes. At higher energies, Ci’s can be computed using perturbation theory due to QCD asymptotic freedom. Thus,

the Wilson coefficients (Ci’s) at an energy R, can be evaluated by first computing them at a high energy scale (MW), which can then be evolved through renormalization group (RG) [21–23] equations:

µ d

dµCk=γ_kiCi (1.37)

to low energy scale R. The Wilson coefficients (Ci’s) at energyR, can thus be written as:

Ck(R) =U_k,i(R, MW)Ci(MW) (1.38) where,

U_k,i(R, M_W) =T_g exp





g(∫R⁾

g(MW)

γ_ki(g^′) β(g^′) dg^′



 (1.39)

is the evolution function which allows to calculateC(R)’s onceC(M_W)’s is known. C(M_W)’s are the Wilson coefficients at higher energy scaleMW. Tg is an ordering criterion defined in [21, 24]. β(g) = R(dg/dR) ≡ dg/d(lnR) is the RG function which governs the evo- lution of g. γ is the anomalous dimension matrix of the operator involved. Keeping the first two terms in the expansions of γ(g) and β(g), in powers ofg, we get:

γ(g) =γ⁽⁰⁾αs

4π +γ⁽¹⁾ α²_s

16π², β(g) =−β0

g³ 16π² −β1

g⁵ (16π²)² Inserting these expansions in equation 1.39, we obtain:

U(R, M_W) = [

1 +α_s(R) 4π J

] [α_s(M_W) αs(R)

]_P[

1−α_s(M_W) 4π J

]

(1.40) where,

P = γ⁽⁰⁾

2β₀, J = P

β₀β1− γ¹ 2β₀.

The Wilson coefficients will, in general, depend on the masses of the particles, which were integrated out.

The calculation of the operators (and hence the matrix elements) on the other hand re- quire non-perturbative calculations, since they involve long distance contributions. These calculations are mostly done using lattice QCD and some model dependent approaches such as QCD sum rules, hadronic sum rules, 1/N expansion (where, N is the number

1.4. Analysis Methods 17