Quantum Chromodynamics

Quantum Chromodynamics∗ Andrey Grozin Institut für Theoretische Teilchenphysik, Karlsruher Institut für Technologie, Karlsruhe, and Budker Institute of Nuclear Physics SB RAS, Novosibirsk Abstract The classical Lagrangian of chromodynamics, its quantization in the perturbation theory framework, and renormalization form the subject of these lectures. Symmetries of the theory are discussed. The dependence of the coupling constant αs on the renormalization scale µ is considered in detail. 1 Introduction Many textbooks are exclusively [1–6] or in part [7–11] devoted to quantum chromodynamics. Quantization of gauge fields is discussed in [12] in detail. Here we’ll follow notation of [11]; many calculational details omitted here can be found in this book. References to original papers will not be given, except a few cases when materials from such papers was directly used in the lectures. Quantum chromodynamics (QCD) describes quarks and their interactions. Hadrons are bound states of quarks and antiquarks rather than truly elementary particles. Quarks have a quantum number called color. We’ll present formulas for an arbitrary number of colors Nc ; in the Nature, Nc = 3. 2 Classical QCD Lagrangian 2.1 Color group SU (Nc) The quark field q i has a color index i ∈ [1, Nc ]. The theory is symmetric with respect to transformations q i → U i j q j or q → Uq , (1) where the matrix U is unitary and has determinant 1: U +U = 1 , det U = 1 . (2) Such matrices form the group SU(Nc ). Quark fields transform according to the fundamental representation of this group. The conjugated quark field q̄i = (q i )+ γ 0 transforms according to the conjugated fundamental representation q̄i → Ui j q̄j ∗ or q̄ → q̄U + , where Ui j = (U i j )∗ . (3) Lectures at Baikal summer school on astrophysics and physics of elementary particles, 3–10 July 2011. 1 The product q̄q ′ is invariant with respect to color rotations: q̄q ′ → q̄U + Uq ′ = q̄q ′ . (4) In other words, δji is an invariant tensor, its components have the same values (1 or 0) in any basis: ′ ′ δji → δji ′ U i i′ Uj j = U i k Uj k = δji . (5) This tensor describes the color structure of a meson. The product of three quark fields εijk q1i q2j q3k (at Nc = 3) is also invariant: ′ ′ εijk q1i q2j q3k → εijk U i i′ U j j ′ U k k′ q1i q2j q3k = det U · εi′ j ′ k′ q1i q2j q3k = εijk q1i q2j q3k . ′ ′ ′ ′ (6) Here εijk is the unit antisymmetric tensor1 . In other words, εijk is an invariant tensor: ′ ′ ′ εijk → εi′ j ′k′ Ui i Uj j Uk k = det U + · εijk = εijk . (7) It describes the color structure of a baryon. The operator with the quantum numbers of an antibaryon has the form εijk q̄1i q̄2j q̄3k → εijk q̄1i q̄2j q̄3k . (8) I. e., εijk is also an invariant tensor. The matrix of an infinitesimal color rotation has the form U = 1 + iαa ta , (9) where αa are infinitesimal parameters, and the matrices ta are called the generators of the fundamental representation of the group SU(Nc ). The properties (2) of matrices U imply that the generators are hermitian and traceless: U + U = 1 + iαa (ta − (ta )+ ) = 1 det U = 1 + iαa Tr ta = 1 ⇒ ⇒ (ta )+ = ta , Tr ta = 0 . (10) The trace Tr ta tb = TF δ ab , (11) where TF is a normalization constant (usually TF = 12 is chosen, but we’ll write formulas with an arbitrary TF ). How many linearly independent traceless hermitian matrices ta exist? The space of hermitian Nc × Nc matrices has dimension Nc2 ; vanishing of the trace is one additional condition. Therefore, the number of the generators ta , which form a basis in the space of traceless hermitian matrices, is equal to Nc2 − 1. 1 In the case of Nc colors it has Nc indices, and the invariant product contains Nc quark fields; a baryon consists of Nc quarks. 2 The commutator [ta , tb ] is antihermitian and traceless, and hence [ta , tb ] = if abc tc , (12) where 1 Tr[ta , tb ]tc (13) iTF are called the structure constants of the group SU(Nc ). Let’s consider the quantities Aa = q̄ta q ′ . They transform under color rotations as f abc = Aa → q̄U + ta Uq ′ = U ab Ab , (14) U + ta U = U ab tb , (15) where and hence 1 Tr U + ta Utb . (16) TF The quantities Aa (there are Nc2 − 1 of them) transform according to a representation of the group SU(Nc ); it is called the adjoint representation. Components of the generators (ta )i j have identical values in any basis: U ab = ′ ′ (ta )i j → U ab U i i′ Uj j (tb )i j ′ = (ta )i j , (17) hence they can be regarded an invariant tensor. The quantities Aa transform under infinitesimal color rotations as Aa → U ab Ab = q̄(1 − iαc tc )ta (1 + iαc tc )q ′ = q̄(ta + iαc if acb tb )q ′ , (18) U ab = δ ab + iαc (tc )ab , (19) where and the generators of the adjoint representation are (tc )ab = if acb . (20) Generators of any representation must satisfy the commutation relation (12). In particular, the relation (ta )dc (tb )ce − (tb )dc (ta )ce = if abc (tc )de . (21) must hold for the generators (20) of the adjoint representation. It can be easily derived from the Jacobi identity [ta , [tb , td ]] + [tb , [td , ta ]] + [td , [ta , tb ]] = 0 (22) (expand all commutators, and all terms will cancel). Expressing all commutators in the left-hand side of (22) according to the formula (12), we obtain
if bdc if ace + if dac if bce + if abc if dce te = 0 , (23) and hence (21) follows. 3 2.2 Local color symmetry and the QCD Lagrangian The free quark field Lagrangian L = q̄(iγ µ ∂µ − m)q (24) is invariant with respect to global color rotations q(x) → Uq(x) (where the matrix U does not depend on x). How to make it invariant with respect to local (gauge) transformations q(x) → U(x)q(x)? To this end, the ordinary derivative ∂µ q should be replaced by the covariant one Dµ q: Dµ q = (∂µ − igAµ )q , Aµ = Aaµ ta . (25) Here Aaµ (x) is the gluon field, and g is the coupling constant. When the quark field transforms as q → q ′ = Uq, the gluon one transforms too: Aµ → A′µ . This transformation should be constructed in such a way that Dµ q transforms in the same way as q: Dµ q → Dµ′ q ′ = UDµ q. Therefore, (∂µ − igA′µ )Uq = U(∂µ − igAµ )q , or ∂µ U − igA′µ U = −igUAµ . We arrive at the transformation law of the gluon field i A′µ = UAµ U −1 − (∂µ U)U −1 . g (26) Infinitesimal transformations of the quark and gluon fields have the form q(x) → q ′ (x) = (1 + iαa (x)ta )q(x) , 1 ab b a Aaµ (x) → A′a µ (x) = Aµ (x) + Dµ α (x) , g (27) where the covariant derivative acting on an object in the adjoint representation is Dµab = δ ab ∂µ − ig(tc )ab Acµ . (28) The expression [Dµ , Dν ]q also transforms as q: [Dµ′ , Dν′ ]q ′ = U[Dµ , Dν ]q. Let’s calculate it: [Dµ , Dν ]q = ∂µ ∂ν q − ig(∂µ Aν )q − igAν ∂µ q − igAµ ∂ν q − g 2Aµ Aν q − ∂ν ∂µ q + ig(∂ν Aµ )q + igAµ ∂ν q + igAν ∂µ q + g 2Aν Aµ q . All derivatives have canceled, and the result is −igGµν q where Gµν = ∂µ Aν − ∂ν Aµ − ig[Aµ , Aν ] = Gaµν ta , Gaµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν (29) is called the gluon field strength. It transforms in a simple way Gµν → UGµν U −1 , Gaµν → U ab Gbµν ; 4 (30) there is no additive term here, in contrast to (26). Now, at last, we are ready to write down the complete QCD Lagrangian. It contains nf kinds (flavors) of quark fields qf and the gluon field: L = Lq + LA . The first term describes free quark fields and their interaction with gluons: X q̄f (iγ µ Dµ − mf )qf . Lq = (31) (32) f The second term is the gluon field Lagrangian: LA = − 1 1 Tr Gµν Gµν = − Gaµν Gaµν . 4TF 4 (33) It is gauge invariant due to (30). In contrast to the photon field Lagrangian in QED, it contains, in addition to terms quadratic in Aaµ , also cubic and quartic terms. The gluon field is non-linear, it interacts with itself. 2.3 Symmetries The QCD Lagrangian is symmetric with respect to translations and Lorentz transformations, as well as discrete transformations2 P , C, T . It is also symmetric with respect to local (gauge) color transformations. The QCD Lagrangian is symmetric with respect to phase rotations of all quark fields: qf → eiα qf ≈ (1 + iα)qf . (34) This U(1) symmetry leads to conservation of the total number of quarks minus antiquarks, i. e. of the baryon charge. If several kinds (flavors) of quarks have equal masses (mf = m), a wider symmetry appears: (35) qf → Uf f ′ qf ′ , where U is an arbitrary unitary matrix (U + U = 1). Any such matrix can be written as U = eiα U0 where det U0 = 1. In other words, the group of unitary transformations is a direct product U(nf ) = U(1) × SU(nf ). Infinitesimal transformations have the form U = 1 + iα + iαa τ a , (36) So called ϑ-term G̃µν Gaµν (where G̃aµν = 21 εµναβ Gaαβ ) is not symmetric with respect to P (and CP ). However, it is the full divergence of a (non gauge-invariant) axial vector. Therefore adding this term (with some coefficient) to the Lagrangian changes nothing in classical theory. In quantum theory, the ϑ-term is inessential in perturbation theory, but it changes the behavior of the theory due to nonperturbative effects, leading to P (and CP ) violation in QCD. Such violations have not been seen experimentally; therefore we shall not discuss the ϑ-term. 2 5 where the SU(nf ) generators are hermitian matrices satisfying the condition Tr τ a = 0. In the Nature, masses of various quark flavors are not particularly close to each other. However, u and d quarks have masses much smaller than the characteristic QCD energy scale (we’ll discuss this scale later). Both of these masses can be neglected to a good accuracy, and the SU(2) symmetry (called isospin) has a good accuracy (of order 1%). The s quark mass is smaller than the characteristic QCD scale but not much so, and the flavor SU(3) symmetry has substantially lower accuracy. Under a stronger assumption that masses of several quark flavors mf = 0, left and right quarks 1 ± γ5 qf = qLf + qRf , qL,R = q, γ5 qL,R = ±qL,R (37) 2 live their own lives without transforming to each other: X X q̄Rf iγ µ Dµ qRf (38) q̄Lf iγ µ Dµ qLf + Lq = f f (the mass term mq̄q = m(q̄L qR + q̄R qL ) transforms left quarks to right ones and vice versa). The theory has a larger symmetry U(nf )L × U(nf )R ; its infinitesimal transformations are qL → (1 + iαL + iαLa τ a )qL , a a qR → (1 + iαR + iαR τ )qL . (39) They can be re-written as a a q → (1 + iαV + iαVa τ a + iαA γ5 + iαA τ γ5 )q , (40) this corresponds to the U(nf )V × U(nf )A symmetry. As already discussed, this symmetry is quite good for u and d quarks (nf = 2), and substantially less accurate if s quark is added (nf = 3). If all quarks are massless, then the Lagrangian contains no dimensional parameters, and it is symmetric with respect to scale transformations xµ → λxµ , Aµ → λ−1 Aµ , q → λ−3/2 q . (41) For a wide class of field theories one can prove that the scale invariance implies invariance with respect to inversion xµ (42) xµ → 2 x (in an infinitesimal neighborhood of each point it is a scale transformation). Performing inversion, then translation by a, and then again inversion produces a special conformal transformation xµ + aµ x2 xµ → . (43) 1 + 2a · x + a2 x2 These transformations, together with scale ones, translations, and Lorentz transformations form the conformal group. The classical massless QCD is invariant with respect to this group. Not all symmetries of the classical theory survive in the quantum one (Table 1). 6 Table 1: Symmetries of massless QCD in classical and quantum theory. Group translations Lorentz conformal SU(Nc ) U(1) SU(nf ) U(1)A SU(nf )A P C T 3 Classical theory Quantum theory anomaly local anomaly spontaneously broken discrete Quantization 3.1 Faddeev–Popov ghosts It is convenient to use the functional integration method to quantize gauge theories. The correlator of two operators O(x) and O(y) (we assume them to be gauge invariant) is written as R RQ a i L d4 x dA (x) e O(x) O(y) 1 1 δ 2 Z[j] µ x,a,µ R R Q , (44) = 2 <T {O(x), O(y)}> = i L d4 x a i Z[j] δj(x) δj(y) j=0 x,a,µ dAµ (x) e where the generation functional Z[j] = Z Y dAaµ (x) ei R (L+jO) d4 x . (45) x,a,µ To make formulas shorter, we’ll consider gluodynamics (QCD without quarks); including quark fields introduces no extra difficulties, one just have to add integration in fermionic (anticommuting) fields. In the case of gauge fields a problem appears: a single physical field configuration is taken into account infinitely many times in the integral. All potentials obtained from a given one by gauge transformations A → AU form an orbit of the gauge group; physically, they describe a single field configuration. It would be nice to include it in the functional integral just once3 To this end, one has to fix a gauge — to require some conditions Ga (AU (x)) = 0 For any A(x) this equation should have a unique solution U(x). I. e., the “surface” G = 0 should intersect each orbit of the gauge group at a single point (Fig. 1). 3 This is necessary in perturbation theory to define the gluon propagator. Gauge fixing is not needed 7 G=0 A AU Figure 1: Gauge fixing. For example, the Lorenz gauge4 Ga (A(x)) = ∂ µ Aaµ (x) . (46) is often used. The axial gauge Ga (A(x)) = nµ Aaµ (x) (with some fixed vector n) and the fixed-point (Fock–Schwinger) gauge Ga (A(x)) = xµ Aaµ (x) are also popular. Let’s define the Faddeev–Popov determinant ∆[A] by the formula Z Y Y −1 ∆ [A] = dU(x) δ(Ga (AU (x))) , (47) x x,a where dU is the invariant integration measure on the (it satisfies the condition Q group a d(U0 U) = dU); for infinitesimal transformations dU = a dα . Near the surface G(AU0 ) = 0, variations of G at infinitesimal gauge transformations are linear in their parameters: δG(A(x)) = M̂ α(x) , so that −1 ∆ [A] = Z Y dα(x)δ(M̂α(x)) = 1/ det M̂ , (48) (49) x i. e, ∆[A] is the determinant of the operator M̂ . For example, for the Lorenz gauge Ga (A(x)) = ∂ µ Aaµ we obtain from (27) 1 1 δGa (x) = ∂ µ Dµab αb (x) ⇒ M̂ = ∂ µ Dµab ; g g (50) in some approaches, e. g., the lattice QCD. QCD in Euclidean space–time (obtained by the substitution t = −itE ) is considered. The oscillating exp iS becomes exp(−SE ) where the Euclidean action SE is positive. Continuous space–time is replaced by a discrete 4-dimensional lattice, the exact gauge invariance is preserved. Random field configurations are generated with probability exp(−SE ); fields belonging to an orbit of the gauge group are generated equiprobably. 4 The solution U of the equation G(AU ) = 0 for this gauge is not unique (Gribov copies), if one considers gauge transformations sufficiently far from the identical one. This problem is not essential for construction of perturbation theory, because it is sufficient to consider infinitesimal gauge transformations. 8 for the axial gauge 1 δ ab µ M̂ = nµ Dµab = n ∂µ , (51) g g because nµ Aaµ = 0 due to the gauge condition. The Faddeev–Popov determinant is gauge invariant: Z Y Y −1 U0 ∆ [A ] = dU δ(Ga (AU0 U (x))) x = Z Y x,a D(U0 U) x Y δ(Ga (AU0 U (x))) = ∆−1 [A] . x,a Now we insert the unit factor (47) in the integrand (45): Z Y Z[J] = dA(x) eiS[A] = x Z Y dU(x) = x dA(x) ∆[A] x x YZ Y dU ! × Y δ(G(AU (x))) eiS[A] x Z Y dA(x) ∆[A] x Y δ(G(A(x))) eiS[A] . (52) x Here the first factor is an (infinite) constant, it cancels in the ratio (44) and may be omitted. We have arrived at the functional integral in a fixed gauge. The only integral which any physicist can calculate (even if awakened in the middle of night) is the Gaussian one: Z 1 ∗ dz ∗ dz e−az z ∼ . a The result is obvious by dimensionality. In the multidimensional case the determinant appears because the matrix M can be diagonalized: Z Y 1 ∗ dzi∗ dzi e−Mij zi zj ∼ . (53) det M i Integration in a fermion (anticommuting) variable c is defined as Z Z dc = 0 , c dc = 1 (hence fermion variables are always dimensionless). From c2 = 0 ∗ e−ac c = 1 − ac∗ c , so that Z ∗ dc∗ dc e−ac c = a . 9 (54) In the multidimensional case Z Y i ∗ dc∗i dci e−Mij ci cj ∼ det M . (55) The functional integral (52) is inconvenient because it contains ∆[A] = det M̂. It can be easily written as an integral in an auxiliary fermion field: Z Y R 4 ∆[A] = det M̂ = dc̄a (x) dca (x) ei Lc d x , Lc = −c̄a M ab cb . (56) x,a The scalar fermion field ca (belonging to the adjoint representation of the color group, just like the gluon) is called the Faddeev–Popov ghost field, and c̄a is the antighost field. Antighosts are conventionally considered to be antiparticles of ghosts, though ca and c̄a often appear in formulas in non-symmetric ways. In the axial gauge (51) ∆[A] = det M̂ does not depend on A, and this constant factor may be omitted; there is no need to introduce ghosts. They can be introduced, of course, but the Lagrangian (56) shows that they don’t interact with gluons, and thus influence nothing. The same is true for the fixed-point gauge. In the generalized Lorenz gauge Ga (A(x)) = ∂ µ Aaµ (x)−ω a (x) we have, up to an inessential constant factor, M̂ = ∂ µ Dµ , and therefore Lc = −c̄a ∂ µ Dµab cb ⇒ (∂ µ c̄a )Dµab cb . (57) A full derivative has been omitted in the last form; note that the ghost and antighost fields appear non-symmetrically — the derivative of c is covariant while that of c̄ is the ordinary one. The generating functional Z Y Y R 4 Z[J] = dAa (x) dc̄a (x) dca (x) δ(∂ µ Aaµ (x) − ω a (x)) ei (LA +Lc +JO)d x (58) x,a x,a a is  on ω (x).  iti Rdoesa nota depend Q gaugea invariant; in particular, 4 ω (x)ω (x) d x : x,a dω (x) with the weight exp − 2a Z[J] = Z Y dAa (x) dc̄a (x) dca (x) ei R (L+JO)d4 x Let’s integrate it in , (59) x,a where the QCD Lagrangian (without quarks) in the covariant gauge L = LA + La + Lc contains 3 terms: the gluon field Lagrangian LA ; the gauge-fixing term La ; and the ghost field Lagrangian Lc , 1 LA = − Gaµν Gaµν , 4 La = −
2 1 ∂ µ Aaµ , 2a 10 Lc = (∂ µ c̄a )Dµab cb . (60) If there are quarks, their Lagrangian Lq (32) should be added, as well as extra integrations in quark fields. In quantum electrodynamics ghosts don’t interact with photons, and hence can be ignored. The Lagrangian (60) obtained as a result of gauge fixing is, naturally, not gauge invariant. However, a trace of gauge invariance is left: it is invariant with respect to transformations 1 δc̄a = − λ+ ∂ µ Aaµ , a δAaµ = λ+ Dµab cb , g δca = − f abc λ+ cb cc , 2 (61) where λ is an anticommuting (fermion) parameter. This supersymmetry (relating boson and fermion fields) is called the BRST symmetry. 3.2 Feynman rules The quark propagator has the usual form = iS0 (p) , p S0 (p) = /p + m 1 = 2 , /p − m p − m2 (62) where the unit color matrix (in the fundamental representation) is assumed. It is not possible to obtain the gluon propagator from the quadratic part of the Lagrangian LA : the matrix which should be inverted is not invertible. The gauge fixing procedure is needed to overcome this problem. In the covariant gauge (60) the quadratic part of LA + La gives the gluon propagator   pµ pν 1 a b = −iδ ab D 0 (p) , 0 . (63) Dµν (p) = 2 gµν − (1 − a0 ) 2 µν µ ν p p p The ghost propagator a p b = iδ ab G (p) , 0 G0 (p) = 1 , p2 (64) as well as the gluon one (63), has the color structure δ ab — the unit matrix in the adjoint representation. The quark–gluon vertex (see (32)) µa = ta × ig0 γ µ (65) has the color structure ta ; otherwise it has the same form as the electron–photon vertex in quantum electrodynamics. 11 The gluon field Lagrangian LA (33) contains, in addition to quadratic terms, also ones cubic and quartic in A. They produce three- and four-gluon vertices. The three-gluon vertex has the form µ1 a1 p1 p3 a3 µ3 µ p2 a2 2 = if a1 a2 a3 × ig0 V µ1 µ2 µ3 (p1 , p2 , p3 ) , (66) V µ1 µ2 µ3 (p1 , p2 , p3 ) = (p3 − p2 )µ1 g µ2 µ3 + (p1 − p3 )µ2 g µ3 µ1 + (p2 − p1 )µ3 g µ1 µ2 . It is written as the product of the color structure if a1 a2 a3 and the tensor structure. To do such a factorization, one has to choose a “rotation direction” around the three-gluon vertex (clockwise in the formula (66)) which determines the order of the color indices in f a1 a2 a3 as well as the order of the indices and the momenta in V µ1 µ2 µ3 (p1 , p2 , p3 ). Inverting this “rotation direction” changes the signs of both the color structure and the tensor one. The three-gluon vertex does remains unchanged — it does not depend on an arbitrary choice of the “rotation direction”. This choice is required only for factorizing into the color structure and the tensor one; it is essential that their “rotation directions” coincide. The four-gluon vertex does not factorize into the color structure and the tensor one — it contains terms with three different color structures. This does not allow one to separate calculation of a diagram into two independent sub-problems — calculation of the color factor and of the remaining part of the diagram. This is inconvenient for writing programs to automatize such calculations. Therefore, authors of several such programs invented the following trick. Let us declare that there is no four-gluon vertex in QCD; instead, there is a new particle interacting with gluons: ⇒ + + . (67) The propagator of this particle doesn’t depend on p: a ν µ i β b = δ ab (g µα g νβ − g µβ g να ) . α 2 (68) In coordinate space it is proportional to δ(x), i. e., this particle does not propagate, and all four gluons interact in one point. Interaction of this particle with gluons has the form b a ν √ β c = if abc × 2g0 g µα g νβ . α µ 12 (69) The sum (67) correctly reproduces the four-gluon vertex following from the Lagrangian LA (33)5 . The number of diagrams increases, but each of them is the product of a color factor and a “colorless” part. Finally, the ghost–gluon vertex has the form µ b c p a = if abc × ig0 pµ . (70) It contains the momentum of the outgoing ghost but not of the incoming one because of the asymmetric form of the Lagrangian (60). In the color structure, the “rotation direction” is the incoming ghost → the outgoing ghost → the gluon. The color factor of any diagram can be calculated using the Cvitanović algorithm. It is described in the textbook [11]. 4 Renormalization 4.1 MS scheme Many perturbation-theory diagrams containing loops diverge at large loop momenta (ultraviolet divergences). Because of this, expressions for physical quantities via parameters of the Lagrangian make no sense (contain infinite integrals). However, this does not mean that the theory is senseless. The requirement is different: expressions for physical quantities via other physical quantities must not contain divergences. Re-expressing results of the theory (which contain bare parameters of the Lagrangian) via physical (i. e., measurable, at least in principle) quantities is called renormalization, and it is physically necessary. Intermediate results of perturbation theory, however, contain divergences. In order to give them a meaning, it is necessary to introduce a regularization, i. e. to modify the theory in such a way that divergences disappear. After re-expressing the result for a physical quantity via renormalized parameters one can remove the regularization. The choice of regularization is not unique. A good regularization should preserve as many symmetries of the theory as possible, because each broken symmetry leads to considerable complications of intermediate calculations. In many cases (including QCD) it happens to be impossible to preserve all symmetries of the classical theory. When a regularization breaks some symmetry, intermediate calculations are non-symmetric (and hence more complicated); after renormalization and removing the regularization, the symmetry of the final result is usually restored. However, there exist exceptions. Some symmetries are not restored after removing the regularization, they are called anomalous. I. e., these symmetries of the classical theory are not symmetries of the quantum theory. 5 One can also prove this equivalence using functional integration (see [13]). We remove the terms quartic in A from LA and introduce an antisymmetric tensor field taµν with the Lagrangian − 21 taµν taµν + ig abc aµν b c √ f t Aµ Aν (producing the Feynman rules (68), (69)). It is easy to calculate the functional integral 2 in this field, and the QCD generating functional with the full LA is reproduced. 13 In the case of gauge theories, including QCD, it is most important to preserve the gauge invariance. For example, the lattice regularization used for numerical Monte-Carlo calculations preserves it. However, this regularization breaks translational and Lorentz invariance (Lorentz symmetry restoration in numerical results is one of the ways to estimate systematic errors). Because of this, the lattice regularization is inconvenient for analytical calculations in perturbation theory. In practice, the most widely used regularization is the dimensional one. The space–time dimensionality is considered an arbitrary quantity d = 4 − 2ε instead of 4. Removing the regularization at the end of calculations means taking the limit ε → 0; intermediate expressions contain 1/εn divergences. Dimensional regularization preserves most symmetries of the classical QCD Lagrangian, including the gauge and Lorentz invariance (d-dimensional). However, it breaks the axial symmetries and the scale (and hence conformal) one, which are present in the classical QCD Lagrangian with massless quarks. The scale symmetry and the flavor-singlet U(1)A symmetry appear to be anomalous, i. e. they are absent in the quantum theory. Now we’ll discuss renormalization of QCD in detail. For simplicity, let all nf quark flavors be massless (quark masses are discussed in Sect. 6). The Lagrangian is expressed via bare fields and bare parameters; in the covariant gauge X 1 1 2 µ a a (∂µ Aaµ (71) L= q̄0i iγ µ Dµ q0i − Ga0µν Gaµν − 0 ) + (∂ c̄0 )(Dµ c0 ) , 0 4 2a0 i where Dµ q0 = (∂µ − ig0 A0µ ) q0 , [Dµ , Dν ]q0 = −ig0 G0µν q0 , A0µ = Aa0µ ta , G0µν = Ga0µν ta , Ga0µν = ∂µ Aa0ν − ∂ν Aa0µ + g0 f abc Ab0µ Ac0ν , b Dµ ca0 = (∂µ δ ab − ig0 Aab 0µ )c0 , c c ab Aab 0µ = A0µ (t ) . The renormalized fields and parameters are related to the bare ones by renormalization constants: 1/2 q0 = Zq1/2 q , A0 = ZA A , a0 = ZA a , g0 = Zα1/2 g (72) (we shall soon see why renormalization of the gluon field and the gauge parameter a is determined by a single constant ZA ). In the MS scheme renormalization constants have the form z1 αs  z22 z21   αs 2 + + +··· (73) Zi (αs ) = 1 + ε 4π ε2 ε 4π In dimensional regularization the coupling constant g is dimensional (this breaks the scale invariance). Indeed, the Lagrangian dimensionality is [L] = d, because the action must be dimensionless; hence the fields and g0 have the dimensionalities [A0 ] = 1 − ε, [q0 ] = 3/2 − ε, [g0 ] = ε. In the formula (73) αs must be exactly dimensionless. Therefore we are forced to introduce a renormalization scale µ with dimensionality of energy: αs (µ) g2 = µ−2ε e−γε . 4π (4π)d/2 14 (74) The name MS means minimal subtraction: minimal renormalization constants (73) contain only negative powers of ε necessary for removing divergences and don’t contain zero and positive powers6 . In practice, the expression for g02 via αs (µ), αs (µ) g02 = µ2ε Zα (αs (µ))eγε , d/2 (4π) 4π (75) is used more often. First we calculate something from Feynman diagrams, results contain powers of g0 ; then we re-express results via the renormalized quantity αs (µ). 4.2 The gluon field The gluon propagator has the structure = + + 0 0 0 −iDµν (p) = − iDµν (p) + (−i)Dµα (p)iΠαβ (p)(−i)Dβν (p) + 0 0 0 (−i)Dµα (p)iΠαβ (p)(−i)Dβγ (p)iΠγδ (p)(−i)Dγν (p) +··· (76) +··· where the gluon self energy iδ ab Πµν (p) is the sum of all one particle irreducible diagrams (which cannot be cut into two disconnected pieces by cutting a single gluon line). This series can be re-written as an equation: 0 0 Dµν (p) = Dµν (p) + Dµα (p)Παβ (p)Dβν (p) . (77) For each tensor of the form  Aµν = A⊥ gµν  pµ pν pµ pν − 2 + A|| 2 p p it is convenient to introduce the inverse tensor   pµ pν pµ pν −1 −1 Aµν = A⊥ gµν − 2 + A−1 , || p p2 satisfying λν A−1 = δµν . µλ A Then the equation (77) can be re-written in the form −1 Dµν (p) = (D 0 )−1 µν (p) − Πµν (p) . (78) In a moment we shall derive the identity Πµν (p)pν = 0 (79) 6 The bar means the modification of the original MS scheme introducing the exponent with the Euler constant γ and the power d/2 instead of 2 in the denominator — these changes make perturbative formulas considerably simpler. 15 which leads to Πµν (p) = (p2 gµν − pµ pν )Π(p2 ) . Therefore, the gluon propagator has the form   1 pµ pν pµ pν gµν − 2 + a0 2 2 . Dµν (p) = 2 2 p (1 − Π(p )) p (p ) (80) (81) There are no corrections to the longitudinal part of the propagator. The renormalized r propagator (related to the bare one by Dµν (p) = ZA (α(µ))Dµν (p; µ)) is equal to   pµ pν pµ pν r r 2 Dµν (p; µ) = D⊥ (p ; µ) gµν − 2 + a(µ) 2 2 . (82) p (p ) The minimal (73) renormalization constant ZA (α) is tuned to make the transverse part of the renormalized propagator r D⊥ (p2 ; µ) = ZA−1 (α(µ)) p2(1 1 − Π(p2 )) finite at ε → 0. But the longitudinal part of (82) (containing a(µ) = ZA−1 (α(µ))a0 ) also must be finite. This is the reason why renormalization of a0 is determined by the same constant ZA as that of the gluon field (72). In quantum electrodynamics, the property (79) follows from the Ward identities, and its proof is very simple (see, e. g., [11]). In quantum chromodynamics, instead of simple Ward identities, more complicated Slavnov–Taylor identities appear; transversality of the gluon self energy (79) follows from the simplest of these identities. Let’s start from the obvious equality <T {∂ µ Aaµ (x), c̄b (y)}> = 0 (single ghosts cannot be produced or disappear, as follows from the Lagrangian (71)). Variation of this equality under the BRST transformation (61) is <T {∂ µ Aaµ (x), ∂ ν Abν (y)}> − a<T {∂ µ Dµac cc (x), c̄b (y)}> = 0 . Using the equation of motion for the ghost field ∂ µ Dµab cb = 0, we arrive at the Slavnov– Taylor identity <T {∂ µ Aaµ (x), ∂ ν Abν (y)}> = 0 . (83) The derivative ∂ ∂ <T {Aaµ (x), Abν (y)}> ∂xµ ∂yν does not vanish: terms from differentiating the θ-function in the T -product remain. These terms contain an equal-time commutator of Aaµ (x) and Ȧbν (y); it is fixed by the canonical quantization of the gluon field, and thus is the same in the interacting theory and in the free one with g = 0: 0 pµ pν Dµν (p) = pµ pν Dµν (p) . 16 Hence (79) follows. The gluon self energy in the one-loop approximation is given by three diagrams (Fig. 2). The quark loop contribution has the structure (80), and can be easily obtained from the QED result. The gluon and ghost loop contributions taken separately are not transverse; however, their sum has the correct structure (80). Details of the calculation can be found in [11]. The result is  g02 (−p2 )−ε G1 2 −4TF nf (d − 2) Π(p ) = (4π)d/2 2(d − 1)   (84) 1 2 + CA 3d − 2 + (d − 1)(2d − 7)ξ − (d − 1)(d − 4)ξ , 4 where ξ = 1 − a0 , G1 = − 2g1 , (d − 3)(d − 4) g1 = Γ(1 + ε)Γ2 (1 − ε) . Γ(1 − 2ε) (85) Figure 2: The gluon self energy at one loop. The transverse part of the gluon propagator (81), expressed via the renormalized quantities αs (µ) (75) and a(µ), is p2 D⊥ (p2 ) = 1 − h g1 αs (µ) −Lε γε 16(1 − ε)TF nf e e 4πε 4(1 − 2ε)(3 − 2ε) 2 − ε(3 − 2ε)a (µ) − 2(3 − 2ε)(1 − 3ε)a(µ) + 26 − 37ε + 7ε 2
where L = log(−p2 /µ2 ). Expanding in ε (eγε g1 = 1 + O(ε2 )), we get    αs (µ) −Lε 13 4 1 2 2 p D⊥ (p ) = 1 + a− CA − TF nf − e 4πε 2 3 3    2 20 9a + 18a + 97 CA − TF nf ε . + 36 9 i CA , r r The result must have the form p2 D⊥ (p2 ) = ZA (αs (µ), a(µ))p2D⊥ (p2 ; µ) where D⊥ (p2 ; µ) is finite at ε → 0. Therefore, at one loop     αs 1 13 4 ZA (αs , a) = 1 − a− CA + TF nf . (86) 4πε 2 3 3 17 4.3 Quark fields The quark propagator has the structure = + + iS(p) = iS0 (p) + iS0 (p)(−i)Σ(p)iS0 (p) + iS0 (p)(−i)Σ(p)iS0 (p)(−i)Σ(p)iS0 (p) + · · · +··· (87) where the quark self energy −iΣ(p) is the sum of all one particle irreducible diagrams (which cannot be cut into two disconnected pieces by cutting a single quark line). This series can be re-written as an equation S(p) = S0 (p) + S0 (p)Σ(p)S(p) ; (88) its solution is 1 . (89) − Σ(p) For a massless quark, Σ(p) = p/ΣV (p2 ) from helicity conservation, and 1 1 . (90) S(p) = 2 1 − ΣV (p ) /p The quark self energy in the one-loop approximation is given by the diagram in Fig. 3. Details of the calculation can be found in [11], the result is S(p) = ΣV (p2 ) = −CF S0−1 (p) g02 (−p2 )−ε d − 2 a0 G1 . (4π)d/2 2 (91) k k+p Figure 3: The quark self energy at one loop. The quark propagator (90), expressed via the renormalized quantities and expanded in ε, is αs (µ) −Lε γε d−2 /pS(p) = 1 + CF e e g1 a(µ) 4π (d − 3)(d − 4) αs (µ) a(µ)e−Lε (1 + ε + · · · ) . = 1 − CF 4πε It must have the form Zq (α(µ), a(µ))/pSr (p; µ) where Sr (p; µ) is finite at ε → 0. Therefore, at one loop αs Zq (α, a) = 1 − CF a . (92) 4πε 18 4.4 The ghost field The ghost propagator is G(p) = p2 1 . − Σ(p2 ) (93) The ghost self energy in the one-loop approximation (Fig. 3) is (see [11]) 1 g 2(−p2 )1−ε G1 [d − 1 − (d − 3)a0 ] . Σ(p2 ) = − CA 0 4 (4π)d/2 (94) Re-expressing the propagator via the renormalized quantities and expanding in ε, we get p2 G(p) = 1 + CA αs (µ) −Lε 3 − a + 4ε e , 4πε 4 and hence Zc (αs , a) = 1 + CA 3 − a αs . 4 4πε (95) k p k+p p Figure 4: The ghost self energy at one loop. 5 Asymptotic freedom In order to obtain the renormalization constant Zα , it is necessary to consider a vertex function and propagators of all the fields entering this vertex. It does not matter which vertex to choose, because all QCD vertices are determined by a single coupling constant g. We shall consider the quark–gluon vertex. This is the sum of all one particle irreducible diagrams (which cannot be separated into two parts by cutting a single line), the propagators of the external particles are not included: µ q p = ig0 ta Γµ (p, p′) , p′ (Λµ starts from one loop). 19 Γµ (p, p′ ) = γ µ + Λµ (p, p′ ) (96) The vertex function expressed via the renormalized quantities should be equal to Γµ = ZΓ Γµr , where ZΓ is a minimal (73) renormalization constant, and the renormalized vertex Γµr is finite at ε → 0. In order to obtain a scattering amplitude (an element of the S-matrix), one should calculate the corresponding vertex function and multiply it by the field renormalization 1/2 constants Zi for each external particle i. This is called the LSZ reduction formula. We shall not derive it; it can be intuitively understood in the following way. In fact, there are no external lines, only propagators. Suppose we study photon scattering in the laboratory. Even if this photon was emitted in a far star (Fig. 5), there is a photon propagator from the star to the laboratory. The bare photon propagator contains the factor ZA . We split 1/2 1/2 1/2 it into ZA · ZA , and put one factor ZA into the emission process in the far star, and 1/2 the other factor ZA into the scattering process in the laboratory. 1/2 ZA Laboratory 1/2 ZA Far star Figure 5: Scattering of a photon emitted in a far star. 1/2 1/2 1/2 The physical matrix element g0 ΓZq ZA = gΓr Zα ZΓ Zq ZA must be finite at ε → 0. 1/2 1/2 Therefore the product Zα ZΓ Zq ZA must be finite. But the only minimal (73) renormalization constant finite at ε → 0 is 1, and hence Zα = (ZΓ Zq )−2 ZA−1 . (97) In QED ZΓ Zq = 1 because of the Ward identities, and it is sufficient to know ZA . In QCD all three factors are necessary. The quark–gluon vertex in the one-loop approximation is given by two diagrams (Fig. 6). In order to obtain ZΓ , it is sufficient to know ultraviolet divergences (1/ε parts) of these diagrams; they don’t depend on external momenta. Details of the calculation can be found in [11], the results for these two diagrams are   3 αs α CA αs α α γ , Λα2 = (1 + a)CA γ , Λ 1 = a CF − 2 4πε 4 4πε and hence   a + 3 αs . Z Γ = 1 + CF a + CA 4 4πε From (98) and (92) we obtain Z Γ Z q = 1 + CA 20 a + 3 αs . 4 4πε (98) Figure 6: The quark–gluon vertex at one loop. The color structure CF has canceled, in accordance with QED expectations. Finally, taking (86) into account, the renormalization constant Zα (97) is   11 4 αs Zα = 1 − CA − TF nf . (99) 3 3 4πε It does not depend on the gauge parameter a, this is an important check of the calculation. It can be obtained from some other vertex, e. g., the ghost–gluon one (this derivation is slightly shorter, see [11]). Dependence of αs (µ) on the renormalization scale µ is determined by the renormalization group equation. The bare coupling constant g02 does not depend on µ. Therefore, differentiating the definition (74) in d log µ, we obtain d log αs (µ) = −2ε − 2β(αs (µ)) , d log µ (100) where the β-function is defined as β(αs (µ)) = 1 d log Zα (αs (µ)) . 2 d log µ (101) For a minimal renormalization constant Zα (αs ) = 1 + z1 αs +··· 4πε we obtain from (101) with one-loop accuracy β(αs ) = β0 αs αs + · · · = −z1 + ··· 4π 4π This means that the renormalization constant Zα has the form Zα (αs ) = 1 − β0 From (99) we conclude that αs +··· 4πε 11 4 CA − TF nf . (102) 3 3 CA = 33 this means that β(αs ) > 0 at small αs , where perturbation theory For nf < 11 4 TF 2 is applicable. In the Nature nf = 6 (or less if we work at low energies where the existence β0 = 21 of heavy quarks can be neglected), so that this regime is realized: αs decreases when the characteristic energy scale µ increases (or characteristic distances decrease). This behavior is called asymptotic freedom; it is opposite to screening which is observed in QED. There the charge decreases when distances increase, i. e. µ decreases. The renormalization group equation with ε = 0, d log αs (µ) = −2β(αs (µ)) , d log µ can be easily solved if the one-loop approximation for β(αs ) is used: 2  αs (µ) d αs (µ) = −2β0 d log µ 4π 4π can be re-written in the form therefore d 4π = 2β0 , d log µ αs (µ) 4π 4π µ′ − = 2β log , 0 αs (µ′ ) αs (µ) µ and finally αs (µ′ ) is expressed via αs (µ) as αs (µ′ ) = αs (µ) . µ′ αs (µ) log 1 + 2β0 4π µ (103) This solution can be written in the form αs (µ) = 2π β0 log µ , ΛMS (104) where ΛMS plays the role of an integration constant (it has dimensionality of energy). If higher terms of expansion of β(αs ) are taken into account, the renormalization group equation cannot be solved in elementary functions. A surprising thing has happened. The classical QCD Lagrangian with massless quarks is characterized by a single dimensionless parameter g, and is scale invariant. In quantum theory, QCD has a characteristic energy scale ΛMS , and there is no scale invariance — at small distances ≪ 1/ΛMS the interaction is weak, and perturbation theory is applicable; the interaction becomes strong at the distances ∼ 1/ΛMS. This is a consequence of the scale anomaly. Hadron masses7 are equal to some dimensionless numbers multiplied by ΛMS ; calculation of these numbers is a non-perturbative problem, and can only be done numerically, on the lattice. 7 except the (n2f − 1)-plet of pseudoscalar mesons, which are the Goldstone bosons of the spontaneously broken SU (nf )A symmetry, and their masses are 0 if nf quark flavors are massless. 22 0.5 July 2009 αs(Q) Deep Inelastic Scattering e+e– Annihilation Heavy Quarkonia 0.4 0.3 0.2 0.1 QCD 1 α s (Μ Z) = 0.1184 ± 0.0007 10 Q [GeV] 100 Figure 7: αs (µ) from various experiments [14]. Values of αs (µ) are extracted from many kinds of experiments at various characteristic energies µ, see [14]. Their µ dependence agrees with theoretical QCD predictions well (Fig. 7). Of course, all known terms of β(αs ) (up to 4 loops) are taken into account here8 . If these results are reduced to a single µ = mZ , they are consistent (Fig. 8); this fact confirms correctness of QCD. 6 Quark masses Until now we considered QCD with massless quarks. With the account of mass, the quark field Lagrangian is Lq = q̄0 (iγ µ Dµ − m0 ) q0 . (105) 8 Decoupling effects which arise at transitions from QCD with nf + 1 flavors one of which is heavy to the low energy effective theory — QCD with nf light flavors are also taken into account. 23 τ-decays (N3LO) Quarkonia (lattice) Υ decays (NLO) DIS F2 (N3LO) DIS jets (NLO) e+e– jets & shps (NNLO) electroweak fits (N3LO) e+e– jets & shapes (NNLO) 0.11 0.12 0.13 αs (Μ Z) Figure 8: αs (mZ ) from various experiments [14]. The MS renormalized mass m(µ) is related to the bare one m0 (appearing in the Lagrangian) as m0 = Zm (α(µ))m(µ) , (106) where Zm is a minimal (73) renormalization constant. The quark self energy has two Dirac structures Σ(p) = p/ΣV (p2 ) + m0 ΣS (p2 ) , (107) because there is no helicity conservation. The quark propagator has the form S(p) = 1 1 = 2 2 /p − m0 − /pΣV (p ) − m0 ΣS (p ) 1 − ΣV (p2 ) 1 . 1 + ΣS (p2 ) /p − m0 1 − ΣV (p2 ) It should be equal Zq Sr (p; µ) where the renormalized propagator Sr (p; µ) is finite at ε → 0. Therefore the renormalization constants are found from the conditions 1 + ΣS Zm = finite , 1 − ΣV (1 − ΣV )Zq = finite , and hence (1 + ΣS )Zq Zm = finite . 24 At −p2 ≫ m2 the mass may be neglected while calculating ΣS (p2 ). In the one-loop approximation, retaining only the 1/ε ultraviolet divergence, we get ΣS = CF (3 + a(µ)) αs (µ) . 4πε (108) Therefore α +··· (109) 4πε The result does not depend on the gauge parameter, this is an important check. The dependence of m(µ) is determined by the renormalization group equation. The bare mass m0 does not depend on µ; differentiating (106) in d log µ, we obtain Zm (αs ) = 1 − 3CF dm(µ) + γm (αs (µ))m(µ) = 0 , d log µ (110) where the anomalous dimension is defined as γm (αs (µ)) = d log Zm (αs (µ)) . d log µ (111) For a minimal renormalization constant (73) we obtain from (111) with one-loop accuracy αs αs γm (αs ) = γm0 + · · · = −2z1 +··· 4π 4π Hence the renormalization constant Zm has the form Zm (αs ) = 1 − γm0 αs +··· 2 4πε From (109) we conclude that αs +··· 4π Dividing (110) by (100) (at ε = 0) we obtain γm (αs ) = 6CF (112) γm (αs ) d log m = . d log αs 2β(αs ) It is easy to express m(µ′ ) via m(µ): ′ m(µ ) = m(µ) exp Z αs (µ′ ) αs (µ) γm (αs ) dαs . 2β(αs ) αs (113) Retaining only one-loop terms in γm (αs ) and β(αs ) we get ′ m(µ ) = m(µ)  αs (µ′ ) αs (µ) 25 γm0 /(2β0 ) . (114) HPQCD 10 Karlsruhe 09 low-moment sum rules, NNNLO, new Babar Kuehn, Steinhauser, Sturm 07 low-moment sum rules, NNNLO Pineda, Signer 06 Υ sum rules, NNLL (not complete) Della Morte et al. 06 lattice (ALPHA) quenched Buchmueller, Flaecher 05 B decays αs2β0 Mc Neile, Michael, Thompson 04 lattice (UKQCD) deDivitiis et al. 03 lattice quenched Penin, Steinhauser 02 Υ(1S), NNNLO Pineda 01 Υ(1S), NNLO Kuehn, Steinhauser 01 low-moment sum rules, NNLO Hoang 00 Υ sum rules, NNLO QWG 2004 PDG 2010 4.1 4.2 4.3 4.4 4.5 4.6 4.7 mb(mb) (GeV) Figure 9: m̄b from various experiments [15]. Bodenstein et. al 10 finite energy sum rule, NNNLO HPQCD 10 lattice + pQCD HPQCD + Karlsruhe 08 lattice + pQCD Kuehn, Steinhauser, Sturm 07 low-moment sum rules, NNNLO Buchmueller, Flaecher 05 B decays αs2β0 Hoang, Manohar 05 B decays αs2β0 Hoang, Jamin 04 NNLO moments deDivitiis et al. 03 lattice quenched Rolf, Sint 02 lattice (ALPHA) quenched Becirevic, Lubicz, Martinelli 02 lattice quenched Kuehn, Steinhauser 01 low-moment sum rules, NNLO QWG 2004 PDG 2010 0.8 0.9 1 1.1 1.2 1.3 mc(3 GeV) (GeV) Figure 10: mc (3 GeV) from various experiments [15]. 26 1.4 Quark masses are extracted from numerous experiments, see the review [15] for b and c. For b quark, the quantity m̄b is usually presented; it is defined as the root of the equation mb (m̄b ) = m̄b (mb (µ) is the MS mass of b quark), see Fig. 9. For c quark this renormalization scale is too low, therefore results for mc (3 GeV) are presented (Fig. 10). Acknowledgments. I am grateful to D. Naumov for inviting me to give the lectures at the Baikal summer school. References [1] B. L. Ioffe, V. S. Fadin, L. N. Lipatov, Quantum Chromodynamics: Perturbative and Nonperturbative Aspects, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 30, Cambridge University Press (2010). [2] T. Muta, Foundations of Quantum Chromodynamics, 3-rd ed., World Scientific (2010). [3] W. Greiner, S. Schramm, E. Stein, Quantum Chromodynamics, 3-rd ed., Springer (2007). [4] F. J. Ynduráin, The Theory of Quark and Gluon Interactions, 4-th ed., Springer (2006). [5] S. Narison, QCD as a Theory of Hadrons, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 17, Cambridge University Press (2004). [6] A. Smilga, Lectures on Quantum Chromodynamics, World Scientific (2001). [7] M. E. Peskin, D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books (1995). [8] M. Srednicki, Quantum Field Theory, Cambridge University Press (2007). [9] L. H. Ryder, Quantum Field Theory, 2-nd ed., Cambridge University Press (1996). [10] K. Huang, Quanks, Leptons and Gauge Fields, 2-nd ed., World Scientific (1992). [11] A. Grozin, Lectures on QED and QCD, World Scientific (2007); hep-ph/0508242. [12] A. A. Slavnov, L. D. Faddeev, Gauge Fields: Introduction to Quantum Theory, 2-nd ed., Perseus Books (1991). [13] A. Pukhov et al., CompHEP: A Package for evaluation of Feynman diagrams and integration over multiparticle phase space, hep-ph/9908288. [14] S. Bethke, The 2009 World Average of αs , Eur. Phys. J. C 64 (2009) 689. 27 [15] K. Chetyrkin et al., Precise Charm- and Bottom-Quark Masses: Theoretical and Experimental Uncertainties, arXiv:1010.6157. 28