Basics of perturbative QCD

Andreas Nyffeler Regional Centre for Accelerator-based Harish-Chandra Research Institute Chhatnag Road, Jhusi Allahabad - 211019, India [email protected]

Lectures at “Advanced School on Radiative Corrections for the LHC”, Saha Institute of Nuclear Physics, Kolkata, India, April 4 - 11, 2011.

Contents

1 The strong interactions: hadrons, and 2

2 QCD Lagrangian and Feynman rules 5

3 Dimensional 9

4 of QCD to one loop 16

5 Renormalization group and running αs 24

6 Selected References 28

1 1 The strong interactions: hadrons, quarks and gluons

Strong interactions

One of the four fundamental forces (interactions) of Nature, besides the electromagnetic, • weak and gravitational interactions.

Binds nucleons (protons, neutrons) inside the atomic nucleus. • Affects strongly interacting particles (hadrons): • – Baryons (half-integer spin): proton, neutron, Λ, Σ, ∆, Ω,... – Mesons (integer spin): π, K, η, ρ, φ, J/ψ, Υ,... Since the Large Hadron Collider (LHC) collides protons on protons, the dynamics of the • strong interactions plays a crucial role in all processes.

Strong interactions are short ranged: range about 1 fm (1 Fermi) = 10−15 m. This leads • to typical cross-sections of 1 fm2 = 10 mb (mb = millibarn). 1 fm corresponds to energies of a few hundred MeV and time-scales (life-times) of 10−24 s. Note: in the following, we will use “natural units” where ~ = c = 1 masses and momenta given in MeV or GeV. → Some useful conversion factors: ~ = 6.582 118 99(16) 10−22 MeV s × ~c = 197.326 9631(49) MeV fm 2 (~c)2 = 0.389 379 304(19) GeV mb

Quantum Chromodynamics (QCD)

Underlying theory of the strong interactions: • [Fritzsch, Gell-Mann, Leutwyler [1], 1973]

Non-abelian of quarks and gluons with three color charges (gauge group • SU(3), Nc = 3).

6 flavors of quarks (Nf = 6): • type Electric charge (e > 0) Quark mass Comment 2 +0.81 u 3 e mu(2 GeV) = 2.49−0.79 MeV MS-scheme 1 +0.75 d e md(2 GeV) = 5.05 MeV MS-scheme − 3 −0.95 1 +29 s e ms(2 GeV) = 101 MeV MS-scheme − 3 −21 2 +0.07 c 3 e mc(mc) = 1.27−0.09 GeV MS-scheme 1 +0.18 b e mb(mb) = 4.19 GeV MS-scheme − 3 −0.06 2 t e mt = 172.0 1.6 GeV Pole mass 3 ± Quark mass values from Particle Data Group (PDG) 2010 [2]. Since quarks have not been observed as free particles (confinement), it is not straightforward to define their masses. For the top quark, which decays before it hadronizes, one usually uses the pole mass. For the other quarks, one employs running masses in the MS-scheme at some scale µR. Note the scale of 2 GeV for the light quarks u, d, s.

2 Hadrons are composite particles, made up of quarks, antiquarks and gluons: • – Baryons qqq (3 valence quarks) ∼ – Mesons qq¯ (1 valence quark + 1 valence anti-quark) ∼ Interaction strength of QCD at typical hadronic energy scale (proton mass): •

αs(1 GeV) 0.5 α 1/137 ≈  ≈ 2 α e /4π0~c: fine-structure constant, electromagnetic coupling. ≡ Non-perturbative phenomena of QCD at low energies: • – Confinement: quarks and gluons are permanently confined into color-neutral hadrons (no free quarks and gluons observed in Nature). If we try to break up a meson into quarks, we have to put more and more energy into the system and at some point a new quark-antiquark pair is created out of the vacuum and we end up with two mesons (string-breaking, schematically):

Source: NIC J¨ulich

Confinement also leads to the finite range of the strong interactions, even though the gluons, which mediate the force, are massless. In fact, the strong force between the nucleons in an atomic nucleus can be described effectively by the exchange of pions (Yukawa).

3 – Structure of hadrons: the intrinsic structure of hadrons, described by hadronic wave functions, form factors and parton distribution functions, is governed by non-per- turbative physics. Can often not be derived from theory and needs to be extracted from data. While the proton can be thought of to consist of three valence quarks (uud), if we “look” closer (at shorter distances, i.e. probe at higher energies), the picture is much more complicated with the creation of virtual quarks, antiquarks and gluons:

Source: NIKHEF

– Spontaneous chiral breaking: pions as Goldstone bosons of spontaneously broken chiral (axial-vector) symmetry. Global for Nf massless flavors:

U(Nf )L U(Nf )R = SU(Nf )V SU(Nf )A U(1)B U(1)A SU(Nf )V U(1)B × × × × → ×

The U(1)A symmetry is anomalous, i.e. broken by quantum effects (triangle in VVA ). h i

– Methods to study QCD at low energies: Lattice QCD, Effective Theories (Chiral Perturbation Theory), Hadronic Models (quark models, resonance models, ...)

Asymptotic freedom [Gross + Wilczek [3]; Politzer [4], 1973]: •

αs(Q) 0 for Q (αs(MZ ) 0.118) → → ∞ ≈

can do perturbation theory (series in αs) for large momenta Q, i.e. at the LHC, for the⇒ hard scattering of quarks and gluons. At short distances (= at large energies), like in deep-inelastic scattering of electrons off protons, the quarks and gluons inside hadrons behave almost like free particles (parton model).

4 2 QCD Lagrangian and Feynman rules

Gauge Symmetry Gauge symmetries play an important role in the construction of Lagrangians for the fundamen- tal forces. Example: (QED) = interaction of electrons and photons (electromagnetic field): µ 1 µν QED = ψ (iγ Dµ me) ψ FµνF L − − 4 where ψ is a 4-component Dirac field (spinor) describing the electron with mass me (Dirac indices suppressed in Lagrangian) and

Dµ = ∂µ ieAµ () − Fµν = ∂µAν ∂νAµ (field strength tensor of gauge field) − Lagrangian invariant under local U(1) gauge transformations (Abelian group):

ψ(x) ψ0(x) = eiω(x)ψ(x) → 0 1 Aµ(x) A (x) = Aµ(x) + ∂µω(x) → µ e

Non-abelian gauge symmetry (Yang-Mills)

Can generalize local gauge transformation to non-abelian groups, e.g. SU(Nc), where ψ trans- a forms in fundamental representation of SU(Nc) and the gauge-field Aµ now carries an index a of the adjoint representation:   ψ1  .  ψ =  .  ψNc

ψ(x) ψ0(x) = U(x)ψ(x) → ˆ a a ˆ0 ˆ † i † Aµ(x) A (x)T A (x) = U(x)Aµ(x)U (x) + U(x)(∂µU (x)) ≡ µ → µ g

iωa(x)T a U(x) = e SU(Nc) ∈ a 2 T , a = 1,...Nc 1: generators of SU(Nc) in fundamental representation. Hermitean and traceless since U †U− = 1 and det U = 1. They form a Lie algebra:

T a,T b = if abcT c

abc ˆ with real and fully antisymmetric structure constants f . Note that Aµ(x) is in the Lie algebra of the group. Normalization of generators in fundamental representation: 1 tr TaTb = δab 2

5 QCD Lagrangian

For NC = 3,Nf = 6 (preliminary version !):   µ 1 ˆ ˆµν QCD = q (iγ Dµ ) q tr FµνF L − M − 2 with  u   d     s  q =    c     b  t

= diag(mu, md, ms, mc, mb, mt) M ˆ Dµ = ∂µ igsAµ − λa Aˆ = Aa T a,T a = , a = 1,..., 8 (λa: Gell-Mann matrices) µ µ 2 ˆ i a a Fµν = [Dµ,Dν] = FµνT gs a a a abc b c F = ∂µA ∂νA + gsf A A (non-Abelian field strength tensor) µν ν − µ µ ν

The covariant derivative Dµ is flavor diagonal and acting only on color indices, e.g. on

 u1  u =  u2  u3

QCD invariant under local SU(3) gauge transformations U(x): L ψ ψ0 = Uψ, ψ = u, . . . , t → ˆ ˆ0 ˆ † i † Aµ Aµ = UAµU + U(∂µU ) → gs 0 † Dµ D = UDµU (Dµ is acting on everything that follows to the right !) → µ ˆ ˆ0 ˆ † Fµν F = UFµνU → µν

6 When one quantizes QCD, one usually employs the Faddeev-Popov trick [5] in the path integral to fix a gauge and to define a . In general, ghost fields will then also appear in the Lagrangian. In this way we arrive at the following Lagrangian, suitable for calculations in perturbative QCD (for simplicity, we write it down for only one flavor ψ; follow largely the notations in Muta [6]):

i µ ij ij j 1 a aµν 1 µ a 2 µ a∗ ab b QCD = ψ iγ D mδ ψ F F (∂ A ) + (∂ χ ) χ L µ − − 4 µν − 2ξ µ Dµ where

ij ij c ij c D = δ ∂µ igs (T ) A (covariant derivative in fundamental rep.) µ − µ ab ab abc c a bc abc = δ ∂µ gsf A (covariant derivative in adjoint rep.: (T ) = if ) Dµ − µ adj − i, j = 1,..., 3 a, b, c = 1,..., 8

The ghost fields have been denoted by χa. These are complex, anti-commuting scalar fields and as for fermions, there is a minus sign for each closed ghost-loop in the Feynman rules.

To read off the Feynman rules, we split the Lagrangian as follows:

QCD = free + int L L L

i µ i free = ψ (iγ ∂µ m)ψ L − 1 a a  µ aν ν aµ 1 µ a ν a ∂µA ∂νA (∂ A ∂ A ) (∂ A )(∂ A ) −4 ν − µ − − 2ξ µ ν µ a∗ a +(∂ χ )(∂µχ )

i a µ j a int = gsψ Tijγ ψ Aµ L 2 gs abc a a bµ cν gs abe cde a b cµ dν f (∂µA ∂νA )A A f f A A A A − 2 ν − µ − 4 µ ν abc µ a∗ b c gsf (∂ χ )χ A − µ

The 4 interactions terms in the gauged fixed Lagrangian are all related by the original gauge symmetry (BRST symmetry after gauge fixing) only 1 independent gs. → In contrast to QED, we now also have vertices involving the self-interactions of 3 and 4 gluons as they carry a . This has very important consequences (confinement, asymptotic freedom).

7 Feynman rules for QCD Quark propagator:

p i i j δ ij p m 6 − Gluon propagator:

k   aµ i kµkν bν δab − gµν (1 ξ) k2 − − k2

Ghost propagator:

k i a b δ ab k2

Quark-gluon vertex:

aµ

a i j igsγµTij

Three-gluon vertex:

a1µ1 a1a2a3 gsf Vµ µ µ (k1, k2, k3) k 1 2 3 ↓ 1

k k 2 3 Vµ µ µ (k1, k2, k3) = (k1 k2)µ gµ µ + (k2 k3)µ gµ µ ր տ 1 2 3 − 3 1 2 − 1 2 3 +(k3 k1)µ2 gµ3µ1 a2µ2 a3µ3 − (all momenta incoming) Four-gluon vertex:

a1µ1 ig2W a1a2a3a4 − s µ1µ2µ3µ4

a µ a µ a1a2a3a4 13,24 14,32 2 2 4 4 W = f f gµ µ gµ µ µ1µ2µ3µ4 − 1 2 3 4 12,34 14,23 + f f gµ1µ3 gµ2µ4 13,42 − 12,34 + f f gµ1µ4 gµ3µ2 a3µ3 − f ij,kl = f aiaj af akala Ghost-gluon vertex: aµ

k abc gsf kµ b ←− c

8 3 Dimensional regularization

The need for regularization

k ← Example: quark self-energy at one loop i j p k p −

Quark propagator: i S (p) = δ , Σ(p) : 1-particle irreducible (1PI) part ij ij p m Σ(p) 6 − − In Feynman gauge (ξ = 1): Z 4 µν d k a i b ( i)g iΣij(p) = (igs)γµT δkl (igs)γνT δab − − (2π)4 ik p k m + i lj k2 + i 6 − 6 − Σij(p) = δijΣ(p) 2 Z 4 µ gs d k γµ(p k + m)γ Σ(p) = CF 6 − 6 i (2π)4 k2((p k)2 m2) − − with the color factor

2 a a Nc 1 (T T )ij = δijCF ,CF = − for SU(Nc) 2NC Integral linearly divergent due to high-momentum region k : | | → ∞ Z k Σ(p) d4k 6 lim K (naive power counting) ∼ k2k2 ∼ K→∞

Ultraviolet (UV) divergence (= short distances) i µ a a j Origin: local interaction vertex gsψ (x)γ TijAµ(x)ψ (x). Product of fields (operators) at same space-time point x.

UV regularization (UV regulator) Mathematical procedure to make a quantum field theory finite (= regular) each step in calculation is well defined, in particular loop integrals which appear in perturbation→ theory are finite. Physical results after renormalization (more on that later) should be unaffected, if the regulator is removed in the end. This has to be shown for each regulator.

Infrared (IR) and collinear divergences There are other types of divergences, if there are massless particles and k 0 (IR divergence) or if two four-vectors k, p become collinear. Those will be discussed in other→ lectures at this School.

9 Dimensional regularization The preferred choice nowadays for calculations in perturbative QCD is dimensional regular- ization. Based on the observation that multiple integrals are more convergent (in UV), if one reduces the number of integrals. For instance, Σ(p) converges in d = 2 space-time dimensions:

d=4 Z k d=2 Z k Σ(p) dk k3 6 dk k 6 UV convergent ∼ k4 → k4 →

In general Z Z 4 d I = d kf(k) Ireg(d) = d kf(k), d < 4 →

Final result Ireg(d) is then analytically continued to d C. 1 ∈ d 4: single poles appear in Ireg(d) (for divergent I). → d−4 Advantages of dimensional regularization:

Preserves Lorentz invariance, translational invariance, gauge invariance, chiral invariance, • unitarity.

Can also be used to regularize IR and collinear divergences. • Disadvantages of dimensional regularization:

Problem with consistent definition of Dirac matrix γ5 (triangle anomaly in VVA ) or of • h i Levi-Civita tensor εµνρσ. Works only for loop integrals in perturbation theory, not non-perturbatively for path • integral, as does lattice regularization.

For a discussion on how to precisely define an integration R ddk for continuous or complex d from the beginning (explicit mathematical construction, consistency and uniqueness), see Chapter 4 in the book “Renormalization” by Collins [8], based on Wilson’s axioms [9] for d-dimensional integration:

R d R d R d 1. Linearity: for all a, b C: d k [af(k) + bg(k)] = a d kf(k) + b d kf(k) ∈ R d −d R d 2. Scaling: for any number s C: d kf(s k) = s d kf(k) ∈ 3. Translation invariance: for any vector k0: R ddkf(k + k0) = R ddkf(k)

10 d-dimensional space-time (integer d for the moment !)

µ = 0, 1, . . . , d 1 − pµ = (p0, p1, . . . , pd−1) gµν = (+, ,..., ) µ −µν − g µ = gµνg = d (Attention !) Z d4k Z ddk (convention) (2π)4 → (2π)d

Dirac matrices in d-dimensions

γµ, γν = 2gµν { }  γµ, µ = 0 (γµ)† = γµ, µ = 1, . . . , d 1 µ µ − µ − γ γµ = g = d, γµγνγ = (2 d)γν µ − tr (γµγν) = 4gµν (convention)

Dimensionful coupling in d-dimensions

Action: Z S = ddx dimensionless dim [ ] = d (mass dimension !) L ⇒ L Kinetic terms:

a a 2  a  d 2 (∂µA ∂νA ) dim A = − ν − µ ⇒ µ 2 d 1 ψ ∂µψi dim [ψi] = − i ⇒ 2 Interaction term:

i µ a a j  a  gsψ γ A T ψ dim [gs] + 2 dim [ψi] + dim A = d µ ij ⇒ µ d dim [gs] = 2 ⇒ − 2 Introduce arbitrary mass scale µ by hand:

2− d gs = g0 µ 2 , g0 dimensionless

11 Some useful formulae Feynman parametrization

1 Z 1 1 1 Z 1 1 x = dx , = 2 dx − AB (xA + (1 x)B)2 AB2 (xA + (1 x)B)3 0 − 0 − Generalization (proof by induction):

n Z 1 n ! Y 1 Γ(α) Y αi−1 δ(1 x) = dxi x − Aαi Qn Γ(α ) i (Pn x A )α i=1 i i=1 i 0 i=1 i=1 i i

αi, i = 1, . . . , n : arbitrary complex numbers n n X X α = αi, x = xi i=1 i=1

Momentum integrals in Minkowski space (if integral converges for some d)

Z ddk k f(k2) = 0 (2π)d µ Z ddk g Z ddk k k f(k2) = µν k2f(k2) (2π)d µ ν d (2π)d

(and similarly for more powers of kµ) Problem with integral that has no (external) mass or momentum scale:

Z ddk 1 = ? (2π)d (k2)α

If we lower d to make it convergent in UV, we get IR divergence and vice-versa. Consistency of dimensional regularization requires:

Z ddk 1 α = 0, for all α C (2π)d (k2) ∈

Momentum integrals in Euclidean space (after Wick rotation)

Z d 2 a d d d K (K ) 1 d Γ(a + )Γ(b a ) 2 +a−b 2 2 d b = d ∆ −d − (2 ) 2 2 π (K + ∆) (4π) Γ(b)Γ( 2 )

For b 0 we get (for generic d and a and non-vanishing ∆): → Z ddK 1 1 0 since 0 (2π)d (K2)−a → Γ(b) →

12 Sketch of calculation of quark self-energy in dimensional regularization

2 4−d Z d µ g0µ d k γµ(p k + m)γ Σd(p) = CF 6 − 6 i (2π)d k2((p k)2 m2) − − Convergence in UV (large k) for d < 3. To simplify consider the case m = 0. Perform the following steps: Combine denominators using Feynman parametrization. • Interchange integrals over momentum k and Feynman parameter x. Allowed since integral • convergent for d < 3. Shift momentum variable k k0 = k xp, ddk0 = ddk. • → − Since R ddk0 k0 f(k02) = 0, the integral is actually only logarithmically divergent. • µ 0 0 2 Perform Wick rotation k = iK0,~k = K~ . Non-singular integral for p < 0. • 0 Use scaling property of dimensional regularization. • d d−1 Some helpful identities (d K = K dKdΩd): • d Z 2π 2 dΩd = d  Γ 2 Z ∞ tp−1 Γ(p)Γ(q) dt p+q = B(p, q) 0 (1 + t) ≡ Γ(p + q) Z 1 dx xp−1(1 x)q−1 = B(p, q) 0 − One obtains: d 2  2  2 −2     g0 p d d d Σd(p) = 2CF p − (d 1)B , Γ 2 (4π)2 6 4πµ2 − 2 2 − 2 This expression can be analytically continued to the whole complex plane for d, p2 C with the exception of the following points: ∈

d  Poles at d = 4, 6, 8,... from Γ 2 2 • − n Γ(z) has simple poles at z = n, n = 0, 1,...; Γ(z) (−1) 1 for z n − ∼ n! z+n ∼ − d −2 Branch cut on the positive real axis in p2-plane from ( p2) 2 • − 1 Performing a Laurent series expansion of Σd(p) near d = 4, using, for  > 0, Γ() =  γ + (), γ = 0.57721 ... (Euler’s constant) and (1 )B(1 , 1 ) = 1 +  + (2), one obtains,− O with Γ(x + 1) xΓ(x), the result: − − − O ≡ 2   2  g0 2 p Σd(p) = CF p γ + 1 ln − + (4 d) (4π)2 6 4 d − − 4πµ2 O − − Logarithmic divergence (effectively) in original integral shows up as simple pole at d = 4. d is “close” to 4, but d = 4 everything is finite (regular) ! Note: in the literature6 → there are several conventions to define dimensions d close to (smaller than) 4 and thus some infinitesimal  > 0. Muta: d = 4 2. Peskin + Schroeder: d = 4 . Sometimes d = 4 + ,  < 0 is used. − −

13 1-loop tensor integrals: Passarino-Veltman functions General tensor integrals of the form

Z d µ1 µm µ1...µm d k k . . . k I (p1, p2,...) = d 2 2 2 2 2 2 (2π) (k m )((k + p1) m )((k + p1 + p2) m ) ... − 1 − 2 − 3 can be reduced to scalar 1-loop integrals (transformations of variables, partial fractions are valid in dimensional regularization). Attention: various conventions are used in the literature, we follow Jegerlehner [10]. Note that everywhere m2 m2 i is implied. → − Steps

1. Covariant decomposition:

µ1...µm µ1 µm 2 2 I (p1, p2,...) = p . . . p I1(p , p1 p2, p ,...) + ... 1 1 1 · 2 in terms of appropriately symmetrized tensor with linearly independent momenta and gµν.

2. Contraction with gµν:

k2 (k2 m2) + m2 m2 = − = 1 + k2 m2 k2 m2 k2 m2 − − −

3. Contraction with piµ:

2 2 2 2 2 2 2 2k p1 = (k + p1) m2 k m1 p1 m2 m1 · − 2 − 2 −2 − − − 2k p1 1 1 p m m · = 1 − 2 − 1 (1)(2) (1) − (2) − (1)(2)

1 where (i) denotes the scalar propagator with mass mi and appropriate momenta as in µ1...µm I (p1, p2,...).

Note: solving the resulting system of equations for the scalar integrals can lead to numerical instabilities for exceptional momentum configurations (on-shell momenta, massless particles), where special care needs to be taken.

Conventions (to conform with Passarino-Veltman [11])

Notation: • Z 16π2 Z ddk d k ≡ i (2π)

n Define invariant functions I1,... using a factor of ( 1) where n = number of • − and factor ( 1) in front of the terms with gµiµk −

Z µ1 µm k . . . k n µ1 µm 2 2 2 2 = ( 1) (p1 . . . pn I1 + ...) (k m ) ((k + p1 + p2 + ... + pn−1) m ) − k − 1 ··· − n

14 1. Tadpoles (One-point functions) m

Z 1 shift Z 1 2 2 = 2 2 = A0(m) p k k m k (k + p) m − −  −  2 2 2 A0(m) = m γ + ln(4π) + 1 ln(m ) − 4 d − − 2 − 2m 2 = + finite, dA0(m) = 4A0(m) + 2m −4 d Z µ Z −µ µ Z µ Z k k p k µ 1 µ 2 2 = 2 − 2 = 2 2 p 2 2 = p A0(m) k (k + p) m k k m k k m − k k m − − | {z− } − =0 Z µ ν k k µ ν µν 2 2 = p p A21 + g A22 k (k + p) m − − Z µ ν Z µ ν (k p) (k p) µ ν k k = − 2 −2 = p p A0(m) + 2 2 k k m − k m Z µ ν Z − 2 Z Z − k k p=0 k 2 1 2 gµν 2 2 = dA22 = 2 2 = 1 +m 2 2 = m A0(m) k m k k m k k k m − − − |{z} − =0 in dim. reg. m2 m2 m4 A21 = A0(m),A22 = A0(m) = A0(m) + ⇒ − d − 4 8

m1

2. Self-energies (Two-point functions) p Z 1 2 = B0(m1, m2, p ) 2 2 2 2 k (k m ) (k + p) m − 1 − 2 m2 2 2 B0(m1, m2, p ) = γ + ln(4π) 4 d − −Z 1  2 2 2  dx ln p x(1 x) + m1(1 x) + m2x i − 0 − − − − 2 = + finite, dB0 = 4B0 2 Z µ 4 d − k µ 2 − = p B1(m1, m2, p ) k (1)(2) 2 1  2 2 2  B1(m1, m2, p ) = A0(m1) + A0(m2) p + m m B0 2p2 − − 1 − 2 Z µ ν k k µ ν µν = p p B21 g B22 k (1)(2) −   2  1 2 2 2 2 1 2 2 p B21 = A0(m2) 2 p + m m B1 m B0 m + m 3p2 − − 1 − 2 − 1 − 2 1 2 − 3   2  1 2 2 2 2 2 2 p B22 = A0(m2) p + m m B1 2m B0 m + m 6 − 1 − 2 − 1 − 1 2 − 3

Similarly for vertex functions / form-factors (three-point functions C0) and box diagrams → (four-point functions D0). → 15 4 Renormalization of QCD to one loop

Renormalization In general, Green’s functions which include loops are divergent, if we remove the regulator (d 4 in dim. reg.). → Important observation: the parameters in the Lagrangian (masses, couplings) and the fields are not observable quantities can redefine masses, couplings and fields in such a way that we get finite results for physical→ quantities (observables), like S-matrix elements, physical masses and physical couplings. A theory is called renormalizable, if only a finite number of parameters or fields needs to be redefined. While redefining parameters explicitly is physically intuitive, it is difficult to keep track for higher loop computations. A systematic approach uses counter-terms and renormalization con- stants Z. The renormalization constants Z are then adjusted in each order of perturbation theory in such a way that Green’s functions are finite. MS and MS renormalization schemes Since there is an ambiguity to define the infinite piece of a Green’s function (from the loop integral), the elimination of divergences is not unique and thus also the finite piece of the Green’s function is not unique renormalization scheme dependence. → 2 Below will use the minimal subtraction (MS) scheme, where only the pole term 4−d is subtracted in the Green’s function to define the Z-factors. In the modified minimal subtraction (MS) 2  scheme, all the terms 4−d γ + ln(4π) , which always appear together, are subtracted. These are mass-independent− renormalization schemes, in contrast to mass-dependent schemes, such as the on-shell scheme (subtraction at p2 = m2) or the momentum-space subtraction (MOM) scheme (subtraction at some off-shell momentum p2 = m2). 6 Bare Lagrangian = Renormalized Lagrangian + Counter-terms i a a Redefine bare fields ψ ,Aµ, χ and bare parameters gs, m, ξ in terms of renormalized quantities: i 1/2 i ψ = Z2 ψr a 1/2 a Aµ = Z3 Arµ a ˜1/2 a χ = Z3 χr

gs = Zggs,r

m = Zmmr

ξ = Z3ξr ˜ Z2,Z3, Z3: quark field, gluon field and ghost field renormalization constants

Zg,Zm: coupling constant and quark mass renormalization constants

Note: to simplify the notation, we will write gr gs,r in the following. ≡ a A remark about ξ = Z3ξr: it is nontrivial that we can choose the same Z3 as for Aµ. Reason: longitudinal part of gluon propagator ( kµkν) is not renormalized (no radiative corrections). Follows from BRST invariance generalized∼ Ward-Takahashi / Slavnov-Taylor identities. → Inserting this into the Lagrangian yields (bare) = r + CT LQCD L L r = r,free+ r,int identical to QCD except that all bare quantities are replaced by renormalized ones.L L L L

16 Counter-term Lagrangian

i µ i i i CT = (Z2 1)ψ iγ ∂µψ (Z2Zm 1)mrψ ψ L − r r − − r r 1 aµ bν ˜ a ∗ b +(Z3 1) A δab (gµν ∂µ∂ν) A + (Z3 1) (χ ) δab ( ) χ − 2 r  − r − r − r i a µ j a gr abc a a bµ cν +(Z1F 1)grψ T γ ψ A (Z1 1) f (∂µA ∂νA )A A − r ij r rµ − − 2 rν − rµ r r 2 gr abe cde a b cµ dν ˜ abc µ a ∗ b c (Z4 1) f f A A A A (Z1 1)grf (∂ χ ) χ A − − 4 rµ rν r r − − r r rµ where 1/2 3/2 2 2 ˜ ˜ 1/2 Z1F = ZgZ2Z3 ,Z1 = ZgZ3 ,Z4 = Zg Z3 , Z1 = ZgZ3Z3 Counter-terms treated as part of interaction Lagrangian, even if they are quadratic in the fields, since (Z 1) contains at least one factor of g2. − r Feynman rules for counter-terms p ←− i j i [(Z2 1)p (Z2Zm 1)mr] δij ⊗ − 6 − −

k ←− 2  aµ bν i(Z3 1)δab kµkν k gµν ⊗ − −

k ←− ˜ 2 a b i(Z3 1)δabk ⊗ − aµ a i(Z1F 1)grT γµ − ij i j ⊗

a1µ1

k ↓ 1 a1a2a3 (Z1 1)grf Vµ1µ2µ3 (k1, k2, k3) k2 ⊗ k3 − ր տ

a2µ2 a3µ3

a1µ1

2 a1a2a3a4 a2µ2 a4µ4 i(Z4 1)g W ⊗ − − r µ1µ2µ3µ4

a3µ3

aµ ˜ abc (Z1 1)grf kµ k − b ←− c ⊗

17 Generalized Ward-Takahashi / Slavnov-Taylor identities

Important observation: the gauge coupling renormalization constant Zg can be determined by using any of the last 4 counter-terms. One has to prove that all 4 ways lead to the same result for Zg using the generalized Ward-Takahashi / Slavnov-Taylor identities following from BRST symmetry and a gauge-invariant regularization, such as dimensional regularization.

If the Zg really are all the same, then one obtains the constraints (Slavov-Taylor identities in narrow sense): Z Z˜ Z Z 1 = 1 = 1F = 4 ˜ Z3 Z3 Z2 Z1

Z1F Compare with QED: Z1F = Z2. In QCD: = 1 Z1F = Z2. Z2 6 ⇒ 6 Slavnov-Taylor identities guarantee universality of renormalized coupling gr, but since the cur- a i a j rent Jµ = ψ γµTijψ is not gauge-invariant, its normalization is not unambiguously fixed Z1F = Z2. Because of these Slavnov-Taylor identities, out of the eight renormalization ⇒ 6 ˜ ˜ constants Z2,Zm,Z3, Z3,Z1F,Z1,Z4, Z1, only five are independent.

Superficially divergent Feynman amplitudes in QCD Power counting shows that Feynman amplitudes in QCD with 3 ddiv = 4 NG (NF + NFP) 0 − − 2 ≥ are superficially divergent in d = 4 dimensions, where

NG = number of external gluon fields

NF = number of external quark fields

NFP = number of external Faddeev-Popov ghost fields

8 cases: (NG,NF,NFP) = (0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 2, 0), (1, 0, 2), (3, 0, 0), (4, 0, 0) First case corresponds to vacuum diagram normalization of generating functional Z[J]. Lorentz invariance forbids (1, 0, 0). → 7 Feynman amplitudes with overall divergences: →

                       

ddiv = 2 ddiv = 1 ddiv = 1

                               

ddiv = 1 ddiv = 0 ddiv = 0 ddiv = 0

Same structure as the 7 counter-terms written earlier ! “Suggests” renormalizability of QCD (overall divergence taken care of by the counter-terms). Explicit calculation ddiv = 0 for all diagrams (logarithmically divergent), due to gauge and Lorentz invariance ! →

18 1-loop calculation of renormalization constants Z

Below we give the results for the renormalization constants Z of QCD at 1-loop order in the MS-scheme (see Muta [6] for details). The renormalized coupling constant in d dimensions is 4−d 2 written as gr = gR µR , where gR is dimensionless.

ab (i) Gluon self-energy Πµν(k)

k

    ←−  = + +    aµ bν

(G) (T) (FP)

+ + ⊗

(F) (CT)

ab 2  2 Π (k) = δab kµkν k gµν Π(k ) µν − 2 "   # 2 gR 1 13 4 2 Π(k ) = CA ξr + TRNf + (Z3 1) + finite terms (4π)2 −2 3 − 3 4 d − | {z } | {z } − (G)+(FP) (F),Nf flavors

Z ddq (T) 0 in dim. reg. (massless tadpole) ∼ q2 ≡ Color factors:

a b acd bcd tr T T = δabTR, f f = δabCA 1 For SU(N ): T = ,C = N c R 2 A c

Gauge invariance: µ ab k Πµν(k) = 0

Generalized Ward-Takahashi identitity ddiv = 0 (only logarithmic divergence). Note: only (G) + (FP) leads to gauge invariant structure.→ (F) itself is gauge invariant. no mass renormalization gluon remains massless ⇒ ⇒

Gluon field renormalization constant Z3 in MS-scheme:

2     MS gR 1 13 4 2 4  Z = 1 CA ξr + TRNf + g 3 − (4π)2 −2 3 − 3 4 d O R −

19 (ii) FP-ghost self-energy Πe ab(k)

k

    ←−  = +   a  b ⊗

 2    ab 2 gR 3 ξr 2 ˜ Πe (k) = δabk CA − + (Z3 1) + finite terms −(4π)2 4 4 d − − k2 no mass renormalization FP-ghost remains massless ∼ ⇒ ⇒ ˜ FP-ghost field renormalization constant Z3 in MS-scheme:

2   ˜ gR 3 ξr 2 4  Z3 = 1 + CA − + g (4π)2 4 4 d O R −

(iii) Quark self-energy Σij(p)

p

    ←−  = +   i  j ⊗

ij Σ (p) = δij [(A mr B p) (Z2Zm 1) mr + (Z2 1) p] + finite terms − 6 − − − 6 2 gR 2 4  A = CF (3 + ξr) + g −(4π)2 4 d O R − 2 gR 2 4  B = CF ξr + g −(4π)2 4 d O R − Color factor: 2 a a Nc 1 (T T )ij = δijCF , for SU(Nc): CF = − 2Nc

Mass and quark field renormalization constants Zm and Z2 in MS-scheme:

2 MS 4  gR 2 4  Z = 1 + A B + g = 1 3CF + g m − O R − (4π)2 4 d O R 2 − MS 4  gR 2 4  Z = 1 + B + g = 1 CF ξr + g 2 O R − (4π)2 4 d O R −

20 abc (iv) Three-gluon vertex Λµνλ(k1, k2, k3)

aµ k ↓ 1

k2 k3

   −→  ←−  = + +    bν cλ

+ + permutations + ⊗

 2      abc abc gR 17 3ξr 4 2 Λ (k1, k2, k3) = igRf Vµνλ(k1, k2, k3) CA + + TRNf + (Z1 1) µνλ − (4π)2 −12 4 3 4 d − − +finite terms

Three-gluon vertex renormalization constant Z1 in MS-scheme

2     MS gR 17 3ξr 4 2 4  Z = 1 CA + + TRNf + g 1 − (4π)2 −12 4 3 4 d O R −

˜ abc 0 (v) Ghost-gluon vertex Λµ (k, p, p )

aµ k ↓ p p′

   ←−  ←−  = + +   ⊗ b  c

 2   ˜ abc 0 abc gR ξr 2 ˜ Λ (k, p, p ) = igRf pµ CA + Z1 1 + finite terms µ − (4π)2 2 4 d − − ˜ Ghost-gluon vertex renormalization constant Z1 in MS-scheme

2 ˜MS gR ξr 2 4  Z = 1 CA + g 1 − (4π)2 2 4 d O R −

21 aij 0 (vi) Quark-gluon vertex ΛFµ (k, p, p )

aµ k ↓ p p′

   ←−  ←−  = + +   ⊗ i  j

 2    aij 0 a gR 3 + ξr 2 Λ (k, p, p ) = gRγµT CA + ξrCF + (Z1F 1) + finite terms Fµ ij (4π)2 4 4 d − −

Quark-gluon vertex renormalization constant Z1F in MS-scheme

2   MS gR 3 + ξr 2 4  Z = 1 CA + ξrCF + g 1F − (4π)2 4 4 d O R −

a1...a4 (vii) Four-gluon vertex Λµ1...µ4 (k1 . . . k4)

a1µ1 a4µ4

ց  ւ  k1  k4    = + +   k2  k3 ր տ

a2µ2 a3µ3

+ + + perm. + ⊗

 2  2  4  2  a1...a4 2 a1...a4 gR Λ (k1 . . . k4) = g W + ξr CA + TRNf + (Z4 1) µ1...µ4 − R µ1...µ4 (4π)2 −3 3 4 d − − +finite terms

Four-gluon vertex renormalization constant Z4 in MS-scheme

2    MS gR 2 4 2 4  Z = 1 + ξr CA + TRNf + g 4 − (4π)2 −3 3 4 d O R −

22 Conclusions All the 1-loop divergences in the 7 superficially divergent Feynman amplitudes can be cancelled by the counter-terms with an appropriate choice of the Z-factors. Proven renormalizability of QCD at 1-loop order →

Note: We did not use gauge symmetry so far (except for ξ = Z3ξr). Can check that

MS ˜MS MS MS 2   Z1 Z1 Z1F Z4 gR 3 + ξr 2 4  = = = = 1 CA + gR ZMS Z˜MS ZMS ZMS − (4π)2 4 4 d O 3 3 2 1 − Slavnov-Taylor identity satisfied at 1-loop order in MS-scheme (dimensional regularization pre- serves gauge invariance), but ZMS = ZMS. 1F 6 2

Gauge coupling renormalization constant Zg in MS-scheme

As mentioned earlier, we have four different ways to calculate Zg, all of them are equivalent owing to the Slavnov-Taylor identities. From a practical point of view, perhaps the easiest way of calculating Zg is to use the definition

Z˜ Z = 1 g ˜ 1/2 Z3Z3 ˜ ˜ where Z1 appears in the ghost-gluon vertex, Z3 in the ghost self-energy and Z3 in the gluon self-energy. Using the relations given above, one obtains

2 g (11CA 4TRNf ) 2 ZMS = 1 R − + g4  g − (4π)2 6 4 d O R − ˜ ˜ MS Note: whereas Z1, Z3 and Z3 all depend on the renormalized gauge parameter ξr, Zg is independent of ξr. This is also true for

2 MS gR 2 4  Z = 1 3CF + g m − (4π)2 4 d O R −

23 5 Renormalization group and running αs Renormalization two-fold ambiguity: → 1. Arbitrariness of renormalization condition: how to define divergent and finite pieces of Green’s functions (= arbitrariness of splitting bare Lagrangian in renormalized La- grangian and counter-terms).

2. Arbitrariness in choosing renormalization scale µR.

Many different expressions for physical quantities. But they all describe one unique physical reality,→ starting from a unique bare Lagrangian: connected by a finite renormalization. Consider coupling and mass renormalization, i.e. relation between renormalized and bare quan- tities (not necessarily using dimensional regularization):

−1 −1 gr = Zg gs, mr = Zm m, at scale µR 0 0 −1 0 −1 0 gr = (Zg) gs, mr = (Zm) m, at scale µR

Physical quantities, such as S-matrix elements, should be the same:

0 0 0 0 S (pi; gr, mr, µR) = S(pi; gr, mr, µR)(pi : fixed set of external momenta)

0 0 0 Transformation from scheme (gr, mr, µR) to (gr, mr, µR)

0 Zg gr = zggr, zg = 0 Zg

0 Zm mr = zmmr, zm = 0 Zm is done by a finite renormalization (in ratios zg, zm the divergent pieces cancel out by construc- tion of Z-factors) physical predictions invariant under finite renormalization (transformations 0 → µR µ form an Abelian group) renormalization group (RG) equations. → R → In practice, we can calculate S-matrix elements only in perturbation theory and truncate the n series after the first few orders, e.g. to order gr :

0 0 0 0 n+1 S (pi; g , m , µ ) S(pi; gr, mr, µR) = g r r R − O r scheme dependence of physical quantities, like cross-sections at the LHC ⇒

24 Renormalization group equations for QCD in MS (MS)-scheme For QCD and using dimensional regularization, we can rephrase the invariance of physical quan- tities under finite renormalization as follows. First we recall that the bare and the renormalized coupling constants acquire a mass dimension in d = 4: 6 4−d 2 gs = g0 µ0

4−d 2 gr = gR µR where g0 and gR are dimensionless coupling constants. The mass scale µ0 for the bare coupling gs is a fixed scale, while the mass scale µR for the renormalized coupling gr is a variable parameter. µR is identified with the renormalization scale in the MS (or MS)-scheme. With the relation between bare and renormalized quantities, we can therefore write

4−d   2 µ0 −1 gR(µR) = Zg(µR) g0 µR

−1 mR(µR) = Zm(µR) m where mR mr (to make the notation similar). The bare parameters≡ g and m are regarded as fixed constants and are free from any dependence on the renormalization scale µR. We therefore get the following set of differential equations for the running of gR and mR:

 4−d  d d 2 µR gs = 0 µR Zg gR µR = 0 (1) dµR ⇒ dµR d d µR m = 0 µR (Zm mR) = 0 (2) dµR ⇒ dµR

  d 4 d µR dZg (1) µR gR = β, β = − gR gR, β function ⇒ dµR − 2 − Zg dµR −   d µR dZm (2) µR mR = mRγm, γm = ⇒ dµR − Zm dµR In QCD, there is a similar equation for the running of the renormalized gauge fixing parameter ξR ξr. Note that the divergences in the Z-factors cancel out β and γm are finite functions, ≡ ⇒ if we remove the regulator (in dim. reg. d 4). Differential equations describe how gR(µR), → mR(µR) (and ξR(µR)) change with µR, such that physical quantities are independent of the renormalization scale µR. In general renormalization scheme: Z = Z(gR(µR), mR(µR), ξR(µR), µR) complicated, cou- pled set of differential equations. In MS (MS) scheme a great simplification→ occurs:

MS MS MS MS β = β (gR), γm = γm (gR)

Equation (1) decouples ! ⇒

25 QCD β-function (in MS-scheme) and asymptotic freedom

  MS 2 2 4  1 11CA 4TRNf Z = 1 A11g + g ,A11 = − g − R 4 d O R (4π)2 6 −     MS 4 d 2 2 MS 5  β (gR) = − gR + 2A11g β (gR) + g 2 R 4 d O R − 3 5  = 2A11g + g , 4 d − R O R − Therefore MS 3 5  1 11CA 4TRNf β (gR) = β0g + g , β0 = − − R O R (4π)2 3 MS In the MS scheme, Zg is always of the form

 2   2 2 ZMS = 1 + A(g ) + B(g ) + ... g R 4 d R 4 d − − Then MS 2 dA(gR) β (gR) = gR dgR

Asymptotic freedom

If β0 > 0, gR(µR) 0 with increasing µR asymptotic freedom ( for large µR we can trust perturbation theory).→ ⇒ → 1 QCD: β0 > 0, if 11CA 4TRNf > 0. For real world, Nc = 3. SU(3) : CA = 3,TR = − 2 ⇒ asymptotically free for Nf 16, i.e. QCD with Nf = 6 flavors as observed so far in Nature, is asymptotically free. ≤ If there are no quarks in theory (pure gluodynamics): Nf = 0 always β0 > 0 always asymptotically free. ⇒ ⇒ Origin of asymptotic freedom: self-interactions of gluons (3-gluon and 4-gluon vertices).

Neglecting higher order terms of gR in the β-function, the differential equation

dgR 3 µR = β0gR dµR − can easily be solved g2 (˜µ ) 2 ( ) = R R gR µR  2  2 µR 1 + gR(˜µR)β0 ln 2 µ˜R whereµ ˜R is some fixed reference scale, e.g.µ ˜R = MZ , where

2 MS gR(MZ )MS α (MZ ) = 0.1184(7) (PDG 2010 [2]) s ≡ 4π

2 Since gR(µR) 0 as µR increases, the approximate solution of the differential equation becomes better and better.→

26 Running αs The β-function in the MS-scheme is now known to four-loop order

3 5 7 9 11 β(gR) = β0g β1g β2g β3g + g − R − R − R − R O R 1 11CA 4TRNf β0 = − (4π)2 3     1 34 2 5 β1 = C 4 CA + CF TRNf (4π)4 3 A − 3       1 2857 3 1415 2 205 2 158 44 2 2 β2 = C C + CACF 2C TRNf + CA + CF T N (4π)6 54 A − 27 A 9 − F 27 9 R f where 2 Nc=3 Nc 1 Nc=3 4 1 CA = Nc = 3,CF = − = ,TR = 2Nc 3 2 The results at 1-loop, 2-loop and 3-loop order were evaluated in Refs. [12], [13], [14], respectively. See Ref. [15] for the 4-loop result β3.

Comparison of the running of αs (4-loop running, 3-loop matching at quark mass thresholds) with experimental data shows excellent agreement and confirms the behavior of asymptotic freedom:

0.5 July 2009 α s(Q) Deep Inelastic Scattering + – 0.4 e e Annihilation Heavy Quarkonia

0.3

0.2

0.1

QCD α s (Μ Z ) = 0.1184 ± 0.0007 1 10 100 Q [GeV]

Source: Bethke [16]

27 6 Selected References

[1] H. Fritzsch, M. Gell-Mann and H. Leutwyler, “Advantages Of The Color Octet Gluon Picture”, Phys. Lett. B 47, 365 (1973).

[2] K. Nakamura et al. [Particle Data Group], “Review of particle physics”, J. Phys. G 37, 075021 (2010).

[3] D. J. Gross, F. Wilczek, “Ultraviolet Behavior of Nonabelian Gauge Theories”, Phys. Rev. Lett. 30, 1343 (1973); “Asymptotically Free Gauge Theories. 1”, Phys. Rev. D8, 3633 (1973); “Asymptotically Free Gauge Theories. 2.”, Phys. Rev. D9, 980 (1974).

[4] H. D. Politzer, “Reliable Perturbative Results for Strong Interactions ?”, Phys. Rev. Lett. 30, 1346 (1973); “Asymptotic Freedom: An Approach to Strong Interactions”, Phys. Rept. 14, 129 (1974).

[5] L. D. Faddeev and V. N. Popov, “Feynman diagrams for the Yang-Mills field”, Phys. Lett. B 25, 29 (1967).

[6] T. Muta, “Foundations of Quantum Chromodynamics”, World Scientific Lecture Notes in Physics, Vol. 78, World Scientific, Singapore, 2010 (3rd edition).

[7] C. G. Bollini, J. J. Giambiagi, “Lowest order divergent graphs in ν-dimensional space”, Phys. Lett. B40, 566 (1972); “Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter”, Nuovo Cim. B12, 20 (1972); G. ’t Hooft, M. J. G. Veltman, “Regularization and Renormalization of Gauge Fields”, Nucl. Phys. B44, 189 (1972); J. F. Ashmore, “A Method of Gauge Invariant Regularization”, Lett. Nuovo Cim. 4, 289 (1972); G. M. Cicuta, E. Montaldi, “Analytic renormalization via continuous space dimension”, Lett. Nuovo Cim. 4, 329 (1972).

[8] J. C. Collins, Renormalization, Cambridge University Press, 1984.

[9] K. G. Wilson, “Quantum field theory models in less than four-dimensions”, Phys. Rev. D7, 2911 (1973).

[10] F. Jegerlehner, “Renormalizing the ”, in the Proceedings of Testing the Standard Model - TASI-90: Theoretical Advanced Study Institute in Elementary Particle Physics, Boulder, Colorado, 3-29 June 1990, pp 476-590. A scanned version can be found here: http://ccdb4fs.kek.jp/cgi-bin/img_index?9107013

[11] G. Passarino, M. J. G. Veltman, “One Loop Corrections for e+e− Annihilation Into µ+µ− in the Weinberg Model”, Nucl. Phys. B160, 151 (1979).

[12] G. ’t Hooft, report at the Marseille Conference on Yang-Mills Fields, 1972 and Refs. [3,4].

[13] W. E. Caswell, “Asymptotic Behavior of Nonabelian Gauge Theories to Two Loop Or- der”, Phys. Rev. Lett. 33, 244 (1974); D. R. T. Jones, “Two Loop Diagrams in Yang-Mills Theory”, Nucl. Phys. B75, 531 (1974); E. Egorian, O. V. Tarasov, “Two Loop Renormal- ization Of The QCD In An Arbitrary Gauge”, Theor. Math. Phys. 41, 863 (1979) [Teor. Mat. Fiz. 41, 26 (1979)].

28 [14] O. V. Tarasov, A. A. Vladimirov, A. Y. Zharkov, “The Gell-Mann-Low Function of QCD in the Three Loop Approximation”, Phys. Lett. B93, 429 (1980); S. A. Larin, J. A. M. Ver- maseren, “The Three loop QCD Beta function and anomalous dimensions”, Phys. Lett. B303, 334 (1993).

[15] T. van Ritbergen, J. A. M. Vermaseren, S. A. Larin, “The Four loop beta function in quantum chromodynamics”, Phys. Lett. B400, 379 (1997); M. Czakon, “The Four-loop QCD beta-function and anomalous dimensions”, Nucl. Phys. B710, 485 (2005).

[16] S. Bethke, “The 2009 World Average of αs(MZ )”, Eur. Phys. J. C 64, 689 (2009).

29