Quantum Field Theory

Theory of Elementary Particles

Ahmed Ali DESY, Hamburg

DESY Summer Student Programme 2009, DESY, Hamburg

August 10 – 14, 2009 Theory of Elementary Particles (page 1) Ahmed Ali DESY, Hamburg August 10 – 14, 2009 Theory of Elementary Particles (page 2) Ahmed Ali DESY, Hamburg Structure of the ﬁrst generation of leptons and quarks in SM

The hypercharge quantum numbers Y are ﬁxed by the relation Q = I + Y • 3 Q are the electric charges (in units of the proton charge): • u +2/3 ν 0 Q = ; Q e = d 1/3 e 1 − − I is the third-component of the weak isospin: • 3 u +1/2 ν +1/2 I = ; I e = 3 d 1/2 3 e 1/2 L − L − I = I = 0 for u , d , e • 3 R R R

August 10 – 14, 2009 Theory of Elementary Particles (page 3) Ahmed Ali DESY, Hamburg Plan of these lectures Introduction to Quantum Field Theory (QFT) • Gauge Invariance and Quantum Electrodynamics (QED) • + + An elementary QED process e e− µ µ− • → Gauge Invariance and Quantum Chromodynamics (QCD) • The QCD running coupling constant α (Q2) • s Applications of perturbative QCD to hard processes • The Electroweak Theory • The Higgs Mechanism • CKM Matrix and CP Violation • Precision Tests of the EW theory • Higgs Boson Physics • Gauge coupling uniﬁcation • Conclusions •

August 10 – 14, 2009 Theory of Elementary Particles (page 4) Ahmed Ali DESY, Hamburg Recommended References

Books Relativistic Quantum Mechanics & Relativistic Quantum Fields • James D. Bjorken, Sidney D. Drell, McGraw-Hill Book Company (1965)

An Introduction to Quantum Field Theory • Michael E. Peskin, Daniel V. Schroeder, ABP, Westview Press (1995)

Modern Quantum Field Theory - A Concise Introduction • Thomas Banks, Cambridge University Press (2008)

Foundations of Quantum Chromodynamics • Taizo Muta, World Scientiﬁc -revised edition (2005)

Dynamics of the Standard Model • John F. Donoghue, Eugene Golowich, Barry R. Holstein, Cambridge University Press (Revised version 1994)

August 10 – 14, 2009 Theory of Elementary Particles (page 5) Ahmed Ali DESY, Hamburg Review Articles A QCD Primer • G. Altarelli, arXiv Preprint hep-ph/0204179 The Standard Model of Electroweak Interactions • A. Pich, arXiv Preprint 0705.4264 [hep-ph] Concepts of Electroweak Symmetry Breaking and Higgs Physics • M. Gomez-Bock, M. Mondragon, M. M¨uhlleitner, M. Spira, P.M. Zerwas arXiv Preprint 0712.2419 [hep-ph] Essentials of the Muon g 2 − • F. Jegerlehner, DESY Report 07-033, arXiv:hep-ph/0703125 (2007) Review papers on the Standard Model Physics • Particle Data Group (URL:http://pdg.lbl.gov/) I. Hinchliﬀe (Quantum Chromodynamics) • J. Erler & P. Langacker (Electroweak Model and Constraints on New Physics) • A. Ceccucci, Z. Ligeti, Y. Sakai (The CKM Quark-Mixing Matrix) • G. Bernardi, M. Carena, T. Junk (Higgs Bosons: Theory and Searches) • B. Foster, A.D. Martin, M.G. Vincter (Structure Functions) •

August 10 – 14, 2009 Theory of Elementary Particles (page 6) Ahmed Ali DESY, Hamburg Quantum Field Theory - Introduction Physical systems can be characterized by essentially two features, roughly speaking, • size (large or small) and speed (slow or fast) Velocity v c: For slowly moving objects (compared to the speed of light), we ≪ • have the Classical (Newtonian) mechanics to describe this motion Velocity v c: For fast moving system (approaching the speed of light), we have → • Relativistic (Einstein’s) mechanics To describe objects at small distances (say, atoms), energy and other attributes (such • as angular momenta) are quantized in units of the Planck’s constant ~ = 0. For 6 quantum systems ~ = 0 moving with non-relativistic velocities v c, we move 6 ≪ from Classical mechanics to Non-relativistic Quantum mechanics For ~ = 0 and v c, we have Relativistic Quantum mechanics (RQM) • 6 → RQM is insuﬃcient as it does not account for the particle production and • annihilation. However, E = mc2 allows pair creation Quantum Mechanics + Relativity + Particle creation and annihilation = Quantum ⇒ • Field Theory Quantum Field Theory Application of Quantum mechanics to dynamical systems of ﬁelds Φ(t, ~x) •

August 10 – 14, 2009 Theory of Elementary Particles (page 7) Ahmed Ali DESY, Hamburg Preliminaries Space-time coordinates & Lorentz scalars Natural units (c = ~ = 1). In these units, the Compton wavelength of the electron • 11 is 1/me 3.86 10− cm and the electron mass is me 0.511 MeV. In the same units,≃ m ×938 MeV and m 172 GeV ≃ p ≃ t ≃ Space-time coordinates (t,x,y,z) are denoted by the contravariant four vector • xµ = (x0, x1, x2, x3) (t,x,y,z)=(t, ~r) ≡

The covariant four-vector xµ is obtained by changing the sign of the space • components x = (x , x , x , x ) = g xν (t, x, y, z)=(t, ~r) µ 0 1 2 3 µν ≡ − − − − µν with the metric gµν = g 10 0 0 0 1 0 0 gµν =  −  0 0 1 0 −  0 0 0 1   −    The inner product, yielding a Lorentz-scalar, is x2 = t2 ~r2 = t2 x2 y2 z2 • − − − − Momentum vectors are deﬁned as: • µ p = (E,px,py,pz) and the inner product (Lorentz scalar) is: p .p = pµp = E E ~p .~p & x.p = tE ~r.~p 1 2 1 2µ 1 2 − 1 2 −

August 10 – 14, 2009 Theory of Elementary Particles (page 8) Ahmed Ali DESY, Hamburg Preliminaries Momentum Operator in Coordinate representation

∂ ∂ 1 pµ = i i , ~ i∂µ ∂x ≡ ∂t i ∇ ≡ µ This transforms as a contravariant four-vector • ∂ ∂ ∂ pµp = = ~ 2 ( )2 = µ µ −∂xµ ∂x ∇ − ∂t − The four-vector potential of the electromagnetic field is defined as • µ µν A = (Φ, A~) = g Aν where Φ = Φ(t, ~x) is Scalar potential; A~ = A~(t, ~x) is Vector potential µν ν Maxwell Equations: ∂µF = eJ • 2 where e is the electric charge ; Sommerfeld Constant α = e 1/137 and J ν is 4π ∼ the electromagnetic current The field strength tensor is defined as F µν = ∂ Aµ ∂ Aν , and • ∂xν − ∂xµ E~ = (F 01, F 02, F 03) B~ = (F 23, F 31, F 12)

August 10 – 14, 2009 Theory of Elementary Particles (page 9) Ahmed Ali DESY, Hamburg Elements of Classical Field Theory Action S and Lagrangian L Action S = t2 Ldt: Time integral of the Langrangian L • t1 Lagrangian L = T V (Kinetic Energy - Potential Energy); L = L(q(t), q˙(t)) • R − In classical mechanics, the Lagrange function is constructed from the generalized • coordinates q(t) and velcocities q˙(t) Hamilton’s Principle of Least Action: Dynamics of the particle traversing a path q(t) • is determined from the condition δS = δ t2 L(q(t), q˙(t))dt = 0 t1 Also in Quantum mechanics, Classical pathR minimises Action [Feynman; see • Feynman Lectures, Vol. II] Lagrangian in field theory is constructed from Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) • ∇ L = d3x Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) L ∇ Z is the Lagrangian density; S = Ldt = d4x Φ(t, ~x), Φ(˙ t, ~x), ~ Φ(t, ~x) • L | {z L } ∇ Minimising the Action, i.e., δS =R 0 leads toRthe Euler-Lagrange Equation of motion • ∂ ∂ ∂ ∂ L L + ~ L = 0 ∂Φ − ∂t ∂Φ˙ ∇∂(~ φ) ∇ Hamiltonian: H = d3x π(x)φ˙(x) = d3x • − L H R h i R August 10 – 14, 2009 Theory of Elementary Particles (page 10) Ahmed Ali DESY, Hamburg Constructing : The free field case Spin 0 field: Φ(t,L ~x) For a free scalar field with mass m, the Lagrangian density is • 1 ∂ 1 Φ = ( Φ)2 (~ Φ)2 m2Φ2 Lfree 2 ∂t − ∇ − 2 The resulting field equation is called the Klein-Gordon equation • ∂ ( + m2)Φ(t, ~x)=0 with = ( )2 ~ 2 ∂t − ∇ 1 Spin 2 field: Ψ(t, ~x) Ψ(t, ~x)=Ψ (t, ~x) is a 4-component spinor (ψ ,ψ ,ψ ,ψ ) • α 1 2 3 4 Lagrangian density of a free Dirac (=spin 1 ) field with mass m • 2 ψ = ψ¯(iγµ∂ m)ψ = ψ¯(i∂/ m)ψ Lfree µ − − γµ are 4 4 matrices. The resulting field equation is called the Dirac equation • × (i∂/ m)ψ(t, ~x) = 0 − Spin 1 field with m = 0: (Example: Electromagnetic field) Aµ(t, ~x) Lagrangian density of a free spin-1 field: A = 1 F F µν • Lfree − 4 µν Yields Maxwell equations in the absence of charge and current densities • µν ∂µF (t, ~x) = 0

August 10 – 14, 2009 Theory of Elementary Particles (page 11) Ahmed Ali DESY, Hamburg Noether’s Theorem Relationship between Symmetries & Conservation Laws in CFT Noether’s theorem concerns continuous transformations on the field φ(x), which in • the infinitesimal form can be written as φ(x) φ′(x) = φ(x)+ α∆φ(x) → This is a symmetry transformation if it leaves the equation of motion (EOM) • invariant, which is ensured if the action S is invariant More generally, one can allow S to change by a surface term, since such a term • leaves the Euler-Lagrange EOM invariant Thus, for some µ(x), the Lagrangian must be invariant up to a 4-divergence • J (x) (x)+ α∂ µ(x) L → L µJ Let us compute explicitly α∆ , and the result is • ∂L ∂ ∂ α∆ = α∂µ L ∆φ + α L ∂µ L ∆φ L ∂(∂µφ) ∂φ − ∂(∂µφ) µ The second term vanishes due to the EOM. Settingh the first termiequal to α∂µ (x) • µ µ ∂ µ J ∂µJ (x)=0 for J (x) = L ∆φ (x) ∂(∂µφ) −J This result states that the current J µ(x) is conserved • Noether’s theorem can also be stated in the form that the charge • Q J 0(x)d3x ≡ all space is constant in time R

August 10 – 14, 2009 Theory of Elementary Particles (page 12) Ahmed Ali DESY, Hamburg Emmy Noether (1882 - 1935)

August 10 – 14, 2009 Theory of Elementary Particles (page 13) Ahmed Ali DESY, Hamburg Examples of conserved currents and charges Example 1: Lagrangian of a complex scalar field with mass m • (φ) = ∂ φ 2 m2 φ 2 L | µ | − | | (φ) is invariant under φ(x) eiαφ(x). Under this transformation: • L → α∆φ(x) iαφ(x); α∆φ∗(x) iαφ∗(x) → → − Conserved Noether currents: • Jµ = i[(∂µφ∗)φ φ∗(∂µφ) µ − µ Adding a term eJµ(x)A (x) to (φ), where A (x) is the electromagnetic • L 3 field, we interpret eJµ(x) as the electromagnetic current density with d xJ0(x) as the conserved electric charge R Example 2: Noether’s theorem applied to space-time transformations • Let us describe the infinitesimal transformation as xµ xµ aµ → − Field Transformation:φ(x) φ(x + a) = φ(x)+ aµ∂ φ(x) • → µ The Lagrangian is also a scalar, so it must transform in the same way • ν µ ′ = + a ∂ (δ ) L → L L µ ν L Noether’s theorem predicts four conserved currents: The stress-energy tensor, also •called the energy-momentum tensor of the field φ(x) µ ∂ µ T L ∂ν φ δ ν ≡ ∂(∂µφ) − L ν The conserved charge associated with time translation is the Hamiltonian • H = T 00(x)d3x = d3x d3x 1 π2 + 1 ( φ)2 + 1 m2φ2 H 2 2 ∇ 2 The conserved charge associated with spatial translations are the momentum • R P i = R T 0id3xR = π∂ φd3x − i R R August 10 – 14, 2009 Theory of Elementary Particles (page 14) Ahmed Ali DESY, Hamburg Field Quantization In QM, the coordinate q(t) and the conjugate coordinate p(t) = ∂L satisfy the • ∂q˙(t) commutation relation (recall, ~ = c = 1): [q(t),p(t)] = i where [A, B] = A.B B.A −

In QFT, the cordinate q(t) is replaced by the corresponding field and its conjugate • field. For the Spin 0 field: Φ(t, ~x), the conjugate field is:

∂ π(t, ~x) = L ∂Φ(˙ t, ~x) Scalar boson ﬁeld quantization

3 Φ(t, ~x), π(t, ~x′) = iδ (~x ~x′) − 1 ∂ For the Spin field: Ψα(t, ~x), the conjugate field is: πα(t, ~x) = ˙ L • 2 ∂Ψα(t,~x) Dirac fermion field quantization

3 Ψ (t, ~x), π (t, ~x′) = iδ (~x ~x′)δ α β − αβ where A, B = A.B+ B.A is the anti-commutator; the use of anti-commutator for fermion{ ﬁelds} instead of commutator for the boson ﬁelds is due to the spin 1/2 nature of the fermions

August 10 – 14, 2009 Theory of Elementary Particles (page 15) Ahmed Ali DESY, Hamburg For the Spin 1 field with m = 0, Aµ(t, ~x) (µ = 0, 1, 2, 3), the conjugate field is: • ∂ πµ(t, ~x) = L ∂A˙ µ(t, ~x) Spin-1 (m = 0) field quantization

µ ν µν 3 A (t, ~x), π (t, ~x′) = g iδ (~x ~x′) − QFT & particle interpretation Free ﬁeld equation for a Scalar ﬁeld with mass m: ( + m2)Φ(t, ~x) = 0 • General solution:

3 2 2 2 i(Et ~p.~x) +i(Et ~p.~x) Φ(t, ~x) dE d pδ(E ~p m ) a(E, ~p)e− − + a†(E, ~p)e − ∝ − − Z a(p)= a(E, ~p) is the ﬁeld operator, a (p) = a (E, ~p) is the hermitian conjugate • † † ﬁeld operator 3 Field quantization: [Φ(t, ~x), π(t, ~x′)] = iδ (~x ~x′) = (with p (E, ~p)): • − ⇒ ≡ (3) a(p), a†(p′) = 2Eδ (~p ~p′); a(p), a(p′) = 0; a†(p), a†(p′) = 0 − h i h i

August 10 – 14, 2009 Theory of Elementary Particles (page 16) Ahmed Ali DESY, Hamburg Fock space of particles Hamilton Operator

3 3 2 2 2 H = d x(πΦ˙ ) = dE d p δ(E ~p m )E a†(p)a(p) − L − − Z Z N(p) a (p)a(p) is the Number Operator • ≡ † N(p) n(p) = n(p) n(p) | i | i n(p) is the number of particles with spin 0, mass m, having an energy between E • and E + dE and momentum between ~p and ~p + d~p, E = + ~p2 + m2

The particle creation a†(p) and annihilation a(p) operators act as follows: • p a†(p) n(p) = n(p) + 1 (n + 1)(p) | i | i a(p) n(p) = n(p) (n 1)(p) | i p | − i They provide the basis for particle productionp and annihilation in QFT • Energy of the ground state is normalized to zero: H 0 = 0 • | i Multiparticle states obeying Bose-Einstein Statistics are built as follows • n1 n n (p ),...,n (p ) (a†(p )) ...(a†(p )) m 0 | 1 1 m m i ∝ 1 m | i

August 10 – 14, 2009 Theory of Elementary Particles (page 17) Ahmed Ali DESY, Hamburg QFT Roadmap Quantization of a classical theory starts with the Field Equations • Knowing these, we seek a Lagrangian which, via Hamilton’s principle, reproduces • them Having the Lagrangian, it is then possible to identify the canonical momenta and • their conjugates, and carry out the quantization procedure, in accordance with the following commutation relations: [p (0),q (0)] = iδ i j − ij [pi(0),pj(0)] = 0

[qi(0),qj(0)] = 0 It is understood from this analogy, we expect Φ(t, ~x) to play the role of the • coordinate qi(t) and ∂Φ(t, ~x) to correspond to q˙i(t); the discrete label i is replaced by the continuous coordinate variable ~x, which in the Heisenberg picture (used for quantization) becomes a function of both space and time

With this step, the ﬁelds Φ(t, ~x) and Π(t, ~x) become operators in a Hilbert space, • operating upon the state vectors

Physical states form a complete set of states in the Hilbert space • For the Dirac particles, the commutatation relations are replaced by the • anticommutation relations

August 10 – 14, 2009 Theory of Elementary Particles (page 18) Ahmed Ali DESY, Hamburg The Klein-Gordon Propagator In the Heisenburg picture, easier to discuss time-dependent quantities, the operators • π(x) and φ(x) are made time-dependent (likewise for π(x) = π(x,t)) iHt iHt φ(x) = φ(x,t) = e φ(x)e− Heisenberg EOM: i ∂ = [ ,H] • ∂t O O Computing the time-dependence of φ(x) and π(x) • d3p 1 ip.x ip.x 0 φ(x) = (2π)3 ape− + ap† e p =Ep , √2Ep | R ∂ π(x,t) = ∂t φ(x,t) Causality requirement: [φ(x), φ(y)] = 0 for (x y)2 < 0 • − The amplitude for a particle to propagate from a point x to a point y is: • 0 φ(x)φ(y) 0 h | | i Each operator φ is a sum of a and a† operators, but only the term • 3 (3) 0 apaq† 0 = (2π )δ (p q) survives h | | i − 3 d p 1 ip.(x y) = D(x y) = 0 φ(x)φ(y) 0 = 3 e− − ⇒ − h | | i (2π) 2Ep Hence, the commutator is: [φ(x), φ(y)] = D(x y) D(y x) • −R − − For (x y)2 > 0, it does not vanish • − For (x y)2 < 0, it vanishes due to a special Lorentz transformation, − • taking (x y) (x y). Thus, the requirement of causality is fulﬁlled: − → − − [φ(x), φ(y)] = 0 for (x y)2 < 0 −

August 10 – 14, 2009 Theory of Elementary Particles (page 19) Ahmed Ali DESY, Hamburg The Klein-Gordon Propagator (Contd.) As [φ(x), φ(y)] is a c-number, [φ(x), φ(y)] = 0 [φ(x), φ(y)] 0 • h | | i Let us compute this quantity • 3 d p 1 ip.(x y) +ip.(x y) 0 [φ(x), φ(y)] 0 = (e− − e − ) h | | i (2π)3 2E − Z p d3p 1 1 ip.(x y) 0 ip.(x y) 0 = e− − p =Ep + e− − p = Ep (2π)3 2E | 2E | − Z p − p 3 0 d p dp 1 ip.(x y) = − e− − (2π)3 2πi p2 m2 x0>y0 Z Z − |{z}

For x0 > y0, the last line is obtained by closing the contour of integration below to • pick up both the poles at p0 = E and p0 = E . p − p For x0 < y0, the contour of integration can be closed above excluding both the • poles, giving zero The last line above together with the prescription of going around the poles deﬁnes • the Retarded Green’s function: D (x y) θ(x0 y0) 0 [φ(x), φ(y)] 0 R − ≡ − h | | i

August 10 – 14, 2009 Theory of Elementary Particles (page 20) Ahmed Ali DESY, Hamburg Feynman propagator for a Klein-Gordon particle

4 d p i ip.(x y) D (x y) = e− − R − (2π)4 p2 m2 Z − One can compute to show that D (x y) is a Green’s function of the • R − Klein-Gordon operator, i.e., (∂2 + m2)D (x y) = iδ(4)(x y) R − − − The p0-integral in D (x y) can be evaluated by selecting four diﬀerent contours. R − • A diﬀerent presciption is due to Feynman

0 Feynman Prescription: poles at p = (Ep iǫ), displaced above and below the • real axis. A convenient way is to write± it as − 4 d p i ip.(x y) DF (x y) = e− − − (2π)4 p2 m2 + iǫ Z − For (x0 > y0), we can perform the p0 integral by closing the contour below, obtaining • D(x y) = 0 [φ(x), φ(y)] 0 − h | | i For (x0 < y0), we can perform the p0 integral by closing the contour above, obtaining • D(y x) = 0 [φ(y), φ(x)] 0 − h | | i

August 10 – 14, 2009 Theory of Elementary Particles (page 21) Ahmed Ali DESY, Hamburg Thus, we have • D(x y) for x0 > y0 D (x y) = − F − D(y x) for x0 < y0 − = θ(x0 y0) 0 [φ(x), φ(y)] 0 + θ(y0 x0) 0 [φ(y), φ(x)] 0 − h | | i − h | | i = 0 T φ(x)φ(y) 0 h | | i The last line above deﬁnes the “time ordering” symbol T • Applying (∂2 + m2) to the last Eq. above, one can verify D (x y) is the F − • Green’s function of the Klein-Gordon operator, and it is called the Feynman Propagartor for a Klein-Gordon particle Likewise, the Feynman Propagator for a free Dirac particle is • 4 d p ip.(x y) p/ + m SF (x y) = e− − − 2π4 p2 m2 + iǫ Z −

SF (x y) satisﬁes the Green’s function equation • − (4) (i /x m)SF (x y) = δ (x y) ∇ − 1 − − In the momentum space: SF (p) = p/ m+iǫ • −

August 10 – 14, 2009 Theory of Elementary Particles (page 22) Ahmed Ali DESY, Hamburg Gauge Invariance & Quantum Electrodynamics (QED)

Lagrangian describing a free Dirac fermion

= i ψ(x)γµ∂ ψ(x) m ψ(x)ψ(x) L0 µ − is invariant under global U(1) transformations L0 U(1) ψ(x) ψ′(x) exp iQθ ψ(x) −→ ≡ { } Qθ an arbitrary real constant; the phase of ψ(x) is a convention-dependent quantity without physical meaning However, no longer invariant if the phase transformation depends on the space-time L0 •coordinate, i.e., under local phase redeﬁnitions θ = θ(x)

U(1) ∂ ψ(x) exp iQθ (∂ + iQ ∂ θ) ψ(x) µ −→ { } µ µ

The ‘gauge principle’ requires that the U(1) phase invariance should hold locally • To achieve this, one introduces a new spin-1 ﬁeld A (x), transforming in such a way • µ as to cancel the ∂µθ term above

U(1) 1 Aµ(x) A′ (x) Aµ(x) ∂µθ −→ µ ≡ − e

August 10 – 14, 2009 Theory of Elementary Particles (page 23) Ahmed Ali DESY, Hamburg and defines the covariant derivative D ψ(x) [∂ + ieQA (x)] ψ(x) µ ≡ µ µ which transforms like the field itself: U(1) D ψ(x) (D ψ)′ (x) exp iQθ D ψ(x) µ −→ µ ≡ { } µ This yields the Lagrangian invariant under local U(1) transformations • i ψ(x)γµD ψ(x) m ψ(x)ψ(x) = eQA (x) ψ(x)γµψ(x) L ≡ µ − L0 − µ Gauge invariance has forced an interaction between the Dirac spinor and the gauge • field Aµ as in Quantum Electrodynamics (QED) For A to be a propagating field, one has to add a gauge-invariant kinetic term • µ 1 µν Kin Fµν (x) F (x) L ≡ −4

where Fµν ∂ν Aµ ∂µAν is the usual electromagnetic ﬁeld strength tensor A mass term≡ for the gauge− ﬁeld, = 1 m2 AµA forbidden due to gauge invariance • Lm 2 γ µ = i ψ(x)γµD ψ(x) m ψ(x)ψ(x) 1 F (x) F µν (x) • LQED µ − − 4 µν yields Maxwell equations (J ν is the fermion electromagnetic current) • LQED ∂ F µν = J ν eQ ψγν ψ µ ≡

August 10 – 14, 2009 Theory of Elementary Particles (page 24) Ahmed Ali DESY, Hamburg Elementary calculation in QED Dirac Wave-functions, Matrices and Algebra A Dirac spinor for a particle of physical momentum p and polarization s is denoted • s s by uα(p), and for the antiparticle, it is vα(p) In each case, the energy E = + p2 + m2 is positive • 0 The general solution of the Dirac equationp can be written as a linear combination of • plane waves. The positive frequency waves are of the form ip.x 2 2 0 ψ(x) = u(p)e− ; p = m ; p > 0 There are 2 linearly independent solutions of u(p) s = 1, 2 • √p.σξs us(p) = √p.σξ¯ s where σµ = (1, σi);σ ¯µ (1, σi) • ≡ ≡ − The negative frequency solutions of the Dirac equation are • ψ(x) = v(p)e+ip.x; p2 = m2; p0 > 0 Dirac equation in terms of the spinor us(p) and vs(p) reads • (p/ m)us(p)=0; (p/ + m)vs(p)=0 − 0 0 In terms of the adjoint spinors u¯ = u†γ and v¯ = v†γ , the Dirac equation reads • u¯s(p)(p/ m)=0;v ¯s(p)(p/ + m)=0 − They are normalized as: u¯rus = 2mδrs; v¯rvs = 2mδrs • − They are orthogonal to each other: u¯r(p)vs(p)=¯v(p)us(p)=0 •

August 10 – 14, 2009 Theory of Elementary Particles (page 25) Ahmed Ali DESY, Hamburg Spin Sums: • = us(p)¯us(p) = p/ + m; = vs(p)¯vs(p) = p/ m s s − Completeness relation is: P P • [us (p)¯us (p) vs (p)¯vs (p)] = δ s α β − α β αβ The γ-matrices in the DiracP equation satify the anticommutation relations • γµγν + γν γµ = 2gµν A representation useful for the electroweak theory is the Weyl representation: • 0 1 0 σi γ0 = , γi = , 1 0 σi 0 − where σ are the Pauli matrices and 1 is a 2 2 unit matrix × Frequently appearing combinations are: • µν i µ µ 5 0 1 2 3 σ = 2 [γ ,γ ]; γ = iγ γ γ γ = γ5 In this representation • 1 0 γ5 = − 0 1 In taking traces in the Dirac space, • we form hermitian conjugates of matrix elements for which, the following relation holds: [¯u(p′, s)Γu(p, s)]† =u ¯(p, s)Γ¯u(p′, s) 0 0 where Γ¯ = γ Γ†γ

August 10 – 14, 2009 Theory of Elementary Particles (page 26) Ahmed Ali DESY, Hamburg Feynman Rules Courtesy: BNL

August 10 – 14, 2009 Theory of Elementary Particles (page 27) Ahmed Ali DESY, Hamburg Rules for Feynman Graphs Differential Cross section Expressions for cross sections are divided in two parts: • 1. The invariant amplitude , which is a Lorentz-scalar and in which physics lies. This is determined by the verticesM (interactions) and propagators from space-time point x1 to x2, given by the Lagrangian of the theory 2. The Lorentz-invariant phase space (LIPS), containing the kinematical factors and the correct boundary conditions for a process Differential cross section dσ for spinless particles and for photons only: e.g., • γ(p )+ γ(p ) P (k )+ ...P (k ) 1 2 → 1 1 n n 3 3 1 1 1 d k1 d kn dσ = ( )( ) 2 ... 3 3 v1 v2 2ωp1 2ωp2 |M| 2ω1(2π) 2ωn(2π) | − | n (2π)4δ4(p + p k ) × 1 2 − i i=1 where ω = ~p 2 + m2, and v and vXare the velocities of the incident particles • p | | 1 2 The expressionp is integrated over all undetected momenta k ...k • 1 n 1 1 The statistical fator S is obtained by including a factor : S = Πi • m! mi! For Dirac particles, the factor 1/2ω is replaced by m/E • p p If desired, polarizations are summed over final and averaged over initial states • August 10 – 14, 2009 Theory of Elementary Particles (page 28) Ahmed Ali DESY, Hamburg + + e e− µ µ− → Some elementary electromagnetic processes based on the lepton-photon interactions • in QED em = +ejemAα L α em ¯ ¯ jα = ψeγαψe + ψµγαψµ + ...

+ + e e− µ µ− @ e → µ @ @R © p1 p @ 4 @ p1 + p2 − @§¤§¤§¤§¤ ¦¥¦¥¦¥@ @ © @R p2 @ p3 − @ @

The above diagram yields the following matrix element (m is the mass of the muon, • µ electron mass is set to zero and p/ = pµγ ) e2 M = [ u( p )γ u(p )] [ u(p )γα u( p )] if 2 2 α 1 3 4 (p1 + p2) − − 2 e β M ∗ = [ u(p )γ u( p )] u( p )γ u(p ) if 2 1 β 2 4 3 (p1 + p2) − − h i

August 10 – 14, 2009 Theory of Elementary Particles (page 29) Ahmed Ali DESY, Hamburg Summing up the spins of the leptons leads to the energy projection operators • ρ(s) ( p) = p/ m ± ± s= This yields X± • 1 e4 M 2 = Sp p/ γα p/ γβ Sp [(p/ + m) γ (p/ m) γ ] if 4 2 1 3 α 4 β 4 | | 4(p1 + p2) − Xλ h i Carrying out the traces and doing the algebra • (s = 4E2 ; t = m2 s cos2 θ ; u = m2 s sin2 θ ) c.m. − 2 − 2 4 1 2 2e 2 2 2 2 2 Mif = (m t) +(m u) 2m s 4 | | s2 − − − λ X This leads to the angular distribution (α = e2/4π and β = v/c) • 2 2 dσ α 2 2 2 α 2 = β 1 + cos θ + (1 β ) sin θ β=1 1 + cos θ dΩ s − | → s yielding the cross section • 4πα2 σ = 3s

dσ + and σ have been checked accurately in e e− experiments • dΩ

August 10 – 14, 2009 Theory of Elementary Particles (page 30) Ahmed Ali DESY, Hamburg Gauge Invariance & Quantum Chromodynamics (QCD)

QCD: A theory of interacting coloured quarks and gluons α Denote by qf a quark ﬁeld of colour α and ﬂavour f. To simplify the equations, let us adopt a vector notation in colour space: qT (q1 , q2 , q3) f ≡ f f f The free Lagrangian for the quarks • = q¯ (iγµ∂ m ) q L0 f µ − f f Xf is invariant under arbitrary global SU(3)C transformations in colour space, α α α β q (q )′ = U q , UU † = U †U = 1 , det U = 1 f −→ f β f The SU(3)C matrices can be written in the form λa U = exp i θa 2 1 a where 2 λ (a = 1, 2,..., 8) denote the generators of the fundamental representation a of the SU(3)C algebra, and θa are arbitrary parameters. The matrices λ are traceless abc and satisfy the commutation relations (f the SU(3)C structure constants, real and antisymmetric) λa λb λc , = i f abc , 2 2 2

August 10 – 14, 2009 Theory of Elementary Particles (page 31) Ahmed Ali DESY, Hamburg Like QED, we require the Lagrangian to be also invariant under local SU(3) • C transformations, θa = θa(x). Need to change the quark derivatives by covariant objects. Since we have now eight independent gauge parameters, eight diﬀerent gauge bosons • µ Ga (x), the so-called gluons, are needed: a µ µ λ µ µ µ D qf ∂ + igs G (x) qf [∂ + igs G (x)] qf ≡ 2 a ≡

with a µ λ µ [G (x)]αβ G (x) ≡ 2 a αβ µ Require that D qf transforms in exactly the same way as the colour-vector qf ; this ﬁxes• the transformation properties of the gauge ﬁelds:

µ µ µ µ µ µ i µ D (D )′ = UD U † , G (G )′ = U G U †+ (∂ U) U † −→ −→ gs

Under an inﬁnitesimal SU(3)C transformation, • a α α α λ β q (q )′ = q + i δθa q , f −→ f f 2 f αβ µ µ µ 1 µ abc µ Ga (Ga )′ = Ga ∂ (δθa) f δθb Gc −→ − gs −

August 10 – 14, 2009 Theory of Elementary Particles (page 32) Ahmed Ali DESY, Hamburg Observations: (i) The gauge transformation of the gluon ﬁelds is more complicated than the one obtained in QED for the photon. The non-commutativity of the SU(3)C matrices gives rise to an additional term involving the gluon ﬁelds themselves

(ii) For constant δθa, the transformation rule for the gauge ﬁelds is expressed in terms of the structure constants f abc; thus, the gluon ﬁelds belong to the adjoint representation of the colour group

(iii) There is a unique SU(3)C coupling gs

(iv) To build a gauge-invariant kinetic term for the gluon ﬁelds, we introduce the corresponding ﬁeld strengths: a µν i µ ν µ ν ν µ µ ν λ µν G (x) [D ,D ] = ∂ G ∂ G + igs [G , G ] Ga (x) , ≡ −gs − ≡ 2 Gµν (x) = ∂µGν ∂ν Gµ g f abc Gµ Gν a a − a − s b c Under a gauge transformation,

µν µν µν G (G )′ = U G U † −→ µν 1 µν a and the colour trace Tr (G Gµν ) = 2 Ga Gµν remains invariant

August 10 – 14, 2009 Theory of Elementary Particles (page 33) Ahmed Ali DESY, Hamburg SU(N) Algebra

SU(N) is the group of N N unitary matrices, UU † = U †U = 1, with det U = 1 • × The generators of the SU(N) algebra, T a (a = 1, 2,...,N 2 1), are hermitian, •traceless matrices satisfying the commutation relations − [T a, T b] = if abc T c f abc are structure constants; they are real and totally antisymmetric The fundamental representation T a = λa/2 is N–dimensional. For N = 2, λa are the usual• Pauli matrices, while for N = 3, they are the eight Gell-Mann matrices 0 1 0 0 i 0 1 0 0 0 0 1 − λ1 = 1 0 0 , λ2 = i 0 0 , λ3 = 0 1 0 , λ4 = 0 0 0      −    0 0 0 0 0 0 0 0 0 1 0 0         0 0 i 0 0 0 0 0 0 1 0 0 − 1 λ5 = 0 0 0 , λ6 = 0 0 1 , λ7 = 0 0 i , λ8 = 0 1 0      −  √   i 0 0 0 1 0 0 i 0 3 0 0 2 −         They satisfy the anticommutation relation 4 λa,λb = δab I + 2dabc λc N N abc IN is the N–dimensional unit matrix; d are totally symmetric in the three indices.

August 10 – 14, 2009 Theory of Elementary Particles (page 34) Ahmed Ali DESY, Hamburg For SU(3), the only non-zero (up to permutations) f abc and dabc constants are • 1 f 123 = f 147 = f 156 = f 246 = f 257 = f 345 = f 367 2 − − 1 1 1 = f 458 = f 678 = √3 √3 2 1 d146 = d157 = d247 = d256 = d344 = d355 = d366 = d377 = − − − 2 1 d118 = d228 = d338 = 2d448 = 2d558 = 2d688 = 2d788 = d888 = − − − − − √3

The adjoint representation of the SU(N) group is given by the (N 2 1) (N 2 1) •matrices (T a) if abc − × − A bc ≡− Relations deﬁning the SU(N) invariants T , C and C • F F A 1 Tr λaλb = 4 T δ , T = F ab F 2 N 2 1 (λaλa) = 4C δ , C = − αβ F αβ F 2N a b acd bcd Tr(TATA) = f f = CA δab CA = N

August 10 – 14, 2009 Theory of Elementary Particles (page 35) Ahmed Ali DESY, Hamburg Gauge fixing and Ghost Fields The fields Gµ have 4 Lorentz degrees of freedom, while a massless spin-1 gluon has 2 • a physical polarizations only. The unphysical degrees of freedom have to be removed The canonical momentum associated with Gµ, Πa (x) δ /δ(∂ Gµ) = Ga , • a µ ≡ LQCD 0 a µ0 vanishes identically for µ = 0. The commutation relation meaningless for µ = ν = 0 Gµ(x), Πν (y) δ(x0 y0) = igµν δ(4)(x y)δ a b − − ab Hence, the unphysical components of the gluon field should not be quantized. This • could be achieved by imposing two gauge conditions, such as G0 = 0 and ~ .G~ = 0. a ∇ a This is a (Lorentz) non-covariant procedure, which leads to an awkward formalism Instead, one can impose a Lorentz–invariant gauge condition, such as ∂ Gµ = 0 by • µ a adding to the Lagrangian the gauge-fixing term ( ξ is the gauge parameter)

1 µ a ν GF = (∂ G )(∂ν G ) L −2ξ µ a The 4 Lorentz components of the canonical momentum are then non-zero and one can • develop a covariant quantization formalism

δ QCD 1 Πa (x) L = Ga g (∂ν Ga) µ µ µ0 µ0 ν ≡ δ(∂0Ga ) − ξ

August 10 – 14, 2009 Theory of Elementary Particles (page 36) Ahmed Ali DESY, Hamburg Since is a quadratic Gµ term, it modiﬁes the gluon propagator: • LGF a 4 ik(x y) µ ν µ ν d k e− − µν k k 0 T [G (x)G (y)] 0 = iδab g + (1 ξ) . h | a b | i (2π)4 k2 + iε − − k2 + iε Z In QED, this gauge-ﬁxing procedure is enough for making a consistent quantization of • the theory. In non-abelian gauge theories, like QCD, a second problem still remains Let us consider the scattering process qq¯ GG, for which the scattering amplitude • (λ) (λ ) → has the general form T = J µµ′ ε ε ′ µ µ′

q G

+ +

q G a) b) c) The probability associated with the scattering process is: •

1 µµ νν (λ) (λ) (λ′) (λ′) J ′ (J ′ )† ε ε ∗ ε ε ∗ P ∼ 2 µ ν µ′ ν′ Xλ Xλ′ This involves a sum over the ﬁnal gluon polarizations

August 10 – 14, 2009 Theory of Elementary Particles (page 37) Ahmed Ali DESY, Hamburg The physical probability T , where only the two transverse gluon polarizations are •considered in the sum, is diﬀerentP from the covariant quantity , which includes a sum PC over all polarization components: > . As only has physical meaning, one PC PT PT should just sum over the physical transverse polarizations to get the right answer However, higher-order graphs such as the ones shown below get unphysical • contributions from the longitudinal and scalar gluon polarizations propagating along the internal gluon lines, implying a violation of unitarity (the two fake polarizations contribute a positive probability)

q q

+ + + . . .

q q a′) b ′) c ′) In QED this problem does not appear because the gauge-fixing condition ∂µA = 0 • µ respects the conservation of the electromagnetic current due to the extra condition θ = 0 , i.e., ∂ J µ = ∂ (eQΨ¯ γµΨ)= 0, and therefore, = . µ em µ PC PT In QCD, C = T stems from the third diagram in the figure above involving a gluon • P 6 P µ self-interaction. The gauge-fixing condition ∂µGa = 0 does not leave any residual invariance. Thus, k J µµ′ = 0 µ 6

August 10 – 14, 2009 Theory of Elementary Particles (page 38) Ahmed Ali DESY, Hamburg A clever solution (due to Feynman; 1963) is to add unphysical ﬁelds, the so-called • ghosts, with a coupling to the gluons so as to exactly cancel all the unphysical contributions from the scalar and longitudinal gluon polarizations

φ qφ q q GGG

q φ q φ q

d)′) d Since a positive fake probability has to be cancelled, one needs fields obeying the wrong • statistics (i.e., of negative norm) and thus giving negative probabilities. The cancellation is achieved by adding to the Lagrangian the Faddeev–Popov term (Fadeev, Popov; 1967), = ∂ φ¯ Dµφa ,Dµφa ∂µφa g f abcφbGµ LFP − µ a ≡ − s c where φ¯a, φa (a = 1,...,N 2 1) is a set of anticommuting (obeying the C − Fermi–Dirac statistics), massless, hermitian, scalar fields. The covariant derivative Dµφa contains the needed coupling to the gluon field. One can easily check that finally = . PC PT Thus, the addition of the gauge-fixing and Faddeev–Popov Lagrangians allows to • develop a simple covariant formalism, and therefore a set of simple Feynman rules

August 10 – 14, 2009 Theory of Elementary Particles (page 39) Ahmed Ali DESY, Hamburg QCD Lagrangian in a Covariant Gauge Putting all pieces together •

= + + LQCD L(q,g) LGF LFP

1 = (∂µGν ∂νGµ)(∂ Ga ∂ Ga ) + q¯α (iγµ∂ m ) qα L(q,g) − 4 a − a µ ν − ν µ f µ − f f f X λa g Gµ q¯α γ qβ − s a f µ 2 f f αβ X g g2 + s f abc (∂µGν ∂νGµ) Gb Gc s f abcf Gµ Gν Gd Ge 2 a − a µ ν − 4 ade b c µ ν

1 = (∂µGa )(∂ Gν) LGF −2ξ µ ν a

= ∂ φ¯ Dµφa ,Dµφa ∂µφa g f abcφbGµ LFP − µ a ≡ − s c

August 10 – 14, 2009 Theory of Elementary Particles (page 40) Ahmed Ali DESY, Hamburg Feynman Rules for QCD Propagators and Vertices

August 10 – 14, 2009 Theory of Elementary Particles (page 41) Ahmed Ali DESY, Hamburg The QCD running coupling constant

The renormalization of the QCD coupling proceeds in a way similar to QED. Owing • to the non-abelian character of SU(3)C , there are additional contributions involving gluon self-interactions (and ghosts)

+ + + + . . .

The scale dependence of α (Q2) is regulated by the so-called β function of a theory • s µ dαs α β(α ); β(α ) = β αs + β ( αs )2 + ... dµ ≡ s s s 1 π 2 π From the above one-loop diagrams, one gets the value of the ﬁrst β function • coeﬃcient [Politzer; Gross & Wilczek; 1973]

2 11 2 Nf 11 NC β1 = TF Nf CA = − 3 − 6 6

The positive contribution proportional to the number of quark ﬂavours Nf is • generated by the q–q¯ loops; the gluonic self-interactions introduce the additional negative contribution proportional to NC

August 10 – 14, 2009 Theory of Elementary Particles (page 42) Ahmed Ali DESY, Hamburg The term proportional to N is responsible for the completely diﬀerent behaviour of • C QCD (NC = 3): β1 < 0 if Nf 16 ≤ 2 The corresponding QCD running coupling, αs(Q ), decreases at short distances: 2 lim αs(Q ) 0 Q2 → →∞ For N 16, QCD has the property of asymptotic freedom f ≤ Quantum eﬀects have introduced a dependence of the coupling on the energy; still • need a reference scale to decide when a given Q2 can be considered large or small

A precise deﬁnition of the scale is obtained from the solution of the β-function • diﬀerential equation. At one loop, one gets π ln µ + = lnΛ 2 QCD β1αs(µ )

where lnΛQCD is an integration constant. Thus,

2 2π αs(µ ) = β ln µ2/Λ2 − 1 QCD Dimensional Transmutation: The dimensionless parameter g is traded against the • s dimensionful scale ΛQCD generated by Quantum eﬀects

August 10 – 14, 2009 Theory of Elementary Particles (page 43) Ahmed Ali DESY, Hamburg For µ Λ , α (µ2) 0, one recovers asymptotic freedom • ≫ QCD s → 2 At lower energies, the running coupling gets larger; for µ ΛQCD, αs(µ ) , • perturbation theory breaks down. The QCD regime α (µ→2) 1 requires → ∞ s ≥ non-perturbative methods, such as Lattice-QCD (beyond the scope of these lectures) Higher Orders Higher orders in perturbation theory are much more important in QCD than in QED, • because the coupling is much larger (at ordinary energies)

N αs k In the meanwhile, the β function is known to four loops β(αs) = i=1 βk( π ) • [van Ritbergen, Vermaseren, Larin [1997]; Caswell; D.R.T. Jones [1974]; Vladimirov, Zakharov [1980]] P

In the MS scheme, the computed higher-order coeﬃcients take the values • (ζ3 = 1.202056903 ...)

51 19 1 5033 325 2 β2 = + Nf ; β3 = 2857 + Nf N ; − 4 12 64 − 9 − 27 f 1 149753 1078361 6508 β4 = − + 3564 ζ3 + ζ3 Nf 128 6 − 162 27 50065 6472 2 1093 3 + + ζ3 N + N 162 81 f 729 f

August 10 – 14, 2009 Theory of Elementary Particles (page 44) Ahmed Ali DESY, Hamburg 2 Measurements of αs(Q ) from jets at HERA/H1 H1 Collaboration (DESY 08-032)

α s from Jet Cross Sections H1 Preliminary α µ 2 2 s( = Q ) for Q < 100 GeV (HERA I) α µ 2 2 s( = Q ) for Q > 150 GeV (HERA I+II) α µ Combined < s( )> (incl., 2-, 3-jet) 0.3 from Q 2 > 150 GeV 2 NLO uncertainty

ssss αααα 0.2

0.1 2 10 Q / GeV 10

August 10 – 14, 2009 Theory of Elementary Particles (page 45) Ahmed Ali DESY, Hamburg Experimental Evidence of Asymptotic Freedom S. Bethke, Summer 2009

0.5 July 2009 α (Q) s 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 10Q [GeV] 100

August 10 – 14, 2009 Theory of Elementary Particles (page 46) Ahmed Ali DESY, Hamburg Nobel Prize for Physics 2004

”for the discovery of asymptotic freedom in the theory of the strong interaction”

David J. Gross H. David Politzer Frank Wilczek 1/3 of the prize 1/3 of the prize 1/3 of the prize USA USA USA

August 10 – 14, 2009 Theory of Elementary Particles (page 47) Ahmed Ali DESY, Hamburg Applications of perturbative QCD to hard processes

+ The simplest hard process is the total hadronic cross-section in e e− annihilation: • + + + R σ(e e− hadrons)/σ(e e− µ µ−) ≡ → → e– q γ, Z Hadrons

e+ q

In the quark-parton model (i.e., no QCD corrections), one has for (i = u,d,c,s,b) • and taking into account only the γ-exchange in the intermediate state (Nc = 3):

2 11 11 R = Nc Q = Nc = i 9 3 Xi In leading order perturbative QCD, the process is approximated by the following • + + σ(e e− hadrons) = σ(e e− qq¯ + qqg¯ ) → → 2 e + q + g + . . . . + γ γ e – q

+ g + + . . . γ γ g

August 10 – 14, 2009 Theory of Elementary Particles (page 48) Ahmed Ali DESY, Hamburg 2 The Photon vacuum polarization amplitude Πem(q )

2 2 + For Ec.m. MZ, e e− hadrons dominated by γ-exchange • ≪ → q e– e– γ γ ~

e+ e+ q

Easier to calculate the so-called Photon vacuum polarization amplitude, Π (q2) • em 2 Πem(q ) is deﬁned by the time-ordered product of two electromagnetic currents • em a Jµ = f Qf q¯f γµqf : P Π (q) = i d4xeiq.x 0 TJ em(x)J em(0) 0 = ( g q2 + q q )Π (q2) µν h | µ ν | i − µν µ ν em Z + σ(e e− hadrons) R + (s) → = 12π ImΠem(s) e e− ≡ σ(e+e µ+µ ) − → −

a Time-ordered product is deﬁned as: T φ(x′)φ(x) = θ(t′ t)φ(x′)φ(x)+ θ(t t′)φ(x)φ(x′) − −

August 10 – 14, 2009 Theory of Elementary Particles (page 49) Ahmed Ali DESY, Hamburg + Re e− (s) in pert. QCD One can compute, in principle, the cross-section of all subprocesses at a given order • in αs + e e− γ∗ qq¯ + qqg¯ + qqgg¯ + qqq¯ ′q¯ + ... → → ′

Nf n 2 αs(s) R + (s) = Q NC 1 + Fn e e−  f   π  f=1 n 1 X  X≥    So far, perturbative series has been calculated to (α3):  • O s F1 = 1 , F2 = 1.986 0.115 Nf , − 2 f Qf F = 6.637 1.200 N 0.005 N 2 1.240 3 f f 2 − − − − 3P f Qf For N = 5 ﬂavours, one has: • f P 2 3 11 αs(s) αs(s) αs(s) 4 R + (s) = 1 + + 1.411 12.80 + (α ) e e− 3 π π − π O s ( )

+ A ﬁt to e e− data for √s between 20 and 65 GeV yields [D. Haidt, 1995] • α (35 GeV) = 0.146 0.030 s ±

August 10 – 14, 2009 Theory of Elementary Particles (page 50) Ahmed Ali DESY, Hamburg + Re e− (s): Data [PDG 2007; W.-M. Yao et al., Journal of Physics, G 33, 1 (2006)]

-2 10 ω φ -3 J/ψ 10 ψ (2 S ) Υ -4 ρ ρ ′ 10 Z -5 10 [mb]

σ -6 10 -7 10 -8 10 1 10 10 2

3 Υ 10 J/ψ ψ (2 S ) Z 10 2 φ R ω 10 ρ ′ 1 ρ -1 10 1 10 10 2 √ s [GeV]

August 10 – 14, 2009 Theory of Elementary Particles (page 51) Ahmed Ali DESY, Hamburg e+e jets − → + Quark jets were discovered in 1975 in e e− annihilation experiment at SLAC • [R.F. Schwitters et al., Phys. Rev. Lett. 35 (1975) 1320; G.G. Hanson et al., Phys. Rev. Lett. 35 (1975) 1609]

+ In QCD, this process is described by e e− qq¯, radiation of soft partons (gluons, → • qq¯ pairs) and subsequent hadronization

One needs a deﬁnition of hadronic jets, e.g., a cone with a minimum fractional • energy and angular resolution (ǫ, δ), or a minimum invariant mass 2 (ymin = mmin(jet)/s) to satisfy the Bloch-Nordsiek and KLN theorems

In (α ) pert. QCD, following Feynman diagrams are to be calculated: • O s e– q γ + +

e+ q

August 10 – 14, 2009 Theory of Elementary Particles (page 52) Ahmed Ali DESY, Hamburg e– q γ G +

e+ q

+ The process e e− 2 jets was ﬁrst calculated in (α ) by Sterman and → O s • Weinberg using the (ǫ, δ)-prescription of jets [G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436]

To (ǫ), (δ): • O O 2 2 αs(Q ) 2 2 2π σ2 jets(ǫ, δ) = σ0[1 + CF 4ln ǫ ln δ 3ln δ + 5 ] − π − − − 3 4πα2 2 2 where σ0 = 3s NC i Qi ; s = Q + Gluon jets were discoveredP in 1979 in e e− annihilation experiment at DESY • [R. Brandelik et al. (TASSO Coll.), Phys. Lett. 86B (1979) 243; D.P. Barber et al. (MARK J Coll.), Phys. Rev. Lett. 43 (1979) 830; and subsequently by Ch. Berger et al. (PLUTO Coll.), Phys. Lett. 86B (1979) 418 and W. Bartel et al. (JADE Coll.), Phys. Lett. 91B (1980) 142]

In (α ), the emission of a hard (i.e., energetic and non-collinear) gluon from a O s • quark leg leads to a three-jet event.

August 10 – 14, 2009 Theory of Elementary Particles (page 53) Ahmed Ali DESY, Hamburg For massless quarks, the differential distribution is [J. Ellis, M.K. Gaillard, G.G. Ross, • Nucl. Phys. B111 (1976) 253] 2 2 2 1 d σ 2αs x + x = 1 2 σ dx dx 3π (1 x )(1 x ) 0 1 2 − 1 − 2 Ei and the kinematics is defined as: xi = 2 √s (i = 1, 2, 3) and x1 + x2 + x3 = 2 Defining a jet by an invariant–mass cut y: • 3 jet s (p + p )2 > ys ( i,j = 1, 2, 3) ⇐⇒ ij ≡ i j ∀ the fraction of 3-jet events is predicted to be: (Li (z) z dξ ln ξ) [See, G. 2 ≡ − 0 1 ξ Kramer, Springer Tracts in Modern Physics, Vol. 102] − R 2αs y 2 y 5 9 2 R3(s, y) = (3 6y) ln +2 ln + 6 y y 3π − 1 2y 1 y 2 − − 2 − − y π2 + 4Li2 1 y − 3 − The corresponding fraction of 2-jet events is given by R = 1 R . The general • 2 − 3 expression for the fraction of n-jet events takes the form (with n Rn = 1): n 2 j αs(s) − (n) αs(Ps) Rn(s, y) = C (y) π j π jX=0

August 10 – 14, 2009 Theory of Elementary Particles (page 54) Ahmed Ali DESY, Hamburg Two- and Three-jet events from Z qq¯ and Z qqg¯ → → [ALEPH Coll., http://aleph.web.cern.ch/aleph/dali/ Z0 examples.html] −

n 2 The jet fractions have a high sensitivity to αs [Rn αs − ]. Measurements at the Z • 2 ∼ peak, from LEP and SLC, using resummed (αs) ﬁts to a large set of shape variables, yields [PDG 2006 ]: O

α (M ) = 0.1202 0.005 s Z ±

August 10 – 14, 2009 Theory of Elementary Particles (page 55) Ahmed Ali DESY, Hamburg Deep Inelastic Scattering and QCD An important class of hard processes is Deep Inelastic Scattering (DIS) oﬀ a Nucleon • ℓ(k) + N(p) ℓ′(k′) + X(p ); ℓ = e±,µ±,ν, ν¯ → X

Kinematics: p is the momentum of the nucleon having a mass M and the • momentum of the hadronic system is pX ; the virtual momentum q of the gauge boson is spacelike 2 2 2 2 2 2 2 Q = q = (k k′) = 4EE′ sin (θ/2); s = (p+k) ; W = p ; ν = (p.q)/M − − − X Useful to define scaling variables; Bjorken-variable x and y • 2 2 2 (k k′) Q Q x = − − = = 2 2 2 2p.(k k′) 2Mν W + Q M − 2 2− 2 p.(k k′) 2Mν W + Q M y = − = = − p.k (s M 2) (s M 2) − − Bjorken limit is defined by: large Q2 and ν such that x is finite •

August 10 – 14, 2009 Theory of Elementary Particles (page 56) Ahmed Ali DESY, Hamburg Cross-sections and Structure functions The cross-section for the process ℓ(k) + N(p) ℓ′(k′) + X(pX ) mediated by • a virtual photon is (M is the nucleon mass; Q2 =→ q2 and q2 is the photon − virtuality) 2 dσ 2M α µν k′ = L Wµν 0 d3k (s M 2) Q4 ′ − Lµν is obtained from the leptonic electromagnetic vertex • µν 1 1 2 L = T r(k/′γµk/γν ) = 2(kµk′ + k′ kν Q gµν ) 2 ν µ − 2

The strong interaction aspects (involving the structure of the nucleon) is contained in • the tensor Wµν , which is deﬁned as

iqx W = dxe < p J †(x)J (0) p > µν | µ ν | Z Structure functions (SF) are deﬁned from the general form of Wµν using Lorentz • invariance and current conservation. For the electromagnetic currents between unpolarized nucleons,one has 2 qµqν 2 p.q p.q W2(ν,Q ) Wµν = (gµν + )W1(ν,Q )+(pµ + qµ)(pν + qν ) − Q2 Q2 Q2 M 2

August 10 – 14, 2009 Theory of Elementary Particles (page 57) Ahmed Ali DESY, Hamburg This leads to the differential cross-section in terms of x and y: • d2σ 2πα2M (s M 2) (1 y) = 2W1(x, y) + W2(x, y) − − 1 dxdy (s M 2)x M 2 xy − − The SFs W and W are related to the absorption cross-sections for virtual • 1 2 transverse (σT ) and longitudinal photons (σ0) ν2 + Q2 W = σ 1 2 T p 4πα Q2 W2 = (σ0 + σT ) 4πα2 ν2 + Q2 One can define a longitudinal structurep function W by • L ν2 + Q2 ν2 W = σ = (1+ )W W L 2 0 2 2 1 p 4πα Q − In the Quark-Parton Model, SFs become scale-invariant in the Bjorken limit • MW (ν,Q2) F (x) = e2q (x) 1 → 1 i i Xi νW = 2xMW = νW (ν,Q2) F (x) = e2xq (x) 2 1 ⇒ 2 → 2 i i Xi qi(x) is the probability of finding charged partons (i.e., quarks) inside the nucleon • having a fractional momentum x

August 10 – 14, 2009 Theory of Elementary Particles (page 58) Ahmed Ali DESY, Hamburg QCD-improved parton model and DIS In QCD, one has also gluons as partons, and one • has eﬃcient radiation of hard gluons from the struck quarks and gluons in the nucleons (α ) corrections to the Quark-Parton Model O s

γ∗ γ∗ 2

+ + . . . . + g

(a) (b)

γ∗ γ∗ g 2

+ + + . . . .

DIS oﬀ a gluon in the nucleon

γ∗

q g

August 10 – 14, 2009 Theory of Elementary Particles (page 59) Ahmed Ali DESY, Hamburg QCD-improved parton model and DIS -Contd. Radiation of gluons produces the evolution of the structure functions. As Q2 • increases, more and more gluons are radiated, which in turn split into qq¯ pairs This process leads to the softening of the initial quark momentum distributions and • to the growth of the gluon density and the qq¯ sea for low values of x. Both effects have been firmly established at HERA In pert. QCD, these effects can be calculated in terms of the evolution governed by • the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi or DGLAP equations DGLAP Equations The quark density evolution equation: • d αs(t) αs(t) qi(x, t) = [qi Pqq] + [g Pqg] dt 2π 2π O O with the notation: 1 q(y, t) x [q P ]=[P q] = dy P ( ) x y · y O O Z Interpretation of this equation: The variation of the quark density is due to the • convolution of the quark density at a higher energy times the probability of finding a quark in a quark (with the right energy fraction) plus the gluon density at a higher energy times the probability of finding a quark (of the given flavour i) in a gluon

August 10 – 14, 2009 Theory of Elementary Particles (page 60) Ahmed Ali DESY, Hamburg The gluon density evolution equation: • d αs(t) αs(t) g(x, t) = [ (qi +q ¯i) Pgq] + [g Pgg] dt 2π 2π i The explicit forms of the splittingX functions canO be derived from theO QCD vertices. • They are universal, i.e., process-independent 4 1 + x2 3 Pqq = + δ(1 x) + (αs) 3 (1 x) 2 − O − + 4 1+(1 x)2 Pgq = − + (αs) 3 x O 1 2 2 Pqg = [x + (1 x) ] + (αs) 2 − O x 1 x 33 2nf Pgg = 6 + − + x(1 x) + − δ(1 x) + (αs) (1 x) x − 6 − O − + The ” + ” distribution is deﬁned as, for a generic non singular weight function f(x) • 1 f(x) 1 f(x) f(1) dx = − dx (1 x) 1 x Z0 − + Z0 − The splitting functions satify the normalization conditions: • 1 1 1 Pqq(x)dx = 0; [Pqq(x)+Pgq(x)]xdx = 0; [2nf Pqg(x)+Pgg(x)]xdx = 0 Z0 Z0 Z0

August 10 – 14, 2009 Theory of Elementary Particles (page 61) Ahmed Ali DESY, Hamburg 2 The proton structure function F2(x, Q ) Particle Data Group 2007 ) 2 1.4 (x,Q 2 F

1.2

0.8

0.6

0.4 H1 ZEUS BCDMS NMC 0.2 SLAC E665

-4 -3 -2 -1 10 10 10 10 x1

August 10 – 14, 2009 Theory of Elementary Particles (page 62) Ahmed Ali DESY, Hamburg Measurements of the proton SF HERA Structure Function Working Group

H1 and ZEUS Combined PDF Fit i 10 7

) x 2 x = 0.000032, i=22 + 2 x = 0.00005, i=21 HERA I e p (prel.) 6 x = 0.00008, i=20 Fixed Target 10 x = 0.00013, i=19 (x,Q r x = 0.00020, i=18 HERA I PDF (prel.) σ x = 0.00032, i=17 10 5 x = 0.0005, i=16 x = 0.0008, i=15 x = 0.0013, i=14 10 4 x = 0.0020, i=13 x = 0.0032, i=12 x = 0.005, i=11 10 3 x = 0.008, i=10 x = 0.013, i=9 x = 0.02, i=8 2 10 x = 0.032, i=7 x = 0.05, i=6 10 x = 0.08, i=5 x = 0.13, i=4 x = 0.18, i=3 1 x = 0.25, i=2 -1 10 x = 0.40, i=1

-2 10 x = 0.65, i=0 -3 10 HERA Structure Functions Working Group April 2008 1 10 10 2 10 3 10 4 10 5 Q 2 / GeV 2

August 10 – 14, 2009 Theory of Elementary Particles (page 63) Ahmed Ali DESY, Hamburg Nobel Prize for Physics 1979

”for their contributions to the theory of the uniﬁed weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current”

Sheldon Lee Glashow Abdus Salam Steven Weinberg 1/3 of the prize 1/3 of the prize 1/3 of the prize USA Pakistan USA Harvard University, International Centre for Harvard University, Lyman Laboratory, Theoretical Physics, Cambridge, MA, USA Cambridge, MA, Trieste, Italy; Imperial USA College London, United Kingdom

August 10 – 14, 2009 Theory of Elementary Particles (page 64) Ahmed Ali DESY, Hamburg Structure of the ﬁrst generation of leptons and quarks in SM

August 10 – 14, 2009 Theory of Elementary Particles (page 65) Ahmed Ali DESY, Hamburg EW Lagrangian Density Lagrangian Density of the SM Electroweak Sector The symmetry group of the EW theory • G = SU(2) U(1) L ⊗ Y Contains four diﬀerent gauge bosons W i(x) (i = 1, 2, 3) and Bµ(x) • µ For simplicity, consider a single family of Quarks and Leptons: • u(x) Quarks: q (x) = , u (x), d (x) L d(x) R R L νe(x) Leptons: ℓL(x) = , νeR(x), eR−(x) e−(x) L For quark family, Lagrangian invariant under local G transformations • (x) =q ¯ (x) iγµD q (x) +u ¯ (x) iγµD u (x) + d¯ (x) iγµD d (x) L L µ L R µ R R µ R Quark covariant derivatives [W (x) (σ /2) W i(x)] • µ ≡ i µ D q (x) [∂ igW (x) ig′y B (x)] q (x) µ L ≡ µ − µ − 1 µ L D u (x) [∂ ig′y B (x)] u (x) µ R ≡ µ − 2 µ R D d (x) [∂ ig′y B (x)] d (x) µ R ≡ µ − 3 µ R σ (i = 1, 2, 3) are the Pauli matrices • i

August 10 – 14, 2009 Theory of Elementary Particles (page 66) Ahmed Ali DESY, Hamburg EW Lagrangian Density Properly normalized kinetic Lagrangian • 1 1 1 1 = B Bµν Tr [W W µν] = B Bµν W i W µν Lkin −4 µν − 2 µν −4 µν − 4 µν i Includes ﬁeld strength tensors • B ∂ B ∂ B ,W (σ /2) W i µν ≡ µ ν − ν µ µν ≡ i µν W i ∂ W i ∂ W i gεijk W jW k µν ≡ µ ν − ν µ − µ ν Explicit form of the SU(2) matrix • L 3 σi i 1 Wµ √2 Wµ− Wµ = Wµ = 3 2 2 √2 W − W µ − µ Term with Wµ in gives rise to the charged-current interaction of • L 1 2 left-handed quarks with charged boson ﬁelds W ± W i W /√2 µ ≡ µ ± µ quarks g ¯ µ + µ (x) = d(x)γ (1 γ5)u(x) W (x)+[¯u(x)γ (1 γ5)d(x)] W −(x) LCC −2√2 − µ − µ Similarly, charged-current interaction of leptons are • leptons g µ + µ (x) = [¯e(x)γ (1 γ5)νe(x)] W (x)+[¯νe(x)γ (1 γ5)e(x)] W −(x) LCC −2√2 − µ − µ August 10 – 14, 2009 Theory of Elementary Particles (page 67) Ahmed Ali DESY, Hamburg EW Lagrangian Density Quarks in have interactions with other two bosons W 3 and B • L µ µ Observed bosons Z and A are their linear combination • µ µ W 3 cos Θ sin Θ Z µ = W W µ B sin Θ cos Θ A µ − W W µ In terms of the masse igenstates, the covariant derivative becomes • g + + 1 2 3 2 gg′ 3 Dµ = ∂µ i (W T +W −T ) i Zµ(g T g′ Y ) i Aµ(T +Y ) µ µ 2 2 2 2 − √2 − − g + g′ − − g + g′ To put this expression in a more useful form, we identify the electron • p p charge e with e = gg′ , and identify the electric charge with 2 2 √g +g′ Q = T 3 + Y Neutral-current Lagrangian can be written as • = + Z LNC LQED LNC

First term is the usual QED Lagrangian [g sin Θ = g′ cos Θ = e] • W W = eA J µ = eA Q (¯eγµe) + Q (¯uγµu) + Q dγ¯ µd LQED − µ em − µ e u d Second term describes the interaction of Z-boson with neutr al • August 10 – 14, 2009 Theory of Elementary Particles (page 68) Ahmed Ali DESY, Hamburg fermionic current Z e µ e ¯ µ NC = ZµJZ = Zµ fγ (vf af γ5) f L −sin(2ΘW ) −sin(2ΘW ) − f X Neutral-current coupling: a = T f and v = T f 1 4 Q sin2 Θ • f 3 f 3 − | f | W fu dνe e 2v 1 8 sin2 Θ 1 + 4 sin2 Θ 1 1 + 4 sin2 Θ f − 3 W − 3 W − W 2af 1 -1 1 -1 Cancellation of Anomaly in the GSW Theory In a gauge theory of left-handed massless fermions coupled with • non-Abelian gauge bosons, the axial-vector current is conserved at the tree level, but has in general an anomalous divergence due to the loop corrections. Deﬁning the axial-vector current as µa ¯ µ 1 γ5 a J = ψγ ( −2 t ψ), 2 p,ν,b; k, λ, c ∂ J µa 0 = g ǫανβλp k . abc h | µ | i 8π2 α β A where abc = tr[ta tb, tc ] A { } If abc = 0, the current is not conserved and we have a divergent • A 6 theory. In the SM, this divergence is miraculously absent, summing over all the fermionic contributions in the loop.

August 10 – 14, 2009 Theory of Elementary Particles (page 69) Ahmed Ali DESY, Hamburg EW Lagrangian Density generates self-interactions among gauge bosons • Lkin Cubic self-interactions • +µν µν + µν + = ie cot Θ W W −Z W − W Z + Z W W − L3g W µ ν − µ ν µ ν +µν µν + µν + + ie W W −A W − W A + A W W − µ ν − µ ν µ ν Quartic self-interactions • 2 e +µ +ν +µ + ν 4g = 2 W Wµ−W Wν− W Wµ W − Wν− L −2 sin ΘW − 2 2 +µ ν + µ ν e cot Θ W W −Z Z W Z W −Z − W µ ν − µ ν 2 +µ ν + µ ν e W W −A A W A W −A − µ ν − µ ν 2 +µ ν + µ ν + µ ν e cot Θ 2W W −Z A W Z W −A W A W −Z − W µ ν − µ ν − µ ν 3g and 4g entail the following vertices • L L + W W +W + W + γ, Z

γ , Z

W − W − W − W − γ, Z

August 10 – 14, 2009 Theory of Elementary Particles (page 70) Ahmed Ali DESY, Hamburg EW Lagrangian Density The Higgs Mechanism φ(+)(x) Consider SU(2) doublet of complex scalar ﬁelds φ(x) = L φ(0)(x) • µ 2 2 (x)=(∂ φ)† ∂ φ V (φ), V (φ) = µ φ†φ + h φ†φ LS µ − S(x) is invariant under global phase transformation: • L iθ φ(x) φ′(x) = e φ(x) → To get ground states, V (φ) should be bounded from below, h > 0 • For the quadratic term, there are two possibilities: • 1. µ2 > 0: trivial minimum 0 φ(0) 0 = 0; massive scalar particle |h | | i| 2. µ2 < 0: inﬁnite set of degenerate states with minimum energy due to the U(1) phase-invariance of (x) LS 3. Choosing a particular solution, the symmetry is broken spontaneously (φ) V V(φ) ϕ 1 |φ| |φ|

ϕ 2

August 10 – 14, 2009 Theory of Elementary Particles (page 71) Ahmed Ali DESY, Hamburg EW Lagrangian Density

Now, let us consider a local gauge transformation; i.e. θi = θi(x) • In the physical (unitary) gauge θi(x) = 0, only one neutral scalar • ﬁeld H(x) (the Higgs ﬁeld) remains in the model The gauged scalar Lagrangian of the Goldstone model (i.e., Scalar • Lagrangian including interactions with gauge bosons) is:

µ 2 2 2 (x)=(D φ)† D φ µ φ†φ h φ†φ [h > 0, µ < 0] LS µ − − Covariant derivative under local SU(2) U(1) transformation • L ⊗ Y D φ(x) [∂ + igW (x) + ig′y B (x)] φ(x) µ ≡ µ µ φ µ Hypercharge of the scalar ﬁeld y = Q T = 1/2 • φ φ − 3 (x) is invariant under local SU(2) U(1) transformations • LS ⊗ The potential is very similar to the one considered earlier. There is an • inﬁnite set of degenerate states, satisfying

µ2 v 0 φ(0) 0 = − |h | | i| s 2h ≡ √2

August 10 – 14, 2009 Theory of Elementary Particles (page 72) Ahmed Ali DESY, Hamburg Choice of a particular ground state makes SU(2) U(1) • L ⊗ Y spontaneously broken to the electromagnetic U(1)QED Now, let us parametrize the scalar doublet as • σi 1 0 φ(x) = exp i θi(x) 2 √ v + H(x) 2 There are four real ﬁelds θi(x) and H(x) • Local SU(2) invariance of Lagrangian allows to rotate away • L any θi(x) dependence. They (θi(x)) are the (would be) massless Goldstone bosons associated with the SSB mechanism In terms of physical ﬁelds • 1 4 S = h v + H + HG2 L 4 L L Pure Higgs Lagrangian has the form • 2 2 1 µ 1 2 2 MH 3 MH 4 H = ∂ H ∂µH M H H H L 2 − 2 H − 2v − 8v2

From 2 , one sees that the vacuum expectation value of the • LHG neutral scalar ﬁeld has generated a quadratic term for the W ± and the Z boson

August 10 – 14, 2009 Theory of Elementary Particles (page 73) Ahmed Ali DESY, Hamburg EW Lagrangian Density Higgs coupling to gauge bosons • 2 +µ 2 1 2 1 2 µ 2 1 2 2 = M W W − 1 + H + H + M Z Z 1 + H + H LHG W µ v v2 2 Z µ v v2

W ±- and Z-bosons acquired masses • Z W − 1 + Z W MZ cos ΘW = MW = gv 2 2 2 2 MZ 2 MW Experimental measurements v H v H • MZ = 91.1875 0.0021 GeV M = 80.398 ±0.025 GeV Z H W + H W ±

Electroweak mixing angle 2 2 M Z M W • Z v2 H W − v2 H sin2 Θ = 1 M 2 /M 2 = 0.223 W − W Z 5 2 Fermi constant GF = 1.166371 10− GeV− gives direct determination • of the electroweak scale or the× Higgs VEV

1/2 − v = √2 GF = 246 GeV h i August 10 – 14, 2009 Theory of Elementary Particles (page 74) Ahmed Ali DESY, Hamburg EW Lagrangian Density Fermion Mass Generation Fermionic mass term = mψψ¯ = m ψ¯ ψ + ψ¯ ψ • Lm − − L R R L is not allowed as it breaks gauge symmetry Gauge-invariant fermion-scalar coupling, or Yukawa couplings, are: • (+) (0) ¯ φ ¯ φ ∗ Y = C1 u,¯ d (0) dR C2 u,¯ d ( ) uR L − L φ − L φ − − φ(+) C (¯ν , e¯) e + h.c − 3 e L φ(0) R C Second term involves a charged-conjugate scalar ﬁeld φ iσ φ∗ • ≡ 2 In physical (unitary) gauge after symmetry breaking, simpliﬁes • LY 1 ¯ Y (x) = [v + H(x)] C1 d(x)d(x) + C2 u¯(x)u(x) + C3 e¯(x)e(x) L −√2 VEV generates masses of the fermions • f md = C1v/√2, mu = C2v/√2, etc. mf Yukawa coupling in terms of fermion masses v • H H = 1 + m uu¯ + m dd¯ + m ee¯ LY − v u d e f

August 10 – 14, 2009 Theory of Elementary Particles (page 75) Ahmed Ali DESY, Hamburg EW Lagrangian Density Flavor Dynamics Experimentally known that there are 6 quark ﬂavors u, d, s, c, b, t, • 3 charged leptons e, µ, τ and 3 neutrinos νe, νµ, ντ Can be organized into 3 families of quarks and leptons • Exist 3 nearly identical copies of the same SU(2) U(1) structure; • L ⊗ Y with varying particles masses

Consider N generations of fermions ν′ , ℓ′ , u′ , d′ (j = 1, 2,...,N ) • G j j j j G The most general Yukawa-type Lagrangian has the form • (+) (0) ¯ (d) φ ¯ (u) φ ∗ Y = u¯j′ , dj′ Cjk (0) dkR′ + u¯j′ , dj′ Cjk ( ) ukR′ L − L φ L φ − Xjk − (+) (ℓ) φ + ν¯′ , e¯′ C ℓ′ + h.c j j L jk φ(0) kR After spontaneous symmetry breaking, in unitary gauge • H = 1 + u¯′ M′ u′ + d¯′ M′ d′ + ¯l′ M′l′ + h.c LY − v L u R L d R L l R (f) Mass matrices are deﬁned by (M′ ) C v/√2 • f jk ≡ jk

August 10 – 14, 2009 Theory of Elementary Particles (page 76) Ahmed Ali DESY, Hamburg EW Lagrangian Density

Mass matrices can be decomposed (M′ ) = S† S U , where S and • f jk f Mf f f f U are unitary and is diagonal, Hermitian and positive definite f Mf In terms of = diag (m , m , m ,...), etc • Mu u c t H = 1 + u¯ u + d¯ d + ¯l l LY − v Mu Md Ml Mass eigenstates are defined by f S f ′ and f S U f ′ • L ≡ f L R ≡ f f R Since ¯f ′ f ′ = ¯f f and ¯f ′ f ′ = ¯f f , there are no flavor-changing neutral • L L L L R R R R currents in the SM [Glashow-Illiopolous-Miani (GIM) mechanism]

For charged quark current: u¯′ d′ = u¯ S S†d u¯ Vd • L L L u d L ≡ L L The Cabibbo-Kobayashi-Maskawa (CKM) matrix V is the unitary • (3 3) matrix; couples any “up-type” quark with all “down-type” × For charged lepton current: ν¯′ l′ =ν ¯ S†l ν¯ l • L L L l L ≡ L L Charged-current Lagrangian density •

g µ µ = W − u¯ γ (1 γ )V d + ν¯ γ (1 γ )ℓ + h.c. LCC − √ µ i − 5 ij j ℓ − 5 2 2 ( " ij ℓ # ) X X

August 10 – 14, 2009 Theory of Elementary Particles (page 77) Ahmed Ali DESY, Hamburg EW Lagrangian Density

General N N unitary matrix is characterized by N 2 real parameters: • G × G G N (N 1)/2 moduli and N (N + 1)/2 phases G G − G G Under the phase redeﬁnitions u eiφi u and d eiθj d , the mixing i → i j → j • i(θj φi) matrix changes as V V e − ; 2N 1 phases are unobservable ij → ij G − The number of physical free parameters in V then gets reduced to • (N 1)2: N (N 1)/2 moduli and (N 1)(N 2)/2 phases G − G G − G − G − Simplest case of two generations N = 2 [1 moduli and no phases] • G cos θ sin θ V = C C sin θ cos θ − C C In case of three generations N = 3, there are 3 moduli and 1 phase • G iδ13 c12 c13 s12 c13 s13 e− iδ13 iδ13 V = s12 c23 c12 s23 s13 e c12 c23 s12 s23 s13 e s23 c13  − − 13 − 13  s s c c s eiδ c s s c s eiδ c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13   Here c cos θ and s sin θ , with i and j being “generation” labels • ij ≡ ij ij ≡ ij The only complex phase in the SM Lagrangian is δ ; only possible • 13 source of CP -violation phenomena in the SM

August 10 – 14, 2009 Theory of Elementary Particles (page 78) Ahmed Ali DESY, Hamburg Cabibbo, Kobayashi and Maskawa

August 10 – 14, 2009 Theory of Elementary Particles (page 79) Ahmed Ali DESY, Hamburg The Cabibbo-Kobayashi-Maskawa Matrix

Vud Vus Vub V V V V CKM ≡  cd cs cb  Vtd Vts Vtb   Customary to use the handy Wolfenstein parametrization • 1 1 λ2 λ Aλ3 (ρ iη) − 2 − V λ(1 + iA2λ4η) 1 1 λ2 Aλ2 CKM ≃ − − 2  Aλ3 (1 ρ iη) Aλ2 (1 + iλ2η) 1 − − −   Four parameters: A, λ, ρ, η • Perturbatively improved version of this parametrization • ρ¯ = ρ(1 λ2/2), η¯ = η(1 λ2/2) − − The CKM-Unitarity triangle [φ = β; φ = α; φ = γ] • 1 2 3

´ ; µ

´¼ ; ¼µ ´½ ; ¼µ

August 10 – 14, 2009 Theory of Elementary Particles (page 80) Ahmed Ali DESY, Hamburg Phases and sides of the UT

V ∗Vtd V ∗ Vcd V ∗ Vud α arg tb , β arg cb , γ arg ub ≡ −V V ≡ − V V ≡ − V V ub∗ ud tb∗ td cb∗ cd

β and γ have simple interpretation • iβ iγ V = V e− , V = V e− td | td| ub | ub|

α deﬁned by the relation: α = π β γ • − − The Unitarity Triangle (UT) is deﬁned by: • iγ iβ Rbe + Rte− = 1

August 10 – 14, 2009 Theory of Elementary Particles (page 81) Ahmed Ali DESY, Hamburg Current determination of VCKM

[ Particle Data Group, 2008] V = 0.97418(27) • | ud| Vus = 0.2255(19) • | | 3 V = (3.93 0.36) 10− • | ub| ± × Unitarity of the Ist Row of VCKM 2 2 2 3 1 ( V + V + V ) = (1 1) 10− − | ud| | us| | ub| ± ×

V = 0.230(11) • | cd| Vcs = 1.04 0.06 • | | ± 3 V = (41.2 1.1) 10− • | cb| ± × Unitarity of the second row of VCKM 1 ( V 2 + V 2 + V 2)=0.136 0.125 − | cd| | cs| | cb| ±

3 Vtd = (8.1 2.6) 10− • | | ± × 3 V = (38.7 0.3) 10− • | ts| ± × V > 0.74 (95%C.L.) | tb| • Conclusion: No Beyond-the-SM Physics in the V • CKM

August 10 – 14, 2009 Theory of Elementary Particles (page 82) Ahmed Ali DESY, Hamburg Current Status of the CKM-Unitarity Triangle

1.5 excluded at CL > 0.95 excluded area has CL > 0.95 γ 1.0 ∆ m d & ∆ m s sin 2 β 0.5 ∆ m d ε α K γ β η 0.0 α V ub SL α −0.5 V ub τ ν ε −1.0 K C K M f i t t e r γ sol. w/ cos 2 β < 0 Moriond 09 (excl. at CL > 0.95) −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 ρ

August 10 – 14, 2009 Theory of Elementary Particles (page 83) Ahmed Ali DESY, Hamburg Precision Tests of the Electroweak Theory: Z PHYSICS

Z boson: mixture of SU(2) and U(1) gauge ﬁelds: 3 Z = B sin θ + W cos θ with tan θ = g′/g − W W W Z – matter interactions:

= g Σ f¯ gf γ gf γ γ f Z L − cos θ V µ − A µ 5 · µ f f h f 2 i 1/2 gV = I3L 2 Q sin θW Rad Cor: current ρf f f − 2 f × ( gA = I3L Rad Cor: sin θeﬀ

PRODUCTION AT LEP1:

e− f 12π ΓeΓf tot cx: σ = 2 2 BW MZ ΓZ × f 3 e f FB asym: AFB = 4 A A Z τ polarization τ e+ f¯ P

3 GF M – partial widths : Γ(Z f f¯)= C Z g2 + g2 → 6√2π V A f 2gV gA – polarization parm.: A = 2 2 gV +gA

August 10 – 14, 2009 Theory of Elementary Particles (page 84) Ahmed Ali DESY, Hamburg RESULTS lineshape : mass width  couplings : branching ratios  asymmetries   experiment MZ = 91 187.5 2.1 MeV ± ⇐ theory ΓZ = 2 495.2 2.3 MeV ⊕ ± sin2 θl = 0.2324 0.0012 eff ± per-mille accuracies: sensitivity to quantum effects b mod. AFB & ALR probing new high scales IMPORTANT CONSEQUENCES within / beyond SM: # ν’s (light) within / beyond SM: top-quark prediction within / beyond SM: unification of gauge couplings

1.) NUMBER OF LIGHT NEUTRINOS 1.) light SM-type ν’s: Γ =Γ + N Γ Z vis ν · νν¯ 1.) light SM-type ν’s: N = 2.985 0.008 ν ± 1.)

August 10 – 14, 2009 Theory of Elementary Particles (page 85) Ahmed Ali DESY, Hamburg + Cross-section e e− hadrons at LEP [LEP Electroweak Working→ Group: http://lepewwg.web.cern.ch/LEPEWWG/)] 10 5 Z

4 − 10 e+e →hadrons Cross-section (pb)

10 3

2 CESR + - 10 DORIS PEP W W PETRA TRISTAN KEKB SLC PEP-II 10 LEP I LEP II 0 20 40 60 80 100 120 140 160 180 200 220 Centre-of-mass energy (GeV)

August 10 – 14, 2009 Theory of Elementary Particles (page 86) Ahmed Ali DESY, Hamburg Pull on the SM from Precision Electroweak Measurements [LEP Electroweak Working Group: http://lepewwg.web.cern.ch/LEPEWWG/)] Measurement Fit |O meas − O fit |/ σ meas (5) 0 1 2 3 ∆α had (m Z ) 0.02758 ± 0.00035 0.02767 m Z [ GeV ] 91.1875 ± 0.0021 91.1874 Γ Z [ GeV ] 2.4952 ± 0.0023 2.4959 0 σ had [ nb ] 41.540 ± 0.037 41.478 R l 20.767 ± 0.025 20.743 0,l A fb 0.01714 ± 0.00095 0.01643 A l (P τ ) 0.1465 ± 0.0032 0.1480 R b 0.21629 ± 0.00066 0.21581 R c 0.1721 ± 0.0030 0.1722 0,b A fb 0.0992 ± 0.0016 0.1038 0,c A fb 0.0707 ± 0.0035 0.0742 A b 0.923 ± 0.020 0.935 A c 0.670 ± 0.027 0.668 A l (SLD) 0.1513 ± 0.0021 0.1480 2 lept sin θ eff (Q fb ) 0.2324 ± 0.0012 0.2314 m W [ GeV ] 80.398 ± 0.025 80.377 Γ W [ GeV ] 2.097 ± 0.048 2.092 m t [ GeV ] 172.6 ± 1.4 172.8 March 2008 0 1 2 3

August 10 – 14, 2009 Theory of Elementary Particles (page 87) Ahmed Ali DESY, Hamburg Determination of the number of Neutrinos at LEP [LEP Electroweak Working Group: http://lepewwg.web.cern.ch/LEPEWWG/)] ] 2ν nb [

3ν had 30 ALEPH σ DELPHI L3 4ν OPAL 20 average measurements, error bars increased by factor 10 10

0 86 88 90 92 94 [ ] Ecm GeV

August 10 – 14, 2009 Theory of Elementary Particles (page 88) Ahmed Ali DESY, Hamburg Important Consequences from LEP Physics

2.) TOP QUARK: (i) existence: Ib = 1 and Ib =0 ; t = isospin-partner to b 3L − 2 3R ⇒

(ii) mass: t ρ =1+∆ρt + ∆ρH 2 ∆ ρt GF m ∼ t +13 b LEP1/SLD: mt = 172.6 10 GeV − 2 l M , sin θ ,G Tevatron: mt = 172.6 1.4 GeV Z eff F → ±

prediction of t mass: successful SM conﬁrmation 2.) [1994 : 173 26 GeV] at quantum level ±

August 10 – 14, 2009 Theory of Elementary Particles (page 89) Ahmed Ali DESY, Hamburg Current Measurements of Mt at the Tevatron and LEP/SLD [From the LEP Electroweak Working Group: http://lepewwg.web.cern.ch/LEPEWWG/] Top-Quark Mass [ GeV ]

CDF 172.5 ± 1.8 D ∅ 172.7 ± 1.8 Average 172.6 ± 1.4 χ 2 /DoF: 9.2 / 10

+ 13.2 LEP1/SLD 172.6 172.6 − 10.2 Γ + 11.6 LEP1/SLD/m W / W 178.7 − 8.6

140 160 180 200 [ ] March 2008 m t GeV In agreement with the indirect measurements through loop eﬀects at LEP1 and SLD: • +13.2 Mt(LEP1/SLD) = 172.6 10.2 GeV −

August 10 – 14, 2009 Theory of Elementary Particles (page 90) Ahmed Ali DESY, Hamburg Nobel Prize for Physics 1999

”for elucidating the quantum structure of electroweak interactions in physics”

Gerardus ’t Hooft Martinus J.G. Veltman 1/2 of the prize 1/2 of the prize the Netherlands the Netherlands Utrecht University Bilthoven, the Netherlands

August 10 – 14, 2009 Theory of Elementary Particles (page 91) Ahmed Ali DESY, Hamburg Precision Tests of the Electroweak Theory: W PHYSICS Central Issues:

(i) Static properties of W ± bosons: mass, width, . . . (ii) Yang-Mills gauge ﬁeld dynamics: = 1 W~ 2 . . . L − 2 h µνi tri-linear couplings: (ii) quattro-linear couplings:

Establish Weyl & Yang / Mills gauge principle as fundamental base for [SM] forces • PRODUCTION: + e + W + e W + ν γ ; Z e − W − L W − e − M = 80.398 0.025 GeV W ± Γ = 2.097 0.048 GeV W ±

August 10 – 14, 2009 Theory of Elementary Particles (page 92) Ahmed Ali DESY, Hamburg Measurement of MW and ΓW

W-Boson Mass[GeV] W-Boson Width[GeV]

TEVATRON± 0.040 TEVATRON 80.430 ± 0.058 2.049

LEP2± 0.033 LEP2 80.376 ± 0.083 2.196

Average± 0.025 Average 80.398 ± 0.048 2.097

2 2 χ/DoF: 1.1 / 1 χ/DoF: 2.1 / 1

− NuTeV± 0.084 pp 80.136 indirect ± 0.057 2.141

LEP1/SLD± 0.032 LEP1/SLD 80.363 ± 0.003 2.091

LEP1/SLD/m 80.363± 0.020 LEP1/SLD/m 2.091± 0.002 t t

80 80.2 80.4 80.6 2 2.2 2.4

m [GeV] March 2008 Γ [GeV] March 2008 W W

August 10 – 14, 2009 Theory of Elementary Particles (page 93) Ahmed Ali DESY, Hamburg Relationship between MW and Mt in the SM [LEP Electroweak working group: http://lepewwg.web.cern.ch/LEPEWWG/]

March 2008 LEP2 and Tevatron (prel.) 80.5 LEP1 and SLD 68 % CL ] GeV

[ 80.4

W m

80.3 [ ] ∆α m H GeV 114 300 1000 150 175 200 [ ] m t GeV

August 10 – 14, 2009 Theory of Elementary Particles (page 94) Ahmed Ali DESY, Hamburg Relationship between MW and Mt in the SM and MSSM [Heinemeyer et al. hep-ph/0604147; Updated by Heinemeyer (2008)]

80.70 experimental errors 68% CL: LEP2/Tevatron (today) Tevatron/LHC 80.60 light SUSY

80.50 MSSM [GeV] W M 80.40 heavy SUSY

M = 114 GeV 80.30 H SM SM = 400 GeV MSSM M H both models 80.20 Heinemeyer, Hollik, Stockinger, Weber, Weiglein ’08 160 165 170 175 180 185 mt [GeV] Fit to the precision electroweak data in the SM & MSSM. The curve labelled heavy •SUSY assumes the supersymmetric parameters are set at 2 TeV

August 10 – 14, 2009 Theory of Elementary Particles (page 95) Ahmed Ali DESY, Hamburg Current Searches of the Higgs Boson Approaching fourth step: 1.) VIRTUAL HIGGS BOSON: H W, Z properties: ρ =1+∆ρt + ∆ρH ∆ρ G M 2 log M 2 H ∼ F Z H

+36 1.) MH = 87 27 GeV and MH < 190 GeV [95% CL] − 1.) [ﬁt probability 36%] ∼ 2.) DIRECT SEARCH: M 114.4 GeV H ≥ 2.) DIRECT SEARCH AT LEP: [1.7σ excess for 115 GeV] ≥ b ALL LEP AND SLC SM OBSERVABLES, EXCEPT AFB, POINT TO LIGHT HIGGS-BOSON MASS Higgs search is a central target of the experiments at Tevatron; current sensitivity to •various production & decay channels restricted by luminosity Discovering Higgs & determining its properties is one of the principal reasons for •building LHC

August 10 – 14, 2009 Theory of Elementary Particles (page 96) Ahmed Ali DESY, Hamburg Direct and Indirect bounds on the Higgs Boson Mass [LEP Electroweak Working Group: http://lepewwg.web.cern.ch/LEPEWWG/)]

m = 160 GeV 6 March 2008 Limit Theory uncertainty ∆α (5) had = 5 0.02758 ±0.00035 0.02749 ±0.00012 4 incl. low Q 2 data

2 3 ∆χ 2

1 Excluded Preliminary 0 30100 300 [ ] m H GeV

August 10 – 14, 2009 Theory of Elementary Particles (page 97) Ahmed Ali DESY, Hamburg SM Higgs X-section at the Tevatron

[T. Hahn et al., hep-ph/0607308 & http://maltoni.home.cern.ch/maltoni/TeV4LHC/]

SM Higgs production

3 10 gg → h TeV II σ [fb ] qq → Wh

10 2 qq → qqh

bb → h 10 qq → Zh

gg,qq → tth

TeV4LHC Higgs working group 1 100 120 140 160 180 200 [ ] m h GeV

August 10 – 14, 2009 Theory of Elementary Particles (page 98) Ahmed Ali DESY, Hamburg SM Higgs decay branching ratios

[Carena & Haber]

August 10 – 14, 2009 Theory of Elementary Particles (page 99) Ahmed Ali DESY, Hamburg CDF and D0 limits on the SM Higgs Boson

Tevatron Run II Preliminary, L=0.9-4.2 fb -1 LEP Exclusion Tevatron Exclusion Expected Observed 10 ±1σ Expected ±2σ Expected 95% CL Limit/SM

1 SM March 5, 2009 100 110 120 130 140 150 160 170 180 190 200 2 mH(GeV/c )

August 10 – 14, 2009 Theory of Elementary Particles (page 100) Ahmed Ali DESY, Hamburg SM Higgs X-section at the LHC

[T. Hahn et al., hep-ph/0607308 & http://maltoni.home.cern.ch/maltoni/TeV4LHC/] SM Higgs production 10 5 LHC σ [fb] gg → h 10 4 qq → qqh

10 3 qq → Wh bb → h gg,qq → tth 10 2 qb → qth qq → Zh TeV4LHC Higgs working group 100 200 300 400 500 [ ] m h GeV

August 10 – 14, 2009 Theory of Elementary Particles (page 101) Ahmed Ali DESY, Hamburg Gauge couplings in the SM The gauge couplings gi (i = 1, 2, 3 for U(1)Y , SU(2)I and SU(3)c) are • energy-dependent due to the renormalization eﬀects d g2 2 µ g (µ) = 3 (11 n ) + O(g5) dµ 3 −16π2 − 3 f 3 d g2 µ g (µ) = 2 (11 n ) + O(g5) dµ 2 −24π2 − f 2 d g2 10 µ g (µ) = 1 ( n ) + O(g5) dµ 1 −4π2 − 9 f 1 To lowest order in the respective couplings • 1 1 1 SM M α (µ)− = α (M)− + b ln( ) i i 2π i µ For n = 6, bSM =(bSM,bSM,bSM) = (41/10, 19/6, 7); M is some reference scale f i 1 2 3 − − Thanks to precise electroweak and QCD measurements at LEP, Tevatron and HERA, • values of the three coupling constants well measured at the scale Q = MZ (5) +25 αs(MZ) = α3(MZ) = 0.1187 0.002 = ΛQCD = 217 23 MeV ± ⇒ − 1 5 1 1 α− (M ) = α− (M ) + α− (M ) = 128.91 0.02 em Z 3 1 Z 2 Z ± α α (M ) = em (M ) with sin2(θ)(M ) = 0.2318 0.0005 2 Z sin2(θ) Z Z ±

August 10 – 14, 2009 Theory of Elementary Particles (page 102) Ahmed Ali DESY, Hamburg Gauge coupling unification in supersymmetric theories Unification of the gauge couplings (of the strong, electromagnetic and weak • interactions) in grand unified theories widely anticipated [Dimopoulos, Raby, Wilczek; Langacker et al.; Ellis et al.; Amaldi et al.;... ] However, unification does not work in SU(5); works in supersymmetric SU(5) but • also in SO(10), as both have an additional scale: ΛS for SUSY phenomenologically of O(1) TeV; grand unification works in SO(10) due to an additional scale M(WR) for the right-handed weak bosons; both cases yielding the Grand Unification Scale Λ = O(3 1016) GeV G × In SUSY, the running of α (µ) changes according to • i

1 1 1 S ΛG α (µ)− = + b ln( ) θ(µ Λ ) i α0(Λ ) 2π i µ − S i G 1 1 Λ + + bSM ln( S ) θ(Λ µ) α0(Λ ) 2π i µ S − i S SM SM SM SM with bi =(b1 ,b2 ,b3 ) = (41/10, 19/6, 7) below the supersymmetric scale, and bS =(bS,bS,bS) = (33/5, 1, 3) above− the− supersymmetric scale i 1 2 3 − The uniﬁed coupling α is related to α (M ) by • G i Z 1 1 αi− (MZ ) = αG− + bitG

1 MG 1 with tG = ln 5.32 and α− = 23.3 2π MZ ∼ G

August 10 – 14, 2009 Theory of Elementary Particles (page 103) Ahmed Ali DESY, Hamburg Gauge coupling uniﬁcation in the SM and MSSM

August 10 – 14, 2009 Theory of Elementary Particles (page 104) Ahmed Ali DESY, Hamburg Summary

Nature does not read papers on free ﬁeld theories. Learn Quantum Field Theory! • Applications of QCD have permeated in practically all the branches of particle and • nuclear physics and there is a lot of scope to make solid contributions in this area

The current experimental and theoretical thrust is on understanding the mechanism • of electroweak symmetry breaking (Higgs mechanism or alternatives)

There exist strong hints for Supersymmetry (gauge coupling uniﬁcation, candidates • for dark matter); Higgs mass stability also requires Supersymmetry

LHC is the next high energy frontier. Let us see what it brings to us! • All those entering the ﬁeld now, my message is: • You are faced with insurmountable oppurtunities! Go, get them!!

August 10 – 14, 2009 Theory of Elementary Particles (page 105) Ahmed Ali DESY, Hamburg Backup Slides

Dimensional regularization • Renormalization: QED •

August 10 – 14, 2009 Theory of Elementary Particles (page 106) Ahmed Ali DESY, Hamburg Regularization of Loop Integrals

General Remarks: In the lowest order (Born Approximation), straightforward to calculate scattering • amplitudes, given the Feynman rules. However, a reliable theory must be stable against Quantum Corrections (Loops)

Loop integrals are often divergent; as QCD is a gauge theory, need to have a • gauge-invariant method of regularizing these divergences; the most popular method is Dimensional Regularization

Conceptually, the dependence on the regulator is eliminated by absorbing it into a • redefinition of the coupling constant, of the quark mass(es), and the wave function renormalization of the quark and gluon fields (Z ,Z ,Z , δm = m m ) 1 2 3 − 0 If we neglect the quark masses (a good approximation for the light quarks u and d), • then QCD has a coupling constant (gs) but no scale. Quantum effects generate a scale, ΛQCD

QCD by itself does not determine ΛQCD, much the same way as it does not • determine the quark masses. These fundamental parameters have to be determined by theoretical consistency and experiments

August 10 – 14, 2009 Theory of Elementary Particles (page 107) Ahmed Ali DESY, Hamburg Dimensional Regularization k q q

a µ ν b

k – q Let us consider the self-energy gluon loop above • 4 µ ν µν 2 d k Tr[γ kγ/ (k/ q/)] i Π (q) = g δabTF − ab − s (2π)4 k2(k q)2 Z − The problem appears in the momentum integration, which is divergent • d4k(1/k2) ∼ → ∞ Regularize the loop integrals through dimensional regularization; the calculation is R • performed in D = 4 2ǫ dimensions. For ǫ = 0 the resulting integral is well deﬁned: − 6 dDk kα(k q)β i q2 ǫ 5 q2gαβ − = − − Γ( ǫ) 1 ǫ + qαqβ (2π)D k2(k q)2 6(4π)2 4π − − 3 2(1 + ǫ) Z − ( ) The ultraviolet divergence of the loop appears at ǫ = 0, through the pole of the • Gamma function

1 2 Γ( ǫ) = γE + (ǫ ) , γE = 0.577215 ... − − ǫ − O

August 10 – 14, 2009 Theory of Elementary Particles (page 108) Ahmed Ali DESY, Hamburg Dimensional Regularization Contd.

Introduce an arbitrary energy scale µ and write • 2 ǫ 2ǫ q 2ǫ 1 2 2 µ − Γ( ǫ) = µ + γE ln 4π +ln( q /µ ) + (ǫ) 4πµ2 − − ǫ − − O Written in this form, one has a dimensionless quantity ( q2/µ2) inside the logarithm. • The contribution of the loop diagram can be written as− Πµν = δ q2gµν + qµqν Π(q2) ab ab − which is ultraviolet divergent, with

ǫ 2 2 4 gsµ 1 2 2 5 Π(q ) = TF + γE ln 4π +ln( q /µ ) + (ǫ) . −3 4π ǫ − − − 3 O While divergent, and hence the self-energy contribution remains undetermined, we • know now how the eﬀect changes with the energy scale

2 2 Fixing Π(q ) at some reference momentum transfer q0, the result is known at any • other scale: 2 2 2 4 gs 2 2 Π(q ) = Π(q ) TF ln(q /q ) 0 − 3 4π 0

August 10 – 14, 2009 Theory of Elementary Particles (page 109) Ahmed Ali DESY, Hamburg Split the self-energy contribution into a divergent piece and a finite q2-dependent • term Π(q2) ∆Π (µ2)+Π (q2/µ2) ≡ ǫ R This separation is ambiguous, as the splitting can be done in many different ways. A • given choice defines a scheme:

2 TF gs 2ǫ 1 5 3π 4π µ ǫ + γE ln(4π) 3 (µ scheme) − 2 − − ∆Π (µ2) = TF gs 2ǫ 1 ǫ  3π 4π µ ǫ (MS scheme) − 2  TF gs µ2ǫ 1 + γ ln(4π) (MS scheme) − 3π 4π ǫ E − 2  TF gs 2 2 3π 4π ln( q /µ ) (µ scheme) − 2 − Π (q2/µ2) = TF gs 2 2 5 R  3π 4π ln( q /µ ) + γE ln(4π) 3 (MS scheme) − 2 − − −  TF gs ln( q2/µ2) 5 (MS scheme) − 3π 4π − − 3 In the µ scheme, Π( µ2) is used to deﬁne the divergent part • − In the MS (minimal subtraction) scheme, one subtracts only the divergent 1/ǫ term • In the MS (modiﬁed minimal subtraction schemes), one puts also the γ ln(4π) E − • factor into the divergent part

2 2 2 Note, the logarithmic q dependence in ΠR(q /µ ) is always the same in these • schemes

August 10 – 14, 2009 Theory of Elementary Particles (page 110) Ahmed Ali DESY, Hamburg Renormalization: QED Recall: A QFT is called renormalizable if all ultraviolet divergences can be absorbed • through a redeﬁnition of the original ﬁelds, masses and couplings Let us consider the electromagnetic interaction between two electrons • e– e–

= q + + +. . .

e– e– At one loop, the QED photon self-energy contribution is: • ǫ 2 2 4 eµ 1 2 2 5 Π(q ) = + γE ln 4π +ln( q /µ ) + (ǫ) −3 4π ǫ − − − 3 O The scattering amplitude takes the form • 2 2 µ e 2 T (q ) J Jµ 1 Π(q ) + ... ∼ − q2 − J µ denotes the electromagnetic fermion current. At lowest order, T (q2) α/q2 ∼ • with α = e2/(4π)

August 10 – 14, 2009 Theory of Elementary Particles (page 111) Ahmed Ali DESY, Hamburg The divergent correction from the quantum loops can be reabsorbed into a • redeﬁnition of the electromagnetic coupling: 2 α0 2 2 2 αR(µ ) 2 2 1 ∆Πǫ(µ ) ΠR(q /µ ) 1 ΠR(q /µ ) q2 − − ≡ q2 − 2 2 α0 2ǫ 1 αR(µ ) = α0 (1 ∆Πǫ(µ )) = α0 1 + µ + Cscheme + ... − 3π ǫ 2 where α e0 , and e is the bare coupling appearing in the QED Lagrangian and 0 ≡ 4π 0 is not an observable

The scattering amplitude written in terms of αR is ﬁnite and the resulting • cross-section can be compared with experiment; thus, one actually measures the renormalized coupling αR

2 2 2 Both αR(µ ) and the renormalized self-energy correction ΠR(q /µ ) depend on µ, • but the physical scattering amplitude T (q2) is µ-independent: (Q2 q2) ≡ − 2 2 2 µ αR(Q ) αR(Q ) T (q ) = 4πJ Jµ 1 + C′ + Q2 3π scheme ···

The quantity α(Q2) α (Q2) is called the QED running coupling • ≡ R

August 10 – 14, 2009 Theory of Elementary Particles (page 112) Ahmed Ali DESY, Hamburg The usual fine structure constant α = 1/137 is defined through the classical • Thomson formula and corresponds to a very low scale Q2 = m2 − e The scale dependence of α(Q2) is regulated by the so-called β function • dα α α 2 µ α β(α); β(α) = β1 + β2 + dµ ≡ π π ··· At the one-loop level, β reduces to its first coefficient • 2 βQED = 1 3 The solution of the differential equation for α is: • 2 2 α(Q0) α(Q ) = 2 1 β1α(Q0) ln(Q2/Q2) − 2π 0

Since β1 > 0, the QED running coupling increases with the energy scale: • 2 2 2 2 α(Q ) > α(Q0) if Q > Q0

The value of α relevant for high Q2, measured at LEP, conﬁrms this: • α(M 2 ) 1/128 Z MS ≃

August 10 – 14, 2009 Theory of Elementary Particles (page 113) Ahmed Ali DESY, Hamburg