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The Nobel Prize in The Nobel Prize in Physics 2004 Physics 2004 Prize Announcement Press Release "for the discovery of asymptotic freedom in the Advanced theory of the strong interaction" Information Information for the

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H. David Politzer Nobel Lecture David J. Gross H. David Frank Wilczek Other Resources Politzer 1/3 of the 1/3 of the 1/3 of the Frank Wilczek prize prize prize Nobel Lecture USA USA USA Interview Other Resources Kavli Institute California Massachusetts

for Theoretical Institute of Institute of Physics, Technology Technology 2003 University of Pasadena, CA, (MIT) The 2004 Prize in: California USA Cambridge, Physics Santa Barbara, MA, USA Chemistry CA, USA Physiology or b. 1941 b. 1949 b. 1951 Medicine Literature Peace

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Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 1 Asymptotic Freedom: History and Implications

Bryan Webber University of Cambridge Cavendish Laboratory

• Quantum Chromodynamics – Quarks – Colour • Asymptotic Freedom – Effective charge – Non-Abelian fields – History • Implications – Confinement – Grand unification? • Conclusions

RP Crease and CC Mann, The Second Creation, Macmillan 1986 L Hoddeson et al., The Making of the Standard Model, CUP 1997

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 2 Quark Model of Hadrons

• In the early 1960’s, many new strongly-interacting particles (hadrons) were discovered, all apparently as ‘fundamental’ as the familiar proton, neutron and π-meson. • In 1964, Gell-Mann and Zweig (independently) noticed that the quantum numbers of all the known hadrons corresponded to those of collections of 2 or 3 spin one- half, fractionally-charged constituents, called quarks by Gell-Mann (Nobel Prize 1969). ddd udd uud uuu ∆ K 0 K + n p ds us udd uud dds uds uus Σ *

du uu,dd ud dds uds uus + π − π 0,η π + Σ Σ 0, Λ Σ dss uss Ξ* su sd dss uss K − K 0 sss Ω Ξ Ξ0

– Zweig (unpublished CERN preprint) called them ‘aces’. – Quarks were regarded as ‘mathematical’ entities because they were not seen individually. – Also they seemed to violate the spin-statistics theorem: some hadrons corresponded to totally symmetric combinations of 3 identical quarks.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 3 Colour

• Han and Nambu (1965, also Greenberg) solved the spin-statistics problem by proposing that quarks have a new “colour” degree of freedom which can take 3 values (red, green, blue), with respect to which the ‘symmetrical’ quark states are antisymmetric. Associated symmetry group SU(3). – Originally thought of as a global symmetry,

0 qa → qa = Uab qb

where a = 1, 2, 3 and the symmetry transformations are represented by constant (i.e. space-time independent) 3×3 unitary matrices U, which mix the colours while preserving the normalization. – Observed states qq¯, qqq are colour singlets. – It was necessary to postulate that non-singlet states (q, qq, (qq¯)8, . . . ) are forbidden. • The notion of quarks as ‘real’ rather than ‘mathematical’ constituents of hadrons was considered quite implausible . . . until amazing experimental data started arriving from the new Stanford Linear Accelerator Centre (SLAC).

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 4 Deep inelastic electron scattering

• The SLAC experiments (1967) probed the structure of the proton by scattering high-energy electrons

• Useful dimensionless variables are:

−q2 Q2 q · p E0 x = = , y = = 1 − 2p · q 2M(E − E0) k · p E

where Q2 = −q2 > 0, M = proton mass, and energies refer to target rest frame. • Elastic scattering has x = 1. Deep inelastic scattering (DIS) means Q2 À M 2 and x < 1.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 5 2 • Structure functions Fi(x, Q ) parametrise target structure as ‘seen’ by virtual photon:

d2σ 8πα2ME 1 + (1 − y)2 = 2xF1 + (1 − y)(F2 − 2xF1) − (M/2E)xyF2 dxdy Q4 2 ·µ ¶ ¸

• Bjorken limit is Q2 → ∞ and p · q → ∞ with x ﬁxed. In this limit structure functions were found to obey approximate Bjorken scaling (1969), i.e. they depend only on dimensionless variable x:

2 Fi(x, Q ) −→ Fi(x)

• Although Q2 varies by two orders of magnitude, in ﬁrst approximation data lie on universal curve.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 6 Implications of Bjorken scaling

• Bjorken scaling implies that virtual photon is scattered by pointlike constituents – partons (Feynman, 1969) — otherwise structure functions would depend on ratio Q/Q0, with 1/Q0 a length scale characterizing size of constituents. • Quantitative study over the next few years established that these pointlike partons had spin one-half and fractional charges, consistent with those expected for the quarks of Gell-Mann and Zweig. – However, the quark were found to carry only about one-half of the momentum of the target proton – The other half must be carried by strongly-interacting, neutral, bosonic constituents. . . • Modern data show weak (logarithmic) scaling violation, understood as a calculable higher-order eﬀect.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 7 x=6.32E-5 (x) x=0.000102 ZEUS NLO QCD fit

10 x=0.000161 x=0.000253 tot. error

-log x=0.0004

em 2 x=0.0005 5 H1 94-00

F x=0.000632 x=0.0008 H1 96/97 x=0.0013 ZEUS 96/97 BCDMS x=0.0021 4 E665 x=0.0032 NMC

x=0.005

x=0.008 3 x=0.013

x=0.021

x=0.032 2 x=0.05

x=0.08

x=0.13 1 x=0.18 x=0.25

x=0.4 x=0.65

0 2 3 4 5 1 10 10 10 10 10 Q2(GeV2) Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 8 Yang-Mills theory

• Back in 1954, Yang & Mills had shown that one could extend the notion of local gauge invariance from the Abelian U(1) symmetry group of QED:

0 ψe(x) → ψe(x) = U(x) ψe(x)

where U(x) = eiφ(x), to a non-Abelian symmetry group:

0 ψa(x) → ψa(x) = Uab(x) ψb(x)

where a = 1, . . .N and U(x) ∈ SU(N) is a space-time dependent N×N unitary matrix, which mixes the N states ψa while preserving the normalization. – As in QED, gauge invariance then requires the existence of a vector gauge field – Quanta of the gauge field would be massless spin-1 particles, analogous to the photon, communicating long-range forces • In 1954, this did not fit with the properties of either strong or weak nuclear interactions. So Yang-Mills theory was ignored. • Even QED was regarded as a sick theory (see later), and quantum field theory went out of fashion for about 15 years. . .

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 9 AND THEN . . .

• The SLAC results prompted renewed interest in a ﬁeld theory of the strong interactions of pointlike quarks. • Fritzsch & Gell-Mann (1971, also Weinberg) proposed that the strong interaction is an SU(3) Yang-Mills gauge theory: Quantum ChromoDynamics. – Quanta of the colour ﬁeld (gluons) would communicate the strong force between quarks – Gluon exchange forces would be attractive in colour singlet states – Gluons could also carry the ‘missing momentum’ of the proton • But why should quarks behave like almost-free particles in DIS? • And why should colour non-singlet states be forbidden?

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 10 Quantum Chromodynamics

• QCD is an SU(3) gauge ﬁeld theory: quarks come in 3 “colours” (a = 1, 2, 3) and the gauge transformations are represented by 3×3 unitary matrices U, which mix the colours while preserving the normalization:

0 qa(x) → qa(x) = Uab(x) qb(x) it·θ(x) Uab(x) = e 8 t · θ ≡ tAθA XA=1

where θA are 8 real parameters and tA, the generators of SU(3), are the 8 linearly independent 3 × 3 traceless hermitian matrices. A B 1 – Conventional normalization is Tr (t t ) = 2δAB – The theory is non-Abelian, i.e., successive gauge transformations do not commute: [tA, tB] = if ABCtC where the structure constants of the gauge group, f ABC, are totally antisymmetric. 4 – t · t = CF I where CF = 3 is the “quark colour charge squared” ABC ABD – A,B f f = CA δCD where CA = 3 is the “gluon colour charge squared” P Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 11 • The QCD Lagrangian density is

1 A Aµν f µ f LQCD = −4FµνF + q¯a [iγ (Dµ)ab − mf δab] qb ﬂavoursX f

A A A ABC B C – Gluon field strength tensor Fµν = ∂µAν − ∂νAµ − gSf Aµ Aν – Covariant derivative (Dµ)ab = ∂µ δab + igS (t · Aµ)ab A – Aµ are the gluon fields (A = 1, . . . , 8) 2 – gS is the QCD coupling; by analogy with QED we define αS ≡ gS/4π • The third term in the gluon field strength tensor is essential for gauge invariance of LQCD. Notice that it would vanish for an Abelian (commuting) gauge group (f ABC = 0). • As a consequence of this term, there are 3-gluon and 4-gluon self-interactions: A,µ B,ν B,ν A,µ q p r C,λ C,λ D,ρ

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 12 Eﬀective Charge

• In QED the observed electron charge is distance-dependent (⇒ momentum transfer dependent) due to vacuum polarization: α eff - - + - - + + - + + 1 - + + - 1 128 - + + 137 + + - - + - - 2 2 mZ q • The one-loop vacuum polarization diagram

is log-divergent and can be regularized by introducing a cut-oﬀ Λ.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 13 • Then the observed coupling at momentum transfer qµ is

q q

2 2 α q α(q ) = αbare 1 + ln + · · · Ã 3π Λ2 ! Thus 2 2 dα α q = + · · · ≡ βQED(α) dq2 3π where the QED β-function βQED(α) > 0. At one-loop order, we ﬁnd

2 2 α(Q ) dα 1 Q dq2 1 Q2 = = ln α2 3π 2 q2 3π m2 Z1/137 Zme e 1 ⇒ α(Q2) = 1 Q2 137 − 3π ln 2 me

2 411π 2 • Therefore α → ∞ at Q = e me and is not defined at higher scales (shorter distances). • This led to the view (Landau, 1954) that QED is not a well-defined theory, and more generally to a lack of confidence in quantum field theory, which lasted for about 15 years.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 14 Asymptotic Freedom

• In QCD there are additional contributions from gluon self-interaction:

qf g

qf g As a consequence, the β-function changes. Now

2 2 2 nf q 11 q αS(q ) = αS,bare 1 + αS ln − CAαS ln − · · · Ã 6π Λ2 12π Λ2 !

2 where nf is the number of quark ﬂavours. Hence βQCD(αS) = β0αS − · · · where 1 β0 = (2nf − 11CA) < 0 12π

• The gluonic contribution to the vacuum polarization reverses the sign of the β-function, so that the 2 strong coupling αS decreases at large q (short distances). This is called asymptotic freedom. It implies that quarks behave as (almost) free particles at short distances, and perturbation theory can be used for hard processes, i.e. processes involving large momentum transfers, such as DIS.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 15 Non-Abelian Vacuum Polarization

∇ · E = g δ3(r) + g (A · E − A · E)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 16 Non-Abelian Vacuum Polarization

∇ · E = g (A · E − A · E)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 17 Non-Abelian Vacuum Polarization

E E

∇ · E = g (A · E − A · E)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 18 Non-Abelian Vacuum Polarization

E E

∇ · E = g δ3(r) + g (A · E − A · E)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 19 The Vacuum as a Magnetic Medium

• In a dielectric, screening of the eﬀective charge corresponds to ε > 1, anti-screening to ε < 1. • Since εµ = 1 for the vacuum, anti-screening implies magnetic susceptibility χ = µ − 1 > 0, i.e. a paramagnetic vacuum. • Vacuum polarization due to quanta of spin S gives a vacuum susceptibility proportional to

1 (−1)2S (2S)2 − 3 · ¸ 1 Thus S = 0 and 2 give diamagnetic (screening) contributions, but gauge bosons give anti-screening. 1 • The existence of large numbers of new spin-0 and/or spin- 2 particles at higher energies could modify or even destroy asymptotic freedom. – This is exactly what happens in supersymmetric extensions of the Standard Model.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 20 History of Asymptotic Freedom

1954 Yang & Mills study vector ﬁeld theory with non-Abelian gauge invariance. 1965 Vanyashin & Terentyev compute vacuum polarization due to a massive charged vector ﬁeld. In our notation, they found 1 21 1 β0 = − = −11 + 12π 2 2 µ ¶ 1 – The 2 comes from longitudinal polarization states (absent for massless gluons) – They concluded that this result “. . . seems extremely undesirable”. 1969 Khriplovich correctly computes the one-loop β-function in SU(2) Yang-Mills theory using the Coulomb gauge CA β0 = (−12 + 1 = −11) 12π

– In Coulomb gauge the anti-screening (–12) is due to an instantaneous Coulomb interaction – He did not make a connection with strong interactions 1971 ’t Hooft correctly computes the one-loop β-function for SU(3) gauge theory but does not publish it. – He wrote it on the blackboard at a conference – His supervisor (Veltman) told him it wasn’t interesting – ’t Hooft & Veltman received the 1999 Nobel Prize for proving the renormalizability of QCD (and the whole Standard Model).

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 21 1972 Fritzsch & Gell-Mann propose that the strong interaction is an SU(3) gauge theory, later named QCD by Gell-Mann 1973 Gross & Wilczek, and independently Politzer, compute and publish the 1-loop β-function for QCD

1 β0 = (2nf − 11CA) 12π

⇒ 2004 Nobel Prize (now that ’t Hooft has one anyway . . . ) 1974 Caswell & Jones compute the 2-loop β-function for QCD 1980 Tarasov, Vladimirov & Zharkov compute the 3-loop β-function for QCD 1997 van Ritbergen, Vermaseren & Larin compute the 4-loop β-function for QCD (∼ 50, 000 Feynman diagrams)

“. . . We obtained in this way the following result for the 4-loop beta function in the MS-scheme:

2 ∂as 2 3 4 5 6 q = −β0a − β1a − β2a − β3a + O(a ) ∂q2 s s s s s

where as = αS/4π and . . .

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 22 11 4 β0 = CA − TF nf 3 3

34 2 20 β1 = C − 4CF TF nf − CATF nf 3 A 3

2857 3 2 205 β2 = C + 2C TF nf − CF CATF nf 54 A F 9

1415 2 44 2 2 158 2 2 − C TF nf + CF T n + CAT n 27 A 9 F f 27 F f

4 150653 44 3 39143 136 β3 = C − ζ3 + C TF nf − + ζ3 A 486 9 A 81 3 µ ¶ µ ¶ 2 7073 656 2 4204 352 +C CF TF nf − ζ3 + CAC TF nf − + ζ3 A 243 9 F 27 9 µ ¶ µ ¶ 3 2 2 2 7930 224 2 2 2 1352 704 +46C TF nf + C T n + ζ3 + C T n − ζ3 F A F f 81 9 F F f 27 9 µ ¶ µ ¶ 2 2 17152 448 424 3 3 1232 3 3 +CACF T n + ζ3 + CAT n + CF T n F f 243 9 243 F f 243 F f µ ¶ abcd abcd abcd abcd dA dA 80 704 dF dA 512 1664 + − + ζ3 + nf − ζ3 N 9 3 N 9 3 A µ ¶ A µ ¶ abcd abcd 2 dF dF 704 512 +n − + ζ3 f N 9 3 A µ ¶ Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 23 Here ζ is the Riemann zeta-function (ζ3 = 1.202 · · · ) and the colour factors for SU(N) are

2 abcd abcd 2 2 1 N − 1 dA dA N (N + 36) TF = , CA = N, CF = , = , 2 2N NA 24

dabcddabcd N(N 2 + 6) dabcddabcd N 4 − 6N 2 + 18 F A = , F F = 2 NA 48 NA 96N

Substitution of these colour factors for N = 3 yields the following numerical results for QCD:

β0 ≈ 11 − 0.66667nf

β1 ≈ 102 − 12.6667nf 2 β2 ≈ 1428.50 − 279.611nf + 6.01852nf 2 3 β3 ≈ 29243.0 − 6946.30nf + 405.089nf + 1.49931nf

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 24 Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 25 QCD Running Coupling

• We have q2 dαS = β (α ) dq2 QCD S • Hence α (Q2) Q2 2 2 S dαS dq Q = = ln 2 β (α ) 2 q2 µ2 ZαS(µ ) QCD S Zµ

2 • In lowest order, βQCD(αS) = −β0αS and thus

1 1 Q2 − = β ln 2 2 0 2 αS(Q ) αS(µ ) µ 2 αS(µ ) ⇒ α (Q2) = S 2 2 2 1 + αS(µ )β0 ln(Q /µ )

• We had to introduce a dimensionful parameter µ to specify the boundary conditions. Alternatively, we can deﬁne ΛQCD (not to be confused with the cut-oﬀ Λ introduced earlier) as the scale at which the (perturbative) solution diverges . . .

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 26 1 Q2 − 0 = β ln 2 0 2 αS(Q ) ΛQCD 1 ⇒ α (Q2) = S 2 2 β0 ln(Q /ΛQCD)

α s ?

0.118 Λ2 2 2 QCD m Z Q • Experimentally the fundamental scale of QCD is

ΛQCD ' 210 § 40 MeV

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 27 • Measurements at diﬀerent scales show clear evidence that αS does run:

0.5 Theory Data NLO NNLO αs(Q) Lattice Deep Inelastic Scattering e+e- Annihilation 0.4 Hadron Collisions Heavy Quarkonia

Λ(5) MS α s (Μ Z ) 251 MeV 0.1215 0.3 QCD 4 213 MeV 0.1184 O(α s ) { 178 MeV 0.1153

0.2

0.1

1 10 100 Q [GeV]

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 28 • Using the formula for the running of αS, measurements at diﬀerent scales can be expressed as 2 measurements of αS(MZ):

DIS [pol. strct. fctn.] DIS [Bj-SR] DIS [GLS-SR] τ-decays [LEP]

xF3 [ν -DIS] F2 [e-, µ-DIS] jets & shapes [HERA] QQ + lattice QCD Υ decays + _ e e [σ had ] + _ e e_ [jets & shapes 22 GeV] e+ e [jets & shapes 35 GeV] + _ e e_ [σhad ] e+ e [jets & shapes 44 GeV] _ e+ e [jets & shapes 58 GeV] pp --> bb X pp, pp --> γ X σ(pp --> jets) Γ(Z 0 --> had.) [LEP] _ e+ e [scaling. viol.] jets & shapes 91.2 GeV [LEP] jets & shapes 133 GeV jets & shapes 161 GeV jets & shapes 172 GeV jets & shapes 183 GeV jets & shapes 189 GeV 0.08 0.10 0.12 0.14 α s (Μ Z )

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 29 • Non-perturbative (lattice) studies of QCD indicate that force between two colour charges (e.g. quarks) becomes constant at large distances, corresponding to a linear potential.

• It follows that quarks (and gluons) cannot be observed as isolated objects and exist only in colour-singlet bound states. Thus QCD explains quark conﬁnement.

q q

q q q q

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 30 Grand Uniﬁcation

• The complete Standard Model covariant derivative is

µ µ µ µ 0 µ D = ∂ + igS t · A + ig I · W + ig Y B

where extra terms represent the SU(2) and U(1) electroweak gauge field interactions. • We would like to write this as µ µ µ D = ∂ + igGUTT · X µ where gGUT is coupling of a Grand Unified Theory with gauge fields Xα, and the gauge group G is a simple Lie group with generators Tα. – Definition: A simple Lie group is one with no invariant subalgebras, i.e. no A such that Xα ∈ A implies [Xα, Xβ] ∈ A for all Xβ ∈ G. – Theorem: G is simple iff

TrR(TαTβ) = NRδαβ

where TrR represents a sum over all states in a representation R of G and NR is a number characteristic of the representation R.

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 31 − • For R = a Standard Model generation, e.g. u, d, e and νe, we have

2 10 TrR(t3t3) = TrR(t8t8) = TrR(I3I3) = 2 , TrR(Y ) = 3

• Hence for grand uniﬁcation we require

5 0 gGUT = gS = g = g r3

• However, this should hold at the GUT energy scale, not at present energies. . .

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 32 Running of Standard Model Couplings

2 2 02 gS g 5 g • Setting α3 = 4π , α2 = 4π , α1 = 3 4π , we have (ignoring Higgs contributions)

2 2 dαi αi 11 1 q = − C1 − C2 dq2 π 12 3 µ ¶ where

A B Tr (T T ) = C1δAB (gauge bosons)

Tr (tatb) = C2δab (fundamental fermions)

• For SU(N), C1 = N, C2 = nf /2, and for ng generations we therefore ﬁnd (now including Higgs contributions)

2 2 dα3 α3 11 1 q = − − ng dq2 π 4 3 µ ¶ 2 2 dα2 α2 11 1 1 q = − − ng − dq2 π 6 3 24 µ ¶ 2 2 dα1 α1 1 1 q = − − ng − . dq2 π 3 40 µ ¶

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 33 • Then running couplings almost meet at scale ∼ 1016 GeV. 60

50 α−1 1 SM 40 α−1 30 −1 α2 20

10 −1 α3 0 2 4 6 8 10 12 14 16 18 log10(E/GeV)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 34 • Adding supersymmetric partners for all Standard Model particles gives much better uniﬁcation, provided superpartner masses are not more than ∼ 1 TeV ⇒ Large Hadron Collider 60

50 −1 α1 SM 40 −1 α 30 −1 α2 20 MSSM

−1 10 α3 0 2 4 6 8 10 12 14 16 18 log10(E/GeV)

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 35 Conclusion: How to Win the Nobel Prize in Physics

• Discover or predict something important • Understand its significance • Publish first • Be ≤ 3 • For theorists: be confirmed by experiment • Live long enough

N.B. All are necessary, but not suﬃcient!

Thanks to Marek Karliner for many helpful suggestions & references

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 36 Iosif Khriplovich Gerhard ’t Hooft

Bryan Webber, Asymptotic Freedom, CPS, 3 Nov 2004 37