Instanton Physics

A. Ringwald

http://www.desy.de/˜ringwald

DESY

QCD at High energies/Frontiers in QCD Sino-German Workshop Beijing/China, September 2004 – Instanton Physics – 1 1. Introduction Standard Model of electroweak • (QFD) and strong (QCD) Electro-weak Gauge Theory (massless) QCD interactions remarkably successfull Violation of B + L Violation of Chirality Q5

e' There are processes that cannot be e _ • _ described by ordinary perturbation Q (>0) uR _d γ∗ u uR c _ [Adler ‘69; Bell,Jackiw ‘69; Bardeen ‘69]] _ Q' theory: c d _ R s _ dR s' t s' _ I _ I sR B+L/Chirality-violating processes t _ sR _b g _ in QFD/QCD ν c + R µ P c Anomalous processes induced by d τ+ R • ANOMALY topological ﬂuctuations of the non- selection rule ∆ (B + L) = -2n g ∆N = -6 ∆Q = 2n ∆N = 8 abelian gauge ﬁelds, notably by CS 5 f CS instantons [Belavin et al. ‘75; ‘t Hooft ‘76]

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 2 Topological gauge field fluctuations and associated anomalous processes • play important role in – QCD in various long-distance aspects: U(1) problem (m m ) [‘t Hooft ‘76] ∗ A η′ ≫ η SU(n ) chiral symmetry breaking [Shuryak ‘82; Diakonov,Petrov ‘86] ∗ f – QFD at high temperatures: [Kuzmin,Rubakov,Shaposhnikov ‘85] Impact on baryon and lepton asymmetries of the universe ∗ Are they directly observable in high energy reactions? • – QFD: Intense studies in early 1990s; inconclusive [AR ‘90; Espinosa ‘90; ...] – QCD: Hard QCD-instanton induced events in deep inelastic scattering ∗ reliably calculable and sizeable rate [Moch,AR,F.Schrempp ‘97; AR,F.Schrempp ‘98] · characteristic final state signature [AR,F.Schrempp ‘94–‘01] Soft· QCD-instanton induced events might be responsible for the ∗ bulk of inelastic processes [E.Levin et al.; Shuryak et al.; F.Schrempp,Utermann ‘02]

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 3 Further content: • 2. Axial anomaly and topology 3. Instanton perturbation theory 4. QCD-instantons at HERA 5. QFD-instantons at VLHC 6. Conclusions

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 4 2. Axial anomaly and topology

In absence of quark masses, QCD Lagrangian [q = column(u,d,s,...)] • 1 n 0 = Ga Gµν + iq¯ γµD q + iq¯ γµD q f LQCD −4 µν a L µ L R µ R | {z } invariant under independent global transformations

G ≡ SU(nf )L ⊗ SU(nf )R ⊗ U(1)L+R ⊗ U(1)L−R chiral symmetry vector axial vector | {z } | {z } | {z }

SU(n ) : q G g q f L,R L,R −→ L,R L,R U(1) U(1) : q G eiθ q L+R ≡ V L,R −→ L,R G iθ G iθ U(1)L R U(1)A : qL e− qL, qR e qR − ≡ −→ −→

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 5 How are these symmetries realized in Nature? •

Chiral symmetry SU(nf )L ⊗ SU(nf )R should be approximately • good in light quark sector (u,d,s), but not seen in hadronic spectrum: – Hadrons can be classiﬁed in SU(3)V ≡ SU(3)L+R representations, but degenerate multiplets with opposite parity do not exist + 0 + 0 0 – Octet of pseudoscalar mesons (π , π−, π , η, K , K−, K , and K¯ ) much lighter than all other hadronic states Ground state of the theory not symmetric under the chiral group: ⇒ SCSB SU(3)L ⊗ SU(3)R −→ SU(3)L+R

and an octet (32 − 1) of pseudoscalar massless bosons appears in the theory; their small masses generated by the quark-mass matrix

What about the axial symmetry? •

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 6

U(1)A problem

If U(1)A unbroken in vacuum in chiral limit all massless hadrons • would have massless partners of opposite⇒ parity not realized in Nature. ⇒

If U(1)A spontaneously broken, then there should be an I = 0 • pseudoscalar Goldstone boson, whose perturbed state should have about the same mass as the pion. Only candidate, η′(958), too heavy: M M . η′ ≫ η

Solution involves U(1)A anomaly and topological (instanton) eﬀects •

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 7

U(1)A is not a symmetry of the theory at the quantum level: • The U(1)A axial current is not conserved due to an axial anomaly: [Adler ‘69; Bell,Jackiw ‘69] α α ∂ (¯qγµγ q) = 2 n ν ; ν s ǫµνρσ Ga Ga s Ga G˜aµν µ 5 f ≡ 16π µν ρσ ≡ 8π µν

ν so-called topological charge density. Can be written as total • divergence

α 1 ν = ∂µK (x) ; K = s ǫ Aα ∂βAγ + gf AβAγ µ µ 4π µαβγ a a 3 abc b c

Kµ so-called Chern-Simons current

For suﬃciently rapidly vanishing ﬁelds can write for integral of anomaly ⇒

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 8 over space-time, d4x ∂ (¯qγµγ q) = 2 n d4x ν 2 n d4x ∂ Kµ, µ 5 f ≡ f µ R R R + + ∞ 3 0 ∞ 3 0 dt ∂t d x qγ¯ γ5q = 2 nf dt ∂t d x K Z Z Z Z −∞ −∞ Q (t= ) Q (t= ) N (t= ) N (t= ) 5 ∞ − 5 −∞ CS ∞ − CS −∞ | {z } | {z } in short

△Q5 = 2 nf △NCS

i.e. in the background of gauge ﬁelds which evolve in time such that their Chern-Simons number N = d3x K0 changes by N the CS △ CS ⇒ fermionic U(1)A charge Q5 changesR by 2 nf NCS △

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 9 Classical gauge fields with zero energy have integer N . • CS There are topological obstructions to deform such zero energy fields • differing in their Chern-Simons numbers smoothly into each other ⇒ separated by energy barrier [Jackiw,Rebbi 176; Callan,Dashen,Gross ‘76]

static field energy Instanton

-2 -1 0 1 2 NCS Instanton describes N = 1 tunneling transition. Associated with • △ CS anomalous violation of axial charge conservation, Q = 2 n . [‘t Hooft ‘76] △ 5 f No reason to expect U(1)A to be a symmetry ⇒ no U(1)A problem. ⇒

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 10 3. Instanton perturbation theory Generalized saddle-points in the Euclidean functional integral formulation • of QCD,

1 S[A,ψ,ψ] = [dA][dψ][dψ] [A,ψ, ψ]e− , hOi Z Z O

S[A,ψ,ψ] Z = [dA][dψ][dψ]e− . Z Perturbation theory: • perturbative QCD: expansion about trivial vacuum solution, i.e. ◦ vanishing gluon ﬁeld and vanishing quark ﬁelds and thus vanishing Euclidean action, S = 0. instanton perturbation theory: generalized saddle-point expansion ◦ of the Euclidean functional integral about non-trivial minima of the Euclidean action with S = 0. 6

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 11 Non-trivial minima ( solutions) have integer Pontryagin index • (topological charge)⇔

4 Q d x ν(x) NCS = 1, 2,..., ≡ Z ≡ △ ± ±

and their action is a multiple of 2π/αs,

4 1 2 π 2 π S d x tr(GµνGµν)= Q = (1, 2,...). ≡ Z 2 αs | | αs ·

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 12

Dominant saddle-point for α 1: Instanton (Q = 1): [Belavin et al. ‘75] ⇒ s ≪ 2 (I) i ρ σµ (x x0) (xµ x0µ) A (x; ρ, U, x )= U − − − U † µ 0 −g (x x )2 (x x )2 + ρ2 − 0 − 0

size ρ, color orientation U, position x0 ◦ localized in Euclidean space and time (“instantaneous”) ◦ (I) Aµ (x; ρ,U, 0) L 4 6 12 ρ 5 = 2 2 4 4 παs · (x + ρ ) L(z,t)3 2 (I) 2 π 1 -0.4 S Aµ = 0 -0.2 ⇒ α -0.4 0z h i s -0.2 0 0.2 t 0.2 0.4 0.4

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 13 Instanton-contribution to vacuum-to-vacuum amplitude Z: • 1 dZ(I) ∞ = dρDm(ρ) dU Z(0) d4x Z Z 0

Size distribution D (ρ) known in instanton perturbation theory • m [‘t Hooft ‘76; Bernard ‘79] α (µ )ln(ρµ ) 1, ρ m (µ ) 1, s r r ≪ i r ≪

in 2-loop renormalization-group invariant form, [Morris et al. ‘85]

nf dnI αs(µr) = D (ρ)= D(ρ) (ρ m (µ ))(ρµ )nf γ0 4π d4x dρ m i r r iY=1

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 14 Reduced size distribution D(ρ): • 2 Nc d 2π 2π αs(µr) −αs(µr) β0+(β1 4 Ncβ0) 4π D(ρ)= 5 e (ρµr) − ρ αs(µr)

2π Clearly non-perturbative, e−αs(µr) • ∝ Power-law behaviour of (reduced) size distribution, •

β 5+ (αs) 11 2 D(ρ) ρ 0− O ; β = N n ∼ 0 3 c − 3 f

dominant contribution to ρ integral generically originates from large ρ ◦ spoils the applicability of instanton perturbation theory, α (1/ρ) 1 ◦ s ≪

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 15 Size distribution basic building block of instanton perturbation theory: • appears in generic instanton contributions to Green’s functions ◦ important to know the region of validity of the perturbative result ⇒ Crucial information from lattice investigations on the topological • structure of the QCD vacuum

[Chu et. al ‘94] Lagrange Density Topological Charge Density

0.03 0.002 4 0.02 0.001 3 15 15 15 2 0.01 0 10 0 10 -0.001 10 5 z 5 5 10 5 10 5 z 10 5 z t 15 15 15 20 t 20 t 20

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 16 Confronting lattice data (n = 0) [Smith,Teper (UKQCD) ‘98] on size dis- • f tribution with perturbative result [AR,F. Schrempp ‘99]:

Instanton perturbation theory reliable for ρ Λ <0.4 ⇒ ∼

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 17 Suppression of instanton contribution to the vacuum-to-vacuum • amplitude for small quark masses ρm 1: i ≪ Axial anomaly: [Adler ‘69; Bell, Jackiw ‘69] ⇐ Any gauge ﬁeld ﬂuctuation with topological charge Q must be accompanied by a corresponding change in axial charge, Q = 2 n Q △ 5 f pure vacuum-to-vacuum transitions vanish in chiral limit ⇒ Green’s functions corresponding to anomalous Q5 violation: [‘t Hooft ‘76] ⇒ g g dR dR dL dL u u R g L g I I u g u g R Q = 1 L Q = -1 g g g g

sR g sL g sR sL main contribution due to instantons ∗ do not suﬀer from any mass suppression ∗

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 18 Simplest example of one light ﬂavour (n = 1): Fermionic two-point function • f ∞ ψ(x )ψ(x ) (I) d4x dρD(ρ) dU (ρ m) S(I)(x , x ; x, ρ, U) h 1 2 i ≃ 1 2 Z Z0 Z

Quark propagator in the I-background S(I),

(I) κ0(x1; ...) κ0†(x2; ...) κn(x1; ...) κn† (x2; ...) S (x1, x2; ...) = + , m m + iλ n=0 n X6 in terms of the spectrum of the Dirac operator in the I-background, which has exactly one right-handed zero mode κ0, [‘t Hooft ‘76] iγ D(I)κ = λ κ ; with λ = 0 and λ = 0 for n = 0, − µ µ n n n 0 n 6 6 for m 0 only the zero mode contribution survives, → ∞ (I) 4 ψ(x )ψ(x ) d x dρD(ρ) dUρκ (x ; x, ρ, U) κ†(x ; x, ρ, U) h 1 2 i ≃ 0 1 0 2 Z Z0 Z

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 19 First principle calculations of instanton contributions only possible for • quantities to which large size instantons do not contribute: – Short-distance coeﬃcient functions in the operator product expansion of two-point functions. [Andrei,Gross ‘78;...;Novikov et al. ‘80;...; Balitsky,Braun ‘93] Problem: No physically relevant two-point function known which receives ∗ contribution solely from instantons. Instanton contribution typically hidden beyond large perturbative → background. – Unique possibility: Anomalous △Q5 = ±2 nf processes in deep-inelastic scattering at large momentum transfer [Moch,AR,F.Schrempp ‘97] In m = 0 limit, receive contributions only from (anti-)instantons. → q

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 20

4. QCD-Instantons at HERA [AR,F.Schrempp ‘94–‘01]

Deep-inelastic scattering variables: Kinematics: S = (e + P )2 e' • e 2 2 2 Q = q = (e e′) − 2 − − q k xBj = Q / (2P q) 2 · yBj = Q / (S xBj) q' W 2 = (q + P )2 = Q2(1/x 1) s Bj − 2 2 2 I W I W sˆ = (q + g) 2 z = xBj (1+s/Q ˆ ) g = z P P Variables of instanton-subprocess: 2 Q 2 = q = (q k)2 ′ − ′ − − x = Q 2/ (2 g q ) ′ ′ · ′ W 2 = (q + g)2 = Q 2(1/x 1) I ′ ′ ′ −

“Fiducial” kinematical region from lattice constraints: [AR,F.Schrempp ‘99;‘01] •

(0) R∗ (nf ) ρ∗Λ <0.4, >1.0 Q′/Λ >30.8, x′ >0.35 MS MS ∼ ρ∗ ∼ ⇒ ∼ ∼

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 21

Instanton-Antiinstanton Estimate [AR,F.Schrempp ‘98]; also [Zakharov ‘90; Khoze,AR ‘91]

∞ ∞ 4π R2 α Ω U, ,... i(p +p ) R 2 p2 (ρ+¯ρ) (I) 4 − g ρρ¯ 1 2 i=1 i σˆp p d R dρ dρ D(ρ)D(ρ) dUe „ « e · − − 1 2 ∼ P q Z Z0 Z0 Z Ingredients: p • 2π/αg p1 – Instanton-size distribution D(ρ) e− ∝ IW IW – Ω U, R2/(ρρ¯),... : p2 p Exponentiation of (1/αg) ﬁnal state gauge ∗ ` ´ O e bosons [AR ‘90;Espinosa ‘90] Anti-instanton-instanton interaction p1 ∗ Is Is p2 General form: [Khoze,AR ‘91; AR,F.Schrempp ‘98] • p 4π F (ǫ) (I) g 2 σˆ e− αg ; with ǫ = √s/Mˆ ∼ sp

“Holy-Grail” function F (ǫ) for ǫ , with g g ց ր

0 < Fg(1) < Fg(0) = 1 .

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 22 Saddle point evaluation: • 4π (I) Γ α Fg(ǫ) σˆ e− ∗ e− g , ∝ ≡ where √s/ˆ (4πM /α ) (QFD) ǫ W W ≡ √s/Qˆ ′ 1/x 1 (QCD)  ≡ ′ − is a scaled cm energy andp [AR ‘02] Fg =1+Ωg(1, ξ )+ ∗ ∂ (ξ 2) Ωg(1, ξ ) 2 (QFD) − ∗ − ∂ξ ∗ ξ =2+ R ∗ | ∗ ρ ( 0 “ ”∗ (QCD)

Increasing ǫ smaller (R/ρ) probed • cross-section⇒ grow due to attractive∗ ⇒ nature of Ωg in perturbative semi- classical regime [AR,F.Schrempp ‘98,‘00] A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 23

e' Event generator QCDINS 2.0: e q k

[Gibbs,AR,F.Schrempp ‘95; AR,F.Schrempp ‘00] q' s I 2 2 Hard subprocess: W I W • – isotropic in q g CM g = z P ′ P – ﬂavour democratic – large parton multiplicity

nq + ng = 2 nf 1+ (1)/αs > 8, h i − O ∼ 2.5

Parton shower (HERWIG) T 2 • E [GeV] 1.5 Hadronization (HERWIG or JET- 1 • 0.5 SET) 3 2 0 1 0 φ -2 0 -1 lab 2 4 η 6 8 -2 lab -3 A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 Pioneering search by H1 collaboration – Instanton Physics – 24

[H1 collab. ‘02] Based on QCDINS 2.0, com- • pared to MEPS and CDM 100 150 H1 200 Events – Excess with instanton-like Events 100 Events 50 100 topology, compatible with 50 instanton signal 0 0 0 0 0.2 0.4 0.6 0.8 1 100 200 300 0 5 10 15 20 ,2 2 n – Statistical signiﬁcant in SphB Qrec [GeV ] B 100 comparison to MEPS 100 H1 Data 96+97 MEPS 100 – Uncertainties in back- CDM Events Events Events QCDINS 50 50 ground simulations? 50 Upper limit on σ ⇒ 0 0 0 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 0 5 10 15 ∆ Et, B [GeV] B Et, Jet [GeV] Data do not exclude cross- • section predicted for small (R /ρ ) >0.5, as long as one probes∗ ∗ small∼ ρ 0.3 fm ∗ A. Ringwald/DESY≪ QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – InstantonSummary Physics – of H1/ZEUS searches at HERA I 25

[H1 collab. ‘02; ZEUS collab. ‘04] Instanton-enriched samples by • cutting on discriminating ob- servables

Large uncertainties due to • diﬀerent event generators

Upper limits on instanton- • induced cross section factor 5 above predictions ∼ HERA II may allow to reach • higher instanton-separation power ǫI/ǫsDIS [F. Schrempp ‘04]

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 26 5. QFD-Instantons at VLHC?

Lattice data as well as H1/ZEUS • limit on small-size QCD-instantons suggest: – II estimate reliable, as long as (R/ρ) 1 – For (∗R/ρ≥ ) < 0.5 1, rapid growth, as implied∗ by ÷Ω, stops. Implications for QFD-instantons: • – (R/ρ) < 0.5 1 corresponds to ǫ< 0.75∗ 1.15÷, √s<ˆ 22 35 TeV – At these÷ energies, parton-par-÷ ton cross-section estimates and bounds reach observable values

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 27

Estimate of σ(IW ) • pp

[AR,Tu (unpublished)] – VLHC: √s 200 TeV PP ≈ 2 1 1 6 10 fb− yr− L ≈ · −3 – Observable, ∼>10 fb, if estimate valid/bound saturated up to √sˆ 28/35 TeV. ≈ Further study worthwhile ⇒

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 28 Phenomenology of QFD-instantons

[AR,F.Schrempp,Wetterich ‘91; Gibbs,AR,Webber,Zadrozny ‘94] No background from perturbative Simulations performed • Standard Model processes by requiring Energy (TeV) nB estimate √sˆ0 (TeV) 17 1/αW 5 – 4 identiﬁed charged e’s or µ’s ≥ 40 (a) 1/αW 18 – ET several TeV 40 (b) LOME 18 ≥ 200 1/αW 18 Event generator HERBVI: • [Gibbs,AR,Webber,Zadrozny ‘94] [Gibbs,Webber ‘95]

– B-violation cannot be established – L-violation veriﬁable: measure

Dℓ = N N + ; ℓ− − ℓ need ∼ 103 events [Gibbs,AR,Webber,Zadrozny ‘94] A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004 – Instanton Physics – 29 6. Conclusions

Opportunities to study instanton-induced hard scattering processes • Need future hadron collider to explore QFD instantons and electroweak • B + L violation – If cross-section exceeds even (10 nb/mb), ﬁrst signs of electroweak sphaleron O production may be/may already have been seen in neutrino nucleon scattering at cosmic ray facilities and neutrino telescopes [Morris,AR ‘94; Fodor,Katz,AR,Tu ‘03; Han,Hooper ‘03]

Hard QCD instanton-induced scattering processes • – can and are being probed presently in DIS at HERA – study of prospects at LHC underway [Carli,Petermann,F.Schrempp] – yield insight into fait of QFD instanton-induced processes at multi-TeV

A. Ringwald/DESY QCD at High Energies/Frontiers in QCD, Beijing/China, September 2004