DOCSLIB.ORG

Lecture 11 Anomalies and Instantons Outline • the Chiral Anomaly: Detailed Discussion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 11 Anomalies and Instantons Outline • the Chiral Anomaly: Detailed Discussion Lecture 11 Anomalies and Instantons Outline • The chiral anomaly: detailed discussion. • Zero-modes and the index theorem. • Triangle diagrams: the qualitative story. • The gauge anomaly. • Anomaly matching. Reading: Terning 7.2, 7.3, 7.4, 7.5. The Chiral Anomaly Setting: massless chiral fermions coupled to a classical gauge field. Comment: for massless particles chiralities make sense { we can focus on two component Weyl spinors ("the upper two components" of a Dirac four-spinor). Lagrangian: ¯ µ Lfermion = i σ¯ Dµ Notation: • ¯ = y for two component spinors. • σ¯µ = (1; −~σ). a a • Dµ = @µ + iBµ with Bµ = BµT a gauge field with values in a Lie algebra. Consider the fermion path integral as function of a background gauge field: R ¯ iSfermion R ¯ −SE Z[Bµ] = D D e = D D e = det(D6 ) : Euclidean continuation: µ µ •6 D = iσ¯E(@µ + iBµ) withσ ¯E = (iI2; −~σ). 4 4 • id x = d xE . R 4 ¯ • SE = − d xE D6 . More precisely: expand , ¯ on complete bases P ¯ P y (x) = n anfn(x) ; (x) = n bngn(x) : Orthogonality and completeness: P y P y n fn(x)fn(y) = δ(x − y)I2 ; n gn(x)gn(y) = δ(x − y)I2 : R 4 y R 4 y Tr d xfn(x)fm(x) = δnm ; Tr d xgn(x)gm(x) = δnm : Choose basis so for some real λn's: Df6 n = λngn : Then the path-integral is a diagonal Gaussian: P R ¯ −SE R Q n λnbnan Q Z[Bµ] = D D e = n;m dandbme = n λn : More on eigenvalues λn and bases ffng, fgng. Issue: the Dirac operator D6 maps positive chirality spinors to negative chirality spinors. Therefore it can have no eigenfunctions: input and output is not even on the same space! Resolution: D6 y D6 is a hermitean operator on the positive chirality space so there is a complete basis of eigenfunctions: y 2 D6 Df6 n = λnfn ; 2 with λn real and non-negative. The operator with opposite ordering has eigenfunctions: y 2 D6 D6 gn = λngn : 1 1 y The two orderings are related as gn = 6Dfn (or fn = 6D gn) so λn λn the operators have the same spectrum except for zero-modes, ie. the eigenvalues λn = 0: y Df6 n = 0 or D6 gn = 0 : The Chiral Current The Noether current for classical symmetry under (x) ! eiα (x), ¯(x) ! e−iα ¯(x). Remark: this is a chiral transformation because (x) is a Weyl fermion (upper two components only). At the quantum level: change variables in the path integral as (x) ! eiα(x) (x) ; ¯(x) ! e−iα(x) ¯(x) : The classical action transforms as R 4 ¯ −iα(x) µ iα R 4 ¯ µ SE ! SE − d xE e iσ¯ (@µe ) = SE − d xEα(x)@µ σ¯ : The classical action is invariant under arbitrary variations so the current is conserved µ µ ¯ µ @µjA = 0 ; jA(x) = σ¯ : The chiral anomaly: quantum correction to classical current conserva- tion. The chiral transformation acts on the basis coefficients (Tr=trace over spin and group indices): P P iα(x) P 0 (x) = n anfn(x) ! n ane fn(x) = n anfn(x) ; 0 R 4 y iα(x) an ! an = Cnmam = Tr d xEfn(x)e fm(x)am : The path integral measure transforms accordingly: Q ¯ −1 Q n;m dandbm ! (detCC) n;m dandbm : where R 4 ¯ −1 −i d xE α(x)A(x) P y y (detCC) = e ;A(x) = Tr n(fn(x)fn(x) − gn(x)gn(x)) : Quantum equation for anomalous current µ @µjA = A(x) : 2 Compute anomaly using completeness (regulate by damping large λN ): X y y A(x) = Tr (fn(x)fn(x) − gn(x)gn(x)) n 2 2 2 2 X y −λn=Λ y −λn=Λ = limΛ!1Tr [fn(x)e fn(x) − gn(x)e gn(x)] n 4 Z d p y 2 y 2 = lim Tr [e−6D 6D=Λ − e−6D6D =Λ ] Λ!1 (2π)4 4 Z d p µ 2 1 1 = lim e−DµD =Λ Tr[ (σµν F )2 − (¯σµν F )2] Λ!1 (2π)4Λ4 2 E µν 2 E µν 2 2 2 Z π p dp 2 1 1 = e−p · µνρλTrF F = TrF F~µν : (2π)4 2 µν ρλ 16π2 µν Explicit form of Dirac operators and Pauli matrices: i D6 y D6 = D Dµ + (σµ σ¯ν − σν σ¯µ )i[D ;D ] = D Dµ − σµν F µ 4 E E E E µ ν µ E µν i D6 D6 y = D Dµ + (¯σµ σν − σ¯ν σµ )i[D ;D ] = D Dµ − σ¯µν F µ 4 E E E E µ ν µ E µν µν ρλ µν ρλ µνρλ Tr[σE σE − σ¯E σ¯E ] = Summary: the operator form of the chiral anomaly: µ 1 ~µν @µjA = A(x) = 16π2 TrFµν F : The important point: current conservation at the classical level is vio- lated by a quantum effect of a very specific form. The Atiyah-Singer Index Theorem Instructive alternative computation of A(x): 1 1 y gn = Df6 n ; fn = D6 gn ; λn λn so contributions to A(x) cancel in pairs except for λn = 0: R 4 R 4 P y y d xA(x) = Tr d x n(fn(x)fn(x) − gn(x)gn(x)) = n − n ¯ : y The number of zero modes of D6 , D6 : n ; n ¯. The Atiyah-Singer index theorem: 1 R 4 ~µν n − n ¯ = 16π2 d xTrFµν F : Mathematical significance: a relation between a topological quantity (the difference in the number of zero-modes cannot change continuously) to an analytical quantity (an integral over a smooth function). Towards the physical significance: there would be no chiral anomaly were it not for the zero modes of the Dirac operator (and those make the path integral vanish). Instantons R 4 µν The Euclidean action S ∼ d xTrFµν F of a gauge field configuration diverges at infinity, unless the gauge field is asymptotically pure gauge y Aµ ! U(x)@µU(x) . µν TrFµν F~ is a total derivative so its integral vanishes except for a surface term at the asymptotic Euclidean 3-sphere S3. This surface term counts the wrapping number ν of the U(x) around the asymptotic S3. Example: G = SU(2) ∼ S3 as a group, so the obvious embedding G ! S3 has winding number ν = 1. Field configurations with winding number have non-trivial field strength in bulk. Their Euclidean action is finite but large for small coupling 2 SE ∼ 1=g . These configurations are localized in Euclidean space, including in time. They are called instantons. The explicit integral over a winding configuration gives 1 R 4 ~ 16π2 d xTrFµν Fρλ = ν : Remark: the winding number ν is just the integral difference of zero modes n − n ¯. The integral above is in the fundamental representation (denoted ) where the gauge matrices U(x) are just the defining matrices. 1 The index T () = 2 in a general representation r is Tr[T aT b] = T (r)δab : The anomaly for a theory with n(r) chiral fermions in representation r: R 4 1 R 4 ~µν P d xA(x) = 16π2 d xTrFµν F = r n(r)2T (r) : The θ-term The gauge theory action generally includes the CP-violating term θYM R 4 µν ~ SCP = − 16π2 d xTrF Fµν : The term is a total derivative but it does not vanish: SCP = −θYMν : The path integral sums over all field configurations, with different in- stanton sectors weighed by the phase factor e−iθYMν . Theories with different θYM are generally physically distinct. Specifically they have different spectrum. Exception: shifting the phase θYM ! θYM + 2πZ is inconsequential. The chiral transformation ! eiα , ¯ ! e−iα ¯ shifts the path integral as: R 4 α R 4 ~µν R iS R iS−iα d xA(x) R iS−i d xTrFµν F D D ¯e ! D D ¯e = D D ¯e 16π2 : Equivalently: P θYM ! θYM − α r nr2T (r) : Example: in pure SYM the gaugino is the only chiral fermion. A chiral rotation λa ! eiαλa is equivalent to θYM ! θYM − 2Nα : because the index of the adjoint is T (Ad) = N. Thus the chiral anomaly breaks the continuous U(1)R in the classical k theory to the Z2N generated by α = N π (with k = 0;:::; 2N − 1). The Euclidean action of instantons is bounded: R 4 2 R 4 µν µν 0 ≤ d xTr(Fµν ± F~µν ) = 2 d x Fµν F ± Fµν F~ ; so R 4 µν R 4 ~µν 2 d xFµν F ≤ d xFµν F = 16π ν : Instantons in fact saturate this bound. Instanton contributions to the Euclidean path integral are suppressed by the characteristic nonperturbative factor: 2 ν − 8π +iθ e g2 : Such effects are qualitatively important because they are the leading contributions to some processes. For example, we would like to break SUSY in a non-perturbative manner, where the breaking is heavily suppressed. Triangle Diagrams Alternative derivation of chiral anomaly: Triangle diagrams. Chiral current (in the interacting theory): µ ¯ ν 5 −ie R d4y =A hjA(x)iint = h (x)γ γ (x) e i0 Expanding to second order, contract all fermions: find triangle diagram with current insertion at one angle and gauge field on two others: The divergence of the chiral current vanishes formally: µ qµhjA(q)iint = 0 (formally) : The reason: explicit integrals for the two diagrams cancel upon linear shifts in the loop momentum variables. The catch: shift of loop momentum is only justified if the regulated integral preserves the shift symmetry.
Load more

Recommended publications
  • Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
    Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings.
    [Show full text]
  • Thermal Evolution of the Axial Anomaly
    Thermal evolution of the axial anomaly Gergely Fej}os Research Center for Nuclear Physics Osaka University The 10th APCTP-BLTP/JINR-RCNP-RIKEN Joint Workshop on Nuclear and Hadronic Physics 18th August, 2016 G. Fejos & A. Hosaka, arXiv: 1604.05982 Gergely Fej}os Thermal evolution of the axial anomaly Outline aaa Motivation Functional renormalization group Chiral (linear) sigma model and axial anomaly Extension with nucleons Summary Gergely Fej}os Thermal evolution of the axial anomaly Motivation Gergely Fej}os Thermal evolution of the axial anomaly Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R > 6= 0 SSB pattern: SUL(Nf ) × SUR (Nf ) −! SUV (Nf ) Anomaly: UA(1) is broken by instantons Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point? Motivation QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Gergely Fej}os Thermal evolution of the axial anomaly Details of chiral symmetry restoration? Critical temperature? Axial anomaly? Is it recovered at the critical point? Motivation QCD Lagrangian with quarks and gluons: 1 L = − G a G µνa + ¯ (iγ Dµ − m) 4 µν i µ ij j Approximate chiral symmetry for Nf = 2; 3 flavors: iT aθa iT aθa L ! e L L; R ! e R R [vector: θL + θR , axialvector: θL − θR ] Chiral symmetry is spontaneuously broken in the ground state: < ¯ > = < ¯R L > + < ¯L R
    [Show full text]
  • Spontaneous Breaking of Conformal Invariance and Trace Anomaly
    Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching ∗ A. Schwimmera and S. Theisenb a Department of Physics of Complex Systems, Weizmann Institute, Rehovot 76100, Israel b Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany Abstract We argue that when conformal symmetry is spontaneously broken the trace anomalies in the broken and unbroken phases are matched. This puts strong constraints on the various couplings of the dilaton. Using the uniqueness of the effective action for the Goldstone supermultiplet for broken = 1 supercon- formal symmetry the dilaton effective action is calculated. N arXiv:1011.0696v1 [hep-th] 2 Nov 2010 November 2010 ∗ Partially supported by GIF, the German-Israeli Foundation for Scientific Research, the Minerva Foundation, DIP, the German-Israeli Project Cooperation and the Einstein Center of Weizmann Institute. 1. Introduction The matching of chiral anomalies of the ultraviolet and infrared theories related by a massive flow plays an important role in understanding the dynamics of these theories. In particular using the anomaly matching the spontaneous breaking of chiral symmetry in QCD like theories was proven [1]. For supersymmetric gauge theories chiral anomaly matching provides constraints when different theories are related by “non abelian” duality in the infrared NS. The matching involves the equality of a finite number of parameters, “the anomaly coefficients” defined as the values of certain Green’s function at a very special singular point in phase space. The Green’s function themselves have very different structure at the two ends of the flow. The massive flows relate by definition conformal theories in the ultraviolet and infrared but the trace anomalies of the two theories are not matched: rather the flow has the property that the a-trace anomaly coefficient decreases along it [2].
    [Show full text]
  • Chiral Anomaly Without Relativity Arxiv:1511.03621V1 [Physics.Pop-Ph]
    Chiral anomaly without relativity A.A. Burkov Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, and ITMO University, Saint Petersburg 197101, Russia Perspective on J. Xiong et al., Science 350, 413 (2015). The Dirac equation, which describes relativistic fermions, has a mathematically inevitable, but puzzling feature: negative energy solutions. The physical reality of these solutions is unques- tionable, as one of their direct consequences, the existence of antimatter, is confirmed by ex- periment. It is their interpretation that has always been somewhat controversial. Dirac’s own idea was to view the vacuum as a state in which all the negative energy levels are physically filled by fermions, which is now known as the Dirac sea. This idea seems to directly contradict a common-sense view of the vacuum as a state in which matter is absent and is thus generally disliked among high-energy physicists, who prefer to regard the Dirac sea as not much more than a useful metaphor. On the other hand, the Dirac sea is a very natural concept from the point of view of a condensed matter physicist, since there is a direct and simple analogy: filled valence bands of an insulating crystal. There exists, however, a phenomenon within the con- arXiv:1511.03621v1 [physics.pop-ph] 26 Oct 2015 text of the relativistic quantum field theory itself, whose satisfactory understanding seems to be hard to achieve without assigning physical reality to the Dirac sea. This phenomenon is the chiral anomaly, a quantum-mechanical violation of chiral symmetry, which was first observed experimentally in the particle physics setting as a decay of a neutral pion into two photons.
    [Show full text]
  • Effective Lagrangian for Axial Anomaly and Its Applications in Dirac And
    www.nature.com/scientificreports OPEN Efective lagrangian for axial anomaly and its applications in Dirac and Weyl semimetals Received: 25 April 2018 Chih-Yu Chen, C. D. Hu & Yeu-Chung Lin Accepted: 9 July 2018 A gauge invariant efective lagrangian for the fermion axial anomaly is constructed. The dynamical Published: xx xx xxxx degree of freedom for fermion feld is preserved. Using the anomaly lagrangian, the scattering cross section of pair production γγ → e−e+ in Dirac or Weyl semimetal is computed. The result is compared with the corresponding result from Dirac lagrangian. It is found that anomaly lagrangain and Dirac lagrangian exhibit the same E B pattern, therefore the E B signature may not serve a good indicator of the existence of axial anomaly. Because anomaly generates excessive right-handed electrons and positrons, pair production can give rise to spin current by applying gate voltage and charge current with depositing spin flters. These experiments are able to discern genuine anomaly phenomena. Axial anomaly1,2, sometimes referred to as chiral anomaly in proper context, arises from the non-invariance of fermion measure under axial γ5 transformation3, it is a generic property of quantum fermion feld theory, massive or massless. Te realization of axial anomaly in condensed matter is frst explored by Nielsen and Ninomiya4. It is argued that in the presence of external parallel electric and magnetic felds, there is net production of chiral charge and electrons move from lef-handed (LH) Weyl cone to right-handed (RH) Weyl cone, the induced anomaly current causes magneto-conductivity prominent. Aji5 investigated pyrochlore iridates under magnetic feld.
    [Show full text]
  • A Note on the Chiral Anomaly in the Ads/CFT Correspondence and 1/N 2 Correction
    NEIP-99-012 hep-th/9907106 A Note on the Chiral Anomaly in the AdS/CFT Correspondence and 1=N 2 Correction Adel Bilal and Chong-Sun Chu Institute of Physics, University of Neuch^atel, CH-2000 Neuch^atel, Switzerland [email protected] [email protected] Abstract According to the AdS/CFT correspondence, the d =4, =4SU(N) SYM is dual to 5 N the Type IIB string theory compactified on AdS5 S . A mechanism was proposed previously that the chiral anomaly of the gauge theory× is accounted for to the leading order in N by the Chern-Simons action in the AdS5 SUGRA. In this paper, we consider the SUGRA string action at one loop and determine the quantum corrections to the Chern-Simons\ action. While gluon loops do not modify the coefficient of the Chern- Simons action, spinor loops shift the coefficient by an integer. We find that for finite N, the quantum corrections from the complete tower of Kaluza-Klein states reproduce exactly the desired shift N 2 N 2 1 of the Chern-Simons coefficient, suggesting that this coefficient does not receive→ corrections− from the other states of the string theory. We discuss why this is plausible. 1 Introduction According to the AdS/CFT correspondence [1, 2, 3, 4], the =4SU(N) supersymmetric 2 N gauge theory considered in the ‘t Hooft limit with λ gYMNfixed is dual to the IIB string 5 ≡ theory compactified on AdS5 S . The parameters of the two theories are identified as 2 4 × 4 gYM =gs, λ=(R=ls) and hence 1=N = gs(ls=R) .
    [Show full text]
  • Π0 Decay Precision-Tests the Chiral Anomaly Harvey B
    NUCLEAR PHYSICS p0 decay precision-tests the chiral anomaly More precise neutral pion lifetime measurements probe quantum symmetry breaking By Harvey B. Meyer point, it was assumed that a symmetry of the errors in quadrature, this amounts to a classical Lagrangian would protect p0 from tension of 1.8 standard deviations versus ost subatomic particles are decaying in the limit of massless up and the combined result of the PrimEx-I and strongly interacting composites down quarks and lead to a longer lifetime. -II experiments, 8.34 (60.13) × 10−17 s. Al- called hadrons. Most hadrons are However, it turns out to be impossible to reg- though this difference could be a statisti- unstable and decay on extremely ularize quantum chromodynamics without cal fluctuation, it provides motivation to short time scales (10−22 s) to lighter breaking that symmetry. Therefore, the latter revisit the theory prediction. The gp pp hadrons. The electrically neutral is not respected by the quantum fluctuations reaction also has a sharp low-energy pre- Mpion, p0, is the lightest hadron and decays of the quantum chromodynamics fields and diction based on the chiral anomaly (6, 7) on a time scale of 10−16 s in 98.8% of cases does not protect p0 from decaying. and is being investigated by the COMPASS into two photons, gg, through the electro- The quantum origin of the symmetry experiment (8). magnetic interaction. Historically, under- breaking leads to an exact prediction for the The gg decay width of p0 has recently standing this time scale presented a major strength of p0 coupling to gg and hence the been evaluated from first principles using Downloaded from challenge to theoreticians.
    [Show full text]
  • Conformal Anomaly Actions for Dilaton Interactions
    EPJ Web of Conferences 80, 00015 (2014) DOI: 10.1051/epj conf/20140001580 C Owned by the authors, published by EDP Sciences, 2014 Conformal anomaly actions for dilaton interactions Luigi Delle Rose1,a, Carlo Marzo1,b, Mirko Serino1,c 1Dipartimento di Matematica e Fisica "Ennio De Giorgi" Universitá del Salento and INFN Lecce Via per Arnesano 73100 Lecce, Italy Abstract. We discuss, in conformally invariant field theories such as QCD with massless fermions, a possible link between the perturbative signature of the conformal anomaly, in the form of anomaly poles of the 1-particle irreducible effective action, and its descrip- tion in terms of Wess-Zumino actions with a dilaton. The two descriptions are expected to capture the UV and IR behaviour of the conformal anomaly, in terms of fundamental and effective degrees of freedom respectively, with the dilaton effective state appearing in a nonlinear realization. As in the chiral case, conformal anomalies seem to be related to the appearance of these effective interactions in the 1PI action in all the gauge-invariant sectors of the Standard Model. We show that, as a consequence of the underlying anoma- lous symmetry, the infinite hierarchy of recurrence relations involving self-interactions of the dilaton is entirely determined only by the first four of them. This relation can be generalized to any even space-time dimension. 1 Introduction Conformal anomalies play a significant role in theories which are classically scale invariant, such as QCD. This type of anomaly manifests as a non-vanishing trace of the vacuum expectation value (vev) of the energy momentum tensor in a metric (gμν) and gauge field (Aμ) background.
    [Show full text]
  • Anomaly Constraint on Massless QCD and the Role of Skyrmions in Chiral Symmetry Breaking
    Prepared for submission to JHEP Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking Yuya Tanizaki RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973 USA E-mail: [email protected] Abstract: We discuss consequences of the ’t Hooft anomaly matching condition for Quantum Chromodynamics (QCD) with massless fundamental quarks. We derive the new discrete ’t Hooft anomaly of massless QCD for generic numbers of color Nc and flavor Nf , and an exotic chiral-symmetry broken phase without quark-bilinear condensate is ruled out from possible QCD vacua. We show that the U(1)B baryon number symmetry is anomalously broken when the (Z2Nf )A discrete axial symmetry and the flavor symmetry are gauged. In the ordinary chiral symmetry breaking, the Skyrmion current turns out to reproduce this ’t Hooft anomaly of massless QCD. In the exotic chiral symmetry break- ing, however, the anomalous breaking of U(1)B does not take the correct form, and it is inconsistent with anomaly matching. This no-go theorem is based only on symmetries and anomalies, and thus has a wider range of applicability to the QCD phase diagram than the previous one obtained by QCD inequalities. Lastly, as another application, we check that duality of N = 1 supersymmetric QCD with Nf ≥ Nc + 1 satisfies the new anomaly matching. arXiv:1807.07666v3 [hep-th] 23 Aug 2018 Contents 1 Introduction1 2 Symmetry of massless QCD4 2.1 Symmetry group of massless Nf -flavor QCD5 2.2 Other equivalent expressions of the flavor symmetry6 2.3 Background
    [Show full text]
  • The Beryllium Anomaly and New Physics
    THE BERYLLIUM ANOMALY AND NEW PHYSICS New Physics at the Intensity Frontier CERN-EPFL-Korea Theory Institute Jonathan Feng, University of California, Irvine 21 February 2017 21 Feb 2017 Feng 1 OUTLINE A. J. Krasznhorkay et al., “Observation of Anomalous Internal Pair Creation in 8Be: A Possible Indication of a Light, Neutral Boson,” 1504.01527 [nucl-ex], PRL 116, 042501 (2016) J. Feng et al., “Protophobic Fifth Force Interpretation of the Observed Anomaly in 8Be Nuclear Transitions,” 1604.07411 [hep-ph], PRL 117, 071803 (2016) J. Feng et al., “Particle Physics Models for the 17 MeV Anomaly in Beryllium Nuclear Decays,” 1608.03591 [hep-ph], PRD 95, 035017 (2017) Jonathan Bart Iftah Susan Jordan Tim Flip Feng Fornal Galon Gardner Smolinsky Tait Tanedo 21 Feb 2017 Feng 2 NEW PHYSICS AT THE INTENSITY FRONTIER • There are currently many outstanding puzzles: neutrino masses, gauge hierarchy, strong CP, flavor, dark matter, baryogenesis, dark energy,... • Some of these motivate searches for new particles and forces at high energies: the energy frontier • But some also motivate searches for new physics that is light, but weakly coupled: the intensity frontier • Of particular interest here are connections to dark matter 21 Feb 2017 Feng 3 DARK MATTER AT THE INTENSITY FRONTIER !!! 21 Feb 2017 Feng 4 DARK SECTORS • All evidence for dark matter is gravitational. Perhaps it’s in a hidden sector, composed of particles with no SM gauge interactions (electromagnetic, weak, strong) Visible Dark Sector Sector • The dark sector may have a rich structure with matter and forces of its own Lee, Yang (1956); Kobsarev, Okun, Pomeranchuk (1966); Blinnikov, Khlopov (1982); Foot, Lew, Volkas (1991); Hodges (1993); Berezhiani, Dolgov, Mohapatra (1995); Pospelov, Ritz, Voloshin (2007); Feng, Kumar (2008);..
    [Show full text]
  • Anomalous Supersymmetry George Katsianis STAG Research Centre, Mathematical Sciences, University of Southampton
    Anomalous Supersymmetry George Katsianis STAG Research Centre, Mathematical Sciences, University of Southampton 1. Symmetries 3. Anomalies 6. Results (for the experts) A symmetry of a physical system is any set of transformations that Sometimes classical symmetries fail to survive the quantization We consider the free massless Wess-Zumino model leave some properties of that system invariant. A simple example is procedure. This means that the original symmetries of the classical Z 1 S = d4x −∂ φ∗∂µφ − (ψγ¯ µ∂ ψ) that of the reflection symmetry of a picture, where the left and theory are not actual symmetries of the full quantum theory. We µ 2 µ right sides look like mirror images of each other. call this phenomenon a quantum anomaly. Since the discovery of the axial anomaly by Adler, Bell and Jackiw in 1969, the anomalies which is invariant under supersymmetry transformations of the form played a central role in theoretical physics. In some cases the δψ ∼ φ , δφ ∼ ψ existence of (gauge) anomalies lead to inconsistencies, so their and U(1) field transformations. The corresponding conserved cancellation help us build viable physical models. In other cases currents are the supercurrent Sµ and the R-current Rµ. (global) anomalies are linked to observable effects and explain We compute the 4-point correlation function of two supercurrents experimental data, since they allow classically forbidden processes and two R-currents T (Sµ(x )S¯ν(x )Rκ (x ) Rλ (x )i. to occur. One such example is the pion decay into two photons. 1 2 3 4 The Feynman diagrams for the connected part of this correlation function have the following form: Source: wild.maths.org 4.
    [Show full text]
  • Ads/CFT Correspondence and Differential Geometry Johanna
    . AdS/CFT Correspondence and Differential Geometry Johanna Erdmenger Max Planck–Institut fur¨ Physik, Munchen¨ 1 Outline 1. Introduction: The AdS/CFT correspondence 2. Conformal Anomaly 3. AdS/CFT for field theories with = 1 Supersymmetry N 4. Example: Sasaki-Einstein manifolds 2 AdS/CFT Correspondence (Maldacena 1997, AdS: Anti de Sitter space, CFT: conformal field theory) Witten; Gubser, Klebanov, Polyakov Duality Quantum Field Theory Gravity Theory ⇔ Arises from String Theory in a particular low-energy limit Duality: Quantum field theory at strong coupling Gravity theory at weak coupling ⇔ Conformal field theory in four dimensions Supergravity Theory on AdS S5 ⇔ 5 × 3 Anti-de Sitter space Anti de Sitter space: Einstein space with constant negative curvature has a boundary which is the upper half of the Einstein static universe (locally this may be conformally mapped to four-dimensional Minkowski space ) Isometry group of AdS5: SO(4, 2) AdS/CFT: relates conformal field theory at the boundary of AdS5 to gravity theory on AdS S5 5 × Isometry group of S5: SO(6) ( SU(4)) ∼ 4 AdS/CFT correspondence Anti-de Sitter space: Einstein space with constant negative curvature AdS space has a boundary 2 2r/L µ ν 2 Metric: ds = e ηµνdx dx + dr Isometry group of (d + 1)-dimensional AdS space coincides with conformal group in d dimensions (SO(d, 2)). AdS/CFT correspondence provides dictionary between field theory opera- tors and supergravity fields 2 φ , ∆ = d + d + L2m2 O∆ ↔ m 2 4 ! Items in the same dictionary entry have the same quantum numbers under superconformal symmetry SU(2, 2 4).
    [Show full text]