DOCSLIB.ORG

Lectures on Chiral Perturbation Theory Brian Tiburzi

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lectures on Chiral Perturbation Theory Brian Tiburzi Lectures on Chiral Perturbation Theory I. Foundations II. Lattice Applications III. Baryons IV. Convergence Brian Tiburzi RIKEN BNL Research Center Chiral Perturbation Theory I. Foundations Low-energy QCD Mesons Solutions of QCD (courtesy of nature) pseudo-scalar Hyperons Kaons Nucleons Baryons positive parity spin-half Pions Ground state: vacuum • Spectrum (and properties) of low-lying hadrons indicative of symmetries ... and symmetry breaking • ChPT is the tool to study such manifestations in low-energy QCD Nf Massless QCD QCD = ψ + YM ψ = ψ D/ψ i L L L L i i=1 • Take two flavors. These will correspond to up and down quarks. • Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry 1 = (1 γ ) projectors ψL,R = L,R ψ PL,R 2 ∓ 5 P ψD/ψ = ψLD/ψ L + ψRD/ψ R • Left- and right-handed fields do not mix: no chirality changing interaction U(2)L U(2)R ψL LψL ⊗ → ψ (L L + R R) ψ ψ Rψ → P P LR R → R Nf Massless QCD QCD = ψ + YM ψ = ψ D/ψ i L L L L i i=1 • Take two flavors. These will correspond to up and down quarks. • Massless QCD Lagrange density obviously has global U(2) flavor symmetry but... Chiral symmetry 1 = (1 γ ) projectors ψL,R = L,R ψ PL,R 2 ∓ 5 P ψD/ψ = ψLD/ψ L + ψRD/ψ R • Left- and right-handed fields do not mix: no chirality changing interaction Vector subgroup U(2)L U(2)R U(2)V ψL VψL ⊗ → ψ V ( L + R)ψ = Vψ ψ Vψ → P P L = R = V R → R Chiral Symmetry of Massless QCD Action invariant under U(2) U(2) = U(1) U(1) SU(2) SU(2) L ⊗ R L ⊗ R ⊗ L ⊗ R U(1) : ψ eiθL ψ L L L 1 iθR iθL 1 iθR iθL → ψ (e + e )+ (e e )γ5 ψ U(1) : ψ eiθR ψ → 2 2 − R R → R iθ Vector subgroup θ = θ = θ ψ e ψ U(1)V L R → iθ5γ5 Axial transformation θ = θ = θ ψ [cos θ5 + iγ5 sin θ5]ψ = e ψ − L R 5 → U(1)A Exercise: Consider a non-singlet axial transformation ψi [exp(iφ τ γ 5)]ijψj → · Is there a corresponding symmetry group of the massless QCD action? Chiral Symmetry of Massless QCD Action invariant under U(2) U(2) = U(1) U(1) SU(2) SU(2) L ⊗ R A ⊗ V ⊗ L ⊗ R • Global symmetries lead to classically conserved currents a a a a J5µ = ψγµγ5ψJµ = ψγµψJLµ = ψLγµτ ψL JRµ = ψRγµτ ψR (Regulated) Theory not invariant under flavor-singlet axial transformation e 2π Nf µν Fµν (x) d =2 ∂µJ5µ(x)=∂µJRµ(x) ∂µJLµ(x)= − − αs N GA (x)GA (x) d =4 − 4π f µνρσ µν ρσ Chiral Anomaly U(1) SU(2) SU(2) V ⊗ L ⊗ R The chiral anomaly obstructs chirally invariant lattice regularization of fermions (see Kaplan’s lectures) Fate of Symmetries in Low-Energy QCD U(1) SU(2) SU(2) V ⊗ L ⊗ R • Chiral pairing preferred by vacuum (non-perturbative ground state) Chiral condensate ψψ = ψ ψ + ψ ψ =0 R L L R • Massless quarks can change their chirality by scattering off vacuum condensate U(1) SU(2) SU(2) U(1) SU(2) V ⊗ L ⊗ R −→ V ⊗ V • Spontaneously broken symmetries lead to massless bosonic excitations Nambu-Goldstone Mechanism Broken generators in coset SU(2) SU(2) /SU(2) L ⊗ R V Number of massless particles? ψ ψ = λδ Chiral Condensate iR jL − ji • Choice for vacuum orientation λ R from Vafa-Witten (P) ∈ • After a chiral transformation ψ ψjL Ljj R† ψ ψj L = λ(LR†)ji iR → ii iR − SU(2) SU(2) SU(2) L ⊗ R −→ V • Describe Goldstone fluctuations of vacuum state with fields δji Σji(x)=δji + ... Σ SU(2) SU(2) /SU(2) → ∈ L ⊗ R V iθ (x) τ iθ (x) τ [L(x)R†(x)] =[e L · e− R · ] θ = θ ji ji L − R 2iφ Σ=e2iφ/f =1+ + ... Transformation properties f Σ LΣR† Σ V ΣV † φ VφV† → → → Exercise: Determine the discrete symmetry properties of the Goldstone modes from the coset’s transformation. Dynamics of Goldstone Bosons: Chiral Lagrangian 2iφ Tr φ =0 1 π0 π+ Σ=e2iφ/f =1+ + ... √2 The Pions φ = 1 0 f φ† = φ π− π − √2 • Build chirally invariant theory of coset field 2 Σ LΣR† f → Σ†Σ=1 = Tr ∂µΣ∂µΣ† Σ† RΣ†L L 8 → • Expand about v.e.v. to uncover Gaußian fluctuations 1 2 1 0 0 + 2 = Tr (∂ φ∂ φ)+ (1/f )= ∂ π ∂ π + ∂ π−∂ π + (1/f ) L 2 µ µ O 2 µ µ µ µ O 3 massless modes • Non-linear theory: interactions between multiple pions at “higher orders” Can treat systematically... Including Quark Masses • We began with massless QCD. Quarks have mass, Higgs makes two very light • Chiral symmetry of action is only approximate: explicit symmetry breaking ∆ = m ψ ψ = m ψ ψ + ψ ψ Lψ q i i q iR iL iL iR i i SU(2) SU(2) SU(2) m /Λ 1 L ⊗ R −→ V q QCD • Need to map ∆ onto ChPT operators breaking symmetry in same way Lψ ∆ = m λ Tr Σ+Σ† Leff − q Σ LΣR† Comments: not chirally invariant → Σ† RΣ†L† → new dimensionful parameter λ 2 included only linear quark mass term mq Perturbing about chiral limit f 2 Chiral Lagrangian χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − 1 8mqλ 1 • Expand up to quadratic order = 4m λ + Tr (∂ φ∂ φ)+ Tr (φφ) Lχ − q 2 µ µ f 2 2 2 2 Pion mass mπ =8mqλ/f • Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom ∂ log ZQCD ( +mq ψψ) ZQCD[mq,...]= e− x ··· = ψψ D··· − ∂mq Low-energy degrees of freedom Effective field theory χ(Σ; mq ) Z [m ,...]= Σ e− x L ZQCD[mq,...] ZχPT[mq,...] χPT q D ≡ Matching From before: ∂ log ZχPT ψψ = = λ Tr Σ + Σ† = 2Nf λ ψiRψjL = λδji − ∂mq − − − λ = λ f 2 Chiral Lagrangian χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − 1 8mqλ 1 • Expand up to quadratic order = 4m λ + Tr (∂ φ∂ φ)+ Tr (φφ) Lχ − q 2 µ µ f 2 2 2 2 2 2 Pion mass m =8mqλ/f f m =2m ψψ (Gell-Mann Oakes Renner) π π q| | • Vacuum energy must be due to chiral condensate (ingredient in our construction) QCD degrees of freedom ∂ log ZQCD ( +mq ψψ) ZQCD[mq,...]= e− x ··· = ψψ D··· − ∂mq Low-energy degrees of freedom Effective field theory χ(Σ; mq ) Z [m ,...]= Σ e− x L ZQCD[mq,...] ZχPT[mq,...] χPT q D ≡ Matching From before: ∂ log ZχPT ψψ = = λ Tr Σ + Σ† = 2Nf λ ψiRψjL = λδji − ∂mq − − − λ = λ f 2 ChPT χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − • Quadratic fluctuations are the approximate Goldstone bosons of SChSB • Quartic terms describe interactions m λ 1 q φ4 (φ∂ φ)2 ∼ f 4 ∼ f 2 µ • Higher-order interactions renormalize lower-order terms m λ 1 m λ m λ ∆m2 q q Λ2 + q log Λ2 +finite π ∼ f 4 k2 + m2 ∼ f 2 f 2 k π power-law divergence logarithmic divergence Absorb in renormalized mass, Renormalization requires new or just use dimensional regularization operator in chiral Lagrangian • ChPT is non-renormalizable (needing infinite local terms to renormalize) 1). low-energy theory, so who cares? 2). must be able to order terms in terms of relevance “power counting” f 2 Power Counting χ = Tr ∂µΣ∂µΣ† mqλ Tr Σ+Σ† L 8 − • Leading-order Lagrangian in expansion in derivatives and quark mass (p2) ∂ pmp2 Low-energy dynamics of pions O µ ∼ q ∼ 1 2 2 4 Propagator p− Vertices ∂ ∂ ,m p Loop integral p k2 + m2 ∼ µ µ q ∼ ∼ π k 4L 2I+2V General Feynman diagram: L loops, I internal lines, V vertices p − ∼ 2L+2 Euler formula L = I V +1 p − ∼ • Loop expansion: one loop graphs require only ( p 4 ) counterterms O Two loop graphs? Exercise: What happens to the power-counting argument in d = 2, 6 dimensions? Do the results surprise you? Why didn’t I ask about d = 3, 5? ( p 4 ) Chiral Lagrangian O • Construct chirally invariant terms out of coset Σ LΣR† → Σ† RΣ†L† → 2 E.g. ( p ) Lagrangian ∂ Σ ∂ Σ† O µ µ • Construct terms that break chiral symmetry in the same way as mass term Simplification: add external scalar field to QCD action ∆ = ψ sψ + ψ s†ψ L L R R L Make the scalar transform to preserve chiral symmetry s LsR† → Giving the scalar a v.e.v. breaks chiral symmetry just as a mass s = m + q ··· 2 E.g. ( p ) Lagrangian Σs† + sΣ† O ( p 4 ) Chiral Lagrangian O Σ LΣR† s LsR† Also impose Euclidean invariance, C, P, T → → Σ† RΣ†L† s† Rs†L† → → 2 2 2 E.g. [Tr Σs† sΣ† ] m [Tr Σ Σ† ] =0 − → q − Easy to generate terms. Care needed to find minimal set. 2 4 = L1[Tr ∂µΣ∂µΣ† ] + L2Tr ∂µΣ∂ν Σ† Tr ∂µΣ∂ν Σ† L 2 mqλ (mqλ) 2 + L3 Tr ∂µΣ∂µΣ† Tr Σ+Σ† + L4 [Tr Σ+Σ† ] f 2 f 4 Lj low-energy constants = Gasser-Leutwyler coefficients, dimensionless { } N.B. these are not Gasser-Leutwyler’s coefficients Complete set of counterterms needed to renormalize one-loop ChPT Additional terms necessary when coupling external fields... Exercise: Determine the effects of strong isospin breaking m = m on the chiral u d Lagrangian.
Load more

Recommended publications
  • 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]
  • Chirality-Assisted Lateral Momentum Transfer for Bidirectional
    Shi et al. Light: Science & Applications (2020) 9:62 Official journal of the CIOMP 2047-7538 https://doi.org/10.1038/s41377-020-0293-0 www.nature.com/lsa ARTICLE Open Access Chirality-assisted lateral momentum transfer for bidirectional enantioselective separation Yuzhi Shi 1,2, Tongtong Zhu3,4,5, Tianhang Zhang3, Alfredo Mazzulla 6, Din Ping Tsai 7,WeiqiangDing 5, Ai Qun Liu 2, Gabriella Cipparrone6,8, Juan José Sáenz9 and Cheng-Wei Qiu 3 Abstract Lateral optical forces induced by linearly polarized laser beams have been predicted to deflect dipolar particles with opposite chiralities toward opposite transversal directions. These “chirality-dependent” forces can offer new possibilities for passive all-optical enantioselective sorting of chiral particles, which is essential to the nanoscience and drug industries. However, previous chiral sorting experiments focused on large particles with diameters in the geometrical-optics regime. Here, we demonstrate, for the first time, the robust sorting of Mie (size ~ wavelength) chiral particles with different handedness at an air–water interface using optical lateral forces induced by a single linearly polarized laser beam. The nontrivial physical interactions underlying these chirality-dependent forces distinctly differ from those predicted for dipolar or geometrical-optics particles. The lateral forces emerge from a complex interplay between the light polarization, lateral momentum enhancement, and out-of-plane light refraction at the particle-water interface. The sign of the lateral force could be reversed by changing the particle size, incident angle, and polarization of the obliquely incident light. 1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; Introduction interface15,16. The displacements of particles controlled by Enantiomer sorting has attracted tremendous attention thespinofthelightcanbeperpendiculartothedirectionof – owing to its significant applications in both material science the light beam17 19.
    [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]
  • Chiral Symmetry and Quantum Hadro-Dynamics
    Chiral symmetry and quantum hadro-dynamics G. Chanfray1, M. Ericson1;2, P.A.M. Guichon3 1IPNLyon, IN2P3-CNRS et UCB Lyon I, F69622 Villeurbanne Cedex 2Theory division, CERN, CH12111 Geneva 3SPhN/DAPNIA, CEA-Saclay, F91191 Gif sur Yvette Cedex Using the linear sigma model, we study the evolutions of the quark condensate and of the nucleon mass in the nuclear medium. Our formulation of the model allows the inclusion of both pion and scalar-isoscalar degrees of freedom. It guarantees that the low energy theorems and the constrains of chiral perturbation theory are respected. We show how this formalism incorporates quantum hadro-dynamics improved by the pion loops effects. PACS numbers: 24.85.+p 11.30.Rd 12.40.Yx 13.75.Cs 21.30.-x I. INTRODUCTION The influence of the nuclear medium on the spontaneous breaking of chiral symmetry remains an open problem. The amount of symmetry breaking is measured by the quark condensate, which is the expectation value of the quark operator qq. The vacuum value qq(0) satisfies the Gell-mann, Oakes and Renner relation: h i 2m qq(0) = m2 f 2, (1) q h i − π π where mq is the current quark mass, q the quark field, fπ =93MeV the pion decay constant and mπ its mass. In the nuclear medium the quark condensate decreases in magnitude. Indeed the total amount of restoration is governed by a known quantity, the nucleon sigma commutator ΣN which, for any hadron h is defined as: Σ = i h [Q , Q˙ ] h =2m d~x [ h qq(~x) h qq(0) ] , (2) h − h | 5 5 | i q h | | i−h i Z ˙ where Q5 is the axial charge and Q5 its time derivative.
    [Show full text]
  • Tau-Lepton Decay Parameters
    58. τ -lepton decay parameters 1 58. τ -Lepton Decay Parameters Updated August 2011 by A. Stahl (RWTH Aachen). The purpose of the measurements of the decay parameters (also known as Michel parameters) of the τ is to determine the structure (spin and chirality) of the current mediating its decays. 58.1. Leptonic Decays: The Michel parameters are extracted from the energy spectrum of the charged daughter lepton ℓ = e, µ in the decays τ ℓνℓντ . Ignoring radiative corrections, neglecting terms 2 →2 of order (mℓ/mτ ) and (mτ /√s) , and setting the neutrino masses to zero, the spectrum in the laboratory frame reads 2 5 dΓ G m = τℓ τ dx 192 π3 × mℓ f0 (x) + ρf1 (x) + η f2 (x) Pτ [ξg1 (x) + ξδg2 (x)] , (58.1) mτ − with 2 3 f0 (x)=2 6 x +4 x −4 2 32 3 2 2 8 3 f1 (x) = +4 x x g1 (x) = +4 x 6 x + x − 9 − 9 − 3 − 3 2 4 16 2 64 3 f2 (x) = 12(1 x) g2 (x) = x + 12 x x . − 9 − 3 − 9 The quantity x is the fractional energy of the daughter lepton ℓ, i.e., x = E /E ℓ ℓ,max ≈ Eℓ/(√s/2) and Pτ is the polarization of the tau leptons. The integrated decay width is given by 2 5 Gτℓ mτ mℓ Γ = 3 1+4 η . (58.2) 192 π mτ The situation is similar to muon decays µ eνeνµ. The generalized matrix element with γ → the couplings gεµ and their relations to the Michel parameters ρ, η, ξ, and δ have been described in the “Note on Muon Decay Parameters.” The Standard Model expectations are 3/4, 0, 1, and 3/4, respectively.
    [Show full text]
  • 1 an Introduction to Chirality at the Nanoscale Laurence D
    j1 1 An Introduction to Chirality at the Nanoscale Laurence D. Barron 1.1 Historical Introduction to Optical Activity and Chirality Scientists have been fascinated by chirality, meaning right- or left-handedness, in the structure of matter ever since the concept first arose as a result of the discovery, in the early years of the nineteenth century, of natural optical activity in refracting media. The concept of chirality has inspired major advances in physics, chemistry and the life sciences [1, 2]. Even today, chirality continues to catalyze scientific and techno- logical progress in many different areas, nanoscience being a prime example [3–5]. The subject of optical activity and chirality started with the observation by Arago in 1811 of colors in sunlight that had passed along the optic axis of a quartz crystal placed between crossed polarizers. Subsequent experiments by Biot established that the colors originated in the rotation of the plane of polarization of linearly polarized light (optical rotation), the rotation being different for light of different wavelengths (optical rotatory dispersion). The discovery of optical rotation in organic liquids such as turpentine indicated that optical activity could reside in individual molecules and could be observed even when the molecules were oriented randomly, unlike quartz where the optical activity is a property of the crystal structure, because molten quartz is not optically active. After his discovery of circularly polarized light in 1824, Fresnel was able to understand optical rotation in terms of different refractive indices for the coherent right- and left-circularly polarized components of equal amplitude into which a linearly polarized light beam can be resolved.
    [Show full text]
  • Chirality-Sorting Optical Forces
    Lateral chirality-sorting optical forces Amaury Hayata,b,1, J. P. Balthasar Muellera,1,2, and Federico Capassoa,2 aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; and bÉcole Polytechnique, Palaiseau 91120, France Contributed by Federico Capasso, August 31, 2015 (sent for review June 7, 2015) The transverse component of the spin angular momentum of found in Supporting Information. The optical forces arising through evanescent waves gives rise to lateral optical forces on chiral particles, the interaction of light with chiral matter may be intuited by con- which have the unusual property of acting in a direction in which sidering that an electromagnetic wave with helical polarization will there is neither a field gradient nor wave propagation. Because their induce corresponding helical dipole (and multipole) moments in a direction and strength depends on the chiral polarizability of the scatterer. If the scatterer is chiral (one may picture a subwavelength particle, they act as chirality-sorting and may offer a mechanism for metal helix), then the strength of the induced moments depends on passive chirality spectroscopy. The absolute strength of the forces the matching of the helicity of the electromagnetic wave to the also substantially exceeds that of other recently predicted sideways helicity of the scatterer. The forces described in this paper, as well optical forces. as other chirality-sorting optical forces, emerge because the in- duced moments in the scatterer result in radiation in a helicity- optical forces | chirality | optical spin-momentum locking | dependent preferred direction, which causes corresponding recoil. optical spin–orbit interaction | optical spin The effect discussed in this paper enables the tailoring of chirality- sorting optical forces by engineering the local SAM density of optical hirality [from Greek χeιρ (kheir), “hand”] is a type of asym- fields, such that fields with transverse SAM can be used to achieve metry where an object is geometrically distinct from its lateral forces.
    [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]
  • Introduction to Flavour Physics
    Introduction to flavour physics Y. Grossman Cornell University, Ithaca, NY 14853, USA Abstract In this set of lectures we cover the very basics of flavour physics. The lec- tures are aimed to be an entry point to the subject of flavour physics. A lot of problems are provided in the hope of making the manuscript a self-study guide. 1 Welcome statement My plan for these lectures is to introduce you to the very basics of flavour physics. After the lectures I hope you will have enough knowledge and, more importantly, enough curiosity, and you will go on and learn more about the subject. These are lecture notes and are not meant to be a review. In the lectures, I try to talk about the basic ideas, hoping to give a clear picture of the physics. Thus many details are omitted, implicit assumptions are made, and no references are given. Yet details are important: after you go over the current lecture notes once or twice, I hope you will feel the need for more. Then it will be the time to turn to the many reviews [1–10] and books [11, 12] on the subject. I try to include many homework problems for the reader to solve, much more than what I gave in the actual lectures. If you would like to learn the material, I think that the problems provided are the way to start. They force you to fully understand the issues and apply your knowledge to new situations. The problems are given at the end of each section.
    [Show full text]
  • Threshold Corrections in Grand Unified Theories
    Threshold Corrections in Grand Unified Theories Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der Fakult¨at f¨ur Physik des Karlsruher Instituts f¨ur Technologie genehmigte DISSERTATION von Dipl.-Phys. Waldemar Martens aus Nowosibirsk Tag der m¨undlichen Pr¨ufung: 8. Juli 2011 Referent: Prof. Dr. M. Steinhauser Korreferent: Prof. Dr. U. Nierste Contents 1. Introduction 1 2. Supersymmetric Grand Unified Theories 5 2.1. The Standard Model and its Limitations . ....... 5 2.2. The Georgi-Glashow SU(5) Model . .... 6 2.3. Supersymmetry (SUSY) .............................. 9 2.4. SupersymmetricGrandUnification . ...... 10 2.4.1. MinimalSupersymmetricSU(5). 11 2.4.2. MissingDoubletModel . 12 2.5. RunningandDecoupling. 12 2.5.1. Renormalization Group Equations . 13 2.5.2. Decoupling of Heavy Particles . 14 2.5.3. One-Loop Decoupling Coefficients for Various GUT Models . 17 2.5.4. One-Loop Decoupling Coefficients for the Matching of the MSSM to the SM ...................................... 18 2.6. ProtonDecay .................................... 21 2.7. Schur’sLemma ................................... 22 3. Supersymmetric GUTs and Gauge Coupling Unification at Three Loops 25 3.1. RunningandDecoupling. 25 3.2. Predictions for MHc from Gauge Coupling Unification . 32 3.2.1. Dependence on the Decoupling Scales µSUSY and µGUT ........ 33 3.2.2. Dependence on the SUSY Spectrum ................... 36 3.2.3. Dependence on the Uncertainty on the Input Parameters ....... 36 3.2.4. Top-Down Approach and the Missing Doublet Model . ...... 39 3.2.5. Comparison with Proton Decay Constraints . ...... 43 3.3. Summary ...................................... 44 4. Field-Theoretical Framework for the Two-Loop Matching Calculation 45 4.1. TheLagrangian.................................. 45 4.1.1.
    [Show full text]
  • The Taste of New Physics: Flavour Violation from Tev-Scale Phenomenology to Grand Unification Björn Herrmann
    The taste of new physics: Flavour violation from TeV-scale phenomenology to Grand Unification Björn Herrmann To cite this version: Björn Herrmann. The taste of new physics: Flavour violation from TeV-scale phenomenology to Grand Unification. High Energy Physics - Phenomenology [hep-ph]. Communauté Université Grenoble Alpes, 2019. tel-02181811 HAL Id: tel-02181811 https://tel.archives-ouvertes.fr/tel-02181811 Submitted on 12 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The taste of new physics: Flavour violation from TeV-scale phenomenology to Grand Unification Habilitation thesis presented by Dr. BJÖRN HERRMANN Laboratoire d’Annecy-le-Vieux de Physique Théorique Communauté Université Grenoble Alpes Université Savoie Mont Blanc – CNRS and publicly defended on JUNE 12, 2019 before the examination committee composed of Dr. GENEVIÈVE BÉLANGER CNRS Annecy President Dr. SACHA DAVIDSON CNRS Montpellier Examiner Prof. ALDO DEANDREA Univ. Lyon Referee Prof. ULRICH ELLWANGER Univ. Paris-Saclay Referee Dr. SABINE KRAML CNRS Grenoble Examiner Prof. FABIO MALTONI Univ. Catholique de Louvain Referee July 12, 2019 ii “We shall not cease from exploration, and the end of all our exploring will be to arrive where we started and know the place for the first time.” T.
    [Show full text]
  • CHIRALITY •The Helicity Eigenstates for a Particle/Anti-Particle for Are
    Particle Physics Michaelmas Term 2009 Prof Mark Thomson Handout 4 : Electron-Positron Annihilation Prof. M.A. Thomson Michaelmas 2009 120 QED Calculations How to calculate a cross section using QED (e.g. e+e– $ +– ): Draw all possible Feynman Diagrams •For e+e– $ +– there is just one lowest order diagram + e e– – + many second order diagrams + … + + e e + +… e– – e– – For each diagram calculate the matrix element using Feynman rules derived in handout 4. Sum the individual matrix elements (i.e. sum the amplitudes) •Note: summing amplitudes therefore different diagrams for the same final state can interfere either positively or negatively! Prof. M.A. Thomson Michaelmas 2009 121 and then square this gives the full perturbation expansion in • For QED the lowest order diagram dominates and for most purposes it is sufficient to neglect higher order diagrams. + + e e e– – e– – Calculate decay rate/cross section using formulae from handout 1. •e.g. for a decay •For scattering in the centre-of-mass frame (1) •For scattering in lab. frame (neglecting mass of scattered particle) Prof. M.A. Thomson Michaelmas 2009 122 Electron Positron Annihilation ,Consider the process: e+e– $ +– – •Work in C.o.M. frame (this is appropriate – + for most e+e– colliders). e e •Only consider the lowest order Feynman diagram: + ( Feynman rules give: e •Incoming anti-particle – – NOTE: e •Incoming particle •Adjoint spinor written first •In the C.o.M. frame have with Prof. M.A. Thomson Michaelmas 2009 123 Electron and Muon Currents •Here and matrix element • In handout 2 introduced the four-vector current which has same form as the two terms in [ ] in the matrix element • The matrix element can be written in terms of the electron and muon currents and • Matrix element is a four-vector scalar product – confirming it is Lorentz Invariant Prof.
    [Show full text]