Lecture 7: N = 2 supersymmetric gauge theory

Size: px
Start display at page:

Download "Lecture 7: N = 2 supersymmetric gauge theory"

Transcription

1 Lecture 7: N = 2 supersymmetric gauge theory José D. Edelstein University of Santiago de Compostela SUPERSYMMETRY Santiago de Compostela, November 22, 2012 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

2 Summary from Lecture 6 We introduced the extended super-poincaré algebra: { } Q I α, Q J β = 2 δ IJ (σ µ ) α β P { µ Q I α, Q J } β = 2 ɛαβ Z IJ where Z IJ = (a a IJ ) T a, and [ T a, T b] = if ab c T c. I, J,... go from 1 to N. The mass of all states is bounded from below, m z (r). The number of states in massless and BPS massive multiplets is 2 N, while that of non-bps massive multiplets is 2 2N. Thus, fields becoming massive due to the Higgs mechanism are BPS. Exact spectrum Since they are short multiplets, their spectrum can be safely extended from weak to strong coupling: it is quantum mechanically exact! José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

3 R-symmetry Recall that N = 1 supersymmetry has a global U(1) R-symmetry. It is an automorphism of the algebra acting non-trivially on the supercharges. If we go back to the extended SUSY algebra, it is immediate to see that, for N > 1, the R-symmetry group is U(N ). The supercharges (fermions) transform in the vector representation, Q I α. For future reference let us recall the R-symmetry group for two cases: For N = 2, the R-symmetry group is U(2) = SU(2) R U(1) R. In particular, for instance, ψ and λ in the N = 2 vector or chiral multiplet, Ψ belong to a doublet of SU(2) R. For N = 4, the R-symmetry group is U(4) = SU(4) R U(1) R (besides, SU(4) R SO(6) R, which happens to be the isometry group of S 5 ). José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

4 N = 2 supersymmetric gauge theory the superfield N > 1 is more constrained than N = 1; it is a particular subcase. The vector superfield Ψ is made of a vector, V, and a chiral, Φ; they belong to the same representation of the non-abelian gauge group. The N = 2 action must be simpler because, being enhanced by SU(2), the R-symmetry relates the fermions of both components of Ψ. In particular, this implies that the superpotential vanishes W (Φ) = 0 Indeed, the N = 1 superfields fit into a chiral N = 2 superfield, Ψ(x, θ, ϑ) = Φ(z, θ) + 2 ϑ α W α (z, θ) 1 2 ϑα ϑ α d 2 θ [ Φ(z iθ σ θ, θ, θ) ] e 2gV (z iθ σ θ,θ, θ) where z µ = x µ + iθ σ µ θ + iϑ σ µ ϑ = y µ + iϑ σ µ ϑ and Wα = 1 8 D 2 e 2V D α e 2V. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

5 N = 2 supersymmetric gauge theory the prepotential The N = 2 superspace notation makes clear that the action is written in terms of a single function, F(Ψ) called the prepotential S = d 4 x L = 1 [ ] 16π Im d 4 x d 2 θ d 2 ϑ F(Ψ) The two remaining functions in N = 1 theory with adjoint superfields, f (Φ) and K (Φ, Φ ), can be written in terms of a single holomorphic function. Indeed, f (Φ) = 2 F(Φ) Φ 2 K (Φ, Φ ) = 1 2i [ Φ e 2V F(Φ) Φ ] h.c. Being holomorphic, it inherits good properties (that you will see later). For instance, f (Φ) is not corrected after 1-loop; thus F(Ψ) isn t either. Formally, the expression above is the most general N = 2 Lagrangian for a supersymmetric gauge theory. In the case of an effective field theory, F is not restricted to be quadratic. It is only constrained by holomorphicity. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

6 N = 2 supersymmetric gauge theory the field content At tree-level, the prepotential reads which has the right scaling dimension. F class (Ψ) := F 0 (Ψ) = 1 2 τ 0 Tr Ψ 2 If we plug it into S and integrate on ϑ, L = 1 [( 16π Im d 2 θ 2 F 0 (Φ) Φ a Φ b W aα W b α + 2 d 2 θ d 2 ( θ Φ e 2gV ) )] a F 0 Φ a where the superfields are in the adjoint of SU(N c ), Φ = Φ a T a, W α = W a α T a, i.e., a, b,... are Lie algebra indices and Tr (T a T b ) = δ ab. Ψ is decomposed in V (A µ, λ) and Φ = (φ, ψ). We refer to φ as the higgs, the Weyl spinors ψ and λ as the higgsino and the gluino, and A µ as the gluon. Therefore, this is just a particular non-abelian gauge theory with scalars and fermions in the adjoint, and certain couplings dictated by SUSY. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

7 N = 2 supersymmetric gauge theory the Lagrangian The previous N = 2 Lagrangian can be written in terms of its components L = 1 ( g0 2 Tr 1 4 F µν F µν + g0 2 θ 0 32π 2 F F µν µν + (D µ φ) D µ φ 1 2 [φ, φ] 2 i λσ µ D µ λ i ψ σ µ D µ ψ i 2 [λ, ψ] φ i ) 2 [ λ, ψ] φ where we used the expression for the bare complexified coupling constant τ 0 = 4πi g θ 0 2π The quadratic prepotential gives the renormalizable microscopic Lagrangian. The theory is asymptotically free β(g) = µ dg ( dµ = g π ) C 2 (G) = g3 3 16π (2 N c) hence, confinement is expected to be present at strong coupling. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

8 N = 2 supersymmetric gauge theory R-symmetry and chiral anomaly This theory has a classical global R-symmetry U(2) R = SU(2) R U(1) R : The SU(2) R rotates the two supercharges as well as the two fermions as doublets. The gauge boson and the higgs are singlets. The U(1) R acts as Φ e 2iα Φ(e iα θ) and V V (e iα θ). Since θ e iα θ, ϑ e iα ϑ, F must have charge 4, i.e., F e 4iα F. U(1) R acts as a chiral symmetry. Thus, it is broken quantum mechanically µ j µ R = N c 8π 2 Tr F µν µν F This dictates the way the 1-loop effective action changes under U(1) R due to Noether s theorem. δl eff = α µ j µ R = α N c 8π 2 Tr F µν µν F José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

9 N = 2 supersymmetric gauge theory full perturbative prepotential Seiberg s non-renormalization theorem (to be discussed) applies: F pert (Φ) = F class (Φ) + F 1 loop (Φ) gives the complete Lagrangian to all orders in perturbation theory. This implies that the full perturbative contribution to the effective action can be obtained by integrating the infinitesimal anomalous variation. Notice that log Tr Φ 2 log Tr Φ 2 + 4iα under U(1) R. At the level of the prepotential, thus, the result is simply F pert (Φ) = 1 2 τ 0 Tr Φ 2 + i N ( ) c Tr Φ 2 4π Tr Φ2 log Λ 2 the coefficient being fixed from the value of the chiral anomaly. Λ is the quantum mechanical dynamically generated scale, as in QCD. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

10 N = 2 supersymmetric gauge theory the fate of the U(1) R The only term in L eff affected by the U(1) R transformation is the θ-term, θ θ + 4N c α since, recall δl eff = α µ j µ R = α N c 8π 2 Tr F µν µν F At the level of the effective action however, notice that, since 1 32π 2 F µν F µν Z (instanton number) a discrete subgroup, Z 4Nc U(1) R, remains a symmetry of the perturbative effective action, k α = 2πi k = 1,, 4N c 4N c This is a general feature of the U(1) R subgroup of U(N ) R. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

11 The classical moduli space, M 0 Let us now focus on the classical vacua of the theory. The potential reads V = 1 2g 2 Tr ( [φ, φ] 2) Unbroken supersymmetry requires that V vanishes in the vacuum. There is a family of vacua parameterized by constant fields φ 0 such that φ 0 and φ 0 commute. One can always rotate it into the Cartan subalgebra, H, φ = φ 0 = N c 1 k=1 a k H k H k are the generators of H and a k are complex numbers. For G = su(2), the Cartan subalgebra is simply generated by the Pauli matrix σ 3 and φ = a σ 3. For a 0, the gauge symmetry group is broken, SU(2) U(1). There is a singular point, a = 0, where the gauge symmetry is unbroken. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

12 The classical moduli space, M 0 φ 0 labels a continuous family of inequivalent ground states that constitute the classical moduli space, M 0. The variables a k are not gauge invariant, in particular under discrete Weyl transformations. Hence they are not faithfull coordinates for M 0. The Weyl group for a Lie algebra is generated by reflections in the roots; in the su(2) case, a σ 3 a σ 3. In general, it acts by conjugation, φ 0 g 1 φ 0 g. Thus, Weyl invariants can be obtained from the characteristic polynomial P Nc (λ) = det (λ φ 0 ) The coefficients of the polynomial (sometimes called W ANc 1 ), P Nc (λ) = λ Nc ū 2 (a k ) λ Nc 2 ū 3 (a k ) λ Nc 3 ū Nc (a k ) label gauge inequivalent vacua and, thus, parameterize faithfully M 0. There is no λ Nc 1 term due to the traceless condition of su(n c ). José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

13 The classical moduli space, M 0 A simple calculation shows that ū 2 (a k ) = 1 2 Tr φ 0 2 ū 3 (a k ) = 1 3 Tr φ 0 3 ū 4 (a k ) = 1 4 Tr φ (Tr φ 0 2 ) 2 These are Casimir operators and, hence, Weyl invariant by construction, ū k (a i ) = 1 k Tr φ 0 k + lower dimensional Casimirs Coming back to SU(2), there is a single (quadratic) Casimir (call ū := ū 2 ), ū(a) = 1 2 Tr φ 0 2 = 1 2 Tr (a σ 3) 2 = a 2 and the characteristic polynomial reads simply P 2 (λ) = λ 2 ū. For ū 0, the gauge symmetry group is broken, SU(2) U(1). There is a singular point, ū = 0, where the gauge symmetry is unbroken. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

14 The classical moduli space, M 0 Figure: M 0 for SU(2) is the complex ū-plane. The origin displays classical symmetry enhancement. Since P 2 (λ) = (λ ū 1/2 )(λ + ū 1/2 ), this is captured by the vanishing locus, Σ 0, of the classical discriminant 0 := (ū 1/2 ( ū 1/2 )) 2 = 4 ū. Another way to define the ū k is through the classical Miura transformation. Namely, factorizing the characteristic polynomial N c P Nc (λ) = (λ e i (a k )) where e i (a k ), i = 1,..., N c, are the eigenvalues of φ 0. i=1 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

15 The classical moduli space, M 0 Expanding P Nc (λ), it is easy to see that ū k (a) = ( 1) k+1 e j1 (a) e jk (a) j 1 =j k which are symmetric polynomials of the eigenvalues e i (a k ), thus manifestly invariant under the Weyl group (which acts by permutation). Consider the SU(3) case (call ū := ū 2 and v := ū 3 ), P 3 (λ) = λ 3 ū λ v = (λ e 1 ) (λ e 2 ) (λ e 3 ) where, of course, ū = ū(a), v = v(a), ū(a) = a a 2 2 a 1 a 2 v(a) = a 1 a 2 (a 1 a 2 ) and e 1 (a) = a 1, e 2 (a) = a 2 and e 3 (a) = a 2 a 1. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

16 The classical moduli space, M 0 Figure: M 0 for SU(3) has two complex coordinates, ū and v. Classical symmetry enhancement is captured by the vanishing locus, Σ 0, of the classical discriminant. 0 := (e 2 e 1 ) 2 (e 3 e 2 ) 2 (e 1 e 3 ) 2 = 4 ū 3 27 v 2 We will focus from now on almost exclusively in the SU(2) case. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

17 The effective prepotential: perturbative part Let us consider for the moment a generic situation, i.e., a vacuum φ 0 = a σ 3 where SU(2) U(1). If we evaluate F class (a), F class (a) = 1 2 τ 0 Tr φ 0 2 = 1 2 τ 0 a a Tr (σ 3 σ 3 ) = τ 0 a 2 It is convenient, to make contact with the SU(N c ) case, F class (a k ) = 1 τ 0 2N c α + + z 2 α + where + are the positive roots, and z α+ := α + φ 0 = z α+ (a k ). In the case of SU(2), z a := 2a, and F class (a) = 1 4 τ 0 z a 2 = τ 0 ū The 1-loop correction can be obtained by plugging φ 0 in the expression above F 1 loop (a) = i ( ) 4π z a 2 2 za log Λ 2 = i ( ) 4ū π ū log Λ 2 José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

18 The effective prepotential: perturbative part Again, this can be similarly written in the SU(N c ) case as ( ) F 1 loop (a k ) = i z 2 α+ log 4π Λ 2 α + + z 2 α + Notice that F 1 loop (a k ) diverges if z α+ (a k ) vanishes. These singularities are in one-to-one correspondence to those of M 0. From the kinetic term of the complex scalar, we see that Tr (D µ φ 0 ) D µ φ 0 = Tr (i[w µ, φ 0 ]) (i[w µ, φ 0 ]) Whenever z α+ (a k ) = 0: = 1 2 α + + z α+ 2 W ±α+ µ W µ ±α+ A pair of charged gauge bosons become massless, and Classically, there is gauge symmetry enhancement. José D. Edelstein (USC) Lecture 7: N = 2 supersymmetric gauge theory 22-nov / 18

Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions

Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions Supersymmetric Gauge Theories, Matrix Models and Geometric Transitions Frank FERRARI Université Libre de Bruxelles and International Solvay Institutes XVth Oporto meeting on Geometry, Topology and Physics:

More information

SUSY QCD. Consider a SUSY SU(N) with F flavors of quarks and squarks

SUSY QCD. Consider a SUSY SU(N) with F flavors of quarks and squarks SUSY gauge theories SUSY QCD Consider a SUSY SU(N) with F flavors of quarks and squarks Q i = (φ i, Q i, F i ), i = 1,..., F, where φ is the squark and Q is the quark. Q i = (φ i, Q i, F i ), in the antifundamental

More information

The Affleck Dine Seiberg superpotential

The Affleck Dine Seiberg superpotential The Affleck Dine Seiberg superpotential SUSY QCD Symmetry SUN) with F flavors where F < N SUN) SUF ) SUF ) U1) U1) R Φ, Q 1 1 F N F Φ, Q 1-1 F N F Recall that the auxiliary D a fields: D a = gφ jn T a

More information

1 Susy in 4d- Notes by Horatiu Nastase A very basic introduction (survival guide)

1 Susy in 4d- Notes by Horatiu Nastase A very basic introduction (survival guide) 1 Susy in 4d- Notes by Horatiu Nastase A very basic introduction (survival guide) 1.1 Algebras 2-component notation for 4d spinors ( ) ψα ψ = χ α (1) C matrix ( ɛαβ 0 C AB = 0 ɛ α β ) ; ɛ αβ = ɛ α β =

More information

Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory

Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory hep-th/9407087, RU-94-52, IAS-94-43 arxiv:hep-th/9407087v1 15 Jul 1994 Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory N. Seiberg Department

More information

Exact Results in D=2 Supersymmetric Gauge Theories And Applications

Exact Results in D=2 Supersymmetric Gauge Theories And Applications Exact Results in D=2 Supersymmetric Gauge Theories And Applications Jaume Gomis Miami 2012 Conference arxiv:1206.2606 with Doroud, Le Floch and Lee arxiv:1210.6022 with Lee N = (2, 2) supersymmetry on

More information

Lecture 12 Holomorphy: Gauge Theory

Lecture 12 Holomorphy: Gauge Theory Lecture 12 Holomorphy: Gauge Theory Outline SUSY Yang-Mills theory as a chiral theory: the holomorphic coupling and the holomorphic scale. Nonrenormalization theorem for SUSY YM: the gauge coupling runs

More information

Symmetries, Groups Theory and Lie Algebras in Physics

Symmetries, Groups Theory and Lie Algebras in Physics Symmetries, Groups Theory and Lie Algebras in Physics M.M. Sheikh-Jabbari Symmetries have been the cornerstone of modern physics in the last century. Symmetries are used to classify solutions to physical

More information

Lecture 5 The Renormalization Group

Lecture 5 The Renormalization Group Lecture 5 The Renormalization Group Outline The canonical theory: SUSY QCD. Assignment of R-symmetry charges. Group theory factors: bird track diagrams. Review: the renormalization group. Explicit Feynman

More information

University of Amsterdam

University of Amsterdam University of Amsterdam MSc Physics Theoretical Physics Master Thesis Dualities in Gauge Theories and their Geometric Realization by S.P.G. van Leuven 5756561 May 2014 60 ECTS February 2013-May 2014 Supervisor:

More information

Lecture 24 Seiberg Witten Theory III

Lecture 24 Seiberg Witten Theory III Lecture 24 Seiberg Witten Theory III Outline This is the third of three lectures on the exact Seiberg-Witten solution of N = 2 SUSY theory. The third lecture: The Seiberg-Witten Curve: the elliptic curve

More information

Lecture 6 The Super-Higgs Mechanism

Lecture 6 The Super-Higgs Mechanism Lecture 6 The Super-Higgs Mechanism Introduction: moduli space. Outline Explicit computation of moduli space for SUSY QCD with F < N and F N. The Higgs mechanism. The super-higgs mechanism. Reading: Terning

More information

Generalized Gaugino Condensation: Discrete R-Symmetries and Supersymmetric Vacua

Generalized Gaugino Condensation: Discrete R-Symmetries and Supersymmetric Vacua Generalized Gaugino Condensation: Discrete R-Symmetries and Supersymmetric Vacua John Kehayias Department of Physics University of California, Santa Cruz SUSY 10 August 23, 2010 Bonn, Germany [1] Generalized

More information

SUSY Breaking in Gauge Theories

SUSY Breaking in Gauge Theories SUSY Breaking in Gauge Theories Joshua Berger With the Witten index constraint on SUSY breaking having been introduced in last week s Journal club, we proceed to explicitly determine the constraints on

More information

Aspects of SUSY Breaking

Aspects of SUSY Breaking Aspects of SUSY Breaking Zohar Komargodski Institute for Advanced Study, Princeton ZK and Nathan Seiberg : arxiv:0907.2441 Aspects of SUSY Breaking p. 1/? Motivations Supersymmetry is important for particle

More information

Functional determinants

Functional determinants Functional determinants based on S-53 We are going to discuss situations where a functional determinant depends on some other field and so it cannot be absorbed into the overall normalization of the path

More information

Chern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee

Chern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of

More information

Quantum Field Theory III

Quantum Field Theory III Quantum Field Theory III Prof. Erick Weinberg April 5, 011 1 Lecture 6 Let s write down the superfield (without worrying about factors of i or Φ = A(y + θψ(y + θθf (y = A(x + θσ θ A + θθ θ θ A + θψ + θθ(

More information

Lecture 8: 1-loop closed string vacuum amplitude

Lecture 8: 1-loop closed string vacuum amplitude Lecture 8: 1-loop closed string vacuum amplitude José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 5, 2013 José D. Edelstein (USC) Lecture 8: 1-loop vacuum

More information

Topological reduction of supersymmetric gauge theories and S-duality

Topological reduction of supersymmetric gauge theories and S-duality Topological reduction of supersymmetric gauge theories and S-duality Anton Kapustin California Institute of Technology Topological reduction of supersymmetric gauge theories and S-duality p. 1/2 Outline

More information

Aspects of N = 2 Supersymmetric Gauge Theories in Three Dimensions

Aspects of N = 2 Supersymmetric Gauge Theories in Three Dimensions hep-th/9703110, RU-97-10, IASSNS-HEP-97/18 Aspects of N = 2 Supersymmetric Gauge Theories in Three Dimensions O. Aharony 1, A. Hanany 2, K. Intriligator 2, N. Seiberg 1, and M.J. Strassler 2,3 1 Dept.

More information

Supersymmetric Gauge Theories in 3d

Supersymmetric Gauge Theories in 3d Supersymmetric Gauge Theories in 3d Nathan Seiberg IAS Intriligator and NS, arxiv:1305.1633 Aharony, Razamat, NS, and Willett, arxiv:1305.3924 3d SUSY Gauge Theories New lessons about dynamics of quantum

More information

Reφ = 1 2. h ff λ. = λ f

Reφ = 1 2. h ff λ. = λ f I. THE FINE-TUNING PROBLEM A. Quadratic divergence We illustrate the problem of the quadratic divergence in the Higgs sector of the SM through an explicit calculation. The example studied is that of the

More information

Roni Harnik LBL and UC Berkeley

Roni Harnik LBL and UC Berkeley Roni Harnik LBL and UC Berkeley with Daniel Larson and Hitoshi Murayama, hep-ph/0309224 Supersymmetry and Dense QCD? What can we compare b/w QCD and SQCD? Scalars with a chemical potential. Exact Results.

More information

Techniques for exact calculations in 4D SUSY gauge theories

Techniques for exact calculations in 4D SUSY gauge theories Techniques for exact calculations in 4D SUSY gauge theories Takuya Okuda University of Tokyo, Komaba 6th Asian Winter School on Strings, Particles and Cosmology 1 First lecture Motivations for studying

More information

Solution Set 8 Worldsheet perspective on CY compactification

Solution Set 8 Worldsheet perspective on CY compactification MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics String Theory (8.821) Prof. J. McGreevy Fall 2007 Solution Set 8 Worldsheet perspective on CY compactification Due: Monday, December 18, 2007

More information

Soft Supersymmetry Breaking in

Soft Supersymmetry Breaking in Soft Supersymmetry Breaking in UMD-PP-99-119 SNS-PH/99-12 Deformed Moduli Spaces, Conformal Theories and N = 2 Yang-Mills Theory arxiv:hep-th/9908085v1 11 Aug 1999 Markus A. Luty Department of Physics,

More information

Lectures on Supersymmetry I

Lectures on Supersymmetry I I Carlos E.M. Wagner HEP Division, Argonne National Laboratory Enrico Fermi Institute, University of Chicago Ecole de Physique de Les Houches, France, August 5, 005. PASI 006, Puerto Vallarta, Mexico,

More information

arxiv:hep-th/ v1 21 May 1996

arxiv:hep-th/ v1 21 May 1996 ITP-SB-96-24 BRX-TH-395 USITP-96-07 hep-th/xxyyzzz arxiv:hep-th/960549v 2 May 996 Effective Kähler Potentials M.T. Grisaru Physics Department Brandeis University Waltham, MA 02254, USA M. Roče and R. von

More information

S-CONFINING DUALITIES

S-CONFINING DUALITIES DIMENSIONAL REDUCTION of S-CONFINING DUALITIES Cornell University work in progress, in collaboration with C. Csaki, Y. Shirman, F. Tanedo and J. Terning. 1 46 3D Yang-Mills A. M. Polyakov, Quark Confinement

More information

A Note on Supersymmetry Breaking. Stephen D.H. Hsu, Myckola Schwetz. Department of Physics Yale University New Haven, CT

A Note on Supersymmetry Breaking. Stephen D.H. Hsu, Myckola Schwetz. Department of Physics Yale University New Haven, CT YCTP-P3-97 A Note on Supersymmetry Breaking Stephen D.H. Hsu, Myckola Schwetz Department of Physics Yale University New Haven, CT 06520-8120 March, 1997 Abstract Using a simple observation based on holomorphy,

More information

Week 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books

Week 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books Week 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books Burgess-Moore, Chapter Weiberg, Chapter 5 Donoghue, Golowich, Holstein Chapter 1, 1 Free field Lagrangians

More information

Anomaly and gaugino mediation

Anomaly and gaugino mediation Anomaly and gaugino mediation Supergravity mediation X is in the hidden sector, P l suppressed couplings SUSY breaking VEV W = ( W hid (X) + W vis ) (ψ) f = δj i ci j X X ψ j e V ψ 2 i +... Pl τ = θ Y

More information

REVIEW. Quantum electrodynamics (QED) Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field:

REVIEW. Quantum electrodynamics (QED) Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field: Quantum electrodynamics (QED) based on S-58 Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field: Noether current of the lagrangian for a free Dirac

More information

3 Representations of the supersymmetry algebra

3 Representations of the supersymmetry algebra 3 Representations of the supersymmetry algebra In this lecture we will discuss representations of the supersymmetry algebra. Let us first briefly recall how things go for the Poincaré algebra. The Poincaré

More information

Instantons in supersymmetric gauge theories. Tobias Hansen Talk at tuesday s Werkstatt Seminar January 10, 2012

Instantons in supersymmetric gauge theories. Tobias Hansen Talk at tuesday s Werkstatt Seminar January 10, 2012 Instantons in supersymmetric gauge theories Tobias Hansen Talk at tuesday s Werkstatt Seminar January 10, 01 References [1] N. Dorey, T. J. Hollowood, V. V. Khoze and M. P. Mattis, The Calculus of many

More information

The Strong Interaction and LHC phenomenology

The Strong Interaction and LHC phenomenology The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules

More information

8.821 F2008 Lecture 5: SUSY Self-Defense

8.821 F2008 Lecture 5: SUSY Self-Defense 8.8 F008 Lecture 5: SUSY Self-Defense Lecturer: McGreevy Scribe: Iqbal September, 008 Today s lecture will teach you enough supersymmetry to defend yourself against a hostile supersymmetric field theory,

More information

Lecture 7 SUSY breaking

Lecture 7 SUSY breaking Lecture 7 SUSY breaking Outline Spontaneous SUSY breaking in the WZ-model. The goldstino. Goldstino couplings. The goldstino theorem. Reading: Terning 5.1, 5.3-5.4. Spontaneous SUSY Breaking Reminder:

More information

Solutions to gauge hierarchy problem. SS 10, Uli Haisch

Solutions to gauge hierarchy problem. SS 10, Uli Haisch Solutions to gauge hierarchy problem SS 10, Uli Haisch 1 Quantum instability of Higgs mass So far we considered only at RGE of Higgs quartic coupling (dimensionless parameter). Higgs mass has a totally

More information

m f f unchanged under the field redefinition (1), the complex mass matrix m should transform into

m f f unchanged under the field redefinition (1), the complex mass matrix m should transform into PHY 396 T: SUSY Solutions or problem set #8. Problem (a): To keep the net quark mass term L QCD L mass = ψ α c m ψ c α + hermitian conjugate (S.) unchanged under the ield redeinition (), the complex mass

More information

The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten

The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten Lecture 4 QCD as a Gauge Theory Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local

More information

Seiberg Duality: SUSY QCD

Seiberg Duality: SUSY QCD Seiberg Duality: SUSY QCD Outline: SYM for F N In this lecture we begin the study of SUSY SU(N) Yang-Mills theory with F N flavors. This setting is very rich! Higlights of the next few lectures: The IR

More information

Supersymmetry Breaking

Supersymmetry Breaking Supersymmetry Breaking LHC Search of SUSY: Part II Kai Wang Phenomenology Institute Department of Physics University of Wisconsin Madison Collider Phemonology Gauge Hierarchy and Low Energy SUSY Gauge

More information

Some Useful Formalism for Supersymmetry

Some Useful Formalism for Supersymmetry Preprint typeset in JHEP style - PAPER VERSION Some Useful Formalism for Supersymmetry Patrick J. Fox Theoretical Physics Department, Fermi National Accelerator Laboratory Batavia, IL 60510,USA pjfox@fnal.gov

More information

The Standard Model and beyond

The Standard Model and beyond The Standard Model and beyond In this chapter we overview the structure of the Standard Model (SM) of particle physics, its shortcomings, and different ideas for physics beyond the Standard Model (BSM)

More information

Non-Supersymmetric Seiberg duality Beyond the Planar Limit

Non-Supersymmetric Seiberg duality Beyond the Planar Limit Non-Supersymmetric Seiberg duality Beyond the Planar Limit Input from non-critical string theory, IAP Large N@Swansea, July 2009 A. Armoni, D.I., G. Moraitis and V. Niarchos, arxiv:0801.0762 Introduction

More information

arxiv:hep-th/ v2 4 Dec 1997

arxiv:hep-th/ v2 4 Dec 1997 CERN-TH/97-316 US-FT-33/97 hep-th/9711132 November, 1997 arxiv:hep-th/9711132v2 4 Dec 1997 MASS PERTURBATIONS IN TWISTED N = 4 SUPERSYMMETRIC GAUGE THEORIES J. M. F. Labastida a,b and Carlos Lozano b a

More information

A Localization Computation in Confining Phase

A Localization Computation in Confining Phase A Localization Computation in Confining Phase Seiji Terashima (YITP) 20 January 2015 at Osaka based on the paper: arxiv:1410.3630 Introduction 2 Analytic computations in QFT are hopeless, but, some exceptions:

More information

Geometry and Physics. Amer Iqbal. March 4, 2010

Geometry and Physics. Amer Iqbal. March 4, 2010 March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries

More information

Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges. Adi Armoni Swansea University

Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges. Adi Armoni Swansea University Two Examples of Seiberg Duality in Gauge Theories With Less Than Four Supercharges Adi Armoni Swansea University Queen Mary, April 2009 1 Introduction Seiberg duality (Seiberg 1994) is a highly non-trivial

More information

N=1 Global Supersymmetry in D=4

N=1 Global Supersymmetry in D=4 Susy algebra equivalently at quantum level Susy algebra In Weyl basis In this form it is obvious the U(1) R symmetry Susy algebra We choose a Majorana representation for which all spinors are real. In

More information

Katrin Becker, Texas A&M University. Strings 2016, YMSC,Tsinghua University

Katrin Becker, Texas A&M University. Strings 2016, YMSC,Tsinghua University Katrin Becker, Texas A&M University Strings 2016, YMSC,Tsinghua University ± Overview Overview ± II. What is the manifestly supersymmetric complete space-time action for an arbitrary string theory or M-theory

More information

Lectures on Supersymmetry Breaking

Lectures on Supersymmetry Breaking UCSD-PTH-07-02 Lectures on Supersymmetry Breaking arxiv:hep-ph/0702069v3 24 Apr 2007 Kenneth Intriligator 1 and Nathan Seiberg 2 1 Department of Physics, University of California, San Diego, La Jolla,

More information

Supersymmetry: Fundamentals. Michael Dine

Supersymmetry: Fundamentals. Michael Dine Supersymmetry: Fundamentals Michael Dine Prospects in Theoretical Physics, Princeton, 2007 Supersymmetry (SUSY) is a symmetry which relates fermions and bosons, i.e. fields with different spin. That such

More information

Nonperturbative Study of Supersymmetric Gauge Field Theories

Nonperturbative Study of Supersymmetric Gauge Field Theories Nonperturbative Study of Supersymmetric Gauge Field Theories Matteo Siccardi Tutor: Prof. Kensuke Yoshida Sapienza Università di Roma Facoltà di Scienze Matematiche, Fisiche e Naturali Dipartimento di

More information

Seiberg-Witten Theory

Seiberg-Witten Theory See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/277813379 Seiberg-Witten Theory RESEARCH JUNE 2015 DOI: 10.13140/RG.2.1.1771.7923 READS 48 1

More information

Lecture 9: RR-sector and D-branes

Lecture 9: RR-sector and D-branes Lecture 9: RR-sector and D-branes José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 6, 2013 José D. Edelstein (USC) Lecture 9: RR-sector and D-branes 6-mar-2013

More information

Supersymmetry Highlights. Kevin Hambleton

Supersymmetry Highlights. Kevin Hambleton Supersymmetry Highlights Kevin Hambleton Outline SHO Example Why SUSY? SUSY Fields Superspace Lagrangians SUSY QED MSSM i Warm Up A Hint Of SUSY... Remember Quantum Simple Harmonic Oscillator... Canonical

More information

g abφ b = g ab However, this is not true for a local, or space-time dependant, transformations + g ab

g abφ b = g ab However, this is not true for a local, or space-time dependant, transformations + g ab Yang-Mills theory Modern particle theories, such as the Standard model, are quantum Yang- Mills theories. In a quantum field theory, space-time fields with relativistic field equations are quantized and,

More information

Theory of Elementary Particles homework VIII (June 04)

Theory of Elementary Particles homework VIII (June 04) Theory of Elementary Particles homework VIII June 4) At the head of your report, please write your name, student ID number and a list of problems that you worked on in a report like II-1, II-3, IV- ).

More information

Introduction to Supersymmetry

Introduction to Supersymmetry Introduction to Supersymmetry 1: Formalism of SUSY M. E. Peskin Maria Laach Herbstschule September, 2004 Among models of electroweak symmetry breaking and physics beyond the Standard Model Supersymmetry

More information

Higher dimensional operators. in supersymmetry

Higher dimensional operators. in supersymmetry I. Antoniadis CERN Higher dimensional operators in supersymmetry with E. Dudas, D. Ghilencea, P. Tziveloglou Planck 2008, Barcelona Outline Effective operators from new physics integrating out heavy states

More information

Some Tools for Exploring Supersymmetric RG Flows

Some Tools for Exploring Supersymmetric RG Flows Some Tools for Exploring Supersymmetric RG Flows Thomas Dumitrescu Harvard University Work in Progress with G. Festuccia and M. del Zotto NatiFest, September 2016 IAS, Princeton Quantum Field Theory and

More information

Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University

Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1

More information

A Renormalization Group Primer

A Renormalization Group Primer A Renormalization Group Primer Physics 295 2010. Independent Study. Topics in Quantum Field Theory Michael Dine Department of Physics University of California, Santa Cruz May 2010 Introduction: Some Simple

More information

A model of the basic interactions between elementary particles is defined by the following three ingredients:

A model of the basic interactions between elementary particles is defined by the following three ingredients: I. THE STANDARD MODEL A model of the basic interactions between elementary particles is defined by the following three ingredients:. The symmetries of the Lagrangian; 2. The representations of fermions

More information

Vacuum Energy and Effective Potentials

Vacuum Energy and Effective Potentials Vacuum Energy and Effective Potentials Quantum field theories have badly divergent vacuum energies. In perturbation theory, the leading term is the net zero-point energy E zeropoint = particle species

More information

Anomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006

Anomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006 Anomaly Kenichi KONISHI University of Pisa College de France, 14 February 2006 Abstract Symmetry and quantization U A (1) anomaly and π 0 decay Origin of anomalies Chiral and nonabelian anomaly Anomally

More information

Dynamical supersymmetry breaking, with Flavor

Dynamical supersymmetry breaking, with Flavor Dynamical supersymmetry breaking, with Flavor Cornell University, November 2009 Based on arxiv: 0911.2467 [Craig, Essig, Franco, Kachru, GT] and arxiv: 0812.3213 [Essig, Fortin, Sinha, GT, Strassler] Flavor

More information

CP n supersymmetric mechanics in the U(n) background gauge fields

CP n supersymmetric mechanics in the U(n) background gauge fields CP n supersymmetric mechanics in the U(n) background gauge fields Sergey Krivonos Joint Institute for Nuclear Research Recent Advances in Quantum Field and String Theory, Tbilisi, September 26-30, 2011

More information

RECENT ASPECTS OF SUPERSYMMETRY BREAKING

RECENT ASPECTS OF SUPERSYMMETRY BREAKING Università degli Studi di Milano Bicocca Facoltà di Scienze Matematiche, Fisiche e Naturali Dipartimento di Fisica G.Occhialini RECENT ASPECTS OF SUPERSYMMETRY BREAKING Tutore: Luciano Girardello Tesi

More information

CP n supersymmetric mechanics in the U(n) background gauge fields

CP n supersymmetric mechanics in the U(n) background gauge fields Preliminaries: and free CP n mechanics CP n supersymmetric mechanics in the U(n) background gauge fields Sergey Krivonos Joint Institute for Nuclear Research Advances of Quantum Field Theory, Dubna, 2011

More information

Nonperturbative Dynamics in Supersymmetric Gauge Theories

Nonperturbative Dynamics in Supersymmetric Gauge Theories Doctoral Thesis Nonperturbative Dynamics in Supersymmetric Gauge Theories N. Maru Department of Physics, Nagoya University, Nagoya 464-8602, Japan E-mail address maru@eken.phys.nagoya-u.ac.jp Abstract

More information

Stability in Maximal Supergravity

Stability in Maximal Supergravity Stability in Maximal Supergravity S. Bielleman, s171136, RuG Supervisor: Dr. D. Roest August 5, 014 Abstract In this thesis, we look for a bound on the lightest scalar mass in maximal supergravity. The

More information

Electroweak and Higgs Physics

Electroweak and Higgs Physics Electroweak and Higgs Physics Lecture 2 : Higgs Mechanism in the Standard and Supersymmetric Models Alexei Raspereza DESY Summer Student Program Hamburg August 2017 Standard Model (Summary) Building blocks

More information

1 4-dimensional Weyl spinor

1 4-dimensional Weyl spinor 4-dimensional Weyl spinor Left moving ψ α, α =, Right moving ψ α, α =, They are related by the complex conjugation. The indices are raised or lowerd by the ϵ tensor as ψ α := ϵ αβ ψ β, α := ϵ α β β. (.)

More information

1. Introduction The holomorphic properties of supersymmetric eld theories [1-3] can be a powerful tool in deriving exact results about them [3-6]. In

1. Introduction The holomorphic properties of supersymmetric eld theories [1-3] can be a powerful tool in deriving exact results about them [3-6]. In hep-th/9408099, RU-94-60, IASSNS-HEP-94/55 Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD N. Seiberg Department of Physics and Astronomy Rutgers University, Piscataway, NJ 08855-0849,

More information

ON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES. Ft. Lauderdale, December 18, 2015 PLAN

ON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES. Ft. Lauderdale, December 18, 2015 PLAN ON ULTRAVIOLET STRUCTURE OF 6D SUPERSYMMETRIC GAUGE THEORIES Ft. Lauderdale, December 18, 2015 PLAN Philosophical introduction Technical interlude Something completely different (if I have time) 0-0 PHILOSOPHICAL

More information

The Phases of QCD. Thomas Schaefer. North Carolina State University

The Phases of QCD. Thomas Schaefer. North Carolina State University The Phases of QCD Thomas Schaefer North Carolina State University 1 Motivation Different phases of QCD occur in the universe Neutron Stars, Big Bang Exploring the phase diagram is important to understanding

More information

Updates in the Finite N=1 SU(5) Model

Updates in the Finite N=1 SU(5) Model Updates in the Finite N=1 SU(5) Model Gregory Patellis Physics Department, NTUA Sven Heinemeyer, Myriam Mondragon, Nicholas Tracas, George Zoupanos arxiv:1802.04666 March 31, 2018 Updates in the Finite

More information

Dynamical SUSY Breaking in Meta-Stable Vacua

Dynamical SUSY Breaking in Meta-Stable Vacua UCSD-PTH-06-03 Dynamical SUSY Breaking in Meta-Stable Vacua arxiv:hep-th/0602239v3 1 Apr 2006 Kenneth Intriligator 1,2, Nathan Seiberg 2 and David Shih 3 1 Department of Physics, University of California,

More information

7 Supersymmetry breaking

7 Supersymmetry breaking 7 Supersymmetry breaking If supersymmetry is at all realized in Nature, it must be broken at low enough energy: we do not see any mass degeneracy in the elementary particle spectrum, at least at energies

More information

t Hooft Anomaly Matching for QCD

t Hooft Anomaly Matching for QCD UCB-PTH-97-3 LBNL-41477 t Hooft Anomaly Matching for QCD John Terning Department of Physics, University of California, Berkeley, CA 9470 and Theory Group, Lawrence Berkeley National Laboratory, Berkeley,

More information

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises SM, EWSB & Higgs MITP Summer School 017 Joint Challenges for Cosmology and Colliders Homework & Exercises Ch!"ophe Grojean Ch!"ophe Grojean DESY (Hamburg) Humboldt University (Berlin) ( christophe.grojean@desy.de

More information

Axions. Kerstin Helfrich. Seminar on Theoretical Particle Physics, / 31

Axions. Kerstin Helfrich. Seminar on Theoretical Particle Physics, / 31 1 / 31 Axions Kerstin Helfrich Seminar on Theoretical Particle Physics, 06.07.06 2 / 31 Structure 1 Introduction 2 Repetition: Instantons Formulae The θ-vacuum 3 The U(1) and the strong CP problem The

More information

A Supergravity Dual for 4d SCFT s Universal Sector

A Supergravity Dual for 4d SCFT s Universal Sector SUPERFIELDS European Research Council Perugia June 25th, 2010 Adv. Grant no. 226455 A Supergravity Dual for 4d SCFT s Universal Sector Gianguido Dall Agata D. Cassani, G.D., A. Faedo, arxiv:1003.4283 +

More information

2-Group Global Symmetry

2-Group Global Symmetry 2-Group Global Symmetry Clay Córdova School of Natural Sciences Institute for Advanced Study April 14, 2018 References Based on Exploring 2-Group Global Symmetry in collaboration with Dumitrescu and Intriligator

More information

Some applications of light-cone superspace

Some applications of light-cone superspace Some applications of light-cone superspace Stefano Kovacs (Trinity College Dublin & Dublin Institute for Advanced Studies) Strings and Strong Interactions LNF, 19/09/2008 N =4 supersymmetric Yang Mills

More information

An Introduction to Dynamical Supersymmetry Breaking

An Introduction to Dynamical Supersymmetry Breaking An Introduction to Dynamical Supersymmetry Breaking Thomas Walshe September 28, 2009 Submitted in partial fulfillment of the requirements for the degree of Master of Science of Imperial College London

More information

A New Regulariation of N = 4 Super Yang-Mills Theory

A New Regulariation of N = 4 Super Yang-Mills Theory A New Regulariation of N = 4 Super Yang-Mills Theory Humboldt Universität zu Berlin Institut für Physik 10.07.2009 F. Alday, J. Henn, J. Plefka and T. Schuster, arxiv:0908.0684 Outline 1 Motivation Why

More information

EDMs from the QCD θ term

EDMs from the QCD θ term ACFI EDM School November 2016 EDMs from the QCD θ term Vincenzo Cirigliano Los Alamos National Laboratory 1 Lecture II outline The QCD θ term Toolbox: chiral symmetries and their breaking Estimate of the

More information

Introduction to Supersymmetry

Introduction to Supersymmetry Introduction to Supersymmetry Unreasonable effectiveness of the SM L Yukawa = y t H 0 t L t R + h.c. H 0 = H 0 + h 0 = v + h 0 m t = y t v t, L t R h 0 h 0 Figure 1: The top loop contribution to the Higgs

More information

The Higgs Mechanism and the Higgs Particle

The Higgs Mechanism and the Higgs Particle The Higgs Mechanism and the Higgs Particle Heavy-Ion Seminar... or the Anderson-Higgs-Brout-Englert-Guralnik-Hagen-Kibble Mechanism Philip W. Anderson Peter W. Higgs Tom W. B. Gerald Carl R. François Robert

More information

Models of Neutrino Masses

Models of Neutrino Masses Models of Neutrino Masses Fernando Romero López 13.05.2016 1 Introduction and Motivation 3 2 Dirac and Majorana Spinors 4 3 SU(2) L U(1) Y Extensions 11 4 Neutrino masses in R-Parity Violating Supersymmetry

More information

Chiral Symmetry Breaking from Monopoles and Duality

Chiral Symmetry Breaking from Monopoles and Duality Chiral Symmetry Breaking from Monopoles and Duality Thomas Schaefer, North Carolina State University with A. Cherman and M. Unsal, PRL 117 (2016) 081601 Motivation Confinement and chiral symmetry breaking

More information

Minimal Supersymmetric Standard Model (MSSM). Nausheen R. Shah

Minimal Supersymmetric Standard Model (MSSM). Nausheen R. Shah Minimal Supersymmetric Standard Model (MSSM). Nausheen R. Shah June 8, 2003 1 Introduction Even though the Standard Model has had years of experimental success, it has been known for a long time that it

More information

+ µ 2 ) H (m 2 H 2

+ µ 2 ) H (m 2 H 2 I. THE HIGGS POTENTIAL AND THE LIGHT HIGGS BOSON In the previous chapter, it was demonstrated that a negative mass squared in the Higgs potential is generated radiatively for a large range of boundary

More information

arxiv: v2 [hep-lat] 23 Dec 2008

arxiv: v2 [hep-lat] 23 Dec 2008 arxiv:8.964v2 [hep-lat] 23 Dec 28, F. Farchioni, A. Ferling, G. Münster, J. Wuilloud University of Münster, Institute for Theoretical Physics Wilhelm-Klemm-Strasse 9, D-4849 Münster, Germany E-mail: k_demm@uni-muenster.de

More information