Discrete R Symmetries and Low Energy Supersymmetry UC Davis, 2011 Michael Dine Department of Physics University of California, Santa Cruz Work with John Kehayias. February, 2011 Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Plan for Today: “New, Improved" Models of Dynamical Supersymmetry Breaking It is often said that SUSY breaking is a poorly understood problem. But much has been known for many years; problem is that models were complicated. Stable, dynamical SUSY breaking requires chiral representations of gauge groups, other special features which are not particularly generic. Model building is hard. All of this changed with work of Intriligator, Shih and Seiberg (ISS): Focus on metastable susy breaking. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Metastable Supersymmetry Breaking Quite generic. First, non-dynamical. O’Raifeartaigh Model: W = X ( λ A 2 − f ) + mAY (1) SUSY broken, can’t simultaneously satisfy ∂ W ∂ X = ∂ W ∂ Y = 0 . (2) E.g. m 2 > f gives � A � = � Y � = 0, � X � undetermined by the classical equations. f is order parameter of susy breaking. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
This model has a continuous "R Symmetry". In accord with a theorem of Nelson and Seiberg, which asserts that such a symmetry is required, generically, for supersymmetry breaking.. In components, using the same labels for the scalar component of a chiral field and the field itself: X → e 2 i α X Y → e 2 i α Y A → A (3) while the fermions in the multiplet have R charge smaller by one unit, e.g. ψ X → e i α ψ X ψ Y → e i α ψ Y ψ A → e − i α ψ A . (4) (For those familiar with superspace, this corresponds to θ → e i αθ d θ → e − i α d θ. ) Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Under an R symmetry, the supercharges and the superpotential transform: α → e − i α ¯ Q α → e i α Q α ¯ W → e 2 i α W . Q ˙ Q ˙ (5) α One loop effects generate a potential for X (Coleman-Weinberg) with minimum at � X � = 0. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
We don’t expect (exact) continuous global symmetries in nature, but discrete symmetries are more plausible. Take a discrete subgroup of the R symmetry, e.g. α = 2 π/ N ; a discrete R symmetry ( Z N ) Allows W = X ( λ A 2 − f ) + mAY + X N + 1 M N − 2 + . . . (6) W → α 2 W (7) (We will assume M ∼ M p ). At low energies the last term is irrelevant, so in this model, there is a continuous R symmetry as an accidental consequence of the discrete symmetries (the model can be the most general consistent with symmetries). Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
One expects that the model has supersymmetric vacua, and it does: X = ( fM N − 2 ) 1 / N + 1 . (8) But the minimum near the origin persists, with positive energy ( ≈ f 2 ), so the susy-breaking vacuum is metastable . Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Retrofitting: Supersymmetry Breaking Made (too?) Easy ISS: A beautiful dynamical example. But for a number of reasons (to which we will return) I will focus on models which are, at first sight, somewhat more ad hoc, but also simpler. "Retrofitting". Would like to generate the scale, f , dynamically. Basic ingredient: dynamical generation of a scale, without susy breaking. Candidate mechanism: gaugino condensation. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Gaugino Condensation Pure susy gauge theory: One set of adjoint fermions, λ . Quantum mechanically: Z N symmetry. 2 π ik � λλ � = N Λ 3 e (9) N breaks discrete symmetry, but not susy. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Retrofitting the O’Raifeartaigh Model Feng, Silverstein, M.D. Take our earlier model, and replace f → Λ 3 M : � W 2 + XA 2 + mYA + X N + 1 W = − 1 � 1 + cX α . (10) 8 π 2 M N − 2 4 M p p At low energies, we can replace the gaugino bilinear by its expectation value as a function of X (i.e. integrate out the massive degrees of freedom): � λλ � = N Λ 3 e − cX NMp ≡ W 0 − f X (11) f = c Λ 3 W 0 = N Λ 3 ; , (12) M p the low energy effective superpotential is (for X ≪ M p ): W = W 0 + X ( A 2 − f ) + XA 2 + mYA , (13) Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
A skeptic can argue that this is all a bit silly: We have introduced a new gauge interaction solely to 1 generate an additional mass scale. We still have a mass parameter M , put into the model by 2 hand. Anything else you might wish to complain about. 3 The rest of this talk will be devoted to confronting these questions. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
A skeptic can argue that this is all a bit silly: We have introduced a new gauge interaction solely to 1 generate an additional mass scale. We still have a mass parameter M , put into the model by 2 hand. Anything else you might wish to complain about. 3 The rest of this talk will be devoted to confronting these questions. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
A skeptic can argue that this is all a bit silly: We have introduced a new gauge interaction solely to 1 generate an additional mass scale. We still have a mass parameter M , put into the model by 2 hand. Anything else you might wish to complain about. 3 The rest of this talk will be devoted to confronting these questions. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
A skeptic can argue that this is all a bit silly: We have introduced a new gauge interaction solely to 1 generate an additional mass scale. We still have a mass parameter M , put into the model by 2 hand. Anything else you might wish to complain about. 3 The rest of this talk will be devoted to confronting these questions. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
The first failing is actually a major success. We have 1 generated a constant in W of the correct order of magnitude to cancel the c.c. Retrofitting almost inevitable(?). Richer dynamics – a simple generalization of gaugino 2 condensation – can account for both scales dynamically. The µ problem of gauge mediation is readily solved in this 3 framework. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
The first failing is actually a major success. We have 1 generated a constant in W of the correct order of magnitude to cancel the c.c. Retrofitting almost inevitable(?). Richer dynamics – a simple generalization of gaugino 2 condensation – can account for both scales dynamically. The µ problem of gauge mediation is readily solved in this 3 framework. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
The first failing is actually a major success. We have 1 generated a constant in W of the correct order of magnitude to cancel the c.c. Retrofitting almost inevitable(?). Richer dynamics – a simple generalization of gaugino 2 condensation – can account for both scales dynamically. The µ problem of gauge mediation is readily solved in this 3 framework. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
The first failing is actually a major success. We have 1 generated a constant in W of the correct order of magnitude to cancel the c.c. Retrofitting almost inevitable(?). Richer dynamics – a simple generalization of gaugino 2 condensation – can account for both scales dynamically. The µ problem of gauge mediation is readily solved in this 3 framework. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Other Consequences of Discrete R Symmetries In “gauge mediation" (lower scale breaking), R symmetries 1 can play a role in suppressing proton decay and other rare processes. We will argue that in “gravity mediation", R symmetries 2 (discrete) are inevitably broken by Planck scale amounts and are not interesting. We will be lead to a general theorem about supersymmetry and R symmetry breaking (Festuccia, Komargodski, and M.D.). Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
Why Retrofitting is (Almost) Inevitable Supergravity and the Cosmological Constant In supergravity theories, the low energy theory is specified by three functions, the superpotential, Kahler potential, and gauge coupling function(s). The potential takes the form V = e K ( φ,φ ∗ ) � F i g i ¯ i − 3 | W | 2 � i F ∗ (14) ¯ where F i = ∂ W + ∂ K W . (15) ∂φ i ∂φ i and g i ¯ i is the Kahler metric. Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
The generalization of the susy order parameter, ∂ W ∂φ i , of 1 globally susy theories, is F i = ∂ W + ∂ K W . ∂φ i ∂φ i If the cosmological constant is to be extremely small, 2 √ |� W �| ≈ 3 | F | M p (16) The gravitino mass is 3 m 3 / 2 = e K / 2 � W � . Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
“Supergravity" Models There is a sense in supergravity theories that a superpotential, W ∼ FM p is “natural". Suppose φ a pseudomodulus, with superpotential W = f M p g ( φ/ M p ) . (17) Polonyi model an example: W = f ( Z + α M p ) . (18) One finds Z ∼ M p , so W ∼ FM p . Michael Dine Discrete R Symmetries and Low Energy Supersymmetry
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