1. Supersymmetry Breaking, lecture 3 Ken Intriligator, UCSD Asian Pacific Winter School 2007 Based on lecture notes prepared with Nathan Seiberg

2. Classical pseudomoduli Theories which break susy at tree-level generally have pseudomoduli. E.g. Goldstino partners. V fields Ms4Ms4 classical pseudomoduli space of susy breaking vacua Physically inequivalent, non-susy vacua!

3. Vacua inequivalent E.g. O'R model. Pseudomodulus X, classically massless. Mass spectrum of the other massive fields depends on the expectation value of X: (Eigenvalues of m 2 matrices) Vacua with different X are physically inequivalent. Quantum effects will lift the classical degeneracy.

4. Pseudomoduli lifted by QM One loop effective potential for pseudomodulus X. Massive scalars and fermions running in loop. Sum over all insertions of X vev. Compute using classical mass spectrum of fields in the loop.

5. Pseudomoduli lifted by QM Coleman- Weinberg potential. comments: non-susy theories also have quartic and quadratic divergent terms; vanish in susy theories:, log divergent term can be absorbed into renormalization of It's X independent.

6. Pseudomodulus lifting in O'R. Compute via Just plug in these classical eigenvalues, to get full potential. It lifts the X degeneracy.

7. Pseudomodulus lifting in O'R. V Ms4Ms4 X Quantum vacuum at origin, X=0. This is where U(1) R is unbroken.

8. Quantum vacuum of O'R. Vacuum at X=0 is stable. Breaks SUSY. Find: expand around min. Can compute. E.g.

9. Quantum O'R vacuum energy Absorb into renormalization of tree-level term:

10. O'R pseudomodulus mass

11. CW potential vs susy effective V If susy splittings are small, can also compute V eff in susy effective field theory. E.g. Integrate out massive fields, with E.g. O'R:

12. CW potential vs susy effective V E.g. O'R Get This way of computing the low-energy effective action only gives up to order |F| 2 vs. the CW V, which generally gets terms at all orders in F. The CW potential is correct. The SUSY effective V method is only valid if higher order F terms are negligible. E.g. in O'R it only reproduces potential for small y, and just leading order in y.

13. Effective V far from origin Far from the origin of pseudo-moduli space, CW potential gives approximately same V as the SUSY effective action: for Rises at large X, if X has positive anomalous dim. Yukawa coupling of X leads to positive anom. dim. No runaways: pseudomoduli V rises for large X (unless it is charged under some gauge interactions)

14. Modified O'R model Recall susy vacua, for pseudo-mod, X, large: Indeed, we should re-compute the mass matrices on the pseudo-moduli space, including the correction. Find tachyonic direction for large X. But for small X, the other fields have non-tachyonic masses and can again be integrated out, via V CW....

15. Meta-stable susy breaking in modified O'R model X Tachyonic direction to susy vacs. Meta-stable state long lived for

16. Recall how false vacua decay By tunnelling, can nucleate a bubble of true vacuum. Like boiling. Bubble expands only if it is big enough (energetically favorable volume effect vs unfavorable surface effect). False vacuum true vac shrinks expands

17. False vacua decay, cont. Decay probability ( Langer,Coleman) The "bounce action" is the Euclidean action of the tunneling trajectory. Turn potential upside down, and compute the classical action of the field config. with b.c.'s of tunneling trajectory. Large action, so long-lived metastable vac, if the barrier is high and/or wide relative to difference in vacuum energy in false vs true vacua. E.g. if barrier is low, then Our example:,.

18. Rank condition susy breaking example V (1) CW lifts the pseudo-moduli space. Up to symmetry transformations, the vacua are at No tachyonic directions: all lifted pseudo-moduli get non-tachyonic masses from the 1-loop V. There are also massless goldstone bosons, and also massless fermions (incl. goldstino).

19. Rank cond. space of vacua Actually a compact moduli space of vacua, the Goldstone boson manifold G/H. (Nontrivial topology. Admits skyrmion topological solitions.) Goldstone bosons stay massless. Can't become tachyonic directions. All other pseudomoduli get non-zero, non-tachyonic masses from 1-loop V. So we have found true local minima of V. Breaks global symms:

20. Relation to R-symmetry Nelson, Seiberg Consider condition for broken supersymmetry, no solution to: This is k conditions on k fields. For generic function W, there would be a solution. Non-R flavor symms do not help, e.g. non-R global U(1) symmetry: Now get k-1 equations for k-1 variables, again generically there would be a solution...

21. Relation to R-symmetry cont. An R-symmetry does help: Now is over-constrained: k equations for k-1 variables. So for generic function W, no solution. Conclude, for generic superpot'ls compatible with symmetries, theories without R-symmetries do not break SUSY, theories with U(1) R-symmetries do. Recall..

22. Our examples R-symmetry and broken SUSY. No R-symm and unbroken SUSY. Can have meta-stable SUSY breaking, even if from approximate R-symmetry. Can break R-symmetry. Can still have susy-breaking with as W is then non-generic.

23. Our examples, cont. Has an R-symmetry, But SUSY is still unbroken. Shows having an R-symmetry does not guarantee broken SUSY. R-symmetry and broken SUSY. No R-symmetry, and unbroken SUSY. Metastable susy breaking vacuum, with approximate R-symmetry.