Supersymmetry Breaking,lecture 3

Ken Intriligator, UCSD

Asian Pacific Winter School 2007

Based on lecture notes prepared with Nathan Seiberg

Classical pseudomoduli

Theories which break susy at tree-level generallyhave pseudomoduli. E.g. Goldstino partners.

V

fields

Ms4

classical pseudomodulispace of susy breaking vacua

Physically inequivalent, non-susy vacua!

Vacua inequivalent

E.g. O'R model. Pseudomodulus X, classicallymassless. Mass spectrum of the other massivefields depends on the expectation value of X:

(Eigenvalues of m2

matrices)

Vacua with different X are physically inequivalent.Quantum effects will lift the classical degeneracy.

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.

Pseudomoduli lifted by QMColeman-Weinbergpotential.

comments: non-susy theories also have quarticand quadratic divergent terms; vanish in susy theories: ,log divergent term can be absorbed into renormalization of It's X independent.

Pseudomodulus lifting in O'R.

Compute via

Just plug in these classical eigenvalues, to get full potential . It lifts the X degeneracy.

Pseudomodulus lifting in O'R.

V

Ms4

X

Quantum vacuum at origin, X=0. This is where U(1)R is unbroken.

Quantum vacuum of O'R.

Vacuum at X=0 is stable. Breaks SUSY. Find:

expand around min.

Can compute. E.g.

Quantum O'R vacuum energy

Absorb into renormalization of tree-level term:

O'R pseudomodulus mass

CW potential vs susy effective VIf susy splittings are small, can also compute Veff

in susy effective field theory. E.g. Integrate outmassive fields, with

E.g. O'R:

CW potential vs susy effective V

E.g. O'R

Get

This way of computing the low-energy effectiveaction only gives up to order |F|2 vs. the CW V,which generally gets terms at all orders in F. TheCW potential is correct. The SUSY effective Vmethod is only valid if higher order F terms arenegligible. E.g. in O'R it only reproduces potentialfor small y, and just leading order in y.

Effective V far from origin

Far from the origin of pseudo-moduli space, CWpotential gives approximately same V as the SUSYeffective 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)

Modified O'R model

Recall susy vacua, for pseudo-mod, X, large:

Indeed, we should re-compute the mass matriceson the pseudo-moduli space, including thecorrection. Find tachyonic direction for large X. But for small X, the other fields have non-tachyonicmasses and can again be integrated out, via VCW....

Meta-stable susy breaking in modified O'R model

X

Tachyonicdirection tosusy vacs.

Meta-stable state long lived for

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 vacshrinks

expands

False vacua decay, cont.

Decay probability (Langer,Coleman)

The "bounce action" is the Euclidean action of thetunneling trajectory. Turn potential upside down, and compute the classical action of the field config.with b.c.'s of tunneling trajectory. Large action, solong-lived metastable vac, if the barrier is high and/orwide relative to difference in vacuum energy in falsevs true vacua. E.g. if barrier is low, then

Our example: , .

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-moduliget non-tachyonic masses from the 1-loop V.There are also massless goldstone bosons,and also massless fermions (incl. goldstino).

Rank cond. space of vacua

Actually a compact moduli space of vacua, theGoldstone boson manifold G/H. (Nontrivial topology. Admits skyrmion topological solitions.)Goldstone bosons stay massless. Can't becometachyonic directions. All other pseudomoduli getnon-zero, non-tachyonic masses from 1-loop V. So we have found true local minima of V.

Breaks global symms:

Relation to R-symmetry Nelson, Seiberg

Consider condition for broken supersymmetry,

no solution to:

This is k conditions on k fields. For generic functionW, there would be a solution. Non-R flavor symmsdo not help, e.g. non-R global U(1) symmetry:

Now get k-1 equations for k-1 variables, againgenerically there would be a solution...

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..

Our examples

R-symmetry and broken SUSY.

No R-symm and unbroken SUSY.

Can have meta-stable SUSYbreaking, even if from approximate R-symmetry.

Can break R-symmetry. Canstill have susy-breaking with as W is then non-generic.

Our examples, cont.

Has an R-symmetry,

But SUSY is still unbroken. Shows having anR-symmetry does not guarantee broken SUSY.

R-symmetry and broken SUSY.

No R-symmetry, and unbroken SUSY.Metastable susy breaking vacuum, withapproximate R-symmetry.