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 fixed point for 3 2 N < F < 3N. Seiberg duality: nontrivial weakly coupled variables with same IR behavior (in the same universality class) as a strongly coupled gauge theory. Familar phases in unfamilar settings: the free electric phase (like QED) and free magnetic phase (logarithmic running of coupling opposite that of QED). Quantum corrections to moduli space for F = N and F = N + 1. Reading: Terning 10.3-10.6

SUSY QCD Matter: each of the F flavors is a fundamental (Φ i, Q i ) and an antifundamental (Φ i, Q i ). Symmetries: SU(N) SU(F ) SU(F ) U(1) U(1) R Φ, Q 1 1 F N F Φ, Q 1-1 F N F

Classical Moduli Space for F N VEV for a single flavor: SU(N) SU(N 1) Generic point in the moduli space (VEV for all flavors): SU(N) completely broken. Parametrization of VEVs Φ = v 1 0... 0..... v N 0... 0, Φ = v 1... v N 0... 0.. 0... 0. Vacua are physically distinct: different VEVs correspond to different masses for the gauge bosons.

Gauge-invariant description mesons and baryons M j i = Φ jn Φ ni, B i1...i N = Φ n1 i 1... Φ nn i N ɛ n 1...n N, B i 1...i N = Φ n 1 i 1... Φ n N i N ɛn1...n N. Definitions extend to superfields so superpartners to mesons and baryons are described as well. ( ) F Constraints needed: M has F 2 components, B and B each have N components; all constructed out of the same 2N F underlying fields. Classical constraints: B i1...i N B j 1...j N = M j 1 [i 1... M j N i N ]. where [ ] denotes antisymmetrization.

Classical Moduli Space for F N Up to flavor transformations, the nonvanishing components are: v 1 v 1... M = v N v N 0, B 1...N = v 1... v N, B 1...N = v 1... v N.... 0 Rank M N. If strictly less than N, then B or B (or both) vanish. If the rank of M is k, then SU(N) is broken to SU(N k) with F k massless flavors. k = N complete breaking.

IR Fixed Points The main subject of the lecture: the low energy effective theory of QCD with F N. F 3N: asymptotic freedom lost. Low-energy effective theory flows to weak coupling. A key claim: for 3 2N < F < 3N there is a non-trivial IR fixed point of the theory. For F just below 3N the IR fixed point is at weak coupling. For smaller F, the fixed point coupling grows. For F just above 3 2N, the fixed point coupling is very strong.

The Banks-Zaks Fixed Point A perturbative treatment: for F just below 3N, the large N limit is under control (the Banks-Zaks fixed point). Exact NSVZ β function: β(g) = g3 16π 2 (3N F (1 γ)) 1 Ng 2 /8π 2. Perturbative expressions for γ (the anomalous dimension of the quark mass term) and the β-function: γ = g2 8π 2 N 2 1 N + O(g4 ), 16π 2 β(g) = g 3 (3N F ) g5 8π 2 ( 3N 2 2F N + F N ) + O(g 7 ).

The Banks-Zaks Fixed Point Take large N with F = 3N ɛn (ɛ fixed in the limit): 16π 2 β(g) = g 3 ɛn g5 8π 2 ( 3(N 2 1) + O(ɛ) ) + O(g 7 ). So: there is a scale invariant fixed point (β = 0) where the first two terms cancel: g 2 = 8π2 3 N N 2 1 ɛ. O(g 7 ) terms are higher order in ɛ, so this limit is under control. Conclusion: without masses, the gauge theory for 3F near N is scaleinvariant for g = g (at large N). This theory has a nontrivial IR fixed point. Physics terminology: this is the non-abelian Coulomb Phase. (The potential is exactly V (R) R 1 due to scale invariance, even though this is an interacting theory).

IR Fixed Points Goal: extend understanding for large N and near weak coupling to all 3 2 < N < 3F. Key ingredients: Any scale-invariant theory of fields with spin 1 is conformally invariant. SUSY + conformal algebra superconformal algebra The R-charge that enters the superconformal algebra is anomaly free, but generally not unique. We denote that R-charge by R sc. Dimensions of the scalar component of gauge-invariant chiral and antichiral superfields are determined by the superconformal algebra: d = 3 2 R sc, for chiral superfields, d = 3 2 R sc, for antichiral superfields.

Chiral Ring Charge of a product of fields is the sum of the individual charges: R sc [O 1 O 2 ] = R sc [O 1 ] + R sc [O 2 ], so for chiral superfields dimensions simply add: D[O 1 O 2 ] = D[O 1 ] + D[O 2 ]. Reminder: generally a product of fields is renormalized in addition to the renormalizations of the individual fields, and this contributes to the anomalous dimension of the composite operator. Terminology: the chiral operators form a chiral ring. Definition of ring: set of elements on which addition and multiplication are defined, with a zero and an a minus operation.

Fixed Point Dimensions The R-symmetry of a SUSY gauge theory is generally ambiguous since we can always form linear combinations with non-anomalous U(1) s leaving the superpotential invariant. For the fixed point of SUSY QCD, R sc is unique since we must have R sc [Q] = R sc [Q]. Our convention: denote the anomalous dimension at the fixed point of the mass operator at the fixed point by γ. So D[M] = D[ΦΦ] = 2 + γ = 3 2 2 F N F = 3 3N F γ = 1 3N F. Consistency check (or alternative derivation): the exact β-function vanishes at the fixed point β 3N F (1 γ ) = 0.

Fixed Point Dimensions For a scalar field in a conformal theory we also have D(φ) 1, with equality exactly when the field is free. Apply to the meson field: D[M] 1 F 3 2 N. Conclusion: the IR fixed point (non-abelian Coulomb phase) is an interacting conformal theory for 3 2 N < F < 3N. Comment: this class of QFTs have no particle interpretation. But anomalous dimensions are physical quantities that can be measured, in principle.

Seiberg Duality (Incorrect) proposition: meson and baryon operators (M, B, and B) describe SUSY QCD at low energy even for F N. Invalidate proposition: Conformal theory: global symmetries unbroken in the low energy theory. t Hooft anomaly matching: anomalies of each global symmetry agrees between low and high energy degrees of freedom. Proposed low energy spectrum does not match anomalies of quarks and gaugino.

Correct proposition: there is an alternate description that invokes a dual SU(F N) gauge theory with a dual gaugino, dual quarks and a gauge singlet dual mesino. Symmetries of dual theory: SU(F N) SU(F ) SU(F ) U(1) U(1) R N N q 1 F N F q 1 N N F N F mesino 1 0 2 F N F Consistency check: this proposal satisfies anomaly matching. This nontrivial change of variables is called Seiberg duality.

Anomaly Matching global symmetry SU(F ) 3 U(1)SU(F ) 2 dual anomaly = anomaly (F N) + F = N N U(1) R SU(F ) 2 U(1) 3 0 = 0 U(1) 0 = 0 U(1)U(1) 2 R 0 = 0 U(1) R F N (F N) 1 2 = N 2 N F F (F N) 1 2 + F 2N F F 1 2 = N 2 2F ( N F F = N 2 1 ( N F F U(1) 3 R = 2N 4 ( U(1) 2 N U(1) R F N ) ( 2(F N)F + F 2N ) F 2 + (F N) 2 1 ) 3 ( 2(F N)F + F 2N F F + N 2 1 2 ) 2 N F F F 2F (F N) = 2N 2 ) 3 F 2 + (F N) 2 1

Mapping of Operators There is a simple map between the moduli spaces of the original and the dual theories: M M, B i1...i N ɛ i1...i N j 1...j F N b j 1...j F N, B i 1...i N ɛ i 1...i N j 1...j F N b j1...j F N, where the dual baryon operators are formed from the dual quarks: b i 1...i F N = φ n 1i 1... φ n F N i F N ɛ n1...n F N, b i1...i F N = φ n1 i 1... φ nf N i F N ɛ n 1...n F N. Notation: φ represents the dual squark and M is the dual meson.

Dual β-function The β-function of the dual theory: β( g) g 3 (3Ñ F ) = g3 (2F 3N). Observation: the dual theory loses asymptotic freedom when F 3N/2. Strength of fixed point coupling: as F decreases from the top of the interval 3 2N < F < 3N, the original theory becomes stronger coupled. The dual theory becomes weaker coupled as F is lowered: when it reaches 3 2N = F it leaves the the conformal regime altogether, and becomes IR free.

Dual Banks Zaks We can understand the behavior for F just above 3 2N by taking large N (and so large F ) in the dual theory with fixed ɛ: F = 3Ñ ɛñ = 3 2 ( 1 + ɛ 6) N. There is a perturbative fixed point at g 2 = 32π2 3Ñ ɛ, λ 2 = 16π2 3Ñ ɛ. Conclusion: there really is an IR fixed point all the way down to 3 2 N = F, at least for large N.

The Dual Superpotential Symmetries of the dual theory are consistent with the superpotential W = λ M j i φ jφ i. This superpotential is mandatory, it must be turned on. Counting degrees of freedom: the M equation of motion removes the color singlet φφ. Only then do the two theories have the same number of degrees of freedom.

An RG Flow Suppose that, near the dual Banks-Zaks fixed point, we fix the coupling g = g but take λ = 0 (instead of λ = λ ). Observation: λ = 0 means the meson M is a free scalar field with dimension 1, since it is a gauge scalar. Assuming that there still is a superconformal fixed point, the dimension of the squark mass operator is determined by the R charge D(φφ) = 3(F Ñ) F = 3N F < 2. RG-flow: the superpotential W = λ Mφφ has dimension D(W ) < 3 so it is a relevant operator. Thus a small coupling λ will grow larger, towards λ. Conclusion: the fixed point at g = g, λ = 0 is unstable. It flows towards the IR fixed point at g = g, λ = λ. So the superpotential is mandatory.

Towards F 3 2 N Reminder: the dimension of the mass operator φφ calculated from the R sc charge for 3N/2 < F < 3N: D(φφ) = 3(F Ñ) F = 3N F (notation unconventional but obvious). (2, 1). Since D( Mφφ) = 3 throughout D( M) = 3(1 N F ) (1, 2). Conclusion: the dual M (a two scalar composite in the original theory) becomes a free scalar field at F 3 2 N.

The Free Magnetic Phase SUSY QCD has an interacting IR fixed point for any 3N/2 < F < 3N. The dual description also has an IR fixed point in this region. For F 3N/2 asymptotic freedom is lost in the dual, so the low energy theory flows to weak coupling: g 2 = 0, λ 2 = 0. For N + 2 F 3 2N the IR is thus a theory of free massless composite gauge bosons, quarks, mesons, and superpartners. This is the free magnetic phase. The special cases F = N and F = N + 1 require new considerations because in these cases there is no dual gauge group SU(F N) and so no dual magnetic description.

Consistency: Integrating out a Flavor Give a mass to one flavor In the dual theory W mass = mφ F Φ F. W d = 1 µ M j i φi φ j + mm F F. Remark: we trade the coupling λ for a scale µ and use the same symbol, M, for fields in the two different theories (This is common practice). λ M = M µ.

Integrating Out a Flavor The equation of motion for M F F is: W d M F F = 1 µ φf φ F + m = 0. Dual squarks have VEVs: φ F φ F = µm. Spectrum: due to the VEV, we have a theory with one less color, one less flavor, and some singlets.

Integrating Out a Flavor SU(F N 1) SU(F 1) SU(F 1) q 1 q 1 M 1 q 1 1 q 1 1 S 1 1 1 Mj F 1 1 M j F 1 1 MF F 1 1 1 Superpotential after symmetry breaking ( ) W eff = 1 µ φ F M j F φ j + φ F Mi F φ i + MF F S + 1 µ M φ φ. Low energy theory: integrate out M j F, φ j, M F i, φ i, M F F, and S.

Result: leaves just the dual of SU(N) with F 1 flavors and a superpotential W = 1 µ M φ φ. Consistency: this is the result if we had just started with one flavor less in the first place.

Consistency Check: Moduli Space The dual description has the same IR behavior as the original theory so, in particular, the dual theory must reproduce the dimension of the moduli space. Complex dimension of moduli space in original theory: 2F chiral multiplets in N or N, modulo N 2 1 complex D-term constraints: 2F N (N 2 1). Classical (incorrect) moduli space in the dual theory: parametrized by the meson VEVs in the F F matrix M since the dual squark VEVs φ, φ vanish due to the M equation of motion from the superpotential W = λ M j i φ jφ i. The complex dimension F 2. Conclusion: the consistency check fails (for now: quantum corrections are important.)

Quantum Moduli Space of Dual Classical reasoning: the superpotential W = λ M j i φ jφ i apparently renders all quarks/squarks massive. So the dual theory is pure SU(F N) SUSY YM, with moduli space parametrized by the F F matrix M j i. The problem: low energy theory is pure SU(F N) SYM with no flavors nonperturbative quantum corrections generate an ADS superpotential there is no quantum vacuum (run-away to large M). Improved analysis: at least F N light quarks/squarks must remain in the SU(F N) theory to avoid the ADS superpotential. Remark: in the original (weakly coupled SU(N)) theory, rank(m) N due to the classical equation of motion. In the dual theory (strongly coupled SU(F N) theory), rank( M) N due to nonperturbative quantum effects.

Dimension of Moduli Space: Recount Squark moduli from 2(F N) squarks in F N (resp. F N) of the SU(F N) gauge group, with D-term constraints enforced: 2(F N)(F N) ( (F N) 2 1 ) = (F N) 2 + 1. Meson moduli: a rank(n), F F matrix can be written with an (F N) (F N) block set to zero so, Equation of motion for M: F 2 (F N) 2. (F N) 2. Upshot: the dimensions of the two moduli spaces match once nonperturbative effects are taken into account, F 2 (F N) 2 + 1 = 2NF N 2 + 1.