The Affleck–Dine–Seiberg superpotential
SUSY QCD Symmetry SU ( N ) with F flavors where F < N SU ( N ) SU ( F ) SU ( F ) U (1) U (1) R F − N Φ, Q 1 1 F F − N Φ, Q -1 1 F Recall that the auxiliary D a fields: jn ( T a ) m D a = g (Φ ∗ jn ( T a ) m ∗ n Φ mj − Φ n Φ mj ) where j = 1 . . . F ; m, n = 1 . . . N , a = 1 . . . N 2 − 1, D -term potential: V = 1 2 D a D a
Classical Moduli Space D-flat moduli space v 1 ... v F ∗ � = � Φ � = � Φ 0 . . . 0 . . . . . . 0 . . . 0 where � Φ � is a N × F matrix generic point in the moduli space SU ( N ) → SU ( N − F ) N 2 − 1 − (( N − F ) 2 − 1) = 2 NF − F 2 of 2 NF chiral s.multiplets only F 2 singlets are massless super Higgs mechanism: vector s.multiplet “eats” a chiral s.multiplet
Light “Mesons” describe F 2 light degrees of freedom in a gauge invariant way by F × F matrix jn Φ ni M j i = Φ where we sum over the color index n M is a chiral superfield which is a product of chiral superfields, the only renormalization of M is the product of wavefunction renormaliza- tions for Φ and Φ
Chiral Symmetries axial U (1) A symmetry is explicitly broken by instantons U (1) R symmetry remains unbroken mixed anomalies between the global current and two gluons U (1) R : multiply the R -charge by the index gaugino contributes 1 · N each of the 2 F quarks contributes (( F − N ) /F − 1) · 1 2 � F − N � 1 A Rgg = N + − 1 2 2 F = 0 F U (1) A : gauginos do not contribute A Agg = 1 · 2 F · 1 2
Spurious Symmetry keep track of selection rules from the broken U (1) A define a spurious symmetry Q → e iα Q Q → e iα Q θ YM → θ YM + 2 Fα holomorphic intrinsic scale transforms as Λ b → e i 2 F α Λ b construct the effective superpotential from: W a , Λ, and M U (1) A U (1) R W a W a 0 2 Λ b 2 F 0 det M 2 F 2( F − N ) det M is only SU ( F ) × SU ( F ) invariant made out of M
Effective Wilsonian Superpotential terms have the form Λ bn ( W a W a ) m (det M ) p periodicity of θ Y M ⇒ only have powers of Λ b (for m = 1 perturbative term b ln(Λ) W a W a because of anomaly) superpotential is neutral under U (1) A and has charge 2 under U (1) R 0 = n 2 F + p 2 F 2 = 2 m + p 2( F − N ) solution is n = − p = 1 − m N − F b = 3 N − F > 0, sensible Λ → 0 limit if n ≥ 0, implies m ≤ 1(because N > F ) W a W a contains derivatives, locality requires m ≥ 0 and integer
Effective Wilsonian Superpotential only two possible terms: m = 0 and m = 1 m = 1 term is field strength term periodicity of θ YM ⇒ coefficient proportional to b ln Λ. m = 0 term is the Affleck–Dine–Seiberg (ADS) superpotential: � � 1 / ( N − F ) Λ 3 N − F W ADS ( N, F ) = C N,F det M where C N,F is renormalization scheme-dependent
Consistency of W ADS : moduli space Consider giving a large VEV, v , to one flavor SU ( N ) → SU ( N − 1) and one flavor is “eaten” by the Higgs mechanism 2 N − 1 broken generators effective theory has F − 1 flavors and 2 F − 1 gauge singlets since 2 NF − (2 N − 1) − (2 F − 1) = 2( N − 1)( F − 1) low-energy effective theory for the SU ( N − 1) gauge theory with F − 1 flavors (gauge singlets interact only through irrelevant operators) running holomorphic gauge coupling, g L � � 8 π 2 µ L ( µ ) = b L ln g 2 Λ L b L = 3( N − 1) − ( F − 1) = 3 N − F − 2 Λ L is the holomorphic intrinsic scale of the low-energy effective theory | Λ L | e iθ YM /b L = µe 2 πiτ L /b L Λ L ≡
Consistency of W ADS : moduli space low-energy coupling should match onto high-energy coupling � µ � 8 π 2 g 2 ( µ ) = b ln Λ at the scale v in DR : 8 π 2 8 π 2 g 2 ( v ) = g 2 L ( v ) � Λ � b = � Λ L � b L v v Λ 3 N − F = Λ 3 N − F − 2 v 2 N − 1 ,F − 1 subscript shows the number of colors and flavors: Λ N − 1 ,F − 1 ≡ Λ L
Consistency of W ADS : moduli space represent the light ( F − 1) 2 degrees of freedom as an ( F − 1) × ( F − 1) matrix � M det M = v 2 det � M + . . . , where . . . represents terms involving the decoupled gauge singlet fields Plugging into W ADS ( N, F ) and using � � 1 / ( N − F ) � � 1 / (( N − 1) − ( F − 1)) Λ 3 N − F Λ 3 N − F − 2 = N − 1 ,F − 1 v 2 reproduce W ADS ( N − 1 , F − 1) provided that C N,F is only a function of N − F
Consistency of W ADS : moduli space equal VEVs for all flavors SU ( N ) → SU ( N − F ) and all flavors are “eaten” from matching running couplings: � � 3( N − F ) � Λ � 3 N − F = Λ N − F, 0 v v we then have Λ 3 N − F = Λ 3( N − F ) v 2 F N − F, 0 So the effective superpotential is given by W eff = C N,F Λ 3 N − F, 0 reproduces holomorphy arguments for gaugino condensation in pure SUSY Yang-Mills
Consistency of W ADS : mass terms mass, m , for one flavor low-energy effective theory is SU ( N ) with F − 1 flavors Matching gauge couplings at m : � Λ � b = � Λ L � b L m m m Λ 3 N − F = Λ 3 N − F +1 N,F − 1 holomorphy ⇒ superpotential must have the form � � 1 / ( N − F ) Λ 3 N − F W exact = f ( t ) , det M where � � − 1 / ( N − F ) Λ 3 N − F t = mM F , F det M since mM F F is mass term in superpotential, it has U (1) A charge 0, and R -charge 2, so t has R -charge 0
Consistency of W ADS : mass terms Taking the limit Λ → 0, m → 0, must recover our previous results with the addition of a small mass term f ( t ) = C N,F + t in double limit t is arbitrary so this is the exact form � � 1 / ( N − F ) Λ 3 N − F + mM F W exact = C N,F F det M
Consistency of W ADS : mass terms F and M j equations of motion for M F F � � 1 / ( N − F ) � � cof( M F Λ 3 N − F F ) ∂W exact − 1 = C N,F + m = 0 ∂M F det M N − F det M � � 1 / ( N − F ) � � cof( M j F F ) Λ 3 N − F ∂W exact − 1 = C N,F = 0 ∂M j det M N − F det M F (where cof( M F i ) is the cofactor of the matrix element M F i ) imply that � � 1 / ( N − F ) Λ 3 N − F C N,F = mM F ( ∗ ) F N − F det M and that cof( M F i ) = 0. Thus, M has the block diagonal form � � � M 0 M = M F 0 F
Consistency of W ADS : mass terms Plugging (*) into the exact superpotential we find � � ( N − F ) / ( N − F +1) C N,F W exact ( N, F − 1) = ( N − F + 1) N − F � � 1 / ( N − F +1) Λ 3 N − F +1 × N,F − 1 det � M ∝ W ADS ( N, F − 1). For consistency, we have a recursion relation: � � ( N − F ) / ( N − F +1) C N,F C N,F − 1 = ( N − F + 1) N − F instanton calculation reliable for F = N − 1 (gauge group is completely broken), determines C N,N − 1 = 1 in the DR scheme C N,F = N − F
Consistency of W ADS : mass terms masses for all flavors Holomorphy ⇒ � � 1 / ( N − F ) Λ 3 N − F j M j + m i W exact = C N,F i det M where m i j is the quark mass matrix. Equation of motion for M � � 1 / ( N − F ) Λ 3 N − F M j ( m − 1 ) j = ( ∗∗ ) i i det M taking the determinant and plugging the result back in to (**) gives � det m Λ 3 N − F � 1 /N Φ j Φ i = M j ¯ ( m − 1 ) j = i i result involves N th root ⇒ N distinct vacua, differ by the phase of M
Consistency of W ADS : mass terms Matching the holomorphic gauge coupling at mass thresholds Λ 3 N − F det m = Λ 3 N N, 0 So W eff = N Λ 3 N, 0 reproduces holomorphy result for gaugino condensation and determines coefficient (up to phase) � λ a λ a � = − 32 π 2 e 2 πik/N Λ 3 N, 0 where k = 1 ...N . Starting with F = N − 1 flavors can derive the correct ADS effective superpotential for 0 ≤ F < N − 1, and gaugino condensa- tion for F = 0 justifies the assumption that there was a mass gap in SUSY YM
Generating W ADS from instantons Recall ADS superpotential W ADS ∝ Λ b/ ( N − F ) instanton effects are suppressed by e − S inst ∝ Λ b So for F = N − 1 it is possible that instantons can generate W ADS SU ( N ) can be completely broken allows for reliable instanton calculation
Generating W ADS from instantons With equal VEVs W ADS predicts quark masses of order ∂ 2 W ADS j ∼ Λ 2 N +1 v 2 N ∂ Φ i ∂ Φ and a vacuum energy density of order � � � � 2 2 � � � � � Λ 2 N +1 � ∂W ADS ∼ � � v 2 N − 1 ∂ Φ i
Generating W ADS from instantons single instanton vertex has 2 N gaugino legs and 2 F = 2 N − 2 quark legs 2N−2 I quark legs connected to gaugino legs by a scalar VEV, two gaugino legs converted to quark legs by the insertion of VEVs fermion mass is generated
Generating W ADS from instantons instanton calculation → quark mass e − 8 π 2 /g 2 (1 /ρ ) v 2 N ρ 2 N − 1 ∼ (Λ ρ ) b v 2 N ρ 2 N − 1 ∼ Λ 2 N +1 v 2 N ρ 4 N m ∼ dimensional analysis works because integration over ρ dominated by ρ 2 = b 16 π 2 v 2 quark legs ending at the same spacetime point gives F component of M , and vacuum energy of the right size can derive the ADS superpotential for smaller values of F from F = N − 1, so we can derive gaugino condensation for zero flavors from the instanton calculation with N − 1 flavors
Recommend
More recommend
