Nonperturbative Study of Supersymmetric Gauge Field Theories

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1 Nonperturbative Study of Supersymmetric Gauge Field Theories Matteo Siccardi Tutor: Prof. Kensuke Yoshida Sapienza Università di Roma Facoltà di Scienze Matematiche, Fisiche e Naturali Dipartimento di Fisica [handout]

2 Gauge theories Study of gauge theories is important: Strong, weak and electromagnetic interactions are believed to be mediated by gauge (vector) bosons. Gauge invariance has become the guiding principle. Clay Millennium prize: Yang-Mills Existence and Mass Gap: Prove that for any compact simple gauge group G, quantum Yang-Mills theory of 4 exists and has a mass gap > 0. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

3 Gauge theories However, study of non-abelian gauge theories is difficult: at low energy the gauge coupling becomes arbitrary strong and quantum perturbation theory breaks down. At some energy scale Λ, characterising the gauge theory, the gauge coupling becomes infinite. The fundamental gauge fields strongly fluctuate and are no longer appropriate degrees of freedom to describe the theory. What happens at energies below the scale Λ has never been derived from first principles. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

4 Supersymmetry? Q boson = fermion Q fermion = boson An equal number of bosonic and fermionic d.o.f. Most general symmetry of the S-matrix (Coleman-Mandula)... Supersymmetry: a regularization! Adding a suitable number of additional fields with the appropriate coefficients helps the convergence of perturbation series. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

5 N = 4 SYM model: a peculiar gauge theory Gauge group: (S)U(N) Coupling constant: g YM (possibly, also a ϑ-angle) Field content: A A µ, λ A αa, X A i a = 1,... 4, i = 1,... 6, A: adjoint colour indices L 1 g 2 YM { tr Fµν 2 i λ A /Dλ A (D µ X i ) C AB i λ A [X i, λ B ] + h. c. + [X i, X j ] 2} (Dipartimento di Fisica) Seminario II anno di dottorato / 18

6 N = 4 SYM model: a peculiar gauge theory Gauge group: (S)U(N) Coupling constant: g YM (possibly, also a ϑ-angle) Field content: A A µ, λ A αa, X A i a = 1,... 4, i = 1,... 6, A: adjoint colour indices L d 2 θ2πıτ 0 tr W α W α + d 2 θd 2 θ 3 Φ i e V Φ i + i=1 + d 2 θw (Φ) + h. c. where W α A µ, λ α1 ; Φ λ α2,3,4, X i ; τ 0 = 4πı and W (Φ) = Φ 1 [Φ 2, Φ 3 ]. g 2 YM + ϑ 2π (Dipartimento di Fisica) Seminario II anno di dottorato / 18

7 N = 4 SYM model: a peculiar gauge theory Highly (super-)symmetric: Maximal amount of supersymmetry in d = 4 (without SUGRA) Conformal symmetry! and others Non-chiral (and massless) matter No perturbative S-matrix it is FINITE at all orders! One-loop β-function: β(g YM ) [ 11 3 C(adj) 2 3 C(λ) 1 6 λ X C(X) ] = N [ ] = 0... and so it is useless! (Dipartimento di Fisica) Seminario II anno di dottorato / 18

8 Towards reality: deformations of N = 4 model N = 1 model: a mass (soft) deformation: + 3 m i Φ 2 i. i=1 mi 0: N = 4 SYM m2,3, m 1 0: N = 2 SYM mi : N = 1 SYM β-deformation: a (complex) marginal deformation of the superpotential: W (Φ) tr Φ 1 [Φ 2, Φ 3 ] β tr [ e ıβ/2 Φ 1 Φ 2 Φ 3 e ıβ/2 ] Φ 1 Φ 3 Φ 2. Preserves N = 1 supersymmetry. Symmetry breaking superpotential: + W (Φ 1 ) = n+1 p=1 g p p Φp 1. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

9 Standard lore Prototype: U(N) N = 2 SYM + W tree (Φ). Gauge: A µ, λ α }{{} W α Matter: φ, ψ α }{{} Φ At classical level: W tree (Φ) = n k=1 g k k + 1 Φk+1 U(N) n U(N i ) i=1 At quantum level, in each SU(N i ) U(N i ): Gaugino condensation: tr λ α λ α S tr W α W α develops a vev. The gauge coupling grows at a dynamically generated scale Λ where SU(N i ) decouple and a mass gap is generated. the low energy gauge group is U(1) n. The photons are IR-free since they reside in the adjoint representation. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

10 Standard lore Prototype: U(N) N = 2 SYM + W tree (Φ). Gauge: A µ, λ α }{{} W α Matter: φ, ψ α }{{} Φ The physical information in the low energy regime is represented by The gaugino condensate: λλ Λ 3, The tension of domain walls, The coupling for the U(1) n fields: τ ij. It is encoded in an effective (Wilsonian) Lagrangian: L eff = d 2 θ2πı τ i S i + d 2 θw pert (N i, w iα, S i, g k ) + h. c. + d 2 θd 2 θ... (Dipartimento di Fisica) Seminario II anno di dottorato / 18

11 The Matrix Model Dijkgraaf and Vafa formulated a suggestive correspondence between 0-dimensional bosonic matrix models and N = 1 supersymmetric gauge theories: N = 1 gauge theory Matrix Model n+1 n+1 g p g p W tree (Φ) = p Φp W (ˆΦ) = p ˆΦ p p=1 p=1 Φ : chiral superfield ˆΦ : ˆN ˆN matrices The superpotential acts as action for the random matrices: Z MM = C ˆN dˆφ exp ˆN g m tr W (ˆΦ) (Dipartimento di Fisica) Seminario II anno di dottorato / 18

12 The Matrix Model - 2 The Matrix Model admits a t Hooft large- ˆN expansion, Z = exp g 0 [g mˆn ] 2g 2Fg ˆNi (S) S i lim g m ˆN ˆN The leading contribution is the planar (g = 0) one. The D-V prescription is: W pert (S i, w α i ; g k ) = i N i F 0 S i i,j τ ij = 2 F 0 S i S j δ ij 1 N i l 2 F 0 S i S j w α i w αj N l 2 F 0 S i S l Perturbative expansion of the (planar) free-energy: F 0 (S) = c i1,...i n S i 1 1 Si n n S Λ 3 (!) i 1,...,i n 0 A Perturbative Window into Non-Perturbative Physics (Dipartimento di Fisica) Seminario II anno di dottorato / 18

13 Determination of C ˆN Consider the N = 1 model in the M.M. setup: S N =1 = ˆN tr {ˆΦ1 [ˆΦ 2, g ˆΦ 3 ] + m 1 m 2 ˆΦ m 2 2 ˆΦ m 3 2 ˆΦ 2 } 3 Z N =1 = e ˆN 2 g m 2 F N =1 = C ˆN dˆφ 1 dˆφ 2 dˆφ 3 e S N =1 and remember the properties of N = 1 and N = 4: lim F N =1 = F N =4 = πıτ 0S 2 m i N C ˆN = ( 3 ˆN ) (2π) 3 e 3/2 gm 2 e πıτ 0 ˆN 2 /N H. Kawai, T. Kuroki, T. Morita, and K. Yoshida, Phys. Lett. B611: (2005). (Dipartimento di Fisica) Seminario II anno di dottorato / 18

14 Validity The M.M. has been tested against a great number of highly non-trivial checks. In particular, it can reproduce: Veneziano-Yankielowicz superpotential for N = 1 SYM, Finite mass effects in N = 1, Instanton series in N = 2 SYM, Seiberg-Witten curve,... cfr. S. Arnone, G. Di Segni, MS, and K. Yoshida, Int. J. Mod. Phys. A22: (2007). (Dipartimento di Fisica) Seminario II anno di dottorato / 18

15 One-loop computation in N = 4 β-deformed The model: a linear combination of deformations of the N = 4 model: Z(β) = e ˆN 2 g m 2 F MM = C ˆN 3 i=1 dˆφ i exp ˆN g m tr { µ 2 ˆΦ M 0 2 (ˆΦ ˆΦ 2 3)+ + W (ˆΦ 1 ) + h 0 ˆΦ1 [ˆΦ 2, ˆΦ 3 ] } We are interested in the case µ 0, 0 M 0 <. Computation proceeds along the following steps: Integration over ˆΦ 2 and ˆΦ 3, Diagonalization of the remaining matrix, ˆΦ, (Ad hoc change of variables), Perturbative expansion around the potential extrema. U(2) U(1) 2 for simplicity. (Dipartimento di Fisica) Seminario II anno di dottorato / 18

16 One-loop computation in N = 4 β-deformed - 2 The outcome is: Z (β) = ( ˆN ) e πıτ 0 ˆN 2 N ˆN 1 [ g 0 ] ˆN h ˆN 1 i=1 dp i ˆN 2 k=1 dq k ˆN 1 i<j sinh α (p i p j ) sinh 1 2 [ 1 α (p i p j ) + ıβ](β β) ˆN 1, ˆN 2 i,k=1 sinh [ 1 α (p i ıq k ) + ] sinh 1 2 [ 1 α (p i ıq k ) + + ıβ](β β) ˆN 2 k<l sin α (q k q l ) sin 1 2 [ 1 α (q k q l ) + β](β β) exp { 1 ˆN1 2 [ i ˆN2 p 2 i + k q 2 k ] ˆN 1 ˆN2 α [ p 3 i + (ıq k ) 3 ]} i k where = φ 1 φ 2 = ln M 0 φ 2h 0 sin β/2 1 M 0 φ 2h 0 sin β/2 2 M 0 0 ln φ 1 φ 2 and 1 α = g 2 m a ˆN (small) Perturbation series in terms of 1 α (Dipartimento di Fisica) Seminario II anno di dottorato / 18

17 One-loop computation in N = 4 β-deformed - 3 The lowest order in 1 α gives the one-loop contribution to F 0 : F (1) 0 lim ˆN gm 2 ˆN ln ( sinh 2 /2 2 sinh( + ıβ/2) sinh( ıβ/2) M 0 0 S 1 S 2 ln ( (φ 1 φ 2 ) 2 (e ıβ/2 φ 1 e ıβ/2 φ 2 )(β β) ) ) ˆN1 ˆN2 from which we can extract the one-loop contribution to the gauge coupling with ( ) τ (1) ij = τ (1) τ (1) = 2 F (1) 0 ln [ (φ 1 φ 2 ) 2 ] S 1 S 2 (e ıβ/2 φ 1 e ıβ/2 φ 2 )(β β) (Dipartimento di Fisica) Seminario II anno di dottorato / 18

18 Conclusions The Matrix Model is a powerful tool in analyzing a wide class of N = 1 gauge theories. It can give insight on the elusive non-perturbative region! What s next: Instanton corrections (beyond 1-loop). Gravitational correction (non-planar graphs). Other (less SUSY) models? (Dipartimento di Fisica) Seminario II anno di dottorato / 18