Aspects of (susy) RG flows, (s)cfts

Transcription

1 Aspects of (susy) RG flows, (s)cfts Ken Intriligator (UC San Diego) KEK Workshop Two topics: (1) based on work with Francesco Sannino and (2) with Clay Cordova and Thomas Dumitrescu

2 What is QFT? perturbation theory around free field Lagrangian theories CFTs + perturbations susy non-lagrangian higher dim l thy compactified unexplored... something crucial for the future?

3 # d.o.f. RG flows UV CFT (+relevant) chutes course graining IR CFT (+irrelevant) ladders E.g. Higgs mass E.g. dim 6 BSM ops

4 RG flows: # d.o.f. Even spacetime d CFTs conf l anom.: (Odd d: entanglment entropy or F-thm.) 2d: Zamolodchikov c-theorem. ht µ µ i ae d + X i Euler density 4d: Cardy; Osborn; Komargodski-Schwimmer: All unitary RG flows must satisfy auv > air. c i I i (Weyl) d/2 6d: Dilaton effective action considered by Sethi-Maxfield, and Elvang et. al., inconclusive for non-susy cases. (2,0) susy: Sethi-Maxfield; Elvang et al; Cordova, Dumitrescu, Yin. (1,0) susy: Cordova, Dumitrescu, KI. (modulo subtleties).

5 UV asymptotic safety? Suppose theory has too much matter, so not asymptotically free in UV. IR theory in free electric phase g IR! 0. UV safe, interacting CFT, completion? with g UV! g? (g? ) = 0? (g) e.g. 4? no (lattice) IR g?? UV g

6 UV asymptotic safety Surprise: recently found in non-susy QCD, D.F. Litim and F. Sannino 14. (g) Banks-Zaks (g) Litim-Sannino UV IR g? g? g IR UV g Theory: SU(Nc) QCD with Nf Dirac fermion flavors +Yukawa coupling to Nf 2 gauge singlet scalars, with also quartic selfinteractions. Nf >(11/2)Nc so not asymp. free. Multiple couplings, each IR free. Interacting CFT via cancellation of 1-loop and higher-loop beta fn contributions. Can be made perturbative, taking Nf just above (11/2)Nc.

7 Susy examples? Warmup: recall example of N=1 susy gauge theory with 3 adjoint matter chiral superfields, with superpotential W = y ijk Tr( i j k ) g N=4: IR y Individually IR free couplings combine to give IR-attractive, interacting SCFT. UV starting point of these RG flows? Needs a UV completion. Not UV safety.

8 susy UV asyp. safety? K.I. and F. Sannino 14. No. at least not nearby, in perturbation theory, in broad classes of non-asymptotically free theories. (g) for susy theories Via imposing IR UV=SCFT g (s) CFT unitarity constraints. } Also J.Wells & S. Martin 00. a-theorem: auv > air. applying a-maximization if needed. We were originally unaware of this excellent, early paper

9 4d N=1 SQFTs Energy-momentum tensor supermultiplet, contains U(1)R current Energy-momentum conservation. X=chiral superfield, supermultiplet of anomalies. Dimension of chiral fields ~ their running R-charges. Exact beta functions ~ linear combinations of R-charges of fields: U(1) R ABJ anomaly. W's R-violation.

10 4d N=1 SCFTs Curved background Euler Weyl 2 Anselmi, Freedman, Grisaru, Johansen. Now vary find The correct R-symmetry locally maximizes a(r). KI, B. Wecht Can use the power of t Hooft anomaly matching.

11 SQCD examples SU(Nc) SQCD with Nf Dirac flavors, non AF: Nf > 3Nc R? R(Q) =R( e Q)= N f Would violate the a-theorem: (Instead UVcompletes to asymp. free Seiberg dual.) a IR a UV? N f N c 2/3 Superconf l U(1)R determined by anomaly free + symm R * a(r) = 3TrR 3 TrR R Also, no asymp safe UV SCFT is possible for W = SQ e Q (S) = 3 2 R(S) =3N c Would violate unitarity (S) 1 for Nf > 3Nc N f See also Martin and Wells.

12 General N=1 cases No N=1 susy theories with non-asymptotically free matter content and W=0 can have a UV-safe SCFT. By a-maximization all such cases would violate the a-theorem: a(r) = 3TrR 3 a IR a UV? TrR 2/3 (W=0), R * Can satisfy a-thm only if some fields have large R charge, far from perturbative limit. Some possible examples via W terms - see Martin & Wells. Various other constraints to check. All satisfied? Do these exist? TBD.

13 Change gears Discuss some work with C. Cordova and T. Dumitrescu Study, and largely classify, possible susy-preserving deformations of SCFTs in various spacetime dims.!! Especially consider 6d susy and SCFT constraints. 6 = the maximal spacetime dim for SCFTs. A growing list of interacting, 6d SCFTs. Yield many new QFTs in lower d, via compactification.

14 susy deform s of SCFTs C. Cordova, T. Dumitrescu, KI, to appear L =Q N Q O long E.g. Kaher potentials in 4d N=1. ( L) > 1 2 N Q + min (O long ) Constrained by SCFT unitarity, bound grows with dim d. Irrelevant. E.g. for 6d N=(1,0) such operators have > = 10. L =Q n top O short ( L) = 1 2 n top + (O short ) Constrained by SCFT unitarity. Short reps already classified, in terms of the superconf l operator at the bottom of the multiplet. We study the Q descendants, looking for lorentz scalar top operators, especially searching for oddballs. Some oddball susy-preserving ops do exist, so we needed to be careful. E.g. 3d, N>3 stress-tensor has mid-level, scalar, susy invt operators: mass terms.

15 susy deformations: C. Cordova, T. Dumitrescu, KI, to appear 6d (2,0): all 16 susy preserving deformations are irrel. least irrelevant operator has dim = 12. 6d (1,0): all 8 susy preserving deformations are irrel. least irrelevant operator has dim = 10. (Also J. Luis, S. Lust.) 5d: all 8 susy preserving deformations are irrel., except for real mass terms associated with global symmetries. Etc

16 6d a-theorem? C. Cordova, T. Dumitrescu, KI Spontaneous conf l symm breaking: dilaton has derivative interactions to give a anom matching Schwimmer, Theisen; Komargodski, Schwimmer 6d case: L dilaton = 1 2 ( )2 b ( )4 + a ( )6 3 6 (schematic) Maxfield, Sethi; Elvang, Freedman, Hung, Kiermaier, Myers, Theisen. Can show that b>0 (b=0 iff free) but b s physical interpretation was unclear; no conclusive restriction on sign of. Clue: observed that, for case of (2,0) on Coulomb branch, a b 2 a

17 Longstanding hunch e.g. Harvey Minasian, Moore 98 Susy multiplet of anomalies: should be able to relate a-anomaly to R-symmetry t Hooft-type anomalies in 6d, as in 2d and 4d. T µ g µ J µ,a R A a R,µ Stress-tensor supermultiplet Sources = bkgrd SUGRA supermultiplet T T µ a? T T 4-point fn with too many indices. Hard to get a, and hard to compute. susy? J µ,a,t µ J,a,T I 8 J,a,T J,a,T Easier to isolate anomaly term, and enjoys anomaly matching

18 (1,0) t Hooft anomalies I origin 8 = 1 4! c 2 2(R)+ c 2 (R)p 1 (T )+ p 2 1(T )+ p 2 (T ) c 2 (R) tr(f SU(2) R ^ F SU(2)R ) p 1 (T ) 1 tr(r ^ R) 8 2 Background gauge fields and metric ( ~ background SUGRA) Recently computed for many (1,0) SCFTs Ohmori, Shimizu, Tachikawa; Ohmori, Shimizu, Tachikawa, Yonekura; Del Zotto, Heckman, Tomasiello, Vafa; Heckman, Morrison, Rudelius, Vafa. E.g. for theory of N small E8 instantons: E N : (,,, ) =(N(N 2 +6N + 3), N 2 (6N + 5), 7 8 N, N 2 ) Ohmori, Shimizu, Tachikawa (Leading N 3 coeff. can be anticipated from Z2 orbifold of AN-1 (2,0) case.)

19 (1,0) on tensor branch t Hooft anomaly matching requires I 8 I origin tensor branch 8 I 8 X 4 X 4 must be a perfect square, match I8 via X4 sourcing B: L GSW S = ib X 4 KI ; Ohmori, Shimizu, Tachikawa, Yonekura X (xc 2 (R)+yp 1 (T )) for some real coefficients x, y then get (,,, ) = (x 2, 2xy, y 2, 0) We show that (1,0) susy relates * ** a = b = 1 (y x) 2 b 2 (See also Chen, Huang, Wen) Adapting an old SUGRA analysis of Bergshoeff, Salam, Sezgin 86 (!).

20 General (1,0) SCFTs More tensors, e.g. N for N small E8 instantons, just iterate:! a = Theory at origin: I origin 8 = 1 4! ( x y) 2 = 16 7 ( + ) > 0 c 2 2(R)+ c 2 (R)p 1 (T )+ p 2 1(T )+ p 2 (T ) Comparing with free hyper or tensor, get relation between conformal and t Hooft anomalies in 6d SCFTs: * *constant on vacua space (no matching mechanism) a origin = 16 7 ( + )+6 7 So e.g. a(e N )= 64 7 N N Using t Hooft anomalies from 7 N Ohmori, Shimizu, Tachikawa

21 Vector multiplet issues Free vector multiplet in d>4 is a unitary SFT: scale but not conformally invt. theory. Subsector of a non-unitary CFT. El-Showk, Nakayama, Rychkov Applying a = 16 7 ( + )+6 7 to free (1,0) vector multiplet gives a(vector) = Negative. (Interestingly, same value was then found by Beccaria and Tseytlin in a non-unitary, higher derivative, conf l vector multiplet.) Get unitary, interacting 6d (1,0) SCFTs from vectors + tensors, with L kin = F 2 + specific matter to cancel gauge anomalies. Seiberg Many examples now, from string/brane/f-theory constructions. We verified for several classes of theories that aorigin >0. Thy on tensor branch, on the other hand, is a SFT and indeed some aaway <0.! Conjecture that aorigin >0, for unitary thys; seems it s a non-trivial requirement.

22 Conclude QFT is vast, expect still much to be found. susy QFTs and RG flows are rich, useful testing grounds for exploring QFT. SCFTs and RG flows are strongly constrained: unitarity, a-thm., etc. Can rule out some things. Exact results for others. Thank you!