Can string theory make predictions for particle physics? Joint Meeting of Paci¯c Region Particle Physics Communities November 1, 2006 W. Taylor, MIT 1 Outline 1. Predictions from string theory 2. Case study: Intersecting brane models 3. Proof of ¯niteness, estimates for NG 4. Distribution of generations 5. Conclusions 2 1. Predictions from string theory Can string theory make predictions for particle physics? If we could do experiments at > 1019 GeV, answer would probably be Yes. String theory predicts (in any macroscopic space-time): |Gravity + gauge ¯elds + matter in D · 11 |SUSY (at some energy scale) But there is a big gap between Planck scale 1019 GeV LHC 104 GeV Better question today: Can string theory make predictions relevant for the LHC, or for other current particle/astro/cosmo expts.? 3 If we are lucky, may see distinctive signals of string theory physics: |KK modes (small extra dimensions) |Cosmic strings |Black holes @ LHC But we probably won't be so lucky. If not, what can we say? Even if |String theory is a correct theory of nature |We knew how to completely de¯ne it |We could calculate properties of con¯gurations to high precision The answer might practically still be no for near-term experiments. Answering the question, however, may now be within reach 4 What is string theory? ² We don't know ² Described in special limits by supergravity, perturbative strings ² Not even a perturbative description exists in the most interesting backgrounds with R-R fluxes, etc. ² No background independent description { but progress via SFT 5 String theory and 4D physics ² String theory in 10D/11D compact 6D Ã! ) 4D by "compacti¯cation" ² Can study compacti¯cations using supergravity (KK) ) 4D GR + gauge ¯elds + matter ² Calabi-Yau: 4D SUSY backgrounds ² Fluxes: |stabilize moduli |expand range of geometries 6 String landscape ² Lots of vacua ² Many CY (1?), flux combos (1) ² Previous decades: Hope for dynamical selection |no hint yet for mechanism ² Now: use to get small cc (Weinberg/eternal inflation/metaverse) 7 Metaverse Multitudes of vacua raise potential problems for predictability Q: what ¯eld theories at scales < TeV come from string compacti¯cations? (\swampland") If all, or \almost all", & vacua uniformly distributed then predictions di±cult Need À 10(# observed digits) vacua for no predictivity (necessary condition) 8 How big is the landscape? ² Early estimates » 10600 from IIB (from CY with » 200 fluxes ) 1000200) ² But in IIA, in¯nite families of highly controlled vacua [de Wolfe/Giryavets/Kachru/WT] |Maybe ¯nite with simple cuto®s though ² May be dominated by 1 numbers of \nongeometric" vacua [Shelton/WT/Wecht] 9 ² No good handle on extent ² Probably most vacua are not in known or controllable corners of ST Di±cult to argue for predictions from discretization ² Program: ¯nd \correlations in corners" Z Y X { Find family of vacua with computable EFT parameters X; Y; Z 10 ² No good handle on extent ² Probably most vacua are not in known or controllable corners of ST Di±cult to argue for predictions from discretization ² Program: ¯nd \correlations in corners" Z Y X { Find family of vacua with computable EFT parameters X; Y; Z { Find constraints/correlations { Compare with other families of vacua 11 2. Case study: Intersecting brane models M. Douglas, WT: hep-th/0606109 General Intersecting Brane Models Branes ) gauge groups, Intersections ) chiral fermions 12 Intersecting Brane Models on a toroidal orientifold ² IIA: Factorizable D6-branes with windings (ni; mi) ² IIB: Magnetized D9-branes with fluxes ² U(N) on multiple branes ² Chiral fermions on intersecting branes 3 I = (nim^ i ¡ n^imi) iY=1 13 6 Most studied orientifold: T =Z2 £ Z2 Z2 £ Z2 orbifold action (z1; z2; z3) ! (¡z1; ¡z2; z3) (z1; z2; z3) ! (z1; ¡z2; ¡z3) Orientifold zi ! z¹i ² Used to construct 3-generation models containing standard model gauge group [Cvetic/Shiu/Uranga, Cvetic/Li/Liu, Cremades/Ibanez/Marchesano] ² Moduli not stabilized but promising arena for real flux compacti¯cation [Marchesano/Shiu, . ] 14 Questions to address: ² How many intersecting brane models on ¯xed orientifold? ¯nite/in¯nite? ² Distribution of gauge group G, # generations, Yukawas, . correlations/constraints? Previous statistical analysis: [Blumenhagen/Gmeiner/Honecker/Lust/Weigand] ² Based on one-year computer search ² Suggested gauge group, # generations are fairly independent ² Suggests » 10¡9 of models have SU(3) £ SU(2) £ U(1) and 3 generations of chiral fermions 15 Outline of basic results Technical advances in hep-th/0606109: ² Analytic proof of ¯niteness ² Focus on models containing G w/ possible extra (\hidden sector") branes Physics results ² Analytic estimates for NG = # models containing G ² G, # generations essentially independent (modulo some number theoretic features, + bound on G, large G ) bounds smaller # generations) 16 3. Proof of ¯niteness, estimates for NG Finding SUSY IBM models » partition problem (Pa; Qa; Ra; Sa) = (T; T; T; T ) = (8; 8; 8; 8) Xa P = n1n2n3 Q = ¡n1m2m3 R = ¡m1n2m3 S = ¡m1m2n3 (8, 8, 8, 8) SUSY conditions (when P; Q; R; S > 0): 1 j k l 1 1 1 + + + = 0; P + Q + R + S > 0 : P Q R S j k l 17 Why proof of ¯niteness is nontrivial Can have negative tadpoles, e.g. n = (3; 1; ¡1); m = (1; 1; ¡1) ) (P; Q; R; S) = (¡3; 3; 1; 1) (8, 8, 8, 8) 3 kinds of branes (up to S4 symmetry): A: ¡ + + +, B: + + 0 0, C: + 0 0 0 18 Proof of ¯niteness: example piece of proof Take A-brane with Sa < 0, 1 1 1 Qa Ra 1 Pa + Qa + Ra + Sa = Pa + + ¡ j k l j k 1 j k Pa + Qa + Ra 2 1 1 ¸ Pa + Qa + Ra 3 µ j k ¶ using 1 1 1 3 + + > x y z x + y + z So for a general con¯guration positive tadpoles give 1 1 1 3 1 1 1 Pa + Qa + Ra + Sa · T (1 + + + ) j k l 2 j k l Xa+ Xa+ Xa+ Xa+ implies, assuming wlog 1 · j · k · l Pa · 6T : Xa+ So negative P 's at most sum to O(T ); similar arg for Q. 19 General results on scaling 1A : » (¡T 3; T; T; T ) (¡T 5; T 3; T; T ) 2A : » 9 ! (¡T 3; T; T; T ) (T 5; ¡T 3; T; T ) = ; Worst scaling » (T 5; T 3; T; T ) ) Finite number of SUSY con¯gurations Also: can estimate numbers of con¯gurations with ¯xed gauge group 20 Counting con¯gurations with gauge subgroup G ² Expect >» O(eT ) solutions to partition problem ² Look at con¯gurations given G undersaturating tadpole ) polynomial number of solutions e.g. U(N) from N A-branes (allow \hidden sector" B, C's) » (¡T 3=N 3; T=N; T=N; T=N) ¼6T 3 N » NA 64(³(3))3N 3 e.g. U(N) £ U(M) from NA + MB A » (¡T 3=N 3; T=N; T=N; T=N); B » (T 3=(N 2M); T=M) T 7 N » O( ) NA+MB N 5M 2 21 Generally, U(N) factor suppressed by 1=N º ; º ¸ 2. e.g., expect 6 7 4 Naa » (T=N) ; Nab » (T=N) ; Nbb » (T=N) ; at T = 8 [U(1) £ U(1)] Naa(8) = 30; 255 Nab(8) = 434; 775 Nbb(8) = 20; 244 at T = 4 [U(2) £ U(2) at T = 8] Naa(4) = 264 Nab(4) = 3; 029 Nbb(4) = 558 Growth as expected (contains extra logs) 22 14 12 10 8 6 ln N(T) 4 2 0 AA combinations AB combinations BB combinations -2 1 2 3 4 5 6 7 8 T (Log of) number of type AA, AB, BB branes for varying T 23 Estimates for NG ² Analytic estimates for # con¯gurations with gauge subgroup G ² E±cient algorithms to scan all valid con¯gurations ² Expect 10 NSU(3)£SU(2)£U(1) » 10 7 NSU(4)£SU(2)£SU(2) » 10 ² Each con¯guration admits » e¢T hidden sectors 24 4. Distribution of generation numbers Generations of chiral fermions between (n; m); (n^; m^ ) branes from 3 I = (nim^ i ¡ n^imi) iY=1 Look at distribution of I ² Intersection #'s for e.g. U(N) £ U(N) » 103=N 5 w/o extra A's generally expect » (T=N)7 ² Mild enhancement of composite I's ² Intersection numbers distributed quite independently ABB^ e.g. for ; T = 3; H(Iab) = H(Ia^b) = 4:553 mutual entropy 2H(Iab) ¡ H(Iab; Ia^b) » 0:085 25 T = 5 T = 6 2000 1500 1000 Appearances of intersection number I 500 0 -30 -20 -10 0 10 20 30 I Frequencies of small intersection numbers 26 Speci¯c models ² For 3-generation SU(3) £ SU(2) £ U(1); SU(4) £ SU(2) £ SU(2) expect O(10 ¡ 100) models ² Parity constraint from image branes Ixy + Ixy0 ´ 0(mod 2) ) odd generations only w/ C branes, discrete B/skew tori ² O(10) models found in previous constructions [Cvetic/Shiu/Uranga, Cvetic/Li/Liu, Cremades/Ibanez/Marchesano] ² We identi¯ed 2 additional models: 4A : n = (3; 1; ¡1); m = (1; 1; ¡1); (P; Q; R; S) = (¡3; 3; 1; 1) 2C : R = 1 2C : S = 1 + two distinct hidden sector A + ¢ ¢ ¢ combinations 27 5. Summary I ² String theory need not make predictions for particle physics below 100 TeV ² We can't de¯ne string theory yet ² The number of suspected solutions is enormous, and growing fast ² Nonetheless, constraints on low-energy physics correlated between calculable corners of the landscape may lead to predictions ² If not, probably need major conceptual breakthrough to have any possibility of predictivity for low-energy particle physics. ² Raison d'etre for string theory: quantum gravity. Remarkably, also gives new insight into gauge theory. Suggests interesting new structures for phenomenological models.