1. Fields and strings in the era of post LHC GCOE Symposium Feb. 12. 2013 Kyoto University Hikaru Kawai

2. LHC gave beautiful results In some sense, they indicate “the worst scenario”. Higgs particle was discovered, but nothing else. Especially, no sign of the SUSY. We need to reconsider the meaning of SUSY.

3. Why did some people like SUSY? The only scientific reason is that it was thought to solve the naturalness problem. Suppose the underlying fundamental theory, such as string theory, has the momentum scale m S and the coupling constant g S. naturalness problem Then, by dimensional analysis and the power counting of the couplings, the parameters of the low energy effective theory are expected as follows:

4. dimension 2 (Higgs mass) dimension 4 (vacuum energy or cosmological constant) naturalness problem (cont.’d) dimension 0 (gauge and Higgs couplings) dimension -2 (Newton constant) unnatural ! → unnatural ! !→

5. SUSY is the symmetry between bosons and fermions Bosons and fermions cancel the UV divergences: However, SUSY must be spontaneously broken at some momentum scale M SUSY, because such symmetry does not exists at the low energy scale. Below the momentum scale, cancellation does not work. + bosons fermions ⇒ + ⇒ ⇒ ⇒

6. Possibility of SUSY as the solution to the naturalness problem Therefore, if M SUSY is close to m H, the Higgs mass is naturally understood, although the cosmological constant is still a big problem. Actually when the Z- and W- bosons were discovered in 1983, there was a strong motivation to expect that M SUSY is close to the weak scale, and the Higgs mass is protected by SUSY. But it turned out not the case. No signal of SUSY was observed around the weak scale. History

7. Present status of SUSY After 30 years the Higgs particle was finally discovered at m H ~ 125 GeV. Furthermore, no signal of new physics is observed below 1 TeV, which indicates that M SUSY >1 TeV, if it ever exists. The inequality suggests that we need to fine tune the Higgs mass parameter at least by factor 100. In other words, the probability that the Higgs mass is naturally explained by SUSY is less than 1%.

8. Possible solutions to the naturalness problem It seems that we would better peruse the solutions other than SUSY. 1.We do not have to mind. We should simply take them as they are. 2. Anthropic principle. The parameters should be such that we can exist. a)The wave function of the universe is a superposition of various worlds having different low energy effective Lagrangians: We are in one world described by one of them, whose parameters must be such that we exist.

9. Anthropic principle. (cont.’d) b)The universe has different parameters place by place. We are sitting at one place, where the parameters are such that we can exist. They are not totally nonsense, but it is difficult to judge what is correct. 3. The parameters are fixed by some non- perturbative effect of quantum gravity/string theory such as Coleman’s baby universe mechanism.

10. Simply accept the observed parameters and analyze to what energy SM is valid Most particle physicists believe that the SM is the low energy effective theory of string theory. Then the question is whether SM or its minor modification is valid to the string scale m S, or completely new physics appears before reaching it. If it is the former case, we have a “desert” in the sense that we have nothing other than the SM physics below m S. In order to examine it, we consider the SM Lagrangian with cutoff momentum Λ, and evaluate its bare parameters. If no inconsistency appears, it means that SM can be valid to Λ. [with Y. Hamada and K. Oda]

11. Bare parameters of the cutoff theory (1) m Higgs =126GeV m top =172 GeV Ytop U(1) SU(2) SU(3) Higgs self coupling Higgs mass 2 log 10 Λ[GeV} No inconsistency arises below the string scale.

12. Bare parameters of the cutoff theory (2) Ytop U(1) SU(2) SU(3) Higgs self coupling Higgs mass 2 log 10 Λ[GeV} Higgs field becomes unstable. m Higgs =126GeV m top =190 GeV

13. Bare parameters of the cutoff theory (3) m Higgs =126GeV m top =150 GeV Ytop U(1) SU(2) SU(3) Higgs self coupling Higgs mass 2 log 10 Λ[GeV} No inconsistency arises, but the Higgs self coupling tends to diverge.

14. Bare parameters of the cutoff theory (4) Ytop U(1) SU(2) SU(3) Higgs self coupling Higgs mass 2 log 10 Λ[GeV} Higgs field becomes unstable. m Higgs =100GeV m top =172 GeV

15. Bare parameters of the cutoff theory (5) m Higgs =150GeV m top =172 GeV Ytop U(1) SU(2) SU(3) Higgs self coupling Higgs mass 2 log 10 Λ[GeV} No inconsistency arises, but the Higgs self coupling tends to diverge.

16. It is possible that the SM is valid to the string scale. In other words, desert seems probable. The experimental value of the Higgs mass seems to be just on the stability bound. Nature seems to like the marginal value. The bare Higgs mass becomes close to zero at the string scale. It implies that SUSY is restored at the string scale. Actually there are many string vacua in which SUSY is spontaneously broken at the string scale. The Higgs self coupling also becomes close to zero at the string scale. It indicates that the Higgs potential becomes almost flat around the string scale, which opens the possibility that the Higgs field plays the roll of inflaton. Summary of the bare parameters

17. Higgs field as inflaton (1) If we allow a fine tuning of the parameters of the SM, the Higgs field can play the role of inflaton. The effective potential of the Higgs field is given by Here the first term on the RHS is determined by the low energy renormalizable theory, that is the standard model, and the other terms are so called Plank suppressed terms that depend on the underlying microscopic theory. c is a constant of order the coupling constants: c ~ 0.1. [with Y. Hamada and K. Oda]

18. Higgs field as inflaton (2) In the case of string theory, we have and typically As we have seen, in some parameter region the Higgs coupling λ(Λ) takes minimum near the Plank scale, whose value is slightly negative. m Higgs =126GeV m top =172 GeV log 10 Λ[GeV }

19. Higgs field as inflaton (3) Then the first term of V eff looks as follows. Therefore if we add the second term, we can obtain a saddle point by tuning one parameter.

20. Higgs field as inflaton (4) Because the logarithmical change of λ in the first term is slow compared to the power behavior of the second term, the saddle point arises near the left zero of λ(Λ), and we can approximate it as Furthermore, at the saddle point Φ is smaller than the Planck scale by about factor 100 so that the higher order Planck suppressed terms are negligible.

21. Higgs field as inflaton (5) Therefore it suffices to consider Then it is easy to see that this function has a saddle point if at and the value there is given by

22. There is a difficulty. From the density perturbation of the early universe, it is desirable to have which indicatesand The former can be satisfied by tuning the parameters in such a way that the Higgs self coupling λ(Λ) is almost tangent to the Λ axis. But the latter becomes hard to satisfy then.

23. There is a difficulty. (cont’d) m Higgs =125GeV m top =171.316 GeV b can be set to the desired value by doing one-parameter tuning, but Λ 0 is too large then. The value of Λ 0 does not change much even if the other couplings are changed.

24. Introduce an extra field In order to change the shape of the function λ(Λ), we introduce an extra scalar field that can be identified with dark matter: Then we can change the position of Λ 0 as we want. m Higgs =125GeV m top =172.895 GeV κ ρ

25. We might have nothing other than the SM particles below the string scale. There is a small room for the other field, and it seems that SM with a little modification is the right theory. It is better not to insist on SUSY but to think about what is really needed. Some parameters such as the cosmological constant, Higgs mass, and strong CP phase are unnatural, but it might not be correct to try to solve them within the framework of field theory. Dark matter is really needed and should be explained in terms of field theory. It is not clear whether inflation should be explained in the field theory context or not, but it is worth trying. Fields in the era of post LHC

26. Gravity is different. Gravity is different from the other interactions in the sense that it is not renormalizable. If the theory is renormalzable, the effects of short distance quantum fluctuations can be absorbed to the redefinition of the parameters of the particles such as mass and coupling constants. It means the picture of point particles is valid even after quantization. ~ ~ mass renormalizationcoupling constant renormalization

27. Gravity is different. (cont’d) On the other hand, if the theory is not renormalizable, short distance quantum fluctuations are too large to absorb to the parameters of the particles. It means that the notion of point particle is no longer valid if we consider the quantum effects of gravity. mass renormalization Not possible coupling constant renormalization Not possible In other words, if we want to include quantum gravity, we need to think about extended objects instead of point particles.

28. In string theory, we consider one-dimensional objects like a rubber band on which some internal degrees of freedom are sitting.,,... spin m 2 0 2 1 standard model graviton Gauge fields Higgs quarks and leptons heavy particles String theory The mass spectra and interactions are completely determined by those freedom.

29. It is believed that they are different vacuum (ground state) of the same theory. String as unification Gravity is automatically contained. Effects of short distance quantum fluctuations are so small that there is no UV divergence. Gauge and matter fields also appear naturally. But their structures depend on the choice of the internal degrees of freedom. There are infinitely many theories that have various space-time dimensions, gauge group, number of generations, …etc., depending on the choice.

30. Non-perturbative effects in string theory vacuum of one theory ⇒ infinitely many strings ＝ vacuum of another theory As we have seen, there are many theories corresponding to the choice of the internal degrees of freedom. However, if infinitely many strings are condensed properly to the vacuum of one theory, it is converted to that of another theory. It is expected that all the theories are different vacua of one theory. ( non-perturbative effects = effects caused by infinitely many objects)

31. Each vacuum is perturbatively stable. Non-pertirbative effects cause transitions. 10D Heterotic E 8 × E 8 10D Type II A 9D theory #1 9D theory #2 4D theory #1 11D M-theory non- perturbative string theory tunneling string condensate String is one. The most important question is whether a unique vacuum exists or not after taking the non-perturbative effects into account. If it is the case, it should be our SM universe, and we should be able to explain all the parameters of SM from one theory.

32. Matrix model We need to find a formulation of string theory which can describe the non-perturbative effects. In other words, we need to find a good definition of string theory. One possibility is the IIB matrix model:

33. Green-Schwartz action in the Schild Gauge Regularization by matrix {, } → [, ] → (1) World sheet regularization Evidences for the IIB Matrix model

34. (2) Loop equation and string field Wilson loop = string field ⇔ creation annihilation operator of loop equation → light-cone string field This can be shown with some assumptions.

35. (3) effective Lagrangian and gravity Integrate out this part. The loop integral gives the exchange of graviton and dilaton.

36. A remarkable feature of the matrix model is that the space-time itself emerges dynamically from the matrix degrees of freedom A μ. (1) A μ as the space-time coordinates mutually commuting A μ ⇒ space-time Emergence of Space-time (2) A μ as non-commutative space-time non-commutative A μ ⇒ NC space-time (3) A μ as derivatives as in the naive large-N reduction There are several mechanisms for the emergence of space-time.

37. Einstein equation follows. If we impose the Ansatz it becomes The classical EOM of the IIB matrix model is The Einstein equation follows from the EOM of the IIB matrix model

38. multiverse in the matrix model Multiverse appears naturally if we consider block diagonal configurations. matrix model ・ Each block represents a universe. ・ quantum gravity 0 0

39. The low energy effective action around the multiverse is not a simple local action but has the multi-local form: Y. Asano, A Tsuchiya, HK Factorized action from IIB matrix model

40. Consider the path integral which involves the summation over topologies, Coleman (‘88) Then there should be a wormhole-like configuration in which a thin tube connects two points on the universe. Here, the two points may belong to either the same universe or the different universe. If we see such configuration from the side of the large universe(s), it looks like two small punctures. But the effect of a small puncture is equivalent to an insertion of a local operator. Summing up wormholes, we obtain the multi-local action.

41. The effective action of quantum gravity/string is given by The path integral is given by Solution to the naturalness broblem? Coupling constants are not merely constant but to be integrated. If is peaked strongly around some values of λ, it means that the coupling constants are dynamically fixed to those values. If not, we have a kind of parallel world consisting of world with different couplings.

42. The matrix model is well defined, and in principle it is possible to determine which case is true. A good point of matrix model There are some attempts to perform Monte Carlo simulations of the IIB matrix model, and they observed an expanding four dimensional universe. [Nishimura et.al.] Although it is in a very primitive level at present, numerical analyses seem to work to examine the vacuum of the matrix model.

43. There might be a desert to the Plank scale. It is not disappointing, but it means that there is no obstruction for connecting the SM physics to the Plank scale one. We can start serious comparison between string theory and the SM model. To do so, it is important to know the precise SM data such as neutrino mass and mixing, and what is really needed besides the SM. Fields and Strins in the era of post LHC