Fields and strings in the era of post LHC

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. naturalness problem Suppose the underlying fundamental theory, such as string theory, has the momentum scale mSand the coupling constant gS. Then, by dimensional analysis and the power counting of the couplings, the parameters of the low energy effective theory are expected as follows:

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

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

6. Possibility of SUSY as the solution to the naturalness problem Therefore, if MSUSYis close to mH,the Higgs massis naturally understood, although the cosmological constant is still a big problem. History Actually when the Z- and W- bosons were discovered in 1983, there was a strong motivation to expect that MSUSY 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.

7. Present status of SUSY After 30 years the Higgs particle was finally discovered at mH~ 125 GeV. Furthermore, no signal of new physics is observed below 1 TeV, which indicates that MSUSY >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. • 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. • 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) • The universe has different parameters place by place. We are sitting at one place, where the parameters are such that we can exist. 3. The parameters are fixed by some non- perturbative effect of quantum gravity/string theory such as Coleman’s baby universe mechanism. They are not totally nonsense, but it is difficult to judge what is correct.

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 scalemS, 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 mS. 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) mHiggs =126GeVmtop =172 GeV No inconsistency arises below the string scale. U(1) SU(2) SU(3) Ytop Higgs self coupling Higgs mass 2 log10 Λ[GeV}

12. Bare parameters of the cutoff theory (2) mHiggs =126GeVmtop =190 GeV Higgs field becomes unstable. SU(3) Higgs mass 2 Ytop SU(2) U(1) log10 Λ[GeV} Higgs self coupling

13. Bare parameters of the cutoff theory (3) mHiggs =126GeVmtop =150 GeV Higgs self coupling No inconsistency arises, but the Higgs self coupling tends to diverge. U(1) SU(2) SU(3) Ytop log10 Λ[GeV} Higgs mass 2

14. Bare parameters of the cutoff theory (4) mHiggs =100GeVmtop =172 GeV Higgs field becomes unstable. Ytop SU(3) SU(2) U(1) Higgs mass 2 log10 Λ[GeV} Higgs self coupling

15. Bare parameters of the cutoff theory (5) mHiggs =150GeVmtop =172 GeV Higgs self coupling No inconsistency arises, but the Higgs self coupling tends to diverge. SU(3) Ytop SU(2) U(1) log10 Λ[GeV} Higgs mass 2

16. Summary of the bare parameters • 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.

17. Higgs field as inflaton (1) [with Y. Hamada and K. Oda] 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 .

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. mHiggs =126GeVmtop =172 GeV log10 Λ[GeV}

19. Higgs field as inflaton (3) Then the first term of Veff 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 λin 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 indicates and The former can be satisfied by tuning the parameters in such a way that the Higgs self coupling λ(Λ) is almost tangent to theΛ axis. But the latter becomes hard to satisfy then.

23. There is a difficulty. (cont’d) mHiggs =125GeVmtop =171.316 GeV b can be set to the desired value by doing one-parameter tuning,but Λ0 is too large then. The value of Λ0 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: mHiggs =125GeVmtop =172.895 GeV Then we can change the position of Λ0 aswe want. ρ κ

25. Fields in the era of post LHC • 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.

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 renormalization coupling 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. In other words, if we want to include quantum gravity, we need to think about extended objects instead of point particles. mass renormalization Not possible coupling constant renormalization Not possible

28. . . . , , spin graviton 2 1 Gauge fields quarks and leptons m2 0 Higgs heavy particles standard model String theory In string theory, we consider one-dimensional objects like a rubber band on which some internal degrees of freedom are sitting. The mass spectra and interactions are completely determined by those freedom.

29. 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. It is believed that they are different vacuum (ground state) of the same theory.

30. Non-perturbative effects in string 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. ⇒ ＝ infinitely many strings vacuum of one theory vacuum of another theory ( non-perturbative effects = effects caused by infinitely many objects)

31. 9D theory #2 9D theory #1 4D theory #1 tunneling 10D Heterotic E8× E8 non-perturbative string theory 10D Type II A 11D M-theory String is one. Each vacuum is perturbatively stable. Non-pertirbative effects cause transitions. string condensate 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. Evidences for the IIB Matrix model (1) World sheet regularization Green-Schwartz action in the Schild Gauge Regularization by matrix { , } → [ , ] →

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

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

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

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

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

40. Coleman (‘88) Consider the path integral which involves the summation over topologies, 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. Solution to the naturalness broblem? The effective action of quantum gravity/string is given by The path integral is given by 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. A good point of matrix model The matrix model is well defined, and in principle it is possible to determine which case is true. 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. Fields and Strins in the era of post LHC • 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.