Supersymmetric Gauge Theory: an overview

Transcription

1 Supersymmetric Gauge Theory: an overview Michael R. Douglas Rutgers and IHES December 4, 2006 Columbia, Department of Physics Abstract We give an overview of supersymmetric gauge theory, both as a theoretical tool which has led to many dramatic advances in our understanding of gauge theory and string theory, and as an extension of the Standard Model of particle physics which might be discovered in upcoming collider experiments. Page 1 of 50

2 1. Our present view of particle physics took form in the early 1970 s, in the Standard Model of particle physics, a remarkably unified framework which at the time described all experiments and observations in fundamental physics not involving gravity. Page 2 of 50

3 1. Our present view of particle physics took form in the early 1970 s, in the Standard Model of particle physics, a remarkably unified framework which at the time described all experiments and observations in fundamental physics not involving gravity. Over the subsequent thirty years, many of its key predictions were verified, such as the existence of the W and Z bosons. New data has refined and even gone beyond the SM, data from collider experiments: Precision measurements of SM couplings Mass bounds and from other observations, especially from astrophysics: Neutrino oscillations Convincing evidence for dark matter and dark energy. But this progress has been hard-won, and everything we have clear evidence for so far can be modeled by variations on the SM, employing the same paradigm of renormalizable quantum field theory. Page 2 of 50

4 Fortunately, many believe we stand on the threshold of a new data-rich era. Besides the spectacular progress in precision cosmological measurements, a new frontier in high energies is opening, with run II at the Tevatron, and especially with the scheduled turn-on of LHC in LHC will produce 7 on 7 TeV proton-proton collisions and give us a good picture of particle physics up to 1 TeV. Page 3 of 50

5 What should we expect to see at these energies? Page 4 of 50

6 What should we expect to see at these energies? There are many speculations about this, which we will come to in due course. But the best current answer to this question is: Page 4 of 50

7 What should we expect to see at these energies? There are many speculations about this, which we will come to in due course. But the best current answer to this question is: We don t know! Page 4 of 50

8 What should we expect to see at these energies? There are many speculations about this, which we will come to in due course. But the best current answer to this question is: We don t know! There is no theoretical idea out there which is as compelling as the inference short-range vector-like interactions spontaneously broken gauge symmetry W, Z bosons. But, there are rather good arguments that we will discover something. And, a wealth of new theoretical ideas has emerged in recent years, mostly flowing from two sources: 1. New possibilities/constraints coming from unification with gravity, especially in the context of string theory. 2.. Page 4 of 50

9 In this talk, we will focus on 2, supersymmetric gauge theory, for several reasons: Supersymmetric gauge theories are a fairly conservative extrapolation of what we have seen so far (actually they are a special case of gauge theory). They have remarkable similarities to (and differences from) conventional gauge theories (like the SM), and can exhibit all of their qualitative physics. But, they are far more accessable to theoretical understanding. We now have analytic results for problems which remain intractable for conventional QFT, and convincing evidence for new behaviors which before were only speculation. Their study leads inevitably towards higher dimensions and string/m theory; in some simple cases they are equivalent, and string theory intuitions are very useful even for understanding physics in four dimensions. can resolve central theoretical puzzles in the SM. There is already circumstantial evidence for (and against...) supersymmetry. Page 5 of 50

10 2. The primary information specifying a gauge theory is its gauge group, a compact Lie group. For the Standard Model, this is SU(3) strong SU(2) weak U(1) E&M Page 6 of 50

11 2. The primary information specifying a gauge theory is its gauge group, a compact Lie group. For the Standard Model, this is SU(3) strong SU(2) weak U(1) E&M Page 6 of 50

12 2. The primary information specifying a gauge theory is its gauge group, a compact Lie group. For the Standard Model, this is SU(3) strong complicated short range SU(2) weak range cm U(1) E&M infinite range Each sector is governed by very similar equations, the Yang- Mills and Maxwell equations. How can they describe such different physics? Page 6 of 50

13 2. The primary information specifying a gauge theory is its gauge group, a compact Lie group. For the Standard Model, this is SU(3) strong complicated short range confining phase SU(2) weak range cm Higgs phase U(1) E&M infinite range Coulomb phase Each sector is governed by very similar equations, the Yang- Mills and Maxwell equations. How can they describe such different physics? As discovered in the 60 s and 70 s, there are three basic phases of gauge theory, (very) loosely analogous to the phases of matter: the Coulomb, Higgs and confining phases. Page 6 of 50

14 2. The primary information specifying a gauge theory is its gauge group, a compact Lie group. For the Standard Model, this is SU(3) strong complicated short range confining phase SU(2) weak range cm Higgs phase U(1) E&M infinite range Coulomb phase Each sector is governed by very similar equations, the Yang- Mills and Maxwell equations. How can they describe such different physics? As discovered in the 60 s and 70 s, there are three basic phases of gauge theory, (very) loosely analogous to the phases of matter: the Coulomb, Higgs and confining phases. Coulomb phase E 1/r 2 Page 6 of 50

15 A prototype for the Higgs (Anderson) phase is Maxwell theory in a BCS superconductor. Lattice interactions between electrons allow them to bind into Cooper pairs, paired electrons of opposite spin φ = e e Such pairs are bosons and thus can Bose-Einstein condense into a coherent state, characterized by a non-zero expectation value φ This expectation value forms an order parameter for breaking of gauge symmetry. Page 7 of 50

16 A prototype for the Higgs (Anderson) phase is Maxwell theory in a BCS superconductor. Lattice interactions between electrons allow them to bind into Cooper pairs, paired electrons of opposite spin φ = e e Such pairs are bosons and thus can Bose-Einstein condense into a coherent state, characterized by a non-zero expectation value φ This expectation value forms an order parameter for breaking of gauge symmetry. Direct consequences of this: gauge boson gets a mass, short range force magnetic field is expelled or confined into strings Page 7 of 50

17 While the phenomena of pairing and condensation are possible in four-dimensional gauge theory, in the Standard Model one instead postulates a scalar field, the Higgs field φ, to play the role of order parameter. This predicts the existence of the Higgs particle, a novel thing: no fundamental scalar particle had been discovered before. Page 8 of 50

18 While the phenomena of pairing and condensation are possible in four-dimensional gauge theory, in the Standard Model one instead postulates a scalar field, the Higgs field φ, to play the role of order parameter. This predicts the existence of the Higgs particle, a novel thing: no fundamental scalar particle had been discovered before. So far, this one hasn t been either. -2 ln(q) Observed Expected background Expected signal + background LEP m H (GeV/c 2 ) Results of Higgs boson search at LEP, Page 8 of 50

19 By now, the confining phase is also familiar. The key idea is that color confinement is analogous to the confinement of magnetic flux in a superconductor into vortices. A hypothetical pair of monopoles M and M in a superconductor, carrying opposite magnetic charge, would be connected by a magnetic flux tube. On trying to separate the pair, the energy of the flux tube would grow with its length. Eventually we would gain energetically by creating a new M M pair, and end up with two charge neutral objects. Page 9 of 50

20 By now, the confining phase is also familiar. The key idea is that color confinement is analogous to the confinement of magnetic flux in a superconductor into vortices. A hypothetical pair of monopoles M and M in a superconductor, carrying opposite magnetic charge, would be connected by a magnetic flux tube. On trying to separate the pair, the energy of the flux tube would grow with its length. Eventually we would gain energetically by creating a new M M pair, and end up with two charge neutral objects. This analogy can be made precise by using electric-magnetic duality: grad E = 4πρ E grad B = 4πρ M grad E = B grad B = E t t Page 9 of 50

21 By now, the confining phase is also familiar. The key idea is that color confinement is analogous to the confinement of magnetic flux in a superconductor into vortices. A hypothetical pair of monopoles M and M in a superconductor, carrying opposite magnetic charge, would be connected by a magnetic flux tube. On trying to separate the pair, the energy of the flux tube would grow with its length. Eventually we would gain energetically by creating a new M M pair, and end up with two charge neutral objects. This analogy can be made precise by using electric-magnetic duality: grad E = 4πρ E grad B = 4πρ M grad E = B grad B = E t t The electric-magnetic dual of the Higgs phase, is a phase in which a magnetically charged scalar field condenses, call it φ. The resulting dual Meissner effect confines color. The order parameter can be either composite, i.e. a bound state of fermionic monopoles, or fundamental in nature. Page 9 of 50

22 So why do the three sectors of the SM realize these different phases? Are there other possible phases? There are two main tools used to answer these questions: the renormalization group, and the effective potential. The perturbative RG explains the basic features of the problem. We need to know the gauge coupling at long distances; if it is weak, the classical equations of motion give a good picture. Page 10 of 50

23 So why do the three sectors of the SM realize these different phases? Are there other possible phases? There are two main tools used to answer these questions: the renormalization group, and the effective potential. The perturbative RG explains the basic features of the problem. We need to know the gauge coupling at long distances; if it is weak, the classical equations of motion give a good picture α log Q (GeV) α = g2 4π U(1) running coupling in the SM. small classical Coulomb or Higgs phase. Page 10 of 50

24 So why do the three sectors of the SM realize these different phases? Are there other possible phases? There are two main tools used to answer these questions: the renormalization group, and the effective potential. The perturbative RG explains the basic features of the problem. We need to know the gauge coupling at long distances; if it is weak, the classical equations of motion give a good picture α log Q (GeV) U(1) and SU(3) running couplings in the SM. If the gauge coupling grows with distance, we cannot use the classical equations of motion. But if it becomes very strong, α >> 1, we can use duality α 1/α. Page 10 of 50 α = g2 4π large Coulomb or confining phase.

25 The choice between Coulomb or other phases can be determined by minimizing the effective potential to determine if an order parameter φ is zero or non-zero: This is the potential for the SM Higgs field. It is very simple because we postulated it to have this form. Page 11 of 50

26 In most gauge theory problems, starting with the effective low energy dynamics of QCD, the effective potential is far more complicated. It is hard to define, and even after defining it, it is too hard to compute. V (φ) Page 12 of 50 Effective potential for supersymmetric SU(3) Yang-Mills theory (as a function of the gaugino condensate order parameter, an uncharged field).

27 Furthermore, there can be first order phase transitions. This is what stops most analytic progress. V (φ) Effective potential for SYM plus non-supersymmetric matter. Page 13 of 50

28 Furthermore, there can be first order phase transitions. This is what stops most analytic progress. V (φ) Effective potential for SYM plus non-supersymmetric matter. One can proceed by combining clever tricks, lattice calculations, etc., and after much work we have a good picture of SM physics. But it is hard to study hypothetical new physics without better tools. Page 13 of 50

29 3. Certain gauge theories, with cleverly chosen matter content and couplings, have supersymmetry. This pairs bosons and fermions: gauge boson G µ g gaugino quark Q q squark lepton L l slepton Higgs H h higgsino Page 14 of 50

30 3. Certain gauge theories, with cleverly chosen matter content and couplings, have supersymmetry. This pairs bosons and fermions: gauge boson G µ g gaugino quark Q q squark lepton L l slepton Higgs H h higgsino These are gauge theories just like the others, raising all of the same questions, and all of the same tools can be applied to their study. However, there are new constraints and tools which can be brought to bear. Page 14 of 50

31 3. Certain gauge theories, with cleverly chosen matter content and couplings, have supersymmetry. This pairs bosons and fermions: gauge boson G µ g gaugino quark Q q squark lepton L l slepton Higgs H h higgsino These are gauge theories just like the others, raising all of the same questions, and all of the same tools can be applied to their study. However, there are new constraints and tools which can be brought to bear. The most basic, is that the couplings of the superpartners, are largely determined in terms of the original couplings. Despite the larger number of particles, these theories are more constrained than ordinary gauge theory. For example, the effective potential cannot take just any form; it must be a sum of squares: V (φ) = W (φ) 2, φ i i where W (φ) is a function called the superpotential. Page 14 of 50

32 V (φ) = i W (φ) 2, φ i This mathematical statement has an important physical consequence: namely, there is a minimal possible vacuum energy, zero. If we consider a zero energy (supersymmetric) vacuum, no competing vacuum can have a lower energy. Thus, we can do perturbation theory, extrapolations, etc., confident that no first order phase transition can spoil our results. Page 15 of 50

33 V (φ) = i W (φ) 2, φ i This mathematical statement has an important physical consequence: namely, there is a minimal possible vacuum energy, zero. If we consider a zero energy (supersymmetric) vacuum, no competing vacuum can have a lower energy. Thus, we can do perturbation theory, extrapolations, etc., confident that no first order phase transition can spoil our results. An even more important physical statement is the (perturbative) non-renormalization theorem. This says that, under certain conditions, the superpotential W (φ) is independent of energy scale the renormalization corrections cancel between bosons and fermions. Page 15 of 50

34 V (φ) = i W (φ) 2, φ i This mathematical statement has an important physical consequence: namely, there is a minimal possible vacuum energy, zero. If we consider a zero energy (supersymmetric) vacuum, no competing vacuum can have a lower energy. Thus, we can do perturbation theory, extrapolations, etc., confident that no first order phase transition can spoil our results. An even more important physical statement is the (perturbative) non-renormalization theorem. This says that, under certain conditions, the superpotential W (φ) is independent of energy scale the renormalization corrections cancel between bosons and fermions. This makes it possible to have an effective potential with flat directions, and a continuously variable choice of vacuum or phase. Phases are distinguished by the expectation value of a scalar field. V (x, y) Page 15 of 50

35 More mathematics: the superpotential W (φ) must be holomorphic. Implicit in this are two statements: 1. The scalar fields are complex variables, φ = Re φ + iim φ. This is sometimes true in general QFT (e.g. the Higgs field; the Cooper pair), and sometimes not. Here, it is forced by the fact that a fermion field (in four dimensions) is complex, so its partner scalar must be too. 2. Regarded as a function of the φ, the function W (φ) is holomorphic, W (φ, φ) φ = 0 where φ = Re φ iim φ. Page 16 of 50

36 More mathematics: the superpotential W (φ) must be holomorphic. Implicit in this are two statements: 1. The scalar fields are complex variables, φ = Re φ + iim φ. This is sometimes true in general QFT (e.g. the Higgs field; the Cooper pair), and sometimes not. Here, it is forced by the fact that a fermion field (in four dimensions) is complex, so its partner scalar must be too. 2. Regarded as a function of the φ, the function W (φ) is holomorphic, W (φ, φ) φ = 0 where φ = Re φ iim φ. This follows from its second role: it determines Yukawa couplings such as those which give mass to the SM quarks and leptons. Here they take the form 2 W (φ) φ φ ψ ψ This is a mass term because it flips the spin of the ψ particle. On the other hand, a term in W like φ 2 φ, would lead to a coupling φψ ψ, which is not even Lorentz invariant. Page 16 of 50

37 If nature is supersymmetric, holomorphy of the superpotential will have direct observable consequences: it forces the supersymmetric Standard Model to have two Higgs fields H 1, H 2, instead of one (and five Higgs particles)! W = H 1 U L u R + H 2 D L d R + H 2 L L l R +... Page 17 of 50

38 If nature is supersymmetric, holomorphy of the superpotential will have direct observable consequences: it forces the supersymmetric Standard Model to have two Higgs fields H 1, H 2, instead of one (and five Higgs particles)! W = H 1 U L u R + H 2 D L d R + H 2 L L l R +... It is also an extremely useful tool for theorists, because of the following mathematical principle: A holomorphic function is determined by its singularities (poles, branch cuts, etc.) And, singularities of functions like the superpotential, have physical meaning this will enable us to infer the effective potential from physical ansätze about the phase structure of the theory Page 17 of The function f(z) = 1 z 1 + (z 2)(z 3).

39 One final mathematical fact: We can consider a holomorphic function f(z) as defining a hypersurface or Riemann surface in four-dimensional real space, the surface y 1 + iy 2 = f(x 1 + ix 2 ). Page 18 of 50 The surface y = 1 + z 2 + z 4 (plotting Re y). Holomorphy of f is then the condition that the surface minimizes its area, as would a soap film, or a string.

40 4. Seiberg-Witten solution If supersymmetry is good, isn t more supersymmetry better? Page 19 of 50

41 4. Seiberg-Witten solution If supersymmetry is good, isn t more supersymmetry better? A very special subset of the supersymmetric theories has N = 2 and N = 4 extended supersymmetry. Page 19 of 50

42 4. Seiberg-Witten solution If supersymmetry is good, isn t more supersymmetry better? A very special subset of the supersymmetric theories has N = 2 and N = 4 extended supersymmetry. Extended supersymmetry is a mixed blessing, as one can easily show that it forbids chiral matter (different properties for left and right helicity fermions), as seen in the SM. On the other hand, for some phenomena this is not important. Page 19 of 50

43 4. Seiberg-Witten solution If supersymmetry is good, isn t more supersymmetry better? A very special subset of the supersymmetric theories has N = 2 and N = 4 extended supersymmetry. Extended supersymmetry is a mixed blessing, as one can easily show that it forbids chiral matter (different properties for left and right helicity fermions), as seen in the SM. On the other hand, for some phenomena this is not important. The simplest example of an N = 2 gauge theory is the SU(2) gauge theory with two gauginos and a massless adjoint scalar field, A µ gauge fields χ 1 χ 2 gauginos φ adjoint scalar fields While the extra gaugino makes little difference, the presence of the massless scalar drastically changes the physics, breaking SU(2) to U(1) gauge symmetry (as in the Georgi-Glashow model). Without supersymmetry, it would be hard to go further. Page 19 of 50

44 1 α log tr φ 2 Because of supersymmetry, we can freely choose the scalar field expectation value tr φ 2. No matter what it is, the vacuum has zero energy. Thus this theory has a moduli space of vacua. In a given vacuum, SU(2) gauge symmetry is broken to U(1) at an energy E = φ. Page 20 of 50

45 1 α log tr φ 2 Because of supersymmetry, we can freely choose the scalar field expectation value tr φ 2. No matter what it is, the vacuum has zero energy. Thus this theory has a moduli space of vacua. In a given vacuum, SU(2) gauge symmetry is broken to U(1) at an energy E = φ. If φ is large, the gauge coupling is still weak at energy E. Thus we can trust a classical analysis, which tells us we are in the Coulomb phase. Page 20 of 50

46 1 α log tr φ 2 Because of supersymmetry, we can freely choose the scalar field expectation value tr φ 2. No matter what it is, the vacuum has zero energy. Thus this theory has a moduli space of vacua. In a given vacuum, SU(2) gauge symmetry is broken to U(1) at an energy E = φ. If φ is large, the gauge coupling is still weak at energy E. Thus we can trust a classical analysis, which tells us we are in the Coulomb phase. If φ is small, the gauge coupling is strong at the energy E, and we cannot trust the classical analysis. So what happens? Page 20 of 50

47 1 α log tr φ 2 Because of supersymmetry, we can freely choose the scalar field expectation value tr φ 2. No matter what it is, the vacuum has zero energy. Thus this theory has a moduli space of vacua. In a given vacuum, SU(2) gauge symmetry is broken to U(1) at an energy E = φ. If φ is large, the gauge coupling is still weak at energy E. Thus we can trust a classical analysis, which tells us we are in the Coulomb phase. If φ is small, the gauge coupling is strong at the energy E, and we cannot trust the classical analysis. So what happens? At first sight, not much. N = 2 supersymmetry guarantees that the scalar must have a U(1) vector boson as partner, so we are always in the Coulomb phase. This theory cannot confine, despite the fact that its coupling seems to run off to infinity! Page 20 of 50

48 1 α log tr φ 2 Because of supersymmetry, we can freely choose the scalar field expectation value tr φ 2. No matter what it is, the vacuum has zero energy. Thus this theory has a moduli space of vacua. In a given vacuum, SU(2) gauge symmetry is broken to U(1) at an energy E = φ. If φ is large, the gauge coupling is still weak at energy E. Thus we can trust a classical analysis, which tells us we are in the Coulomb phase. If φ is small, the gauge coupling is strong at the energy E, and we cannot trust the classical analysis. So what happens? At first sight, not much. N = 2 supersymmetry guarantees that the scalar must have a U(1) vector boson as partner, so we are always in the Coulomb phase. This theory cannot confine, despite the fact that its coupling seems to run off to infinity! Actually, before getting to infinity, complicated nonperturbative effects (instantons) become important. What to do? Page 20 of 50

49 The key is to consider the masses of the lightest charged particles. Besides the W bosons of broken gauge symmetry, there are monopoles and dyons. These are all BPS states, the lightest possible states of a given charge. Page 21 of 50

50 The key is to consider the masses of the lightest charged particles. Besides the W bosons of broken gauge symmetry, there are monopoles and dyons. These are all BPS states, the lightest possible states of a given charge. We know how masses behave at very weak coupling, since that can be found by solving the classical Yang-Mills equations. Page 21 of 50

51 The key is to consider the masses of the lightest charged particles. Besides the W bosons of broken gauge symmetry, there are monopoles and dyons. These are all BPS states, the lightest possible states of a given charge. We know how masses behave at very weak coupling, since that can be found by solving the classical Yang-Mills equations. We have to guess how they behave at strong coupling. The guess is that somewhere in the space of vacua, a monopole can become massless. This can be motivated by thinking about the related N = 1 supersymmetric theory, in which we make the scalar (and one gaugino) massive. In this theory, using our guess, one can determine the effective potential for the monopole field. One sees that the monopoles condense and this theory confines, precisely according to the dual Meissner effect picture. Page 21 of 50

52 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

53 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

54 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

55 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

56 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

57 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

58 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

59 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

60 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

61 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

62 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

63 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

64 This is nice, but we have to connect up the different vacua somehow, to get anywhere. does this, by telling us that BPS masses must be holomorphic functions. And, it turns out that N = 2 susy allows a very useful trick: they can be described by a Riemann surface, in this case a torus: The lengths of the loop α and β are the mass of the W and monopole respectively. The variation of BPS masses with moduli is encoded in a family of Riemann surfaces, one for each point in the moduli space. Page 22 of 50

65 Since the Riemann surfaces are holomorphic, if we know what singularities these masses can have, we can infer the whole family of surfaces. It turns out not to make sense if there is just one special point at which the monopole becomes massless. The only sensible extrapolation to strong coupling is to postulate two new singularities. At one, the monopole becomes massless. At the other, a dyon becomes massless. At this second point, the N = 1 theory has a second vacuum in which dyons condense, realizing oblique confinement. Page 23 of 50

66 Since the Riemann surfaces are holomorphic, if we know what singularities these masses can have, we can infer the whole family of surfaces. It turns out not to make sense if there is just one special point at which the monopole becomes massless. The only sensible extrapolation to strong coupling is to postulate two new singularities. At one, the monopole becomes massless. At the other, a dyon becomes massless. At this second point, the N = 1 theory has a second vacuum in which dyons condense, realizing oblique confinement. Putting all this together, one can solve for all the tori, and all the masses. Page 23 of 50

67 All this can be generalized to SU(N) theories, by using higher genus Riemann surfaces. One finds that SU(N) supersymmetric Yang-Mills theory has N vacua, generalizing the 2 for SU(2), at which various dyons condense. Page 24 of 50

68 All this can be generalized to SU(N) theories, by using higher genus Riemann surfaces. One finds that SU(N) supersymmetric Yang-Mills theory has N vacua, generalizing the 2 for SU(2), at which various dyons condense. And the picture of color confinement works, but with many surprises, having to do with the presence of very light scalars. In this theory, the mesons are not q q, but can also be qφ q, qφφ q, and so on. In fact, the vacuum breaks color permutation symmetry, and the different color quarks are attached by N 1 different confining flux tubes k-strings, with different string tensions: T k = mλ sin πk N These are the strings coupling to Wilson loops with charge k under the Z N center. They are something like bound states of k elementary flux tubes. k = 1, 2,..., N 1. Q Q Q a. b. Q Q Q l L Page 24 of 50

69 The sine law for the k-string tensions appears to be true even in non-supersymmetric gauge theory, as found by lattice calculation: (From Del Debbio et al, hep-th/ ) R(k,N) 2.0 S(k,4) C(k,4) S(k,6) C(k,6) Large-N limit N=4 MC N=6 MC 1.5 Page 25 of k

70 For many years, the SW solution remained a brilliant guess, confirmed mostly by the beauty of the larger picture into which it fits. But recently it was proven by Nikita Nekrasov, who summed the instanton expansion This was made possible by considering the noncommutative version of the gauge theory, in which coordinates do not commute: [x µ, x ν ] = θ µν. This theory was formulated by Alain Connes and Mark Rieffel as a tool for studying noncommutative geometry. With Connes, Albert Schwarz, and Chris Hull, we showed that it arises from simple limits of M theory and string theory, giving strong evidence for its consistency. As then shown by Nekrasov, Schwarz, Gross, and others, instantons and other solitons are far simpler in this theory, making such computations possible. Page 26 of 50

71 5. Having confirmed the conventional picture of the phase structure of gauge theory, we can go on to ask: are there new phases? Since there are no first order transitions, we should look for second order transitions. These are signalled at least by a massless field, and in general by critical phenomena. A field theory at the critical point shows scaling behavior: correlation functions behave simply under overall rescaling of distances. It is associated with a fixed point of the RG, and a zero of the beta function: α Page 27 of log Q (GeV) Running coupling with an IR fixed point.

72 There is a rich classification of two dimensional critical theories, and they certainly exist in three dimensions, starting with the standard liquid-gas critical point, but they are not so easy to find in four dimensions. For a long time, the only known examples were N = 4 supersymmetric gauge theory, and certain large N gauge theories. But, let us consider the example of N = 2 gauge theory with group SU(3). The SW curve is a genus two Riemann surface, α β2 2 β α 1 1 If two α cycles shrink to zero size, this again corresponds to confinement by monopole condensation. Almost everything else one can do corresponds to oblique confinement. Page 28 of 50

73 There is a rich classification of two dimensional critical theories, and they certainly exist in three dimensions, starting with the standard liquid-gas critical point, but they are not so easy to find in four dimensions. For a long time, the only known examples were N = 4 supersymmetric gauge theory, and certain large N gauge theories. But, let us consider the example of N = 2 gauge theory with group SU(3). The SW curve is a genus two Riemann surface, α β2 2 β α 1 1 If two α cycles shrink to zero size, this again corresponds to confinement by monopole condensation. Almost everything else one can do corresponds to oblique confinement. But what if the neck in the middle gets long? After a conformal rescaling, this is the same as two intersecting cycles shrinking to zero. Page 28 of 50

74 This leads to simultaneously massless electrons and monopoles, at the Argyres-Douglas point. In fact this cannot be described by a conventional (Lorentz invariant) lagrangian, and one cannot even think of the electrons and monopoles as particles. But it is still a sensible quantum field theory, a new type of quantum disorder. Page 29 of 50

75 This leads to simultaneously massless electrons and monopoles, at the Argyres-Douglas point. In fact this cannot be described by a conventional (Lorentz invariant) lagrangian, and one cannot even think of the electrons and monopoles as particles. But it is still a sensible quantum field theory, a new type of quantum disorder. Some years later, there is a long catalog of such fixed point theories, obtained by starting with gauge theory and tuning the couplings. Most are supersymmetric but there is evidence for many nonsupersymmetric theories as well. Another simple class of fixed point theories is SU(N c ) supersymmetric QCD with N f flavors of quarks, in the conformal window 3 2 N c < N f < 3N c. For example, N c = 3 and N f = 6. Page 29 of 50

76 In general, many different UV theories can have the same IR fixed point. Here, a magnetic theory with gauge group SU(N f N c ) is dual to super-qcd. It has the same hadronic spectrum, but constructed from a different quark model. One could tell the difference by doing a e + e annihilation thought experiment: one would see a highly anomalous threshold at which cross sections decreased, because one is crossing over to a UV theory with a small gauge coupling. R Electric theory Magnetic theory t 2500 From Murayama et al, hep-th/ Here N c = 7, N f = 20. Page 30 of 50

77 6. Back in the 60 s, when Gell-Mann and Zweig postulated quarks, the question was immediately asked: are these just mathematical fictions, useful for describing the spectrum, or are they actually particles? Now let us ask, is the Seiberg-Witten curve just a mathematical trick, useful for describing the solution of N = 2 gauge theory, or could it have some sort of physical reality? Since each point in four-dimensional space-time can come with its own curve, could it be that, in some sense, this gauge theory generates two extra dimensions of space? Page 31 of 50

78 6. Back in the 60 s, when Gell-Mann and Zweig postulated quarks, the question was immediately asked: are these just mathematical fictions, useful for describing the spectrum, or are they actually particles? Now let us ask, is the Seiberg-Witten curve just a mathematical trick, useful for describing the solution of N = 2 gauge theory, or could it have some sort of physical reality? Since each point in four-dimensional space-time can come with its own curve, could it be that, in some sense, this gauge theory generates two extra dimensions of space? No: this theory is asymptotically free, so at high energies it describes free quarks and gluons in four dimensions. Page 31 of 50

79 6. Back in the 60 s, when Gell-Mann and Zweig postulated quarks, the question was immediately asked: are these just mathematical fictions, useful for describing the spectrum, or are they actually particles? Now let us ask, is the Seiberg-Witten curve just a mathematical trick, useful for describing the solution of N = 2 gauge theory, or could it have some sort of physical reality? Since each point in four-dimensional space-time can come with its own curve, could it be that, in some sense, this gauge theory generates two extra dimensions of space? No: this theory is asymptotically free, so at high energies it describes free quarks and gluons in four dimensions. But, there is a sense in which the converse is true. There is a six dimensional theory, such that taking two of the dimensions in the shape of the SW curve, we get the gauge theory. Its only definition at present is within string theory, the theory of a five-brane in M theory. Page 31 of 50

80 Re V Im S D W D Re V M Re S Σ In this picture, we see the two extra dimensions of a fivebrane, arranged to reproduce SU(2) gauge theory. The structure of the SW curve arises when the five-brane tries to minimize its surface area. The W bosons and monopoles are realized as strings (left) and membranes (right), respectively. From Henningson and Yi, hep-th/ Page 32 of 50

81 By now many other higher dimensional contexts have been found in which supersymmetric gauge theories can be usefully embedded: Page 33 of 50

82 By now many other higher dimensional contexts have been found in which supersymmetric gauge theories can be usefully embedded: Geometric engineering (Vafa, Ooguri and collaborators) taking limits of Calabi-Yau compactification of M theory, gauge theory solutions can be derived using mirror symmetry. Page 33 of 50

83 By now many other higher dimensional contexts have been found in which supersymmetric gauge theories can be usefully embedded: Geometric engineering (Vafa, Ooguri and collaborators) taking limits of Calabi-Yau compactification of M theory, gauge theory solutions can be derived using mirror symmetry. Branes on orbifolds and intersecting branes (Douglas, Moore, Greene, Morrison, Berkooz, Leigh) provides chiral matter, a necessary ingredient in all brane world constructions of the Standard Model. Page 33 of 50

84 By now many other higher dimensional contexts have been found in which supersymmetric gauge theories can be usefully embedded: Geometric engineering (Vafa, Ooguri and collaborators) taking limits of Calabi-Yau compactification of M theory, gauge theory solutions can be derived using mirror symmetry. Branes on orbifolds and intersecting branes (Douglas, Moore, Greene, Morrison, Berkooz, Leigh) provides chiral matter, a necessary ingredient in all brane world constructions of the Standard Model. AdS/CFT (Maldacena) N = 4 super Yang-Mills is dual to IIb string theory on AdS 5 S 5. Page 33 of 50

85 By now many other higher dimensional contexts have been found in which supersymmetric gauge theories can be usefully embedded: Geometric engineering (Vafa, Ooguri and collaborators) taking limits of Calabi-Yau compactification of M theory, gauge theory solutions can be derived using mirror symmetry. Branes on orbifolds and intersecting branes (Douglas, Moore, Greene, Morrison, Berkooz, Leigh) provides chiral matter, a necessary ingredient in all brane world constructions of the Standard Model. AdS/CFT (Maldacena) N = 4 super Yang-Mills is dual to IIb string theory on AdS 5 S 5. Gopakumar-Vafa duality: a related duality between wrapped branes and magnetic flux. They provide many senses in which supersymmetric gauge theory is just one limit of the web of dualities of string/m theory. Page 33 of 50

86 7. Can we use these reformulations of gauge theory to address the outstanding problems of strongly coupled QCD? So far, the most promising approach is to start with the AdS/CFT correspondence, which provides a simple dual theory (10d supergravity) for N = 4 super Yang-Mills. Physical phenomena related to the RG are translated into variation of couplings with respect to an extra radial dimension u, such as a warp factor ds 2 = du2 u 2 + u2 ( dt 2 + d x 2). We can then explicitly break N = 4 susy to N = 1 or even N = 0. In the supergravity language, this corresponds to changing the boundary conditions, in a small way at high energies, but which can drastically change the IR behavior. Page 34 of 50

87 Qualitative phenomena of gauge theory can be read off from properties of the supergravity solution. For example, confinement corresponds to a lower bound on the warp factor (or Euclidean event horizon). One can directly see phase transitions: deconfinement, chiral symmetry restoration following from the existence of multiple supergravity solutions. One can also get quantitative properties such as the spectrum of glueballs, but so far only in the large t Hooft coupling limit λ = g 2 N >> 1, which looks rather different from realistic gauge theory physics. One hope for going beyond this is the idea that N = 4 SYM (and perhaps some N < 4) corresponds to an integrable string theory (Minahan and Zarembo 2002). This idea has seen much recent progress (Beisert, Staudacher, Maldacena, etc...). Page 35 of 50

88 One can also study collective phenomena, such as hydrodynamic properties of the finite temperature theory. For example, the N = 4 theory at finite temperature T is dual to a Schwarzschild black hole in AdS with Hawking temperature T. There are hopes that this case is similar enough to QCD to serve as a useful model for the quark-gluon plasma studied at RHIC. Examples of such computations include The shear viscosity η (Son et al). By standard results, this is related to the two-point function of the stress-energy tensor. But, by AdS/CFT, this two-point function is related to the absorption cross section of the black hole, in other words the area of the event horizon. Page 36 of 50

89 One can also study collective phenomena, such as hydrodynamic properties of the finite temperature theory. For example, the N = 4 theory at finite temperature T is dual to a Schwarzschild black hole in AdS with Hawking temperature T. There are hopes that this case is similar enough to QCD to serve as a useful model for the quark-gluon plasma studied at RHIC. Examples of such computations include The shear viscosity η (Son et al). By standard results, this is related to the two-point function of the stress-energy tensor. But, by AdS/CFT, this two-point function is related to the absorption cross section of the black hole, in other words the area of the event horizon. By Bekenstein-Hawking, this is proportional to the entropy S. Thus, we get a relation η S 4πk B, which is empirically true, and may be roughly saturated for QCD. Page 36 of 50

90 One can also study collective phenomena, such as hydrodynamic properties of the finite temperature theory. For example, the N = 4 theory at finite temperature T is dual to a Schwarzschild black hole in AdS with Hawking temperature T. There are hopes that this case is similar enough to QCD to serve as a useful model for the quark-gluon plasma studied at RHIC. Examples of such computations include The shear viscosity η (Son et al). By standard results, this is related to the two-point function of the stress-energy tensor. But, by AdS/CFT, this two-point function is related to the absorption cross section of the black hole, in other words the area of the event horizon. By Bekenstein-Hawking, this is proportional to the entropy S. Thus, we get a relation η S 4πk B, which is empirically true, and may be roughly saturated for QCD. The jet quenching parameter which models energy loss of jets in a plasma (Liu, Rajagopal and Wiedemann). Page 36 of 50

91 8. The landscape of string theory While string/m theory is a remarkably unified framework, it appears to have many, many solutions with at least a passing resemblance to observed physics. Already in the late 80 s one had compactifications to four dimensions which realized the SM gauge group and matter; mathematical work suggests around different solutions. Page 37 of 50

92 8. The landscape of string theory While string/m theory is a remarkably unified framework, it appears to have many, many solutions with at least a passing resemblance to observed physics. Already in the late 80 s one had compactifications to four dimensions which realized the SM gauge group and matter; mathematical work suggests around different solutions. More recently, one can study the effective potential, and phase structure of string theory, using the field theory effective potentials we discussed. This can be justified in various ways, related by duality: A string compactification can contain branes wrapped about cycles of the internal manifold. Each group of N wrapped branes leads to an SU(N) supersymmetric Yang-Mills theory. It can also contain generalized magnetic flux which contributes to the potential as V = B 2. Page 37 of 50

93 8. The landscape of string theory While string/m theory is a remarkably unified framework, it appears to have many, many solutions with at least a passing resemblance to observed physics. Already in the late 80 s one had compactifications to four dimensions which realized the SM gauge group and matter; mathematical work suggests around different solutions. More recently, one can study the effective potential, and phase structure of string theory, using the field theory effective potentials we discussed. This can be justified in various ways, related by duality: A string compactification can contain branes wrapped about cycles of the internal manifold. Each group of N wrapped branes leads to an SU(N) supersymmetric Yang-Mills theory. It can also contain generalized magnetic flux which contributes to the potential as V = B 2. Either way, one can show that a compactification with K cycles will have on the order of N K distinct vacuum configurations, for some small N 10. Page 37 of 50

94 A typical Calabi-Yau manifold used in string compactification has 100 s of cycles. So each of the previous solutions, in fact represents N K distinct phases or vacua. (!?!) As pointed out by Bousso and Polchinski (2000), this can lead to a solution of the cosmological constant problem. Page 38 of 50

95 A typical Calabi-Yau manifold used in string compactification has 100 s of cycles. So each of the previous solutions, in fact represents N K distinct phases or vacua. (!?!) As pointed out by Bousso and Polchinski (2000), this can lead to a solution of the cosmological constant problem. We now have good observational evidence for a dark energy, probably a cosmological constant, of magnitude Λ = (0.71 ± 0.02)Ω, where Ω is the critical density beyond which the universe would eventually recollapse. Page 38 of 50

96 A typical Calabi-Yau manifold used in string compactification has 100 s of cycles. So each of the previous solutions, in fact represents N K distinct phases or vacua. (!?!) As pointed out by Bousso and Polchinski (2000), this can lead to a solution of the cosmological constant problem. We now have good observational evidence for a dark energy, probably a cosmological constant, of magnitude Λ = (0.71 ± 0.02)Ω, where Ω is the critical density beyond which the universe would eventually recollapse. Compared to the other scales of fundamental physics, this is incredibly small, Λ M 4 P lanck M 4 low E susy. And, to the extent that we can compute Λ in a string theory vacuum, all evidence is that it will come out roughly comparable to one of these larger scales. Page 38 of 50

97 A typical Calabi-Yau manifold used in string compactification has 100 s of cycles. So each of the previous solutions, in fact represents N K distinct phases or vacua. (!?!) As pointed out by Bousso and Polchinski (2000), this can lead to a solution of the cosmological constant problem. We now have good observational evidence for a dark energy, probably a cosmological constant, of magnitude Λ = (0.71 ± 0.02)Ω, where Ω is the critical density beyond which the universe would eventually recollapse. Compared to the other scales of fundamental physics, this is incredibly small, Λ M 4 P lanck M 4 low E susy. And, to the extent that we can compute Λ in a string theory vacuum, all evidence is that it will come out roughly comparable to one of these larger scales. But if string theory had solutions which otherwise agree with the SM, and have uniformly distributed cosmological constants, we can expect at least one solution to realize its seemingly fine-tuned value. One can even argue that we must live in such a universe (Weinberg, Banks, Linde 1988). Page 38 of 50

98 At present, this is the only generally accepted solution of the cosmological constant problem, so we have to take it seriously. But, it opens Pandora s box. We might be unhappy if we discovered that string theory had solutions which look like the SM. After all, the total accuracy with which all SM parameters have been measured, including the c.c., is only about Even future discoveries are unlikely to increase this too dramatically. So how can we test string theory? Page 39 of 50

99 At present, this is the only generally accepted solution of the cosmological constant problem, so we have to take it seriously. But, it opens Pandora s box. We might be unhappy if we discovered that string theory had solutions which look like the SM. After all, the total accuracy with which all SM parameters have been measured, including the c.c., is only about Even future discoveries are unlikely to increase this too dramatically. So how can we test string theory? There are many ideas (a subject for another talk), in many directions: Non-field theoretic physics (e.g. large extra dimensions) No-go theorems in string theory (e.g. variation of α EM ) Perhaps if we understood the number and distribution of vacua, we would find that some possibilities are highly preferred. Over the last few years, work on counting the solutions of string theory, finding the distribution of predictions they make, and looking for an a priori principle which might prefer a solution, has become a very active topic of research. It will require new physical insights, and long-term and in-depth interaction with mathematicians, such as has been cultivated for some years now at Rutgers, Stanford, and many other places. Page 39 of 50

100 9. Discovery of supersymmetry Many physicists are betting that supersymmetry will be discovered at LHC. The original argument for this dates to the late 70 s: The scale of electroweak symmetry breaking, say M Z = 91 GeV, is set by the parameters in the Higgs potential, say m. These parameters will obtain renormalization corrections at all scales up to a fundamental cutoff scale Λ, so we expect m Λ. The only obvious candidates for Λ, such as the Planck scale M p GeV, are far higher (the hierarchy problem ). The only general class of theories we know of in which m << Λ are supersymmetric field theories. In this case, m M SUSY, the scale of supersymmetry breaking. Page 40 of 50

101 9. Discovery of supersymmetry Many physicists are betting that supersymmetry will be discovered at LHC. The original argument for this dates to the late 70 s: The scale of electroweak symmetry breaking, say M Z = 91 GeV, is set by the parameters in the Higgs potential, say m. These parameters will obtain renormalization corrections at all scales up to a fundamental cutoff scale Λ, so we expect m Λ. The only obvious candidates for Λ, such as the Planck scale M p GeV, are far higher (the hierarchy problem ). The only general class of theories we know of in which m << Λ are supersymmetric field theories. In this case, m M SUSY, the scale of supersymmetry breaking. Of course, since no superpartners have been detected, supersymmetry is broken at the energies we have studied so far. But if supersymmetry is involved in solving the hierarchy problem, it is broken at low energy, predicting superpartner masses and couplings well within range of LHC. Page 40 of 50

102 Over the years, a certain amount of circumstantial evidence has built up for low energy supersymmetry: The RG extrapolated couplings of the three SM gauge groups become equal at a single high energy scale, suggesting a common origin in a grand unified theory. However, in light of precision coupling measurements, this does not work in the SM. But it works if we postulate the minimal additional supersymmetric matter (the MSSM): Page 41 of 50

103 Over the years, a certain amount of circumstantial evidence has built up for low energy supersymmetry: The RG extrapolated couplings of the three SM gauge groups become equal at a single high energy scale, suggesting a common origin in a grand unified theory. However, in light of precision coupling measurements, this does not work in the SM. But it works if we postulate the minimal additional supersymmetric matter (the MSSM): Standard Model extrapolated gauge couplings Page 41 of 50

104 Over the years, a certain amount of circumstantial evidence has built up for low energy supersymmetry: The RG extrapolated couplings of the three SM gauge groups become equal at a single high energy scale, suggesting a common origin in a grand unified theory. However, in light of precision coupling measurements, this does not work in the SM. But it works if we postulate the minimal additional supersymmetric matter (the MSSM): MSSM extrapolated gauge couplings Page 41 of 50

105 Circumstantial evidence for low energy supersymmetry: Coupling unification in the MSSM. The conjectured significance of the GUT scale GeV is loosely supported by other numerical coincidences: m ν M 2 Z /M GUT (the seesaw mechanism) H inflation M GUT in the simplest models of inflation. There is very good evidence that dark matter makes up about 25% of the energy density of the universe. Considerations of early cosmology rule out most proposals for the particles which could make it up. One of the few windows is for particles (WIMPs) with mass around 50 GeV and roughly weak strength interactions. These would be the expected properties of the lightest supersymmetric particle (LSP) in many versions of the MSSM. Page 42 of 50

106 Circumstantial evidence against low energy supersymmetry: strongly favors a low Higgs mass. This is because the coupling in the Higgs potential which determines this mass, is also determined by supersymmetry. At zeroth (tree) order this leads to the (false) relation M H M Z = 91 GeV (for the lightest Higgs). Page 43 of 50

107 Circumstantial evidence against low energy supersymmetry: strongly favors a low Higgs mass. This is because the coupling in the Higgs potential which determines this mass, is also determined by supersymmetry. At zeroth (tree) order this leads to the (false) relation M H M Z = 91 GeV (for the lightest Higgs). In fact, radiative corrections raise the actual value of M H, and adjusting parameters can help as well. But consistency with the present bound m H > 114 GeV already forces the MSSM into a corner of parameter space. The Higgs mass should be far below the existing bound 219 GeV from precision measurements. Page 43 of 50

108 Circumstantial evidence against low energy supersymmetry: strongly favors a low Higgs mass. This is because the coupling in the Higgs potential which determines this mass, is also determined by supersymmetry. At zeroth (tree) order this leads to the (false) relation M H M Z = 91 GeV (for the lightest Higgs). In fact, radiative corrections raise the actual value of M H, and adjusting parameters can help as well. But consistency with the present bound m H > 114 GeV already forces the MSSM into a corner of parameter space. The Higgs mass should be far below the existing bound 219 GeV from precision measurements. In general, the Standard Model works too well: many versions of the MSSM predict subtle corrections (such as flavor changing neutral currents) which would already have been seen. Neither objection is conclusive, but given the fact that the MSSM is no simpler than and in some ways more complicated than the SM, are cause for doubt. Page 43 of 50

109 Of course, the fact that supersymmetry is a necessary part of superstring theory is another strong point in its favor. BUT string theory does not require low energy supersymmetry. One can come up with scenarios in which supersymmetry is broken at high energy, and the hierarchy problem is solved in other ways: technicolor there is no Higgs, rather technipions. composite quarks and leptons large extra dimension models (brane worlds) other field theories with better RG behavior ( little Higgs, fixed point theories, quiver/deconstruction,...) fine tuning (anthopic, entropic or otherwise). Page 44 of 50

110 Of course, the fact that supersymmetry is a necessary part of superstring theory is another strong point in its favor. BUT string theory does not require low energy supersymmetry. One can come up with scenarios in which supersymmetry is broken at high energy, and the hierarchy problem is solved in other ways: technicolor there is no Higgs, rather technipions. composite quarks and leptons large extra dimension models (brane worlds) other field theories with better RG behavior ( little Higgs, fixed point theories, quiver/deconstruction,...) fine tuning (anthopic, entropic or otherwise). To say anything about which alternatives string theory prefers, we need to understand breaking mechanisms Stringy naturalness; the question of how many vacua realize each candidate mechanism, and how likely each one is. While the latter topic is in its infancy, we start to see what it would mean to form a picture of the landscape, and use it to make predictions. Page 44 of 50

111 10. Some arguments suggest, and for all we know it may turn out, that string theory actually favors high scale supersymmetry breaking. To give the gist of the argument, it is that both scenarios (low and high scale breaking) have problems with stringy naturalness: breaking at a high scale E: need to tune the Higgs mass M H << E, with likelihood M 2 H /E for E M GUT GeV. breaking at a low scale E 10M H 1 TeV. While this makes a low Higgs mass natural, it requires many other tunings to solve the µ term, flavor changing neutral currents, and other problems. It may turn out that the likelihood of this is far smaller than If high scale supersymmetry breaking is realized in nature, then many of the traditional collider physics signatures of supersymmetry (missing energy, light colored squarks, etc.), will NOT be realized. Nevertheless, many high scale models have other distinctive collider physics signatures. Page 45 of 50

112 A good example, motivated by the string theory landscape, is the split supersymmetry proposal of Arkani-Hamed, Dimopoulos, Giudice and Romanino In this proposal, Fermionic superpartners (gauginos and Higgsinos) are light. Bosonic superpartners (squarks and sleptons) are heavy. Page 46 of 50

113 A good example, motivated by the string theory landscape, is the split supersymmetry proposal of Arkani-Hamed, Dimopoulos, Giudice and Romanino In this proposal, Fermionic superpartners (gauginos and Higgsinos) are light. Bosonic superpartners (squarks and sleptons) are heavy. At first this may seem arbitrary, and according to previous thinking, unnatural. But it might turn out to be natural in string theory! And, from a bottom-up point of view, it has many nice features: The neutralinos can provide a WIMP dark matter candidate. GUT unification works almost as well as low scale susy theories. The worst problems of the generic MSSM (FCNC and other disagreements with precision measurements) are caused by the bosonic superpartners, so are solved or mitigated. The expectations for M H 170 GeV. rise from GeV to 140 Furthermore, gaugino couplings still satisfy supersymmetry relations, so this model, and many other new variations on the MSSM, emerging from the study of the stringy landscape, are testable. Page 46 of 50