T-reflection and the vacuum energy in confining large N theories

Transcription

1 T-reflection and the vacuum energy in confining large N theories Aleksey Cherman! FTPI, University of Minnesota! with Gokce Basar (Stony Brook -> U. Maryland),! David McGady (Princeton U.),! and Masahito Yamazaki (IAS -> IPMU) arxiv: , arxiv:

2 Vacuum energy Empty space is never truly empty! Quantum field theory implies that quantum fluctuations fill space with vacuum energy. Often only energy differences matter, so vacuum energy is not important. But vacuum energy can have observable consequences! If we put a QFT in a box, the vacuum (Casimir) energy induces a dependence on the box size. Vacuum energy gravitates, and plays a dramatic role in cosmology. The expansion of the universe is accelerating! Nobel Prize, 2011 So how much vacuum energy do we get, given a QFT?

3 Vacuum energy puzzle Vacuum energy V0 determined by the spectrum of a system: Renormalization required, as always with quantum fluctuations! In a (weakly-coupled) QFT with a mass gap, expect mass of heaviest particle species So in the Standard Model of particle physics, we estimate

4 Vacuum energy puzzle Vacuum energy V0 determined by the spectrum of a system: Renormalization required, as always with quantum fluctuations! In a (weakly-coupled) QFT with a mass gap, expect mass of heaviest particle species So in the Standard Model of particle physics, we estimate Arguably the biggest failure of a wellmotivated estimate in the history of physics! accelerating expansion of universe

5 Ways to escape V0 ~ Lmax estimate (1) Supersymmetry: But if SUSY is a property of our universe, it has to be broken Same big problem as SM (2) Conformal symmetry No mass scales means that after renormalization V0 = 0. But conformal symmetry broken in SM - world is full of mass scales!

6 Ways to escape V0 ~ Lmax estimate (3) Declare that V0 is anthropically fine-tuned If V0 ~ Lmax, we wouldn t be here to be puzzled, so declare that V0 is very carefully tuned to be tiny, without any symmetry reason Many find this philosophically distasteful Amounts to giving up on getting any deep understanding of the issue. (4) Modify long-distance behavior of gravity Extremely difficult to do without getting into trouble with theoretical self-consistency and observational constraints

7 Ways to escape V0 ~ Lmax estimate in QFT Those are the only current options.

8 Ways to escape V0 ~ Lmax estimate in QFT Those are the only current options. The situation is desperate - any new examples where V0 ~ Lmax estimate is evaded should be interesting Could there be other symmetries that constrain V0?

9 Preview of the punchline Basar, AC, McGady, Yamazaki! arxiv: How can the standard estimate go wrong? QFTs with mass gap: mass of heaviest particle species Conformal symmetry: no gap, no scales SUSY: term by term cancellation due to (-1) F

10 Preview of the punchline Basar, AC, McGady, Yamazaki! arxiv: How can the standard estimate go wrong? QFTs with mass gap: mass of heaviest particle species Conformal symmetry: no gap, no scales SUSY: term by term cancellation due to (-1) F Our question: what if there is no heaviest particle? There are quantum field theories with an infinite number of narrow particle modes with increasing masses. N = confining gauge theories weakly-coupled string theories Limit of QCD

11 Outline (1) Introduce a technical tool, T-reflection symmetry: Formal property present in wide variety of QM and QFT systems. T-reflection symmetry broken by shifts of the vacuum energy! (2) Explain most surprising prediction - and its explicit verification. 4D N = confining gauge theories with adjoint matter on S 3 x S 1 are calculable when S 3 is small. They enjoy T-reflection symmetry and in this case T-reflection turns out to imply that their renormalized vacuum energy vanishes.

12 T-reflection Basar, AC, McGady, Yamazaki! arxiv: Consider a thermodynamic partition function: What happens if we formally send T to -T? At first, this looks like a crazy question. Sum clearly diverges for β < 0, so would need to define an appropriate continuation/regularization of Z(-β) to make it well-posed. Turns out there is actually a very natural definition of Z(-β) But directly checking behavior of Z(T) under T- reflection seems to require exact partition function There are few systems where checking what happens if T goes to -T is practical - but they do exist, so let s look at some of them!

13 Fermionic harmonic oscillator Start with one of simplest QM systems, a two-level system: So if and only if V0 = -1/2 ω, we have cosh is even in β! Looks rather trivial - can always choose V0 in an arbitrary two level system to make T-reflection work!

14 Three-level systems Random 3-level system: Clearly for any choice of V0 So T-reflection symmetry cannot be present in all systems with more than two levels. Could it be present in some systems with an infinite number of levels? Worth looking at examples

15 Bosonic harmonic oscillator sinh is odd in β! Observe that, if and only if V0 = 1/2 ω, we have T-reflection symmetry only present if vacuum energy set to its correct `quantum value in the working examples.

16 2D quantum field theory So T-reflection symmetry holds if and only if V0 = -1/(24L). Then

17 Deeper look at T-reflection in QFT In quantum field theory (in a spatial box V) can formally write Classical and quantum contributions to V0 Thermal modes Expression needs UV regularization to make it precise! With a UV regulator understood, natural way to define Z(-β): This can be written in a more illuminating form

18 Deeper look at T-reflection in QFT versus means that T-reflection symmetry will hold if

19 Examples of QFTs with T-reflection symmetry With our current primitive understanding of T- reflection, need to have exact Z(T) to check it All free QFTs we looked at have it - from 2D to 4D, in flat space and on S 3 xs 1 Not restricted to conformal theories, works for free massive scalar QFTs T-reflection also works for all 2d minimal model CFTs, and partition functions of assorted supersymmetric gauge theories strongly interacting! T-reflection symmetry also present in confining large N gauge theories on S 3 xs 1

20 But what is T-reflection symmetry good for? If anything, T-reflection symmetry actually seems to make the vacuum energy puzzle worse! Insisting on T-reflection => no ability to dial classical part of V0 Fine-tuning of V cl 0 gives the only known way to deal with the vacuum energy puzzle in QFT, so it seems useless To see why things are not so grim, consider confining non-abelian gauge theories.

21 N = confining gauge theories Real-world QCD, and other confining gauge theories, have no obvious control parameters Strongly coupled at low energies, generally intractable except using e.g. numerics t Hooft 1974: number of colors N is a control parameter! Theory remains confining for any N, with a mass gap L. At N =, mesons and glueballs become stable and non-interacting. Infinite number of mesons and glueballs! For our N = 3 world, 1/N expansion looks quite reasonable! Actually solving N = theory is still only a dream for QCD.

22 Vacuum energy in N = confining QFTs What should we expect for V0? makes no sense! Still have a mass gap L, so a natural guess is If we want to check this, two obvious issues (1) Confining theories tend to be strongly-coupled! How can we hope to calculate V0 in any nonsupersymmetric confining gauge theory? (2) V0 even more divergent than usual due to number of particles Is there a sensible definition of V0 at N =? There is a setting where both issues can be addressed!

23 N = gauge theories on S 3 xs 1 Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk! 2003 Asymptotically-free QFTs become weakly-coupled if S 3 radius R << L -1 Gauss law constraint on S 3 implies that only color-singlet states are part of Hilbert space Flux lines can t end at spatial infinity as on R 3. Physical excitations associated to single-trace and multi-trace operators Weakly-coupled R L << 1 large N theory turns out to be confining! Mass gap L =1/R Unbroken center symmetry when S 1 is large (low temperature) ~N 0 free energy for low T Infinite number of stable particles with increasing masses Spectrum is exactly calculable when R L > 0

24 N = partition function We need to figure out the energies and degeneracies of the states In R L << 1 limit theory described by excitations of an infinite number of adjoint harmonic oscillators, with a singlet constraint At N = single-trace states do not interact, so space of multi-trace states is just Fock space of single-trace states Means we can think of single-trace partition function as the single-particle partition function, while including multi-trace states = including multi-particle states So we need to count the single-trace states!

25 N = single-trace YM partition function Rough estimate for pure YM theory zv is a partition function counting the distinct oscillators associated to Maxwell gauge field on S 3 Rough estimate energies and degeneracies of massless vector field on S 3 Estimate does not correctly deal with repetitions of oscillators

26 N = single-trace YM partition function Exact single-trace partition function for large N pure YM theory Full (grand canonical) partition function can be written as

27 T-reflection for N = YM theory What does T-reflection predicts for the vacuum (Casimir) energy? This is a shocking result. z-function regularization T-reflection symmetry is telling us that the renormalized N = vacuum energy of pure YM theory vanishes. Is it some strange coincidence, true only for pure YM?

28 T-reflection for N = adjoint-matter gauge theories Have also checked what happens for theories with adjoint fermions and scalars added zs, zf are odd in β T-reflection continues to work with. Seems to be a generic feature of N = confining theories on S 3 xr! But this is a quite unexpected result and cries out for a check by other methods!

29 Regularizing and renormalizing V0 We now compute V0 directly. heat-kernel type regularization So the renormalized N = V0 can be calculated from Have used this to compute V0 analytically and numerically

30 Singularities on [0,1] tied to Hagedorn behavior Others are more mysterious (to us) Re x Im x YM Theory Z ST [x = e 1/(μR) ] has singularities

31 Regularized vacuum (Casimir) energy Typical form of regularized V0 in a 4D QFT with on S 3 : Valid if R is dominant IR scale number of spacetime dimensions renormalized Casimir vacuum energy As usual, divergences absorbed by local counterterms

32 The results We evaluated V0 from ZST analytically and numerically. Result for 4D N = confining theories with adjoint matter: Leading divergence is μ 2, not μ 4. Confining N = theories believed to have dual free string theory description Could this be connected of 2D nature of string worldsheet?

33 The results Basar, AC, McGady, Yamazaki! arxiv: We evaluated V0 from ZST analytically and numerically. Result for 4D N = confining theories with adjoint matter: The renormalized vacuum energy vanishes: Did not need to fine tune to get the result; happens for any ns, nf Hard to see how this could be an accident Now trying to understand what symmetry could be responsible.

34 Conclusions How can the standard vacuum energy estimate go wrong? QFTs with mass gap: mass of heaviest particle species Conformal symmetry: no gap, no scales SUSY: term by term cancellation due to (-1) F

35 Conclusions How can the standard vacuum energy estimate go wrong? QFTs with mass gap: mass of heaviest particle species Our question: what if there is no heaviest particle? Calculated V0 in 4D N = confining theories with adjoint matter Is there some emergent symmetry in confining N = theories (and their string duals) behind this result? Coleman-Mandula-type no-go theorems do not apply at N =, since the N = S-matrix vanishes