On the Perturbative Stability of des QFT s

Transcription

1 On the Perturbative Stability of des QFT s D. Boyanovsky, R.H. arxiv: PPCC Workshop, IGC PSU 22

2 Outline Is de Sitter space stable? Polyakov s views Some quantum Mechanics: The Wigner- Weisskopf Method WW in de Sitter Space Conclusions and Further Directions

3 .. Is de Sitter Space Stable? Why worry? 46 Union2 data-set µ µ -µ empty Λ-CDM.. Redshift

4 Why wouldn t de Sitter space be stable? Polyakov: IR behavior of QF s is such that interactions produce so many particles that the in and out vacua become inequivalent. This is the ``adiabatic catastrophe There are some calculations that claim to bear this out. ALL use some kind of S-matrix type argument: many particles H I vacuum = vacuum decay rate But is this how it works out?

5 The Wigner-Weisskopf Method: QM and QFT If you want to know what happens to a quantum state, do quantum mechanics! H = H + H I i t (t) I = H I (t) (t) I (t) I = X n C n (t) n iċn(t) = X m n H I (t) m C m (t)

6 The Wigner-Weisskopf Method: QM and QFT If you want to know what happens to a quantum state, do quantum mechanics! H = H + H I i t (t) I = H I (t) (t) I (t) I = X n C n (t) n iċn(t) = X m n H I (t) m C m (t)

7 In general, an infinite dimensional mess! But s pose that at some order in the interaction one state is only connected to a subset of states. A { } A κ κ A κ H I A A H I κ FIG. : Transitions in first order in. Then we can restrict ourselves to this sector and set up a simpler set of equations C (t) = i Ċ A (t) = Z t Z t H I (t ) A C A (t ) dt (t, t ) C A (t ) dt (t, t )= X A H I (t) H I (t ) A C A () =, C () = How do we solve this integro-differential equation?

8 The Markovian Approximation The kernel is perturbatively ``slow. S pose it s mostly constant over the time range under consideration. Z t dt (t, t )C A (t )= ) C A (t) C A () exp Z t Z t dt (t, t ) C A (t) dt (t, t ) We can systematize this approximation as a consistent expansion in derivatives of the coefficient of A W (t, t )= Z t Z t (t, t )dt ) (t, t ) C A (t ) dt = W (t, t) C A (t) (t, t )= d dt W (t, t ), W (t, ) = R t dt W (t, t ) d dt C A(t ) {z } 4 th order W (t, t )= Z t Z t Z t W (t, t )dt, W (t, ) = W (t, t ) d dt C A(t ) dt = W (t, t) ĊA(t)+ (t, t ) C A (t ) dt = W (t, t) C A (t) W (t, t) ĊA(t)+

9 Finally Ċ A (t)[ W (t, t)] + W (t, t)c A (t) = C A (t) =e i R t E(t )dt, E(t) = iw (t, t) W (t, t) iw (t, t)[+w (t, t)+ ] In the Markovian approximation C (t) = i Z t dt H I (t ) A exp i Z t dt E(t )! st order PT matrix elt C (t) = i Z t dt H I (t ) A WW goes beyond PT and is valid at LATE times

10 Does this actually work? Look at case where interaction is time independent. C A (t) = I( ) = Z Z d 2 i e i t I( ) i d ( ) + E A i C (t) = i H I A Z t e i(e E A)t C A (t ) (t, t )= X A H I 2 e i(e A E )(t t ) Z d ( ) e i(e A )(t t ) ( )= X A H I 2 (E ) Late time evolution determined by pole nearest the real axis. With no interaction, pole is at origin. C A (t) Z A e i Z A = Er A t e +z A z A r A 2 t I( ) E A z A + i E A = P z A = P Z Z A =2 (E A ) A 2 d ( ) E A d ( ) (E A ) 2 EA r = Z A E A r A = Z A A

11 How does the Markovian approximation do in this case? E(t) = i Z t dt (t, t )= Z d ( ) h i E A e i( E A )t Z t A(t) = B(t) = dt E(t )=ta(t) Z Z ib(t) d ( ) (E A ) apple sin( E A )t ( E A )t t P Z d ( ) (E A ) 2 [ cos( E A )t] t t (E A )+P d ( ) (E A ) Z d ( ) (E A ) 2 This gives the same result as the exact answer!

12 Some Flat space QFT results ex. H I (t) = Z d 3 x : 4 ( x, t) : Normal ordering connects vacuum to 4-particle state at leading order κ H I H I κ κ H I (t, t )= Z ( )= 2 V d ( )e i (t t ) 3Y Z i= P d 3 3 k i ( i= k i + k tot ) (2 ) 3 6 k k 2 k 3 k tot C (t) =e z e i E = z = 2 V 2 V Z Z 3Y i= E t 3Y i= =2 ( = ) = d 3 k i (2 ) 3 6 k k 2 k 3 k tot d 3 k i (2 ) 3 6 k k 2 k 3 k tot P3 i= k i + k tot P3 i= k i + k tot 2 Note that vacuum is stable, as Nature intended!

13 ex 2. H I (t) =M Z d 3 x ( x, t) 2 ( x, t) Take phi to be massive and chi massless κ H I H I κ κ H I C (t) =e z e i E = z = P P Z Z E t d vac( ) d vac( ) 2 vac(t, t )= Z vac ( )=VM 2 Z d vac ( ) d 3 p (2 ) 3 Z d 3 k ( E p k k + p ) (2 ) 3 2E p 2k 2 k + p Suppose we take all fields massless (t, t )= E = im 2 V (4 ) 4 (t t i ) 3,! + M 2 V 2(4 ) 4 2, z = M 2 V 2(4 ) 4

14 WW in de Sitter Space Consider CONFORMALLY COUPLED scalar in (Poincare patch of) DeS Do the conformal rescaling of the field to make it look like flat space QFT ( x, )=a( ) ( x, ) S = 2 Z d 3 xd ( 2 h 2 i ( ) 2 M 2 ( ) 2 + ga (4 p) ( ) p ) Now quantize using BD mode functions in a comoving box ( x, )= p V X k ha k g (k; ) e i k x + a k g (k; ) e i k x i g (k; )= 2 i 2 p H (2) (k ) a k BD = Interaction Hamiltonian is H I ( )= g ( H ) 4 p Z d 3 x : p ( x, ):

15 Now just use the WW formalism as in flat space, now using BD Fock space states Z C ( )= i d H I ( ) A C A ( ) Z C A ( )= d (, (, )= X A H I ( ) H I ( ) A C A ( )=, C ( )= ) Markovian Approximation C A ( )=e W (, )= R W (, ) d Z (, )d Now go back and revisit our previous examples

16 ex. Z H I ( )= d 3 x : 4 ( x, ): Just like flat space in terms of rescaled field. Vacuum is stable, just as in flat space Disagrees with Higuchi who uses st order PT to calculate decay rate. Resolution: Having a non-zero matrix elt of interaction between vacuum and 4-particle state does NOT mean vacuum decays. It gets dressed up instead! Adiabatic turn on gives exact ground state e = U (, ) = Z X = i H I (t) e z = X e 2 t dt

17 ex 2. H I ( )= M ( H) Z d 3 x 3 ( x, ) = (, )= M 2 V H 2 Z d 3 p (2 ) 3 im 2 V (4 ) 4 H 2 h i Z i 3 d 3 k e i(p+k+ k+ p )( (2 ) 3 2p 2k 2 k + p ) C ( ) ' e i ( ) e z ( ) " M 2 V ( ) = 2(4 ) 4 H 2 ( ) 2 " # M 2 V z ( ) = 2(4 ) 4 H ln # Now there IS vacuum decay; wave function renormalization is time dependent and grows as we go into the far IR We can make contact with the flat space case here in a particular renormalization scheme where UV cutoff is constant in PHYSICAL coords. ( H ) = constant z = M 2 V phys ( ) 2(4 ) 4 It gets harder and harder to overlap dressed state with the bare one as the universe expands

18 Non-Pertubative Screening Mechanism? Recall C ( )=e i R d E( ), E( )= iw (, ) W (, ) Can W (, ) become secular? If it did, maybe it could cancel late time behavior of the wavefunction renormalization. This would be like finding a dressed state with very small, but NON-ZERO overlap with the bare state; no adiabatic catastrophe.

19 Some Results for Minimally Coupled Fields Boyanovsky has followed up this work with the minimally coupled case. He finds some interesting (or odd, depending!) results. For a cubic theory, a condensate forms i.e. the field develops a late time expectation value and a dynamical mass!

20 [ (η) = +i λ H η dη η η d 3 x χ 3 ( x, η )] [ ] [ ] χ( y, η) =3i λ η dη [ ] d 3 x χ( y, η), χ( x, η ) χ 2 ( x, η ), H η η [ ] χ( y, η) = 3 λ 6 π H ν [ ( η ) ln η η 3 ] dz z z3 H () ν (z) 2

21 h (~y, )i = [ + ] M 2 = H p 3 apple = M 2 3H 2 2 H 3 This is weird; the cubic interaction doesn t turn over in flat space. What does this say about the two-point function?

22 DRG Resummation of super-hubble Fluctuations

23 DRG Resummation of Secular Growth Two types of secular logs coming from quantum corrections ln =ln a( ) a( ) DeS inv. broken by a beginning of inflation ln( k )=ln a( k) a( ),a( k)= k H These show up in higher order corrections even in De S

24 In-In formalism In cosmology we need to calculate time dependent expectation values O(t) Tr( (t)o(t)) = Tr( (t )U (t, t )O(t)U(t, t )) This corresponds to a path integral defined + contour on a closed time contour contour τ τ

25 Field content doubled { +, } C = = + Times on - contour + contour are later than those on + contour contour τ τ 3 Green s functions h C (x) C (y)i = ig C (x, y) h C (x) (y)i = G R (x, y), h (x) C (y)i = G A (x, y)

26 L = g 2 gµ µ 2 m2 + R 2 4! 4 L( c, )= g g µ µ C 4! 4 3 C + C 3 +c.t. G R(k,, 2) ( 2 ) H2 3 ( 3 G C(k,, 2) H2 2k 3 +O((k ) 2 ) 3 2 ) +O((k ) 2 )

27 The vertices are

28 Loop Corrections and Secular behavior Let s go to the tadpole graph τ τ 2 + τ τ 2 IR cutoff is unphysical; should be replaced by physical scale L due to missing physics

29 For logs, IR cutoff from UV calculation can give us full dependence on L In our case, the choice is whether L depends on time or not. mass term: L is time indep Pre-inflationary physics: L time dep

30 Now use this to correct propagator G C (k, ) = H2 2k 3 H2 2k 3 apple + 3(2 ) 2 ln µ IR apple + 3(2 ) 2 ln ( k )+ ln (µl) ln ( k ) + How to resum the secular terms? How can we fix L?

31 Let s recall how the RG works:. Compute -loop corrected coupling (µ) = (µ )+b 2 (µ )ln µ µ valid for (µ ), (µ )ln(µ/µ ) 2. Differentiate then integrate wrt subtraction point (µ) = (µ ) b ln µ µ, valid for Domain of validity has been extended What is the time dependent analog?

32 Another (Easier) Secular problem: Damped SHO in PT ÿ + y = ẏ, Exact solution: y(t) =y e 2 t cos t r 2 4 +! Perturbative solution y(t) =y e it 2 t t2 + i 2 8 t +c.c. + non secular

33 DRG Resummation y = A( )Z( ) Z( ) =+ z ( )+ 2 z 2 ( )+ z ( ) = 2, z 2( ) = 2 i 8 8 Coefficients chosen to cancel secular behavior at a time tau y(t, ) =A( ) e it 2 (t )+ 2 8 (t )2 + i 2 8 +c.c. + non (t ) + secular Now demand tau independence: DE for A(tau)

34 A( ) =A() exp( dy(t, ) d =) 2 + i 2 8 ) Finally use arbitrariness of tau to set tau=t y(t) =A() exp 2 t + i t +c.c

35 Example: If y(t) =c + f(t)+o 2 the DRG improvement is y(t) =ce f(t) +O 2 G C (k, ) = H2 2k 3 Now let s work this on the two G C (k, ) = H2 2k 3 exp = H2 2k 3 point function: apple + k ah = ln (µl) 3(2 ) 2 ln (µl)ln( k )+ 3(2 ) 2 apple + ln (µl)ln( k ) ( + ) 3(2 ) 2 +O( 2 )

36 Finally, try a cubic theory. Does the IR regulating physics look like a mass? h G C (k,, ) =G 2 C(k,, )exp 9H 2 apple (2 ) 2 ln3 ( k )+ 4 H 2 ln2 ( k )+... Not a mass which would have been no surprise since potential is ill behaved. However, there are hints of NP generation of a condensate from WW analysis.

37 Discussion and Further Questions We have a formalism that resums late time behavior and can track the evolution of quantum states. In des with conformally coupled scalars, we can see examples of vacuum decay. However, other examples just show state dressing, NOT decay. It s really time dependent wavefunction renormalizations that diverge at late time/large physical volume that drive the decay of the state. Decay of one particle states has its own interesting story (Read our paper!) How to go beyond leading Markovian approx? Mass generation in massless MINIMALLY coupled theory? Relationship to DRG results?