Renormalized CFT/ Effective AdS. JiJi Fan Princeton University Work in progress

Transcription

1 Renormalized CFT/ Effective AdS JiJi Fan Princeton University Work in progress

2 Outline Motivation and goal Prescription Examples Conclusion and open questions

3 AdS/CFT correspondence: Maldacena conjecture `97 N=4 conformal SYM g s =g 2 YM Type IIB string theory on AdS 5 S 5 R 4 =4πg s N l 4 s Phenomenological works motivated by the duality: Randall-Sundrum models AdS/QCD bottom-up AdS/CMT

4 Rules of thumb for model-building Copied from Csaki, Hubisz and Meade 05

5 But unlike the original N=4/Type IIB duality, the phenomenological duality Finite N Non-supersymmetric

6 The phenomenological duality Finite N Non-supersymmetric Higher order corrections Instability (won t consider further) e.g.: Horowitz-Orgera-Polchinski instablity `07;

7 Goal: assuming existence of an effective AdS field theory/cft correspondence, understand how bulk interactions renormalize CFT 2-point correlator beyond the leading order in N. More specifically, calculate anomalous dimensions of single- and double-trace operators arising from bulk interactions. E.g.: scalar single-trace operator: = d 2 + d m2 + γ

8 Aside: definition of single- and double-trace operators Given a matrix-valued field Φ(N N hermitian matrix), Single-trace operators: Double-trace operators: Example: TrΦ n (TrΦ n ) 2 Scalar operators Single-trace operators: Double-trace operators: TrF 2 ψ ψ (TrF 2 ) 2 (ψ ψ) 2

9 More on the effective AdS/CFT correspondence Conjecture: (Heemskerk, Penedones, Polchinski and Sully `09; Fitzpatrick, Katz, Poland, Simmons-Duffin `10) Sufficient conditions for CFT to have a local bulk dual a mass gap: all single-trace operators of spin greater than 2 have parametrically large dimension. More specifically, m KK ~ 1/R; R: AdS radius String states m s ~ 1/l s ~ λ 1/4 /R; a small parameter such as 1/N M 5 ~ N 2/3 /R

10 m planck m s 5D AdS EFT 1/R 0

11 Setup AdS d+1 : metric ds 2 = z2 0 + d i=1 dx2 i scalar field theory with contact interactions z 2 0 CFT: at the bottom of the spectrum, only one single-trace scalar operator O(x) order in 1/N, the primary operators appearing in the O O operator product expansion (OPE) are the double-trace operators Double-trace operator from OPE O O O n,l (x) O( µ µ ) n ( ν1 ν2 νl )O(x), (2.7) where parameters n and l denote the twist n: twist and the spin of l: spin the operator. At the zeroth order in 1/N, the double-trace operator dimension is n,l =2 +2n + l. Once the bulk interactions are turned on, the CFT two-point function would be renormalized. For the sinlge-trace operator, the leading order correction to the bare two-point correlator can be obtained from the non-zero leading order expansion of the boundary 1-PI action ɛ 2(d ) φ 0 (x)φ 0 (0)e S bulk(φ) 1PI non local, (2.8)

12 Boundary action To make precise sense of the AdS/CFT, needs to introduce a radial regulator ε: z 0 ε > 0 S ɛ = Sɛ local S = S ɛ + + Sɛ non local d d x z 0 >ɛ dz 0 L bulk

13 S local ɛ Consist of all (high-dimensional) local counter-terms to restore conformal invariance. Requiring the total action S is independent of ε, holographic RGE of the boundary local operators at classical level: ɛ S local ɛ = z=ɛ d d x(π z φ gl) = d d xh Lewandowski, May and Sundrum `02 ; Heemskerk and Polchinski `10; Faulkner, Liu and Rangamani `10.

14 S non local ɛ Encodes the correlators of the dual field theory. The ansatz: exp ɛ Generating function of CFT Boundary value of bulk field φ 0 O CF T = Z ɛ (φ 0 ) Boundary partition function O(x)O(0) = ɛ 2(d ) φ 0 (x)φ 0 (0) 1PI nonlocal Normalization

15 S non local ɛ Continued Example: free AdS scalar theory with Dirichlet b.c: (mixed momentum-position rep.) S ɛ = 1 2 = 1 2 d d kd d k δ( k + k )ɛ d+1 φ 0 (ɛ, k) z0 φ(z 0, k ) z0 =ɛ d d kd d k δ( k + k )ɛ d+1 φ 0 ( k)( z0 K(z 0, k ) z0 =ɛφ 0 ( k )) O( k)o( k ) φ(z 0, k) = K(z 0, k)φ 0 ( k) K(z 0, ( z0 ) d/2 K ν (kz 0 ) k) = ɛ K ν (kɛ) Bulk to boundary propagator

16 Prescription After turning on bulk interactions, CFT gets renormalized O R (x) =ZO(x) Calculate correction to the two-point O(x)O(0) (1) from AdS : correction to the 1PI boundary action ɛ 2(d ) φ 0 (x)φ 0 (0) 1PI non local Insertions of bulk interaction

17 The renormalized two-point functions are Finite after taking the regulator away O R (x)o R (0) = Z 2 O(x)O(0) ɛ 0 Z factor absorbs the ε dependence, more specifically, log ε dependence γ ɛ log Z ɛ

18 A toy example: mass perturbation δm 2 φ 2 Exact solution: exact = d 2 + (d 2 ) 2 + m 2 + δm 2 = 0 + δm2 2ν + O((δm2 ) 2 ) ν = d m2

19 δm 2 φ 2 δm 2 A bit of information beforehand: Main ingredient for calculating anomolous dimensions from contact interactions In this representation, Eq. 3.2 turns into dz0 d d z 1 φ 0 (x)φ 0 (0) z0 d+1 2 δm2 φ(z 0, z) 2 1PI nonlocal = δm 2 dz 0 1 d d ( ) k Kν (kz 0 ) 2 ɛ z 0 ɛ d (2π) d e ik x (3.4) K ν (kɛ) = 2δm 2 dz 0 d d ( ) k k 2ν Γ(1 ν) z 0 (2π) d ɛ2ν d e ik x + (3.5) 2 Γ(1 + ν) = + ɛ 2ν d log ɛ 2δm2 π d/2 ɛ Γ( ) Γ( d/2) x 2 +, (3.6) where in the third line we expand the Bessel functions for small arguments kz, kɛ 1 and keep only the leading logarithmic divergent term in the last two lines. Notice that with the bulk-boundary correlator we chose, the two point function at the classical level is normalized as [16] O(x)O(0) (0) = 2ν d/2 Γ( ) x 2. (3.7)

20 δm 2 φ 2 δm 2 A bit of information beforehand: Main ingredient for calculating anomalous dimensions from contact interactions dz0 d d z φ 0 (x)φ 0 (0) z d+1 0 = + ɛ 2ν d log ɛ 2δm2 π d/2 Z 2 1 = δm2 ν γ = ɛ log Z ɛ 1 2 δm2 φ(z 0,z) 2 1PI nonlocal Γ( ) Γ( d/2) x 2 + log ɛ + finite terms = δm2 2ν Compared to the exact solution: exact = d 2 + (d 2 ) 2 + m 2 + δm 2 = 0 + δm2 2ν + O((δm2 ) 2 )

21 On the CFT side, O(x)O(0) = Z 2 1=2γ log Λ 1 x = 1 (1 2γ log( x Λ)) 2( +γ) x 2 γ = log Z log Λ UV/IR duality Λ 1 ɛ γ ɛ log Z ɛ

22 Example 2: mass perturbation again -- toy model of integrating out heavy states V = 1 2 m2 1φ m2 2χ 2 δm 2 φχ, δm 2 m 2 1 m 2 2 Again one could obtain the exact solution: ( m 2 1 δm 2 ) Diagonalize mass matrix δm 2 m 2 2 = d 2 + d m2 γ 1 = (δm2 ) 2 m 2 1 m ν 1

23 Example 2: continued φ δm 2 δm 2 χ φ φ 0 (x)φ 0 (0) (1) = dz0 z d+1 0 = 2(δm 2 ) 2 γ 1 = (δm2 ) 2 m 2 1 m2 2 dw 0 w0 d+1 d d k dz 0 1 K ν1 (kz 0 ) z 0 ɛ d K ν1 (kɛ) K ν 2 (kz 0 ) ɛ 1 2ν 1 (2π) d e ik x K(z 0,k)G(z 0,w 0 ; k)k(w 0,k) z0 ɛ dw 0 I ν2 (kw 0 ) K ν 1 (kw 0 ) w 0 K ν1 (kɛ)

24 Example 2:Continue V = 1 2 m2 1φ m2 2χ 2 δm 2 φχ, δm 2 m 2 1 m 2 2 γ 1 = (δm2 ) 2 m 2 1 m ν 1 δm 2 = (δm2 ) 2 φ δm 2 δm 2 χ φ γ 1 = δm 2 2ν 1 m 2 2 δm 2 φ

25 Contact interaction φ 4 Single-trace operator ɛ d ɛ dz 0 z 0 Momentum loop integration is divergent γ = δm2 2ν δm 2 = α/z0 0 d d k (2π) d ( ) 2 Kν (kz 0 ) e ik x µ K ν (kɛ) 2 α/z0 0 φ 4 Position-dependent cutoff d d p (2π) d G(z 0,z 0 ; p) d d p (2π) d G(z 0,z 0 ; p) δm 2 d d p (2π) d G(z 0,z 0 ; p)

26 Power countings of the divergences: Flat space: Warped: (d d+1 p) 1 p 2 Λ d 1 (d d p)g(z 0,z 0 ; p) (d d p)k ν (pz 0 )I ν (pz 0 ) (d d p) 1 p Λd 1 Could also regulate the momentum loop with Pauli-Villas Final answer is scheme-dependent

27 order in Double-trace 1/N, the primary operators appearing operators in the O O operator product expansion (OPE) are the double-trace operators O n,l (x) O( µ µ ) n ( ν1 ν2 νl )O(x), (2.7) where parameters n and l denote the twist and the spin of the operator. At the zeroth order O(x)O(0) = c O in 1/N, the double-trace operator dimension is n,l n,l =2 +2n + l. Why should we care about these composite CFT operators? Once the bulk interactions are turned on, the CFT two-point function would be renormalized. For the sinlge-trace operator, the leading order correction to the bare two-point correlator can They be obtained encode from information the non-zero about leading scattering order expansion in AdS of the boundary 1-PI action ɛ 2(d ) φ 0 (x)φ 0 (0)e S bulk(φ) 1PI non local, (2.8) where integration over the AdS space is involved. As we will see from examples, generally this integration is divergent and a cutoff in the radial Double-trace direction as z 0 = ɛ is necessary for regulating the integration. On the Anomalous CFT side, wedimension introduce the of double-trace wavefunction renormalization operators and factor OPE Z which relates renormalized coefficients, operator O R the to the two bare sets oneof O: data, contain all the dynamical information of CFT at O(1/N 2 ) The renormalized correlation function O R (x) =ZO(x). (2.9)

28 More recently, Anomolous dimension of double-trace gives important information about bulk locality Heemskerk, Penedones, Polchinski and Sully `09; Fitzpatrick, Katz, Poland, Simmons-Duffin `10 RGE of boundary local operators flow (classical level) multitrace-trace Heemskerk and Polchinski `10; Faulkner, Liu and Rangamani `10 Possible phenomelogical applications of double-trace deformation

29 Turning on φ 4 O n (x) O( µ µ) n O(x) Would be renormalized n =2 +2n + γ(n) Single-trace operator dim.

30 Assume ϕ 0 2 sources the double-trace, calculate φ 2 0(x)φ 2 dz0 d d z 0(0) z d+1 0 µ 4! φ(z 0,z) 4 1PI non local Trick: Instead of using the single-particle propagator, use the two-particle propagator φ 2 (x) φ 2 (0) = n 1 N 2 n Single-particle propagator: G (x, 0) Two-particles propagator: G n (x, 0) Analog of partial-wave decomposition

31 µφ 4 µ µ = n d = 2 γ(n) = µ 8π n 1, d = 4 γ(n) = µ (n + 1)( + n 1)(2 + n 3) 16π 2 (2 +2n 1)(2 +2n 3) Agrees with Heemskerk, Penedones, Polchinski and Sully `09; Fitzpatrick, Katz, Poland, Simmons-Duffin `10 Santa Barbara group: calculate 4-point function and project onto each individual two-particle partial wave; Boston group: in global AdS, calculate perturbations of the dilatation operator using the old-fashioned perturbation theory.

32 Physical interpretation d = 2 γ(n) = µ 8π n 1, d = 4 γ(n) = µ (n + 1)( + n 1)(2 + n 3) 16π 2 (2 +2n 1)(2 +2n 3) n 1 φ 4 Λ d 3 γ(n) n d 3

33 Physical interpretation φ 4 Λ d 3 γ(n) n d 3 O Λ p A E p F 4 A E 4 Eg: Euler-Heisenberg Lagrangain γ(n) A n E

34 Three cases AdS UV b.c IR b.c CFT Dirichlet b.c Regular at z 0 Standard quantization Mixed Neuman/ Dirichlet b.c Regular at z 0 > d 2 CFT w/ a double-trace deformation: CFT UV CFT IR Dirichlet b.c z 0 = z IR Spontaneous breaking of CFT

35 CFT with double-trace deformation CFT: L CF T + fo 2 Dual to mixed boundary condition: Witten 01 ; UV : f =0 = d/2 ν [O 2 ] <d relevant perturbation IR : f fφ(k)+ɛ z0 φ(k) z0 =ɛ =0 CFT UV + = d/2+ν CFT IR Some possible pheno applications: (unsuccessful) attempt to explain QCD confinement D.B. Kaplan, Lee, Son, Stephanov 09; Non-susy theory w/ natural light scalar: Strassler 03; Split SUSY: Sundrum 09;

36 Repeat calculation with the new boundary conditions and corresponding propagators For instance, for the toy example of mass perturbation, UV f 0 γ = δm2 2ν, IR f γ = δm2 2ν, = d 2 (d 2 ) 2 + m 2 + δm 2 = 0 δm2 2ν + O((δm2 ) 2 )

37 Spontaneously broken CFT Supposing the existence of a Wilsonian scheme, the renormalization of CFT should not be sensitive to the interior boundary condition. Impose an IR cutoff surface with Dirichlet b.c., for mass perturbations, the answers do not change!

38 dz T, of which the two-point function doesn t 0 d z 1 in order φ need to follow the simple power scaling behavior Specifically, to 0 (x)φ subtract we 0 (0) impose CFT anomolous dimensions that are independent of the IR condition. the IR. More specifically, at smallzmomentum d+1 an Dirichlet 2 δm2 φ(z boundary 0, z) 2 1PI nonlocal condition at z = z or large separation x, there could IR. Then one could We would not proceed along 0 be poles parametrize the boundary-bulk propagator this line here as but just comment on that for mass perturbations pearing in in the the two point functions corresponding to discrete tower of KK modes. In general, = bulk, δm 2 thedzanomalous 0 1 d d ( ) kdimensions Kν (kz 0 ) 2 of operators are indeed insensitive to the IR physics e would expect following Eq our prescription. to be modified and correspondingly our prescription to be modified K(z order to subtract CFT anomolous dimensions 0, ( z0 ) ɛ z 0 ɛ d (2π) d e ik x (3.4) K ν (kɛ) d/2 K ν (kz 0 )+ai ν (kz 0 ) k)= Specifically, we impose an Dirichlet ɛthatboundary are K ν independent (kɛ)+ai condition ν (kɛ) of the, at z IR= condition. z IR. Then one (5.1) could = 2δm would not proceed along this line here but just comment that for mass perturbations Spontaneously 2 dz 0 d d ( ) k k 2ν Γ(1 ν) parametrize the boundary-bulk ɛ z 0 (2π) broken propagator d ɛ2ν d CFT (continued) e ik x + (3.5) as the bulk, where the we anomalous still demand dimensions lim z0 ɛk(z of operators 0, 2 Γ(1 + ν) k) = 1. are Parameter indeed insensitive a is fixed to bythe their IRphysics boundary conditions whose = specific + ɛ 2ν d value log ɛis 2δm2 Γ( ) lowing our prescription. K(z Impose an IR cutoff 0, ( z0 ) irrelevant here. For d/2 example K ν (kz 0 )+ai 1.1, with ν (kz 0 this ) k)= surface with Dirichlet b.c., for mass Specifically, we impose an Dirichlet boundary condition ɛ K ν at (kɛ)+ai z = z ν (kɛ), modified propagator, π d/2 Γ( d/2) x 2 +, (3.6) Eq. 3.6 becomes (5.1) IR. Then one could where in the third rametrize the boundary-bulk perturbations, line we expand the propagator Bessel as answers functions do not change! where we still demand lim z0 ɛk(z 0, for small arguments kz, kɛ 1 and keep only the k) = 1. Parameter a is fixed by the IR boundary conditions whose specific φ leading 0 (x)φ 0 logarithmic dz0 d d z (0) divergent 1 K(z 0, zvalue d+1 ( 2 z0 is δm2 ) φ(zterm 0,z) 2 in 1PI the last two lines. nonlocal Notice that with the bulk-boundary irrelevant d/2 K ν (kzhere. 0 )+ai For ν (kz example 0 ) k)= ɛ K ν (kɛ)+ai ν (kɛ), 1.1, with this modified (5.1) propagator, 0 correlator we chose, the two point function at the Eq. 3.6 becomes ( ) classical level is normalized = + ɛ 2ν d as log ɛ 2δm2 [16] Γ( ) π d/2 ere we still demand φ lim dz0 z0 ɛk(z 0, 0 (x)φ 0 (0) k) d d Γ( d/2) +2a Γ( ) x 2 +, Γ( d/2 + 1)Γ(ν)Γ( ν) = z 1 1. Parameter a is fixed by the IR boundary condins whose specific value is irrelevant z0 d+1 (0) here. O(x)O(0) 2 δm2 = 2ν φ(z 0,z) Γ( ) 2 1PI nonlocal (5.2) ( For π d/2 example Γ( 1.1, d/2) with x 2. (3.7) this modified propagator, ). 3.6 becomes which now = contains + ɛ 2ν d an logirɛ 2δm2 dependentγ( ) factor. But π d/2 Γ( d/2) +2a the normalization Γ( ) of the two point x 2 function To keep two-point correlator finite at order δm 2, we have +, also changes Γ( d/2 + 1)Γ(ν)Γ( ν) dz0 d d z 1 φ 0 (x)φ 0 (0) z0 d+1 2 δm2 φ(z 0,z) 2 1PI (5.2) nonlocal O(x)O(0) which now ( (0) = 2ν ( ) Z 2 1= δm2 Γ( ) log ɛ + ) = + ɛ 2ν d log ɛ 2δm2 contains an Γ( ) π d/2 IR dependent Γ( π d/2 Γ( d/2) +2a d/2) factor. +2a finite terms. Γ( ) x 2 (3.8) ν, (5.3) Γ( ) But the d/2 normalization + 1)Γ(ν)Γ( ν) of the two point function By using alsoeq. changes 2.11, we got x 2 +, Γ( d/2 + 1)Γ(ν)Γ( ν) γ (5.2) O(x)O(0) (0) = 2ν = ɛ ( log Z = δm2 ) ɛ Γ( ) 2ν, π ich now contains an IR dependent factor. d/2 Γ( d/2) +2a Γ( ) (3.9) x 2, (5.3) which agrees with the leading order correction Γ( d/2 + 1)Γ(ν)Γ( ν) But the from normalization 11 expansion of of the the exact two point result. function o changes Example 1.2 2ν ( ) Γ( ) Γ( )

39 For contact interactions, however, Single-trace: the loop momentum integration is dependent of the particular choice of the bulk propagator; UV divergences: determined by short-distance physics; unaffected by the IR boundary; Finite correction: sensitive to the interior b.c.. Double-trace: similar to mass perturbation, independent of IR condition

40 Conclusion I present a simple prescription to calculate the anomalous dimensions of CFT operators from bulk interactions. The key ingredient is to use the radial position as the regulator.

41 Open questions RGE of local boundary operators beyond leading order in N: Lewandowski `04; However, part of his answer does not look Wilsonian-like as it depends on the IR b.c. s; Understand correction to non-rg quantity, e.g., OPE coefficients Theory with gauge interactions and fermions.

42 Thank you!

43 Boundary action Cutoff surface Sliding brane ε 0 ε z 0 S ɛ = Sɛ local S = S ɛ + + Sɛ non local d d x z 0 >ɛ dz 0 L bulk

44 double-trace perturbation 2 O 2 as [26] K(k,z 0 )= ( ). (4.3) ( ) fψ(kɛ)+ ψ(kɛ) n f = f = fɛ 2ν d/2 Γ(1 fɛ ν) 2ν d/2 Γ(1 ν) 2π. (4.2) For a free theory, the two point 2π correlator parametrized Γ(. by ) f, in the(4.2) momentum space is Γ( ) The bulk-to-boundary propagator in this case turns oundary propagator O(k)O(k in this ) (0) f case = turns ɛ d δout d (k + tok be ψ(kɛ) out to be ) ψ(kz 0 ) z d/2 fψ(kɛ)+ ψ(kɛ) n ψ(kz 0 ) z d/2 0 K ν (kz 0 ), 0 K ν (kz 0 ), = ɛ d δ d (k + k ) ψ(kz 1 ( 0 ) K(k,z ψ(kz 0 ) 0 )= ), (4.4) K(k,z 0 )= fɛ fψ(kɛ)+ ψ(kɛ) n. 2ν 2πd/2 Γ(1 ν) Γ( ) + ɛ 2ν (4.3) ( k Γ( ν) fψ(kɛ)+ ψ(kɛ) n. 2 )2ν Γ(ν) (2ν) (4.3) For where a free explicit theory, ɛ the dependences two pointare correlator kept. Starting parametrized at the by UVf, with in the f = momentum 0, the CFTspace UV is isin the ory, the alternative two point quantization correlator parametrized as one can by see f, from in the Eq. momentum 4.4. Flowing space down is to the IR, f grows as O(k)O(k ) (0) )O(k double-trace ) (0) f = ɛ d δ d perturbation f = ɛ d δ d (k + k ψ(kɛ) ) (k + k isψ(kɛ) relevant. As ) fψ(kɛ)+ ψ(kɛ) f in the n deep IR, we recover the Dirichlet boundary condition and fψ(kɛ)+ ψ(kɛ) the two-point n correlator in the standard quantization up to some = ɛ d δ d (k + k 1 ) ( ), (4.4) overall constant = ɛ d δ d normalizations. (k + k 1 ) ( fɛ ) 2ν 2πd/2 Γ(1 ν), (4.4) Now we add a mass fɛperturbation 2ν 2πd/2 Γ(1 ν) δm 2 φ Γ( ) + 2 ɛ 2ν ( k Γ( ) + ɛ 2ν ( k Γ( ν) Γ( ν) 2 )2ν Γ(ν) (2ν) 2 )2ν Γ(ν) in the bulk. The correction to the (2ν) correlator is where explicit ɛ dependences are kept. Starting at the ( UV with f = 0, the CFT ) t ɛ dependences O(k)O(k are kept. ) (1) Starting at the UV with f = 0, the CFT UV is in the f = ɛ d δ d (k + k )δm 2 dz 0 ψ(kz 0 ) 2 UV is in the alternative quantization as one can see from Eq Flowing down to the IR, f grows as ɛ z antization double-trace as oneperturbation can see fromis Eq. relevant Flowing down 0 to fψ(kz IR, 0 )+ ψ(kz f grows 0 as ) n As f in the deep IR, we recover the Dirichlet perturbation is relevant. As f in the deep IR, we recover the Dirichlet dition and the = two-point ɛ d δ d (k + correlator k )δm 2 dz 0 2ν 1 Γ(ν)(kz 0 ) ν +2 ν 1 Γ( ν)(kz 0 ) ν 2 boundary condition and the two-point correlator in the standard quantization up to some overall constant normalizations. in the standard quantization up to some ɛ z 0 fɛ ν π d/2 2 ν Γ(1 ν)γ(ν), (4.5) Γ( nt normalizations. ) k ν 2 ν Γ(1 ν)(kɛ) ν Now we add a mass perturbation δm 2 φ 2 in the bulk. The correction to the correlator is dd a mass perturbation δm 2 φ 2 in the bulk. The correction ( to the correlator is ) O(k)O(k ) (1) ( ) O(k ) (1) f = ɛ d δ d f = ɛ d δ d (k + k )δm 2 dz 0 ψ(kz 0 ) 2 (k + k )δm 2 dz 0 ψ(kz 0 ) 2 ɛ z 0 fψ(kz 0 )+ ψ(kz 0 ) n ɛ z 0 fψ(kz 0 )+ ψ(kz 10 0 ) n = ɛ d δ d (k + k )δm 2 dz 0 2ν 1 Γ(ν)(kz 0 ) ν +2 ν 1 Γ( ν)(kz d δ d (k + k )δm 2 dz 0 2ν 1 Γ(ν)(kz 0 ) ν +2 ν 1 Γ( ν)(kz 0 ) ν 2 0 ) ν 2 ɛ z 0 fɛ ν π d/2 2 ν Γ(1 ν)γ(ν), (4.5) ɛ z 0 fɛ ν π d/2 2 ν Γ(1 ν)γ(ν) Γ( ) k ν 2 ν Γ(1 ν)(kɛ), (4.5) ν Γ( ) k ν 2 ν Γ(1 ν)(kɛ) ν