The two-point function of noncommutative φ 4 4 -theory

Transcription

1 The two-point function of noncommutative φ 4 4 -theory Raimar Wulkenhaar Münster, Germany Mathematisches Institut der Westfälischen Wilhelms-Universität (ased on arxiv: with Harald Grosse and work in progress with Harald Grosse and Vincent Rivasseau)

2 Introduction The Standard Model is a perturatively renormalisale quantum field theory. Scattering amplitudes can e computed as formal power series in coupling constants such as e The first terms agree to high precision with experiment. The radius of convergence in e 2 is zero! We are far away from understanding the Standard Model (see e.g. confinement).

3 Refined summation techniques (e.g. Borel) may estalish reasonale domains of analyticity. Unfortunately, this also fails for QED due to the Landau ghost prolem. It is expected to work for non-aelian gauge theories ecause of asymptotic freedom. But these theories are too complicated.

4 Refined summation techniques (e.g. Borel) may estalish reasonale domains of analyticity. Unfortunately, this also fails for QED due to the Landau ghost prolem. It is expected to work for non-aelian gauge theories ecause of asymptotic freedom. But these theories are too complicated. QFT s on noncommutative geometries may provide toy models for non-perturative renormalisation in four dimensions.

5 φ 4 4-theory on Moyal space with oscillator potential action functional S[φ] = ( 1 d 4 x 2 φ ( +Ω 2 x 2 + µ 2) φ + λ ) 4 φ φ φ φ (x) Moyal product defined y Θ and x := 2Θ 1 x parameters: µ 2,λ R +, Ω [, 1], redef n φ Z 1 2φ, Z R + renormalisale as formal power series in λ [Grosse-W.] means: well-defined perturative quantum field theory β-function vanishes to all orders in λ for Ω = 1 [Disertori-Gurau-Magnen-Rivasseau] means: model is elieved to exist non-perturatively Up to the sign of µ 2, this model arises from a spectral triple.

6 The matrix asis of noncommutative Moyal space (f g)(x)= R d R d dy dk (2π) d f(x+ 1 2Θ k) g(x+y) eiky central oservation (in 2D): f := 2e 1 θ (x2 1 +x2 2 ) f f = f left and right creation operators applied to f lead to f mn (ρ,ϕ) = 2( 1) m m! ( n! eiϕ(n m) 2 θ ρ ) n me ρ 2 θ L n m m ( 2 θ ρ2 ) -product ecomes simple matrix product: f mn f kl = δ nk f ml, d 2 x f mn (x) = det(2πθ)δ mn ( + Ω 2 x 2 + µ 2 )f mn = ( µ θ (1+Ω2 )(m+n+1) ) f mn (x) 2 θ (1 Ω2 ) ( mn f m 1,n 1 + (m+1)(n+1) f m+1,n+1 )

7 The 4D-action functional for Ω = 1 expand φ(x) = m 1,m 2,n 1,n 2 φ mn f m1 n 1 (x 1, x 2 )f m2 n 2 (x 3, x 4 ) matrices (φ mn ) m,n N 2 M Λ with cut-off Λ in matrix size Λ correlation functions generated y partition function ( ) Z[J] = N m,n N 2 dφ mn exp ( S[φ] + tr(φj) ) Λ

8 The 4D-action functional for Ω = 1 expand φ(x) = m 1,m 2,n 1,n 2 φ mn f m1 n 1 (x 1, x 2 )f m2 n 2 (x 3, x 4 ) matrices (φ mn ) m,n N 2 M Λ with cut-off Λ in matrix size Λ correlation functions generated y partition function ( ) Z[J] = N m,n N 2 dφ mn exp ( S[φ] + tr(φj) ) Λ We are interested in N 2 Λ N2. Correlation functions ill-defined unless S[φ] is a suitaly divergent function of Λ: S[φ] = 1 2 φ mnh mn φ nm + V(φ) m,n N 2 Λ H mn = Z ( µ 2 are + m + n ), V(φ) = Z 2 λ 4 m,n,k,l N 2 Λ with m = m 1 + m 2 and divergent µ are [Λ,λ], Z[Λ,λ]. There is no separate Λ-dependence in λ! φ mn φ nk φ kl φ lm

9 Ward identity inner automorphism φ UφU of M Λ infinitesimally φ mn φ mn + i k N 2 Λ(B mk φ kn φ mk B kn ) not a symmetry of the action, ut translation invariance of the measure Dφ = m,n N 2 dφ mn gives Λ = δw = 1 ( Dφ δs + δ ( ) ) tr(φj) e S+tr(φJ) iδb a Z iδb a iδb a = 1 Dφ ( ) (H n H an )φ n φ na + (φ n J na J n φ na ) Z n where W[J] = ln Z[J] generates connected functions e S+tr(φJ)

10 Ward identity inner automorphism φ UφU of M Λ infinitesimally φ mn φ mn + i k N 2 Λ(B mk φ kn φ mk B kn ) not a symmetry of the action, ut translation invariance of the measure Dφ = m,n N 2 dφ mn gives Λ = δw = 1 ( Dφ δs + δ ( ) ) tr(φj) e S+tr(φJ) iδb a Z iδb a iδb a = 1 Dφ ( ) (H n H an )φ n φ na + (φ n J na J n φ na ) Z n where W[J] = ln Z[J] generates connected functions δ δj nm perturation trick φ mn { ( ( δ = n (H n H an ) 2 δj n δj an + ( exp V ( ) ) δ δj e 1 2 Pp,q JpqH 1 δ J na } pq J qp δj n J n c )) δ δj an e S+tr(φJ)

11 Interpretation The insertion of a special vertex Va ins := (H an H n )φ n φ na n into an external face of a rion graph is the same as the difference etween the exchanges of external sources J n J na and J an J n Z( a ) a a..... =..... Z( a )G ins [a]... = G... G a... a a.....

12 Two-point Schwinger-Dyson equation Γ a = a a = a q a + a a p + a p a vertex is Z 2 λ, connected two-point function is G a : first graph equals Z 2 λ q G aq

13 Two-point Schwinger-Dyson equation Γ a = a a = a q a + a a p + a vertex is Z 2 λ, connected two-point function is G a : first graph equals Z 2 λ q G aq in other two graphs we open the p-face and compare with insertion into connected two-point function; it inserts 1 either into one-particle reducile line 2 or into 1PI function: a a a G ins [ap] = a p p = a p p + a p amputation of G a : last two graphs together equal Z 2 λ p G 1 a Gins [ap] p p a

14 Result (using G 1 a = H a Γ a ): Γ a = Z 2 λ p ( ) G ap + G 1 a Gins [ap] ( 1 H p Γ p + = Z 2 λ p 1 H ap Γ ap ( G ap G 1 G p G ) a a Z( p a ) 1 (Γ p Γ a ) ) H p Γ p Z( p a ) = Z 2 λ p This is a self-consistent functional equation for Γ a. It is non-linear and singular. Its singular part at (a, = ) already appeared in [Disertori-Gurau-Magnen-Rivasseau].

15 Result (using G 1 a = H a Γ a ): Γ a = Z 2 λ p ( ) G ap + G 1 a Gins [ap] ( 1 H p Γ p + = Z 2 λ p 1 H ap Γ ap ( G ap G 1 G p G ) a a Z( p a ) 1 (Γ p Γ a ) ) H p Γ p Z( p a ) = Z 2 λ p This is a self-consistent functional equation for Γ a. It is non-linear and singular. Its singular part at (a, = ) already appeared in [Disertori-Gurau-Magnen-Rivasseau]. We perform the renormalisation directly in the SD-equation for Γ a. The Z -factors are essential for that. Taylor: Γ a = Zµ 2 are µ2 + (Z 1)( a + ) + Γ ren a G 1 a = H a Γ a = a + + µ 2 Γ ren a Γ ren = and ( Γren ) = determine µ 2 are and Z.

16 Integral representation We replace discrete indices a,, p N 2 y continuous indices a,, p (R + ) 2, and sums y integrals. This captures the Λ ehaviour of the discrete version (or defines another interesting field theory). The mass-renormalised Schwinger-Dyson equation depends only on the length a = a 1 + a 2. Partial derivatives a i needed to extract Z are equal. Therefore, Γ ren a depends only on a and. Hence, (Λ) (R +) 2 dp 1 dp 2 f( p ) = Λ p d p f( p )

17 Mass renormalisation = sutraction at : (Z 1)( a + ) + Γ ren a Λ ( Z = λ p d p + p +µ 2 Γ ren + p Z + p +µ 2 Γ ren p Γ ren p Γren a + p a Z 2 a + p +µ 2 Γ ren ap Z p+µ 2 Γ ren p Γ ren p p Z 2 + Z p +µ 2 Γ ren p )

18 Mass renormalisation = sutraction at : (Z 1)( a + ) + Γ ren a Λ ( Z = λ p d p + p +µ 2 Γ ren + p Z + p +µ 2 Γ ren p Γ ren p Γren a + p a Z 2 a + p +µ 2 Γ ren ap Z p+µ 2 Γ ren p perturative solution depends on comination change of variales Γ ren a =: µ2 1 αβ (1 α)(1 β) α= to extract Z Γ ren p p Z 2 + Z p +µ 2 Γ ren p ) a 1+a and Λ 1+Λ a =: µ 2 α 1 α, =: β µ2 1 β, p =: ρ µ2 1 ρ, ( 1 1 ), Λ =: µ 2 ξ G αβ 1 ξ a= a i = α

19 Theorem [Grosse-W., 29] The renormalised planar connected two-point function G αβ of self-dual n.c. φ 4 4-theory satisfies (and is determined y) ( 1 α ( G αβ = 1 + λ Mβ L β βy ) + 1 β ( Mα L α αy ) 1 αβ 1 αβ + (1 α)(1 β) (G αβ 1)Y α(1 β) ( ) Lβ +N αβ N α 1 αβ 1 αβ + 1 β ( Gαβ ) (Mα ) ) 1 L α +αn α 1 αβ G α with α,β [, 1) and L α := N αβ := dρ G αρ G ρ 1 ρ dρ G ρβ G αβ ρ α M α := dρ α G αρ 1 αρ Y = lim α M α L α α

20 Discussion The previous integral equation for G αβ is non-perturatively defined. It is non-linear and singular (at 1). Nonlinearity and singularity can e resolved in perturation theory. Then: Is the perturation series analytic at λ =?

21 Discussion The previous integral equation for G αβ is non-perturatively defined. It is non-linear and singular (at 1). Nonlinearity and singularity can e resolved in perturation theory. Then: Is the perturation series analytic at λ =? Non-perturative approach: There are methods for singular ut linear integral equations (Riemann-Hilert prolem). Non-linearity treatale y implicit function theorem (or Nash-Moser theorem), ut singularity is prolematic.

22 Discussion The previous integral equation for G αβ is non-perturatively defined. It is non-linear and singular (at 1). Nonlinearity and singularity can e resolved in perturation theory. Then: Is the perturation series analytic at λ =? Non-perturative approach: There are methods for singular ut linear integral equations (Riemann-Hilert prolem). Non-linearity treatale y implicit function theorem (or Nash-Moser theorem), ut singularity is prolematic. If we could solve the equation for G αβ, then all other n-point function should result from a hierarchy of Ward-identities and Schwinger-Dyson equations which are linear (and inhomogeneous) in the highest-order function.

23 Theorem The renormalised planar 1PI four-point function Γ αβγδ of self-dual n.c. φ 4 4-theory satisfies (and is determined y) Γ αβγδ = λ ( 1 (1 α)(1 γδ)(g αδ G γδ ) G γδ (1 δ)(α γ) + ρ dρ (1 β)(1 αδ)g βρg δρ Γ ρβγδ Γ ) αβγδ (1 βρ)(1 δρ) ρ α ( G αδ +λ (M β L β Y)G αδ + dρ G αδg βρ (1 β) (1 δρ)(1 βρ) 1 ) + ρ dρ (1 β)(1 αδ)g βρ (1 βρ)(1 δρ) (G ρδ G αδ ) (ρ α)

24 Theorem The renormalised planar 1PI four-point function Γ αβγδ of self-dual n.c. φ 4 4-theory satisfies (and is determined y) Γ αβγδ = λ Corollary ( 1 (1 α)(1 γδ)(g αδ G γδ ) G γδ (1 δ)(α γ) + ρ dρ (1 β)(1 αδ)g βρg δρ Γ ρβγδ Γ ) αβγδ (1 βρ)(1 δρ) ρ α ( G αδ +λ (M β L β Y)G αδ + dρ G αδg βρ (1 β) (1 δρ)(1 βρ) 1 ) + ρ dρ (1 β)(1 αδ)g βρ (1 βρ)(1 δρ) (G ρδ G αδ ) (ρ α) Γ αβγδ = is not a solution! We have a non-trivial (interacting) QFT in four dimensions!

25 Perturative solution We look for an iterative solution G αβ = n= λn G (n) αβ. This involves iterated integrals laelled y rooted trees. Up to O(λ 3 ) we need α I α := dx = ln(1 α), 1 αx I α := α I x dx 1 αx = Li 2(α) + 1 ( ) 2 ln(1 α) 2 I α = dx α I x I ( x 1 αx = 2 Li 3 α ) 1 α α I x ( I α = dx 1 αx = 2Li 3 α ) 2Li 3 (α) ln(1 α)ζ(2) 1 α + ln(1 α)li 2 (α) + 1 ( ) 3 ln(1 α) 6

26 In terms of I t and A := 1 α 1 β 1 αβ, B := { } G αβ = 1 + λ A(I β β) + B(I α α) 1 αβ : + λ 2{ A ( ) ( βi β βi β αab (Iβ ) 2 ) 2βI β + I β + B ( αi α αi ) ( α βba (Iα ) 2 ) 2αI α + I α + AB ( (I α α) + (I β β) + (I α α)(i β β) + αβ(ζ(2) + 1) )} + λ 3{ AW β + αab ( U β + I α I β + I α I β) + αa 2 BV β + BW α + βba ( ) U α + I β I α + I β I α + βb 2 AV α + AB ( T β + T α I β (I α ) 2 I α (I β ) 2 ) 6I α I β + AB 2( (1 α)(i α α) + 3I αi β + I β I α + I β (I α ) 2) + BA 2( (1 β)(i β β) + 3I α I β + I α I β + I α (I β ) 2)} + O(λ 4 )

27 where T β := βi β βi β + (I β β), U β := βi β (I β ) 3 + βi β I β + 2I β I β + βζ(2)i β 2βζ(3) 2(I β ) 2 + β(i β ) 2 + I β + βi β + 2I β β 2, V β := βi β β 2 I β 2β 2 ζ(3) + 2βI β + (1 β) ( 2βI β 3Iβ 2 + 3βI β 3I β + β ), W β := (I β βζ(2)) 1 2 I β I β β β I β I 3 β + 2βI βζ(2) 3β 2 ζ(2) (I β) 2 (I β β) 1 2 (I β β) 1 2 β2 Remark: I β β βx β = dx 1 βx (optimal family of iterated integrals not yet determined)

28 Beyond perturation Ansatz (suggested y perturation, ut consistent in general) G αβ = 1 + ( ) 1 α 1 αβ β 2 G β + ( 1 β ) 1 αβ α 2 G α + ( )( 1 α 1 β 1 αβ 1 αβ) αβgαβ coupled system of integral equations for G α, G αβ

29 Beyond perturation Ansatz (suggested y perturation, ut consistent in general) G αβ = 1 + ( ) 1 α 1 αβ β 2 G β + ( 1 β ) 1 αβ α 2 G α + ( )( 1 α 1 β 1 αβ 1 αβ) αβgαβ coupled system of integral equations for G α, G αβ The 1 inserted into M α produces λ ln(1 α) in G α which spreads everywhere 1 1 G α = 1+α 2 G α ecomes singular at some < α(λ) < 1 for any λ <. This could e cancelled y a common zero of M α L α +αn α, which is hard to control.

30 New strategy To avoid ln(1 α) we need G α = 1+S α with lim α 1 S α = This additional condition distinguishes one special value of λ = λ at which we want to prove existence of the theory.

31 New strategy To avoid ln(1 α) we need G α = 1+S α with lim α 1 S α = This additional condition distinguishes one special value of λ = λ at which we want to prove existence of the theory. ansatz G αβ = 2 αg α βg β + T αβ with lim α 1 T αβ = lim α 1 S α = equivalent to 1 = λ ( 1 + λ 2 dρ ρ2 S ρ 1 ρ 3 dρ ρ 2 S ρ + dρ ρt ρ ) (*) (*) is intrinsically non-perturative insert (*) ack into equation for G α = 1 + S α :

32 S α + λ 1 + λ 2 = λ 1 + λ 2 dρ K(α,ρ)S ρ ( (1 α)y + (1 α)s α Y dρ 1 α ( (1 α) 2 (1 αρ) 3ρT αρ ρ(1 α)t ρ )) with K(α,ρ) = 1 ρ 5 (1 α) 3 (1 α)(1 ρ)2 + 1 ρ (1 αρ) 3 (1 αρ) 3 ρ 2 (1 α) 3 1 αρ ρ2 Y = dρ ρ 2 S ρ dρ ρt ρ integral operator K is unounded, rhs non-linear

33 S α + λ 1 + λ 2 = λ 1 + λ 2 dρ K(α,ρ)S ρ ( (1 α)y + (1 α)s α Y dρ 1 α ( (1 α) 2 (1 αρ) 3ρT αρ ρ(1 α)t ρ )) with K(α,ρ) = 1 ρ 5 (1 α) 3 (1 α)(1 ρ)2 + 1 ρ (1 αρ) 3 (1 αρ) 3 ρ 2 (1 α) 3 1 αρ ρ2 Y = dρ ρ 2 S ρ dρ ρt ρ integral operator K is unounded, rhs non-linear ut K is ounded on functions vanishing polynomially at 1: dρ K(α,ρ)(1 ρ) ν Kν (1 α) ν, < ν 1

34 We put S α = (1 α)g(α) and K(α,ρ) = (1 ρ) (1 α) K(α,ρ) Lemma (id + λ 1+ λ 2 K) : C([, 1]) C([, 1]) is invertile for λ < 3 7, with λ 1 + λ 2 ( id + λ 1 + λ 2 K) 1 λ λ

35 We put S α = (1 α)g(α) and K(α,ρ) = (1 ρ) (1 α) K(α,ρ) Lemma (id + λ 1+ λ 2 K) : C([, 1]) C([, 1]) is invertile for λ < 3 7, with λ 1 + λ 2 ( id + λ 1 + λ 2 K) 1 λ λ We put T αρ = and define recursively g = g n+1 = λ 1+ λ 2 ( id + λ K 1+ λ 2 ) 1(lgn 1 )( ) dρ ρ 2 (1 ρ)g n (ρ) with l(α) = 1 α

36 Proposition Let T αρ = and λ < 12 =.218. Then: 55 1 The sequence (g n ) is uniformly convergent to g = lim n g n C([, 1]). 2 S α = (1 α)g(α) C ([, 1]) is the unique solution of our integral equation, with S α λ (12 55 λ )(12 31 λ ) 6 λ (1 α) 55 λ (1 α) 6

37 Proposition Let T αρ = and λ < 12 =.218. Then: 55 1 The sequence (g n ) is uniformly convergent to g = lim n g n C([, 1]). 2 S α = (1 α)g(α) C ([, 1]) is the unique solution of our integral equation, with S α λ (12 55 λ )(12 31 λ ) 6 λ (1 α) 55 λ (1 α) 6 3 G α (1 α)(1 + α 55 6 λ α2 ) > for all α [, 1[. Equation for T αβ is regular in first approximation!

38 Proposition Let T αρ = and λ < 12 =.218. Then: 55 1 The sequence (g n ) is uniformly convergent to g = lim n g n C([, 1]). 2 S α = (1 α)g(α) C ([, 1]) is the unique solution of our integral equation, with S α λ (12 55 λ )(12 31 λ ) 6 λ (1 α) 55 λ (1 α) 6 3 G α (1 α)(1 + α 55 6 λ α2 ) > for all α [, 1[. Equation for T αβ is regular in first approximation! With careful discussion of signs, this extends to 6 17 < λ 6 5.

39 The next steps 1 estalish differentiaility of S α to control dρ S ρ S α ρ α 2 interpret equation for T as recursion T n+1 (T n, S n ) with T = and S from Proposition 3 compute T 1 and re-iterate (g n ) for smaller λ 4 iterate the procedure

40 The next steps 1 estalish differentiaility of S α to control dρ S ρ S α ρ α 2 interpret equation for T as recursion T n+1 (T n, S n ) with T = and S from Proposition 3 compute T 1 and re-iterate (g n ) for smaller λ 4 iterate the procedure Vision The resulting function G αβ solves the original prolem only for 1 1 λ = 1 2 dρ ρ2 S 1 ρ 1 ρ + 3 dρ ρ 2 S ρ dρ ρt ρ This equation defines the value of λ at which there is a realistic chance to prove non-perturative existence of the theory.