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

The two-point function of noncommutative φ 4 4 -theory Raimar Wulkenhaar Münster, Germany Mathematisches Institut der Westfälischen Wilhelms-Universität (ased on arxiv:99.1389 with Harald Grosse and work in progress with Harald Grosse and Vincent Rivasseau)

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 2 1 137. 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).

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.

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.

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 2 + 2 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 )

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

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)

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.....

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

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

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.

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 )

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 )

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

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

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.

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)

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!

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

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 )

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) + 1 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)

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.

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.

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 :

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 = 1 + 3 dρ ρ 2 S ρ dρ ρt ρ integral operator K is unounded, rhs non-linear

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 1 7 3 λ

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 1 7 3 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 )( 1 + 3 ) dρ ρ 2 (1 ρ)g n (ρ) with l(α) = 1 α

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 12 43 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!

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.