Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

Abstract

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati, and Moretti for vector valued free fields. As a by-product of such classification, we provide a mathematically rigorous proof that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

Appendices

Hadamard Expansion for the Parametrix of E

Goal of this appendix is to give a finer description of the local structure of a parametrix P associated with the elliptic operator E, introduced in Eq. (4). Let $[N;b]=(\Sigma ,M;\psi ,\gamma ,g)\in \mathrm {Obj}(\mathsf {Bkg})$ be arbitrary but fixed. In the following, we will be considering convex, geodesic neighbourhoods of $\Sigma$, but at the same time we will be concerned about their image under the action of $\psi$ which is smooth, but not necessarily proper. Hence, whenever we consider convex, geodesic neighbourhoods of a point, we are implicitly constructing them as follows: For any $x\in \Sigma$, consider $\psi (x)\in M$ and any convex, geodesic neighbourhood $U\subset M$ centred at this point. Being $\psi$ smooth, $\psi ^{-1}(U)$ is an open subset of $\Sigma$ centred at x. If this is not a convex, geodesic neighbourhood, then consider an open subset, which we identify with ${\mathcal {O}}$, which has this property. In addition $\psi ({\mathcal {O}})$ is a subset of U and hence any two points therein are connected by a unique geodesic of (M, g).

We summarize our results in the following proposition:

Proposition 51

Let $(N;b)=(\Sigma ,M,\gamma ,g,\psi )\in {\text {Obj}}(\mathsf {Bkg})$ and let $E:\Gamma (\psi ^*TM)\rightarrow \Gamma (\psi ^*T^*M)$ be the elliptic operator defined in (4). For $\lambda >0$ let $H, H_\lambda \in \mathrm {S}\Gamma _{\mathrm {c}}(\psi ^*T^*M^{\boxtimes 2})'$ be the Hadamard parametrices associated with background data (N; b) and $(N;b_\lambda )$ respectively—cf. Remark 10 and Definition 4. It holds

$$\begin{aligned} H^{bc}_\lambda (x)-H^{bc}(x)=-2\log (\lambda )V^{bc}(x)\,, \end{aligned}$$

(66)

where $V\in \Gamma (\mathrm {S}^{\otimes 2}\psi ^*T{\mathcal {O}})$ is constructed out the background data $(\psi ,\gamma ,g)$—cf. Eq. (69)—and it satisfies $[V]^{bc}(x):=g^{bc}(\psi (x))$.

Proof

Let ${\mathcal {O}}$ be a geodesically convex neighbourhood of $\Sigma$. We begin by recalling the construction of the so-called Hadamard parametrix associated to the restriction to ${\mathcal {O}}$ of E on the background data (N; b). This is defined as the bi-distribution $H\in \mathrm {S}\Gamma _{\mathrm {c}}(\psi ^*T^*{\mathcal {O}}^{\boxtimes 2})'$ whose integral kernel reads [G98, Mor99a, Mor99b]

$$\begin{aligned} H^{bc}(x,x^\prime ):=V^{bc}(x,x^\prime )\log \frac{\sigma (x,x^\prime )}{\ell _H^2}:=\sum _{n\ge 0}V_n^{bc}(x,x^\prime )\sigma (x,x^\prime )^n\log \frac{\sigma (x,x^\prime )}{\ell _H^2}\,, \end{aligned}$$

(67)

where $\sigma (x,x^\prime )$ denotes the halved squared geodesic distance between $x,x^\prime \in {\mathcal {O}}$, while $\ell _H\in {\mathbb {R}}$ is an arbitrary reference length, which will play no rôle in the proof. Before focusing on the tensor coefficients $V_n^{bc}(x,x^\prime )$, observe that Eq. (67) defines H in terms of the so-called Hadamard expansion which is a formal power series in $\sigma$. Hence, with a slight abuse of notation, we have left implicit both the existence of a suitable cut-off which ensures convergence of (67) and the necessity or replacing $\sigma$ with a regularized counterpart $\sigma +i\epsilon$, which controls the singularity as $x=x^\prime$. Neither the cut-off nor the regularization will play a rôle in our analysis.

We focus now on the remaining unknowns, the tensor coefficients $V_n^{bc}$ of (67). Recalling that both HE and EH coincide with the identity operator up to smooth terms, it holds locally that

$$\begin{aligned} \nonumber (EH)^c_a =&\sum _{n\ge 0}E(V_n)^c_{\phantom {c}a}\sigma ^n\log \frac{\sigma }{\ell _H^2}\\ \nonumber&+\sum _{n\ge 0}\bigg [ ng_{ab}V_n^{bc}\big (\Delta _\gamma \sigma +2(n-1)\big )+ 2ng_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi V_n)^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta \bigg ]\sigma ^{n-1}\log \frac{\sigma }{\ell ^2_H}\\&+\sum _{n\ge 0}\bigg [ 2g_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi V_n)^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta +g_{ab}V_n^{bc}\big (\Delta _\gamma \sigma -2+4n\big ) \bigg ]\sigma ^{n-1}\,, \end{aligned}$$

(68)

where we omitted for notational simplicity the explicit dependence on $(x,x^\prime )$ and where we exploited the identity $\gamma ^{\alpha \beta }(\mathrm {d}\sigma )_\alpha (\mathrm {d}\sigma )_\beta =2\sigma$, see e.g. [PPV11]. To ensure that $EH-{\text {Id}}_{\Gamma _{\mathrm {c}}(\psi ^*T^*M)}\in \Gamma (\psi ^*TM\boxtimes \psi ^*T^*M)$, the coefficients multiplying $\log \sigma$ and $\sigma ^{-1}$ ought to vanish. This leads to the following hierarchy of equations for $V_n^{bc}$:

$$\begin{aligned} 2g_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi V_0)^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta +g_{ab}V_0^{bc}(\Delta _\gamma \sigma -2)&=0 \end{aligned}$$

(69a)

$$\begin{aligned} E(V_{n-1})^c_{\phantom {c}a}+2ng_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi V_n)^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta +ng_{ab}V_n^{bc}\big (\Delta _\gamma \sigma +2(n-1)\big )&=0\,. \end{aligned}$$

(69b)

Notice that the latter is a system of transport equations which can be solved recursively once we provide initial conditions for the tensors $V_n^{bc}$. The customary choice for the initial data is to consider the limit $x\rightarrow x^\prime$ of Eq. (69). Denoting with $[A](x):=\lim _{x\rightarrow x^\prime }A(x,x^\prime )$ the coinciding point limit of a generic smooth bi-tensor—cf. Remark 8—we get

$$\begin{aligned}{}[E(V_0)^c_{\phantom {c}a}]+2[g_{ab}V_1^{bc}]=0\,,\qquad [E(V_{n-1})^c_{\phantom {c}a}]+2n^2[g_{ab}V_n^{bc}]=0\,, \end{aligned}$$

(70)

where we used the identities

$$\begin{aligned}{}[\sigma ]=0\,,\qquad [(\mathrm {d}\sigma )_\alpha ]=0\,,\qquad [(\nabla ^\Sigma \circ \nabla ^\Sigma \sigma )_{\alpha \beta }]=\gamma _{\alpha \beta }\,. \end{aligned}$$

(71)

Notice that the equations in (70) specify initial data for $V_n^{bc}$ for all $n\ge 1$, leaving us only with an arbitrariness in the choice of the initial datum for $V_0$. In order for $EP-{\text {Id}}\,,PE-{\text {Id}}\in \Gamma (\psi ^*TM\boxtimes \psi ^*T^*M)$, we fix

$$\begin{aligned}{}[V_0^{bc}]=g^{bc}\,. \end{aligned}$$

(72)

We now consider the Hadamard parametrix $H_\lambda$ associated with E and background data $(N;b_\lambda )$. Once again we have

$$\begin{aligned} H^{bc}_\lambda (x,x')=\sum _{n\ge 0}V_{\lambda ,n}^{bc}(x,x')\sigma _\lambda (x,x')^n\log \sigma _\lambda (x,x')\,, \end{aligned}$$

where $\sigma _\lambda$ is the halved squared geodesic distance built out of the metric $\lambda ^{-2}\gamma _{\alpha \beta }$. The smooth tensors $V_{\lambda ,n}^{bc}$ satisfy the system (69) with background data $(N;b_\lambda )$. Observe that Eq. (69a) is invariant under scaling $\gamma _{\alpha \beta }\rightarrow \lambda ^{-2}\gamma _{\alpha \beta }$ because $\sigma _\lambda =\lambda ^{-2}\sigma$. Together with the initial conditions $[V_0]^{bc}=[V_{\lambda ,0}]^{bc}=g^{bc}$ this entails $V_{\lambda ,0}=V_0$. By induction it easily follows from the scaling behaviour of Eq. (69b) that $V_{\lambda ,n}=\lambda ^{2n}V_n$. Therefore

$$\begin{aligned} H_\lambda ^{bc}-H^{bc}&= \sum _{n\ge 0}V_{\lambda ,n}^{bc}\lambda ^{-2n}\sigma ^n\big (\log \sigma -2\log \lambda \big )- \sum _{n\ge 0}V_n^{bc}\sigma ^n\log \sigma = -2\log (\lambda )V^{bc}\,. \end{aligned}$$

Using the initial condition 72, Eq. (66) follows. $\quad \square$

The Peetre-Slovák Theorem

In this section we recall succinctly the Peetre-Slovák theorem as well as all ancillary definitions. For more details, we refer to [NS14] and especially to [KM16, Appendix A], to which this appendix is inspired. In the following $E{\mathop {\rightarrow }\limits ^{\pi _E}} B, F{\mathop {\rightarrow }\limits ^{\pi _F}}B$ are smooth bundles over a smooth manifold B, while $J^rE$ denotes the r-jet bundle over B for $r\in {\mathbb {N}}$—refer to [KMS93] for definitions and properties.

Definition 52

A map $D:\Gamma (E)\rightarrow \Gamma (F)$ is a called a differential operator of globally bounded order $r\in {\mathbb {N}}$ if there exists a smooth map $d:J^rE\rightarrow F$ such that $\pi _F\circ d=\pi _{J^rE}$ and

$$\begin{aligned} D(\varepsilon )=d(j^r\varepsilon )\qquad \forall \varepsilon \in \Gamma (E)\,, \end{aligned}$$

(73)

where $j^r\varepsilon \in \Gamma (J^rE)$ denotes the r-jet extension of $\varepsilon$.

Definition 53

A map $D:\Gamma (E)\rightarrow \Gamma (F)$ is called a differential operator of locally bounded order if for all $x_0\in B$ and for all $\varepsilon _0\in \Gamma (E)$, there exists

• 1. an open subset $U\subseteq B$ containing $x_0$ and with compact closure,

• 2. an integer $r\in {\mathbb {N}}$, as well as a neighbourhood $Z^r\subseteq J^rE$ of $j^r\varepsilon _0(U)$ such that $\pi _{J^rE}Z^r=U$,

• 3. a smooth map $d:Z^r\rightarrow F$ such that $\pi _F\circ d=\pi _{J^rE}$

so that

$$\begin{aligned} D(\varepsilon )(x)=d(j^r\varepsilon )(x)\,, \end{aligned}$$

(74)

for all $x\in U$ and $\varepsilon \in \Gamma (E)$ with $j^r\varepsilon (U)\subseteq Z^r$.

The Peetre-Slovák’s Theorem gives a sufficient condition for a map $D:\Gamma (E)\rightarrow \Gamma (F)$ to be a differential operator of locally bounded order.

In addition recall that, denoting with $\pi _d:B\times {\mathbb {R}}^d\rightarrow B$ the canonical projection to B, the pull-back bundle $\pi _d^*E{\mathop {\rightarrow }\limits ^{\pi _{\pi _d^*E}}}B\times {\mathbb {R}}^d$ is the smooth bundle defined by

$$\begin{aligned} \pi ^*E:=\{(s,x,e)\in {\mathbb {R}}^d\times B\times E|\;\pi _E(e)=\pi _d(s,x)\}\simeq {\mathbb {R}}^d\times E\,. \end{aligned}$$

(75)

Denoting with $\pi _{d,E}$ the projection $\pi _{d,E}:\pi _d^*E\rightarrow E$, each smooth section $\zeta \in \Gamma (\pi _d^*E)$ induces a smooth family of sections $\{\zeta _s\}_{s\in {\mathbb {R}}^d}$ in $\Gamma (E)$ defined by $\zeta _s(x):=\pi _{d,E}\zeta ((s,x))$ which, in turn, depends smoothly on the parameter $s\in {\mathbb {R}}^d$.

Definition 54

Let $d\in {\mathbb {N}}$ and let $\{\zeta _s\}_{s\in {\mathbb {R}}^d}$ be a smooth family of sections in $\Gamma (E)$ induced by a smooth section $\zeta \in \Gamma (\pi _d^*E)$. We say that $\{\zeta _s\}_{s\in {\mathbb {R}}^d}$ is a smooth compactly supported d-dimensional family of variations if there exists a compact $K\subseteq B$ such that $\zeta (s,x)=\zeta (s',x)$ for all $x\notin K$ and for all $s,s'\in {\mathbb {R}}^d$.

Definition 55

A map $D:\Gamma (E)\rightarrow \Gamma (F)$ is called weakly-regular if, for all $d\in {\mathbb {N}}$ and for all smooth compactly supported d-dimensional families of variations $\{\zeta _s\}_{s\in {\mathbb {R}}^d}$—see Definition 54—$\psi _s:=D\zeta _s$ is a smooth compactly supported d-dimensional family of variations.

Theorem 56

(Peetre-Slovák): Let $D:\Gamma (E)\rightarrow \Gamma (F)$ be a smooth map such that

• for all $\varepsilon \in \Gamma (E)$ and for all $x\in B$, $D\varepsilon (x)$ depends only on the germ of $\varepsilon$ at $x\in B$, i.e. $(D\varepsilon )(x)=(D{\widetilde{\varepsilon }})(x)$ for all ${\widetilde{\varepsilon }}\in \Gamma (E)$ which coincides with $\varepsilon$ in a neighbourhood of x;

• D is weakly regular as per Definition 55.

Then D is a differential operator of locally bounded order as per Definition 52.

Fulfilment of the Perturbative Agreement

In this section we comment on the principle of perturbative agreement (PPA for short) for the model we have introduced in Definition 20.

The PPA has been introduced in [HW05] as a further constraint on the structure of Wick powers—see also [DHP16, Za15]. Loosely speaking, it requires that a theory associated with a quadratic perturbation $E_s$ of the elliptic operator E introduced in Eq. (4) should yield to an algebra ${\mathcal {A}}_s[N;b]$ compatible with the unperturbed algebra ${\mathcal {A}}[N;b]$. Here $E_s-E\in \Gamma _{\mathrm {c}}(\mathrm {S}^{\otimes 2}\psi ^*T^*M)$ is a smooth and compactly supported (1-dimensional) family of variations. The compatibility between ${\mathcal {A}}_s$ and ${\mathcal {A}}$ is in the sense of formal power series in s—cf. Definition 60.

As pointed out in [Za15] the PPA is important in our setting because, among other things, it ensures that the renormalization group flow technique we applied in Sect. 3 does not depend on the splitting ${\mathcal {L}}={\mathcal {L}}_{\mathrm {free}}+{\mathcal {L}}_{\mathrm {int}}$. A complete discussion of the PPA is not within the scopes of this paper—for a complete discussion in the Riemannian setting see [DDR19]. In the present appendix we provide a brief resumé of the content of the PPA, proving that there exists a family of Wick powers as per Definition 38 which fulfils it—cf. Proposition 62.

In what follows $E_s$ will always denote a smooth and compactly supported (1-dimensional) family of variations—cf. Definition 54—of the elliptic operator E defined as per equation (4). In particular $E_s$ is elliptic for all s. Notice that, for the sake of simplicity, we are assuming that $E_s-E\in \Gamma _{\mathrm {c}}(\mathrm {S}^{\otimes 2}\psi ^*T^*M)$ is a differential operator of order at most 1. This is actually enough for our setting see however [DDR19, HW05] for completeness.

Formulation of the PPA. In order to formulate the PPA a few preliminary definitions are in due order. First of all we need a linear isomorphism $R_s:{\text {Par}}[N;b]\ni P\rightarrow P_s\in {\text {Par}}_s[N;b]$ between the space of parametrices ${\text {Par}}_s[N;b]$ associated with $E_s$ and those of E. The construction of this map is rather standard, see [DDR19] for further details and [DD16, DHP16, HW05, Za15] for the corresponding map in the Lorentzian setting. For what concerns the PPA, we just need the perturbative expansion of $R_s$ as a formal power series in s up to a smooth remainder. Let $P\in {\text {Par}}[N;b]$; since $Q_s:=E_s-E$ is compactly supported,

$$\begin{aligned} E_s= E+Q_s= E({\text {Id}}+PQ_s)-SQ_s\,, \end{aligned}$$

where $S\in \Gamma (\psi ^*TM\boxtimes \psi ^*T^*M)$ is such that $PE-{\text {Id}}_{\Gamma _{\mathrm {c}}(\psi ^*T^*M)}=S$—cf. equation (12). We consider the map $R_{[[s]]}:\Gamma _{\mathrm {c}}(\psi ^*T^*M)\rightarrow \Gamma (\psi ^*TM)[[s]]$ defined by

$$\begin{aligned} R_{[[s]]}\omega :=\sum _{n\ge 0}(-PQ_s)^nP\omega \,. \end{aligned}$$

(76)

This map can be interpreted as a perturbative expansion (up to a smooth remainder) of a well-defined isomorphism $R_s:{\text {Par}}[N;b]\rightarrow {\text {Par}}_s[N;b]$—cf. [DD16, DDR19, DHP16, HW05].

The second ingredient we need is a $*$-isomorphism $\beta _s:{\mathcal {A}}_{\text {reg}}[N;b]\rightarrow {\mathcal {A}}_{s,\text {reg}}[N;b]$. Here the subscript $_{\text {reg}}$ denotes the algebra generated by regular local functionals, namely those with smooth functional derivatives of all orders. We will not enter into the details of this construction, however, we give the explicit form for $\beta _s$:

$$\begin{aligned} (\beta _s F)(P_s):=\exp \big [\Upsilon _{P_s-P}\big ]F[P]\qquad \forall F\in {\mathcal {A}}[N;b]\,. \end{aligned}$$

(77)

This can be extended to a map $\beta _{[[s]]}:{\mathcal {A}}[N;b]\rightarrow \Gamma ({\mathcal {E}}[N;b])[[s]]$ with values in the algebra of formal power series in s with coefficients in $\Gamma ({\mathcal {E}}[N;b])$—cf. Definitions 18-20. The expansion is possible since, on account of Eq. 76), $P_s-P=P_{[[s]]}-P+R=\sum _{n\ge 1}(-PG_s)^nP+R$ has a well-defined coinciding point limit—here R is a smooth remainder. Therefore $\beta _{s}$ is well-defined at each order in s. As explained in [DDR19, DHP16] the map $\beta _{[[s]]}$ can be interpreted as an extension of the expansion in formal power series of $\beta _s$.

We focus on Wick powers, strengthening the smoothness requirement of Definition 38 by allowing also variations of the elliptic operator $E_s$.

Definition 57

Let $d,n\in {\mathbb {N}}$ and let $(N;b_s)\in {\text {Obj}}(\mathsf {Bkg})$ be such that $\{b_s=(\psi ,\gamma _s,g_s)\}_{s\in {\mathbb {R}}^d}$ is a smooth, compactly supported d-dimensional family of variations of $b=(\psi ,\gamma ,g)$ as per Definition 54. Moreover let $E_{t,s}$ be a smooth and compactly supported n-dimensional family of variations of the elliptic operator $E_s$ constructed out of the background data $b_s$ as per Eq. 4). For all smooth families $\{P_{t,s}\}_{s\in {\mathbb {R}}^d}$ where $P_{t,s}\in {\text {Par}}_t(N,b_s)$ is a parametrix for $E_{t,s}$ for all $s\in {\mathbb {R}}^d$ and $t\in {\mathbb {N}}^n$, let ${\mathcal {U}}_k\in \Gamma _{\mathrm {c}}(\pi _{d+n}^*\mathrm {S}^{\otimes k}\psi ^*T^*M)'$ be the distribution defined by

$$\begin{aligned} {\mathcal {U}}_k(\chi \otimes \omega _1):= \int _{{\mathbb {R}}^{d+n}}\mathrm {d}s\mathrm {d}t\,\Phi ^k_t[N;b_s](\omega _1,P_{t,s},0)\chi (s,t)\,, \end{aligned}$$

(78)

where $\omega _1\in \mathrm {S}\Gamma _{\mathrm {c}}^{k,1}[N;b],\chi \in C^\infty _{\mathrm {c}}({\mathbb {R}}^{d+n})$. Here $\Phi ^k_t[N;b_s]$ denotes the (k-th) Wick power associated with the background data $(N;b_s)$ and with the elliptic operator $E_{s,t}$. If $\text {WF}({\mathcal {U}}_k)=\emptyset$, we call the family of Wick powers $\Phi ^\bullet$ smooth.

Remark 58

Loosely speaking Definition 57 requires a suitable smoothness of $\Phi ^\bullet$ with respect both to the background data b and to the variation of the elliptic operator E. For certain models—like the scalar field cf. [DDR19]—the variations of the background data exhaust all possible variations of the associated elliptic operator E. In this situation the smoothness as per Definition 57 coincides with the one required in Definition 38.

Remark 59

A smooth family of parametrices $P_{t,s}\in {\text {Par}}_t[N;b_s]$ can be constructed by setting $P_{t,s}:=R_{t}P_s$ where $P_s\in {\text {Par}}[N;b_s]$ is a smooth family of parametrices for $E_s$—cf. Remark 39.

From now on $\Phi ^\bullet$ will denote a family of Wick powers as per Definition 38 satisfying the smoothness requirement of Definition 57. Notice that the family $:\!{\Phi ^\bullet }\!:$ defined in Example 45 satisfies such smoothness requirement.

Definition 60

Let $E_s$ denote a smooth and compactly supported (1-dimensional) family of variation—cf. Definition 54—of the elliptic operator E defined as per Eq. 4. We say that the family of Wick powers $\Phi ^\bullet$ satisfies the principle of perturbative agreement (PPA) if for all $k\ge 2$, $n\in {\mathbb {N}}\cup \lbrace 0\rbrace$, $P\in {\text {Par}}[N;b]$, $\omega _m\in \mathrm {S}\Gamma _{\mathrm {c}}^{k,m}[N;b]$ it holds

$$\begin{aligned} \frac{\mathrm {d}^n}{\mathrm {d}s^n}\Phi ^k_s[N;b](\omega _m,P_s)\bigg |_{s=0}= \frac{\mathrm {d}^n}{\mathrm {d}s^n}\beta _{s}(\Phi ^k[N;b])(\omega _m,P_{s})\bigg |_{s=0}\,. \end{aligned}$$

(79)

Remark 61

A direct computation shows that the PPA is satisfied if and only if equation (79) holds true for $n=1$—cf. [DDR19]. Moreover, on account of the lack of renormalization ambiguities for $m\ge 2$—cf. Remarks 15–27—the PPA is fulfilled whenever it holds for $m=1$.

We state the main result of this appendix.

Proposition 62

The family $:\!{\Phi ^\bullet }\!:$ defined in Eq. (43) satisfies the PPA as per Definition 60 with respect to a family of variations $E_s$ of the elliptic operator E such that $E_s-E\in \Gamma _{\mathrm {c}}(\mathrm {S}^{\otimes 2}\psi ^*T^*M)$ is a differential operator of order at most 1.

Proof

Observe that on account of Theorems 44 and 45 we may write for all $k\ge 2$ and $\omega \in \mathrm {S}\Gamma _{\mathrm {c}}^{k,1}[N;b]$

$$\begin{aligned} \Phi ^k_s[N;b](\omega _1)=\; :\!{\Phi ^k}\!:_s[N;b](\omega _1)+ \sum _{\ell =0}^{k-2}:\!{\Phi ^\ell }\!:_s\big (c_{s,k-\ell }[N;b]\lrcorner \omega _1\big )\,, \end{aligned}$$

where $:\!{\Phi ^k}\!:$ is defined as in Example 42 while $c_{s,\ell }\in \Gamma ^{\ell ,1}[N;b]$ satisfies the hypothesis of Theorems 44–45. Moreover, $c_{s,\ell }$ is a smooth and compactly supported family of variation and we set $c_\ell :=c_{s,\ell }|_{s=0}$.

Our aim is to show that $:\!{\Phi ^\bullet }\!:$ satisfies Eq. (79). In particular we shall impose Eq. (79) for a generic family $\Phi ^\bullet$ of Wick powers. This will constraint the coefficient $c_{s,\ell }$ defined above, in particular we shall prove that Eq. (79) implies that we can choose $c_{s,\ell }=0$, that is, $\Phi ^\bullet =:\!{\Phi ^\bullet }\!:$. We first consider the case $k=2$. Setting $\delta :=\frac{\mathrm {d}}{\mathrm {d}s}\big |_{s=0}$, by direct inspection it holds

$$\begin{aligned} \delta [\Phi ^2[N;b](\omega _1,P)]&= \delta \big [:\!{\Phi ^2}\!:[N;b](\omega _1,P)+C_{s,2}[N;b](\omega _1)\big ]\\ {}&= \Upsilon _{\delta (W_P)}:\!{\Phi ^2}\!:[N;b](\omega _1,P)+\delta \big [C_2[N;b](\omega _1))\big ]\,, \end{aligned}$$

where $C_2[N;b](\omega _1):=\int _\Sigma \langle c_2[N;b],\omega _1\rangle$. Similarly the first order in s in the right hand side of Eq. (79) reads

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}s}\beta _s\Phi ^k[N;b](\omega _1,P_s)\bigg |_{s=0}= \Upsilon _{\delta (P)}\Phi ^2[N;b](\omega _1,P)= \Upsilon _{\delta (P)}:\!{\Phi ^2}\!:[N;b](\omega _1,P)\,, \end{aligned}$$

where we exploited Eqs. (76)–(77). Equation (79) entails

$$\begin{aligned} \delta \big [C_2[N;b](\omega _1))\big ]= -\Upsilon _{\delta (W_P)+\delta (P)}:\!{\Phi ^2}\!:[N;b](\omega _1,P)= \langle [\delta (H)],\omega _1\rangle \,, \end{aligned}$$

where we used Eq. (40) and Remark 10. Therefore the PPA for $k=2$ can be fulfilled if

$$\begin{aligned} \delta (c_2)=-[\delta (H)]\,. \end{aligned}$$

(80)

Since $\delta (H)$ is local and covariant the above equation can be considered as a definition of the coefficient $c_2$—indeed, it respects all requirement of Theorems 44 and 45. In particular $c_{2,s}$ is the solution to the ODE $\delta (c_{2,s})=-[\delta (H_s)]$.

For the general case we compute once again the right and the left hand side of Eq. (79) using Eq. (50). The final result is

$$\begin{aligned} 0=&\sum _{\ell =0}^{k-2}:\!{\Phi ^\ell }\!:[N;b](\delta (c_{k-\ell }[N;b])\lrcorner \omega _1,P)\\ {}&- \Upsilon _{[\delta (H)]}\bigg [:\!{\Phi ^k}\!:[N;b](\omega _1,P)+\sum _{\ell =0}^{k-2}:\!{\Phi ^\ell }\!:[N;b](c_{k-\ell }[N;b]\lrcorner \omega _1,P)\bigg ]\,. \end{aligned}$$

Equation (40) leads to the following generalization of Eq. (80)

$$\begin{aligned} \delta (c_{k-\ell })=-(\ell +2)(\ell +1)[\delta (H)][\otimes ]c_{k-2-\ell }\qquad 2\le \ell \le k-3\,. \end{aligned}$$

Once again this can be used to define inductively the coefficients $c_{\ell }$.

We now prove that $[\delta (H)]=0$, which implies that the choice $c_{s,\ell }=0$ leads to a family of Wick powers satisfying the PPA. We recall that we are considering families of variations $E_s$ such that $E_s-E\in \Gamma _{\mathrm {c}}(\mathrm {S}^{\otimes 2}\psi ^*T^*M)$ is a differential operator of order at most 1. For $\varphi \in \Gamma (\psi ^*TM)$ we can write locally

$$\begin{aligned} (E_s\varphi -E\varphi )_a=(A_s)^{\alpha }_{\phantom {\alpha }ab}(\nabla ^\psi \varphi )^b_\alpha +(T_s)_{ab}\varphi ^b\,, \end{aligned}$$

(81)

where $(A_s)^{\alpha }_{\phantom {\alpha }ab},(T_s)_{ab}$ are suitable smooth tensors. This implies that the Hadamard parametrix $H_s$ associated with $E_s$ has the form

$$\begin{aligned} H_s=\sum _{n\ge 0}V_{s,n}\sigma ^n\log \sigma \,, \end{aligned}$$

where $\sigma$ does not depend on s since so it does the principal symbol of $E_s$ [G98]. The tensors $V_{s,n}\in \mathrm {S}\Gamma (\psi ^*TM^{\boxtimes 2})$ satisfies a hierarchy of transport equations analogous to system (69). In particular $V_{s,0}$ satisfies

$$\begin{aligned}&2g_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi V_{s,0})^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta + g_{ab}V_{s,0}^{bc}(\Delta _\gamma \sigma -2) \nonumber \\&\quad + (A_s)^\alpha _{\phantom {\alpha }ab}V^{bc}_{s,0}(\mathrm {d}\sigma )_\alpha =0\qquad [V_{s,0}]^{bc}=g^{bc}\,. \end{aligned}$$

It follows that $\delta (V_0)$ satisfies the transport equation

$$\begin{aligned}&2g_{ab}\gamma ^{\alpha \beta }(\nabla ^\psi \delta (V_0))^{bc}_{\phantom {bc}\alpha }(\mathrm {d}\sigma )_\beta + g_{ab}\delta (V_0)^{bc}(\Delta _\gamma \sigma -2)\\&\quad + \delta (A)^\alpha _{\phantom {\alpha }ab}V^{bc}_0(\mathrm {d}\sigma )_\alpha =0\qquad [\delta (V_0)]=0\,, \end{aligned}$$

where we exploited that $A_s|_{s=0}=0$. The transport equation implies that $\delta (V_0)$ is a function of $\sigma$, moreover, smoothness and the initial condition implies that $\delta (V_0)=O(\sigma )$ as $\sigma \rightarrow 0^+$. Therefore

$$\begin{aligned} \delta (H)=\sum _{n\ge 0}\delta (V_n)\sigma ^n\log \sigma =O(\sigma \log \sigma )\,. \end{aligned}$$

$\square$

For completeness we provide a concise proof of the fact that the renormalization group flow does not depend on the splitting ${\mathcal {L}}={\mathcal {L}}_{\mathrm {free}}+{\mathcal {L}}_{\mathrm {int}}$ whenever one exploits a family of Wick powers $\Phi ^\bullet$ satisfying the PPA as per Definition 60.

Proposition 63

Let ${\mathcal {L}}={\mathcal {L}}_{\mathrm {free}}+{\mathcal {L}}_{\mathrm {int}}$ be the splitting of the Lagrangian density as per Eq. (54) where ${\mathcal {L}}_{\mathrm {free}},{\mathcal {L}}_{\mathrm {int}}$ are defined as per Eqs. (55, 56). Let ${\mathcal {L}}={\mathcal {L}}_{s,\mathrm {free}}+{\mathcal {L}}_{s,\mathrm {int}}$ be another splitting such that ${\mathcal {L}}_{s,\mathrm {free}}(\varphi )=\frac{\nu ^2}{2}\langle \varphi ,E_s\varphi \rangle$ where $E_s$ is a family of variations of the elliptic operator E such that $E_s-E\in \Gamma _{\mathrm {c}}(\mathrm {S}^{\otimes 2}\psi ^*T^*M)$ is a differential operator of order at most 1. Finally let $\Phi ^\bullet$ a family of Wick powers as per Definition 38 satisfying the PPA as per Definition 60. Let ${\mathcal {R}}_{\mathrm {int},\lambda }[\Phi ^\bullet ],{\mathcal {R}}_{s,\mathrm {int},\lambda }[\Phi ^\bullet _s]$ the local and covariant observables defined by

$$\begin{aligned} S_\lambda {\mathcal {L}}_{\mathrm {int}}[\Phi ^\bullet ]= {\mathcal {L}}_{\mathrm {int}}[\Phi ^\bullet ] +{\mathcal {R}}_{\mathrm {int},\lambda }[\Phi ^\bullet ]\,,\qquad S_\lambda {\mathcal {L}}_{s,\mathrm {int}}[\Phi ^\bullet _s]= {\mathcal {L}}_{s,\mathrm {int}}[\Phi ^\bullet _s] +{\mathcal {R}}_{s,\mathrm {int},\lambda }[\Phi ^\bullet _s]\,, \end{aligned}$$

where ${\mathcal {L}}_{\mathrm {int}}[\Phi ^\bullet ]$ has been defined in Eq. (57)—${\mathcal {L}}_{s,\mathrm {int}}[\Phi ^\bullet _s]$ is defined similarly.

Then we have ${\mathcal {R}}_\lambda ={\mathcal {R}}_{s,\lambda }$, that is, the analytic form of the renormalization group flow associated with ${\mathcal {L}}_{\mathrm {int}}$ and ${\mathcal {L}}_{s,\mathrm {int}}$ is the same.

Proof

Given $\Phi ^\bullet$, we introduce the local and covariant free Lagrangian densities ${\mathcal {L}}_{\mathrm {free}}:C^\infty _{\mathrm {c}}\rightarrow {\mathcal {A}}$, ${\mathcal {L}}_{\mathrm {s,free}}:C^\infty _{\mathrm {c}}\rightarrow {\mathcal {A}}_s$ defined by

$$\begin{aligned} {\mathcal {L}}_{\mathrm {free}}[\Phi ^\bullet ][N;b](f)&\,{:}{=}\,\frac{\nu ^2}{2}\Phi ^2[N;b](\theta _{\mathrm {free}}[N;b] f\mu _\gamma )\,,\\ {\mathcal {L}}_{s,\mathrm {free}}[\Phi ^\bullet ][N;b](f)&\,{:}{=}\,\frac{\nu ^2}{2}\Phi ^2[N;b](\theta _{s,\mathrm {free}}[N;b] f\mu _\gamma )\,, \end{aligned}$$

where $(N;b)\in {\text {Obj}}(\mathsf {Bkg})$, $f\in C^\infty _{\mathrm {c}}(\Sigma )$ while $\theta _{\mathrm {free}}[N;b],\theta _{s,\mathrm {free}}[N;b]$ are defined by

$$\begin{aligned} \theta _{\mathrm {free}}[N;b](\varphi )=\langle \varphi ,E\varphi \rangle \,,\qquad \theta _{s,\mathrm {free}}[N;b](\varphi )=\langle \varphi ,E_s\varphi \rangle \,. \end{aligned}$$

Actually $\theta _{\mathrm {free}},\theta _{s,\mathrm {free}}\in \Gamma (\mathrm {S}^{\otimes 2}J_\infty \psi ^*TM)$ are 2-symmetric forms over the jet bundle over $\psi ^*TM$—see [KMS93] for further details. Notice that ${\mathcal {L}}_{\mathrm {s,free}}[\Phi _s^\bullet ]$, ${\mathcal {L}}_{s,\mathrm {int}}[\Phi _s^\bullet ]$ depend on the chosen splitting of the Lagrangian ${\mathcal {L}}$—i.e. they depend on s. However, the sum ${\mathcal {L}}_{s,\mathrm {free}}[\Phi _s^\bullet ]+{\mathcal {L}}_{s,\mathrm {int}}[\Phi _s^\bullet ]$ depends on s only via $\Phi ^\bullet _s$. Since $\Phi ^\bullet$ satisfies the PPA we have $\Phi ^\bullet _s\sim \beta _s\Phi ^\bullet$ and

$$\begin{aligned} {\mathcal {L}}_{s,\mathrm {free}}[\Phi _s^\bullet ] +{\mathcal {L}}_{s,\mathrm {int}}[\Phi _s^\bullet ]=: {\mathcal {L}}[\Phi ^\bullet _s]\sim \beta _s{\mathcal {L}}[\Phi ^\bullet ]= \beta _s{\mathcal {L}}_{\mathrm {free}}[\Phi ^\bullet ]+ \beta _s{\mathcal {L}}_{\mathrm {int}}[\Phi ^\bullet ]\,. \end{aligned}$$

Under scaling the local and covariant total Lagrangian ${\mathcal {L}}[\Phi ^\bullet ]$ defines an s-independent renormalization group term ${\mathcal {R}}_{\mathrm {tot},\lambda }$ via $S_\lambda {\mathcal {L}}[\Phi ^\bullet ]={\mathcal {L}}[\Phi ^\bullet ]+{\mathcal {R}}_{\mathrm {tot},\lambda }[\Phi ^\bullet ]\,.$ The s-independence of ${\mathcal {R}}_{\mathrm {tot},\lambda }$ is a direct consequence of the PPA: indeed we have

$$\begin{aligned} S_\lambda {\mathcal {L}}[\Phi _s^\bullet ]&= {\mathcal {L}}[\Phi _s^\bullet ]+{\mathcal {R}}_{s,\mathrm {tot},\lambda }[\Phi _s^\bullet ]\\\ S_\lambda {\mathcal {L}}[\Phi _s^\bullet ]&\sim S_\lambda \beta _s{\mathcal {L}}[\Phi ^\bullet ]= \beta _s{\mathcal {L}}[\Phi ^\bullet ]+ \beta _s{\mathcal {R}}_{\mathrm {tot},\lambda }[\Phi ^\bullet ]\\ {}&\sim {\mathcal {L}}[\Phi _s^\bullet ]+ {\mathcal {R}}_{\mathrm {tot},\lambda }[\Phi _s^\bullet ]\,, \end{aligned}$$

which implies ${\mathcal {R}}_{s,\mathrm {tot},\lambda }={\mathcal {R}}_{\mathrm {tot},\lambda }$—here we used that $S_\lambda$ and $\beta _s$ commute. In order to show that ${\mathcal {R}}_{s,\mathrm {int},\lambda }={\mathcal {R}}_{\mathrm {int},\lambda }$ we will show that ${\mathcal {R}}_{s,\mathrm {free},\lambda }={\mathcal {R}}_{\mathrm {free},\lambda }=0$, that is, the free local and covariant Lagrangian has no anomalous scaling. Without loss of generality we can consider $\Phi ^\bullet =:\!{\Phi ^\bullet }\!:$. This is due to the fact that ${\mathcal {L}}$ is at most quadratic in the field and that a different prescription of $\Phi ^2$ would differs from $:\!{\Phi ^2}\!:$ by a scaling invariant term $C_2$—cf. Theorem 45—which does not contribute to the renormalization flow. Using the definition of $:\!{\Phi ^2}\!:$—cf. Eq. 45—we arrive at

$$\begin{aligned} {\mathcal {L}}[:\!{\Phi ^\bullet }\!:][N;b](f,P,\varphi ):= \frac{\nu ^2}{2}\int _\Sigma \big (\langle \varphi ,E\varphi \rangle +[EW_P]\big )f\mu _\gamma \,, \end{aligned}$$

where $W_P$ has been introduced in Remark 10. Following the proof of Theorem 46 the anomalous scaling of ${\mathcal {L}}[:\!{\Phi ^\bullet }\!:]$ is computed by exploiting the identity $W_{P,\lambda }=W_P-2\log (\lambda )V$—cf. Proposition 51. However, the proof of Proposition 51 shows that $EV=0$ which implies that $[EW_P]$ is scaling invariant. $\quad \square$