Swiftcomp Braided Composite

and Higgs Centre for Theoretical Physics, Edinburgh, U.K.

Abstract

The homotopy algebraic formalism of braided noncommutative field theory is used to define the explicit example of braided electrodynamics, that is, U(1) gauge theory minimally coupled to a Dirac fermion. We construct the braided L∞-algebra of this field theory and obtain the braided equations of motion, action functional and conserved matter current. The modifications of the electric charge conservation law due to the braided noncommutative deformation are described.

We develop a braided generalization of Wick’s theorem, and use it to compute correlation functions of the braided quantum field theory using homological perturbation theory.

Our

putative calculations indicate that the braided theory does not contain the non-planar Feynman diagrams of conventional noncommutative quantum field theory, and that correlators do not exhibit UV/IR mixing.

Keywords: Drinfel’d twists, L∞-algebras, braided symmetries, quantum electrodynamics, homological per-

∞-Algebras And Classical Field Theory

. Braided L∞-algebras and braided field theory . Braided Wick’s theorem .

Braided homological perturbation theory . .

∞-Algebra Of Electrodynamics

. Braided L∞-algebra of electrodynamics . Braided electrodynamics .

Braided Batalin–Vilkovisky Functional

. Braided homological perturbation theory .

Photon Self-Energy At One-Loop

. Fermion self-energy at one-loop .

Ntroduction

Homotopical methods based on L∞-algebras and A∞-algebras have been playing an increasingly significant role in our understanding of the algebraic and kinematic structures inherent in scattering amplitudes and correlation functions of quantum field theory; for an incomplete sample of recent works see e.g. and references therein. At a given order of perturbation theory, these can be calculated in a purely algebraic fashion without resorting to canonical quantization or path integral techniques: the quantum Batalin–Vilkovisky (BV) formalism gives an explicit homological homological perturbation theory is discussed in e.g. .

In this paper we are interested in what these techniques can teach us about noncommutative quantum field theory. Noncommutative field theories arise in many scenarios as effective theories; see [15, 16] for early reviews of the subject as well as its relevance in open string theory with B- fields. Their L∞-algebra formulation at the classical level was discussed in ; see for a recent exposition. There are two main problems that this paper aims to address.

The standard noncommutative quantum field theories are famously plagued by the notorious problem of ‘UV/IR mixing’, which is related to the appearance of non-planar loop diagrams in perturbation theory . The naive ultraviolet regulator θ provided by the noncommutative defor-

Θ ≪1. Uv/Ir Mixing Occurs In A Loop Correlator

when the regularisation entangles ultraviolet and infrared regimes: an ultraviolet cutoff Λ induces an effective infrared cutoff Λ0 = 1/θ Λ. While non-planar graphs are generically well-defined, they can lead to uncontrollable divergences when inserted as subgraphs into higher order graphs. These divergences increase with the order of perturbation theory, and all correlation functions are affected and diverge. As a consequence, the quantum field theory cannot be renormalized.

For noncommutative ϕ4-theory with the Moyal–Weyl star-product, the UV/IR mixing problem can be cured by adding a background harmonic oscillator potential to the free part of the classical action functional. The quantum field theory is then covariant under Fourier transformation of the fields , which renders the interchange of ultraviolet and infrared regimes a symmetry. This is the celebrated Grosse–Wulkenhaar model , which is renormalizable to all orders in perturbation theory. But it is not understood how to achieve this in gauge theories.

In this paper we explore a new approach to renormalizable noncommutative quantum field theory by instead modifying the path integral directly, rather than the classical action functional. Our approach is firmly rooted in the homological techniques developed by : By deforming the L∞-structure of a field theory to a braided L∞-algebra, one constructs field theories which are covariant under the action of a triangular Hopf algebra of symmetries, with braided noncommutative fields.

Quantum correlation functions are then computed via a braided deformation of the BV formalism and homological perturbation theory: This is called braided quantum field theory. The renormalization properties of braided quantum field theory turn out to be very different, and UV/IR mixing seems to be less severe and maybe even absent. This is due in part to the absence of non- planar diagrams, which we demonstrate explicitly in the example of noncommutative ϕ4-theory.

Our formulation of braided quantum field theory realises a special case of Oeckl’s approach [24, 25] (see also ), which formulates the path integral of ordinary quantum field theory in a purely algebraic language and then generalizes it to ‘braided spaces’ of fields which are objects in the braided monoidal representation category of a quasi-triangular Hopf algebra. This algebraic ap- proach is based on normalised Gaussian integration over braided spaces, which leads to a braided generalization of Wick’s theorem. In this way braided quantum field theory follows the traditional path integral approach, going from Gaussian path integrals via perturbation theory to Feynman also having the advantage of going beyond the classes of theories immediately covered by Oeckl’s approach: our formalism also naturally treats theories with gauge symmetries.

For Moyal–Weyl twists, our systematic treatment of noncommutative ϕ4-theory makes precise some older claims from the literature, many of which were reached through somewhat ad hoc lines of reasoning and constructions. Based on Oeckl’s computation of deformed Green’s functions for scalar fields in terms of undeformed ones , braided deformations of free quantum field commutation and anti-commutation relations, i.e. of the oscillator algebras of creation and annihilation operators, were suggested in several works, see e.g. . In particular, from this it was argued in that noncommutative quantum field theory with braided symmetry dispels with UV/IR mixing in S-matrix elements of scalar field theories. These treatments employ braided tensor products, which also twist the propagators of the field theory, in contrast to our approach.

N Fact, Our

more systematic calculations generally reach somewhat different conclusions: braided quantum field theory is not the same as its undeformed counterpart . The second main goal of this paper is to understand the homotopy algebraic approach to quan- tization of braided field theories with gauge symmetries. As a first step towards understanding the more elaborate non-abelian gauge theories, here we undertake a detailed study of the simplest example of a U(1) gauge theory coupled to a Dirac fermion. We call this theory braided quantum electrodynamics (QED); a preliminary investigation of this model was announced in .

This

theory is markedly different from that of the standard noncommutative QED, whose photon field is self-interacting, contrary to the photon of braided QED. We develop the classical braided field theory in detail, and in particular demonstrate how the homotopy Noether identity associated to the U(1) gauge symmetry naturally implies the electric charge conservation law. In the quantized theory we demonstrate the absence of non-planar diagrams as well as UV/IR mixing in one-loop two-point correlators. This is also in agreement with earlier calculations which found that no UV/IR mixing occurs in S-matrix elements of U(1) gauge theory coupled to matter. The implications of this for the renormalizability of braided QED is left for future work.

Outline. A central purpose of this paper is to demonstrate how to compute correlation functions for perturbative braided quantum field theory using braided quantum L∞-algebras (equivalently the braided BV formalism), and in particular to present explicit expressions for correlation functions obtained through these algebraic techniques. The structure of this paper is as follows.

In Section 2 we briefly review braided L∞-algebras and their application in developing a novel particular class of examples of noncommutative field theories, called braided field theory. We further describe their perturbative quantization by developing a braided version of Wick’s theorem and applying the BV formalism. We treat the noncommutative ϕ4-theory in some detail, demonstrating the absence of UV/IR mixing in the one-loop self-energy.

In Section 3 we present the explicit example of braided electrodynamics. We first construct the corresponding braided L∞-algebra, and then use it to formulate the action functional and the equations of motion. Using the braided Noether identity we find the associated conserved matter current and discuss the modifications to the electric charge conservation law.

In Section 4 we explore the perturbative expansion of quantized braided electrodynamics defined by the braided Wick’s theorem and homological perturbation theory. In particular, we find that there are no non-planar Feynman diagrams as well as no UV/IR mixing through explicit computa- tions of the vacuum polarization and the fermion self-energy at one-loop.

Two appendices at the end of the paper contain some technical details which are used in the main text: Appendix A briefly reviews the basics of twist deformations that are used in our constructions of braided field theories, while Appendix B summarises our conventions for Dirac spinors.

We are grateful to Paolo Aschieri, Martin Cederwall, Branislav Jurčo, Den- joe O’Connor, Biljana Nikolić, Christian Sämann, Miša Toman, Francesco Toppan and Guillaume for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 39083149) for hospitality and support during part of this work. The work of M.D.C., N.K. and V.R. is supported by Project 451-03-47/2023-01/200162 of the Serbian Ministry of Education, Science and Technolog- ical Development. The work of M.D.C and R.J.S. was partially supported by the Croatian Science Foundation Project IP-2019-04-4168. The work of R.J.S. was supported by the Consolidated Grant ST/P000363/1 from the UK Science and Technology Facilities Council.

∞-Algebras And Classical Field Theory

Let us begin by briefly recalling the definition of a classical L∞-algebra. An L∞-algebra L is a

Z-Graded Real Vector Space L = L

k∈Z Lk equipped with graded antisymmetric multilinear maps

A1 ⊗· · · ⊗An 7−→ℓn(A1, . . . , An)

for each n ≥1, which have degree |ℓn| = 2 −n. The graded antisymmetry translates to ℓn(. . , a, a′, . ) = −(−1)|a| |a′| ℓn(. , a′, a, . ) ,

(2.1)

where |a| denotes the degree of a homogeneous element a ∈L. We write ℓ:= {ℓn}n≥1 for the collection of all multilinear maps, which are also called multibrackets. The n-brackets ℓn are required to fulfill infinitely many homotopy relations, for each n ≥1. The

= 0

says that underlying any L∞-algebra L is a cochain complex (L, ℓ1):

says that the differential ℓ1 is a graded derivation with respect to the 2-bracket ℓ2, or in other words that ℓ2 is a cochain map. The third homotopy relation

says that the graded Jacobi identity for ℓ2 is violated by a cochain homotopy determined by ℓ3, and so on for n > 3. In applications to Lagrangian field theory, one additionally asks that an L∞-algebra L is en- dowed with a graded symmetric non-degenerate bilinear pairing ⟨−, −⟩: L ⊗L →R which is cyclic

In The Sense That

⟨a0, ℓn(a1, a2, . . , an)⟩= ± ⟨an, ℓn(a0, a1, . , an−1)⟩ for all n ≥1. Here and in the following we write ± for the sign factors determined by the grading of the elements ai involved through the Koszul sign rule.

It was shown in (see for a review) that any classical field theory with irreducible gauge symmetries (that is, with independent gauge transformations) can be completely encoded in 4-term L∞-algebras L with underlying graded vector space

= L0 ⊕L1 ⊕L2 ⊕L3 .

Given a gauge parameter c ∈L0 and a dynamical field A ∈L1, the gauge variations are given by

(2.2)

where the ellipses designate higher brackets involving tensor powers A⊗n for n ≥3, which are not needed in the applications considered in this paper. The equations of motion FA = 0 in L2 are

(2.3)

Under gauge transformations (2.2) the homotopy Maurer–Cartan equations transform covariantly δcFA = ℓ2(c, FA) + ℓ3(c, FA, A) + · · · .

(2.4)

For the purposes of this paper, we may assume for simplicity that the algebra of gauge variations closes off-shell (that is, when FA̸ = 0). We further assume that the brackets of L satisfy

(2.5)

for all n ≥1, c1, c2 ∈L0, A1, . . , An ∈L1 and A+ ∈L2. Then the homotopy relations imply that the closure relation for the gauge algebra has the form

(2.6)

where [δc1, δc2]◦:= δc1 ◦δc2 −δc2 ◦δc1 is the commutator of gauge variations. The Noether identities in L3 corresponding to the gauge symmetry are encoded by

(2.7)

which vanishes identically as a consequence of the homotopy relations on A⊗n for all n ≥1. The action functional of a Lagrangian field theory can be written via a symmetric non-degenerate bilinear pairing ⟨−, −⟩: L ⊗L →R of degree −3 which makes L into a cyclic L∞-algebra. Then the equations of motion FA = 0 follow from varying the homotopy Maurer–Cartan action functional

(2.8)

since cyclicity implies δS(A) = ⟨FA, δA⟩. Cyclicity also implies

(2.9)

so that gauge invariance of the action functional δcS(A) = 0 is then equivalent to the Noether identities dAFA = 0. Note that ℓ1(A) is associated with the free equations of motion, whereas ℓn(A⊗n) for n ≥2 correspond to interaction vertices in the Lagrangian. In this formalism, free fields (that is, solutions to the linearised equations of motion) are in the cohomology H•(L) of the underlying cochain complex (L, ℓ1).

Braided L∞-Algebras And Braided Field Theory

Starting from a suitable classical L∞-algebra L , using Drinfel’d twist deformation techniques one can construct a braided L∞-algebra L ⋆in the sense of (see for a review). A brief review of the Drinfel’d twist deformation formalism is presented in Appendix A, while more details can be found in . Let v := Γ(TM) be the Lie algebra of vector fields on a manifold M, and let Uv be its enveloping algebra. For a twist F ∈Uv[[ν]]⊗Uv[[ν]], we write F = fα ⊗fα and F−1 = ¯fα ⊗¯fα for its inverse. The corresponding triangular R-matrix is R = F21 F−1 =: Rα ⊗Rα, with inverse

R−1 = R21 = Rα ⊗Rα.1

To apply the twist deformation formalism, we start from a classical L∞-algebra L whose un-

Derlying Graded Vector Space L = L

k∈Z Lk is a Z-graded (left) Uv-module and the n-brackets ℓn : L⊗n →L are equivariant maps, that is, they all commute with the action of v = Γ(TM) on L via the trivial coproduct ∆. Given any Drinfel’d twist F ∈Uv[[ν]] ⊗Uv[[ν]], we deform the brack-

Ets ℓn To Twisted Brackets ℓ⋆

n which commute with the action of Γ(TM) on L[[ν]] via the twisted coproduct ∆F. Following the standard prescription (A.6), we set ℓ⋆

(2.10)

for n ≥2, where a ⊗⋆a′ := F−1(a ⊗a′) = ¯fα(a) ⊗¯fα(a′) for a, a′ ∈L[[ν]]. These define multilinear

Maps ℓ⋆

n : L[[ν]]⊗n →L[[ν]] which are braided graded antisymmetric:

. The first and second homotopy relations are unchanged with respect to the corresponding clas- sical homotopy relations; that is, the braided L∞-algebra L ⋆still has underlying cochain complex

(L[[Ν]], ℓ1) And ℓ⋆

2 is again a cochain map. The third homotopy relation is given by

(2.11)

We observe that the non-trivial braiding now appears in this relation, which says that the braided

Is Violated By The Cochain Homotopy ℓ⋆

3. If L is a cyclic L∞-algebra with a Uv-invariant inner product, then its cyclic structure ⟨−, −⟩: L ⊗L →R is twist deformed to a new inner product ⟨−, −⟩⋆: L[[ν]] ⊗L[[ν]] →R[[ν]] defined by ⟨a1, a2⟩⋆:= ⟨¯fα(a1),¯fα(a2)⟩.

(2.12)

In general, graded symmetry of the cyclic pairing ⟨−, −⟩implies that the twisted pairing ⟨−, −⟩⋆is

⟨A0 , ℓ⋆

n(a1, a2, . . , an)⟩⋆= ± ⟨Rα0 Rα1 · · · Rαn−1(an) , ℓ⋆ n(Rα0(a0), Rα1(a1), . , Rαn−1(an−1))⟩⋆. However, for applications to field theory, we have to restrict to compatible Drinfel’d twists that

⟨A2, A1⟩⋆= (−1)|A1| |A2| ⟨A1, A2⟩⋆

for all homogeneous a1, a2 ∈L[[ν]]. In this case, L ⋆becomes a strictly cyclic braided L∞-algebra. Following the classical case, a braided field theory is built as a noncommutative deformation of a classical field theory which is completely defined in terms of its braided L∞-algebra.

Et

1Throughout this paper, repeated upper and lower indices are always implicitly summed over. L ⋆= (L[[ν]], ℓ⋆) be a 4-term braided L∞-algebra, obtained by twist deformation of an L∞-algebra L = (L, ℓ) which organises the symmetries and dynamics of a classical field theory. For a gauge parameter c ∈L0[[ν]], we define the braided gauge variation2 of a dynamical field A ∈L1[[ν]] by

(2.13)

Braided covariant dynamics is described by the equations of motion F ⋆

(2.15)

for all gauge parameters c ∈L0[[ν]]. The braided gauge transformations obey the off-shell closure relation in terms of the braided

(2.16)

Corresponding to the braided gauge symmetry, a suitable combination of the braided homotopy

(2.17)

Unlike the classical Noether identity (2.7), the braided Noether identity (2.17) is no longer linear

In The Equations Of Motion F ⋆

A and contains inhomogeneous terms involving brackets of the fields A themselves. This is related to the violations of the Bianchi identities in braided gauge theories . In the classical limit ν = 0, where R = 1 ⊗1, the braided homotopy formulas (2.13)–(2.17) all reduce to the classical formulas in (2.2)–(2.7).

For a Lagrangian field theory, using the (strictly) cyclic inner product one can define an analogue of the homotopy Maurer–Cartan action functional for the braided field theory as

(2.18)

whose variational principle yields the braided equations of motion F ⋆

(2.19)

for all c ∈L0[[ν]] and A ∈L1[[ν]]. However, unlike the classical case, the braided Noether identity for the braided gauge symmetry cannot be derived from the variational principle, because braided gauge variations and Euler–Lagrange variations behave very differently, see .

Note that the free fields of braided field theory are unchanged from the classical field theory: they are still the degree 1 elements of the cohomology H•(L[[ν]]) of the underlying cochain complex (L[[ν]], ℓ1). Only the interaction vertices, corresponding to the higher brackets ℓ⋆

N For N ≥2, Are

modified by the braided noncommutative deformation. 2In general, one can define both left and right braided gauge transformations. In this paper we focus only on left braided gauge transformations for simplicity. More details can be found in .

Braided Wick’S Theorem

L∞-algebras are the natural algebraic structure underlying the Batalin–Vilkovisky (BV) formal- ism , which may be used for the quantization of classical field theories. Similarly, we expect that braided L∞-algebras should be related to a braided generalization of BV quantization. This was described explicitly in for field theories with finitely many degrees of freedom, and applied to an example of braided fuzzy scalar field theory; the braided BV formalism is also discussed in .

One of the purposes of the present paper is to formulate braided quantum field theory and study its features in a simple example of a continuum field theory with gauge symmetry. In order to set up and illustrate the general framework for this, we consider here the simple example of scalar field theory on d-dimensional Minkowski spacetime R1,d−1. We show how to recover, using heuristic field theory arguments, Oeckl’s approach to (symmetric) braided quantum field theory which relies upon a braided generalization of Wick’s theorem based on purely algebraic arguments [24, 25]. This result will then be substantiated in Section 2.4, where we develop the braided BV quantization of scalar field theory.

The action functional for a free real scalar field ϕ on R1,d−1 with mass m is given by

Ddx = Dx0 ∧Dx1 ∧· · · ∧Dxd−1

is the standard volume form on Minkowski spacetime and □is the d’Alembertian operator. The differential of the underlying abelian L∞-algebra L0 is given by the Klein–Gordon operator as ℓ1 = −(□+ m2). Since there are no gauge symmetries, only L1 = L2 = Ω0(R1,d−1) are non-trivial and given by two copies of the space of functions (regarded as 0-forms) on R1,d−1.

The L∞-algebra L0 is completely described by the 2-term cochain complex

(2.21)

concentrated in degrees 1 and 2, where the square brackets indicate shifts of the cohomological degree.3 Elements of the degree 1 cohomology of this complex H1(L) = ker(□+ m2) are the states ϕ(0) ∈Ω0(R1,d−1) that solve the Klein–Gordon equation. Elements ϕ+ ∈L2 correspond to the BV antifields of the physical fields ϕ ∈L1.

The cyclic structure of degree −3 is given by the non-zero inner product

(2.22)

for ϕ ∈L1 and ϕ+ ∈L2, which is cyclic because the Klein–Gordon operator is formally self-adjoint with respect to this inner product. Interaction vertices are incorporated by including the non-zero

(2.23)

for n ≥2, λn ∈R and ϕ1, . . , ϕn ∈L1, and writing the homotopy Maurer–Cartan action functional (2.8). The homotopy relations follow trivially for degree reasons, while cyclicity of (2.22) with respect to ℓn is a trivial consequence of commutativity of pointwise multiplication of functions.

3For any vector space W and integer p ∈Z, elements of W[p] are of degree −p. The perturbative n-point functions of the free quantum field theory are defined by the formal

Normalized Functional Integral

Gn(x1, . . , xn)(0) := ⟨0|T[ϕ(x1) · · · ϕ(xn)]|0⟩(0)

(2.24)

where T implements the time-ordered product of fields, and Z is a normalization factor such that ⟨0|0⟩(0) = 1. This is non-zero only when n = 2k is even, and Wick’s theorem expresses it as the

⟨0|T[Φ(Xσ(2A−1)) Φ(Xσ(2A))]|0⟩(0) ,

where Sn denotes the symmetric group of all permutations of degree n. The free two-point functions

(2.25)

for ϵ ∈R>0. In the following we will drop the i ϵ-prescription in the Feynman propagators, and

(2Π)D ,

in order to simplify the presentation. The braided noncommutative deformation follows the twist formalism discussed in Section 2.2. The classical scalar field theory is Poincaré invariant (but not diffeomorphism invariant), so its L∞-algebra L0 consists of modules and equivariant brackets for the universal enveloping algebra Uiso(1, d −1) ⊂Uv of the Poincaré algebra iso(1, d −1).

Hence We Have To Restrict To Twists

F ∈Uiso(1, d−1)[[ν]]⊗Uiso(1, d−1)[[ν]]. For simplicity, we will work with abelian twists, for which R = F21 F−1 = F−2; the standard Moyal–Weyl twist and also the angular twist of are examples of such twists. We will further restrict to abelian twists F ∈Uiso(d −1)[[ν]] ⊗Uiso(d −1)[[ν]] constructed from the spatial isometries of Rd−1 ⊂R1,d−1, as this simplifies some of the analysis in the quantum field theory, such as the treatment of time-ordering, as well as avoiding potential issues with unitarity. For definiteness, and for the sake of illustration, let us choose here the Moyal–Weyl

(2.26)

where (θij) is a (d −1)×(d −1) antisymmetric real-valued matrix, and ∂i =

∂Xi ∈Γ(Trd−1) For

i = 1, . . , d −1 are vector fields generating spatial translations in R1,d−1.

Since L ⋆

0 = L0, and since the twist F is compatible with the cyclic inner product (2.22), the free braided scalar field theory is unchanged from its commutative version. In particular, because ⟨−, −⟩⋆= ⟨−, −⟩, the action functional (2.20) is unchanged:

S0⋆(Φ) = 1

2 ⟨ϕ, ℓ1(ϕ)⟩⋆= S0(ϕ) . Interaction vertices are included by twisting the brackets (2.23) to

N(Φ1, . . . , Φn) = Λn Φ1 ⋆· · · ⋆Φn ,

for n ≥2, λn ∈R and ϕ1, . . , ϕn ∈L1[[ν]], and writing the braided version of the homotopy Maurer–Cartan action functional (2.18). It follows that, at the classical level, even the standard noncommutative scalar field theory is organised by a braided L∞-algebra . However, the braided symmetry makes a crucial difference in the quantum field theory, particularly in the application of a braided Wick expansion, rather than the standard one, and in the different interaction vertices arising from the braided symmetry.

Correlation functions in the interacting quantum field theory will be discussed in Section 2.4 below. Given this braided L∞-algebra structure, we would now like to define the free braided n-point functions. While the traditional path integral and canonical quantization methods are not readily available for braided quantum field theory, we can quantize the theory in a purely algebraic fashion using modern techniques from homological algebra, as we explain in Section 2.4. Here we shall define them operationally by a heuristic noncommutative deformation of the Feynman representation of (2.24) in the following way, which also leads to a purely algebraic prescription, while at the same time elucidating the physical meaning of the braiding.4 Firstly, whereas in conventional noncommutative field theory the functional integral would still be taken over the commutative space of fields Ω0(R1,d−1), in braided quantum field theory the domain of integration is a ‘braided space’ of fields Ω0

⋆(R1,D−1) , That Is, The Algebra Of Fields With

the star-product ⋆which can be thought of as endowing them with ‘braided statistics’. The braided

G⋆

n(x1, . . , xn)(0) = ⟨0|T[ϕ(x1) ⋆· · · ⋆ϕ(xn)]|0⟩(0)

,

with the star-product canonically extended to the tensor product ϕ⊗n ∈Ω0(R1,d−1)×n



ϕ(x1) · · · ϕ(xn) . More generally, the twist (2.26) can be lifted to the space of functionals of the fields as

,

where Π is the conjugate momentum to the field ϕ. Secondly, the integration measure D⋆ϕ is taken to be UFiso(d−1)-invariant, so that the operation

⋆

defines a UFiso(d −1)-equivariant map. This implies that the twist can be factored out of the functional integration and taken to act on the n-point function of the commutative scalar field theory, which may then be expanded using the usual Wick theorem.

Altogether, for the non-vanishing correlation functions we prescribe the simple expression

C=1

⟨0|T[ϕ(xσ(2c−1)) ϕ(xσ(2c))]|0⟩(0) .

(2.27)

This formula will be derived in a more precise way in Section 2.4. Unravelling the combinatorics of the formula (2.27) reveals the following general statement of

The Braided Wick Theorem:

4Since the free braided field theory is the same as its commutative counterpart, this can also be formulated via the operator formalism in the Dyson representation of (2.24). • The braided n-point function is defined by replacing the pointwise products of fields in the commutative n-point function with star-products.

• Only nearest neighbouring fields can contract. To contract we therefore have to first permute fields, which introduces corresponding R-matrices. • After the contractions we use simplifications to remove some R-matrices, such as the identities from Appendix A together with relativistic invariance of the commutative two-point functions.

• The final result is the braided Wick theorem, which agrees with the results from . Note that no star-products will appear between contractions, due to iso(d −1)-invariance of the commutative two-point functions: noncommutativity enters only through the permutations of fields, which produce R-matrices. Let us give a few explicit examples to illustrate how the theorem works.

(2.29)

where the noncommutative phase factor vanishes due to the antisymmetry of θij. Thus the two-point function remains unchanged, as expected. This same result is explained in . In the following we

∂

∂xia for brevity. Four-point function.

G⋆

4(x1, x2, x3, x4)(0) = ⟨0|T[ϕ(x1) ⋆ϕ(x2) ⋆ϕ(x3) ⋆ϕ(x4)]|0⟩(0)

(2.30)

Applying the first biderivative operation from (2.27) results in

,

which follows from (2.25). The remaining biderivative operations from (2.27) follow in a similar

(2.31)

From (2.31) we recognise the appearance of the inverse R-matrix and we can finally write

G⋆

4(x1, x2, x3, x4)(0) = ϕ1 ϕ2 ϕ3 ϕ4 + ϕ1 Rα(ϕ3) Rα(ϕ2) ϕ4 + ϕ1 ϕ4 ϕ2 ϕ3 .

(2.33)

where in the first equality we used the R-matrix identities (A.4), in the second equality we used Uiso(d −1)-invariance of the two-point function, and in the last equality we used the normalization (A.2) of the twist; this can also be checked by explicitly computing the left-hand side of (2.33) using (2.25).

Six-point function. As a final illustration of our statement of the braided Wick theorem, we

G⋆

6(x1, x2, x3, x4, x5, x6)(0) = ⟨0|T[ϕ(x1) ⋆ϕ(x2) ⋆ϕ(x3) ⋆ϕ(x4) ⋆ϕ(x5) ⋆ϕ(x6)]|0⟩(0)

(2.34)

Applying (2.27), a calculation similar to that for the four-point function gives the slightly cumber-

G⋆

6(x1, . . , x6)(0) = ϕ1 ϕ2 ϕ3 ϕ4 ϕ5 ϕ6 + ϕ1 ϕ2 ϕ3 Rα(ϕ5) Rα(ϕ4) ϕ6

+ Φ1 Φ2 Φ3 Φ6 Φ4 Φ5 + Φ1 Rα(Φ3) Rα(Φ2) Φ4 Φ5 Φ6

+ ϕ1 Rα(ϕ3) Rα(ϕ2) Rβ(ϕ5) Rβ(ϕ4) ϕ6 + ϕ1 ϕ6 ϕ2 ϕ5 ϕ3 ϕ4

+ Φ1 Rα Rβ(Φ4) Rα(Φ2) Rσ(Φ5) Rσ Rβ(Φ3) Φ6

+ ϕ1 Rα(ϕ5) ϕ2 ϕ3 Rα(ϕ4) ϕ6 + ϕ1 Rα Rβ(ϕ4) Rα(ϕ2) ϕ6 Rβ(ϕ3) ϕ5 + ϕ1 Rα(ϕ5) ϕ2 Rβ(ϕ4) Rβ Rα(ϕ3) ϕ6 + ϕ1 Rα(ϕ5) Rα(ϕ2) ϕ6 ϕ3 ϕ4 + ϕ1 ϕ6 ϕ2 ϕ3 ϕ4 ϕ5 + ϕ1 ϕ6 ϕ2 Rα(ϕ4) Rα(ϕ3) ϕ5 .

(2.35)

In the commutative limit, when the R-matrix reduces to the identity operator, our results (2.32) and (2.35) reduce to the well-known expressions for the four-point and the six-point functions in standard free scalar field theory. The noncommutative deformation enters through the permutations of fields before contracting them.

Braided Homological Perturbation Theory

Following we now explain how to compute correlation functions of the interacting braided scalar field theory via the technique of ‘homotopy transfer’. We start from the cohomology H•(L ⋆

Of The Abelian L∞-Algebra L ⋆

0 , which describes the classical vacua of the free scalar field theory on R1,d−1. This is also an abelian L∞-algebra, and from (2.21) it follows that it is also concentrated in degrees 1 and 2, given by the solution space H1(L) = ker(ℓ1) of the massive Klein–Gordon equation □ϕ + m2 ϕ = 0 and the space H2(L) = coker(ℓ1) of on-shell Maurer–Cartan expansions. The

□+ M2

[−2] . To describe correlation functions in the path integral framework, we need to define a Uiso(d−1)- equivariant projection p : L →H•(L) of degree 0 and a Uiso(d−1)-invariant contracting homotopy h : L →L of degree −1. For this, we denote the scalar Feynman propagator G : Ω0(R1,d−1) →

K2 −M2 ,

where ˜G(k) are the eigenvalues of the Green operator G when acting on plane wave eigenfunctions

□+ M2

◦G = idΩ0(R1,3) . If we were to compute scattering amplitudes, then the first component of the projection p(1) : L1 →H1(L) should be taken to be the projection idΩ0(R1,3) −G ◦ℓ1 to on-shell states. However, for the purposes of computing correlation functions, we should project to the trivial vacuum ϕ = 0.

This is a consequence of the fact that correlators can be equivalently computed by Wick rotating to Euclidean signature, where the kernel and cokernel of the kinetic operator ℓ1 become trivial, and this was already used in Section 2.3. Similarly, whereas the second component p(2) : L2 →H2(L) could be taken to be the natural projection induced by the quotient map to coker(ℓ1), we use the

P(1) = 0 = P(2) ,

or more accurately we restrict the cochain complex of H•(L ⋆ 0 ) to its trivial subspaces. With these choices, the only non-vanishing component of the contracting homotopy h(2) : L2 → L1 is given by the propagator h(2) = G. Explicitly

(2.36)

for ϕ+ ∈L2. We apply the braided homological perturbation theory developed by . For this, we need to extend the maps p and h to the space of functionals on L; in this paper we will only com- pute correlation functions of polynomial observables, hence we restrict to the braided symmetric

P(Φ1 ⊙⋆· · · ⊙⋆Φn) = 0 ,

along with a contracting homotopy H : SymRL →SymRL through

(2.37)

for all φa ∈L, with a = 1, . . , n; we used Uiso(d −1)-invariance of h in (2.37) which trivializes the actions of R-matrices. Note that on generators the twisted symmetric product ⊙⋆is braided

Graded Commutative:

φa ⊙⋆φb = (−1)|φa| |φb| Rα(φb) ⊙⋆Rα(φa) . We perturb the free differential ℓ1 to the ‘quantum’ differential

Qδ = ℓ1 + Δ

on L, where the formal Uiso(d −1)-invariant perturbation δ will be specified below. The braided extension of the homological perturbation lemma then constructs the perturbed projection map

−1 Δ H ,

which in the classical case gives the path integral . We thus define the n-point correlation functions of the braided quantum field theory by

G⋆

n(x1, . . , xn) = ⟨0|T[ϕ(x1) ⋆· · · ⋆ϕ(xn)]|0⟩⋆:= Pδ(δx1 ⊙⋆· · · ⊙⋆δxn)

(2.38)

where δxa(x) := δ(x−xa) are Dirac distributions supported at the insertion points xa of the physical field ϕ ∈L1. Because only P(1) = 1 is non-zero, this is a function in Ω0(R1,d−1)×n

[[Ν]].5 We Are

interested in two perturbations δ of ℓ1. Free theory. The free braided scalar field theory of Section 2.3 is recovered from the perturbation

Δ = I ℏ∆Bv ,

where ∆BV : SymRL →(SymRL) is the braided BV Laplacian defined by

± ⟨Φa, Rαa+1 · · · Rαb−1(Φb)⟩⋆Φ1 ⊙⋆· · · ⊙⋆Φa−1

⊙⋆Rαa+1(φa+1) ⊙⋆· · · ⊙⋆Rαb−1(φb−1) ⊙⋆φb+1 ⊙⋆· · · ⊙⋆φn ,

(2.39)

for all φ1, . . , φn ∈L. The BV Laplacian satisfies the two key properties (∆BV)2 = 0 and ∆BV ◦ℓ1 = −ℓ1 ◦∆BV which guarantee that Q0 = ℓ1 + i ℏ∆BV is a differential, (Q0)2 = 0.

Since the braided BV Laplacian contracts fields pairwise and lowers the symmetric algebra degree from n to n −2, it is clear that in this case the correlation functions (2.38) vanish unless n = 2k is even, in which case the free braided 2k-point functions are then defined by

G⋆

2k(x1, . . , x2k)(0) = ⟨0|T[ϕ(x1) ⋆· · · ⋆ϕ(x2k)]|0⟩(0)

⋆

:= (i ℏ∆BV H)k(δx1 ⊙⋆· · · ⊙⋆δx2k) .

(2.40)

It is not difficult to check using (2.22), (2.36), (2.37) and (2.39) that (2.40) reproduces the braided Wick expansion (2.27) by iterating the basic operation

Φa Rαa+1 · · · Rαb−1(Φb) Δx1 ⊙⋆· · · ⊙⋆Δxa−1

⊙⋆Rαa+1(δxa+1) ⊙⋆· · · ⊙⋆Rαb−1(δxb−1) ⊙⋆δxb+1 ⊙⋆· · · ⊙⋆δx2k , 5As usual in quantum field theory, the amputated correlation functions are distributions in position space, and so should be properly defined by smearing them with suitable test functions, as done in . Here we follow the more conventional physics practice of defining ‘localized’ correlation functions.

Φa Φb := ⟨0|T[Φ(Xa) Φ(Xb)]|0⟩(0) = −I ℏg(Xa −Xb)

for the free propagator, and the Koszul sign factors are trivial for antifields φa = δxa ∈L2; we used the fact that the only non-zero pairings are ⟨δxa, G(δxb)⟩⋆= G(xa −xb). For the two-point function one finds immediately the free propagator

G⋆

2(x1, x2)(0) = i ℏ∆BV H(δx1 ⊙⋆δx2) = −i ℏG(x1 −x2) = −i ℏ

Φ1 Φ2 Δx3 ⊙⋆Δx4 + Φ1 Rα(Φ3) Rα(Δx2) ⊙⋆Δx4

+ ϕ1 Rα Rβ(ϕ4) Rα(δx2) ⊙⋆Rβ(δx3) + ϕ2 ϕ3 δx1 ⊙⋆δx4

I ℏ∆Bv H(Δxa ⊙⋆Δxb) = Φa Φb

we then find that the four-point function is given by

G⋆

4(x1, x2, x3, x4)(0) = (i ℏ∆BV H)2 (δx1 ⊙⋆δx2 ⊙⋆δx3 ⊙⋆δx4)

= 1

2 ϕ1 ϕ2 ϕ3 ϕ4 + ϕ1 Rα(ϕ3) Rα(ϕ2) ϕ4 + ϕ1 Rα(ϕ3) ϕ2 Rα(ϕ4)

. We now employ the same R-matrix manipulations as in Section 2.3. Using

we conclude that the second and third terms are equal; this also follows abstractly from the trian- gular Hopf algebra identity (SF ⊗idUiso(d−1)) R = R−1 = R21 and Uiso(d −1)-invariance of the two-point functions. From the identity (2.33) we see that the last two terms are also equal.

G⋆

4(x1, x2, x3, x4)(0) = ϕ1 ϕ2 ϕ3 ϕ4 + ϕ1 Rα(ϕ3) Rα(ϕ2) ϕ4 + ϕ1 ϕ4 ϕ2 ϕ3 , which agrees with (2.32), and also with [11, Equation (5.40)]. It is straightforward, if lengthy, to extend this calculation to compute the six-point function (2.35), and also to higher order correlation functions. In this way we recover the braided Wick theorem of Section 2.3.

Interacting theory. An interacting scalar field theory on R1,d−1 is captured by the perturbation

Δ = I ℏ∆Bv + {Sint, −}⋆,

where the operator {Sint, −}⋆is constructed in the following way. For definiteness, we consider braided λ ϕ4-theory in four dimensions, but the methods easily extend to scalar field theories with arbitrary polynomial interactions in any dimension. This amounts

To Extending L ⋆

0 to a non-abelian braided L∞-algebra L ⋆with the single non-vanishing higher

(Φ1, Φ2, Φ3) = −Λ Φ1 ⋆Φ2 ⋆Φ3

for all ϕ1, ϕ2, ϕ3 ∈L1. The braided version of the homotopy Maurer–Cartan action functional (2.18)

(2.41)

which is just the standard noncommutative scalar λ ϕ4-theory .

Bracket ℓ⋆

3 to give a new braided L∞-algebra on H•(L), called the minimal model of the braided L∞-algebra L ⋆, and at the same time “quantize” it. For the computation of interacting correlation functions, we extend the braided L∞-algebra structure on L[[ν]] to (SymRL) ⊗L[[ν]] via the non-zero brackets

,

for a1, a2, a3 ∈SymRL and ϕ1, ϕ2, ϕ3 ∈L1[[ν]]; again we write ± for the Koszul sign factors determined by the gradings of the elements involved in all operations. Similarly, the cyclic structure

(2.42)

for a1, a2 ∈SymRL, ϕ ∈L1[[ν]] and ϕ+ ∈L2[[ν]]. The antibracket is the braided graded Poisson bracket {−, −}⋆: SymRL ⊗SymRL →

{Φa, Φb}⋆= ⟨Φa, Φb⟩⋆= ± {Rα(Φb), Rα(Φa)}⋆

for φa ∈L, and extending this to all of SymRL as a braided graded Lie bracket which is a braided graded derivation on SymRL in each of its slots. For example {φ1, φ2 ⊙⋆φ3}⋆= ⟨φ1, φ2⟩⋆⊙⋆φ3 ± Rα(φ2) ⊙⋆⟨Rα(φ1), φ3⟩⋆.

(2.43)

The antibracket is compatible with the differential ℓ1, extended as a graded derivation to all of SymRL, as a consequence of cyclicity of the inner product ⟨−, −⟩⋆.

Braided Bv Laplacian Through

∆BV(a1 ⊙⋆a2) = ∆BV(a1) ⊙⋆a2 + (−1)|a1| a1 ⊙⋆∆BV(a2) + {a1, a2}⋆,

(2.44)

for all a1, a2 ∈SymRL. Via Fourier transformation, we introduce the basis of plane waves ek(x) = e−i k·x for L1 and the

K(X) = E−K(X) = E I K·X

for L2. These bases are dual with respect to the inner product (2.22), in the sense that

Where Throughout We Use

R1,3 d4x e± i k·x = (2π)4 δ(k) .

(2.45)

where k · θ p := ν kµ θµλ pλ = −p · θ k, while the action of the inverse R-matrix on them is given by R−1(ek ⊗ep) = Rα(ek) ⊗Rα(ep) = e i k·θ p ek ⊗ep .

(2.46)

Using this basis we now define the contracted coordinate functions ξ ∈(SymRL) ⊗L[[ν]] in degree 1 by the formal Uiso(3)-invariant expression

. Using the braided Maurer–Cartan action functional (2.41), we define the the interacting part of the BV action functional Sint ∈SymRL in degree 0 by the Uiso(3)-invariant element

(2.47)

As discussed in , this satisfies the classical master equation

{Sint, Sint}⋆= 0 ,

and it is annihilated by the braided BV Laplacian, ∆BV(Sint) = 0. As a consequence, the operator

Qint = ℓ1 + I ℏ∆Bv + {Sint, −}⋆

is a differential, (Qint)2 = 0, which describes the correlation functions in terms of a braided quantum L∞-algebra. Calculating (2.47) explicitly using (2.22), (2.45) and (2.46) we find

K1,...,K4

e i k2·θ k3+i k2·θ k4+i k3·θ k4 ⟨ek1 ⊗ek1 , (ek2 ⊙⋆ek3 ⊙⋆ek4) ⊗ℓ⋆

K1,...,K4

V (k1, k2, k3, k4) ek1 ⊙⋆ek2 ⊙⋆ek3 ⊙⋆ek4 .

(2.48)

coincides with the vertex of the standard noncommutative λ ϕ⋆4

(2.49)

under interchange of any pair of neighbouring momenta, and also the cyclic symmetry

(2.50)

which follows from momentum conservation. The interacting correlation functions of the braided quantum field theory are now given by

G⋆

n(x1, . . , xn)int = ⟨0|T[ϕ(x1) ⋆· · · ⋆ϕ(xn)]|0⟩int

(2.51)

The interaction terms are computed by using the non-zero pairings ⟨eka, G(δxb)⟩⋆= e i ka·xb ˜G(ka) = ⟨G(eka), δxb⟩⋆,

(2.52)

together with the braided derivation property (2.43) of the antibracket and the symmetry properties (2.49)–(2.50) of the interaction vertex V (k1, k2, k3, k4) to get the basic operation

K1,...,K4

V (k1, . . , k4) e i k1·xa ˜G(k1) δx1 ⊙⋆· · · ⊙⋆δxa−1 ⊙⋆ek2 ⊙⋆ek3 ⊙⋆ek4 ⊙⋆δxa+1 ⊙⋆· · · ⊙⋆δxn . The operator {Sint, −}⋆inserts four legs ek1, . , ek4 and contracts ek1 with an external leg δxa, at each of the n insertion points x1, . , xn. Thus it changes the symmetric algebra degree from n to n + 2, and so it follows again that only n-point correlation functions with n = 2k even are non-vanishing, in which case the sum in (2.51) starts from p = k because only P(1) = 1 is non-zero.

The correlation function (2.51) is a formal power series in the parameters ℏand λ. The order λ0 contribution is just the free 2k-point function (i ℏ∆BV H)k(δx1 ⊙⋆· · · ⊙⋆δx2k) discussed earlier. A general order λl contribution will involve a mixture of loop corrections and both connected as well as disconnected parts; these require at least k + l braided Wick contractions i ℏ∆BV H in order to produce a non-vanishing result. It follows that the loop expansion parameter is κ := ℏλ: an l-loop contribution to the 2k-point function is weighted by the factor ℏk κl. We represent terms in the perturbative expansion (2.51) using standard Feynman diagrammatic techniques; the Feynman

Rules Are Depicted In Figure 1.6

6See for a detailed description of how to arrange the computation of (2.51) in terms of a diagrammatic calculus.

K4

Figure 1: Diagrammatic representation of the propagator (left) and interaction vertex (right) for braided

Λ Φ4

4-theory. Two-point function at one-loop. As an explicit example, let us compute the first non-trivial

Correction G⋆

2(x1, x2)(1) (at order λ) to the free two-point function, which from (2.51) is given by

(2.53)

The free four-point functions in (2.53) are evaluated by using the braided Wick expansion (2.32)

And The Pairing (2.52) Together With

⟨eka, G(ekb)⟩⋆= ˜G(ka) (2π)4 δ(ka + kb) .

= −ℏ2 ⟨ek2, G(ek3)⟩⋆⟨ek4, G(δx2)⟩⋆+ ⟨ek2, Rα(G(ek4))⟩⋆⟨Rα(ek3), G(δx2)⟩⋆

= −ℏ2 (2Π)4 ˜G(K2)

δ(k2 + k3) ˜G(k4) e i k4·x2 + δ(k2 + k4) ˜G(k3) e i k3·x2 e−i k3·θ k4

= −ℏ2 (2Π)4 ˜G(K4)

δ(k3 + k4) ˜G(k2) e i k2·x1 + δ(k2 + k4) ˜G(k3) e i k3·x1 e−i k2·θ k3

(2.55)

We now substitute (2.54) and (2.55) into (2.53) using the braided symmetry (2.49) of the in- teraction vertex, and resolve the delta-functions. After relabelling momenta, the noncommutative phase factors in (2.48) are all unity for the relevant momentum combinations (k1, k2, k3, k4) given by (k1, k2, −k2, −k1) and (k1, −k1, k2, −k2), and one finds that all six contributions are the same.

Altogether the one-loop contribution to the two-point function is given by

(2.56)

This result is independent of the deformation parameter and coincides with the classical two-point function (at ν = 0), including the correct sign and overall combinatorial factor. We can recognise the more traditional form by relating the exact two-point function, includ- ing all loop corrections, to the dressed propagator in momentum space using the usual Fourier

P2 −M2 −Π⋆(P) ,

where Π⋆(p) is the self-energy which is given by a sum over all one-particle irreducible (1PI) dia-

(2.57)

which leads to the standard one-loop mass renormalization in λ ϕ4

-Theory, Represented By The Usual

tadpole diagram in Figure 2.

Figure 2: In Braided Λ Φ4

4-theory, the one-loop self-energy receives a contribution only from a planar tadpole diagram. This result is analogous to the one obtained by for braided scalar field theory on the fuzzy torus. In particular, it shows that there is no UV/IR mixing in the two-point function at one-loop order, in contrast to the standard noncommutative quantum field theory , and it seemingly implies the absence of non-planar Feynman diagrams in perturbation theory.

This Appears To

be a consequence of the braided symmetries of the interaction vertex due to the braided L∞- algebra structure, through its interplay with the braided Wick theorem. It would be interesting to understand to what extent this surprising feature of braided quantum field theory persists at higher loop orders and in higher point correlation functions. However, having now introduced the main tools from homological algebra, we instead move on to the main theory of interest in the present paper.

Lassical Electrodynamics

The field content of electrodynamics consists of a U(1) gauge field A = Aµ(x) dxµ ∈Ω1(R1,3) coupled to a massless7 Dirac spinor ψ on four-dimensional Minkowski spacetime R1,3. We use the

∂

∂xµ , where xµ are coordinates on R1,3 with t = x0 the time direction. The infinitesimal U(1) gauge variations are given by

Δc ¯Ψ = −I ¯Ψ C ,

7The zero mass restriction is done for simplicity only, in order to illustrate the general construction. There is no problem with including a mass term for the Dirac spinor as well, and indeed we shall do so later on in Section 4. where c ∈Ω0(R1,3) is the infinitesimal gauge parameter, e is the electric charge of the fermion, and ¯ψ = ψ† γ0 is the conjugate Dirac spinor with γµ the Dirac matrices in four dimensions (see Appendix B for conventions and identities).

The action functional invariant under these gauge transformations is

(3.1)

where we introduced the field strength tensor F = dA ∈Ω2(R1,3) whose components are given by

Fµν Dxµ ∧Dxν = 1

2 (∂µAν −∂νAµ) dxµ ∧dxν. The corresponding equations of motion are

i γµ = 0 . Corresponding to the U(1) gauge symmetry is the electric matter current J = Jµ dxµ ∈Ω1(R1,3)

(3.2)

The gauge field A minimally couples to this current and the continuity equation8 ∂µJµ A≈0 follows from the equation of motion for A. It gives the conserved electric charge

(3.3)

enclosed by any spatial volume B ⊆R3 ⊂R1,3 at fixed time with volume form dB. The conservation law together with Gauss’ theorem imply that the time variation of QB is cancelled by the net current through the spatial surface ∂B bounding the volume B.9 From the modern perspective of generalized global symmetries , the conserved electric charge is a ‘zero-form symmetry’, which is implemented by codimension one defects B in the field theory, whose natural charged objects are the quanta of the local fermion field ψ. Electrodynamics also possesses a one-form magnetic symmetry, implemented by codimension two defects Σ, with two-form

Current Ef = 1

2 g ϵµνλσ F λσ dxµ ∧dxν ∈Ω2(R1,3). Its conservation law ∂µ eF µν = 0 is equivalent to the Bianchi identity for the field strength F, or alternatively to the absence of magnetic monopoles. The natural objects that are charged under this symmetry are the non-local ’t Hooft line defects, of magnetic charge g obeying the Dirac–Zwanziger quantization condition e g ∈2π Z which ensures mutual locality of the electric and magnetic charges. The corresponding conserved quantity is the

(3.4)

through a spatial surface Σ ⊂R3. For an open surface, by Stokes’ theorem this is equivalently

∂Σ A Over The Loop ∂Σ Bounding Σ. For A Closed

surface, the magnetic flux vanishes. The global one-form symmetry can be gauged by minimally coupling the current eF to a back- ground two-form field b ∈Ω2(R1,3) with the U(1) gauge transformation δλb = dλ for λ ∈Ω1(R1,3).

This shifts (3.1) by the action functional of a BF-type topological field theory:

We Use The Notation

Φ≈to indicate an equality which holds on-shell when the equation of motion for a field Φ is imposed. When currents are conserved in this sense, we refer to them as ‘weakly conserved’. 9A more general Lorentz invariant definition uses a codimension one hypersurface B ⊂R1,3. The corresponding charge QB is then conserved in the normal direction to B.

Integrating the source term by parts, the shifted action becomes

−G

R1,3 A ∧db . This shows that the field strength H = db ∈Ω3(R1,3) of the fixed external field b is electrically charged under the gauged zero-form symmetry, thereby introducing a background which modifies

Jµ

b = q ¯ψ γµ ψ + g ϵµνλσ ∂νbλσ . By Gauss’ theorem, the electric charge (3.3) is then shifted to QB,b = QB + g

∂B B By The Flux Of

the two-form b through the boundary of the spatial volume B. It will prove useful later on, in our general twist deformation formalism, to rewrite this well- known field theory in a basis independent form. Let d be the exterior differential, ∗H the Hodge star operator induced by the Minkowski metric, and δ = ∗H d ∗H the corresponding codifferential.

Et Us Introduce A Background Field V Defined By

V = γµ γν γλ γ5 dxµ ∧dxν ∧dxλ = i ϵµνλσ γσ dxµ ∧dxν ∧dxλ ,

(3.5)

where in the second equality we used the Dirac matrix identity (B.1). This is a gauge invariant closed three-form valued in the endomorphism algebra End(S ) of the complex spinor representation S of Spin(1, 3) (see Appendix B). It can be used to rewrite the spinor action as

(3.6)

Expanding all the forms, one arrives at the usual covariant Dirac action10

Sψ =

R1,3 d4x ¯ψ i γµ (∂µψ −i q Aµ ψ) . The full set of equations of motion can now be written in the form

∞-Algebra Of Electrodynamics

The graded vector space L = L0 ⊕L1 ⊕L2 ⊕L3 of the L∞-algebra underlying electrodynamics is

(3.8)

10The covariant action for a Dirac spinor in four dimensions in an arbitrary basis is usually written as

Where E = Ea

µ γa dxµ is a vierbein one-form valued in End(S ) . This action can also describe a Dirac spinor in curved spacetime, where the gauge field A is then the spin connection ω. Since we work in a fixed background spacetime, we simply group the three flat vierbeins e ∧e ∧e and the chirality matrix γ5 into a single background field V .

We arrange the physical fields A ∈L1 and their duals A+ ∈L2 as

(3.9)

The elements A+ correspond to antifields in the BV formalism containing the equations of motion



. Elements c ∈L0 correspond to ghosts in the BV–BRST formalism related to the gauge parameters, while elements c+ ∈L3 correspond to their antifields containing the Noether identities dAFA = 0.

,

ℓ2(A, A+) = i ¯ψ+ ψ + i ¯ψ ψ+ .

(3.11)

All higher brackets ℓn with n ≥3 vanish. We checked explicitly that these brackets satisfy the homotopy relations, and thus define an L∞-algebra L (which is a differential graded Lie algebra in this case).

The underlying cochain complex of the L∞-algebra L is

∗H (D ¯Ψ ∧V ) And I /∂Ψ := −1

6 ∗H (V ∧dψ). The cohomology H•(L) of this complex is given by the free fields in the kernel of ℓ1. This defines the minimal model for the L∞-algebra L , which is an L∞-algebra H•(L ) quasi-isomorphic to L with underlying cochain complex

. In particular, fields (A(0), ψ(0), ¯ψ(0)) in the degree 1 cohomology

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

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