Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution, I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applie...
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| description | Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution, I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory that is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank > 2.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 094, 14 pages
Renormalization in Combinatorially Non-Local
Field Theories: the BPHZ Momentum Scheme
Johannes THÜRIGEN ab
a) Mathematisches Institut, Westfälische Wilhelms-Universität Münster,
Einsteinstr. 62, 48149 Münster, Germany
E-mail: johannes.thuerigen@uni-muenster.de
URL: http://www.uni-muenster.de/mathphys/en/u/thuerigen/
b) Institut für Physik/Mathematik, Humboldt-Universität zu Berlin,
Unter den Linden 6, 10099 Berlin, Germany
Received February 28, 2021, in final form October 24, 2021; Published online October 27, 2021
https://doi.org/10.3842/SIGMA.2021.094
Abstract. Various combinatorially non-local field theories are known to be renormalizable.
Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory.
In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the
Connes–Kreimer Hopf algebra applies to any combinatorially non-local field theory which
is renormalizable. This algebraic method improves the understanding of known results in
noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new
explicit perturbative calculations of amplitudes in tensorial field theories of rank r > 2.
Key words: non-local field theory; renormalization; Hopf algebras; multiple polylogarithms
2020 Mathematics Subject Classification: 05C10; 16T05; 16T30; 81T15; 81T18; 81T32
Dedicated to Dirk Kreimer
on the occasion of his 60th birthday
1 Introduction
Since Dirk Kreimer’s seminal work [29] it is well known that a Hopf algebra of divergent Feynman
diagrams is the universal structure underlying renormalization in perturbative quantum field
theory [16, 17, 30]. A classical example is BPHZ momentum subtraction as used for the general
proof of renormalization [7, 27, 55]. In [53] I have shown that the Connes–Kreimer Hopf algebra
describes renormalization not only in field theories with point-like interactions but also with
full generality in combinatorially non-local field theories (cNLFT) such as non-commutative and
matrix field theory [23, 28, 54], tensorial field theories [2, 3] and group field theory [13, 20, 37].
So far, explicit calculations of renormalized amplitudes in cNLFT depending on external
kinematic variables are only known for noncommutative field theory and its matrix-field rep-
resentation using the BPHZ momentum scheme [6, 22, 28]. In fact, exact solutions for matrix
field theory have been found recently [11, 21, 41] and the perturbative results are mainly used
as a check of consistency [22, 28]. For other examples of cNLFT, renormalizability has been
proven for various field theories with tensorial interactions [2, 3, 4, 13, 14, 15] but no explicit
results of amplitudes, let alone of full renormalized correlation functions, are known.1 Using the
This paper is a contribution to the Special Issue on Algebraic Structures in Perturbative Quan-
tum Field Theory in honor of Dirk Kreimer for his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Kreimer.html
1Exact solutions similar to matrix field theory [41] have been found for a super-renormalizable rank-3 ten-
sor field in [42]; due to the restriction to a single tensorial interaction and renormalization even of convergent
amplitudes, a closer look reveals that this is still the mentioned matrix theory in disguise.
mailto:johannes.thuerigen@uni-muenster.de
http://www.uni-muenster.de/mathphys/en/u/thuerigen/
https://doi.org/10.3842/SIGMA.2021.094
https://www.emis.de/journals/SIGMA/Kreimer.html
2 J. Thürigen
Hopf-algebraic structure of renormalization here I will give new results for explicit amplitudes
of perturbative tensorial field theory.
Explicit perturbative calculations in matrix field theory use a version of Zimmermann’s forest
formula [55] adapted to ribbon graphs [22]. While agreement with exact non-perturbative solu-
tions shows that this implementation of the BPHZ momentum scheme is perfectly valid, it uses
a very peculiar definition of subdiagrams which are not themselves ribbon graphs, thus strictly
speaking in a different class of diagrams. Here I will show how BPHZ momentum subtraction
can be defined on any cNLFT in Kreimer’s way using the Hopf algebra of divergent diagrams and
a subtraction operator as a Taylor expansion which is a Rota–Baxter operator on the algebra of
amplitudes. This gives two results: First, it reproduces in a very natural way the known pertur-
bative calculations in matrix field theory [22, 28]. Second, it provides the algorithm to calculate
renormalized amplitudes in any cNLFT in a straightforward way. I will demonstrate this for the
example of tadpole and sunrise diagrams in field theory with tensor-invariant interactions.
The physical meaning of amplitudes in cNLFT is not as obvious as for scattering events of
high energy particles described by standard-model quantum field theories. Matrix field theory
can be understood as a representation of field theory of elementary particles on non-commutative
(Moyal-deformed) spacetime which still allows for such scattering interpretation. On the other
hand, one main application of theories with tensor fields of higher rank r > 2 are models of
quantum gravity as their perturbative series is a sum over r-dimensional combinatorial (pseudo)
manifolds [24] and one can build models which have gravitational amplitudes on such discrete
manifolds as lattice gauge theories [13, 20, 37], known as spin foam models [43]. Then, the
main challenge is to recover continuous D = 4 dimensional spacetime in some critical regime of
the theory [38, 46] in the sense of continuous random geometries [1, 33, 36]; to this end, non-
perturbative results like [44] are necessary. However, also perturbation theory might be the right
starting point towards this goal, either by identification of appropriate linear combinations of
amplitudes in the perturbative series related to non-perturbative structures as found for exam-
ple via topological recursion [10, 11] or exploiting the Hopf-algebraic structure of perturbative
renormalization leading to Kreimer’s combinatorial Dyson–Schwinger equations [5, 12, 31, 32].
Thus, the explicit perturbative results presented here might also be seen as a starting point to
investigate the non-perturbative regime of cNLFT.
2 Feynman rules in cNLFT
In the following I introduce the general structure of renormalization as captured by a Hopf alge-
bra [29, 30]. In parallel, I explain its application to combinatorially non-local field theory. This
is largely based on [53] where one can find the precise details of combinatorial and algebraic de-
finitions and statements. Here, the emphasis is on the practical use for perturbative calculations
in field theory.
A perturbative expansion in coupling constants is a formal power series labelled by Feynman
diagrams. Feynman rules associate to each diagram Γ an amplitude AΓ, i.e., the coefficient in
this power series. This can be understood as a map from diagrams to amplitudes
A : G → A, Γ 7→ A(Γ) = AΓ.
In the perturbative series these amplitudes are summed and for renormalization it will also
be necessary to multiply them which demands an algebra A of amplitudes. This structure is
mirrored in a Q-algebra G generated by the set of diagrams in which multiplication is simply
the disjoint union of diagrams, mG(Γ1 ⊗ Γ2) := Γ1 ⊔ Γ2. Both algebras are unital with unit
uG : Q → G, q 7→ q1 where 1 ∈ G is the empty diagram and uA maps to the amplitude A(1) = 1.
The map A must then be an algebra homomorphism.
Renormalization in Combinatorially Non-Local Field Theories 3
A combinatorially non-local field theory (cNLFT) is characterized by a pairing of arguments
at interaction vertices which leads to strand graphs as Feynman diagrams (called 2-graphs
in [53]2). That is, order-n interactions are of the form
Sia[Φ] = λγ
∫ n∏
i=1
dpppi
∏
(ia,jb)
δ
(
pai − pbj
) n∏
i=1
Φ(pppi), (2.1)
where the field Φ depends on r arguments ppp =
(
p1, . . . , pr
)
of which each pa is in a d-dimensional
manifold. For simplicity I will consider pa ∈ Rd throughout this paper. The product over pairs
(ia, jb) means that for each argument pai there is a convolution in terms of the Dirac distribution
δ
(
pai − pbj
)
with exactly one other argument pbj . Diagrammatically, these convolutions cannot be
fully captured by a vertex in a graph but it is necessary to add a second layer called strands.
For example,
∼=
is the diagrammatic representation of an interaction vertex of n = 4 fields (red) with r = 3
arguments each (green) which are convoluted pairwise (blue dots); this is equivalent to, on the
right-hand side of the equation, a vertex (black) with half-edges (red) and additional edges
(green) adjacent to these half-edges which characterize the interaction vertex by a graph. Thus,
interactions are labelled in general by vertex graphs γ and so are their coupling constants λγ .
A set of graphs Gv
1 determines the diagrammatics of a given theory.
Feynman diagrams follow as usual by Wick contraction of the fields according to a quadratic
Gaussian part of the action. The arguments pa of the field Φ are not necessarily ordered, i.e.,
the quadratic part might pair the r arguments according to an arbitrary permutation σ2,
3
S0(Φ) =
1
2
∫
dpppidpppjΦ(pppi)
∑
σ2
r∏
a=1
δ
(
pai − p
σ2(a)
j
)( r∑
a=1
|pai |2ζ + µ
)
Φ(pppj), (2.2)
where µ is a mass parameter and the scaling 2ζ of the propagator is in general parametrized by
a ζ > 0, usually ζ ∈ ]0, 1]. This leads to the general class of strand graphs defined in [53] under
the name of “2-graphs” and based on [39, 52]. Still, in many theories this permutation is fixed.
For example, in Hermitian matrix field theory [28, 54], the strand graphs are combinatorial maps
(ribbon graphs ) and σ2 is fixed by the orientation of the vertices, e.g., in
Γ =
1 2
3
4
5
6 7 ∼=
1 2
3
4
5
6 7 ∼=
1 2
3
4
7
6
5
(2.3)
the pairing of strands adjacent to half edges i ∈ {1, 2, 3, 4, 5, 6, 7} along the edges {1, 2}, {3, 5}
and {4, 6} is determined by the orientation of the vertices (1), (234) and (576) (see Fig. 2
2I thank an unknown referee for pointing out the possible confusions with the notion of a “two-graph” as
defined by G. Higman [48, 51]. For this reason I will here use “strand graph” for what is defined as “2-graph”
in [53].
3Note that for a given permutation this quadratic form might not be positive which, however, is necessary to
yield a Gaussian measure. With the sum
∑
σ2
over an appropriate set of such permutations σ2 the quadratic
form becomes positive and thus Gaussian. I thank one of the referees for pointing this out.
4 J. Thürigen
in [53] for further details). Following a strand through the strand graph defines a face which
is an internal face if the strand is closed and external otherwise. In the example Γ there is
one internal face (3564) and one external face (764212357). The external structure of a strand
graph Γ is the vertex graph γ obtained by deleting all internal edges and internal faces which is
the boundary γ = ∂Γ.
The above Gaussian and interaction parts of the action, equations (2.2) and (2.1), lead to
an identification of variables pa along strands and thus to the following Feynman rules: For
a strand graph Γ, the amputated amplitude AΓ is an integral consisting of
1) a Lebesgue integral
∫
Rd dqf for each internal face f ∈ F int
Γ , integrating over the variable
identifying qf = pai for all strands a at half edges i along f ,
2) a propagator factor 1∑r
a=1 |pai |2ζ+µ
for each internal edge e = {i, j} ∈ EΓ,
3) a coupling constant λγv for each vertex v ∈ VΓ with vertex graph γ = γv and
These rules define the algebra homomorphism A : G → A to the algebra A of (formal) integral
functions with rational functions as integrands,
AΓ : {pf}f∈F̃ext
Γ
7→ AΓ({pf}) :=
∏
v∈VΓ
λγv
∏
f∈F int
Γ
∫
Rd
dqf
∏
{i,j}∈EΓ
1∑r
a=1 |pai |2ζ + µ
.
They depend on external variables pf for each external face f ∈ F̃ext
Γ adjacent to at least one
internal edge, given again by identifying pf = pai for all half edges i and adjacent strands a
along f . Importantly, only faces in this subset F̃ext ⊂ Fext of external faces play a role: If a face
is not adjacent to any internal edge, there is no propagator in which the corresponding variable
occurs and the amputated amplitude AΓ does not depend on it. This is in contrast to local field
theory where loop amplitudes always depend on all external variables.
Multiplication in the algebra A of amplitudes is given by multiplication of integrands. That
is, the product of two amplitudes A,B ∈ A with
A(p1, . . . , pm) = cA
∫ ∏
i
dqiIA(p1, . . . , pm, q1, . . . , qk)
and
B(r1, . . . , rn) = cB
∫ ∏
j
dsjIB(r1, . . . , rn, s1, . . . , sl),
where the integrands IA and IB are rational functions is
(A ·B)(pa, rb) := cAcB
∫ k∏
i=1
dqi
l∏
j=1
dsjIA(p1, . . . , pm, q1, . . . , qk)IB(r1, . . . , rn, s1, . . . , sl).
This is relevant since some external variables pa of the amplitude A might coincide with some
integration variables sj of B, i.e., pa = sj for some a ∈ {1, . . . , n} and j ∈ {1, . . . , l}, and vice
versa. Accordingly, these are not external variables of the product A ·B.
To give an example of an amplitude, for Γ in equation (2.3) there is one external variable p
for the external face and one integration variable q for the internal face leading to the amputated
amplitude
AΓ(p) = A
(
q
p
)
(p) = λ λ2
∫
Rd
dq
1
2|p|2ζ + µ
1(
|p|2ζ + |q|2ζ + µ
)2 .
Renormalization in Combinatorially Non-Local Field Theories 5
For d ≥ 2ζ this integral does not converge. To render the algebra of formal amplitudes A an
algebra of well defined functions it is necessary to introduce a regularization cutoff Λ, that is
consider the algebra AΛ of the same integrals but integrating all internal variables over the
domain |q|2ζ ≤ Λ2ζ ,
AΛ
Γ({pf}) :=
∏
v∈VΓ
λγv
∏
f∈F int
∫
Λ
dqf
∏
{i,j}∈E
1∑r
a=1 |pai |2ζ + µ
. (2.4)
For a well-defined limit Λ → ∞ renormalization is needed.
3 BPHZ renormalization using the Hopf algebra
of strand graphs
To renormalize a divergent amplitude one subtracts a counter term which exactly eliminates
the divergent part and leaves a finite result. To this end one needs a measure of how much
a divergent amplitude is divergent. A useful starting point is the power ωΓ of the asymptotic
scaling in the cutoff Λ of the regularized amplitude (2.4), i.e., AΛ
Γ ∼ ΛωΓ . A simple way to
determine this quantity is to count the number of integrals and propagators in the integration
variables in the amplitude. For (2.4) this gives the superficial degree of divergence
ωsd(Γ) = dFΓ − 2ζEΓ +
∑
v∈VΓ
ω(γv), (3.1)
where FΓ = |F int
Γ | is the number of internal faces, EΓ = |EΓ| is the number of (internal) edges
and a scaling λγ ∼ Λω(γ) of the couplings with vertex graph γ = γv for each vertex v ∈ VΓ
in the strand graph Γ is assumed in addition. A strand graph Γ is superficially divergent
iff ωsd(Γ) ≥ 0. These properties are specific to a given combinatorially non-local field theory T
which is defined by a set of interactions given by its vertex graphs Gv
1 together with their scaling
weights ω : Gv
1 → R, the propagators with scaling 2ζ and the dimension d.
The theorem of Bogoliubov and Parasiuk [7] with proof completed by Hepp [27] and further
improvements by Zimmermann leading to the forest formula [55] states that in a renormaliz-
able local quantum field theory all divergences can be renormalized by counter terms for each
superficially divergent one-particle irreducible (1PI) diagram. Kreimer [29] has further signif-
icantly improved the understanding of the combinatorics of the forest formula (in particular,
and renormalization in much more general) showing that it has an algebraic structure given by
a Hopf algebra of divergent Feynman diagrams. In [53] I have shown that this Hopf algebra
exists not only in field theories with point-like interactions but also for theories with any com-
binatorially non-local interactions, in this way generalizing and improving similar statements
for some specific examples [45, 49, 50]. Here I demonstrate how this works explicitly using the
BPHZ momentum scheme.
The challenge to define the right counter term to renormalize a divergent amplitude lies in the
nested structure of divergent subdiagrams. In cNLFT, just like in local quantum field theory [9],
a strand graph Θ is a subgraph Θ ⊂ Γ of a strand graph Γ iff it has a subset of edges EΘ ⊂ EΓ,
including the adjacent strands, and the same structure otherwise [53]. A primitively divergent
strand graph is then one which has no proper subgraphs which are superficially divergent. For
a primitively divergent strand graph Γ one defines the renormalized amplitude Ar(Γ) directly
by a subtraction operation which is a linear operator R : A → A as
Ar : G → Ar(Γ) := (A−R ◦A)(Γ). (3.2)
The explicit form of R is exactly what specifies a particular renormalization scheme. For con-
creteness, let us here define R for the BPHZ momentum scheme of cNLFT:
6 J. Thürigen
Definition 3.1 (BPHZ momentum scheme). The operator R : AΛ → AΛ on the unital com-
mutative algebra AΛ of functions of the form AΛ
Γ , equation (2.4), with degree of divergence
ω
(
AΛ
Γ
)
:= ωsd(Γ), equation (3.1), is given by the multivariate Taylor expansion Tω of order
ω = ω
(
AΛ
Γ
)
,
R
[
AΛ
Γ
]
({pf}) :=
(
Tω
{pf}A
Λ
Γ
)
({pf}) =
∑
|⃗k|≤ω(AΛ
Γ)
1
k⃗!
∂ |⃗k|AΛ
Γ∏
f ∂p
kf
f
(
{pf = 0}
) ∏
f∈F̃ext
Γ
p
kf
f , (3.3)
where the sum is over multi-indices k⃗ of length |⃗k| =
∑
f∈F̃ext
Γ
kf . Note that the underlying
strand-graph structure is only used to encode the form of the function A({pf}) = AΛ
Γ({pf})
but the definition of R is otherwise independent of it (any Λ-regularized integral has a degree ω
given by its scaling Λω).
For divergent diagrams Γ with subdivergences the whole counter term is a nested sum over
counter terms for the various superficially divergent subgraphs according to Zimmermann’s forest
formula [55]. These terms are products of counter terms of subdiagrams Θ ⊂ Γ and amplitudes
of remaining diagrams in which the subdiagram is shrunken to a vertex, that is contractions Γ/Θ.
For strand graphs Θ ⊂ Γ the definition of such contraction Γ/Θ is that there is a vertex for each
connected component of Θ and all internal edges and internal strands (and thus internal faces)
of Θ are deleted (see [53, Definition 3.2] for further detail). This defines a coproduct ∆ on the
algebra of superficially divergent strand graphs Hf2g
T of a given renormalizable theory T as
∆: Hf2g
T → Hf2g
T ⊗Hf2g
T , ∆Γ :=
∑
Θ⊂Γ,Θ∈Hf2g
T
Θ⊗ Γ/Θ.
For this it is crucial that Hf2g
T is closed under contractions which is satisfied due to the property
of renormalizability ω(∂Γ) = ωsd(Γ) − δΓ where the Γ-dependent part δΓ needs to satisfy δΓ =
δΘ + δΓ/Θ for all subgraphs Θ ⊂ Γ [53]. Together with the counit ϵ : Hf2g
T → Q defined as the
projector to strand graphs without any edges, Hf2g is a coalgebra and due to compatibility with
multiplication also a bialgebra [53, Proposition 4.2].
To implement the forest formula, the crucial object is an inverse to the algebra homomorphism
of amplitudes A : Hf2g → A with respect to the convolution product, i.e.,
ϕ ∗ ψ := mA ◦ (ϕ⊗ ψ) ◦∆
for arbitrary such homomorphisms ϕ, ψ : Hf2g → A. Such inverse is a natural consequence
if Hf2g is a Hopf algebra, i.e., has a coinverse, also called antipode, S : Hf2g → Hf2g defined by
S∗id = id∗S = u◦ϵ. In fact, one can show that the bialgebraHf2g
T of superficially divergent strand
graphs in T is a Hopf algebra [53, Theorem 5.1] using the possibility to express S recursively as
S = −(S ∗ P ),
where P = id − u ◦ ϵ is the projector to the augmentation ideal, i.e., all strand graphs in Hf2g
T
with at least one edge [9, 35]. This works because Hf2g is graded with respect to the number of
edges. As a direct consequence, the inverse of an algebra homomorphism ϕ is Sϕ = S ◦ ϕ. Both
facts can now be used to recursively define the counter term
Sa
r : Hf2g → A, (3.4)
Sa
r(Γ0) = 1, Sa
r(Γ) = −R
[(
Sa
r ∗A ◦ P
)
(Γ)
]
= −
∑
Θ⊊Γ,Θ∈Hf2g
T
R
[
Sa
r(Θ)A(Γ/Θ)
]
Renormalization in Combinatorially Non-Local Field Theories 7
for any strand graph Γ0 without edges, and any Γ in the augmentation ideal. Then, the renor-
malized amplitude is simply
Ar := Sa
r ∗A.
The counter term map Sa
r is an algebra homomorphism, in particular Sa
r(Γ1Γ2) = Sa
r(Γ1)S
a
r(Γ2),
if the subtraction map R is a Rota–Baxter operator, that is satisfies
R[AB] +R[A]R[B] = R[R[A]B +AR[B]].
Proposition 3.2. The BPHZ momentum subtraction operator R = Tω (Definition 3.1) is
a Rota–Baxter operator.
Proof. The statement is equivalent to Proposition 9.1 in [19]. There, the subtraction operator
is defined on the integrands of the amplitudes which are ignorant about the degree ω with the
consequence that each specific R depends on ω and it defines therefore a Rota–Baxter family
(which still is very close to a Rota–Baxter operator by Proposition 9.2 therein). As R is defined in
Theorem 3.1 via the scaling Λω of Λ-regularized integrals, this construction is not necessary.4 ■
4 BPHZ renormalization in tensorial Φ4
2,2 and Φ4
1,3 theory
To demonstrate how BPHZ renormalization works, I give the example of tadpole and sunrise
amplitudes in the just renormalizable complex tensorial field theories of rank r = 2 and r = 3.
A tensorial Φn
d,r theory is a cNLFT of a scalar field Φ with r arguments in a d-dimensional
manifold with tensor-invariant interactions of maximal order n (degree of the vertex) [2]. An
interaction is tensor-invariant if its vertex graph γ is r-regular and edge-coloured, i.e., each vertex
in γ is adjacent to exactly r edges with distinct labels c = 1, 2, . . . , r [8]. This corresponds
to fixing the position of each argument in the field and pairing only arguments in the same
position. As a consequence, also the strand graphs Γ labelling the perturbative expansion can
be represented as coloured graphs,5 now with r + 1 colours. In the following I will consider
a complex field Φ, just for the reason that the vertex graphs then have to be furthermore
bipartite which simplifies the theory space further. Thus, the action of such theory is
S[Φ, Φ̄] =
(
r∏
c=1
∫
Rd
dpc
)
Φ̄(ppp)
(
r∑
c=1
|pc|2ζ + µ
)
Φ(ppp) +
∑
γ∈Gv
1
λγ Trγ
(
Φ, Φ̄
)
,
where Trγ denotes the convolution of the field, detailed in equation (2.1), according to a given
graph γ in the set Gv
1 of the theory’s interaction vertex graphs.
With weight ω(γv) = dr − dv(dr − 2ζ)/2 for each dv-valent vertex with vertex graph γv, the
superficial degree of divergence in tensorial Φn
d,r theory is [2, 53]
ωsd(Γ) = dr −
dr − 2ζ
2
V∂Γ − d
(
2ωg
Γ − 2ωg
∂Γ
(r − 1)!
+K∂Γ − 1
)
,
where dr = d(r − 1) is the equivalent of dimension compared to local field theory, K∂Γ is the
number of connected components of the boundary graph and ωg ∈ N is the Gurau degree of
a coloured graph [25, 26]. It is a quantity generalizing the genus g in the r = 2 case, though not
4I thank one of the referees for pointing out this subtlety.
5Strictly speaking, this is only true for theories without multi-trace vertices, that is vertices whose vertex graphs
have more than one connected component. Otherwise, the full coloured strand-graph structure is necessary [53].
8 J. Thürigen
a topological invariant. Crucially, the overall Gurau degree is conserved under the coproduct [45]
such that renormalizability is given as ωsd(Γ) = ω(∂Γ)−δΓ with δΓ = d
2ωg
Γ−2ωg
∂Γ
(r−1)! . One can further
show that ωg
Γ ≥ ωg
∂Γ [3] such that the whole term in brackets is always greater or equal to zero
and renormalizable theories are determined by the first two terms. Thus, a tensorial Φn
d,r theory
is just renormalizable if n = 2dr
dr−2ζ .
Just-renormalizable quartic theories are those with dr = d(r − 1) = 4ζ. The tensorial equiv-
alent to standard local ϕ44 theory with quadratic propagator ζ = 1 are therefore theories with
dr = 4, that is theories with (d, r) = (4, 2), (2, 3) or (1, 5). Let us here simplify explicit renor-
malization even further and chose ζ = 1/2. The two just renormalizable theories are then
tensorial Φ4
2,2 and Φ4
1,3 theory. The divergence degree in these two theories is
ωsd(Γ) = dF − E = 2− 1
2
V∂Γ −
{
2 (2gΓ +K∂Γ − 1) for Φ4
2,2 theory,
(ωg
Γ − g∂Γ +K∂Γ − 1) for Φ4
1,3 theory.
In both cases there is only a single type of marginal quartic interaction given by the graph
and c for c = 1, 2, 3 respectively. It is not necessary to include the quartic double-trace
interaction.
For these two cases I will present the explicit BPHZ-renormalized amplitudes AΓ for the
tadpole and sunrise diagrams. Complex Φ4
2,2 theory is very similar to the quartic Hermitian
matrix field theory known as the Grosse-Wulkenhaar model or quartic Kontsevich model [23,
28, 54], i.e., has the same power counting and the same set of divergent diagrams with identical
amplitudes; for this theory I simply reproduce known results [28] with the Hopf algebra method.
The rank r = 3 tensorial theory is one of the first for which renormalizability was proven [4]. In
this case the calculations presented here are new explicit results of amplitudes in tensorial field
theory.
The tadpole in Φ4
2,2 theory
The only primitive 2-point amplitude in Φ4
2,2 theory is the one for the tadpole strand graph.
There are two versions, ∼= c with colour c = 1 and ∼= c with c = 2 when
using the convention that in the representation via combinatorial maps and one has
always the external variable p1 associated to the face of colour c = 1 on the upper face and p2
respectively on the lower face. Its divergence degree is ωsd( ) = ωsd( ) = 2−1 = 1. There
are no subdivergences and thus by definition (2.4)
Ar
( )
=
(
Sa
r ∗A
)( )
= mA
(
Sa
r ⊗A
)
∆
( )
= mA(S
a
r ⊗A)
(
c ⊗ + ⊗
)
= A
( )
+ Sa
r
( )
A ( ) = A
( )
−R
[
A
( )]
since A ( ) = 1 as there are no edges in a single vertex. This calculation works in general
for any primitive diagram and thus validates equation (3.2). Explicit calculation in the BPHZ
momentum scheme according to Definition 3.1 gives then
Ar
( )
(p1) ≡ Ar
(
p1
q2 )
= λ
(
1− T 1
p1
) ∫
R2
dq2
1
|p1|+ |q2|+ µ
Renormalization in Combinatorially Non-Local Field Theories 9
= λ
∫ ∞
0
2π|q2|d|q2|
(
1
|p1|+ |q2|+ µ
− 1
|q2|+ µ
+
|p1|
(|q2|+ µ)2
)
= 2πµλ
(
(p̃1 + 1) log(p̃1 + 1)− p̃1
)
(4.1)
with p̃1 = |p1|/µ. The tadpole amplitude for colour c = 2 gives the same with p2 instead of p1.
This result is exactly the same as in Hermitian matrix field theory [28].
Tadpole diagrams in Φ4
1,3 theory
In Φ4
1,3 theory, the tadpole has not only versions for each colour but also two possibilities where
to have the edge. In particular, the two differ in degree of divergence (using ωsd(Γ) = F − E),
ωsd
(
c
)
= 2− 1 = 1 and ωsd
(
c
)
= 1− 1 = 0.
The renormalized amplitude of the first one for c = 1 is
Ar
(
p1
)
= λ
1
(
1− T 1
p1
) ∫
R
dq2
∫
R
dq3
1
|p1|+ |q2|+ |q3|+ µ
= λ
1
∫ ∞
0
2dq2
∫ ∞
0
2dq3
(
1
|p1|+ q2 + q3 + µ
− 1
q2 + q3 + µ
+
|p1|
(q2 + q3 + µ)2
)
= 4µλ
1
(
(p̃1 + 1) log (p̃1 + 1)− p̃1
)
with p̃1 = |p1|/µ. Up to a constant it is the same as in the r = 2 matrix case, equation (4.1).
This is not surprising as the external variable is the same and the two internal faces with
one-dimensional integration variables q2 and q3 behaves effectively like one face with a two-
dimensional variable.
The first amplitude specific to a higher rank r = 3 is
Ar
( q1
p2, p3
)
= λ
1
(
1− T 0
p2,p3
) ∫
R
dq1
1
|q1|+ |p2|+ |p3|+ µ
= λ
1
∫ ∞
0
2dq1
(
1
q1 + |p2|+ |p3|+ µ
− 1
q1 + µ
)
= −2λ
1
log(p̃23 + 1) (4.2)
with p̃23 = (|p2| + |p3|)/µ. In the full 1PI 2-point Green’s function one has a sum over the
different colours c = 1, 2, 3 for which in general the couplings λ c are distinct.
The sunrise diagram in Φ4
2,2 theory
The strand-graph version of the sunrise diagram is the first example which shows the Hopf-
algebraic formulation of Zimmermann’s forest formula in its full beauty. In complex Φ4
2,2 theory
the sunrise combinatorial map corresponds to the colouring
∼=
c c
and has ωsd
( )
= 2 · 2− 3 = 1.
(The other choice of colouring of this strand graph corresponds to a non-planar map of cylinder
topology which is not divergent, ωsd = 0 − 3 = −3.) The sunrise strand graph has three 1PI
subgraphs of which only two are divergent,
ωsd
( c c )
= ωsd
(
c c
)
= 2 · 1− 2 = 0,
10 J. Thürigen
ωsd
( c c )
= 0− 2 = −2. (4.3)
As they do not have any divergent 1PI subgraphs, they are primitives. Accordingly, the coprod-
uct in Hf2g is
∆
( c c )
= c c ⊗
c c
+
c c
⊗
c
+ c c ⊗ c +
c c
⊗
which yields the renormalized amplitude
Ar
( p1
p2
q1
q2
)
= A
( p1
p2
q1
q2
)
+ Sa
r
( p1
q1
q2
)
A
(
p2
q1
)
+ Sa
r
( q2
p2
q1
)
A
(
p1
q2 )
+ Sa
r
( p1
p2
q1
q2
)
.
In the last term, Sa
r acts on a divergent diagram which is not primitive such that according
to (3.4) there is a nontrivial sum
Sa
r
( p1
p2
q1
q2
)
= −R
[
A
( p1
p2
q1
q2
)
−R
[
A
( p1
q1
q2
)]
A
(
p2
q1
)
−R
[
A
( q2
p2
q1
)]
A
(
p1
q2 )]
. (4.4)
Putting everything together and applying R = Tω one arrives at the finite integral
Ar(p1, p2) = λ2
(
1− T 1
p1,p2
) ∫
R2
dq1
∫
R2
dq2
(
1
|p1|+ |q2|+ µ
1
|q1|+ |q2|+ µ
1
|q1|+ |p2|+ µ
+
1
|q1|+ |p2|+ µ
(
−T 0
p1,q1
) 1
|p1|+ |q2|+ µ
1
|q1|+ |q2|+ µ
+
1
|p1|+ |q1|+ µ
(
−T 0
q2,p2
) 1
|q1|+ |q2|+ µ
1
|q2|+ |p2|+ µ
)
= λ2
4π2µ
p̃1 + p̃2 + 1
[
p̃1p̃2ζ2 + (p̃1 + p̃2 + 1)
∑
i=1,2
(
(p̃i + 1) log(p̃i + 1)− p̃i
)
−
∏
i=1,2
(p̃i + 1) log(p̃i + 1) +
∑
i=1,2
p̃i(p̃i + 1)Li2(−p̃i)
]
involving the polylogarithm Li2 and the zeta value ζ2 = π2/6 and again p̃i = |pi|/µ. In general,
any kind of multiple polylogarithms are to be expected due to the form of the integral. The
result reproduces equation (D.9) in [28]. As the diagram is symmetric under colour change, the
amplitude is symmetric under exchange of p1 and p2, i.e., Ar(p2, p1) = Ar(p1, p2).
Renormalization in Combinatorially Non-Local Field Theories 11
The sunrise diagram in Φ4
1,3 theory
Also in Φ4
1,3 there is only one divergent sunrise strand graph out of four possibilities to join
two quartic vertices by three edges (without tadpoles). This diagram has only a logarithmic
divergence,
ωsd
( c c )
= 3− 3 = 0.
Out of the three 1PI subgraphs, in analogy to (4.3), only one is divergent which is
ωsd
( c c )
= 2− 2 = 0.
Thus, in rank r = 3 there is one counter term less than in the r = 2 case,
Ar(p1, p2, p3) = Ar
( p1
p2, p3
q2, q3
q1
)
= A
( p1
p2, p3
q2, q3
q1
)
+ Sa
r
( p1
q2, q3
q1
)
A
(
p1
)
+ Sa
r
( p1
p2, p3
q2, q3
q1
)
.
Expanding these counter terms in R expressions analogously to (4.4) and using the explicit
subtraction operation R = Tω, this amplitude is
Ar(p1, p2, p3) = λ2
1
(
1− T 0
p1,p2,p3
) ∫
R
dq1
1
|q1|+ |p2|+ |p3|+ µ
×
(
1− T 0
p1,q1
) ∫
R
dq2
∫
R
dq3
1
|q1|+ |q2|+ |q3|+ µ
1
|p1|+ |q2|+ |q3|+ µ
= λ2
1
6
1 + p̃1 + p̃23
[
log(p̃23 + 1)
(
(p̃1 + 1) log(p̃1 + 1)− p̃1 − p̃23 − 1
)
− log(p̃1)(p̃1 + 1) log(p̃1 + 1) + p̃23
(
Li2
(
− 1
p̃23
)
+
1
2
log2(p̃23) + 2ζ2
)
+ (p̃1 + 1)
(
Li2
(
1
p̃1 + 1
)
+
1
2
log2(p̃1 + 1)− ζ2
)]
, (4.5)
where again p̃1 = |p1|/µ and p̃23 = (|p2| + |p3|)/µ. Note that the Taylor operator T 0
p1,p2,p3 in
the first line acts on the whole expression which is following. One might use shuffle relations of
multiple polylogarithms to further simplify the terms of weight two in p̃1 + 1 and p̃23.
Interestingly, the calculation of the sunrise diagram in the r = 3 theory is simpler than
in the r = 2 case due to the lack of overlapping divergences. This is an instance of the well
known breaking of the combinatorial symmetry of matrix field theory which xleads to asymptotic
freedom in quartic tensorial field theories [47]. In general, the expectation is that amplitudes
in tensorial field theory can always be expanded in multiple polylogarithms for which efficient
algorithms exist for computation [40].
5 Outlook
The main purpose of this contribution has been to demonstrate how the Connes–Kreimer Hopf
algebra [16, 17, 29, 30] can be used for explicit calculations in cNLFT. Feynman diagrams in
12 J. Thürigen
cNLFT are strand graphs generalizing the usual Feynman graphs of point-like interactions by
adding a second layer of strands along which the degrees of freedom in cNLFT propagate. The
crucial difference to standard quantum field theory is that the external structure of such dia-
grams is not just given by the number of legs of a given field type but is captured by graphs.
Nevertheless, with the right definitions, the Hopf-algebraic structure relying on the usual co-
product of subgraph contraction is exactly the same as for local quantum field theory [53]. Thus,
it directly gives a convenient algorithm to renormalize Green’s functions in cNLFT, such as non-
commutative field theory and matrix field theory [23, 28, 54] as well as higher rank tensorial
and group field theories [2, 3, 13, 20, 37].
While the Hopf algebra unveils the structure of renormalization independent of a specific
scheme, here I have demonstrated how it works for BPHZ momentum subtraction [7, 27, 55]
in two simple examples of tensorial field theory on
(
Rd
)⊗r
. As a first example, in Φ4
2,2 theory,
i.e., complex matrix field theory on a 2-dimensional configuration space, the tadpole and sunrise
diagram have the same amplitude as in the Grosse–Wulkenhaar model [23, 54] and the Hopf-
algebraic calculations reproduce the known results [28]. Second, as a first example of a field
theory of higher rank r > 2, I have calculated the amplitudes for the same diagrams in Φ4
1,3
theory. As to be expected, these calculations give first hints that also in tensorial field theories
the algebra of amplitudes is spanned by multiple polylogarithms. A deeper understanding of
this space of amplitudes for tensorial theories would be an interesting research topic in its own.
The renormalized amplitudes of the tensorial versions of the tadpole and sunrise diagrams give
examples showing that the UV regime of tensorial field theories [2] is richer than the large-N
limit of the corresponding tensor models [25, 26]. The latter is completely described by so-
called melonic diagrams, i.e., diagrams which have vanishing Gurau degree ωg = 0. There have
been considerable efforts to find random geometries beyond the melonic regime, for example by
enhancement of subleading classes of diagrams [18, 33, 34]. In tensorial field theories, melonic
diagrams are the most divergent but not the only divergent ones. In fact, the tadpole and
sunrise amplitudes (4.2) and (4.5) are examples of divergent, non-melonic amplitudes. Thus,
in renormalized quantities they contribute as much as the melonic contributions. Such effects
make tensorial field theory an interesting candidate to find new regimes of the theory beyond
the melonic one.
Acknowledgments
I would like to thank A. Hock and R. Wulkenhaar for discussions on the forest formula in
matrix field theory, as well as the referees for very valuable comments and suggestions. I am
deeply grateful to D. Kreimer for advice and support, in particular in developing the research
project this work is part of which is funded by the Deutsche Forschungsgemeinschaft (DFG, Ger-
man Research Foundation) under the project number 418838388. Further support comes from
the DFG under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster:
Dynamics–Geometry–Structure.
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1 Introduction
2 Feynman rules in cNLFT
3 BPHZ renormalization using the Hopf algebra of strand graphs
4 BPHZ renormalization in tensorial Phi42,2 and Phi41,3 theory
5 Outlook
References
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| id | nasplib_isofts_kiev_ua-123456789-211433 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T04:35:18Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Thürigen, Johannes 2026-01-02T08:31:56Z 2021 Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme. Johannes Thürigen. SIGMA 17 (2021), 094, 14 pages 1815-0659 2020 Mathematics Subject Classification: 05C10; 16T05; 16T30; 81T15; 81T18; 81T32 arXiv:2103.01136 https://nasplib.isofts.kiev.ua/handle/123456789/211433 https://doi.org/10.3842/SIGMA.2021.094 Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution, I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory that is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank > 2. I would like to thank A. Hock and R. Wulkenhaar for discussions on the forest formula in matrix field theory, as well as the referees for their very valuable comments and suggestions. I am deeply grateful to D. Kreimer for advice and support, in particular in developing the research project this work is part of, which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the project number 418838388. Further support comes from the DFG under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme Article published earlier |
| spellingShingle | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme Thürigen, Johannes |
| title | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme |
| title_full | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme |
| title_fullStr | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme |
| title_full_unstemmed | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme |
| title_short | Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme |
| title_sort | renormalization in combinatorially non-local field theories: the bphz momentum scheme |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211433 |
| work_keys_str_mv | AT thurigenjohannes renormalizationincombinatoriallynonlocalfieldtheoriesthebphzmomentumscheme |