# One-loop algebras and fixed flow trajectories
in adjoint multi-scalar gauge theory

Nadia Flodgren  and Bo Sundborg<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the one loop renormalisation of 4d $SU(N)$ Yang-Mills theory with $M$
adjoint representation scalar multiplets related by $O(M)$ symmetry. General
$M$ are of field theoretic interest, and the 4d one loop beta function of the
gauge coupling $g^{2}$ vanishes for the case $M=22$, which is intriguing for
string theory. This case is related to D3 branes of critical bosonic string
theory in $D=22+4=26$. An RG fixed point could have provided a definition for
a purely bosonic AdS/CFT, but we show that scalar self-couplings $\lambda$
ruin one-loop conformal invariance in the large $N$ limit. There are real
fixed flows (fixed points of $\lambda/g^{2}$) only for $M\geq 406$, rendering
one-loop fixed points of the gauge coupling and scalar couplings incompatible.

We develop and check an algebraic approach to the one-loop renormalisation
group which we find to be characterised by a non-associative algebra of
marginal couplings. In the large $N$ limit, the resulting RG flows typically
suffer from strong coupling in both the ultraviolet and the infrared. Only for
$M\geq 406$ fine-tuned solutions exist which are weakly coupled in the
infrared.

††institutetext: The Oskar Klein Centre for Cosmoparticle Physics & Department
of Physics,
Stockholm University,
AlbaNova, 106 91 Stockholm, Sweden

##  1 Introduction

Two classes of renormalisation group (RG) fixed points have been found
amenable to perturbative treatment: infrared (IR) fixed points of the Caswell-
Banks-Zaks type Caswell:1974gg ; Banks:1981nn  and ultraviolet (UV) fixed
points of the Litim-Sannino or asymptotically safe type Litim:2014uca ;
Litim:2015iea . Both classes of fixed points involve gauge theory, and it
seems that non-Abelian gauge theory is needed for weak coupling UV consistency
Coleman:1973sx ; Bond:2018oco . In both cases, perturbative control is
achieved through Veneziano type limits with a large number of colours $N$ and
fermion flavours $N_{F}$. We study a similar, but purely bosonic theory, in
order to see explicitly what happens in the absence of fermions.

To select scalar matter we appeal to simplicity: since a gauge theory is
needed to improve short-distance behaviour, and the gauge bosons belong to the
adjoint representation, it is simplest to consider scalars which also
transform in the adjoint representation. We however allow for $M$ such scalar
multiplets. Again, we choose the simplest versions of such theories, which are
also symmetric under global $O(M)$ transformations.

In addition to their simplicity, adjoint scalar gauge theories are similar to
gauge theories derived from D3 branes of string theory, which have been
studied before in the hope of finding non-supersymmetric conformal field
theories. An early example is a non-supersymmetric D3-brane theory
Klebanov:1999ch  based on type 0 string theory. For that theory, and other
similar theories based on ‘orbifolding’ $\mathcal{N}=4$ super-Yang-Mills
theory Kachru:1998ys ; Lawrence:1998ja ; Bershadsky:1998cb  the gauge coupling
and single trace beta functions vanish on a line of fixed points for large
$N$, but it was soon realised that conformal invariance is ruined by double
trace couplings, unless their values are complex Tseytlin:1999ii ;
Adams:2001jb ; Dymarsky:2005uh ; Pomoni:2009joh . The present work will
display both similarities and differences between our simple bosonic model and
the superstring-related models.

Our present perspective is however complementary and almost opposite. We study
gauge theories with purely scalar matter because they are in a sense the worst
and most generic case for quantum properties of a gauge theory.

Methodologically, we use specialised standard expressions Machacek:1984zw  and
compute by hand, but we have also fixed normalisations so as to apply the
Mathematica package RGBeta Thomsen:2021ncy  and compare results. We find
complete agreement.

We also develop a new (to us) algebraic way of expressing the one loop RG
equations. The simple basic observation is that the renormalisable four point
couplings form a closed system determining their own RG flow. The one loop
terms are second order and define a multiplication between linear combinations
of couplings. The algebra defined this way turns out to be commutative and
nonassociative, and completely encodes the one loop RG. Although we hope that
this algebraic picture can be extended to higher loops, we have not checked
this, and do not make any such claims111Of course, for the case of a single
scalar with a $\lambda\phi^{4}$ interaction, the algebra is associative, and
higher loops are usefully expressed in powers of $\lambda$.. Already the
present one loop formulations opens new perspectives on multi-coupling RG
equations and the structure of their RG flows, which we explain in the body of
the paper.

We search for fixed points of the RG flow for $M=22$, a case of special
interest, since the gauge coupling beta function vanishes. This has been know
for a long time, and is actually related to special properties of the bosonic
string in $D=26=4+22$ dimensions Curtright:1981wv ; Fradkin:1982kf ;
Nepomechie:1983cq ; Dubovsky:2018vde . It is even noted in Fradkin:1982kf
that the scalar coupling is renormalised. Now we can finally fill a gap in the
analysis and determine non-trivial fixed points of the full set of couplings.
None of the roots of the equations are real however, and all one-loop fixed
point theories are therefore non-unitary. Instead requiring real fixed points,
we find fixed points of relative couplings which represent ‘fixed flows’ for
the original scalar couplings, ie flows where all couplings run as the gauge
coupling $g^{2}$. Such fixed flows however only appear for real couplings if
$M\geq 406$, corresponding to a large number of dimensions in a string
interpretation.

In conclusion, we have developed an algebra description of the RG at one loop
which could be interesting to extend to higher orders. We have also
conclusively ruled out simultaneous real fixed points for all couplings in the
multi-scalar models we explore. There are no theories in this class that are
under one loop control in the UV, and only isolated cases that are under
control in the IR.

The paper is organised as follows. Section 2 reviews scalar gauge theories,
their marginal interaction terms and standard expressions for their one loop
beta functions. Then, we rephrase the general structure in terms of a
commutative, but generically non-associative algebra. The algebraic
reformulation reappears in other sections but is not necessary for
appreciating the results. In section 3, the focus is multi-adjoint scalar
gauge theory and the invariant polynomials or tensor structures that describe
its interactions. A special subsection 3.4 is devoted to the resulting
algebra, and also exemplifies the significance of various properties of the
algebra. The expressions for the beta functions, and different checks of them,
are then given in section 4. The actual RG equations, their qualitative
structure and the characterisation of their fixed points and fixed flows are
collected in section 5. We finish with a short general discussion and an
outlook in the concluding section 6.

##  2 Quartic scalar interactions in $D=4$, beta functions and algebras

We begin by reviewing scalar gauge theories, their marginally relevant
interaction terms and beta functions. Then, we rephrase this structure
algebraically. We introduce a _marginal algebra_ , which encodes the
renormalisation group flow of classically scale invariant theories, and
reflect its properties.

###  2.1 Quartic interactions and beta functions

We focus on classically scale invariant scalar gauge theories in four
dimensions. Hence, we only include kinetic terms, quartic interaction terms
and minimal coupling to non-Abelian pure Yang-Mills theory. Alternatively, we
can regard this set of theories as high energy limits of models which also
include masses and cubic couplings which are dimensionful.

The interaction terms in the Lagrangian are quartic polynomials, which are
automatically symmetrised when written in terms of general reducible
representations. We employ symbolic multi-indices
${\bar{A}\bar{B}\bar{C}\bar{D}}$ etcetera, which include indices labelling the
individual irreducible representations as well as the corresponding gauge
indices. Alternatively, we may view these indices as the indices of a single,
typically reducible, representation of the gauge group.

With this understanding, the total Lagrangian including Yang-Mills term in
terms of the vector potential $A$, minimally coupled scalars and the explicit
form of the quartic interaction term reads

$\mathcal{L}=\mathcal{L}_{YM}[A]+\mathcal{L}_{0}[\phi,A]-\frac{1}{4!}\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}\phi_{\bar{A}}\phi_{\bar{B}}\phi_{\bar{C}}\phi_{\bar{D}}\
.$ (1)

The quartic interaction tensor $\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}$ is
taken to be totally symmetric in the multi-indices. When constructing the
$\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}$ tensor from specific invariant
polynomials below we get specific symmetry factors for each invariant.

The one-loop beta functions of these theories are special cases of standard
results collected for example in Machacek:1984zw ; Luo:2002ti :

$\begin{split}&\frac{d}{dt}{\lambda}_{\bar{A}\bar{B}\bar{C}\bar{D}}=\mu\frac{d}{d\mu}\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}=\beta_{\bar{A}\bar{B}\bar{C}\bar{D}}=\frac{1}{(4\pi)^{2}}\beta^{I}_{\bar{A}\bar{B}\bar{C}\bar{D}}\\\
&\beta^{I}_{\bar{A}\bar{B}\bar{C}\bar{D}}=\Lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}^{2}-3g^{2}\Lambda^{S}_{\bar{A}\bar{B}\bar{C}\bar{D}}+3g^{4}A_{\bar{A}\bar{B}\bar{C}\bar{D}}\end{split}$
(2)

Here, $\mu$ is the scale, and we think of $t=\log\mu$ as the “time” in the RG
equations, running from the IR to the UV. The different terms arise from
different classes of Feynman diagrams, with different group theory factors

$\begin{split}&\Lambda^{2}_{\bar{A}\bar{B}\bar{C}\bar{D}}=\frac{1}{8}\sum_{\text{perms}}\lambda_{\bar{A}\bar{B}\bar{E}\bar{F}}\lambda_{\bar{E}\bar{F}\bar{C}\bar{D}}\\\
&\Lambda^{S}_{\bar{A}\bar{B}\bar{C}\bar{D}}=\sum_{i}C(i)\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}\\\
&A_{\bar{A}\bar{B}\bar{C}\bar{D}}=~\frac{1}{8}\sum_{\text{perms}}\\{\theta^{E},\theta^{F}\\}_{\bar{A}\bar{B}}\\{\theta^{E},\theta^{F}\\}_{\bar{C}\bar{D}},\end{split}$
(3)

where the sum over $i$ is a sum over four external legs, $C(i)$ being the
corresponding quadratic Casimir operator. Similarly, the sum over permutations
is also a sum over permutations over the four external legs,
$\theta^{C}_{\bar{A}\bar{B}}$ being a possibly reducible representation of the
gauge Lie algebra and indices $A,B,C,\dots$ are Lie algebra indices. Clearly
it is important in the use of the group theory expressions to use the same
normalisation as in Machacek:1984zw . We have concluded that the normalisation
of matrices $T_{A}$ in the fundamental representation should be

$\operatorname{Tr}T_{A}T_{B}=T_{R}\delta_{AB}=\frac{1}{2}\delta_{AB}\ .$ (4)

From this normalisation all values of Casimir operators follow, eg the
quadratic Casimir in the adjoint representation

$C_{2}(G)=N\ .$ (5)

for the $G=SU(N)$ gauge group.

###  2.2 Couplings and algebras of couplings

We now introduce the _marginal algebra_ of marginal four point couplings to
describe the one loop renormalisation of the theories of interest to us. We
have found it an illuminating way to do the calculations and describe the
structures in the theory, but the algebra is not essential to understand our
results and we will also describe them in more conventional terms below.

The idea behind the algebra originates in renormalisation theory. In a
renormalisable theory there is a finite number of couplings $\lambda$ that are
renormalised. The renormalisation group (RG) couples their evolution under a
change of scale $\mu$ in non-linear renormalisation group equations.
Importantly, the RG equations are determined by beta functions
$\beta(\lambda)$ which are given by specific sums of Feynman diagrams. Each
diagram in this perturbation theory contributes with a monomial in the
marginal couplings and the total RG equation defines flows $\lambda(t)$ in the
space of couplings.

Ignoring higher orders, since we are focusing on the one loop approximation,
and thinking about the marginal couplings $\lambda$ as a vector space
$\mathcal{A}$, the RG equations give a flow in this vector space, determined
by a second order expression from the Feynman diagrams, symbolically

$\beta_{\lambda}(\lambda)=\frac{1}{(4\pi)^{2}}\left(P_{2}(\lambda)+P_{1}(\lambda)+P_{0}\right)\\\
,$ (6)

where the subscript $q$ in $P_{q}$ denotes the order in $\lambda$. Note that
the polynomials $P_{q}$ take values in the vector space of couplings. This
structure naturally gives rise to an algebra with a product constructed as

$\lambda\diamond\kappa\equiv\frac{1}{2}\left(P_{2}(\lambda+\kappa)-P_{2}(\lambda)-P_{2}(\kappa)\right)\
,$ (7)

to isolate the cross-terms and become linear in $\lambda$ and $\kappa$
separately. Recall that an algebra means a vector space equipped with a
product satisfying the distributive law. From now on, we can therefore regard
the space $\mathcal{A}$ of couplings as an algebra. By its definition, we see
that the product is commutative, but note that there is no guarantee for it to
be associative.

Let us now spell out how the algebra picture works concretely with the
standard RG equations cited above.

Suppose that we have a complete basis $g_{\bar{A}\bar{B}\bar{C}\bar{D}}^{k}$
of symmetric tensors with corresponding coupling constants $\lambda_{k}$ such
that the general symmetric couplings (1) and their beta functions (2) are
captured by

$\begin{split}\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}=\lambda_{k}g^{k}_{\bar{A}\bar{B}\bar{C}\bar{D}}\end{split}$
(8)
$\begin{split}\beta_{\bar{A}\bar{B}\bar{C}\bar{D}}=\beta_{k}g^{k}_{\bar{A}\bar{B}\bar{C}\bar{D}}.\end{split}$
(9)

The beta functions at one-loop order

$\begin{split}\beta_{\bar{A}\bar{B}\bar{C}\bar{D}}&=\beta_{k}g^{k}_{\bar{A}\bar{B}\bar{C}\bar{D}}\\\
&=\frac{1}{(4\pi)^{2}}\beta^{I}_{\bar{A}\bar{B}\bar{C}\bar{D}}=\frac{1}{(4\pi)^{2}}(\Lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}^{2}-3g^{2}\Lambda^{S}_{\bar{A}\bar{B}\bar{C}\bar{D}}+3g^{4}A_{\bar{A}\bar{B}\bar{C}\bar{D}})\\\
\end{split}$ (10)

can then be written in terms of the $g_{\bar{A}\bar{B}\bar{C}\bar{D}}^{k}$
basis as

$\begin{split}(4\pi)^{2}\beta_{k}=(\lambda_{m}\lambda_{n}c^{mn}_{k}-3g^{2}C_{1}\lambda_{k}+3g^{4}d_{k}),\\\
\end{split}$ (11)

where $C_{1}=4C_{2}(G)$ for four point vertices in the adjoint
representation.The right hand side has been expanded in the complete basis.
The coefficients $c^{mn}_{k}$ in the quadratic first term capture the whole
quadratic contribution because the basis is assumed to be closed under the
renormalisation group flow, the linear second term is a real multiple of the
tree-level coupling because the wave-function renormalisation is diagonal in
this model, and $d_{k}$ represents the RG flow of scalar couplings induced by
other interactions, here the gauge coupling. The $c^{mn}_{k}$ act as structure
constants which encode how a second order expression in the coupling constants
$\lambda_{m}$ and $\lambda_{n}$ enter in the beta function for $\lambda_{k}$.
This is the origin of the algebra description.

The algebra product

$\begin{split}\left(\lambda\diamond\kappa\right)_{\bar{A}\bar{B}\bar{C}\bar{D}}\equiv\frac{1}{8}\sum_{\text{perms}}\lambda_{\bar{A}\bar{B}\bar{E}\bar{F}}\
\kappa_{\bar{E}\bar{F}\bar{C}\bar{D}},\end{split}$ (12)

may be taken as the practical definition, which replaces (7) by realising that
equations (6), (2) and (3) effectively equates $P_{2}$ with $\Lambda^{2}$. The
sum over permutations refers to all permutations of the four external legs,
which guarantees that the product is totally symmetric, just as its two
factors $\lambda$ and $\kappa$. For the basis couplings

$\begin{split}\left(g^{m}\diamond
g^{n}\right)_{\bar{A}\bar{B}\bar{C}\bar{D}}=\frac{1}{8}\sum_{\text{perms}}g^{m}_{\bar{A}\bar{B}\bar{E}\bar{F}}g^{n}_{\bar{E}\bar{F}\bar{C}\bar{D}}=c^{mn}_{k}g^{k}_{\bar{A}\bar{B}\bar{C}\bar{D}}\end{split}$
(13)

which may be written abstractly as a product $g^{m}\diamond
g^{n}=c^{mn}_{k}g^{k}$ by suppressing all field indices. Since coupling
tensors $g_{\bar{A}\bar{B}\bar{C}\bar{D}}^{k}$ form a linear space and the
product is distributive the component version is manifestly a representation
of an abstract algebra $\mathcal{A}$. In this representation the $\diamond$
product is commutative by the total symmetry and the sum over permutations.
The coefficients $c^{mn}_{k}$ completely characterise the product. By
construction the basis spans the space of symmetric 4-point couplings that is
engaged by the renormalisation group flow, generated by the beta functions

$\begin{split}(4\pi)^{2}\beta_{k}=(\lambda_{m}g^{m}\diamond\lambda_{n}g^{n})_{k}-3g^{2}C_{1}\lambda_{k}+3g^{4}d_{k}).\\\
\end{split}$ (14)

We have established a form for one-loop RG equations for multiple couplings.
It will be helpful to have a general understanding of transformation
properties under general coordinate transformations in the space of couplings.
We only have to remember that the beta functions are not scalars, but vectors
in the space of couplings, by their definition (2). Hence, the beta functions
transform as

$\beta_{i}(\lambda)=\mu\frac{d\lambda_{i}}{d\mu}(\lambda)=\mu\frac{\partial\lambda_{i}}{\partial\kappa_{j}}\frac{d\kappa_{j}}{d\mu}(\kappa)=\frac{\partial\lambda_{i}}{\partial\kappa_{j}}\beta_{j}(\kappa)\
.$ (15)

The coefficients in the second order expression (14) will then also transform
as tensors in the space of coupling constants. The transformations

$\begin{split}&\lambda_{k}\to\kappa_{k}=R_{k}^{{\
}m}\lambda_{m},\lambda_{m}=R^{k}_{{\ }m}\kappa_{k}\\\ &R_{k}^{{\ }m}R^{k}_{{\
}n}=\delta^{m}_{n}=R_{m}^{{\ }k}R^{n}_{{\ }k}\\\ \end{split}$ (16)

yield

$\begin{split}&d_{j}\to
d_{j}(\kappa)=d_{i}\frac{\partial\kappa_{j}}{\partial\lambda_{i}}=d_{i}R_{j}^{{\
}i}\\\
&\lambda_{m}\lambda_{n}c^{mn}_{k}\to\lambda_{m}\lambda_{n}c^{mn}_{i}\frac{\partial\kappa_{k}}{\partial\lambda_{i}}=\lambda_{m}\lambda_{n}c^{mn}_{i}R_{k}^{{\
}i}=\kappa_{m}\kappa_{n}R^{m}_{{\ }i}R^{n}_{{\ }j}c^{ij}_{l}R_{k}^{{\ }l}\\\
&c^{mn}_{k}\to c^{mn}_{k}(\kappa)=R^{m}_{{\ }i}R^{n}_{{\
}j}c^{ij}_{l}R_{k}^{{\ }l}.\\\ \end{split}$ (17)

##  3 Adjoint multi-scalar gauge theory invariants and their algebra

We introduce adjoint multi-scalar gauge theory and represent its interactions
in two different but related ways, as invariant polynomials and as symmetric
tensor structures. We then discuss the resulting algebra $\mathcal{A}$ of
marginal couplings before and after the large $N$ limit. Finally, we detail
algebraic properties of $\mathcal{A}$ with physical interpretations for some
of them. Specifically, we note some simplifications in the large $N$ limit.

###  3.1 Adjoint multi-scalar gauge theory

For multi-scalar gauge theories with all scalars in the adjoint representation
of the gauge group, _all_ fields belong to the adjoint representation. A
simple description is then obtained by writing all fields, including the
vector potentials, as $N\times N$ matrices and constructing gauge invariants
from traces of products of matrices. We choose to study $M$ adjoint scalar
multiplets of an $SU(N)$ gauge group. The adjoint representation is real,
permitting an $O(M)$ symmetry to act on the scalar multiplets. In effect, the
index $a$ labelling the scalar $SU(N)$ matrix fields $\Phi_{a}$ transforms in
the fundamental representation of $O(M)$.

The Lagrangian of the theory may be summarised as

$\mathcal{L}=-\frac{1}{4}\operatorname{Tr}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\operatorname{Tr}D^{\mu}\Phi_{a}D_{\mu}\Phi_{a}-\frac{1}{4!}\operatorname{Tr}P_{4}(\Phi)\
,$ (18)

where $F$ is the Yang-Mills field strength and $D$ is the gauge covariant
derivative. $P_{4}$ is a homogeneous fourth order polynomial in $M$ variables,
which we will take to be $O(M)$ invariant. It is in general a sum over four
independent invariants which we describe below. The different ways to write
the scalars are related by

$\Phi_{a}=\phi_{aA}T_{A}=\phi_{\bar{A}}T_{A}\ ,$ (19)

which also explains how we resolve the multi-indices $\bar{A}$.

Then the unique quadratic invariant

$\mathcal{O}_{2}=\operatorname{Tr}\Phi_{a}\Phi_{a}=\delta_{ab}\operatorname{Tr}\phi_{aA}T_{A}\phi_{bB}T_{B}=\frac{1}{2}\delta_{ab}\delta_{AB}\phi_{aA}\phi_{bB}$
(20)

illustrates what will be a recurring theme below. Invariants may be
represented either as invariant polynomials,
$\operatorname{Tr}\Phi_{a}\Phi_{a}$, or as invariant tensor structures,
$\frac{1}{2}\delta_{ab}\delta_{AB}$.

###  3.2 Invariant polynomials

We now turn to the invariants needed to describe an $O(M)$ and $SU(N)$
symmetric potential. By squaring $\mathcal{O}_{2}$ and ordering it to remove
short-distance singularities we can define a quartic double trace operator
$\operatorname{Tr}\Phi_{a}\Phi_{a}\operatorname{Tr}\Phi_{b}\Phi_{b}$. We will
first discuss more general renormalisable terms in the potential written as
quartic _polynomials_ of the fields, and then in terms of the $SU(N)$ and
$O(M)$ _tensor structures_ which summarise the coefficients of the
polynomials.

The list of invariant quartic polynomials includes
$\operatorname{Tr}\Phi_{a}\Phi_{a}\operatorname{Tr}\Phi_{b}\Phi_{b}$ and
$\operatorname{Tr}\Phi_{a}\Phi_{b}\operatorname{Tr}\Phi_{a}\Phi_{b}$, which
are both double trace expressions - sums of squares of $SU(N)$ traces. There
are also two different single trace monomials. The full space of invariants is
spanned by the operators

$\operatorname{Tr}\Phi_{a}\Phi_{a}\Phi_{b}\Phi_{b},\operatorname{Tr}\Phi_{a}\Phi_{b}\Phi_{a}\Phi_{b},\operatorname{Tr}\Phi_{a}\Phi_{a}\operatorname{Tr}\Phi_{b}\Phi_{b},\operatorname{Tr}\Phi_{a}\Phi_{b}\operatorname{Tr}\Phi_{a}\Phi_{b}\
.$ (21)

The double trace polynomials are important for several reasons, primarily
because they decompose, as above, into factors of quadratic polynomial
invariants of the gauge group $SU(N)$. The large $N$ scaling of single and
double trace operators is quite different, and leads to important differences
in the large $N$ limit Pomoni:2009joh . Double trace operators appear as
corrections to single trace potentials, but are still significant for large
$N$. We shall see this in in more detail below. In particular, we will find
that the algebra of marginal operators differentiates clearly between single
trace and double trace operators.

###  3.3 Invariant tensor structures

To each invariant polynomial corresponds an invariant tensor structure, which
we label by the same symbols. It is obtained by expressing the $\Phi_{a}$ in
terms of the Lie algebra basis and identifying the coefficients in the quartic
polynomial of $\phi_{aA}$. Conversely, we may start from the completely
symmetric tensor structures

$\begin{split}{g}^{1s}&\equiv\delta_{ab}\delta_{cd}(\Tr(T_{A}T_{B}T_{C}T_{D})+\Tr(T_{A}T_{D}T_{C}T_{B})+\Tr(T_{A}T_{B}T_{D}T_{C})+\Tr(T_{A}T_{C}T_{D}T_{B}))\\\
&+\delta_{ac}\delta_{bd}(\Tr(T_{A}T_{B}T_{D}T_{C})+\Tr(T_{A}T_{C}T_{D}T_{B})+\Tr(T_{A}T_{C}T_{B}T_{D})+\Tr(T_{A}T_{D}T_{B}T_{C}))\\\
&+\delta_{ad}\delta_{bc}(\Tr(T_{A}T_{C}T_{B}T_{D})+\Tr(T_{A}T_{D}T_{B}T_{C})+\Tr(T_{A}T_{B}T_{C}T_{D})+\Tr(T_{A}T_{D}T_{C}T_{B}))\\\
{g}^{1t}&\equiv\delta_{ab}\delta_{cd}(\Tr(T_{A}T_{C}T_{B}T_{D})+\Tr(T_{A}T_{D}T_{B}T_{C}))+\delta_{ac}\delta_{bd}(\Tr(T_{A}T_{B}T_{C}T_{D})+\Tr(T_{A}T_{D}T_{C}T_{B}))\\\
&+\delta_{ad}\delta_{bc}(\Tr(T_{A}T_{B}T_{D}T_{C})+\Tr(T_{A}T_{C}T_{D}T_{B}))\\\
{g}^{2s}&\equiv\delta_{ab}\delta_{cd}\delta_{AB}\delta_{CD}+\delta_{ac}\delta_{bd}\delta_{AC}\delta_{BD}+\delta_{ad}\delta_{bc}\delta_{AD}\delta_{BC}\\\
{g}^{2t}&\equiv\delta_{ab}\delta_{cd}(\delta_{AC}\delta_{BD}+\delta_{AD}\delta_{BC})+\delta_{ac}\delta_{bd}(\delta_{AB}\delta_{CD}+\delta_{AD}\delta_{CB})+\delta_{ad}\delta_{bc}(\delta_{AB}\delta_{DC}+\delta_{AC}\delta_{DB}),\\\
\end{split}$ (22)

where we have suppressed field indices. The polynomials follow from
contracting the tensor structures above with the scalar fields $\phi_{aA}$.
The superscript 1 and 2 denote single trace and double trace objects,
respectively, while we use $s$ to symbolise scalar product and $t$ to
symbolise tensor products of neighbouring $O(M)$ vectors. The notation may
become clearer in the expressions for the corresponding invariant polynomials

$\displaystyle\frac{1}{4!}g^{1s}(\Phi)\equiv\frac{1}{4!}{g}^{1s}_{\bar{A}\bar{B}\bar{C}\bar{D}}\phi_{\bar{A}}\phi_{\bar{B}}\phi_{\bar{C}}\phi_{\bar{D}}$
$\displaystyle=\frac{1}{2}\operatorname{Tr}\Phi_{a}\Phi_{a}\Phi_{b}\Phi_{b}$
(23)
$\displaystyle\frac{1}{4!}g^{1t}(\Phi)\equiv\frac{1}{4!}{g}^{1t}_{\bar{A}\bar{B}\bar{C}\bar{D}}\phi_{\bar{A}}\phi_{\bar{B}}\phi_{\bar{C}}\phi_{\bar{D}}$
$\displaystyle=\frac{1}{4}\operatorname{Tr}\Phi_{a}\Phi_{b}\Phi_{a}\Phi_{b}$
$\displaystyle\frac{1}{4!}g^{2s}(\Phi)\equiv\frac{1}{4!}{g}^{2s}_{\bar{A}\bar{B}\bar{C}\bar{D}}\phi_{\bar{A}}\phi_{\bar{B}}\phi_{\bar{C}}\phi_{\bar{D}}$
$\displaystyle=\frac{1}{8}\operatorname{Tr}\Phi_{a}\Phi_{a}\operatorname{Tr}\Phi_{b}\Phi_{b}$
$\displaystyle\frac{1}{4!}g^{2t}(\Phi)\equiv\frac{1}{4!}{g}^{2t}_{\bar{A}\bar{B}\bar{C}\bar{D}}\phi_{\bar{A}}\phi_{\bar{B}}\phi_{\bar{C}}\phi_{\bar{D}}$
$\displaystyle=\frac{1}{4}\operatorname{Tr}\Phi_{a}\Phi_{b}\operatorname{Tr}\Phi_{a}\Phi_{b}\
,$

which comply with the form (1), and the factors in front of the trace
expressions are symmetry factors obtained from the number of terms in the
totally symmetric tensors (22). Any invariant tensor structure
$\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}$ with total symmetry may then be
expanded in the basis of invariant tensor structures:

$\lambda_{\bar{A}\bar{B}\bar{C}\bar{D}}=\lambda_{k}g^{k}_{\bar{A}\bar{B}\bar{C}\bar{D}}\
.$ (24)

The most important factor above is actually the $1/4!$ which establishes the
connection to standard beta function results (2). Then, any change of tensor
structure basis will translate to a different algebra and different coupling
constants and beta functions, but always respecting the equations in section
2.

###  3.4 Algebra

The commutative but non-associative algebra $\mathcal{A}$ defined above in
equation (13) can be computed by any method for multiplying tensors,
contracting their indices and identifying the resulting objects222In practice,
we used a graphical method, inspired by birdtrack calculus, Cvitanovic:1976am
; Cvitanovic:2008zz  . Our results have been checked and compared by RGBeta
which is Mathematica packge, Thomsen:2021ncy ..

The single trace tensor multiplication table reads

$\begin{split}g^{1s}\diamond
g^{1s}&=\frac{1}{2}(M+3)g^{2s}+\frac{1}{2}g^{2t}+\frac{1}{2}N(M+3)g^{1s}\\\
g^{1t}\diamond g^{1t}&=\frac{1}{8}(M+2)g^{2t}+\frac{1}{2}Ng^{1s}\\\
g^{1s}\diamond
g^{1t}&=\frac{1}{2}g^{2s}+\frac{1}{2}g^{2t}+\frac{1}{2}Ng^{1s}+Ng^{1t}\ ,\\\
\end{split}$ (25)

explicitly demonstrating that double trace tensor structures are produced.
This means that the double trace operators are needed for renormalisability.
They are generated by one loop corrections.

In contrast, the algebra of double trace tensor structures closes:

$\begin{split}g^{2s}\diamond g^{2s}&=(M(N^{2}-1)+8)g^{2s}\\\ g^{2t}\diamond
g^{2t}&=(2M+2N^{2}+6)g^{2t}+12g^{2s}\\\ g^{2s}\diamond
g^{2t}&=2(M+N^{2})g^{2s}+6g^{2t}\ ,\end{split}$ (26)

which means that it forms a sub-algebra.

Finally, the product of single trace and double trace tensor structures
generates linear combinations of single trace and double trace structures,

$\begin{split}g^{1s}\diamond g^{2s}&=(M+1)Ng^{2s}+6g^{1s}\\\ g^{1s}\diamond
g^{2t}&=2(M+3)g^{1s}+2Ng^{2s}+Ng^{2t}+8g^{1t}\\\ g^{1t}\diamond
g^{2s}&=Ng^{2s}+6g^{1t}\\\ g^{1t}\diamond
g^{2t}&=2(M+1)g^{1t}+Ng^{2t}+4g^{1s}.\\\ \end{split}$ (27)

Interestingly, these mixed products simplify considerably in the the large $N$
limit, where only double trace operators survive. Next, we describe this
limit.

The standard large $N$ limit amounts to taking $N$ large while keeping
rescaled ’t Hooft couplings fixed, in our case $\lambda_{1S}=N\lambda_{1s}$,
$\lambda_{1T}=N\lambda_{1t}$ and $\lambda_{2S}=N^{2}\lambda_{2s}$,
$\lambda_{2T}=N^{2}\lambda_{2t}$. We have thus adopted the convention to use
capital $S$ and $T$ symbols in place of $s$ and $t$ whenever they describe a
’t Hooft coupling or a corresponding large $N$ rescaled operator, tensor or
polynomial. Hence, the rescalings

$\begin{split}g^{1S}&=\frac{1}{N}{g}^{1s}\\\ g^{1T}&=\frac{1}{N}{g}^{1t}\\\
g^{2S}&=\frac{1}{N^{2}}g^{2s}\\\ g^{2T}&=\frac{1}{N^{2}}g^{2t},\end{split}$
(28)

preserve the decomposition (24), making the large $N$ choice of variables a
simple change of base. We then find the large $N$ algebra directly by studying
the algebra of the rescaled operators, minding the transformation properties
(17) and dropping subleading terms:

$\begin{split}g^{1S}\diamond
g^{1S}&=\frac{1}{2}(M+3)g^{2S}+\frac{1}{2}g^{2T}+\frac{1}{2}(M+3)g^{1S}\\\
g^{1T}\diamond g^{1T}&=\frac{1}{8}(M+2)g^{2T}+\frac{1}{2}g^{1S}\\\
g^{1S}\diamond
g^{1T}&=\frac{1}{2}g^{2S}+\frac{1}{2}g^{2T}+\frac{1}{2}g^{1S}+g^{1T}\\\
\end{split}$ (29)

$\begin{split}g^{2S}\diamond g^{2S}=&Mg^{2S}\\\ g^{2T}\diamond
g^{2T}=&2g^{2T}\\\ g^{2S}\diamond g^{2T}=&2g^{2S}\\\ \end{split}$ (30)

$\begin{split}g^{1S}\diamond g^{2S}=&(M+1)g^{2S}\\\ g^{1S}\diamond
g^{2T}=&2g^{2S}+g^{2T}\\\ g^{1T}\diamond g^{2S}=&g^{2S}\\\ g^{1T}\diamond
g^{2T}=&g^{2T}\\\ \end{split}$ (31)

The algebra clearly simplifies in the large $N$ limit. As promised above, the
mixed single trace-double trace products are all double trace tensor
structures. From the previous result that double trace-double trace products
yield double trace structures, we conclude that _all_ products of a double
trace structure with anything else yields a double trace structure in the
large $N$ limit. In algebra terminology, this amounts to the statement that
the double trace subalgebra is in fact an algebra _ideal_. Then, one may
consistently mod out the double trace operators and construct a quotient
algebra, which essentially is the algebra of the single trace operators,
ignoring any double trace terms. Physically, the beta functions, of the single
trace operators are independent of the double trace couplings, and the single
trace RG equations form a closed dynamical system. See (32) below. The
quotient algebra controls the RG equations of $\lambda_{1S}$ and
$\lambda_{1T}$ independently of $\lambda_{2S}$ and $\lambda_{2T}$. As noted
previously, the double trace sector of a large $N$ gauge theory is still
significant Adams:2001jb ; Dymarsky:2005nc ; Dymarsky:2005uh ; Pomoni:2009joh
since a running double trace coupling may prevent conformal fixed points or
may lead the four point coupling to a region where the potential is unstable.
There is also a smaller ideal spanned by $g^{2S}$ inside the double trace
ideal. We then expect its quotient algebra to generate an RG flow independent
of $\lambda_{2S}$, as can be confirmed by inspecting (32) below.

Adding $g^{1S}$ to the double trace ideal still generates a sub-algebra, but
not an ideal, due to the $g^{1T}$ term in $g^{1S}\diamond g^{1T}$. This means
that a model with vanishing $\lambda_{1T}$ is consistent with the RG flow, and
that there is no quotient algebra controlling the RG evolution of
$\lambda_{1T}$ independently of $\lambda_{2S},\lambda_{2T}$ and
$\lambda_{1S}$. Furthermore, the tensor $g^{1T}$ does not form a sub-algebra
by itself, much less an ideal, making the RG equations of
$\lambda_{2S},\lambda_{2T}$ and $\lambda_{1S}$ dependent on $\lambda_{1T}$.

We conclude this section by some other algebra related observations, without
spelling out their physical implications here. Non-associative algebras with
various properties, some mentioned below, are studied in their own right
MR1375235 . In particular, commutative non-associative algebras like
$\mathcal{A}$ have been used in the mathematics literature (starting with
MR0132743 ) for classifying systems of ordinary differential equations.
Algebraic properties are directly related to the properties of the dynamical
system defined by the algebra, which in our case would be the RG equations.

Commutative and non-associate algebras are not prominent in the theoretical
physics toolkit, but Jordan algebras, with a $\circ$ product satisfying the
Jordan property

$a^{2}\circ(a\circ b)=a\circ(a^{2}\circ b),$

where $a^{2}\equiv a\circ a$, have been studied now and again. Another related
property is power associativity, which means that $a^{m}$ is unambiguously
defined and $a^{m}\circ a^{n}=a^{m+n}$ holds for all positive integers $m$ and
$n$.

The large $N$ double trace ideal is power associative, and has a unit
element333Note that this element is not a unit element in the full algebra.
$\mathbb{I}=g^{2T}/2$. These facts imply that it is actually a Jordan algebra,
and it is simple enough that we can give a complete characterisation. The
element $\mathbb{E}_{1}\equiv g^{2S}/M$ is idempotent, ie
$\mathbb{E}_{1}\diamond\mathbb{E}_{1}=\mathbb{E}_{1}$. We also find that
$\mathbb{E}_{2}\equiv\mathbb{I}-\mathbb{E}_{1}$ is idempotent. These elements
are then complementary projection operators, which means that this two-
dimensional Jordan algebra decomposes into the direct sum
$\mathbb{R}\oplus\mathbb{R}$.

We have found the full algebra $\mathcal{A}_{SU(N)}$ corresponding to
symmetric $SU(N)$ adjoint scalar gauge theory to neither satisfy the Jordan
property nor power associativity. It only satisfies the weaker _flexible_
identity

$a\diamond(b\diamond a)=(a\diamond b)\diamond a$

for arbitrary elements $a$ and $b$.

##  4 Adjoint multi-scalar gauge theory beta functions

We can now find the complete beta function of the scalar couplings, and write
it down in the large $N$ limit from our expressions for the algebra. We then
discuss the relation to a simple bosonic D brane model, and then obtain its
beta function both from the algebra and by a direct calculation. The D brane
model is found to require completion to the full multi-adjoint gauge theory.
All results agree with each other and with calculations in the Mathematica
package RGBeta.

###  4.1 Complete one loop beta functions from the algebra

As seen in section 2 the algebra $\mathcal{A}$ directly reflects the one-loop
beta functions (14), which can be read off from its structure constants, which
are extracted from the $\diamond$ products (25), (26) and (27) or (29), (30)
and (31). We find the large $N$ beta functions in terms of large $N$ scalar ’t
Hooft couplings:

$\begin{split}\beta_{1S}&=\frac{1}{16\pi^{2}}\bigg{(}\frac{M+3}{2}\lambda_{1S}\lambda_{1S}+\frac{1}{2}\lambda_{1T}\lambda_{1T}+\lambda_{1T}\lambda_{1S}-12Ng^{2}\lambda_{1S}+6N^{2}g^{4}\bigg{)}\\\
\beta_{1T}&=\frac{1}{16\pi^{2}}\bigg{(}2\lambda_{1T}\lambda_{1S}-12Ng^{2}\lambda_{1T}\bigg{)}\\\
\beta_{2S}&=\frac{1}{16\pi^{2}}\bigg{(}M\lambda_{2S}\lambda_{2S}+\frac{(M+3)}{2}\lambda_{1S}\lambda_{1S}+4\lambda_{2S}\lambda_{2T}+2\lambda_{2S}\lambda_{1T}+2(M+1)\lambda_{2S}\lambda_{1S}+4\lambda_{2T}\lambda_{1S}\\\
&+\lambda_{1T}\lambda_{1S}-12Ng^{2}\lambda_{2S}+6N^{2}g^{4}\bigg{)}\\\
\beta_{2T}&=\frac{1}{16\pi^{2}}\bigg{(}2\lambda_{2T}\lambda_{2T}+\frac{1}{2}\lambda_{1S}\lambda_{1S}+\frac{M+2}{8}\lambda_{1T}\lambda_{1T}+2\lambda_{2T}\lambda_{1T}+2\lambda_{2T}\lambda_{1S}+\lambda_{1T}\lambda_{1S}\\\
&-12Ng^{2}\lambda_{2T}+6N^{2}g^{4}\bigg{)}.\\\ \end{split}$ (32)

Note that the $\lambda_{1T}$ coupling can self-consistently be set to zero.
The reason is twofold. First, the gauge boson loops do not induce running
coupling with a $g^{1T}$ structure. Second, the tensors corresponding to the
other couplings form a closed algebra, and will not generate a $g^{1T}$ term
at one loop. We indeed find below that all the most interesting features of
the RG flow occur at $\lambda_{1T}=0$.

###  4.2 A D-brane model

The adjoint multi-scalar gauge theories we study are related to string theory
D-branes DaiLeighPolchinski1989 , which are defects where open strings can
end. The low energy effective theory for open strings on $N$ D-branes is
$U(N)$ Yang-Mills theory with $M$ adjoint scalars representing fluctuations in
dimensions transverse to the defect. There are also typically massless
fermions required by supersymmetry. In many cases, but not all, with
insufficient supersymmetry, there are tachyons in the string theory, which
render it unstable. Purely bosonic string theory suffers from this problem,
which is why we call the corresponding gauge theories D-brane models rather
than limits of the theory on the D-brane. Although it is possible that the
theory becomes meta-stable in the field theory limit we do not make such
claims, but use the formal limit without tachyons as an interesting example.

The classical scale invariant four point coupling for the scalars in these
models is completely analogous to the Yang-Mills four point coupling, and
given by dimensional reduction of higher-dimensional Yang-Mills theory. In the
maximally supersymmetric case, the coupling receives no quantum corrections
and scale invariance is preserved. The potential has flat directions but is
otherwise positive Witten1996 . In more general cases, these D brane models
are still renormalisable in four dimensions, but will be corrected by loops
and a scale will typically be introduced through dimensional transmutation.
Generically, the potential lacks flat directions and is stable or unstable
corresponding to favouring $N$ branes staying together or moving apart,
respectively. Since the classical behaviour is marginally stable, we expect
the first quantum corrections, which we study here, to be decisive.

The classical four-point coupling is modelled on the Yang-Mills four-point
coupling, and reads

$\begin{split}\lambda_{aAbBcCdD}&=-\frac{\lambda}{6}[\delta_{ac}\delta_{bd}f_{AB}^{E}f_{CD}^{E}+\delta_{bc}\delta_{ad}f_{BA}^{E}f_{CD}^{E}+\delta_{ca}\delta_{bd}f_{CB}^{E}f_{AD}^{E}\\\
&+\delta_{dc}\delta_{ba}f_{DB}^{E}f_{CA}^{E}+\delta_{cb}\delta_{ad}f_{CA}^{E}f_{BD}^{E}+\delta_{dc}\delta_{ab}f_{DA}^{E}f_{CB}^{E}]\\\
&=-\frac{\lambda}{3}\big{[}(\Tr(T_{A}T_{B}T_{C}T_{D})+\Tr(T_{A}T_{D}T_{C}T_{B}))(2\delta_{ac}\delta_{bd}-\delta_{ab}\delta_{cd}-\delta_{ad}\delta_{bc})\\\
&+(\Tr(T_{A}T_{B}T_{D}T_{C})+\Tr(T_{A}T_{C}T_{D}T_{B}))(2\delta_{ad}\delta_{bc}-\delta_{ab}\delta_{cd}-\delta_{ac}\delta_{bd})\\\
&+(\Tr(T_{A}T_{C}T_{B}T_{D})+\Tr(T_{A}T_{D}T_{B}T_{C}))(2\delta_{ab}\delta_{cd}-\delta_{ad}\delta_{bc}-\delta_{ac}\delta_{bd})\big{]}\\\
&=\frac{\lambda}{3}(g^{1s}-2g^{1t})\end{split}$ (33)

in terms of the Lie algebra structure coefficients $f^{A}_{BC}$ from
$\left[T_{B},T_{C}\right]=if^{A}_{BC}T_{A}$.

###  4.3 D-brane model beta functions from the general theory

Focusing on leading terms for large $N$, the D-brane model’s beta functions
follow from the general beta function (32) by specialising the four point
coupling to (33). Alternatively, the standard expressions (2) and (3) may be
used directly, or we use the general beta functions (9) from the components
(14) and the products (25) with $\lambda_{i}$ from the D-brane model
interaction (33):

$\begin{split}\mathbf{\beta}^{I}_{\bar{A}\bar{B}\bar{C}\bar{D}}&=g^{1s}\bigg{(}\frac{N(M+3)\lambda^{2}}{18}-4Ng^{2}\lambda+6g^{4}N\bigg{)}+g^{1t}\bigg{(}\frac{-4N\lambda^{2}}{9}+8Ng^{2}\lambda\bigg{)}\\\
&+(g^{2s}+g^{2t})\bigg{(}\frac{(M-1)\lambda^{2}}{18}+6g^{4}\bigg{)},\end{split}$
(34)

which to this order yields

$\begin{split}\beta_{1s}&=\frac{1}{16\pi^{2}}\bigg{(}\frac{N(M+3)\lambda^{2}}{18}-4Ng^{2}\lambda+6g^{4}N\bigg{)}\\\
\beta_{1t}&=\frac{1}{16\pi^{2}}\bigg{(}\frac{-4N\lambda^{2}}{9}+8Ng^{2}\lambda\bigg{)}\\\
\beta_{2s}&=\frac{1}{16\pi^{2}}\bigg{(}\frac{(M-1)\lambda^{2}}{18}+6g^{4}\bigg{)}\\\
\beta_{2t}&=\frac{1}{16\pi^{2}}\bigg{(}\frac{(M-1)\lambda^{2}}{18}+6g^{4}\bigg{)}.\\\
\end{split}$ (35)

Double trace tensor structures are generated, and the two single trace
couplings run at different speeds. This means that the bosonic D brane model
is not consistent with the RG flow and has to be extended to four coupling
constants. In the end, a specific RG trajectory could describe a gauge theory
descending from string theory, but which particular trajectory is not a pure
field theory question. Furthermore, the general one loop RG flows we find
below in section 5 do not support the idea that a tree level coupling (33) can
describe the model over a substantial range of scales.

We have also obtained the same beta functions (35) directly, using standard
birdtrack methods Cvitanovic:1976am ; Cvitanovic:2008zz , thereby partially
validating our algebra approach. The result also agrees with a calculation in
the Mathematica package RGBeta Thomsen:2021ncy 444We thank Fedor Popov for
alerting us to a discrepancy between an RGBeta calculation and an earlier
edition of our calculation. The results and methods are now thoroughly cross-
validated..

##  5 One loop RG flow in adjoint multi-scalar gauge theory

We can now discuss the physics of the RG flow and how it follows from the
algebra of invariants. We consider the one loop RG flow equations and the
partial decoupling of the gauge coupling by focusing on ratios of scalar and
gauge couplings. Then we describe fixed points of these ratios, which are
fixed flows of the original scalar couplings. The full structure of real fixed
flows is surveyed for positive $M$ in the large $N$ limit. A hierarchical
structure of the RG equations in this limit simplifies the investigation. In
particular, the RG flow of the single trace couplings is independent of double
trace equations. We also briefly describe complex fixed points and flows.

###  5.1 The algebra and the RG flow

The beta functions above describe the RG flow, but it should be noted that the
gauge coupling $g^{2}$ and the corresponding ’t Hooft coupling $Ng^{2}$ are
usually not constant but running. The running gauge coupling is taken into
account simply by a new choice of relative couplings $\mu_{i}$ with RG
equations that form an autonomous system of first order equations. The idea to
focus on the fixed points of the relative couplings, called _fixed flows_ was
advocated a while ago Giudice:2014tma  and has a long history Gross:1973ju ;
Cheng:1973nv . For fixed flows scalar couplings are locked relative to each
other and the gauge coupling. The complete system of one loop RG equations
includes the gauge coupling beta function, which we write in terms of
$\beta_{g^{2}}$ or $\beta_{Ng^{2}}$:

$\beta_{g}=-\frac{g^{3}}{16\pi^{2}}\frac{22-M}{6}C_{2}(G)$ (36)
$\beta_{g^{2}}=-2g\frac{g^{3}}{16\pi^{2}}\frac{22-M}{6}C_{2}(G)$ (37)
$\beta_{Ng^{2}}=-\frac{22-M}{48\pi^{2}}(Ng^{2})^{2},$ (38)

where we used $C_{2}(G)=N$. As sketched above, we disentangle the dependence
of the RG equations on the gauge coupling for non-critical $M\neq 22$ by
considering the relative (large $N$) scalar couplings
$\mu_{I}\equiv\frac{\lambda_{I}}{Ng^{2}}$ which yields their beta functions

$\begin{split}\beta_{1S}(\mu)&=\frac{Ng^{2}}{16\pi^{2}}\bigg{(}\frac{M+3}{2}\mu_{1S}\mu_{1S}+\frac{1}{2}\mu_{1T}\mu_{1T}+\mu_{1T}\mu_{1S}-(\frac{M-22}{3}+12)\mu_{1S}+6\bigg{)}\\\
\beta_{1T}(\mu)&=\frac{Ng^{2}}{16\pi^{2}}\bigg{(}2\mu_{1T}\mu_{1S}-(\frac{M-22}{3}+12)\mu_{1T}\bigg{)}\\\
\beta_{2S}(\mu)&=\frac{Ng^{2}}{16\pi^{2}}\bigg{(}M\mu_{2S}\mu_{2S}+\frac{(M+3)}{2}\mu_{1S}\mu_{1S}+4\mu_{2S}\mu_{2T}+2\mu_{2S}\mu_{1T}+2(M+1)\mu_{2S}\mu_{1S}+4\mu_{2T}\mu_{1S}\\\
&+\mu_{1T}\mu_{1S}-(\frac{M-22}{3}+12)\mu_{2S}+6\bigg{)}\\\
\beta_{2T}(\mu)&=\frac{Ng^{2}}{16\pi^{2}}\bigg{(}2\mu_{2T}\mu_{2T}+\frac{1}{2}\mu_{1S}\mu_{1S}+\frac{M+2}{8}\mu_{1T}\mu_{1T}+2\mu_{2T}\mu_{1T}+2\mu_{2T}\mu_{1S}+\mu_{1T}\mu_{1S}\\\
&-(\frac{M-22}{3}+12)\mu_{2T}+6\bigg{)}.\\\ \end{split}$ (39)

Together with the gauge coupling beta function (38) these beta functions give
a complete set of RG equations. Note that the one loop RG equations have the
form $\frac{d\mu}{dt}=\alpha f(\mu),\frac{d\alpha}{dt}=-b\alpha^{2}$, which
allows them to be completely separated by a new choice of RG time given by
$d\tau=\alpha dt$ into $\frac{d\mu}{d\tau}=f(\mu)$ and
$\frac{d\alpha}{d\tau}=-b\alpha$. This means that the RG flow phase portrait
in $\mu_{i}$ space is entirely independent of the flow of the gauge coupling
$\alpha$ and forms a separate dynamical system. We investigate this system by
regarding the gauge coupling as constant.

###  5.2 Real or complex fixed flows

We now search for fixed flows in the large $N$ limit by setting the beta
functions of relative couplings to zero, $\beta_{i}(\mu)=0$. At one loop these
equations are four second order polynomial equations in four variables which
simplify for large $N$ because the the single trace equations based on the
large $N$ beta functions (39) only involve the single trace couplings. The
double trace flows then also simplify considerably for fixed single trace
flows. This is because the flow of the double trace couplings is parametrised
by the single trace couplings, which are then constant. Our procedure to
analyse the RG flow makes use of this hierarchical structure. The actual flow
pattern is then determined by number $M$ of adjoint scalars.

In practice, the phase portraits of the single trace flow are of primary
importance and can be analysed directly, then yielding different phase
portraits of double trace flows for each single trace fixed flow. We focus
especially on the completely IR or UV stable flows.

The single trace fixed flow equations are also hierarchically arranged and
easy to solve, first solving for $\mu_{1T}$ in terms of $\mu_{1S}$, and then
solving the resulting second order equation. We find that there are two pairs
of single trace fixed flow (FF) roots, $\mu^{FF1\pm}$ and $\mu^{FF2\pm}$:

$\begin{array}[]{clr}\mu_{1S}^{FF1\pm}&=\frac{M+14\pm\sqrt{M^{2}-80M-128}}{3(M+3)},&\mu_{1T}^{FF1\pm}=0\\\
&&\\\
\mu_{1S}^{FF2\pm}&=\frac{M+14}{6},&\mu_{1T}^{FF2\pm}=-\frac{M+14}{6}\pm\frac{\sqrt{-M^{3}-26M^{2}-140M-40}}{6}.\end{array}$
(40)

None of the second pairs of fixed flow roots are real for the physical $M>0$.
The first pair of fixed flow roots is more interesting: There are two single
trace fixed flows, given by the first pair of roots $\mu^{FF1\pm}$ for integer
$M\geq 82$. An important conclusion is that there are _no real one-loop fixed
points_ for this class of theories. The fixed flow equations above are fixed
point equations only for $M=22$, which is the conditions for the gauge
coupling to not flow at one loop order. That $M=22$ is outside the range which
allows real fixed flows means that there are no real fixed points to this
order.

Here, we pause the general search of real fixed flows to temporarily consider
complex roots. Even if complex coupling constants in general break unitarity
complex roots of the fixed point equations are interesting in their own right.
They define complex conformal field theories Gorbenko:2018ncu , and similarly,
the fixed flows at complex values of the roots above are complex fixed flows.
Such flows give characteristic flow patterns for real couplings in the
vicinity of the complex roots Gorbenko:2018ncu ; Faedo:2021ksi , as we will
see an example of in figure 2. Allowing for complex roots, all the four single
trace fixed flows (40) make sense, and each one of them gives rise to a second
order equation for $\mu_{2T}$. Finally, each of these roots yields a second
order equation for $\mu_{2S}$. Altogether, there are generically 16 complex
roots arranged hierarchically, and for each set of complex roots, the complex
conjugate couplings also solve the equations. Specifically, the case $M=22$
yields complex valued fixed points, and in contrast to related examples
engineered from $\mathcal{N}=4$ supersymmetric Yang-Mills theory, for which
only double trace couplings are complex, one of the single trace couplings are
complex in this purely bosonic case.

(a) Single trace flow. Fixed flows determine parameters for double trace
flows.

(b) Double trace flow, coarse grained view, Details are resolved in figure 3.

Figure 1: RG flows for $M=425$. UV stable fixed flows are marked with purple
circles, IR stable fixed flows with red circles and fixed flows of mixed
stability with green circles. They are connected by (green) separatrix flows.
Cyan flows and regions mark other models for which the one loop approximation
also makes sense in the IR.

Returning to the search for real fixed flows, the RG flows of the single trace
couplings change qualitatively between $M=81$ and $M=82$, by the appearance of
two real roots. In the lower range of $M$ there are only complex fixed flows,
while in the upper range there are four special flows: a UV stable fixed flow,
an unstable fixed flow, a separatrix flow between the two fixed flows, and one
between the unstable fixed flow and infinity. Figure 1(a) illustrates a
typical phase portrait for single trace flows with $M\geq 82$. Note that all
the theories described by these phase portraits are IR free in terms of the
gauge coupling. Despite the IR weakness of the gauge couplings, the scalar
single trace couplings generally grow in the IR, with the exception of these
four special flows. The single trace couplings and the gauge coupling are
asymptotically IR free for these four flows, such that the one loop
approximation is expected to hold.

The final verdict on the consistency of the one loop approximation in the IR
is settled by a study of the double trace phase portraits. First, the single
trace fixed flow roots determine quadratic fixed flow equations for the double
trace couplings. There are no completely real double trace roots for $M\leq
405$. The qualitative behaviour is illustrated in figure 2. For
$M=406,\dots,427$ and the UV stable single trace root $\mu^{FF1-}$, there are
four fixed flows at real couplings, while for $M\geq 428$ there are two groups
of four real fixed flow solutions, one for each of the two real single trace
fixed flows.

(a) Overview.

(b) Upper flow feature.

(c) Lower flow feature.

Figure 2: Double trace flows without fixed flows for $M=82$. Compare the
coarse grained view of figure 1(b) and closeups of the flows around its fixed
flows given below in figure 3. The features above represent the presence of
two complex conjugate fixed flows, which coalesce between $M=405$ and $M=406$
and separate again for large $M$ into the pairs of fixed points of figure 3.

The double trace phase portraits are all qualitatively captured by figures
1(b) and 3 where the four fixed flows form tall, narrow parallelograms. In
figure 1(b) the parallelogram is not resolved and the general flow inside it
is well represented by a separatrix. Except for the parallelogram/separatrix
preventing any flows to pass through, the phase portraits 2 for $M\leq 405$
and 1(b) for $M\geq 406$ are qualitatively similar. The obstacle to the flow
for $M\geq 406$ however permits a whole class of trajectories to have
asymptotically IR free limits, characterised by the IR stable double trace
fixed flow in the phase portrait 1(b). The region covered by these flows has
been coloured cyan.

For $406\leq M\leq 427$, the only IR free double trace flows correspond to the
UV stable single trace fixed flows. If the single trace flow is not exactly on
this fixed point the IR limit will be strongly coupled, or end up on the
separatrix connecting to the fixed flow of mixed stability in the IR. Since
there are no double trace fixed flows associated with this single trace fixed
flow in this range of $M$, its overall IR free status is undermined. Thus, for
$406\leq M\leq 427$ only the UV stable (IR unstable!) single trace fixed flow
leads to IR free limits with the double trace flows illustrated by figure
1(b). In contrast, for $M\geq 428$, all the four specially marked fixed flows
and separatrices in figure 1(a) are compatible with such IR free limits.

(a) Upper fixed flow details.

(b) Lower fixed flow details.

Figure 3: Close-ups of double trace flows for $M=425$ with fixed flows marked
by circles. Two almost vertical separatrices connecting the upper and lower
fixed flows have not been drawn. Together, the four fixed flows and the four
separatices form a parallelogram through which no flows can pass. In the
coarse grained view of figure 1(b), it appears as a separatrix.

We now summarise the big picture. Since we have only found fixed flows for
$M\geq 406$, where the gauge coupling grows indefinitely in the UV, the one
loop analysis yields no theory which is under control in the UV. Leaving short
distance consistency aside, each trajectory which asymptotes to a fixed flow
in the IR provides a model which is under perturbative control in the IR. For
the fixed flows and bounded separatrices discussed above, the scalar sector is
under complete control in the one-loop approximation, but the indefinitely
growing gauge coupling ruins the one-loop approximation in the UV. Additional
flows which are under control in the IR are coloured cyan in figure 1.

For full IR consistency, the effective potential corresponding to the IR limit
of the theory should be stable. We have not performed this analysis, but note
that an unstable potential in the IR is likely to correspond to a Coleman-
Weinberg symmetry breakdown Coleman:1973jx . Many of the fixed flow couplings
are actually positive and multiply positive semi-definite polynomials in
$\Phi_{a}$, suggesting that at least some of the fixed flows represent stable
potentials giving stable symmetric vacua. Such fixed flows then represent
fully consistent symmetric theories in the IR.

##  6 Conclusion

###  6.1 Discussion

We have found the one-loop RG flow to be of limited validity. Only for very
exceptional trajectories can the one-loop approximation be valid
asymptotically in the infrared, and no trajectories are under control
asymptotically in the ultraviolet.

A fixed point of all couplings could have connected multiscalar adjoint gauge
theory to a bosonic D3-brane model of potential use in a bosonic AdS/CFT
correspondence through its large $N$ limit. At least our naive choice of gauge
theory seems to fail. First, the initially promising coincidence of the
critical dimension 26 for the bosonic string and the vanishing of the one-loop
gauge coupling beta function in $D=4$ for $M=22=26-4$555On 3-branes in 26
dimensions live four-dimensional field theories with 22 scalar multiplets
corresponding to fluctuations in 22 transverse dimensions. does not survive
the corrections from interactions: The one-loop beta functions of the scalar
self-interactions have no zero for $M\leq 405$. Second, one might have thought
that the gauge theory D brane model in $M$ transverse dimensions could have
had a chance to describe something similar to supersymmetric D branes in the
large $M$ limit. We see no such signs. Allowing for running couplings, the
one-loop flows of couplings seem to favour D brane interactions that are very
different from the tree level interactions that govern supersymmetric D
branes. There is no sign that the single trace flows derived from (32) and
displayed in figure 1(a) retain any similarity to the tree level coupling over
any reasonable range of scales. If these gauge theory models are related to D
branes, the relation is different, or D brane physics is different from what
we have assumed here. This may not come as a surprise, since we have neglected
the tachyon instability that is present in a full string theory description of
bosonic D branes, but the mismatch with the one loop RG evolution provides a
another perspective on bosonic D branes.

Conformal invariance is ruined in our case, just as in the orbifolded gauge
theories discussed in the introduction, unless we consider $M=22$, and accept
complex couplings at the fixed points, ie non-unitarity complex CFTs. A new
phenomenon is that not even single trace fixed point couplings are protected
from being complex in our models. A basic question raised in Dymarsky:2005uh ;
Pomoni:2009joh  however remains. Conformal symmetry, or unitarity, is broken
in all non-supersymmetric large $N$ conformal field theories with adjoint
scalars that have been studied. Is a principle at work here, or have we just
not found the right example yet?

Methodologically, we have introduced a non-associative algebra representation
of the one loop renormalisation group. The results of this paper were computed
in terms of the algebra but also checked both manually and with the RGBeta
Mathematica package to standard calculations from general expressions
Machacek:1984zw .

###  6.2 Outlook

A more general validity of perturbative RG trajectories would require higher
order corrections, but corrections could also reinforce the limitations of
perturbation theory. It would be interesting to find out the fate of the
renormalisation group phase portraits.

For the D3-brane interpretation of the adjoint scalar gauge theories, this
study points to an impasse. It seems unlikely that higher corrections balance
the one loop terms to yield a scalar self interaction that resembles the tree
level interaction in a substantial part of the RG flow. A study of the
effective potential rather than the four point coupling is however more
relevant for the infrared flow, and could evade our negative conclusions.
Another intriguing avenue would be to study the complex one loop fixed points
for $M=22$ and regard them as a starting point for non-unitary bosonic
D-branes and complex holography Faedo:2019nxw  in a complex version of
AdS/CFT.

We have limited the present study to the RG evolution of the couplings, but a
stability analysis of the potentials at fixed points is definitely warranted
for all $M$. Even if the the RG trajectories are unstable in the infrared,
vacuum stability can be investigated. Again, the effective potential is
important. RG trajectories leading to instability of the naive vacuum should
correspond to the instability of the effective potential to radiative symmetry
breakdown in the Coleman-Weinberg mechanism Coleman:1973jx , but it would be
interesting to see this in detail.

Finally, we are curious about the prospects of non-associative algebras
illuminating the RG group in other systems, as well as to higher orders.
Generalising our approach to other systems seems straightforward, since we
merely rephrased well established equations in an algebraic language. The
question is more if the algebraic formulation, or its application to dynamical
systems leads to new insights. Perhaps, special quantum field theories instead
illuminates mathematics. Generalising the non-associative algebras to higher
orders seems much more challenging, but also potentially more rewarding.

###### Acknowledgements.

We wish to thank Igor Klebanov, Fedor Popov, Francesco Sannino, Anders Eller
Thomsen and Arkady Tseytlin for helpful correspondence. The work of B.S. is
supported by the Swedish research council VR, contract DNR-2018-03803.

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#  Kaonic lead feasibility measurement at DA$\Phi$NE to solve the charged kaon
mass discrepancy

D. Bosnar<EMAIL_ADDRESS>L. Abbene  C. Amsler  F. Artibani  M. Bazzi  M.
Bragadireanu  A. Buttacavoli  M. Cargnelli  M. Carminati  A. Clozza  F. Clozza
G. Deda  L. De Paolis  R. Del Grande  K. Dulski  L. Fabbietti  C. Fiorini  I.
Friščić  C. Guaraldo  M. Iliescu  M. Iwasaki  A. Khreptak  M. Makek  S. Manti
J. Marton  P. Moskal  F. Napolitano  S. Niedźwiecki  H. Ohnishi  K. Piscicchia
A. Scordo  F. Sgaramella  D. Sirghi  F. Sirghi  M. Skurzok  M. Silarski  A.
Spallone  K. Toho  M. Tüchler  O. Vasquez Doce  J. Zmeskal  C. Curceanu

###### Abstract

An HPGe detector equipped with a transistor reset preamplifier and readout
with a CAEN DT5781 fast pulse digitizer was employed in the measurement of
X-rays from kaonic lead at the DA$\Phi$NE $e^{+}e^{-}$ collider at the
Laboratori Nazionali di Frascati of INFN. A thin scintillator in front of a
lead target was used to select kaons impinging on it and to form the trigger
for the HPGe detector. We present the results of the kaonic lead feasibility
measurement, where we show that the resolution of the HPGe detector in regular
beam conditions remains the same as that without the beam and that a
satisfactory background reduction can be achieved. This measurement serves as
a test bed for future dedicated kaonic X-rays measurements for the more
precise determination of the charged kaon mass.

###### keywords:

Charged kaon mass , Kaonic atoms , HPGe detector , Fast pulse digitizer

††journal: Nucl. Instr. Meth. Phys. Res. A

[a1]organization=Department of Physics, Faculty of Science, University of
Zagreb,addressline=Bijenička c.32, city=Zagreb, postcode=10000,
country=Croatia

[a2]organization=Dipartimento di Fisica e Chimica - Emilio Segrè, Università
di Palermo, addressline=Viale Delle Scienze, Edificio 18, city=Palermo,
postcode=90128, country=Italy

[a3]organization=Stefan-Meyer-Institut für Subatomare
Physik,addressline=Kegelgasse 27, city=Vienna, postcode=1030, country=Austria

[a4]organization=Laboratori Nazionali di Frascati INFN,addressline=Via E.
Fermi 54, city=Frascati, postcode=00044, country=Italy

[a5]organization=Horia Hulubei National Institute of Physics and Nuclear
Engineering (IFIN-HH),addressline=Reactorului 30, city=Magurele,
postcode=077125, country=Romania

[a6]organization=Politecnico di Milano, Dipartimento di Elettronica,
Informazione e Bioingegneria and INFN Sezione di Milano, addressline=Via
Giuseppe Ponzio 34, city=Milano, postcode=20133, country=Italy

[a7]organization=Excellence Cluster Universe, Technische Universität München
Garching,addressline=Boltzmann str. 2, city=Garching, postcode=85748,
country=Germany

[a8]organization=RIKEN,addressline=2-1 Hirosawa, Wako, Saitama, city=Tokyo,
postcode=351-0198, country=Japan

[a9]organization=Faculty of Physics, Astronomy, and Applied Computer Science,
Jagiellonian University,addressline=Łojasiewicza 11, city=Krakow,
postcode=30-348, country=Poland

[a10]organization=Centre for Theranostics, Jagiellonian University
,addressline=Kopernika 40, city=Krakow, postcode=31-501, country=Poland

[a11]organization=Research Center for Electron Photon Science (ELPH), Tohoku
University,,addressline=1-2-1 Mikamine,Taihaku-ku, city=Sendai,
postcode=982-0826, country=Japan

[a12]organization=Centro Ricerche Enrico Fermi - Museo Storico della Fisica e
Centro Studi e Ricerche “Enrico Fermi”,addressline=Via Panisperna 89A,
city=Roma, postcode=00184, country=Italy

##  1 Introduction

The current value of the charged kaon mass, $m_{K}=493.677\pm 0.016$
MeV/$c^{2}$, was determined as a weighted average of six measurements with
very different uncertainties [1]. The two most precise measurements [2, 3],
which largely determine this value, differ by 60 keV and have uncertainties of
approximately 10 keV. This led to a large scaling factor on the error and,
consequently, the accuracy on the charged kaon mass (32 p.p.m.) is one order
of magnitude worse than the accuracy on the charged pion mass (1.3 p.p.m.).

The value of the charged kaon mass is important in non-perturbative QCD where
its uncertainty has a large influence on the K-N scattering lengths, and
consequently on the kaon-nucleon sigma terms, which reflect the degree of
chiral symmetry breaking [4]. The accuracy of the charged kaon mass also
determines the systematic uncertainty of the $D^{0}$-meson mass [5] which, in
turn, has an influence on the determination of the masses of the excited charm
mesons [1] and mixing parameters in the $D^{0}$-$\overline{D}^{0}$ system [5].
It is therefore imperative to resolve the charged kaon mass discrepancy and it
is sufficient to have the same precision as the previous two most precise
measurements to substantially improve its accuracy [6].

One of the most precise determinations of the charged kaon mass can be
obtained by using the measurements of the X-rays emitted from kaonic atoms [1,
2, 3]. In this approach, one electron in the atom is replaced by a negative
kaon and a kaonic hydrogen-like atom is formed. The kaon cascades to lower
states of the kaonic atom by emitting X-rays and is eventually absorbed by the
nucleus. By measuring the energies of the emitted X-rays and comparing them
with the theoretically calculated values for some chosen kaon mass, which is
varied until agreement is obtained, the charged kaon mass can be determined.
It is important to choose X-rays from the cascading spectrum for which the
influence of strong interaction between the nucleus and the kaon is totally
negligible, and the screening of the inner electrons does not contribute
significantly. One also includes important corrections such as the finite
nuclear-charge distribution, vacuum polarization, electron screening and
parallel non-circular transitions, as well as higher-order relativistic
corrections and static polarization of the orbiting particle and the nucleus,
see e.g. [2].

In this context we measured the X-rays from kaonic lead with an HPGe detector
at the DA$\Phi$NE e+e- collider at the Laboratori Nazionali di Frascati (INFN-
LNF) to study the feasibility of X-ray measurements from targets suitable for
the determination of the charged kaon mass. This was performed in parallel
with SIDDHARTA-2 measurements, which investigate X-rays from light kaonic
atoms [7].

DA$\Phi$NE provides low momenta kaons and, in contrast to previous
measurements which mostly used energetic kaons, a degrader is not needed to
slow down the kaons, and there are no secondary hadronic particles in the
beam. However, there is background caused by e+ and e- beam losses and their
interactions with the surroundings, as well as the background originating from
the kaons absorbed by the nuclei in the lead target. The latter background
type cannot be eliminated, but according to GEANT4 simulations, its
contribution can be neglected for our purpose.

Since there are no detailed simulations of e+ and e- beam losses, the
background they cause can only be determined by measurements in the DA$\Phi$NE
hall in real beam conditions. The aim of this study is to experimentally
assess the impact of the event rate seen by the HPGe detector on its energy
resolution, and to investigate the possibilities to reject the background
coming from the e+ and e- beams to further reduce the total background.

To process the high rates we used an HPGe detector with a transistor reset
preamplifier (TRP) whose signals were fed into one of the four inputs of a
CAEN DT5781 fast pulse digitizer unit. To reject the background from the e+
and e- beams we employed a thin scintillation detector in front of the lead
target for the detection of kaons impinging on it. In addition, the transit
times of the kaons and the minimum ionizing particles (MIPs) from the e+e-
interaction point to the scintillation detector were also measured and used
for the reduction of background.

The results show that the resolution of the HPGe detector in the beam
condition remains the same as that with the beam off. The number of collected
events and the observed signal-to-background ratio for X-rays from kaonic lead
also show that a precise determination of the charged kaon mass is feasible in
a reasonable time.

In the next section we describe the HPGe setup at DA$\Phi$NE. The third
section presents the results of the measurement of the X-rays from kaonic lead
and we conclude with a discussion of the results and an outlook.

##  2 The HPGe setup at DA$\Phi$NE

The HPGe setup was installed together with the SIDDHARTA-2 setup which
measures X-rays from kaonic transitions in gaseous targets with SDD detectors,
and whose primary goal is to measure X-rays from kaonic deuterium at
DA$\Phi$NE [8].

Kaons originate from the decays of $\phi$ mesons produced in e+ and e-
annihilations at the interaction point (IP) of the DA$\Phi$NE e+e- collider
[9]. The momenta of kaons are slightly higher toward the inner (boost) than
toward the outer (anti-boost) side of the DA$\Phi$NE ring. Their mean momenta
are 141 MeV/c and 113 MeV/c, respectively.

The target and SDD detectors of the SIDDHARTA-2 setup are placed above the
e+e- IP, Figure 1, left. Two plastic scintillators (80 x 40 x 2 mm3 Saint-
Gobain Crystals BC-408), SC1 and SC2, which serve as a luminosity monitor in
the SIDDHARTA-2 measurements [10], are placed on opposite sides of the IP in
the e+e- beam plane, with the long side parallel to the beams. The signals
from each scintillator are read by fast R4998 Hamamatsu PMTs coupled to 6 cm
fish-tail plastic light guides on both 40 mm sides of the scintillator.

A lead target of the same size as the scintillation detector SC1 and with a
thickness of 1.5 mm was placed immediately behind. Its distance from the IP
was 78 mm and it completely stopped the entering kaons. They were selected by
applying a threshold on the signals from SC1. To enhance their identification
and reduce the background, the trigger signal for the HPGe detector was formed
as a coincidence between SC1, SC2 and the DA$\Phi$NE radio-frequency (RF)
signal ($\simeq$368 MHz) divided by 4, using a CAEN N455 coincidence unit
module. The usage of the RF/4 signal is a consequence of the hardware
limitation, given that the constant fraction discriminator units (Ortec 935
CFD) are restricted to working with signals below 200 MHz.

The HPGe detector is a p-type detector produced by Baltic Scientific
Instruments. The active part is a cylinder with a base diameter of 59.8 mm and
a height of 59.3 mm. The front side of the cylinder could be positioned at a
minimal distance of 155 mm from the lead target, a constraint posed by the
geometry of the SIDDHARTA-2 setup, and the measurement was performed at this
position. The active part of the HPGe detector was shielded by 5 cm thick lead
bricks, Figure 1, right. The front brick (not shown in the figure) was 2.5 cm
thick with a circular hole of 4 cm in diameter in the center, for the X-rays
to reach the active part of the detector from the lead target.

Figure 1: The location of the HPGe detector in the SIDDHARTA-2 setup, left: A)
the HPGe detector, B) the SIDDHARTA-2 setup, C) beam pipes, D) shielding.
Details of the HPGe setup, right: a) beam pipe, b) scintillation detector SC1,
c) lead target, d) active part of the HPGe detector, e) lead shielding with
holder.

A schematic representation of the HPGe detector system and electronics for the
data acquisition is shown in Figure 2.

Signals from the TRP were fed into a CAEN DT5781 fast pulse digitizer through
the CAEN AC coupler A386. The DT5781 unit has 4 signal inputs with 100 MS/s
14-bit waveform digitizers and is capable of selecting coincidence events
which come in the selected inputs in the predetermined coincidence time
window. In the measurement of kaonic lead X-rays, signals from the TRP were
put in coincidence with the trigger signals, which opened a broad coincidence
gate of 2.5 $\mu$s.

Since it is not possible to entirely discriminate kaons from MIPs with a
threshold on the scintillator signal height, their transit times from the IP
to the scintillation detector SC1 were measured as well. An ORTEC 566 time-to-
amplitude-converter (TAC) was used for this purpose. The start signal for the
TAC was given by the trigger signal. As the stop signal, a signal produced by
a CAEN N93B dual mean timer unit was used, into which discriminated signals
(by ORTEC 935 CFDs) from both sides of the scintillation detector SC1 were
fed. The output signals from the TAC were put into the DT5781 unit (by using
the CAEN A386 AC coupler) and were taken in coincidences with the
corresponding events in the HPGe. These transit times were used in the off-
line analysis to additionally discriminate between the kaons and the MIPs.

The CAEN MC2 Analyzer software was used for data acquisition, on-line analysis
and control. In particular, it was used for the reconstruction of signal
amplitudes in the HPGe detector with the trapezoid algorithm [11] tuned for
high rates, and for the determination of the signal occurrence time, which can
be done with a resolution of 10 ns. These data were stored in binary format
for off-line analysis, which was performed using the CERN Root software [12].

Figure 2: Schematics of the HPGe detector system (not to scale, dimensions are
given in the text) and electronics for the data acquisition.

##  3 Measurement of kaonic lead X-rays with the HPGe detector at DA$\Phi$NE

The measurement was performed during the SIDDHARTA-2 experiment in June 2023.
The total integrated luminosity measured with the SIDDHARTA-2 luminosity
monitor was 39.4 pb-1.

The calibration of the energy scale of the digitizer was done by using a 1
$\mu$Ci 133Ba source. This was also used to determine and monitor the energy
resolution of the detector during the whole period of the measurement.

Figure 3: The spectrum of $\gamma$-rays from the 133Ba source with beams off. The inset shows the peak at 302.9 keV fitted with a Gaussian and a linear function for the background estimation. The energy resolution is 4.39 $\pm$ 0.02 keV (FWHM). Table 1: The 133Ba lines in terms of peak, positions and energy resolutions obtained from a fit of the energy-calibrated spectrum, and residuals (R). 133Ba line | Peak position | Resolution (FWHM) | R (10-4)
---|---|---|---
(keV) | (keV) | (keV) |
80.9971 | 80.994 $\pm$ 0.006 | 3.74$\pm$ 0.01 | 0.38
276.398 | 276.41 $\pm$ 0.01 | 4.30 $\pm$ 0.02 | -0.36
302.853 | 302.868 $\pm$ 0.008 | 4.39 $\pm$ 0.02 | -0.49
356.017 | 355.893 $\pm$ 0.005 | 4.582 $\pm$ 0.007 | 3.48
383.851 | 383.76 $\pm$ 0.01 | 4.70 $\pm$ 0.03 | 2.42

A spectrum of 133Ba taken in the DA$\Phi$NE hall with beams off at the
beginning of the measurement and in the same detector configuration as for
kaonic lead X-rays is shown in Figure 3. The energy resolution at 302.9 keV
was 4.39 $\pm$ 0.02 keV (FWHM), the inset of Figure 3, and it remained stable
during the measurement. This line is of special interest since it is the
closest to the ($9\rightarrow 8$) transition line in kaonic lead. The
positions and energy resolutions of the other peaks from 133Ba obtained from a
fit of the energy-calibrated spectrum, as well as their relative deviations
with respect to the nominal values (residuals, R), are shown in Table 1.

In the measurement of kaonic lead X-rays, signals from the HPGe were taken in
coincidence with signals from the scintillation detector SC1 and with the
corresponding signals from the TAC by using the DT5781 unit. The spectrum of
events seen by the HPGe detector in coincidence with the scintillation
detector SC1 within a broad coincidence gate of 2.5 $\mu s$ is shown in Figure
4, left.

Figure 4: The energy spectrum seen by the HPGe detector with a 2.5 $\mu$s
coincidence gate, left, and the spectrum with a cut of 400 ns on the time
difference between signals in the scintillation detector SC1 and HPGe, right.

In the off-line analysis, a narrower cut of 400 ns on the time difference
between signals in the scintillator and HPGe determined by the digitizer was
imposed. An example of this time correlation is shown in Figure 5 while the
spectrum obtained with this cut is shown in Figure 4, right.

Figure 5: Time correlation between the signals in the scintillation detector
SC1 and HPGe, determined by the digitizer (time resolution of 10 ns).

A further reduction of the background was obtained by using the transit times
of the kaons and MIPs from the IP to the scintillation detector SC1, measured
by the TAC and expressed in channels of the digitizer (Figure 6). The transit
time difference between the MIPs and kaons is approximately 2.7 ns. The 8 peak
structures, 4 peaks from the kaons and 4 from the MIPs, are a consequence of
the usage of the RF/4 signal [10, 13].

Figure 6: Spectrum of the transit times (expressed in channels of the
digitizer) of the kaons and MIPs from the IP to the scintillation detector
SC1.

Figure 7 shows the final energy spectrum seen by the HPGe detector after
applying all the cuts.

Figure 7: The spectrum seen by the HPGe detector after applying all cuts. The
inset shows the peak at 292.47 $\pm$ 0.17 keV with a fit done by a Gaussian
and a linear function for the background, the energy resolution is 3.97 $\pm$
0.49 keV (FWHM).

Clear peaks are visible at 208.92 $\pm$ 0.17 keV, 292.47 $\pm$ 0.17 keV and
427.07 $\pm$ 0.24 keV which come from the (10$\rightarrow$9),
(9$\rightarrow$8) and (8$\rightarrow$7) transitions in the kaonic lead,
respectively. The peak at 154.2 $\pm$ 1.2 keV, from the (11$\rightarrow$10)
transition, is less pronounced.

Besides the peak at 511 keV which comes from positron annihilation, there are
peaks which come from the transitions in ordinary lead with which the
sensitive part of the HPGe detector was shielded. Expected peaks are at 72.80
keV and 74.96 keV, which are not resolved due to the resolution of the
detector, and the peak at 84.94 keV.

There are also visible peaks at 356.0 keV and 81.0 keV which come from the
133Ba source, which was in front of the sensitive part of the HPGe detector
during the measurement. The other peaks from 133Ba are less pronounced in this
spectrum and are more visible in the spectrum shown in Figure 4, left, where
no cuts were applied.

##  4 Discussion and outlook

The inset of Figure 7 shows the peak at 292.47 $\pm$ 0.17 keV from
(9$\rightarrow$8) transitions in kaonic lead. The energy resolution is 3.97
$\pm$ 0.49 keV. Compared with the resolution of 4.39 $\pm$ 0.02 keV at 302.9
keV of 133Ba in the measurement with the beams off, Table 1, it can be
concluded that there is no worsening of the resolution in measurements in the
full beam conditions at DA$\Phi$NE.

This feasibility measurement was performed with the HPGe detector whose energy
resolution at the 302.9 keV 133Ba line was 4.39 keV in the configuration with
the DT5781 fast pulse digitizer. This is worse than the standard resolution
(approximately 1.3 keV, FWHM) of such detectors at this energy and in the same
configuration.

Assuming the standard resolution of the HPGe detector, we made an estimation
of the required number of events in the (9$\rightarrow$8) transition peak to
reach the 10 keV accuracy on the charged kaon mass. In a simple approach, we
used the Rydberg formula for kaonic lead atoms, with the kaon-nucleus reduced
mass, to obtain the relation between the precision of the transition and the
accuracy of the charged kaon mass. To reach the 10 keV accuracy, the required
precision of the (9$\rightarrow 8$) transition needs to be approximately 6 eV.
This estimation is also confirmed by the preliminary results of a more
sophisticated calculation based on the self-consistent multiconfiguration
Dirac-Fock method [14, 15], which is in progress. Approximately 8500 events in
the (9 $\rightarrow$ 8) transition peak are needed to reach the 10 keV
accuracy on the charged kaon mass, assuming a detector with a resolution of
1.3 keV at 292 keV.

The total number of events in the (9 $\rightarrow$ 8) peak in our measurement
is 770 $\pm$ 65\. This implies that a total integrated luminosity of 435 pb-1
is needed to achieve the required accuracy by using only the (9 $\rightarrow$
8) transition in kaonic lead.

Table 2: Summary of the clearly visible X-rays from kaonic lead. K--Pb transition | Peak position | Resolution (FWHM) | Number of events
---|---|---|---
| (keV) | (keV) |
10 $\rightarrow$ 9 | 208.92 $\pm$ 0.17 | 3.68 $\pm$ 0.42 | 584 $\pm$ 30
9 $\rightarrow$ 8 | 292.47 $\pm$ 0.17 | 3.97 $\pm$ 0.49 | 770 $\pm$ 65
8 $\rightarrow$ 7 | 427.07 $\pm$ 0.24 | 4.37 $\pm$ 0.54 | 457 $\pm$ 45

However, there are clearly visible peaks at 208.92 keV and 427.07 keV with 584
$\pm$ 30 and 457 $\pm$ 45 events in the peak, respectively (Table 2), which
can also be used to determine the charged kaon mass more precisely.

##  5 Acknowledgements

We thank C. Capoccia from LNF-INFN for his contribution in designing and
building the holder for the HPGe detector and the holder for the lead target.
We also thank the DA$\Phi$NE staff for the excellent working conditions
provided and for their support.

Part of this work was supported by the Croatian Science Foundation under
project IP-2022-10-3878; the EU STRONG-2020 project (grant agreement No.
(824093); the Austrian Science Fund (FWF): [P24756-N20 and P33037-N]; the EU
Horizon 2020 project under the MSCA G.A. 754496; Ministero degli Affari Esteri
e dela Cooperazione Internazionle, EXOTICA project, PO21MO03; the Polish
Ministry of Science and Higher Education grant No. 7150/E-338/M/2018; the
Polish National Agency for Academic Exchange (grant No. PPN/BIT/2021/1/00037);
the SciMat and qLife Priority Research Areas budget under the program
Excellence Initiative - Research University at the Jagiellonian University.
This publication was made possible through the support of Grant 62099 from the
John Templeton Foundation. The opinions expressed in this publication are
those of the authors and do not necessarily reflect the views of the John
Templeton Foundation.

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