#  Quadrupole instability of static scalarized black holes

Burkhard Kleihaus  Institut für Physik, Universität Oldenburg, Postfach 2503,
D-26111 Oldenburg, Germany  Jutta Kunz  Institut für Physik, Universität
Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany  Tim Utermöhlen  Institut
für Physik, Universität Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
Emanuele Berti  Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, MD 21218 USA

###### Abstract

The addition of a Ricci coupling to Einstein-scalar-Gauss-Bonnet theories
makes general relativity a cosmological attractor. Previous work considered a
quadratic coupling function with two independent coupling constants in such
theories and showed that static, spherically symmetric, spontaneously
scalarized black holes are radially stable beyond a critical value of the
Ricci coupling constant. Here we demonstrate that these black holes are
affected by a quadrupole instability which leads to two new branches of
static, axially symmetric scalarized black holes. We discuss the properties of
these solutions and provide embedding diagrams.

###### pacs:

04.50.-h, 04.70.Bw, 97.60.Jd

_Introduction._ In general relativity (GR), the no-hair theorems highly
restrict the allowed black hole (BH) solutions and their properties Chrusciel
_et al._ (2012); Herdeiro and Radu (2015). Real scalar fields, for instance,
cannot lead to scalar hair. The situation is different in generalized theories
of gravity intended to amend various shortcomings of GR (see, e.g., Berti  _et
al._ (2015); Saridakis  _et al._ (2021)).

A particularly attractive and well-studied class of theories contains a Gauss-
Bonnet (GB) quadratic term coupled to a real scalar field. Such Einstein-
scalar-Gauss-Bonnet (EsGB) theories lead to second order equations of motion
and do not feature ghosts. In the low-energy limit of string theory, the
scalar field corresponds to a dilaton Gross and Sloan (1987); Metsaev and
Tseytlin (1987). In these Einstein-dilaton-Gauss-Bonnet theories BHs always
carry scalar hair Kanti  _et al._ (1996); Kleihaus  _et al._ (2011), and the
vacuum BHs of GR are no longer solutions.

However, EsGB theories allowing for different (non-dilatonic) coupling
functions $f(\phi)$ of the scalar field to the GB term admit GR BHs as special
solutions of the field equations. For example, theories with coupling
functions quadratic in $\phi$ feature BH solutions with scalar hair that
reduce to GR solutions for small couplings. These “spontaneously scalarized”
BH solutions arise from a tachyonic instability of the GR BHs, due either to
strong curvatures or to large spins Doneva and Yazadjiev (2018); Silva  _et
al._ (2018); Antoniou  _et al._ (2018a, b); Cunha  _et al._ (2019); Collodel
_et al._ (2020); Dima  _et al._ (2020); Hod (2020); Herdeiro  _et al._ (2021);
Berti  _et al._ (2021).

Static, spherically symmetric, spontaneously scalarized BHs “branch off” the
Schwarzschild BH solutions below a critical value of the mass (for fixed
coupling constant), where the tachyonic instability produces a zero mode of
the Schwarzschild BHs. For masses lower than this critical value, the
Schwarzschild BHs possess an unstable radial mode Blázquez-Salcedo  _et al._
(2018); Silva  _et al._ (2019); Macedo  _et al._ (2019) and a new branch of
scalarized BHs (the “fundamental branch”) emerges. The stability of scalarized
BHs under radial perturbations depends on the specific choice of the coupling
function and of the scalar field potential Blázquez-Salcedo  _et al._ (2018);
Silva  _et al._ (2019); Macedo  _et al._ (2019).

As shown in Refs. Blázquez-Salcedo  _et al._ (2018, 2020a, 2020b), linear mode
stability applies to (most of) the fundamental scalarized branch of the
extended scalar-tensor GB theories considered in Ref. Doneva and Yazadjiev
(2018). In contrast, for a purely quadratic coupling function,
$f(\phi)=\phi^{2}/2$, the fundamental scalarized branch is radially unstable
everywhere Blázquez-Salcedo  _et al._ (2018). The fundamental branch can be
made partially (radially) stable by including higher order terms or a
potential. Then the radial instability sets in at a minimum of the scalarized
BH mass Silva  _et al._ (2019); Macedo  _et al._ (2019).

Recently, a partially radially stable fundamental branch of scalarized BHs was
found in another interesting scenario Antoniou  _et al._ (2021a, b, 2022):
Einstein-scalar-Gauss-Bonnet-Ricci (EsGBR) theories, which include also a term
coupling the scalar field with the Ricci scalar via a quadratic coupling
function (but with a different coupling constant). EsGBR theories are well
motivated from a cosmological point of view, since they allow GR to be a
cosmological attractor Antoniou  _et al._ (2021a): no fine-tuning of the
scalar field in the early Universe is needed in order to have a vanishing
scalar field at late times.

Here we revisit scalarized BH solutions for the EsGBR action Antoniou  _et
al._ (2021a, b, 2022)

$\displaystyle{\cal S}=\frac{1}{16\pi}$ $\displaystyle\int$
$\displaystyle\mathrm{d}^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\partial_{\mu}\phi)^{2}\right.$
(1) $\displaystyle\left.-\frac{\phi^{2}}{2}\left(\frac{\beta}{2}R-\alpha
R_{\mathrm{GB}}^{2}\right)\right]$

with a real scalar field $\phi$, coupling constants $\alpha$ and $\beta$, and
GB invariant $R^{2}_{\rm
GB}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$ .

We show that scalarized BHs in EsGBR theories have an intriguing new feature:
radially stable, static, spherically symmetric BHs on the fundamental
scalarized branch develop a quadrupole instability below a critical value of
the BH mass. At this critical value of the mass the spherically symmetric
solutions possess a zero mode from where two branches of static, but only
axially symmetric, BH solutions arise. Therefore, stability under radial
perturbations may not necessarily imply linear mode stability. We present the
domain of existence of the new branches of BH solutions and study their
physical properties.

_General framework._ Variation of the action (1) with respect to the metric
yields the generalized Einstein equations

$\displaystyle
E_{\mu\nu}=G_{\mu\nu}-\frac{1}{2}T_{\mu\nu}^{\mathrm{(eff)}}=0\,,$ (2)

where the effective stress-energy tensor

$\displaystyle T_{\mu\nu}^{\mathrm{(eff)}}=T_{\mu\nu}^{(\phi)}-2\alpha
T_{\mu\nu}^{(\mathrm{GB})}+\beta T_{\mu\nu}^{(\mathrm{R})}$ (3)

has contributions from the scalar, GB and Ricci terms, respectively. Variation
with respect to the scalar field yields the generalized Klein-Gordon equation

$\displaystyle\nabla^{2}\phi-\left(\frac{\beta}{2}R-\alpha
R_{\mathrm{GB}}^{2}\right)\phi=0\,.$ (4)

The latter features the effective mass

$m^{2}_{\rm eff}=\frac{\beta}{2}R-\alpha R^{2}_{\rm GB}\,,$ (5)

which allows for spontaneous scalarization and for the attractive cosmological
features of the model.

We find solutions corresponding to static, axially symmetric spacetimes by
imposing the ansatz Kleihaus and Kunz (1997, 2001)

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
be^{F_{0}}dt^{2}+e^{F_{1}}\left(dr^{2}+r^{2}d\theta^{2}\right)$ (6)
$\displaystyle+$ $\displaystyle e^{F_{2}}r^{2}\sin^{2}\theta d\varphi^{2}\,,$

with “quasi-isotropic" radial coordinate $r$, auxiliary function
$b=\left(1-\frac{r_{\rm H}}{r}\right)^{2}$ (here $r_{\rm H}$ is the isotropic
horizon radius), and three unknown metric functions $F_{0}$, $F_{1}$, $F_{2}$.
These metric functions $F_{i}$ $(i=0,\,1,\,2$) and the scalar field $\phi$
depend only on the coordinates $r$ and $\theta$.

Based on considerations of symmetry, regularity and asymptotic flatness of the
solutions, we impose the following set of boundary conditions at spatial
infinity, at the horizon and on the symmetry axis, respectively:
$F_{i}(\infty)=0\quad(i=0,\,1,\,2),\,\quad\phi(\infty)=0$;
$\partial_{r}F_{0}(r_{\rm H})=1/r_{\rm H}$, $\partial_{r}F_{1}(r_{\rm
H})=-2/r_{\rm H}$, $\partial_{r}F_{2}(r_{\rm H})=-2/r_{\rm H}$,
$\partial_{r}\phi(r_{\rm H})=0$; and
$\partial_{\theta}F_{i}|_{\theta={0,\pi}}=0$ $(i=0,\,1,\,2)$,
$\partial_{\theta}\phi|_{\theta={0,\pi}}=0$.

Assuming the scalar field to be even under parity, a series expansion at
infinity leads to $\phi=Q/r+\dots$, where $Q$ represents the scalar charge.
The BH mass $M$ can be found from the metric function $g_{tt}=-1+2M/r+\dots$.
Since all functions are even, we can limit calculations to one quadrant only
and impose the boundary conditions $\partial_{\theta}F_{i}|_{\theta=\pi/2}=0$
$(i=0,\,1,\,2)$, $\partial_{\theta}\phi|_{\theta=\pi/2}=0$ on the equatorial
plane.

The horizon metric determines the Hawking temperature Wald (1984)

$\displaystyle T_{\rm H}=\frac{1}{2\pi r_{\rm H}}e^{(F_{0}-F_{1})/2}$ (7)

and the horizon area

$\displaystyle A_{\rm H}=2\pi r_{\rm H}^{2}\int_{0}^{\pi}d\theta\sin\theta
e^{(F_{1}+F_{2})/2}\,.$ (8)

The entropy, however, differs from the GR result, where it is simply a quarter
of the horizon area Wald (1984). For EsGBR BHs there are additional
contributions Wald (1993), and the entropy is found as the following integral
over the spatial cross section of the horizon:

$\displaystyle S=\frac{1}{4}\int_{\Sigma_{\rm
H}}d^{2}x\sqrt{h}\left[1-\frac{\phi^{2}}{2}\left(\frac{\beta}{2}-2\alpha\tilde{R}\right)\right]\,,$
(9)

where $h$ is the determinant of the induced metric on the horizon and
$\tilde{R}$ is the corresponding scalar curvature.

_Numerical Results._ The numerical approach we employed previously for EsGB
BHs Collodel  _et al._ (2020); Berti  _et al._ (2021) is based on the finite
difference solver FIDISOL/CADSOL W. Schönauer and R. Weiß (1989); M. Schauder,
R. Weiß, and W. Schönauer . For EsGBR BHs, we supplemented this solver by a
spectral solver. In both cases the unknown functions
$(F_{0},F_{1},F_{2},\phi)$ are obtained for given values of the coupling
constants $\alpha$ and $\beta$ and of the horizon radius $r_{\rm H}$ by
solving the chosen set of partial differential equations (PDEs) subject to the
prescribed boundary conditions. In particular, we employ the linear
combinations of the Einstein equations
$E_{t}^{t},E_{\varphi}^{\varphi},E_{\varphi}^{t}$ and
$E_{r}^{r}+E_{\theta}^{\theta}$ for the PDEs yielding the metric functions,
and we treat the equations involving $E_{r}^{r}-E_{\theta}^{\theta}$ and
$E_{r}^{\theta}$ as constraints. Introducing a compactified radial variable
$x=1-r_{\rm H}/r$, we map the interval $[r_{\rm H},\infty)$ to the finite
interval $[0,1]$. We then discretize the equations on a nonequidistant grid in
the variables $x$ and $\theta$ whose range is $0\leq x\leq 1$ and
$0\leq\theta\leq\pi/2$. The resulting estimated numerical error is of order
$10^{-3}$ or less.

Figure 1:  Branches of scalarized BHs for $\beta=2$ and 5: scaled scalar
charge $Q/\sqrt{\alpha}$ vs scaled mass $M/\sqrt{\alpha}$ for the fundamental
spherically symmetric fundamental branches (dotted blue) and the axially
symmetric branches (red) in the insets. The critical bifurcation points
$P_{2}$ are shown by black dots.

We begin our discussion by recalling the fundamental scalarized static
spherically symmetric BH branches Antoniou  _et al._ (2021b, 2022). When
$\beta$ exceeds a critical value of about 1.15, radially stable BHs arise from
the bifurcation point with the Schwarzschild BHs (see e.g. Fig. 2 of Ref.
Antoniou  _et al._ (2022)). Analogous to EsGB BHs Silva  _et al._ (2019);
Macedo  _et al._ (2019), for smaller values of $\beta$ these branches feature
a minimum of the mass, where the radial instability sets in. For larger values
of $\beta$ the branches are radially stable, and they terminate with a
solution such that a certain radicand in the horizon expansion of the scalar
field vanishes Kanti  _et al._ (1996). In Fig. 1 we show the scaled scalar
charge $Q/\sqrt{\alpha}$ versus the scaled mass $M/\sqrt{\alpha}$ for two
values of the coupling, $\beta=2$ and $\beta=5$. These results agree with
those of Ref. Antoniou  _et al._ (2022).

However, unlike previously assumed, radial stability along these branches does
not imply general mode stability. At the locations marked by black dots in
Fig. 1 the scalarized BHs gain a zero mode with respect to quadrupole
deformations, and we find two branches of axially symmetric BH solutions (red
dashed and red dotted lines in the insets).

Figure 2:  Domain of existence of axially symmetric scalarized BHs: scaled
scalar charge $Q/\sqrt{\alpha}$ vs scaled mass $M/\sqrt{\alpha}$ for the
endpoints of the two branches (black, determined by the vanishing of the
respective discriminant) together with the critical points of the fundamental
spherically symmetric branch (blue, determined by the zero mode). The black
star indicates the onset of the quadrupole instability at $\beta=1.3458$. The
numbers correspond to selected values of $\beta$. The inset shows a zoom of
the spherically and axially symmetric branches for $\beta=2$.

Let us denote the bifurcation points of the axial branches by $P_{2}$. The
blue line in Fig. 2 shows these bifurcation points, starting from the critical
value $\beta_{\rm cr}=1.3458$ where they first appear. We note that for
$\beta_{\rm cr}\leq\beta\lesssim 1.52$ the onset of instability occurs on the
radially unstable part of the corresponding spherically symmetric branch,
whereas for $\beta\gtrsim 1.52$ it occurs on the radially stable part. We also
show the endpoints of the two axial branches. These are denoted by
“$\mathrm{dis}=0$,” since in the horizon expansion for the solutions a
discriminant vanishes, and the existence of real BH solutions requires a
positive sign Berti  _et al._ (2021). Therefore the region between the blue
line and the two black lines represents the domain of existence of the two
axial branches for $M/\sqrt{\alpha}>0.95$.

The inset in Fig. 2 is a zoom-in of the axial branches for $\beta=2$. Starting
at the bifurcation with the spherical branch, the upper branch (oblate, red
dashed line) curiously features a small portion where the BH mass decreases,
but then it bends towards larger BH masses. On the contrary, the lower branch
(prolate, red dotted line) immediately moves in the direction of larger BH
masses. This curious feature is lost for larger values of $\beta$.

Figure 3:  Embedding of the horizon of scalarized BHs for $\beta=5$: critical
spherical BH ($\alpha=13.79$), endpoints of upper ($\alpha=14.05$) and lower
($\alpha=13.45$) axial branch for fixed circumferential radius of the horizon.

At the endpoints of the axial branches, their deformation becomes maximal. The
deformation of the two axial branches is illustrated in Fig. 3, where we show
embeddings of the BH horizons at the endpoints of the two axial branches for
$\beta=5$ and compare them with the spherical horizon. Clearly, along the
lower (dotted) axial branch the deformation becomes prolate, while along the
upper (dashed) axial branch it becomes oblate.

Figure 4:  Scaled entropy $S/(4\pi\alpha)$ (left panel) and scaled Hawking
temperature $16\pi T_{H}\sqrt{\alpha}$ (right panel) versus the scaled mass
$M/\sqrt{\alpha}$ for the spherically symmetric fundamental branches (dotted
blue) of scalarized BHs with $\beta=2$ and $\beta=5$. Black dots mark the
critical bifurcation points. The prolate and oblate axially symmetric branches
(red dotted and red dashed lines, respectively) are shown in the insets.

In Fig. 4 we turn to thermodynamic properties, plotting the scaled entropy
$S/(4\pi\alpha)$ (left panel) and the scaled Hawking temperature $16\pi
T_{H}\sqrt{\alpha}$ (right panel) versus the scaled mass $M/\sqrt{\alpha}$ for
the spherical branches with $\beta=2$ and $\beta=5$.

Let us focus first on the left panel. The entropy of the axial branches for
$\beta=5$ (shown in the insets) is very close to the entropy along the
spherical solutions, but a high magnification (bottom right inset) uncovers
small deviations: the prolate lower branch (dotted) has slightly higher
entropy than the spherical branch, while the oblate upper branch (dashed) has
slightly smaller entropy. Therefore the BHs on the prolate branch are, if only
slightly, entropically favored. The closeness of the entropy of the spherical
and axial branches is somewhat surprising, because the scaled horizon area
$A_{\rm H}/\alpha$ of the spherical branch is considerably larger than the
area of the axial branches away from the bifurcation for BHs with the same
scaled mass $M/\sqrt{\alpha}$.

The right panel of Fig. 4 shows that the temperature $T_{\rm H}\sqrt{\alpha}$
of the scalarized BHs is a decreasing function of $M/\sqrt{\alpha}$, as in the
case of Schwarzschild BHs. Both axial branches possess larger $T_{\rm
H}\sqrt{\alpha}$ than the spherical branch. The temperature of the axially
symmetric BHs is larger than the temperature of spherically symmetric BHs on
the radially stable branch, while it is smaller than the temperature of
spherically symmetric BHs on the radially unstable branch (not shown in the
figure). For large enough values of $\beta$ (including the value 2, shown in
the top right inset), there is still a part of the branch of oblate BHs
(dashed red line) where the temperature increases with mass. This feature
seems to disappear for $\beta\gtrsim 3$.

_Conclusions._ We have investigated spontaneously scalarized, static BH
solutions in EsGBR theories with quadratic coupling functions. These
cosmologically motivated theories allow for branches of static spherically
symmetric BHs that are (in part) stable with respect to radial perturbations
Antoniou  _et al._ (2021a, b, 2022). We have found that radial stability does
not, in general, imply perturbative mode stability. In particular, we have
demonstrated the occurrence of a quadrupole instability along the spherical
branches for a wide range of coupling constants.

Two distinct, axially deformed branches of BHs arise starting from a zero mode
of the spherical BHs with respect to quadrupole deformations. The lower branch
is prolate and entropically favored, while the upper branch is oblate and
entropically disfavored relative to the spherical branch, although the
difference in entropy is very small (the differences in horizon area and
temperature are larger). These static axial branches of BHs represent new
counterexamples to Israel’s theorem Israel (1968).

The occurrence of a quadrupole instability might be attributed to the presence
of the Ricci coupling. However, we cannot exclude the possibility that such an
instability may occur in simpler EsGB theories with higher order terms in the
coupling function or with a nonzero potential Silva  _et al._ (2019); Macedo
_et al._ (2019). This is an interesting topic for further study.

We have considered the possibility of a dipole instability of EsGBR BHs,
because Schwarzschild BH instabilities in EsGB theories involve first for the
monopole (i.e., the radial instability) and then the dipole, before the
quadrupole instability occurs Collodel  _et al._ (2020). However, our
numerical investigations did not reveal any dipole instability.

A perturbative study of the mode stability of the new axial branches will be
technically challenging, and their nonlinear evolution is particularly
interesting. Before tackling these difficult questions, it should be possible
to understand various other physical properties of these solutions, such as
their shadow. The axial symmetry of the static spacetimes should lead to
deformations of the shadow, but without distinction with respect to co- and
counter-rotation. A comparison with observations could set constraints on the
coupling constants of the theory Akiyama  _et al._ (2019).

The static axially symmetric branches can be set into rotation, just like the
spherically symmetric branches. This implies interesting overlapping domains
of existence for rotating BHs in EsGBR theories, that will be presented
elsewhere B. Kleihaus, J. Kunz, T. Utermöhlen, and E. Berti .

_Acknowledgments._ B.K. and J.K. gratefully acknowledge support by the DFG
Research Training Group 1620 Models of Gravity and DFG project Ku612/18-1.
E.B. is supported by NSF Grants No. AST-2006538, PHY-2207502, PHY-090003 and
PHY-20043, and NASA Grants No. 19-ATP19-0051, 20-LPS20-0011 and 21-ATP21-0010.
This research project was conducted using computational resources at the
Maryland Advanced Research Computing Center (MARCC). This work has received
funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 690904. This
research project was conducted using computational resources at the Maryland
Advanced Research Computing Center (MARCC).

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# A heuristic study of prime number distribution

Carlos Ros Pérez

###### Abstract

This work consists of a heuristic study on the distribution of prime numbers
in short intervals. We have modelled the occurrence of prime numbers such
intervals as a counting experiment, as a result, we have provided an
experimental validation and an extension of the Montgomery and Soundararajan
conjecture. This is a reduced version of my bachelor’s thesis presented at the
University of Valencia.

##  1 Introduction

“It is evident that the primes are randomly distributed but,unfortunately, we
don’t know what ‘random’ means.”

—R. C. Vaughan

At the end of the eighteenth century, at the age of 15, Gauss empirically
found that the density of primes in the neighborhood of an integer n can be
estimated by $\frac{1}{\log n}$ [1], leading him to conjecture the celebrated
prime number theorem111$f(x)\sim g(x)$ means$f(x)$ is asymptotically
equivalent to $g(x)$,i.e., $f(x)\sim
g(x)\rightarrow\underset{x\to\infty}{\lim}\frac{f(x)}{g(x)}=1$:

$\pi(x)\sim\frac{x}{\log x}$ (1)

This conjecture was demonstrated independently and at the same time by
Hadamard [2] and de la Valée Poussin [3].

In the first half of the twentieth century, Cramér tried a revolutionary
approach consists in considering the prime numbers as random variables,
considering that, for a given a number N, there is a chance of $\frac{1}{\log
N}$ of being prime, therefore a probability of $1-\frac{1}{\log N}$ of being
composed, and the different numbers are treated as independent events. This
kind of approach has many shortcomings, such as considering that the
probability of one number and the following are primes is different from zero,
or provide a non-vanishing probability that an even number is prime.
Nevertheless, it has had many successes, for instance leading to the
asymptotic limit: $\underset{p_{n}\leq x}{\max}(p_{n+1}-p_{n})\sim\log^{2}x$,
known as the Crámer conjecture [4]. That conjecture can be supplemented by the
Hardy–Littlewood Prime k-tuples Conjecture. To understand what it is about, it
is necessary to introduce the concept of k-tuple. We may define a k-tuple
($\wp_{k}$) simply as a set of k distinct non-negative integers. We can define
the following function:

$\mathcal{L}(\wp_{k})=\prod_{p}\frac{p^{k-1}(p-\nu_{\wp_{k}}(p))}{(p-1)^{k}}$
(2)

Where the product is extended over all prime numbers and where
$\nu_{\wp_{k}}(p)$ is the number of distinct residue classes modulo p
represented by elements of $\wp_{k}$ 222 In other words, if we consider the
list formed by all the remainders left by dividing the different numbers in
$\wp_{k}$ by p, the number residue classes modulo p occupied by $\wp_{k}$
would correspond to the number of different elements in the former list. Let
$\wp_{k}=\left\\{h_{1},h_{2},...,h_{k}\right\\}$, we denote as
$\pi(x;\wp_{k})$ the number of positive integers $n\leq x$ such that
$\left\\{n+h_{1},n+h_{2},...,n+h_{k}\right\\}$ are prime numbers. Now we can
enunciate the Hardy–Littlewood k-tuples conjecture [5]:

$\pi(x;\wp_{k})\sim\mathcal{L}(\wp_{k})\frac{x}{\log^{k}x}$ (3)

Now the definition of $\nu_{\wp_{k}}$ makes sense, because if the elements of
$\wp_{k}$ occupy all the residue classes modulo p, resulting in
$\mathcal{L}(\wp_{k})=0$, then at least one number of the list
$\left\\{n+h_{1},n+h_{2},...,n+h_{k}\right\\}$ must be divisible by p, forcing
$\pi(x;\wp_{k})$ to be zero.

In 1975 Gallagher, starting from the Hardy–Littlewood k-tuples Conjecture,
demonstrated the following result [6]:

$\\#\left\\{integers\;x\leq N\;/\;\pi(x+\lambda\log x)-\pi(x)=k\;\right\\}\sim
N\lambda^{k}\frac{e^{-\lambda}}{k!}$ (4)

Which is a remarkable result because it states that the distribution of the
number of integers (x) such that there are k prime numbers between x and
x+$\lambda\log x$ asymptotically tends to the well-known Poisson distribution.
This result can be generalized to assess the distribution of primes in some
intervals. Mongomery and Soundararajan [7] demonstrated333In the article they
work with the second Chebyshev function $\psi(x)$ instead of with $\pi(x)$, we
can relate its moments in the following way:
$M_{k}(\pi,N)=\frac{M_{k}(\psi,N)}{(\log N)^{k}}$ that the Cramér model
correctly predicts the distribution of primes in intervals of length $h(N)$
where $h\asymp\log N$ 444$f(x)\asymp g(x)\rightarrow f(x)=\mathcal{O}(g(x))$
and $g(x)=\mathcal{O}(f(x))$ with mean and variance $\sim\frac{h}{\log N}$,
this range is called "microscopic" scale [8]. Nevertheless, it fails out of
that range, which led them to conjecture that if $(\log N)^{1+\delta}\leq
h\leq(N^{1-\delta}$)555In particular, we can set $\delta\rightarrow\infty$,
then the condition becomes $\frac{h}{\log N}\sim\infty$ and $\frac{h}{N}\sim
0$, what is called "mesoscopic" scale, then the distribution is normal with
mean $\sim\frac{h}{\log N}$ and variance
$\sim\frac{h}{\log^{2}N}\log\frac{N}{h}$.

In this work we are going to study the distribution of the prime numbers on
short scales with the aid of the methods developed by Sanchis [9].

##  2 Computational model

The main purpose of this work is to conduct a counting experiment on prime
numbers in short intervals to the study their distribution. In particular, we
are going to compute the mean $\langle p\rangle$ and variance $\sigma_{p}$ of
the number of primes in certain intervals, supplying an experimental test of
the predictions of Mongomery and Soundararajan (hereinafter cited as MS).
Furthermore, we are going to look for an empirical formula for $\sigma_{p}$ in
intervals with finite N.

We are going to work with number intervals around an integer N. These
intervals are divided into m subsets of length h, see figure 1 for a
diagrammatic view. Let p be a random variable symbolising the number of prime
numbers in a subset. Given intervals with fixed h and m, we are going to study
the normalized variance $w$ as we vary N.

Figure 1: Interval around an integer number N (taken from [9])

$w=\frac{\sigma^{2}_{p}}{\langle p\rangle}=\frac{\langle p^{2}\rangle-\langle
p\rangle^{2}}{\langle p\rangle}$ (5)

With $\langle p\rangle=\frac{\sum_{k=1}^{m}p_{k}}{m}$, where $p_{k}$ is the
number of primes in the k-th subset. In order to parametrize $w$ we must
recall the characteristic scales [8]:

  1. 1.

Microscopic scale: When $h\asymp\log N$. On this scale the mean and the
variance tend asympotically to $\frac{h}{\log N}$

  2. 2.

Mesoscopic scale: When $\frac{h}{\log N}\rightarrow\infty$ and
$\frac{h}{N}\rightarrow 0$. On this scale $\langle p\rangle\sim\frac{h}{\log
N}$ and $\sigma_{p}^{2}\sim\frac{h}{\log^{2}N}\log\frac{N}{h}$.

  3. 3.

Macroscopic scale: When $h(N)>>>N$. On this scale there is no known
distribution law.

In this work, we are going to employ fixed values of h such that $h<<<N$
666Quantitatively, the biggest h that we are employing is $h=5\cdot 10^{4}$
and the smallest N is $N=10^{9}$, therefore we never reach the macroscopic
scale. The definitions of the scales are asymptotic limits, however, for
finite and fixed h, we may consider that an interval is mesoscopic if
$\frac{h}{\log N}>>1$ is verified, otherwise, we would consider it
microscopic. We shall take into account several facts. For fixed h,
$N\rightarrow\infty$ leads to $\frac{h}{\log N}\rightarrow 0$, hence, we
expect a Poisson distribution in that limit, thus
$\underset{N\rightarrow\infty}{\lim}w=1$. In addition, for large values of h,
we may consider a value for the mean and the variance similar to the predicted
by MS in the asymptotic limit ($\frac{h}{\log N}$ and
$\frac{h}{\log^{2}N}\log\frac{N}{h}$ respectively), therefore $w\sim
1-\frac{\log h}{\log N}$. We can try to model the normalized variance as:

$w=1-\frac{b(h,m)}{\log N}$ (6)

It is noteworthy to mention that we are not going to propose a model for the
mean of the distribution because, as we will see to see in our results, the
asymptotic value $\langle p\rangle\sim\frac{h}{\log N}$ predicts it with an
astonishing accuracy, even for N as small as $10^{9}$. Nevertheless, if we had
chosen $w=1-\frac{\log h}{\log N}$, it would have led to inaccurate results.
The reason is that the MS correction is not providing a formula to compute the
variance and the mean for large h, but an asymptotic limit for those
quantities. Thus, we are measuring the mean and variance of an unknown
distribution and expecting, based on the MS correction, that they are similar
to $\frac{h}{\log N}$ and $\frac{h}{\log^{2}N}\log\frac{N}{h}$ respectively,
in our range of $h$ and $N$. This means that for large N and h we may assume
$b(h,m)=\log h$. In this context, we consider h large if $\frac{h}{\log
N}>>>1$.

On the other hand, a subinterval of length $h=1$ may contain either one prime
or none at all, implying $p_{k}\in\\{0,1\\}$, and therefore $\langle
p\rangle=\frac{\sum_{k=1}^{m}p_{k}}{m}=\frac{\sum_{k=1}^{m}p_{k}^{2}}{m}=\langle
p^{2}\rangle$. Consequently, for $h=1$, $w=1-\langle p\rangle$. If we choose
$h=1$ we may consider that we are at the microscopic scale, therefore, a
suitable choice of $\langle p\rangle$ would be a deviation from $\frac{1}{\log
N}$, we may parametrize it as $\langle p\rangle=\frac{C}{\log N}+D$,with C an
arbitrary constant that has been include to extend the generality of the
formula, but we expect, and the result shall confirm, that $C=1$. On the other
hand, a Poisson distribution is expected in the limit $N\rightarrow\infty$,
which implies $w=1$, hence $D=0$.

To sum it all up, we expect $b(1,m)=C$ and $b(h,m)=\log h$ for large h. If we
combine both limits, the simplest estimate for $b(h,m)$ could be
$b(h,m)=C+\alpha(h)\log h$, with
$\underset{h\rightarrow\infty}{\lim}\alpha(h)=1$. The expression of
$\alpha(h)$ will be estimated in base of the empirical results. Therefore, the
normalized variance would read:

$w=1-\frac{C+\alpha(h)\log{h}}{\log{N}}$ (7)

As in [9] we are going to make the simulations with intervals with three
different values of m: $m=10^{3}$, $m=10^{4}$ and $m=10^{5}$, using different
values of h for every m, and varying N for every pair $(h,m)$.

###  2.1 Error analysis

As every empirical procedure, our calculations are not exempt from errors. In
this section we aim to estimate the error associated with the number of primes
in each interval. In order to assess the value of the errors, we are going to
consider the probability distribution given by the asymptotic limits of the MS
conjecture, i.e., a normal distribution characterized by $\mu=\frac{h}{\log
N}$ and $\sigma^{2}=\frac{h}{(\log N)^{2}}\log\frac{N}{h}$. In this work we
are computing the statistical estimators from the value of the random variable
$p$ in the different subsets and associating the result to an interval of
fixed N. Nevertheless, we are not considering the N associated to each subset
but the N associated to the interval. Therefore we are inferring a systematic
error in the results. Considering that the length of the interval is $\Delta
N=m\cdot h$. It is possible to assess the value of this error in the normal
approximation as the difference between the maximum and the minimum value of
the mean of primes in the interval.

$\Delta\mu=\frac{h}{\log(N-\frac{\Delta N}{2})}-\frac{h}{\log(N+\frac{\Delta
N}{2})}=\frac{2h\tanh^{-1}(\frac{\Delta N}{2N})}{\log(N-\frac{\Delta
N}{2})\log(N+\frac{\Delta N}{2})}$ (8)

Where we have used $\tanh^{-1}(x)=\frac{1}{2}\log(\frac{1+x}{1-x})$. By Taylor
expanding equation (8) around $N\rightarrow\infty$, keeping the first-order
terms and substituting $\Delta N=m\cdot h$ we obtain

$\Delta\mu=\frac{mh^{2}}{N(\log N)^{2}}$ (9)

We are led to the following relative systematic error:

$\epsilon_{sys}=\frac{\Delta\mu}{\mu}=\frac{mh}{N\log N}$ (10)

We can obtain the systematic error of the variance using the same procedure,
by considering the value of the variance in the asymptotic limit we obtain:

$\Delta\sigma_{p}^{2}=\frac{h\log(\nicefrac{{\left(N-\frac{\Delta
N}{2}\right)}}{{h}})}{\log(N-\frac{\Delta
N}{2})^{2}}-\frac{h\log(\nicefrac{{\left(N+\frac{\Delta
N}{2}\right)}}{{h}})}{\log(N+\frac{\Delta N}{2})^{2}}\approx\frac{h\Delta
N\left(2\log(\nicefrac{{N}}{{h}})-\log(N)\right)}{\log(N)^{3}N}$ (11)

We can easily compute the relative statistical error of $\langle p\rangle$
given that we are doing a counting experiment related to a normal random
variable with standard deviation$\sigma=\sqrt{\frac{h}{(\log
N)^{2}}\log\frac{N}{h}}$.

$\epsilon_{stat}=\frac{\sigma(p)}{\mu}=\sqrt{\frac{1}{h}\log\frac{N}{h}}$ (12)

The range in which we are interested is $\frac{h}{\log N}>>>1$ and
$\frac{h}{N}<<<1$, therefore it is a suitable scope to keep errors small.
Since we are restricted to the computing power of an average computer, we
cannot approach these limits, so errors are not negligible. Quantitatively;
for $m=10^{5}$, $h=10^{3}$ and $N=10^{10}$; the relative systematic and
statistic errors are $\epsilon_{sys}=0.04\%$ and $\epsilon_{stat}=12\%$.

The parameter m is related to the number of measures of primes carried out,
hence, the bigger m, the more accurate the data will be. While keeping a low
systematical error.

N | $10^{9}$ | $10^{10}$ | $10^{11}$ | $10^{12}$ | $10^{13}$ | $10^{14}$
---|---|---|---|---|---|---
Empirical mean | $120,655\pm 1.455\pm 8.665$ | $108,568\pm 0.118\pm 8.466$ | $98,724\pm 0.009\pm 8.259$ | $90,483\pm 8.053$ | $83,517\pm 7.854$ | $77,536\pm 7.663$
| Theoretical asymptotic
---
mean
$120,637$ | $108,574$ | $98,703$ | $90,478$ | $83,518$ | $77,553$
Empirical variance | $67,380\pm 0.356$ | $65,103\pm 0.037$ | $62,712\pm 0.004$ | $60,048$ | $57,531$ | $55,300$
| Theoretical asymptotic
---
variance
$75,091$ | $71,681$ | $68,213$ | $64,858$ | $61,688$ | $58,730$

Table 1:  Comparison between empirical and theoretical parameters for an
interval with $h=2500$ and $m=10^{5}$

(a) $m=10^{3}\;fits$

(b) $m=10^{4}\;fits$

(c) $m=10^{5}\;fits$

Figure 2: Fits of $w$ for different h and a) $m=10^{3}$ b) $m=10^{4}$ c)
$m=10^{3}$

##  3 Results

A comparison between the theoretical asymptotic and empirical values for the
mean and variance of a sample interval is shown in Table 1. See that the
systematic error has been neglected in the last columns.

It is clear that the MS conjecture predicts the correct values of the mean,
therefore no further analysis is needed. Nonetheless, it is worth mentioning
that the statistical error is overestimated, this could be a consequence of
using big amount of points to compute each parameter.

b(h,m) | m=$10^{3}$ | m=$10^{4}$ | m=$10^{5}$ | $\alpha(h,m)$ | m=$10^{3}$ | m=$10^{4}$ | m=$10^{5}$
---|---|---|---|---|---|---|---
h=1 | 0.996 $\pm$ 0.004 | 0.998 $\pm$ 0.003 | 0.999 $\pm$ 0.001 | h=1 | - | - | -
h=$10$ | 2.819 $\pm$ 0.069 | 2.919 $\pm$ 0.037 | 2.946 $\pm$ 0.012 | h=$10$ | 0.846 $\pm$ 0.030 | 0.833 $\pm$ 0.016 | 0.845 $\pm$ 0.005
h=$10^{2}$ | 5.941 $\pm$ 0.079 | 5.929 $\pm$ 0.041 | 5.911 $\pm$ 0.011 | h=$10^{2}$ | 1.071 $\pm$ 0.017 | 1.070 $\pm$ 0.009 | 1.071 $\pm$ 0.002
h=$10^{3}$ | 8.343 $\pm$ 0.093 | 8.305 $\pm$ 0.035 | 8.288 $\pm$ 0.011 | h=$10^{3}$ | 1.062 $\pm$ 0.013 | 1.057 $\pm$ 0.005 | 1.055 $\pm$ 0.002
h=$10^{4}$ | 10.503 $\pm$ 0.081 | 10.581 $\pm$ 0.029 | 10.635 $\pm$ 0.016 | h=$10^{4}$ | 1.031 $\pm$ 0.009 | 1.037 $\pm$ 0.003 | 1.046 $\pm$ 0.002
h=$10^{5}$ | 13.041 $\pm$ 0.076 | 12.937 $\pm$ 0.036 | 12.8487 $\pm$ 0.037 | h=$10^{5}$ | 1.045 $\pm$ 0.007 | 1.036 $\pm$ 0.003 | 1.029 $\pm$ 0.003

Table 2: Fits for b(h,m) and $\alpha(h,m)$

(a) $\alpha(\log h)\;with\;m=10^{3}$

(b) $\alpha(\log h)\;with\;m=10^{4}$

(c) $\alpha(\log h)\;with\;m=10^{5}$

Figure 3: Fits of $\alpha(\log h)$ for a) $m=10^{3}$ b) $m=10^{4}$ c)
$m=10^{3}$

The results for the normalized variance can be seen in Figure 2. The linear
behaviour of $w$ is manifest, validating equation (6). From the fitted values
of $b(h,m)$ we can infer C and $\alpha(h)$ from the equation (7). All the
values of $b(1,m)$ are coherent with 1, thence $C=1$. From Table 2 and Figure
3 we can see, as we expected, that
$\underset{h\rightarrow\infty}{\lim}\alpha(h)=1$. A simple way to achieve this
result is by parametrizing
$\alpha(h)=\frac{\alpha_{1}+f(h)}{\alpha_{2}+f(h)}$. See that there are
oscillations for $h<200$, nevertheless, these values are not included in the
fit since they do not belong to the mesoscopic scale, since, on such a scale,
$\frac{h}{\log N}\approx\infty$. The error bars do not appear in Figure 3
because they are negligible for $h>200$, as can be seen in Table 2. We have to
bear in mind that the dependence in m is just a matter of precision, but it
can not explicitly appear in the formulas.

From Table 2 we expect $\alpha(h)$ to be a monotonically decreasing function,
with $f(h)$ being a monotonically increasing function. To verify both
conditions it is necessary that $\alpha_{1}>\alpha_{2}$. We can rewrite
$\alpha(h)$ as
$\alpha(h)=\frac{\alpha_{1}+(f(h)^{-1})^{-1}}{\alpha_{2}+(f(h)^{-1})^{-1}}$,
doing a Taylor expansion in $f(h)^{-1}$ around $h\rightarrow\infty$ we get:

$\alpha(h)=1+\frac{\alpha_{1}-\alpha_{2}}{f(h)}+\mathcal{O}(\frac{1}{(f(h))^{2}})$
(13)

Comparing Equation (13) with Equation (7) we obtain a variance
$\sigma_{p}\approx\\\
\approx\frac{h}{\log^{2}N}(\log(\nicefrac{{N}}{{\left(h^{\nicefrac{{(\alpha_{1}-\alpha_{2})}}{{f(h)}}+1}\right)}})-1)$.
Taking into account that $\sigma_{p}\sim\frac{h}{\log^{2}N}\log\frac{N}{h}$ it
is reasonable to expect $f(h)=\log(h)$, as
$\log\left(h^{\nicefrac{{c}}{{\log(h)}}}\right)=c$. In addition, logarithms
are closely related with prime numbers [10].

Apart from $\alpha(h)=\frac{\alpha_{1}+\log{h}}{\alpha_{2}+\log{h}}$, in
appendix A we have studied the parametrizations $\alpha_{\rm
I}(h)=\frac{\alpha_{1}+\log{h}}{\log{h}}$ and $\alpha_{\rm
II}(h)=\frac{\log{h}}{\alpha_{2}+\log{h}}$. Assessing the goodness of fit of
each parametrization we have concluded that the best-fitting one is
$\alpha_{\rm I}(h)=\frac{\alpha_{1}+\log{h}}{\log{h}}$. See Apendix A for a
detailed statistical analysis and Table (3) for the results of the fit.
Identifying $B=1+\alpha_{1}-\alpha_{2}=1.414\pm 0.004$ and substituting the
expression (13) in (7) we obtain:

$w=\frac{\log\nicefrac{{N}}{{h}}}{\log N}+\frac{B}{\log N}$ (14)

The variance can be obtained by multiplying equation (14) by the mean
($\nicefrac{{h}}{{\log N}}$), which, in the asymptotic limit, tends to:

$\underset{h\rightarrow\infty}{\lim}\sigma^{2}=\frac{h}{\log^{2}N}\log\frac{N}{h}$
(15)

Which is the result predicted by the MS conjecture.

m | $10^{3}$ | $10^{4}$ | $10^{5}$
---|---|---|---
$B$ | $1.420\pm 0.023$ | $1.408\pm 0.010$ | $1.414\pm 0.004$
Table 3: Fits for $B=1+\alpha_{1}$. Using
$\alpha_{\rm I}(h)=\frac{\alpha_{1}+\log{h}}{\log{h}}$

##  4 Conclusion

This project aimed to verify experimentally the MS conjecture and to provide
an empirical formula to extend its range of application on the mesoscopic
scale. We have obtained a formula that predicts the variance of the
distribution of prime numbers on such scale for large but finite N.

$\sigma^{2}=\frac{h}{\log^{2}N}(\log\frac{N}{h}+B)$ (16)

This formula is completely consistent with the asymptotic limit of the MS
conjecture. We have computed $B=1.414\pm 0.004$ using the parameters of the
best-fitting model. It is noteworthy to mention that, in all parametrizations,
$B$ is coherent with $-B^{\prime}=\gamma+\log{2\pi}-1\approx 1.415$, where
$\gamma$ is the Euler–Mascheroni constant, which arises in the calculation of
the moments of the distribution of prime numbers in some scales [7]. See that
we have obtained the same result as Sanchis [9] using a different
parametrization of $\alpha(h)$.

All the calculations have been done using _Mathematica_ and have been limited
by the computation capacity of an average computer. For that reason, we have
worked with N less than $10^{14}$. In order to verify the range of validity of
the equation (16), it would be necessary to carry out calculations with bigger
N, that would require a larger computational capacity. Nevertheless, in the
range in which we have been able to work, we have provided an experimental
result that supports the validity of the MS conjecture.

##  Appendix A Statistical analysis of the parametrizations $\alpha(h)$

In the parametrization of $\alpha(h)$ we are only interested in the behaviour
at large h, hence the only significant parameter, according to equation (13),
will be $B=1+\alpha_{1}-\alpha_{2}$. To assess the goodness of fit of every
parameterization we are going to employ the Pearson’s $\chi^{2}$ test to
compute the p-value [11]. The p-value would give us an estimation of the
degree of confidence in the null hypothesis, in this case, that hypothesis
corresponds to consider that $\alpha(h)$ is well fitted by the
parametrization. We have used 16 points for $m=10^{3}$, 14 points for
$m=10^{4}$ and 20 points for $m=10^{5}$. Let $\nu$ be the degrees of freedom
of the model, that may be computed as the number of points minus the number of
parameters. In the tables of this appendix we have shown values of the
different parameters used in the different parametrizations and the reduced
chi-square ($\chi_{\nu}^{2}=\nicefrac{{\chi^{2}}}{{\nu}}$) and p-value for
each one:

m | $10^{3}$ | $10^{4}$ | $10^{5}$
---|---|---|---
$\alpha_{1}$ | $-0,475\pm 1.364$ | $0.571\pm 0.821$ | $0.666\pm 0.305$
$\alpha_{2}$ | $-0.848\pm 1.291$ | $0.154\pm 0.776$ | $0.238\pm 0.289$
$B$ | $1.373\pm 1.878$ | $1.597\pm 1.129$ | $1.428\pm 0.420$
Table 4: Fits for $B=1+\alpha_{1}-\alpha_{2}$. Using
$\alpha_{\rm III}(h)=\frac{\alpha_{1}+\log{h}}{\alpha_{2}+\log{h}}$ m | $10^{3}$ | $10^{4}$ | $10^{5}$
---|---|---|---
$B$ | $1.420\pm 0.023$ | $1.408\pm 0.010$ | $1.414\pm 0.004$
Table 5: Fits for $B=1+\alpha_{1}$. Using
$\alpha_{\rm I}(h)=\frac{\alpha_{1}+\log{h}}{\log{h}}$ m | $10^{3}$ | $10^{4}$ | $10^{5}$
---|---|---|---
$B$ | $1.399\pm 0.021$ | $1.386\pm 0.009$ | $1.392\pm 0.004$
Table 6: Fits for $B=1-\alpha_{2}$. Using
$\alpha_{\rm II}(h)=\frac{\log{h}}{\alpha_{2}+\log{h}}$ $\chi_{\nu}^{2}$ | m=$10^{3}$ | m=$10^{4}$ | m=$10^{5}$ | p-value | m=$10^{3}$ | m=$10^{4}$ | m=$10^{5}$
---|---|---|---|---|---|---|---
$\alpha_{\rm I}$ | 1.683 | 1.400 | 1.063 | $\alpha_{\rm I}$ | 4.6 % | 15 % | 38%
$\alpha_{\rm II}$ | 1.718 | 1.447 | 1.351 | $\alpha_{\rm II}$ | 4 % | 13% | 14%
$\alpha_{\rm III}$ | 1.904 | 1.521 | 1.078 | $\alpha_{\rm III}$ | 2 % | 11 % | 37 %
Table 7: $\chi_{\nu}^{2}$ and p-value for all the parametrizations

See that, for $\alpha_{\rm III}(h)$, there is no contradiction in the fact of
having different values of $\alpha_{1}$ and $\alpha_{2}$ for different m. As
stated before, we are only interested in the difference
$\alpha_{1}-\alpha_{2}$, that is coherent between the different sets
characterized by $m$ in each of the three parametrizations. From Table (7) we
can see that the best-fitting parametrization is $\alpha_{\rm
I}(h)=\frac{\alpha_{1}+\log{h}}{\log{h}}$. We can infer the value $B=1.414\pm
0.004$.

## References

  * [1] L. J. Goldstein. A history of the Prime Number Theorem. The American Mathematical Monthly, Vol. 80, No. 6 (Jun. - Jul., 1973), pp. 599-615.
  * [2] J. Hadamard, Sur la distribution des zéros de la fonction $\zeta(s)$ et ses conséquences arithmétiques, Bull. Soc. Math. France, 1896.
  * [3] C.-L. de la Vallée-Poussin, Recherches analytiques sur la théorie des nombres premiers, Annales de la Société Scientifiques de Bruxelles, 1896, pp. 183-256.
  * [4] Granville, Andrew. Harald Cramer and the distribution of prime numbers. Athens, Georgia (U.S.A), (n.d.).
  * [5] Scott Funkhouser, Daniel A. Goldston, Andrew H. Ledoan, Irregularities in the Distribution of Prime Numbers, Springer International Publishing AG, Springer Nature, 2018
  * [6] P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), 4–9, Corrigendum, 28 (1981), 86
  * [7] H. Montgomery and K. Soundararajan, Commun. Math. Phys. (2004) 252: 589, arXiv:math/0409258 [math.NT]
  * [8] K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], p. 13.
  * [9] M.A. Sanchis-Lozano, A heuristic study of the distribution of primes in short and not-so-short intervals arXiv:1804.07659 [math.NT]
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  * [11] Barlow, R. J. (2013). Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences. Hoboken: Wiley. p. 106.


[a,†]Ignasi Rosell

# Heavy states and electroweak effective approaches

Antonio Pich  Juan José Sanz-Cillero

###### Abstract

The existence of a mass gap between the Standard Model (SM) and possible new
states encourages us to use effective field theories. Here we follow the non-
linear realization of the electroweak symmetry breaking: the electroweak
effective theory (EWET), also known as Higgs effective field theory (HEFT) or
electroweak chiral Lagrangian (EWChL). At short distances an effective
resonance Lagrangian which couples the SM states to bosonic and fermionic
resonances is considered. After integrating out the resonances and assuming a
well-behaved high-energy behavior, we estimate or bound purely bosonic low-
energy constants in terms of only resonance masses. Current experimental
information on these low-energy constants allows us to constrain the high-
energy resonance masses.

##  1 Introduction

The success of the Standard Model (SM) has been confirmed by the LHC and it
implies the existence of a mass gap between SM fields and possible new-physics
(NP) states. Therefore, effective field theories (EFTs) are very convenient in
order to study systematically low-energy data to search for traces of NP
scales, the so-called bottom-up approach. Whereas the low-energy constants
(LECs) of these EFTs are free-parameters and contain information about heavy
scales, the structure of the effective Lagrangian relies on the particle
content (SM particles here), the symmetries and the power counting. The
symmetries and the power counting depend on the mechanism to introduce the
Higgs. Here we follow the most general case, the electroweak (EW) effective
theory (EWET): the non-linear realization of the electroweak symmetry breaking
is taken and, consequently, no assumptions are made about the relation between
the Higgs and the EW Goldstones and a “chiral” (generalized momenta) expansion
is observed. It is worth noting that the so-called SM effective field theory
(SMEFT) is a particular case of the EWET. At higher energies possible heavy
resonances are included by following the same scheme, i.e., same symmetries
and power expansion.

Accordingly, we work with two EFTs: the EWET at long distances, which includes
only the SM fields, and an EW resonance theory at shorter distances, which
contains in addition heavy resonances. After integrating out the resonances
both Lagrangians can be matched and the EWET LECs are determined in terms of
resonance parameters. Assuming a good high-energy behavior in the resonance
theory allows us to imagine this effective Lagrangian as a bridge between the
EWET and the unknown underlying theory and also to reduce the number of
unknown resonance parameters. Note that the main focus of this work is to
constrain NP scales by using current experimental bounds on EWET LECs and our
determinations of these LECs in terms of only a few resonance parameters [1,
2].

##  2 Theoretical framework

The EWET Lagrangian is organized following an expansion in generalized momenta
[3, 4, 5]:

$\displaystyle\mathcal{L}_{\mathrm{EWET}}$ $\displaystyle=$
$\displaystyle\sum_{\hat{d}\geq
2}\,\mathcal{L}_{\mathrm{EWET}}^{(\hat{d})}\,,$ (1)

where the chiral dimension $\hat{d}$ indicates the infrared behavior at low
momenta [3]. The building blocks and their related power-counting rules can be
found in Refs. [1, 5]. In view of the current experimental information, the
relevant bosonic part of the LO EWET Lagrangian reads

$\displaystyle\Delta\mathcal{L}_{\mathrm{EWET}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{4}\,\left(1+\frac{2\,\kappa_{W}}{v}h+\frac{c_{2V}}{v^{2}}\,h^{2}\right)\langle\,u_{\mu}u^{\mu}\,\rangle\,,$
(2)

where $h$ is the Higgs field, $u_{\mu}$ the tensor containing one covariant
derivative of the EW Goldstones and $\langle\cdots\rangle$ indicate an $SU(2)$
trace. $\kappa_{W}$ and $c_{2V}$ parametrize the $hWW$ and $hhWW$ couplings,
respectively. Taking into account again the available experimental
information, the relevant part of the purely bosonic NLO EWET Lagrangian is
given by [5]:

$\Delta\mathcal{L}_{\mathrm{EWET}}^{(4)}=\frac{\displaystyle\mathcal{F}_{1}}{\displaystyle
4}\,\langle\,{f}_{+}^{\mu\nu}{f}_{+\,\mu\nu}-{f}_{-}^{\mu\nu}{f}_{-\,\mu\nu}\,\rangle+\frac{\displaystyle
i\,\mathcal{F}_{3}}{\displaystyle
2}\,\langle\,{f}_{+}^{\mu\nu}[u_{\mu},u_{\nu}]\,\rangle+\mathcal{F}_{4}\langle\,u_{\mu}u_{\nu}\,\rangle\,\langle\,u^{\mu}u^{\nu}\,\rangle+\mathcal{F}_{5}\langle\,u_{\mu}u^{\mu}\,\rangle\,\langle\,u_{\nu}u^{\nu}\,\rangle,$
(3)

where $f^{\mu\nu}_{\pm}$ introduce the field-strength tensors. In the SM,
$\kappa_{W}=c_{2V}=1$ and $\mathcal{F}_{1,3,4,5}=0$. $\mathcal{F}_{1}$,
$\mathcal{F}_{1,3}$ and $\mathcal{F}_{1,3-5}$ contribute, respectively, to the
$S$ oblique parameter, the trilinear and the quartic gauge couplings.

Since we are interested only in the resonance contributions to the bosonic
${\cal O}(p^{4})$ EWET LECs, only ${\cal O}(p^{2})$ operators with up to one
bosonic resonance $R$ are required [5]. The relevant $CP$-even resonance
Lagrangian reads [1, 5]:

$\displaystyle\qquad\Delta\mathcal{L}_{\mathrm{RT}}\,=\,$
$\displaystyle\frac{v^{2}}{4}\\!\left(\\!1\\!+\\!\frac{2\,\kappa_{W}}{v}h\\!+\\!c_{2V}\,h^{2}\\!\right)\\!\langle\,u_{\mu}u^{\mu}\,\rangle+\langle\,V_{\mu\nu}\left(\frac{F_{V}}{2\sqrt{2}}f_{+}^{\mu\nu}+\frac{iG_{V}}{2\sqrt{2}}[u^{\mu},u^{\nu}]+\frac{\widetilde{F}_{V}}{2\sqrt{2}}f_{-}^{\mu\nu}\right)\,\rangle\phantom{\bigg{(}}$
$\displaystyle\quad+\langle\,A_{\,\mu\nu}\left(\frac{F_{A}}{2\sqrt{2}}f_{-}^{\mu\nu}+\frac{\widetilde{F}_{A}}{2\sqrt{2}}f_{+}^{\mu\nu}+\frac{i\widetilde{G}_{A}}{2\sqrt{2}}[u^{\mu},u^{\nu}]\right)\,\rangle+\frac{c_{d}}{\sqrt{2}}\,S_{1}\langle\,u_{\mu}u^{\mu}\,\rangle\,.$
(4)

By integrating out the resonance fields in (4), the EWET Lagrangian of (3) is
recovered and, consequently, the EWET LECs are determined in terms of
resonance parameters [1, 5],

$\displaystyle\mathcal{F}_{1}$ $\displaystyle=-\frac{\displaystyle
F_{V}^{2}-\widetilde{F}_{V}^{2}}{\displaystyle 4M_{V}^{2}}+\frac{\displaystyle
F_{A}^{2}-\widetilde{F}_{A}^{2}}{\displaystyle 4M_{A}^{2}}\,,$
$\displaystyle\mathcal{F}_{3}$ $\displaystyle=-\frac{\displaystyle
F_{V}G_{V}}{\displaystyle
2M_{V}^{2}}-\frac{\displaystyle\widetilde{F}_{A}\widetilde{G}_{A}}{\displaystyle
2M_{A}^{2}}\,,$ $\displaystyle\mathcal{F}_{4}$
$\displaystyle=\frac{\displaystyle G_{V}^{2}}{\displaystyle
4M_{V}^{2}}+\frac{\displaystyle{\widetilde{G}_{A}}^{2}}{\displaystyle
4M_{A}^{2}}\,,$ $\displaystyle\mathcal{F}_{5}$
$\displaystyle=\frac{\displaystyle c_{d}^{2}}{\displaystyle
4M_{S_{1}}^{2}}-\frac{\displaystyle G_{V}^{2}}{\displaystyle
4M_{V}^{2}}-\frac{\displaystyle{\widetilde{G}_{A}}^{2}}{\displaystyle
4M_{A}^{2}}\,.$ (5)

Figure 1: $\mathcal{F}_{1}$, $\mathcal{F}_{3}$, $\mathcal{F}_{4}$ and
$\mathcal{F}_{4}+\mathcal{F}_{5}$ as functions of the corresponding resonance
mass ($M_{V}$, $M_{A}$ or $M_{S_{1}^{1}}\,v/c_{d}$) [1]. The 95% CL
experimentally-allowed regions given in Table 1 are covered by the green
areas, and they are further extended by dashed green bands accounting for our
estimation of the one-loop running uncertainties. If the prediction of the
LECs depend on both $M_{V}$ and $M_{A}$, the gray and/or brown regions cover
all possible values for $M_{A}>M_{V}$. If the 2nd WSR has been assumed, we
indicate it explicitly in the plot, with the corresponding lines for the limit
$M_{A}=M_{V}$ (orange), $M_{A}=1.1\,M_{V}$ (blue), $M_{A}=1.2\,M_{V}$ (red)
and $M_{A}\to\infty$ (dark gray). In the case without the 2nd WSR, the
prediction for $\mathcal{F}_{1}$ is given by the gray and brown regions. In
case of using only the even-parity operators, it is indicated.

As stressed previously, high-energy constraints allow us to reduce the number
of resonance parameters. For the case at hand, the following short-distance
constraints have been used [1, 5]:

  1. 1.

Well-behaved two-Goldstone vector form factor (VFF). Assuming that this form
factor vanishes at high energies, the following constraint is found:

$v^{2}\,-\,F_{V}\,G_{V}\,-\,\widetilde{F}_{A}\,\widetilde{G}_{A}\,=\,0\,.$ (6)

  2. 2.

Weinberg Sum Rules (WSRs). The $W^{3}B$ correlator is an order parameter of
the EWSB. In asymptotically-free gauge theories it vanishes at short distances
as $1/s^{3}$, implying two superconvergent sum rules [6]: the 1st WSR
(vanishing of the $1/s$ term) and the 2nd WSR (vanishing of the $1/s^{2}$
term). While the 1st WSR is supposed to be also fulfilled in gauge theories
with nontrivial ultraviolet (UV) fixed points, the validity of the 2nd WSR
depends on the particular type of UV theory considered [7]. The 1st and 2nd
WSRs read, respectively,

$F_{V}^{2}+\widetilde{F}_{A}^{2}-F_{A}^{2}-\widetilde{F}_{V}^{2}\,=\,v^{2}\,,\qquad\quad
F_{V}^{2}M_{V}^{2}+\widetilde{F}_{A}^{2}M_{A}^{2}-F_{A}^{2}M_{A}^{2}-\widetilde{F}_{V}^{2}M_{V}^{2}=0.$
(7)

The use of (6) and (7) in (5) lets us get the following determinations or
bounds, which are displayed in the plots of Figure 1 [1]:

  1. 1.

$\mathcal{F}_{1}$. Assuming both WSRs, $\mathcal{F}_{1}$ can be determined in
terms of only resonance masses; contrary, discarding the 2nd WSR, we only find
a bound in terms of only resonance masses [1]:111For this bound one needs to
assume that $M_{A}>M_{V}$ and, in the case of considering $P$-even and also
$P$-odd operators, $F_{A}^{2}>\widetilde{F}_{A}^{2}$. Both hypothesis seem to
be reasonable working assumptions.

$\mathcal{F}_{1}\Big{|}^{\mathrm{1st\,\&\,2nd\,WSRs}}\,=\,-\displaystyle\frac{v^{2}}{4}\left(\frac{1}{M_{V}^{2}}\\!+\\!\frac{1}{M_{A}^{2}}\right)\,,\qquad\qquad\mathcal{F}_{1}\Big{|}^{\mathrm{1st\,WSR}}\,<\,\displaystyle\frac{-v^{2}}{4M_{V}^{2}}\,.$
(8)

These results are shown in the top-left plot in Figure 1. The dark gray curve
shows the upper bound of (8). Therefore, and if only the 1st WSR is obeyed,
the whole region below this line (gray and brown areas) would be theoretically
allowed. If one accepts moreover the validity of the 2nd WSR,
$\mathcal{F}_{1}$ is determined as a function of $M_{V}$ and $M_{A}$, see (8),
being the dark gray curve the limit $M_{A}\to\infty$. The values of
$\mathcal{F}_{1}$ for some representative axial-vector masses are shown in the
red ($M_{A}=1.2\,M_{V}$), blue ($M_{A}=1.1\,M_{V}$) and orange ($M_{A}=M_{V}$)
curves; being the orange line the lower bound
$\mathcal{F}_{1}=-v^{2}/(2M_{V}^{2})$. This range of $M_{A}\sim M_{V}$
corresponds actually to the most plausible scenario [8]. Then, and if both
WSRs are followed, only the gray region would be theoretically allowed if one
assumes that $M_{A}>M_{V}$.222In the case with only $P$-even operators,
following both WSRs implies $M_{A}>M_{V}$, but this is only a reasonable
assumption if one considers also $P$-odd operators.

  2. 2.

$\mathcal{F}_{3}$. In this case the WSRs are not relevant. Assuming only
$P$-even operators, (6) implies a determination in terms of only resonance
masses, whereas the inclusion of $P$-odd operators reduces this determination
to a bound [1]:333For this bound one needs to assume that $M_{A}>M_{V}$ and
$F_{V}G_{V}>0$, reasonable working assumptions.

$\mathcal{F}_{3}\Big{|}^{\mathrm{P{\text{-}}even}}\,=\,-\displaystyle\frac{v^{2}}{2M_{V}^{2}}\,,\qquad\qquad\mathcal{F}_{3}\Big{|}^{\mathrm{P{\text{-}}even\,\&\,P{\text{-}}odd}}\,<\,-\displaystyle\frac{v^{2}}{2M_{A}^{2}}\,.$
(9)

The only $P$-even-operators prediction of (9) is shown by the black curve in
the top-right plot of Figure 1. Adding $P$-odd operators, we have the upper
bound of (9), which is represented by the same curve but this time with
$M_{R}=M_{A}$. Therefore, the whole region below this line (gray area) would
be allowed in the most general case.

  3. 3.

$\mathcal{F}_{4}$. In order to get a determination in terms of only resonance
masses, one needs to consider a well-behaved two-Goldstones VFF, both WSRs and
only $P$-even operators [1],

$\mathcal{F}_{4}\Big{|}^{\mathrm{1st\,\&\,2nd\,WSRs;\,P{\text{-}}even}}\,=\,\displaystyle\frac{v^{2}}{4}\left(\frac{1}{M_{V}^{2}}\\!-\\!\frac{1}{M_{A}^{2}}\right)\,.$
(10)

We show it in the bottom-left panel in Figure 1. The upper bound (dark gray
curve) is obtained at $M_{A}\to\infty$. Thus, the theoretically allowed region
is the gray area below that curve. The values of $\mathcal{F}_{4}$ for some
representative axial-vector masses are shown in the red ($M_{A}=1.2\,M_{V}$),
blue ($M_{A}=1.1\,M_{V}$) and orange ($M_{A}=M_{V}$) curves again.

  4. 4.

$\mathcal{F}_{4}+\mathcal{F}_{5}$. No constraints are required in this case,
since directly from (5) one gets [1]

$\mathcal{F}_{4}+\mathcal{F}_{5}\,=\,\displaystyle\frac{c_{d}^{2}}{4M_{S^{1}_{1}}^{2}}\,.$
(11)

This clean prediction is shown by the black curve in the bottom-right plot of
Figure 1.

##  3 Phenomenology

LEC | Ref. | Data | LEC | Ref. | Data
---|---|---|---|---|---
$0.89$$\,<\,$ |  $\kappa_{W}$ $\,<\,$ | $1.13$ | [9] | LHC |  $-0.06$$\,<\,$ |  $\mathcal{F}_{3}$ $\,<\,$ | $0.20$ | [12] | LEP & LHC
$-1.02$$\,<\,$ |  $c_{2V}$ $\,<\,$ | $2.71$ | [10] | LHC |  $-0.0006$$\,<\,$ |  $\mathcal{F}_{4}$ $\,<\,$ | $0.0006$ | [13] | LHC
$-0.004$$\,<\,$ |  $\mathcal{F}_{1}$ $\,<\,$ | $0.004$ | [11] | LEP |  $-0.0010$$\,<\,$ |  $\mathcal{F}_{4}+\mathcal{F}_{5}$ $\,<\,$ | $0.0010$ | [13] | LHC
Table 1: Experimental constraints on purely bosonic EWET LECs, at 95% CL [1].

Experimental constraints on $\mathcal{F}_{1,3,4,5}$ are reported in Table 1
[1] and are shown in the plots of Figure 1 as green areas. While these
experimental constraints have been obtained at lower scales, the tree-level
predictions of (5) are supposed to apply at a scale around the resonance
masses. Thus, it is convenient to estimate the one-loop running uncertainties
[14] due to these different scales and they are indicated in Figure 1 with the
dashed green bands that enlarge the experimentally allowed regions. They
depend on $\kappa_{W}$ and $c_{2V}$, which are also given in Table 1.

The principal result of Figure 1 is that the resonance mass scale is pushed to
the TeV range, $M_{R}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2\,$TeV, in
agreement with Ref. [8], where we estimated theoretically the $S$ and $T$
oblique parameters at NLO. Other minor results are obtained [1]: the $S$
oblique parameter provides the most precise LEC determination
($\mathcal{F}_{1}$), implying the constraints
$M_{V,A}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$ TeV (95% CL); the triple
gauge couplings give a weaker limit on $\mathcal{F}_{3}$, translating into the
lower bound $M_{V,A}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0.5$ TeV (95% CL);
assuming both WSRs and considering only $P$-even operators, the bounds on
$\mathcal{F}_{4}$ constrain the mass of the vector resonance to
$M_{V}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$ TeV if $M_{A}/M_{V}>1.1$ (95%
CL); the limit on $\mathcal{F}_{4}+\mathcal{F}_{5}$ implies that the singlet
scalar resonance would have a mass
$M_{S_{1}}\stackrel{{\scriptstyle>}}{{{}_{\sim}}}2$ TeV (95% CL), taking a
$S_{1}WW$ coupling close to the $hWW$ one ($c_{d}\sim v$). Thus, experimental
constraints start already to be competitive and, once new data be available,
more precise information will be obtained using this kind of approaches.

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