††thanks: This work was completed with the support of the NLAGA project

# The Wade Formula and the oil price fluctuation: An Optimal Control Theory
approach

Pierre Mendy  Babacar M. Ndiaye  Diaraf Seck  Idrissa Ly  Laboratory of
Mathematics of Decision and Numerical Analysis.Cheikh Anta Diop University. BP
45087, 10700. Dakar, Senegal<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(November 21, 2022)

###### Abstract

By considering the Wade Formula, we propose a model to study the evolution of
the oil price per barrel. Our model shows that the policy of diversification
of the energy is to be supported. This model is proposed to see how it is
possible to control parameters so that the oil price should decrease.

###### keywords:

Wade Formula, optimal control, options, volatility, oil prices

##  1 Introduction

The rapid increase in oil prices between 2002 and 2008 and their sharp decline
in the second half of 2008 and 2014, (see [1], [2]), and the consequence of
covid19 and Ukraine conflic on oil price, ([3], [4], [5]), has renewed the
interest in the causes of oil price fluctuation [figure 1, curve(d)], and the
effects of energy prices on the macroeconomy.
A large of studies prove that the oil fluctuations have a considerable
consequence in economic activity ([6],[11],[7], [8],[9], [10], [13], [15],
[14]).
While the oil price shock is asymmetric between oil exporters countries and
oil importers countries ([12], [16], [17]), this asymmetric oil shock had
inspired the Wade Formula.
Between 2005 and 2006, the full professor in Economics Abdoulaye Wade, former
President of the Republic of Senegal, proposed a formula related to the
evolution of the oil price per barrel. This formula translates the super-
profits generated by the oil companies (selling in the world) and countries
which produce oil.
At first, from the Wade Formula, we describ in section 2, the model. We
resolve and show a necessary optimal condition which could give some hints for
the orientations of the energy policies in section 3. We propose some
simulations and interpretations of the obtained results, section 4. As
conclusion in section 5, we present some variations of the obtained results
and politics implications.

##  2 Model description

The formula should be stated as follows: Let be the time $t\in[T_{0},T]$,
where $T_{0}$ is the initial state. Since it is possible to find the structure
of the oil price/barrel without financial speculation, President Wade studied
and proposed a reasonable price (win-win price) $P_{0}=29\$$. For more
details, see [18].
The Wade Formula for super profit is defined by the equation (2.1)

$(p_{t}-29)q_{t}=S_{t}$ (2.1)

where $p_{t}$ represents the price which is applied in the world for buying a
barrel of oil, $q_{t}$ is the quantity of oil bought by the countries, and
$S_{t}$ is the super profit for the oil companies or the super coast supported
by the countries which don’t produce oil.
It is important to remark that this formula is a conservation law.

##  3  Model resolution: Optimal control approach

In this section, we give a necessary optimal condition to minimize the super
coasts, denoted111For more details in optimal control theory see [19], [20],
[21], [22], $S(t).$
A reasonable model to describe the evolution of the reserves of oil in the
world can be written as follows:

$\frac{dR}{dt}=-v(t)+\alpha(t)R(t),t\geq 0$ (3.1)

where $\alpha(t)\in[0,1[$ is the rate of the increase of oil in the world,
$R(t)$ are the reserves and $v(t)$ is the demand in the world at time $t$.
In the following, we are going to take $\alpha(t)=\alpha$ a real constant. But
the theorem below is satisfied even if $\alpha$ depends on the time $t$.

######  Theorem 3.1.

Let $a(t)$ be the demand of the country which doesn’t produce oil, then a
necessary optimal condition to minimize the super coasts for a country which
doesn’t product oil is:

$a^{*}(t)=\displaystyle\frac{c_{0}\exp(-\alpha t)}{2(p(t)-29)^{2}}.$

###### Proof.

Let $T$ be a fixed time.

$\displaystyle\min\int_{0}^{T}S(t)^{m}dt,\quad m\in\mathbb{N}$ (3.2)
$\displaystyle s.c\left\\{\begin{array}[]{lclcl}\frac{dR}{dt}&=&-v(t)+\alpha
R(t)&on\,\,[0,T]&\\\ R(0)&=&Q\end{array}\right.$ (3.5)

where, $Q$ is the initial stock of the reserves,

$S(t)=(p(t)-29)a(t),$ $v=a+w,$

where $a$ is the demand of country which doesn’t produce oil.
Let’s take the Hamiltonian defined by:

$H(R(t),a(t),\lambda(t),t)=S(t)^{m}+\lambda(t)(-v(t)+\alpha R(t))$ (3.6)

and $\lambda(t)$ being the Pontryagin multiplier.
We have the necessary optimal conditions:

$\displaystyle\frac{\partial H}{\partial R}$ $\displaystyle=$
$\displaystyle\alpha\lambda(t)=-\frac{d\lambda}{dt}$
$\displaystyle\frac{\partial H}{\partial a}$ $\displaystyle=$ $\displaystyle
0$

The condition $\frac{\partial H}{\partial a}=0$ is equivalent to
$mS(t)^{m-1}(p(t)-29)-\lambda=0.$
This implies that:

$S(t)=\left(\frac{\lambda}{m(p(t)-29)}\right)^{\frac{1}{m-1}}$

Since:

$-\frac{d\lambda}{dt}=\alpha\lambda(t),$

then $\lambda(t)=c_{0}\exp(-\alpha t)$ where $c_{0}$ is a real constant.
Finally:

$S(t)=\left(\frac{c_{0}\exp(-\alpha t)}{m(p(t)-29)}\right)^{\frac{1}{m-1}}$

If $m=2,$ we obtain:

$\displaystyle S(t)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{c_{0}\exp(-\alpha
t)}{2(p(t)-29)}\,\,\mbox{and by the following equality:}$ $\displaystyle S(t)$
$\displaystyle=$ $\displaystyle(p(t)-29)a(t)\mbox{ we have:}$ $\displaystyle
a{*}(t)$ $\displaystyle=$ $\displaystyle\displaystyle\frac{c_{0}\exp(-\alpha
t)}{2(p(t)-29)^{2}}\qed$

∎

###  3.1 Initial condition variation

We assume that $R(t)$ is Lipschitzian, so there exists a flow defined by:

$\begin{cases}X\rightarrow X\;\text{space of phases}\\\
x_{0}\rightarrow\phi_{t}(x_{0})=x(t)\end{cases}$

with $X=\mathbb{R}^{n}$, for n= 1,

$\begin{cases}x(t)\rightarrow R(t)\\\ x_{0}\rightarrow Q\end{cases}$

We consider a period of horizon h on the time interval $[t_{0},t_{0}+h]$ and
$Q\in[Q_{0},Q^{*}]$ with $Q_{0}=Q(t0)$ and $Q^{*}=Q(t_{t0+h})$.
We define:

$Q_{k}=Q_{0}+\frac{k}{t_{0}+h}(Q^{*}-Q_{0})$

The initial conditions vary with the variants of $k\in[t_{0},t_{0}+h]$.

###  3.2 Terminal condition variation

We still consider equations (3.3) and (3.4) and rewrite them by varying the
terminal conditions. Let $s\in[0,T]$. Let $t=T-s$, $Y(s)=R(T-s)$. The
optimization problem becomes:

$min\int_{0}^{T}[S(T-s)]^{m}ds$

$sc\begin{cases}\frac{dY(s)}{ds}=-v(T-s)+\alpha Y(s)\\\ Y(0)=R_{T}\end{cases}$

where $R_{T}$ is a real value sequence on $[Y_{0},...,Y^{*}]$ with
$Y_{0}=Y(t_{0})$ and $Y^{*}=Y(t_{0}+h)$.

$R_{T}(k)=Y_{0}+\frac{k}{t_{0}+h}\left(Y^{*}-Y_{0}\right)$
$S(T-s)=\left(P(T-s)-29\right)a(T-s)$

In the next section we are going to use the curve below. This represents the
plot of the result of the necessary optimal condition. The horizontal axis
being the values of the prices while the vertical axis is the values of the
demand $a^{*}$.

##  4 Numerical resolution and results interpretations

The simulation parameters are as follows. In the graphs of figure 1, we have
taken as constant c the average of the percentage rate increase of the world
fossil reserves. In figure 2, we consider the evolution of the rate of the
resources over the study period in the numerical resolution of our optimal
control problem The curves in Figures 1-2 are the results of the optimal
control resolution. The result could be interpreted as follows:

  1. 1.

Between 1980 and 2021, [1d], there were two periods where the average price of
oil was above 29 dollars, 1980 to 1983 and 2004 to 2021, and a period where it
was below 29 dollars. In the first period (1980 to 1983), the average price of
a barrel of oil fluctuated between 30 dollars and 37 dollars. It peaked at 39
dollars in 1981 due to the Reagan cut taxes. Two other causes of the high
fluctuation of the average barrel price are the Iran embargo in 1980 and the
end of the recession in 1982.

  2. 2.

During the second period (2004-2021), the average price of a barrel of oil
fluctuated between about 35 dollars in 2004 and 103 dollars in 2011. It
reached a peak of 127 dollars in 2008 during the financial crisis. Other
factors that explain the high fluctuation during this period are Hurricane
Katrina in 2005, Bernanke becomes Fred Chair in 2006, the banking crisis in
2007, the great recession in 2009, Iran threatening the Straits of Hormuz, the
increase of the 15 % dollars , the US shale oil increased 2015, the decrease
of the dollars in 2016, the OPEC cut oil supply to keep prices stable and
demand reduction of the pandemic since 2020.

  3. 3.

From 1984 to 2003, the average price of a barrel oscillated between 14 dollars
in 1986 and 27 dollars in 2003. During the Gulf War, the peak was reached with
a barrel at almost 33 dollars . Other events during this period allowed this
fluctuation are the prices doubled, the recession and the war in Afghanistan
in 2002. The barrel reached the lowest price of 10 dollars in 1999.

  4. 4.

The curves [1c] and [1d] have the shape of the sigmoid function. The world’s
demand for fossil energy cannot grow in an unlimited way. With climate change
and the strong fluctuations of the oil price, many countries are diversifying
their energy sources toward renewable energies. Fossil fuel reserves are also
limited despite the discovery of new deposits.

  5. 5.

The evolution of the Super profit (curve c fig1) is consecutive to the
differential between the price 29 and the average price oil with a phase of
strong fall between 1984 and 2003

  6. 6.

The curves [2a] and [2b] have the shape of the sigmoid function. The world’s
demand for fossil energy cannot grow in an unlimited way. With climate change
and the strong fluctuations of the oil price, many countries are diversifying
their energy sources toward renewable energies. Fossil fuel reserves [2c] are
also limited despite the discovery of new deposits.

  7. 7.

The countries which don’t product oil have to diversify their energy sources
as soon as possible. It would be a good orientation even for the countries
which product oil. Because the resources will disappear at a time $T^{*}$.

  8. 8.

The strategies to develop the research in the other type of energy and their
production have to be encouraged. This assertion is justified in the following
sense: when we plot the necessary optimal condition (see Figure1 a)222The same
interpretation can be done for the real data [Figure 1b)], it is easy to see
that, if the price of the barrel goes far from 29 dollars, $a^{*}$ decreases.
And then the super coast $S(t)$ is minimized. But this quantity will not
suffice for the consumption in the country because of growth of the needs in
energy.

(a) Simulated data $(c0=\%\bar{\alpha}=0.2$, m=2)

(b) Evolution of world demand oil

(c) S(t) Real data from from International Energy Agency (IEA)

(d) Oil price data from EIA

Figure 1: Wade Curve

(a) oil wold demande optimal control result

(b) Objective optimal control function

(c) Reserves real data from IEA

Notes: The horizontal lines in Fig. 2 represent the maximum price of a barrel
of oil during the different crises (Subprime around $146, covid19 before the
Ukrainian war, $85 and Russo-Ukrainian war around $124).The vertical lines are
the beginning of the different cries

##  5 Concluding and Remarks

The Wade Formula is also satisfied if one day in the future the 29 dollars are
not reasonable like a win win price. It suffices to replace 29 by $P_{0}(t)$
where $P_{0}(t)$ could satisfy the following system of equation and without
financial speculation $\frac{dP_{0}(t)}{dt}=f(i(t))-\mu P_{0}(t)$ and
$P_{0}(t)=i(t)+pr$. The index $i$ is the investment per barrel, $\mu\in[0,1[$
is the depreciation rate, $pr$ represents the reasonable profit and $f$ is a
function to be determined.

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# Electroweak Sphaleron in a Magnetic field

Jaakko Annala<EMAIL_ADDRESS>Department of Physics and Helsinki
Institute of Physics, PL 64 (Gustaf Hällströmin katu 2), FI-00014 University
of Helsinki, Finland  Kari Rummukainen<EMAIL_ADDRESS>Department of Physics and Helsinki Institute of Physics, PL 64 (Gustaf
Hällströmin katu 2), FI-00014 University of Helsinki, Finland

###### Abstract

Using lattice simulations we calculate the rate of baryon number violating
processes, the sphaleron rate, in the Standard Model with an external
(hyper)magnetic field for temperatures across the electroweak crossover,
focusing on the broken phase. Additionally, we compute the Higgs expectation
value and the pseudocritical temperature. The electroweak crossover shifts to
lower temperatures with increasing external magnetic field, bringing the onset
of the suppression of the baryon number violation with it. When the
hypermagnetic field reaches the magitude $B_{Y}\approx 2T^{2}$ the crossover
temperature is reduced from $160$ to $145$ GeV. In the broken phase for small
magnetic fields the rate behaves quadratically as a function of the magnetic
flux. For stronger magnetic fields the rate reaches a linear regime which
lasts until the field gets strong enough to restore the electroweak symmetry
where the symmetric phase rate is reached.

##  I Introduction

The results from the ATLAS and CMS experiments at the LHC are in complete
agreement with the Standard Model of particle physics: a Higgs boson with a
mass of $\approx 125$ GeV has been discovered Aad  _et al._ (2012); Chatrchyan
_et al._ (2012), and no evidence of beyond-the-Standard-Model physics has been
observed. If the electroweak-scale physics is fully described by the Standard
Model, then the electroweak symmetry breaking transition in the early Universe
was a smooth crossover from the symmetric phase at $T>T_{c}$, where the
expectation value of the Higgs field was approximately zero, to the broken
phase at $T<T_{c}$ where it is finite, reaching the value $246/\sqrt{2}$ GeV
at zero temperature.

The infrared problems inherent in high-temperature gauge theories Linde
(1979); Gross  _et al._ (1981) make the physics nonperturbative. The overall
nature of the transition was resolved already in the 1990s using lattice
simulations Kajantie  _et al._ (1996a); Gurtler  _et al._ (1997); Csikor  _et
al._ (1999); Rummukainen  _et al._ (1998), which indicated that the transition
is first order with Higgs masses $\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}72$ GeV, and
crossover otherwise. More recently, the precise thermodynamics of the
crossover at the physical Higgs mass was analyzed in Ref. D’Onofrio and
Rummukainen (2016) (see also Laine and Meyer (2015)), and e.g. the crossover
temperature was determined to be $T_{c}=159.6\pm 1.5$ GeV.

The chiral anomaly of the electroweak interactions lead to the non-
conservation of the baryon and the lepton number ’t Hooft (1976). In
electroweak baryogenesis scenarios Kuzmin  _et al._ (1985); Rubakov and
Shaposhnikov (1996) the baryon number of the Universe arises through processes
at a first order electroweak phase transition, and a smooth crossover makes
these ineffective. Thus, in electroweak baryogenesis the origin of the baryon
asymmetry must be due to beyond-the-Standard-Model physics (for reviews, see
e.g. Di Bari (2022); Morrissey and Ramsey-Musolf (2012)).

The Chern-Simons (CS) number for weak SU(2) gauge is defined as

$\displaystyle N^{W}_{\rm CS}(t)$
$\displaystyle\equiv\frac{g^{2}}{32\pi^{2}}\int_{0}^{t}\mathrm{d}t\int\mathrm{d}^{3}x\epsilon_{\alpha\beta\gamma\delta}\textrm{Tr}\,F^{\alpha\beta}F^{\gamma\delta}\
,$ (1)

where $F^{\alpha\beta}$ is the field strength tensor of SU(2), $g$ is the
SU(2) gauge coupling and $\epsilon_{\alpha\beta\gamma\delta}$ is the totally
antisymmetric tensor. Analogous to SU(2), the hypercharge U(1) CS number is
given by

$N^{Y}_{\rm
CS}(t)\equiv\frac{g^{\prime}{}^{2}}{32\pi^{2}}\int_{0}^{t}\mathrm{d}t\int\mathrm{d}^{3}x\epsilon_{\alpha\beta\gamma\delta}B^{\alpha\beta}B^{\gamma\delta}\
,$ (2)

where $g^{\prime}$ is the hypercharge gauge coupling. The chiral anomaly
couples the baryon and lepton numbers to the change in the Chern-Simons
numbers as

$\Delta B=\Delta L=3\Delta N_{\rm CS}\ ,$ (3)

where

$\displaystyle N_{\rm CS}(t)\equiv N^{W}_{\rm CS}(t)-N^{Y}_{\rm CS}(t)\ .$ (4)

For SU(2), the Chern-Simons number is topological, and there exists infinitely
many classically equivalent but topologically distinct vacua that cannot be
continuously transformed into one another without crossing an energy barrier.
The sphaleron is a saddle point finite energy solution of the classical field
equations separating two topologically distinct vacua Manton (1983);
Klinkhamer and Manton (1984). The CS number is an integer for vacuum field
configurations and a half integer $N^{W}_{\rm CS}=\tfrac{1}{2}+n,\
n\in\mathbb{Z}$ for sphaleron configurations Klinkhamer and Manton (1984).

In contrast, the U(1) field has trivial topology and without an external
(hyper)magnetic field its CS number in vacuum vanishes. However, in an
external magnetic field the vacuum is degenerate with respect to the U(1) CS
number which can obtain any value in contrast to the SU(2) case where it is an
integer Giovannini and Shaposhnikov (1998); Joyce and Shaposhnikov (1997).
This can lead to baryon and lepton number change on its own Giovannini and
Shaposhnikov (1998); Joyce and Shaposhnikov (1997); Figueroa and Shaposhnikov
(2018); Figueroa  _et al._ (2019); Kamada and Long (2016); Kamada (2018).

Close to thermal equilibrium the evolution of the CS number is diffusive and
is described by a diffusion constant known as the sphaleron rate

$\Gamma=\lim_{V,t\to\infty}\frac{\langle N_{\rm CS}(t)^{2}\rangle}{Vt}\ .$ (5)

In the absence of hypermagnetic fields the contribution of the U(1) can be
neglected due to it having little effect on the form of the phase transition
Kajantie  _et al._ (1996a); D’Onofrio and Rummukainen (2016); Laine and
Rummukainen (1999) and the sphaleron rate Klinkhamer and Manton (1984);
Kleihaus  _et al._ (1991); Klinkhamer and Laterveer (1992); Kunz  _et al._
(1992). In this framework the sphaleron rate has been studied extensively with
analytical and numerical lattice methods. The general behavior is that in the
broken phase the rate is suppressed by the energy of the sphaleron
$\Gamma_{\text{brk}}\sim\alpha_{W}^{4}T^{4}e^{-E_{sph}/T}$ Arnold and McLerran
(1987); Moore (1999) and in the symmetric phase the rate is unsuppressed
behaving as $\Gamma_{\text{sym}}\sim\ln(1/\alpha_{W})\alpha_{W}^{5}T^{4}$
Arnold  _et al._ (1997); Bodeker (1998); Arnold  _et al._ (1999a); Moore
(2000a); Moore and Rummukainen (2000); Bodeker  _et al._ (2000), where
$\alpha_{W}=g^{2}/(4\pi)$. In the Standard Model with the physical Higgs mass
the sphaleron rate was recently measured using lattice simulations across the
crossover from the symmetric phase to deep in the broken phase D’Onofrio  _et
al._ (2014). The temperature where the transitions decouple, i.e. the baryon
number freezes, was found to be $\approx 132$ GeV, substantially below the
crossover temperature $\approx 160$ GeV.

The presence of a U(1) hypercharge magnetic field can affect both the
thermodynamics of the crossover and the sphaleron rate. Large scale magnetic
fields exist in the Universe which may have primordial origin, see e.g.
reviews Subramanian (2016); Vachaspati (2021). Primordial magnetic fields
could have been generated before the electroweak transition corresponding to
hypermagnetic fields before the transition which turn into the
U(1)${}_{\text{em}}$ magnetic fields after the transition. The magnitude of
such fields are largely unconstrained Durrer and Neronov (2013). However, see
Kamada  _et al._ (2021) for recent stronger constraints at larger scales.

When the U(1) is taken into account the spherical symmetry of the sphaleron
reduces to axial symmetry and the sphaleron has a magnetic dipole moment. (It
has been shown to be formed from magnetic monopole-antimonopole pair and a
loop of electric current Hindmarsh and James (1994).) Thus the minimum energy
of the sphaleron can be lowered by an external magnetic field. In a small
external field analytical estimates give a simple dipole interaction $\Delta
E_{sph}=-\vec{B}_{ext}\cdot\vec{\mu}_{sph}$ Comelli  _et al._ (1999). In
addition the form of the phase transition is modified by an external magnetic
field Kajantie  _et al._ (1999) which has an effect on the sphaleron rate
through the transition.

At zero temperature the classical sphaleron energy has been computed on the
lattice for a wide range of magnetic field values Ho and Rajantie (2020). The
situation is complicated by the appearance of Ambjorn-Olesen phase for large
magnetic field values. At a critical field value $B_{c1}=m_{W}^{2}/e$ the
ground state becomes a nontrivial vortex structure and at a second critical
value $B_{c2}=m_{H}^{2}/e$ the electroweak symmetry is restored Ambjorn and
Olesen (1988, 1989, 1990); Chernodub  _et al._ (2022). The sphaleron energy is
found to decrease until at the second critical field value when the symmetry
is restored the energy vanishes Ho and Rajantie (2020). At finite temperature
around the electroweak scale previous studies have not been able to find the
aforementioned vortex phase Kajantie  _et al._ (1999).

Elaborate methods have been developed to compute the sphaleron rate on the
lattice accurately Bodeker (1999); Moore (1999, 2000b); D’Onofrio  _et al._
(2012). We employ the dimensionally reduced effective theory of the Standard
Model and perform the first dynamical simulations of the sphaleron rate which
includes the U(1) field and compute the sphaleron rate for different
magnitudes for the external hypermagnetic field over the electroweak crossover
with focusing on the behavior in the broken phase.

The structure of the paper is as follows. In Sec. II we describe the effective
theory and its lattice formulation that we will use in our simulations. In
Sec. III the methods used to measure the sphaleron rate from the lattice is
described. In Sec. IV we present the results and finally in Sec. V we
conclude.

##  II Effective three-dimensional theory

In our simulations we use dimensionally reduced three-dimensional effective
theory of the Standard Model. The method of dimensional reduction is made
possible due to the fact that in finite temperature the fields are naturally
expressed in terms of three-dimensional Matsubara modes having thermal masses
around $\pi T$. This and the fact that Standard Model couplings are
sufficiently small around the electroweak scale gives rise to a parametric
hierarchy of scales in the Euclidean path integral $\pi T$, $gT$, $g^{2}T$,
called superheavy, heavy and light scales respectively. This allows us to
integrate out the superheavy and heavy modes by well defined perturbative
methods. All the fermionic modes are integrated out since their Matsubara
frequencies $\omega^{f}=(2n+1)\pi T$ are all proportional to $\pi T$. In
addition all temporal bosonic modes $\omega^{b}=2\pi nT$, $n\neq 0$ are also
integrated out. Thus we are left with a 3d purely bosonic effective theory
with the soft scales $g^{2}T$. The soft scales have to be studied regardless
with nonperturbative methods due to the infrared problem in thermodynamics of
Yang-Mills fields Linde (1980). The resulting (super-)renormalizable
Lagrangian reads

$\displaystyle L=$
$\displaystyle\frac{1}{4}\textrm{Tr}\,F_{ij}F_{ij}+\frac{1}{4}B_{ij}B_{ij}$
$\displaystyle+(D_{i}\phi)^{\dagger}D_{i}\phi+m_{3}^{2}\phi^{\dagger}\phi+\lambda_{3}(\phi^{\dagger}\phi)^{2},$
(6)

where

$\displaystyle F_{ij}$
$\displaystyle=\partial_{i}A_{j}-\partial_{j}A_{i}-g_{3}[A_{i},A_{j}],~{}~{}~{}A_{i}=\mbox{${\frac{1}{2}}$}\sigma_{a}A_{i}^{a}$
$\displaystyle B_{ij}$ $\displaystyle=\partial_{i}B_{j}-\partial_{j}B_{i}$ (7)
$\displaystyle D_{i}$
$\displaystyle=\partial_{i}+ig_{3}A_{i}+ig^{\prime}_{3}B_{i}/2.$

Here $A_{i}$,$B_{i}$ are the 3d SU(2) and U(1) gauge fields; $g_{3}$,
$g_{3}^{\prime}$ are the dimensionful SU(2) and U(1) couplings and $\phi$ is a
complex scalar doublet.

The dimensionful parameters of the 3d theory
$g_{3},g_{3}^{\prime},\lambda_{3},m_{3}^{2}$ are mapped to Standard Model
parameters $\alpha_{S},G_{F},m_{H},m_{W},m_{Z},m_{t}$ and the temperature $T$
via a perturbatively computable functions. All the details of the construction
of the effective 3d theory and the mapping of parameters can be found in Refs.
Farakos  _et al._ (1995, 1994); Kajantie  _et al._ (1996b). The accuracy of
the 3d effective theory has been estimated to be $\sim 1\%$ Jakovac  _et al._
(1994); Farakos  _et al._ (1994, 1995); Kajantie  _et al._ (1996b); Laine
(1999).

We choose the SU(2) coupling $g_{3}^{2}$ to set the scale and use a set of
dimensionless couplings defined by

$x\equiv\frac{\lambda_{3}}{g_{3}^{2}},\qquad
y\equiv\frac{m_{3}^{2}}{g_{3}^{4}},\qquad z\equiv\frac{g_{3}^{\prime
2}}{g_{3}^{2}}.$ (8)

The three parameters and the scale are plotted in Fig. 1 in the relevant
temperature range (a code for computing these parameters can be found in
zenodo Annala (2023)). As seen from the plot only the parameter $y$ varies
significantly over temperature and it is the natural choice for the
temperature variable of the system. In D’Onofrio and Rummukainen (2016) the
crossover temperature (defined as the peak of the susceptibility of the Higgs
condensate) was found to be few GeV below the temperature where $y=0$. We find
$y=0$ at $T=162.9$ GeV which is slightly different from D’Onofrio and
Rummukainen (2016) due to using an updated value for the top mass.

Figure 1: The temperature dependence of the dimensionless effective 3d theory
parameters using the Standard model parameters $G_{F}=1.1663788\times
10^{-5}\,\text{GeV}^{-2}$, $m_{H}=125.25$ GeV, $m_{Z}=91.1876$ GeV,
$m_{W}=80.377$ GeV, $m_{t}=172.69$ GeV, $\alpha_{S}=0.1179$ Workman and Others
(2022).

###  II.1 Lattice action

The 3d effective theory in purely bosonic and straightforward to put on the
lattice. For convenience we write the Higgs field as

$\Phi=\frac{1}{g_{3}^{2}}\biggl{(}(\tilde{\phi})(\phi)\biggr{)}\equiv\frac{1}{g_{3}^{2}}\left(\begin{array}[]{cc}\phi_{2}^{*}&\phi_{1}\\\
-\phi_{1}^{*}&\phi_{2}\end{array}\right),$ (9)

which transforms under the SU(2)$\times$U(1) gauge transformation as

$\Phi(x)\to G(x)\Phi e^{-i\theta(x)\sigma_{3}},$ (10)

where $\sigma_{3}$ is the third Pauli matrix and $G(x)$ is an element of
SU(2). Now the lattice action that corresponds to the continuum theory (6) can
be written as

$\displaystyle S$
$\displaystyle=\beta_{G}\sum_{x}\sum_{i<j}[1-\mbox{${\frac{1}{2}}$}\textrm{Tr}\,P_{ij}]+\beta_{Y}\sum_{x}\sum_{i<j}\mbox{${\frac{1}{2}}$}\alpha_{ij}^{2}$
$\displaystyle-\beta_{H}\sum_{x}\sum_{i}\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}(x)U_{i}(x)\Phi(x+i)e^{-i\alpha_{i}(x)\sigma_{3}}$
(11)
$\displaystyle+\beta_{2}\sum_{x}\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}(x)\Phi(x)+\beta_{4}\sum_{x}\big{[}\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}(x)\Phi(x)\big{]}^{2},$

where

$\displaystyle P_{ij}(x)$
$\displaystyle=U_{i}(x)U_{j}(x+\hat{i})U^{\dagger}_{i}(x+\hat{j})U^{\dagger}_{j}(x),$
(12) $\displaystyle\alpha_{ij}(x)$
$\displaystyle=\alpha_{i}(x)+\alpha_{j}(x+\hat{i})-\alpha_{i}(x+\hat{j})-\alpha_{j}(x).$
(13)

Here $U_{i}(x),\alpha_{i}(x)$ are the SU(2) and noncompact U(1) link
variables, respectively, and $P_{ij},\alpha_{ij}$ are their corresponding
plaquettes. The lattice parameters
$\beta_{G},\beta_{Y},\beta_{H},\beta_{2},\beta_{4}$ are related to the
continuum parameters by perturbatively computable functions computed in Laine
and Rajantie (1998). In addition we employ the partial $O(a)$ improvements on
these relations Moore (1997, 1998). Notably,
$\beta_{G}=4/(g_{3}^{2}a)+0.6674...$ with $a$ being the lattice spacing. Rest
of the lengthy relations can be found in Appendix A.

Now the lattice observable
$\langle\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}\Phi\rangle$ is
related to the $\overline{\mbox{\rm MS}}$ renormalized 3d continuum value
$\langle\phi^{\dagger}\phi\rangle$ by Laine and Rajantie (1998)

$\displaystyle\frac{\langle\phi^{\dagger}\phi\rangle}{g_{3}^{2}}=$
$\displaystyle
Z_{g}Z_{m}\Big{[}\langle\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}\Phi\rangle-\frac{\Sigma\beta_{G}}{8\pi}$
$\displaystyle-\frac{3+\bar{z}}{16\pi^{2}}\Bigl{(}\log(3\beta_{G}/2)+0.6679...\Bigr{)}\Big{]},$
(14)

where $Z_{g},Z_{m}$ and $\bar{z}$ are defined in Appendix A. Finally, the 3d
expectation value is related to the physical SM Higgs expectation value $v$ as

$v^{2}/T^{2}=2\langle\phi^{\dagger}\phi\rangle/T.$ (15)

###  II.2 Hypermagnetic field on the lattice

A flux of magnetic field perpendicular to the $x_{3}$ axis,

$g_{3}^{\prime}\Phi_{B}=\int\mathrm{d}x_{1}\mathrm{d}x_{2}B_{12}(x),$ (16)

can be imposed to the lattice by modifying the periodic boundary conditions of
the U(1) link variables $\alpha_{i}$ Kajantie  _et al._ (1999). Requiring the
action to be periodic quantizes the total flux $g_{3}^{\prime}\Phi_{B}/2=2\pi
n_{b}$, $n_{b}\in\mathbb{N}$. Without this restriction there would be boundary
defects and the translational invariance would be lost. One possible way to
add a flux of magnitude $g_{3}^{\prime}\Phi_{B}/2=2\pi n_{b}$ is by modifying
the boundary conditions as

$\alpha_{1}(n_{1},0,n_{3})-\alpha_{1}(n_{1},L_{2},n_{3})=2\pi
n_{b}\delta_{n_{1},1}\ ,$ (17)

for each $n_{3}$ in a lattice with extent $L_{1}L_{2}L_{3}$. We define a
dimensionless parameter describing the average magnetic flux density as

$b\equiv\frac{g_{3}^{\prime}B_{Y}^{3d}}{g_{3}^{4}}=\frac{4\pi
n_{b}}{L_{1}L_{2}}\left(\frac{1}{g_{3}^{2}a}\right)^{2}\ ,$ (18)

where $B_{Y}^{3d}\equiv\Phi_{B}/(L_{1}L_{2})$ is the magnetic flux density
which is related to the four-dimensional density as $B^{3d}_{Y}\simeq
B_{Y}^{4d}/\sqrt{T}+\mathcal{O}(g^{\prime}{}^{2})$. The dimensionless
parameter then relates to the $4d$ flux approximately as
$B_{Y}^{4d}=(g^{\prime}/g^{4})bT^{2}+\mathcal{O}(g^{\prime}{}^{3})$. We cannot
use the effective 3d theory to simulate arbitrarily large magnetic fields due
to the external magnetic field affecting higher dimensional operators
invalidating the effective theory, so we require $b\ll 2\pi^{2}$ Kajantie  _et
al._ (1999).

##  III Measuring the Sphaleron rate

###  III.1 Real time evolution

In itself the effective 3d theory (6) does not describe dynamical phenomena,
such as the sphaleron process. As shown by Arnold, Son and Yaffe Arnold  _et
al._ (1997), the classical equations of motion suffer from ultraviolet
divergences which prevent taking the continuum limit on the lattice.

However, in SU(2) gauge theory the dynamics of the soft modes ($k\lesssim
g^{2}T$), which are relevant for sphaleron transitions, are fully overdamped
and to leading logarithmic accuracy $1/\ln(1/g)$ the evolution can be
described by Langevin equation with Gaussian noise $\xi^{a}_{i}$ (in $A_{0}=0$
gauge) Bodeker (1998, 1999); Arnold  _et al._ (1999b, a):

$\displaystyle\partial_{t}A_{i}$
$\displaystyle=-\frac{1}{\sigma_{el}}\frac{\partial H}{\partial
A_{i}}+\xi^{a}_{i}\ ,$ (19)
$\displaystyle\langle\xi^{a}_{i}(x,t)\xi^{b}_{j}(y,t^{\prime})\rangle$
$\displaystyle=2\sigma_{el}T\delta_{ij}\delta^{ab}\delta^{3}(x-y)\delta(t-t^{\prime})\
,$ (20)

where $H/T=S$ with $S$ defined in (II.1). Here $\sigma_{el}\simeq 0.9239T$
Arnold and Yaffe (2000) is the non-Abelian color conductivity of SU(2).

It can be shown that any diffusive field update algorithm, for example the
heat bath update, is equivalent to Langevin evolution Moore (2000a). This is
advantageous because heat bath update is computationally much more efficient.
The Langevin time $t$ for SU(2) can be related to performing $n$ full random
order heat bath update sweeps as $\Delta t=\tfrac{1}{4}\sigma_{el}a^{2}n$ and
the leading corrections are observed to be small Moore (2000a); Moore and
Rummukainen (2001). The heat bath approach enables us to take a well-defined
continuum limit on the lattice.

The Higgs field evolves parametrically much faster than the SU(2) gauge field
Moore and Rummukainen (2001). Thus, the Higgs field almost equilibrates in the
background of the instantaneous SU(2) field. This can be achieved by updating
the Higgs field much more often than the SU(2) field. We use a mixture of
overrelaxation and heat bath updates, see Kajantie  _et al._ (1996c) for
details of the algorithms used. We increased the number of Higgs updates until
the lattice observables of interest stayed constant resulting to around 50
more Higgs updates per gauge field update (similarly as in Gould  _et al._
(2022)).

Finally, in the broken phase the U(1) field also evolves faster than the SU(2)
gauge field for wavelengths relevant for sphaleron transitions. The size of
the sphalerons $\sim(g^{2}T)^{-1}$ is given by the SU(2) dynamics. Because the
U(1) gauge coupling $g^{\prime 2}$ is much smaller than the SU(2) coupling
$g^{2}$, the U(1) modes with wavelength $\lambda\sim(g^{2}T)^{-1}$ behave as
weakly coupled nondamped modes evolving with timescale $\tau\approx\lambda$.
This is in contrast to the overdamped SU(2) evolution with timescale
$\propto(\lambda^{2})$. Thus, on the lattice the sphaleron rate should be
independent of the U(1) update rate provided it is frequent enough in
comparison with the SU(2) updates. Indeed, we have tested this behavior with a
few simulations with different heat bath update frequencies for the U(1) field
and found no significant effect to the sphaleron rate, as seen from Fig. 2. In
our final analysis we use equal update frequency for SU(2) and U(1) fields.

Figure 2: Comparing the obtained sphaleron rate with different U(1) update
frequencies. On the y axis the ratio of U(1) updates per SU(2) update. The
dependence is observed to be negligible with our statistical accuracy.

We note that in the broken phase the magnetic field remains unscreened and
very long-range magnetic fields evolve very slowly in comparison with other
fields (magnetohydrodynamics). These modes have very small effect on the
sphalerons, and indeed if there is no external (hyper)magnetic field the
contribution from the U(1) sector is usually ignored D’Onofrio  _et al._
(2012, 2014); Moore (1999).

In the symmetric phase the SU(2) and U(1) Chern-Simons numbers are effectively
decoupled and evolve independently. The U(1) Chern-Simons number is not
topological, and there is no characteristic length scale for its evolution
when the external magnetic field is present. It is not clear how to accurately
capture the full quantum dynamics in numerical lattice simulations in this
case. However, this is not a problem for the analysis of the sphaleron rate in
the broken phase, and, as will be discussed in Sec. IV, the effect of the U(1)
remains subleading in comparison with the SU(2) rate in the symmetric phase.

###  III.2 Calibrated cooling

Topology is not well defined on a discrete lattice, and a naive discretization
of the CS number leads to ultraviolet noise which ruins the sphaleron rate
measurement. However, for sufficiently fine lattice spacing the sphaleron is
large in lattice units with a length scale of order $1/(g^{2}T)$ Moore and
Rummukainen (2000). This makes it possible to use methods which filter out the
ultraviolet noise and allows us to accurately integrate the CS number. One of
these methods is the calibrated cooling Ambjorn and Krasnitz (1997); Moore
(1999), which we employ here with the modification that we use gradient flow
for all fields and integrate both the SU(2) and the U(1) CS numbers.
Crucially, in the broken phase we track the difference of the CS numbers (4).
Periodically we cool all the way to the vacuum and check that the vacuum-to-
vacuum integration result is close to an integer and remove any residuals in
order to avoid the accumulation of errors.

Parametrizing the SU(2) links as
$U_{i}(x)=\exp[i\theta^{a}_{i}(x)\sigma^{a}/2]$ the gradient flow can be
written as

$\displaystyle\frac{\partial U_{i}(x)}{\partial\tau}$
$\displaystyle=-i\frac{\sigma^{a}}{2}U_{i}(x)\frac{\partial
S}{\partial\theta^{a}(x)}\ ,$ (21)
$\displaystyle\frac{\partial\alpha_{i}(x)}{\partial\tau}$
$\displaystyle=-\frac{\partial S}{\partial\alpha_{i}(x)}\ ,$ (22)
$\displaystyle\frac{\partial\Phi(x)}{\partial\tau}$
$\displaystyle=-\frac{\partial S}{\partial\Phi(x)}\ ,$ (23)

where $\tau$ is the flow time. Evolving the fields with the gradient flow
equations removes ultraviolet fluctuations smoothing the fields with a
smoothing radius related to the flow time by $r=\sqrt{6\tau}a$ in three
dimensions Lüscher (2010).

With these methods we can integrate the CS number accurately from a real time
trajectory generated by heat bath updates. In the symmetric phase the SU(2) CS
number diffuses rapidly between vacua. In the broken phase the SU(2) and U(1)
gauge fields mix and the diffusion of the difference of the Chern-Simons
numbers, $N_{\rm CS}=N_{\rm CS}^{W}-N_{\rm CS}^{Y}$, slows down dramatically,
jumping between integer values. This can be seen in Fig. 3, where the CS
number is measured somewhat below the crossover temperature. Interestingly,
the SU(2) and U(1) Chern-Simons numbers are not suppressed individually, only
their difference is. This is precisely the quantity which couples to the
baryon and lepton number.

Finally, from the real time trajectory we can compute the sphaleron rate. We
use the cosine transform method described in Moore and Turok (1997).

Figure 3: Real time CS trajectory in the broken phase at $T=153$ GeV in an
external magnetic field $b=0.196$. SU(2) $N_{\rm CS}$ trajectory (yellow),
U(1) $N_{\rm CS}$ trajectory (red) and their difference (blue). It is clear
that the difference becomes frozen at low temperatures.

###  III.3 Multicanonical method

Near the crossover temperature we measure the sphaleron rate using the real-
time simulation methods discussed above, but deep in the broken phase the rate
gets strongly suppressed and normal methods become impractical. At any
reasonable amount of simulation time only few transitions take place, if any.
Thus in the broken phase we have to use special multicanonical methods to
compute the rate. Details of the method can be found in Moore (1999); Moore
and Rummukainen (2001); D’Onofrio  _et al._ (2012). The computation consists
of two parts. The multicanonical method is used to measure the probabilistic
suppression of the sphaleron at the height of the potential barrier, i.e. the
$N_{\rm CS}$ distribution between two integer vacua $P(N_{\rm CS})$. In a
nonzero magnetic field we need to use the $N_{\rm CS}$ given by (4) so that in
a vacuum it is an integer. Dynamical simulations are performed to compute the
rate of tunneling over the top of the barrier. The tunneling rate is computed
by measuring $|\Delta N_{\rm CS}/\Delta t|$ from dynamical simulations when
the trajectory crosses the sphaleron barrier $N_{\rm CS}=\tfrac{1}{2}$. This
needs to be compensated by a dynamical prefactor
$\mathrm{d}=\sum_{\text{traj}}\delta_{\text{tunnel}}/(N_{\text{cross}}N_{\text{traj}})$
where $\delta_{\text{tunnel}}=0$ if the trajectory does not get to a new
vacuum and $\delta_{\text{tunnel}}=1$ if it does, and $N_{\text{cross}}$ is
the number of times $N_{\rm CS}$ crosses the barrier. This is needed due to
the fact that the dissipative update is noisy which can result in multiple
crossing of the barrier in a one trajectory. With these ingredients the
sphaleron rate is given by

$\Gamma=\frac{P(|N_{\rm CS}-\tfrac{1}{2}|<\tfrac{\epsilon}{2})}{\epsilon
V}\left\langle\left|\frac{\Delta N_{\rm CS}}{\Delta
t}\right|\right\rangle\mathrm{d}\ ,$ (24)

where $\epsilon\ll 1$ (we used $\epsilon=0.04$).

##  IV Results

We investigated the lattice spacing dependence of the sphaleron rate with an
external hypermagnetic field for a few temperatures in the symmetric and
broken phase. The parameters were chosen such that the external magnetic field
had the same value, see Table 1.

$4/(ag_{3}^{2})$ | $~{}~{}~{}V/a^{3}~{}~{}~{}$ | $~{}~{}~{}n_{b}~{}~{}~{}$ | $~{}~{}~{}b~{}~{}~{}$
---|---|---|---
5.6 | $16^{3}$ | 2 | 0.196
8 | $16^{3}$ | 1 | 0.196
10 | $20^{3}$ | 1 | 0.196
12 | $24^{3}$ | 1 | 0.196
Table 1: Lattice spacings, volumes, magnetic flux and magnitude of the
external magnetic field used when investigating the lattice spacing
dependence.

In the symmetric phase and close to the crossover in the broken phase the
lattice spacing dependence on the sphaleron rate is small, see the top most
plot in Fig. 4. This is similar to what was observed in previous studies
without U(1) D’Onofrio  _et al._ (2012).

In the broken phase for large lattice size and small lattice spacing even the
multicanonical method becomes very inefficient, making the measurement of the
Chern-Simons number evolution impractical at large lattices. This prevents us
from obtaining sufficient range in lattice spacings for a reliable continuum
limit deep in the broken phase. Nevertheless, our limited results show only a
mild lattice spacing dependence, as shown in Fig. 4. In the following most of
our results have been obtained at single lattice spacing $g_{3}^{2}a=1/2$.

Similar inefficiency was noted in previous works where the U(1) field was
omitted D’Onofrio  _et al._ (2012). In our case the problem appears to be
worse, presumably due to the additional noise of the combined SU(2) and U(1)
Chern-Simons number observable.

Figure 4: Sphaleron rate with few different lattice spacings
$4/(ag_{3}^{2})=5.6,8,10,12$ in an external magnetic field $b=0.196$, see
Table 1. Top most plot at $T=157$ GeV close to the crossover using a normal
simulation. Bottom plot at $T=150$ GeV deep in the broken phase with
multicanonical simulation. Both showing a linear and constant fits.

We investigated the finite volume effects on the sphaleron rate in an external
magnetic field with $b=0.196$ for a few different volumes $L^{3}a^{3}$ with
$L=8/g_{3}^{2},13.9/g_{3}^{2},16/g_{3}^{2}$. The chosen temperatures were in
the symmetric and in the broken phase near the crossover so that we could
still use nonmulticanonical simulations. Similar to the previous studies we do
not observe systematic volume dependence above $L=8/g_{3}^{2}$. In pure SU(2)
theory it was found that $L=8/g_{3}^{2}$ is close to the smallest volume where
the finite size effects are negligible Moore and Rummukainen (2000).

Due to the small observed lattice spacing dependence and no significant finite
size effects at $L=8/g_{3}^{2}$, we present the results for the lattice
parameters $g_{3}^{2}a=1/2$, $V=16^{3}a^{3}$ when deep in the broken phase
where we need to use the multicanonical simulations. With these parameters the
lattice is still small enough for us to get reliable measurements of the CS
number. This enables us to get good statistics with reasonable computational
effort. Due to the magnetic field flux being quantized as (18) the flux quanta
are quite large for small volumes and we can only obtain a few different
values of the magnetic flux for the multicanonical simulations. Thus in
addition we present results for the lattice parameters $g_{3}^{2}a=1/2$,
$V=32^{3}a^{3}$ using nonmulticanonical simulations as deep as possible in to
the broken phase.

For all nonmulticanonical runs we simulated $2\times 10^{6}$ time steps; and
for all multicanonical simulations we generated $12\times 10^{3}$ trajectories
and generated $\sim 3\times 10^{6}$ realizations to estimate the CS number
distribution $P(N_{CS})$.

###  IV.1 Zero magnetic field

Let us first present the results for zero external magnetic field since we
find slightly different results as in previous works. We measure the sphaleron
rate from simulations with and without the dynamical hypercharge U(1) field.
We do not observe any systematic difference between the results, see Fig. 5.
This justifies the omission of the U(1) field when there is no external
magnetic field, as done e.g. in D’Onofrio  _et al._ (2014). Below we discuss
results with the U(1) field included.

The Higgs field expectation value is observed to be very close to the
perturbative result Kajantie  _et al._ (1996c); Laine and Meyer (2015) even
without taking a continuum limit, see $b=0$ points in Fig. 6. For the Higgs
expectation value it is straightforward to check the continuum limit because
the Chern-Simons number measurement can be omitted. We measured the Higgs
expectation value on lattice spacings $g_{3}^{2}a=1/2$, $1/3$ and $1/4$ on a
few temperature values and found the continuum limit to match the perturbative
result.

In the symmetric phase the measured sphaleron rate is approximately constant,
with the value

$\Gamma_{\text{sym.}}/T^{4}=(6.23\pm 0.05)\times 10^{-7}\approx(13.9\pm
0.1)\alpha_{W}^{5}\ ,$ (25)

with $\alpha_{W}\approx 0.03389$ at the electroweak scale.111The numerical
factor in front of $\alpha_{W}^{5}$ includes contributions from logarithmic
factors $\ln\alpha_{W}$ Bodeker (1998). This form is presented for easier
comparisons with earlier work. In the broken phase the rate is well fitted by
a pure exponential and we obtain

$\ln(\Gamma_{\text{brk.}}/T^{4})=(0.86\pm 0.01)T/\text{GeV}-(153.1\pm 0.9)\ .$
(26)

Figure 5: Sphaleron rate in the absence of the external magnetic field in
theories with and without the U(1) field. Data points marked with squares are
obtained using the multicanonical method. Fits are performed using the data
that include U(1). The perturbative line is from Burnier  _et al._ (2006) with
their nonperturbative correction removed.

Using the linear fit we can estimate when the sphaleron processes freeze out.
This happens when the Hubble rate $H(T)$ becomes comparable to the sphaleron
rate $\Gamma(T_{*})/T_{*}^{3}=\alpha(v/T_{*})H(T_{*})$. The function
$\alpha(v/T_{*})$ (where $v$ is the Higgs expectation value) is well
approximated by a constant $\alpha=0.1015$ in the relevant temperature range.
Furthermore, $H(T)^{2}=g_{*}\pi^{2}T^{2}/(90M_{\text{Pl}}^{2})$ where the
effective number of degrees of freedom is well approximated by $g_{*}=106.75$
over the electroweak scale. The Hubble rate is seen in Fig. 5 as the green
line. With these we find the freeze-out temperature $T_{*}=133.5\pm 0.97$ GeV.

The freeze-out temperature is slightly higher than the value obtained in Ref.
D’Onofrio  _et al._ (2014). The differences are due to using an updated value
for the top mass and the fact that the previous simulations did not fully
implement the partial $O(a)$ improvement of the lattice parameters (see
Appendix A). Because neither of the computations have been able to obtain a
reliable continuum limit, the lack of improvement has an effect on the final
results.222We note that the analysis of the thermodynamics of the Standard
Model crossover in Ref. D’Onofrio and Rummukainen (2016) implements the
continuum limit, but the sphaleron rate is not measured.

The main effect of both the improvement and the updated top mass is to
effectively reduce the parameter $y$. The new top mass changes $y$ by an
approximately constant shift $0.02$, whereas the partial $O(a)$ improvement
modifies $y$ in a temperature-dependent manner, so that close to the
pseudocritical temperature at $T\approx 160$ GeV the effect is small but at
$T\sim 140$ GeV it reduces $y$ by $\sim 0.09$. The net effect is that the
sphaleron rate at $b=0$, Fig. 5, reaches a given value at slightly higher
temperatures than in Ref. D’Onofrio  _et al._ (2014): for example, the almost-
symmetric phase value $\ln\Gamma/T^{4}=-16$ is reached at $T=158.8$ GeV in
D’Onofrio  _et al._ (2014) and we obtain $T=159.4$ GeV here. Deep in the
broken phase the shift is slightly larger, value $\ln\Gamma/T^{4}=-30$ is
obtained at $T=141.9$ GeV and $T=143.1$ GeV, in D’Onofrio  _et al._ (2014) and
here respectively. This difference is well within estimated systematic errors.

###  IV.2 Nonzero magnetic field

Let us now look at the results for nonzero external magnetic field. We ran
simulations with $g_{3}^{2}a=1/2$ and volume $V=16^{3}a^{3}$ with magnetic
flux quantum $n_{b}=0,1,2,3,4$ yielding $b$ in range from $b=0$ to $0.785$
with a step of $\Delta b=0.196$. To get a smaller step size for $b$ we
additionally performed simulations with $g_{3}^{2}a=1/2$ and volume
$V=32^{3}a^{3}$ (without multicanonical simulations due to problems discussed
above) with magnetic flux quantum $n_{b}=0,1,2,3,4$ and
$6,8,10,12,14,16,18,20,22,24$ yielding a range $b=0$ to $1.178$ with $\Delta
b=0.049$.

The form of the electroweak crossover is changed by the external magnetic
field. This can be clearly seen from plotting the Higgs expectation value
against the temperature with different magnitudes for the magnetic field, see
Fig. 6. The crossover temperature can be seen to shift to smaller
temperatures.

Figure 6: Higgs expectation value with different values for the magnetic
field, with $V=32^{3}a^{3}$. The lines are added for clarity, they are not
fits. The expectation value becomes negative in the symmetric phase due to
additive renormalization factors, see (14). The gray contours are the zero
magnetic field symmetric Laine and Meyer (2015) and broken phase Kajantie  _et
al._ (1996c) perturbative results.

To get a better picture on the effect of the magnetic field on the crossover
let us look at the susceptibility of the Higgs field. We define the crossover
or pseudocritical temperature $T_{c}$ as the location of the maximum in the
dimensionless susceptibility

$\chi_{\phi^{\dagger}\phi}(T)=VT\left\langle\left[(\phi^{\dagger}\phi)_{V}-\left\langle\phi^{\dagger}\phi\right\rangle\right]^{2}\right\rangle$
(27)

where $(\phi^{\dagger}\phi)_{V}=1/V\int_{V}\phi^{\dagger}\phi$ is the volume
average. We use the interpolating function defined in D’Onofrio and
Rummukainen (2016) to estimate the location of the peak. The susceptibility
with different magnitudes of the magnetic field are shown in Fig. 7. From this
we clearly see that the pseudocritical temperature is shifted to smaller
temperatures and the crossover region gets wider in the sense of widening the
peak of the susceptibility. The pseudocritical temperature against the
magnitude of the magnetic field can be seen in Fig. 8. With small field
magnitudes it behaves quadratically after which it quickly reaches linear
regime. At $b=1$ ($B_{Y}^{4d}\approx 2T^{2}$) the crossover temperature has
decreased from $160$ down to $145$ GeV.

Figure 7: The dimensionless susceptibility with different magnitudes for the
magnetic field, with $V=32^{3}a^{3}$. The lines are from fitting the
interpolating function (defined in D’Onofrio and Rummukainen (2016)) to the
data. Figure 8: Pseudocritical temperature against the magnitude of the
magnetic field.

Let us finally look at how the sphaleron rate is affected by a non-zero
external magnetic field.

We measure the SU(2) and U(1) diffusion rates separately, and in Fig. 9 we
show an example of the behavior of the rates through the crossover at
$b=0.884$. The U(1) diffusion rate $\Gamma_{Y}/T^{4}$ is seen to stay constant
through the transition. In the high-temperature symmetric phase $N_{\rm
CS}^{W}$ and $N_{\rm CS}^{Y}$ evolve independently, and the evolution of
$N_{\rm CS}^{W}-N_{\rm CS}^{Y}$ (with rate $\Gamma/T^{4}$) is slightly faster
than the evolution of each of the components alone. In the broken phase the
Chern-Simons numbers are strongly correlated, and the pure SU(2) rate
$\Gamma_{W}$ is no longer strongly suppressed but reaches a plateau at small
temperatures. Only the physically relevant combination $N_{\rm CS}^{W}-N_{\rm
CS}^{Y}$ becomes frozen. The dashed vertical line at $T=145$ GeV in Fig. 9 is
the point where the measured rate matches the one from the pure SU(2) case.
This is seen to happen systematically around $2$ GeV below the pseudocritical
temperature regardless of the magnitude of the magnetic flux $b$.

Figure 9: Example of diffusion rates of the pure SU(2) CS number, pure U(1) CS
number and their difference (4) for magnetic field magnitude $b=0.884$. Black
dotted vertical line is the pseudocritical temperature.

The full diffusion rate (4) is plotted against the temperature for different
values of the magnetic field in Fig. 10. (For clarity, we do not plot all
values of $b$ that were simulated.) In the symmetric phase the SU(2) sphaleron
rate is unaffected by the presence of the external magnetic field and the data
is compatible with the $b=0$ case in Eq. (25). However, the U(1) rate
increases with increasing $b$, and so does the physically relevant $\Gamma$.
This is discussed in more detail below.

In the broken phase for small field values the slope at which the rate drops
is compatible with the slope obtained from the $b=0$ fit. For larger magnetic
field values we do not have enough data to verify this with confidence, but as
shown in Fig. 10 the suppression of the rate continues to drop at
approximately the same rate as at $b=0$, only the temperature is shifted to
lower values. The shift in temperature is roughly according to the shift in
the pseudocritical temperature, as seen in Fig. 8. For the largest magnetic
field we simulate, $b=1.178$ ($B_{Y}^{4d}/T^{2}\approx 2.3$) the sphaleron
rate suppression is shifted approximately to $22$ GeV lower temperatures from
the $b=0$ case.

Figure 10: Sphaleron rate against temperature. Circle data points are from
$V=32^{3}a^{3}$, diamond data points from $V=16^{3}a^{3}$ and square data
points from multicanonical simulations. Grey dotted lines have the same slope
as the fit of $b=0$ (black) and are shifted according to the shift of the
pseudocritical temperature seen in Fig. 8. Black horizontal line is the $b=0$
symmetric rate fit.

The change in the sphaleron rate when the external field is increased can be
understood to arise from two effects: the Higgs field expectation value
decreases, and the sphaleron interacts with the field through its magnetic
dipole moments. Both of these effects reduce the sphaleron barrier. To isolate
the effects arising from the sphaleron dipole moment from the effects of
changing Higgs expectation value we plot the sphaleron rate against the Higgs
expectation value in Fig. 11. It can be observed that at larger magnetic field
values the Higgs expectation value can become quite large before the onset of
the suppression of the rate.

Finally, in Fig. 12 we show how the sphaleron rate depends on the magnetic
field at constant Higgs expectation value. This enables the comparison with
the semianalytical results in Ref. Comelli  _et al._ (1999), where the change
in the Higgs expectation value was neglected. The rates at constant Higgs
expectation value are obtained by interpolating the data shown in Fig. 11. For
small field values the rate behaves quadratically until around $b\simeq 0.2$
it reaches a linear regime. The linear regime ends when the field gets strong
enough to start restoring the electroweak symmetry where the rate eventually
reaches the $b=0$ SU(2) symmetric phase value (25), see left plot in Fig. 12.

Qualitatively similar behavior is seen when plotting the sphaleron rate with
constant temperature, see Fig. 13. At small magnetic fields the change in
$\ln\Gamma$ is proportional to $b^{2}$, turning into approximately linear
behavior at intermediate $b$ until finally reaching the symmetric phase value
where the rate flattens to constant. Comparing Figs. 6 and 11, we can observe
that the “restoration” of the rate happens before the Higgs field is fully
restored.

Figure 11: Sphaleron rate against the Higgs expectation value. The Higgs
expectation value can get quite large before the rate gets suppressed.

Figure 12: Left: the rate with constant Higgs expectation value. Vertical
dotted line corresponds to the magnetic field value where the expectation
value $\langle\phi^{\dagger}\phi\rangle/T=0.21$ is obtained at the pseudo
critical temperature (i.e. to the right of the line we are getting in to the
symmetric phase). The horizontal gray line is the symmetric $b=0$ sphaleron
rate (25). Right: comparing the difference of the rate $\Delta\ln\Gamma/T^{4}$
from simulations (black) to the analytical estimate (orange). We also plot the
energy difference of the sphaleron configuration (blue) obtained from the
analytical computation, see Appendix B.

Figure 13: How the rate changes with the magnitude of the magnetic field with
constant temperature $T=144$ GeV on the left and $T=155$ GeV on the right. The
vertical dotted line is the value of $b$ where the constant temperature of the
plot is the pseudocritical temperature. The horizontal gray line is the
symmetric $b=0$ sphaleron rate (25).

To compare the simulation results to a semianalytical estimate we did the
analysis presented in Ref. Comelli  _et al._ (1999) (where they used
nonphysical Higgs mass) but now with Standard Model parameters. Details of the
computation can be found in Appendix B. From the analytical computation we get
the sphaleron energy as a function of the external magnetic field. Assuming
that for small fields the change in energy is due to a simple dipole
interaction $\Delta E=-\vec{\mu}_{\text{sph}}\cdot\vec{B}^{4d}_{c}$ and that
the change to the rate
$\Delta\ln\Gamma/T^{4}\equiv\ln\Gamma(b)/T^{4}-\ln\Gamma(b=0)/T^{4}$ is purely
due to the change in energy, the change in the rate is approximately

$\Delta\ln\Gamma/T^{4}\sim\ln\left[\frac{\sinh(\Delta E/T)}{\Delta
E/T}\right].$ (28)

In Ref. Comelli  _et al._ (1999) it was assumed that the change in $\ln\Gamma$
is directly proportional to the change of the minimum energy for the
sphaleron, i.e. $\Delta\ln\Gamma/T^{4}\propto\Delta E/T$. Our result in Eq.
(28) takes into account the random orientations of the magnetic dipoles at
finite $T$. At small fields Eq. (28) gives $\Delta\ln\Gamma/T^{4}\sim(\Delta
E/T)^{2}/6$, and turns into $\sim$ linear behavior at larger field values.

For small external field values Eq. (28) is close to what we obtain from our
simulations, however, the simple dipole approximation quickly becomes invalid,
see Fig. 12. Our results above indicate that at small magnetic fields the
dominant effect on the sphaleron rate arises from the magnetic dipole moment
of the sphaleron, and the change in the Higgs expectation value is subleading.

Figure 14: Left: diffusion rate of the U(1) CS number with different external
magnetic fields (blue) compared with the results from classical simulations
and expected rate from magnetohydrodynamics Figueroa and Shaposhnikov (2018);
Figueroa  _et al._ (2019). The shown rate is computed at $T=168$ GeV, however,
we do not find any systematic temperature dependence with the temperatures
simulated. Right: pure SU(2) rate (gray) and the full rate (black) in the
symmetric phase. The horizontal line is the $b=0$ SU(2) symmetric rate fit
(25) and the blue dashed line is the sum of the latter and the fit from the
left plot. The SU(2) rate $\Gamma_{W}/T^{4}$ stays approximately constant with
increased magnetic field.

Finally, let us look at the behavior of the sphaleron rate in the symmetric
phase. Here the SU(2) rate does not show any systematic dependence on the
magnetic field, see right plot in Fig. 14. Only the U(1) rate is affected by
the magnetic field in the symmetric phase. Despite the ambiguities associated
with the U(1) field evolution in the symmetric phase as discussed in Sec.
III.1, we investigate the U(1) rate in our simulations and the dependence on
the magnetic field fits very well the expected $B_{4d}^{2}$ behavior Figueroa
and Shaposhnikov (2018), see Fig. 14. As seen in Fig. 9 the U(1) rate
$\Gamma_{Y}/T^{4}$ is approximately constant over temperature and we obtain a
fit $\Gamma_{Y}/T^{4}=(0.5\pm 0.01)\times 10^{-3}g^{\prime}{}^{6}B_{4d}^{2}$
(with $g^{\prime}{}^{2}\simeq 0.12237$). Comparing this to results obtained
from classical simulations of U(1)-Higgs theory (scalar QED) performed in
Figueroa and Shaposhnikov (2018); Figueroa  _et al._ (2019) our rate is $\sim
4$ times slower; comparing with magnetohydrodynamics our rate is $\sim 3$
times faster. Given the ambiguities in the update algorithm the qualitative
agreement between the results is good.

##  V Conclusion

Using lattice simulations of an effective 3d theory of the Standard Model we
have computed the baryon violation (sphaleron) rate over the electroweak
crossover deep into the broken phase with an external magnetic field. Both the
baryon violation rate and the form of the electroweak crossover is changed due
to an external magnetic field. We have argued that the fully dissipative
Langevin-type update is accurate to leading logarithmic order in $g_{W}^{2}$
in the broken phase.

For zero external field we computed the rate with and without the U(1) fields
included and found no difference between the results. The zero external field
results differ slightly from previous results D’Onofrio  _et al._ (2014), see
(25) and (26) for our results. The difference is due to us using an updated
value for the top mass (which affects the mapping between the physical and the
effective 3d theory parameters) and the fact that the previous computations
did not fully implement the partial $O(a)$ improvement. Nevertheless, the
difference is well within the uncertainties of the calculation.

The baryon violation rate is affected by an external magnetic field due to
multiple factors. The sphaleron has a dipole moment and its energy can be
lowered. With an external field also the U(1) contributes to the baryon
violating rate. With an external field the combination of the SU(2) and U(1)
CS numbers couple to the baryon violating current and in the broken phase it
is precisely this combination that gets suppressed.

To get a picture of how the electroweak transition is affected by an external
magnetic field we computed the Higgs expectation value and its susceptibility.
As the magnetic field is increased, the crossover shifts to lower temperatures
and the transition region broadens. This shifts onset of the suppression of
the sphaleron rate to lower temperatures. In the broken phase the rate
increases with the external magnetic field. For small fields it increases
quadratically before switching over to a linear regime. The linear regime
stops after the field becomes strong enough to restore the electroweak
symmetry where the rate reaches the symmetric phase value.

For small external fields we performed a semianalytical computation for the
sphaleron energy in an external field (following Comelli  _et al._ (1999)) and
used a simple dipole approximation to estimate the change in the sphaleron
rate. For small field values the semianalytical result and our simulations are
in relatively good agreement, see Fig. 12. This shows that for small fields
the sphaleron dipole moment has the biggest effect on the rate. However, for
larger fields the simple dipole approximation quickly becomes invalid and
nonlinear effects become important.

In the symmetric phase the SU(2) and U(1) Chern-Simons numbers evolve
independently. There are ambiguities in how to perform real-time lattice
simulations of the U(1) field evolution in the symmetric phase. However, from
our simulations we find no significant effect of the magnetic field on the
pure SU(2) rate which behaves as $\propto T^{4}$ and is compatible with the
zero external field value (25). The U(1) part of the rate is found to increase
with the magnetic field with the expected behavior $\propto B_{4d}^{2}$.

Full results of our simulations are available as tables at Zenodo Annala
(2023).

## Acknowledgments

The authors acknowledge the support from the Academy of Finland Grants No.
345070 and No. 319066. Part of the numerical work has been performed using the
resources at the Finnish IT Center for Science, CSC.

##  Appendix A Continuum to lattice parameters with improvements

We use the partial $O(a)$ improvements computed in Moore (1997, 1998) (these
are partial since there is an additive correction to the parameter $y$ that
has not been computed to date). We choose the desired values of $x,y,z$ and
$g_{3}^{2}a$ that we want to simulate and then compute the relevant
counterterms given by

$\displaystyle Z_{g}^{-1}$
$\displaystyle=1+\frac{g_{3}^{2}a}{4\pi}\left(\frac{\pi}{3}+6\xi+\frac{\Sigma}{24}\right),$
(29) $\displaystyle Z_{b}$
$\displaystyle=1+z\frac{g_{3}^{2}a}{4\pi}\left(\frac{\pi}{3}-\frac{\xi}{12}+\frac{\Sigma}{24}\right),$
(30) $\displaystyle Z_{m}^{-1}$
$\displaystyle=1+\frac{g_{3}^{2}a}{4\pi}\left[(9-24x+3z)\frac{\xi}{4}+(3+z)\frac{\Sigma}{24}\right],$
(31) $\displaystyle\delta x$
$\displaystyle=\frac{g_{3}^{2}a}{4\pi}\Big{\\{}\Big{[}1-6x(3+z)+48x^{2}+\tfrac{1}{2}(1+z)^{2}\Big{]}\frac{\xi}{4}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-x(3+z)\frac{\Sigma}{12}\Big{\\}},$
(32)

where $\Sigma=3.175911...\ $ and $\xi=0.152859...\ $ are constants. We then
construct the lattice action with the relations between the continuum
parameters and the lattice parameters
$\beta_{G},\beta_{Y},\beta_{H},\beta_{2},\beta_{4}$ Laine and Rajantie (1998)

$\displaystyle\beta_{Y}$ $\displaystyle=\frac{\beta_{G}}{\bar{z}}\
,\beta_{H}=\frac{8}{\beta_{G}},~{}\beta_{4}=\frac{\beta_{H}^{2}}{\beta_{G}}\bar{x},$
(33) $\displaystyle\frac{\beta_{2}}{\beta_{H}}$
$\displaystyle=3+\frac{8\bar{y}}{\beta_{G}^{2}}-(3+12\bar{x}+\bar{z})\frac{\Sigma}{4\pi\beta_{G}}$
$\displaystyle-\frac{1}{2\pi^{2}\beta_{G}^{2}}\bigg{[}\left(\frac{51}{16}-\frac{9\bar{z}}{8}-\frac{5\bar{z}^{2}}{16}+9\bar{x}-12\bar{x}^{2}+3\bar{x}\bar{z}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\times\big{(}\ln(\tfrac{3}{2}\beta_{G})+0.09\big{)}$
$\displaystyle+4.9-0.9\bar{z}+0.01\bar{z}^{2}+5.2\bar{x}+1.7\bar{x}\bar{z}\bigg{]},$
(34)

using the modified parameters

$\displaystyle\beta_{G}$
$\displaystyle=\frac{4}{g_{3}^{2}a}Z_{g}^{-1}=\frac{4}{g_{3}^{2}a}+0.6674...\
,$ (35) $\displaystyle\bar{x}$ $\displaystyle=\frac{x+\delta x}{Z_{g}},\
\bar{y}=y\frac{Z_{m}}{Z_{g}^{2}},\ \bar{z}=z\frac{1}{Z_{g}Z_{b}}\ .$ (36)

Then the lattice observables are related to the continuum values with the
parameters $x,y,z$ by a multiplicative correction (and possible
renormalization factors). For example, the lattice observable
$\langle\mbox{${\frac{1}{2}}$}\textrm{Tr}\,\Phi^{\dagger}\Phi\rangle$ is
related to the $\overline{\mbox{\rm MS}}$ renormalized 3d continuum value
$\langle\phi^{\dagger}\phi\rangle$ by Eq. (14).

##  Appendix B Small field analytical estimate

In this appendix we present details on the analytical computation in a small
external field. We follow the analysis performed in Comelli  _et al._ (1999).

When the U(1) field is included the sphalerons spherical symmetry is reduced
into an axial symmetry. With the physical value for the weak mixing angle
$\theta_{W}$ the angular dependence of the solution is found to be mild Kunz
_et al._ (1992) at zero magnetic field. The expansion parameter, with external
magnetic field $B^{4d}_{c}$, is effectively $\theta_{W}B^{4d}_{c}/gv^{2}$ and
the angular dependence becomes relevant for larger magnetic fields. For small
fields it suffices to use simpler ansatz that is spherically symmetric
Klinkhamer and Laterveer (1992). The ansatz depends on four functions
$f(\xi),f_{0}(\xi),f_{3}(\xi),h(\xi)$ of dimensionless radial coordinate
$\xi\equiv gvr$, where $v(T)$ is the temperature dependent Higgs expectation
value. The energy functional of the sphaleron using the ansatz (see Klinkhamer
and Laterveer (1992)Comelli  _et al._ (1999)) in a constant external
hypermagnetic field $B^{4d}_{c}$ is $E=E_{0}-E_{\text{dip}}$ with

$\displaystyle E_{0}$ $\displaystyle=\frac{4\pi
v}{g}\int_{0}^{\infty}\mathrm{d}\xi\Bigg{[}\frac{8}{3}f^{\prime}{}^{2}+\frac{4}{3}f_{3}^{\prime}{}^{2}+\frac{1}{2}\xi^{2}h^{\prime}{}^{2}+\frac{4g^{2}}{3g^{\prime}{}^{2}}f_{0}^{\prime}{}^{2}$
$\displaystyle+\frac{8}{3\xi^{2}}\left\\{2f_{3}^{2}(1-f)^{2}+[f(2-f)-f_{3}]^{2}+\frac{g^{2}}{g^{\prime}{{}^{2}}}(1-f_{0})^{2}\right\\}$
$\displaystyle+\frac{h^{2}}{3}\left\\{(f_{0}-f_{3})^{2}+2(1-f)^{2}\right\\}+\frac{\lambda}{4g^{2}}\xi^{2}(h^{2}-1)^{2}\Bigg{]}\
,$ (37)

and

$\displaystyle E_{\text{dip}}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}\xi\frac{8\pi}{3gg^{\prime}v}\left[-2\xi
f_{0}^{\prime}+2(1-f_{0})\right]B^{4d}_{c}\ .$ (38)

The field equations for the ansatz functions turn out as

$\displaystyle
h^{\prime\prime}+\frac{2}{\xi}h^{\prime}-\frac{2h}{3\xi^{2}}\left[2(1-f)^{2}+(f_{0}-f_{3})^{2}\right]-\frac{\lambda}{g^{2}}(h^{2}-1)h=0,$
$\displaystyle
f^{\prime\prime}+\frac{1-f}{\xi^{2}}\left[2f(f-2)+2f_{3}+2f_{3}^{2}\right]+\frac{1}{4}(1-f)h^{2}=0,$
$\displaystyle
f_{0}^{\prime\prime}+\frac{2(1-f_{0})}{\xi^{2}}-\frac{g^{\prime}{}^{2}}{4g^{2}}(f_{0}-f_{3})h^{2}=0,$
$\displaystyle
f_{3}^{\prime\prime}-\frac{2}{\xi^{2}}\left[3f_{3}+f(f-2)(1+2f_{3})\right]-\frac{h^{2}}{4}(f_{3}-f_{0})=0.$
(39)

The ansatz functions are subject to the following boundary conditions

$\displaystyle f,h\to 1,\ f_{3},f_{0}\to 1-\frac{\sin
2\theta_{W}\xi^{2}}{8gv^{2}}B^{4d}_{c},~{}~{}\text{as}\ \xi\to\infty\ ,$
$\displaystyle f,h,f_{3}\to 0,f_{0}\to 1,~{}~{}\text{as}\ \xi\to 0\ .$ (40)

It is convenient to make a change of variables

$g_{i}(\xi)=f_{i}(\xi)+\sin 2\theta_{W}\frac{\xi^{2}}{2gv^{2}},$ (41)

for $i=0,3$, so that the boundary conditions for the new functions at infinity
are simply $g_{i}\to 1$ as $\xi\to\infty$. Furthermore, we use a change of
variables $x\equiv\xi/(3+\xi)$ which maps $\xi\to\infty$ to $x\to 1$. Finally
we use the standard model values for the parameters $\lambda,g,g^{\prime}$ at
the electroweak scale.

With the above we have all the ingredients to compute the change to the
sphaleron energy for small fields in the spherical approximation. From the set
of coupled differential equations (B) we solve the functions $f,h,g_{0},g_{3}$
numerically using a fourth order collocation method implemented in Virtanen
_et al._ (2020) . Equations (B) are divergent at the boundaries and thus we
solve the system only in range $[\epsilon_{1},1-\epsilon_{2}]$ where the
$\epsilon_{i}$ are small offsets. Despite the divergences in the equations the
solutions are completely regular near the boundaries and we just linearly
extrapolate them to boundary values. The accuracy of this simple procedure is
sufficient for our comparison purposes. A set of solved functions is plotted
in Fig. 15 for zero magnetic field and for one example of a nonzero magnetic
field. Typical range of $[\epsilon_{1},1-\epsilon_{2}]$ is $[0.0006,0.913]$,
but even large variations of these values does not significantly change the
solutions or the energy computed from them.

Figure 15: Examples of numerical solutions for the functions $f,h,g_{0},g_{3}$
for zero and non-zero magnetic field.

The energy of the sphaleron configuration is obtained by numerically
integrating over the energy functional while omitting the constant external
magnetic field terms which would make the expression divergent. The energy as
a function of the magnetic field is plotted in Fig. 16.

Figure 16: Energy of the sphaleron configuration (computed using the spherical
approximation) in terms of the magnitude of the magnetic field.

Assuming that the change in energy $\Delta E\equiv E(B=0)-E(B)$ is due to a
simple dipole interaction $\Delta
E=-\vec{\mu}_{\text{sph}}\cdot\vec{B}^{4d}_{c}$ and that the change of the
rate $\Gamma$ is only due to this energy difference $\Gamma\sim\exp(\Delta
E/T)\Gamma_{0}$, where $\Gamma_{0}$ is the rate without magnetic field.
Averaging over the space of orientations for the dipole the change to the rate
is roughly

$\displaystyle\Delta\ln\Gamma/T^{4}$
$\displaystyle\sim\ln\left\\{\int\frac{\mathrm{d}\Omega}{4\pi}\exp\left[-\frac{\mu_{\text{sph}}B^{4d}_{c}}{T}\cos\theta\right]\right\\}$
$\displaystyle\simeq\ln\frac{\sinh(\Delta E/T)}{\Delta E/T}\ .$ (42)

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