$B^{\pm}\rightarrow\pi^{\pm}\ell^{+}\ell^{-}$ | $|V_{L5b}V_{L5d}^{*}+V_{R4b}V_{R4d}^{*}|<2.6\times 10^{-4}$
$B^{0}\rightarrow\mu^{+}\mu^{-}$ | $|V_{L5b}V_{L5d}^{*}-V_{R4b}V_{R4d}^{*}|<2.2\times 10^{-4}$
$Z\rightarrow b\bar{b}$ | $|V_{L5b}|^{2}+0.18|V_{R4b}|^{2}<1.0\times 10^{-3}$
Table 9:  Limits on the mixing of the SM three families with the extra vector-
like weak doublet $q_{L4},q{R4}$ and the weak singlets $u_{L5},u_{R5}$,
$d_{L5},d_{R5}$. See the text for constraints obtained from neutral mesons
mixing and $Z$ decay into hadrons.

Regarding neutral mesons mixing, tree level and loop level contributions from
both LH and RH currents are present, including mixed contributions. By taking
$M_{b^{\prime\prime}}=M_{t^{\prime\prime}}=M_{q}=M$ for simplicity, the new
additional terms in mixing mass of $\bar{K}^{0}\rightarrow K^{0}$ transition
would be:

$\displaystyle M_{12}^{\text{new}}=\frac{1}{3}f^{2}_{K}m_{K^{0}}$
$\displaystyle\bigg{\\{}\frac{G_{F}}{\sqrt{2}}\left[(V_{L5d}^{*}V_{L5s})^{2}+(V_{R4d}^{*}V_{R4s})^{2}-27.5(V_{R4d}^{*}V_{R4s})(V_{L5d}^{*}V_{L5s})\right]+$
$\displaystyle\;\frac{G_{F}^{2}}{16\pi^{2}}M^{2}\Big{[}(V_{L5d}^{*}V_{L5s})^{2}+2(V_{R4d}^{*}V_{R4s})^{2}-54.9(V_{R4d}^{*}V_{R4s})(V_{L5d}^{*}V_{L5s})+$
$\displaystyle\qquad\qquad\;+\left(|V_{L5u}|^{2}V_{Lus}+V_{L5u}^{*}V_{L5c}+V_{L5u}^{*}V_{L5t}V_{Lts}\right)^{2}\Big{]}\bigg{\\}}$
(908)

(where we neglected terms growing logarithmically with the mass of the extra
quarks) and for $D^{0}$ mesons system:

$\displaystyle M_{12\,\text{new}}^{D}=\frac{1}{3}f^{2}_{D}m_{D^{0}}$
$\displaystyle\bigg{\\{}\frac{G_{F}}{\sqrt{2}}\left[(V_{L5u}^{*}V_{L5c})^{2}+(V_{R4u}^{*}V_{R4c})^{2}-3.65(V_{R4u}^{*}V_{R4c})(V_{L5u}^{*}V_{L5c})\right]+$
$\displaystyle\;\frac{G_{F}^{2}}{16\pi^{2}}M^{2}\Big{[}(V_{L5u}^{*}V_{L5c})^{2}+2(V_{R4u}^{*}V_{R4c})^{2}-7.3(V_{R4u}^{*}V_{R4c})(V_{L5u}^{*}V_{L5c})+$
$\displaystyle\qquad\qquad\;+\left(|V_{L5d}|^{2}V_{Lcd}^{*}+V_{L5d}^{*}V_{L5s}+V_{L5d}^{*}V_{L5b}V_{Lcb}^{*}\right)^{2}\Big{]}\bigg{\\}}$
(909)

and similarly for neutral B-mesons systems. Therefore, in this case there can
also be cancellations. However, also constraints on flavour conserving
processes should be taken into account. The results on $Z$-boson decay rate
into hadrons give the bound (see sections 4.4, 5.3, 6.3):

$\displaystyle|V_{R4u}|^{2}+|V_{R4c}|^{2}+0.50(|V_{R4s}|^{2}+|V_{R4d}|^{2}+|V_{R4b}|^{2})+$
$\displaystyle\qquad+2.72(|V_{L5d}|^{2}+|V_{L5s}|^{2}+|V_{L5b}|^{2})+2.24(|V_{L5u}|^{2}+|V_{L5c}|^{2})<4.4\times
10^{-3}$ (910)

Figure 24:  Excluded area in the $|V_{R4u}|$ and $|V_{R4s}|$ parameter space.
The blue area is excluded by experimental constraints from $Z$-boson decay
rate into hadrons (6.5) using $|V_{L5d}|^{2}+|V_{L5u}|^{2}=1.1\times 10^{-3}$.
The blue curve indicates where the bound is shifted if the condition
$|V_{L5d}|^{2}+|V_{L5u}|^{2}=1.5\times 10^{-3}$ (904) is used. The black curve
stands for the solution $|V_{R4u}^{*}V_{R4s}|=0.99(36)\times 10^{-3}$ in eq.
(904).

Figure 24 shows the parameter space in a simplified case in which
$y_{41}^{d}=y_{25}^{d}=y_{42}^{u}=y_{25}^{u}=0$ in eq. (903), corresponding to
$V_{R4d}=V_{L5s}=V_{R4c}=V_{L5c}=0$. The blue area is excluded by the
condition (6.5) using the conservative choice
$|V_{L5d}|^{2}+|V_{L5u}|^{2}=1.1\times 10^{-3}$ for the solution in eq. (904).
The solution $|V_{R4u}^{*}V_{R4s}|=0.99(36)\times 10^{-3}$ in eq. (904) is
indicated by the black curve. In this scenario with
$V_{R4d}=V_{L5s}=V_{R4c}=V_{L5c}=0$, the constraint from neutral $D$-mesons
mixing would give (367):

$\displaystyle|V_{Lub^{\prime\prime}}V_{Lcb^{\prime\prime}}^{*}|\approx|V_{L5d}|^{2}|V_{cd}|<4.6\times
10^{-4}\left[\frac{1\,\text{TeV}}{M_{b^{\prime\prime}}}\right]$
$\displaystyle|V_{L5d}|<0.046\left[\frac{1\,\text{TeV}}{M_{b^{\prime\prime}}}\right]^{1/2}$
(911)

Kaon mixing would receive a real contribution at loop level giving the
constraint (568):

$\displaystyle|V_{Lt^{\prime\prime}d}V_{Lt^{\prime\prime}s}^{*}|\approx\left||V_{L5u}|^{2}V_{Lus}+V_{L5u}V_{L5t}^{*}V_{Lts}\right|<5.2\times
10^{-4}\left[\frac{1\,\text{TeV}}{M_{b^{\prime\prime}}}\right]$ (912)

which together with the constraints from B-mesons decays and mixing gives:

$\displaystyle|V_{L5u}|\lesssim
0.049\left[\frac{1\,\text{TeV}}{M_{t^{\prime\prime}}}\right]^{1/2}$ (913)

This means that, in order to have $|V_{L5d}|^{2}+|V_{L5u}|^{2}>1.1\times
10^{-3}$ (904), it should be for example $M_{b^{\prime\prime}}\approx
M_{t^{\prime\prime}}<4.1$ TeV.

##  7 Conclusion

Present situation of the determination of the CKM matrix elements is very
intriguing. From one side, there is a controversy in the determination of
$|V_{us}|$. Namely, given the present experimental accuracy in kaon leptonic
and semileptonic decays and present theoretical precision in the calculation
of form factors (lattice QCD), there is about $3\sigma$ tension between the
$|V_{us}|$ values extracted from $K\ell 3$ (determination A) and $K\mu 2$
(determination B) decays, as discussed in section 2. Although the recent high
precision calculations in the kaon decays $K\ell 3$ Seng:2021boy  and $K\mu 2$
Martinelli  demonstrate that the radiative corrections should not be
responsible for it, this tension per sè cannot be considered that alarming
since still there can be some theoretical loopholes in the interpretation of
the lattice QCD results, or perhaps some unfixed systematics in the
measurements of the kaon decay rates. In this case, the real value of
$|V_{us}|$ could be near the average $A+B$ between the two above
determinations.

On the other side, the recent calculations of the short distance radiative
corrections in the neutron $\beta$-decay with improved hadronic uncertainties
Seng ; Marciano2  leads to high accuracy in the determination of $|V_{ud}|$
from superallowed $0^{+}-0^{+}$ nuclear transitions (determination C). The
obtained $|V_{ud}|$ value, in combination with the determinations of
$|V_{us}|$, indicates towards a violation of the CKM unitarity at about
$4\sigma$ level. Namely, there appears a significant deficit $\delta_{\rm
CKM}$ in the first row unitarity, as given by Eq. (11).

These anomalies, if confirmed with future high precision data, would indicate
towards some new physics beyond the Standard Model. In Ref. Belfatto:2019swo
we pointed out that the anomalies could be originated by mixing of ordinary
light quarks to some extra vector-like quarks with masses at the TeV scale. In
this paper we gave more detailed study of these scenarios, analyzing one by
one the implications of vector-like quarks in the weak isosinglet or
isodoublet representations. By introducing a weak isosinglet quark of the up
($U$-type) or down ($D$-type) quarks, one can explain the CKM unitarity
anomaly, i.e. deficit of the first row unitarity (11), but the tension between
the two determinations of $|V_{us}|$ cannot be explained in this case.
However, if the latter discrepancy is taken seriously one has to look for a a
solution addressing the whole situation. By introducing a weak isodoublet
($Q$-type) of quarks one could potentially explain both anomalies.

However, there are strong phenomenological limits on the hypothetical vector-
like quarks: participation of vector-like quark species in the quark mixing
induces flavor changing effects at the tree level as well as via box diagrams
involving heavy species. In fact, we show that the latter give a bigger
contribution than the tree level one if the masses of heavy quarks are larger
than 3 TeV or so. However, tree level effects are independently testable in
$Z$-boson physics since $Z$-boson acquires small flavor non-diagonal couplings
with quarks while also its flavor-diagonal couplings get isospin violating
contributions which can be confronted with the limits on many observables
related to $Z$-boson physics.

Two approaches to the above anomalies can be considered. The incompatibility
between two determinations A and B from kaon physics may be attributed to some
uncertainties which perhaps will disappear with more precise calculations.
Neglecting this problem, one can focus instead on the unitarity violation
problem, i.e. on the unitarity deficit which emerges by confronting the
average $A+B$ value of $|V_{us}|$ obtained from kaons, with the value of
$|V_{ud}|$ extracted from superallowed nuclear transitions.

The just mentioned scenarios involving extra isosinglets quarks are on this
line. Namely, the CKM problem can be solved provided that the first family of
quarks has a mixing $|V_{ub^{\prime}}|\simeq 0.04$ with extra $D$-type quark
$b^{\prime}$, or alternatively the same size mixing $V_{t^{\prime}d}$ with
$U$-type quark $t^{\prime}$. For this scenario the most severe limits come
from the flavor changing phenomena, but comparable limits emerge from the
precision data on $Z$-boson decays. There still remains some available
parameter space, although not very large. In fact, we show that constraints
from $K^{0}-\overline{K}^{0}$ and $D^{0}-\overline{D}^{0}$ systems become more
stringent for larger masses of $b^{\prime}$ or $t^{\prime}$. In particular,
e.g. for $M_{b^{\prime}}>1.5$ TeV they would exclude the possibility of having
extra mixing as large as $|V_{ub^{\prime}}|\sim 0.04$ if the unitarity deficit
is due to a down-type species or, in the case of up-type extra quark,
$M_{t^{\prime}}>2.5$ TeV would exclude the possibility of having the extra
mixing $|V_{t^{\prime}d}|\sim 0.04$ (see sections 4.5, 5.4). Therefore, we
claim that if the CKM unitarity anomaly is due to the mixing with extra
isosinglet ($U$ or $D$ type) quarks, then “4th family" states $b^{\prime}$ or
$t^{\prime}$ should be discovered with a mass below a couple of TeV or so.

As we noted above, both $V_{us}$ and CKM unitarity anomalies can be solved by
introducing $Q$-type extra quarks having substantial mixings with both 1st and
2nd families of the normal quarks. In fact, kaon semileptonic decays $K\ell 3$
measure the quark vector current coupling to $W$-boson while leptonic decays
$K\mu 2$ measure the axial current coupling. While both vector and axial
couplings should be identical in the SM frames where $W$-boson couples solely
to LH quarks, their difference can be induced via the mixing with weak
isodoublet $Q$-type quarks. The latter in fact induces some small $W$-boson
couplings with the RH quarks which can be at the origin of the discrepancy
between the two determinations A and B of $|V_{us}|$ element. However, we show
that this solution with only one $Q$-type species is fully excluded by the
flavor changing limits together with $Z$ boson physics and electroweak low
energy observables (section 6.4).

This brings us to conclude that the full solution cannot be achieved by
introducing the only one species of extra $Q$-type quarks: it should be
complemented by isosinglet quarks of $U$ or/and $D$ type, or by another
isodoublet species. In this case somewhat larger available parameter space can
be found with the present limits on flavor-changing phenomena and on anomalous
$Z$-couplings. However, these scenarios are testable with the future
experimental limits (section 6.5). In particular, in the scenario with two
isodoublets, flavour changing limits can be softened, and then the masses of
the extra quarks would not be strictly constrained, apart from a
perturbativity bound (Yukawa couplings less than $1$), which implies $M_{q}<7$
TeV or so. However, in this case the main issue would come from limits in
$Z$-boson physics. In the case of one extra isodoublet with isosinglets (up
or/and down type), also constraints from flavour changing phenomena should be
considered. We conclude that if the CKM anomalies are due to extra vector-like
quarks, then “4th family" quarks should be discovered with masses of few TeV,
and anomalous $Z$-couplings should be detected by improving the experimental
precision.

For the minimality reasons, we did not consider extra vector-like species of
leptons. However, if they also exist at the TeV scale and have the same size
mixings with the electron and muon, then the experimental limits from the
flavor changing phenomena involving leptons or limits on $Z$-boson flavor-
nondiagonal couplings as $Z\overline{e}\mu$ would be much more stringent.

###### Acknowledgements.

The work of B.B. was supported by the ERC research grant NEO-NAT no. 669668.
The work of Z.B. was supported in part by Ministero dell’Istruzione,
Università e della Ricerca (MIUR) under the program PRIN 2017, grant no.
2017X7X85K “The Dark Universe: A Synergic Multimessenger Approach", and in
part by Shota Rustaveli National Science Foundation (SRNSF) of Georgia, grant
DI-18-335 “New Theoretical Models for Dark Matter Exploration".

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redshift in the energy of A by a factor of
$\frac{1}{\left(1+\frac{g\ell}{c^{2}}\right)}$ (Figure 5-(b)). Conversely, the
observer in rocket A will observe a blue shift in B’s energy by a factor of
$\left(1+\frac{g\ell}{c^{2}}\right)$. Indeed, when a photon emitted from A
reaches B, observer B measures a redshift in the wavelength of the photon.
Suppose rockets are accelerated via photon emission and by the impulse
generated by photon momentum. Then, observer at $S_{B}$ observes that the
energy gained by rocket A decreases and consequently its acceleration
decreases to $\frac{g}{1+\frac{g\ell}{c^{2}}}$. This is equal to the proper
acceleration of B. Therefore, according to observer $S_{B}$ the accelerations
of rockets A and B are equal to $\frac{g}{1+\frac{g\ell}{c^{2}}}$. On the
other hand, observer at $S_{A}$ observes that the acceleration of rocket B
increases due to blue shift. According to observer $S_{A}$ the accelerations
of rockets A and B are equal to $g$. As a result, according to the observer on
both rockets, the accelerations of the rockets are equal and the proper
distance $\ell$ remains constant during the acceleration.

We have seen that such a rigid reference frame is always definable with
respect to an accelerating observer A or B, provided that
$\ell<\frac{c^{2}}{g}$. Similarly, we can think of each point between rockets
A and B as material points modeled by accelerating rockets. Thus we can
construct a rigid rod connecting points A and B. Rigid rods constructed in
this way can be used to model the axes of the accelerating $S^{\prime}$ frame.
However, defining an accelerating rigid frame of reference covering such a
finite region with respect to the inertial observer $S_{0}$ is problematic. As
we have already shown, such an accelerated frame does not maintain its rigid
structure with respect to the inertial frame $S_{0}$, but it flows. In the
context of classical relativity, unless we abandon the assumption of the
continuity of acceleration, such an accelerating rigid frame can only be
defined with respect to an inertial frame if its size is infinitesimal. Some
authors have ignored this fact and claimed that the equivalence principle is
invalid [54]. The point that is missed in such assertions, is that a rigid
acceleration is not uniform with respect to an inertial frame. Indeed,
relative to the inertial frame $S_{0}$, the acceleration of the frame
$S^{\prime}$ is not uniform, but different points on its axis have different
accelerations. Therefore, the equivalence principle can only be applied in the
infinitesimal neighborhood of a point of the $S^{\prime}$ frame. In such an
infinitesimal neighborhood, the equivalence principle is unambiguously valid.

For the Kottler-Møller metric (2.6) describing the rigidly accelerating
$S^{\prime}$ frame, the Riemann curvature tensor is $R_{\beta\sigma\mu\nu}=0$.
Therefore, it can be concluded that the metric (2.6) does not describe a
gravitational field. On the other hand, although the metric (2.6) does not
represent a uniformly accelerating frame, it converges to a metric describing
uniform proper acceleration in an infinitesimal neighborhood. The metric
representing such a uniform proper acceleration is given by [54]

$\displaystyle
ds^{2}=-e^{(2gx^{\prime}/c^{2})}c^{2}{dt^{\prime}}^{2}+{dx^{\prime}}^{2}$
(4.39)

where the coordinates $y^{\prime}$ and $z^{\prime}$ perpendicular to the
acceleration direction are neglected. This metric reduces to the Kottler-
Møller metric (2.6) in the limit $x^{\prime}\rightarrow 0$. It can be shown
that the Riemann curvature tensor for metric (4.39) is nonzero:

$\displaystyle
R_{1001}=R_{0110}=\frac{g^{2}}{c^{4}}e^{2gx^{\prime}/c^{2}}=-R_{1010}=-R_{0101}$
(4.40)

Therefore it describes a gravitational field. What we can say about the (2.6)
metric is that it describes the gravitational field locally, but not globally.

We have seen that gravitational redshift occurs in a way that compensates for
the difference between the observations of accelerated and inertial observers.
The inertial frame $S_{0}$ observes that the accelerating $S^{\prime}$ frame
is flowing. According to $S_{0}$, more energy is transferred to the fast-
flowing regions of $S^{\prime}$ than to the slow-flowing regions. But the
$S^{\prime}$ frame does not observe such a flow due to the nature of rigid
acceleration. Thus, the energy supplied by the flow is not observed in the
accelerating frame, which leads to a redshift in energy. Therefore, the ”flow”
observed by the inertial frame is the origin of the gravitational redshift. In
the context of classical relativity, this flow is considered to be a
continuous quantity. However if we take the acceleration as discrete bounces,
as we argued in the previous subsection, the flow becomes discontinuous.
Suppose the acceleration occurs in instantaneous $\Delta v$ steps with time
intervals $\Delta t_{a}$ satisfying the condition $\Delta t_{a}\geq\ell_{0}/c$
in the accelerating reference frame (Figure 4). In such a case, since the
acceleration on the time scale $\Delta t_{a}$ is a completed acceleration,
regions of size $\sim\ell_{0}$ on the flow are represented by inertial frames.
Accordingly, if we restrict ourselves to this time scale, gravitational
redshift and other general relativistic effects do not arise, but we remain in
a world where only special relativity applies. Indeed, for an observation of
the $S_{0}$ inertial frame in the time interval $\Delta t_{a}$, there is no
change in the velocity of the flow, so there is no energy transfer to
different regions of the flow. As a result, no gravitational redshift
occurs.222222Another way to see this is to consider the Doppler shift observed
by $S_{0}$ between two separate points on the accelerating frame. Consider two
points on the accelerating frame, spaced $\ell\lesssim\ell_{0}$. In this case,
when a light wave emitted from the rear point reaches the front point, the
wavelength does not undergo a shift due to acceleration. This is because the
light wave reaches the front point before the $\Delta v$ bounce occurs. On the
other hand, for a long time interval such that $\Delta t>>\Delta t_{a}$, a
large number of consecutive acceleration steps ($\Delta v$ bounces) occurred.
In this case, the acceleration becomes continuous as it is averaged over many
small discrete acceleration steps. We argue that gravitational redshift and
general relativity emerge in this way, that is, the spacetime continuum exists
only in the average sense. We admit that our claim that general relativity
emerged in this way is a speculation in need of further arguments. However,
when we consider that the equivalence principle is a corner stone of general
relativity, we dare to make such a speculation. Obviously, the equivalence
principle is a necessary but not a sufficient principle for general relativity
[55]. The equivalence principle provides a local equivalence of the
acceleration and the gravitational field. On the other hand, the equivalence
principle is insufficient to determine the global properties of the spacetime
manifold. Nevertheless, our dynamical model predicts some implications for
general relativity. If acceleration is of a discontinuous nature, the
application of the equivalence principle implies that the gravitational field
is also discontinuous. For local flatness of the spacetime manifold, an
infinitesimal region is not needed, but in a very small but still finite
region of size $\sim\ell_{0}$ the metric reduces to the Minkowski metric. In
regions of size $\ell>>\ell_{0}$, we speculate that the spacetime metric
evolves from the Minkowski metric by gradually changing in small discrete
steps.

##  5 Conclusions

In this paper, we have approached the theory of relativity within the
framework of a dynamical model that we developed inspired by quantum field
theory. Our approach provides a dynamical explanation for relativistic
phenomena such as length contraction, time dilation and clock
desynchronization, which Einstein’s principle approach leaves in the dark. It
should be noted that our approach differs from that of Lorentz and others
[57], which includes aether models. In our view, the strength of Einstein’s
approach is that there is no single preferred frame of reference in the
universe that can be called the aether frame in which the laws of physics are
defined. On the other hand, when quantum theory is involved, we should be open
to the fact that different frames of reference may temporarily have the status
of preferred frame. We have argued that there is an initial inertial frame of
reference, and this initial frame is the ”preferred“ frame for all other
inertial frames that have accelerated from that initial inertial frame by a
dynamical acceleration process. On the other hand, this initial frame of
reference is not unique, but different initial frames of reference may exist
at different points in spacetime. If we consider the initial reference frame
as a classical frame, the initial synchronization on it is endowed by quantum
decoherence. In other words, the universe freely chose the ”preferential”
synchronization on this initial frame of reference. In the paper, we also use
the term ”preferred” in a second meaning. Our simultaneous interaction
hypothesis (hypothesis (i)) implies that the center-of-mass frame of the
accelerating system is in some sense a preferred frame. This is in the sense
that the interactions occur simultaneously in this frame and is in fact
necessary for rigid acceleration to take place. We then claimed that
hypothesis (i) can be derived from the more fundamental hypothesis
$(\text{i}^{\prime})$, which states that interacting particles interact
simultaneously with force carriers in the center-of-mass frame of the
interacting particles. Quantum field theory deals with relativity from the
perspective of the principle approach. Therefore, the question remains
unanswered as to which frame of reference the fundamental interactions occur
simultaneously. For someone who takes the principle approach strictly, this
might even be a meaningless question. However, within the framework of our
dynamical approach, this question is a meaningful question and requires that
the center-of-mass system of the interacting particles is a preferred frame.
It can be said that nature chooses this frame as the frame in which the
interactions take place simultaneously and makes it preferred. It should be
noted that the notions of preferred frames in our model do not violate the
principle of relativity.232323As we have discussed before, the choice of clock
synchronization of the initial inertial frame is arbitrary in terms of
physical laws. On the other hand, the initial frame is also the preferred
frame for state vector reduction. But the preferentiality here is related to a
part of reality that we do not have access to and does not lead to a change in
the measurement results. Accordingly, the principle of relativity is not
violated. Nevertheless, if there are some laws of physics that determine the
reduction of the state vector, it can be said that the principle of relativity
is violated for these laws. The notion of preferentiality, which we use for
the frame of reference in which interactions are simultaneous with respect to
it, is also not contrary to the principle of relativity. Such a
preferentiality property is Lorentz invariant, although not manifestly Lorentz
invariant. But it requires an extension of the language of relativity
theory.242424If we consider the theory of relativity as an axiomatic system,
we can imagine that we form sentences with grammatical rules from the alphabet
of this system. A strict adherent of the principle approach might argue that
sentences that make sense in the context of relativity theory should contain
only invariant quantities, and that, if we make a formalization of relativity,
grammatical rules should take this fact into account. According to this view,
sentences such as ”this rod is 3 cm long” or ”these two events happened
simultaneously” are meaningless in the context of relativity since they are
observer dependent. Thus, it can be said that if a complete formalization is
made for relativity, grammatical rules should be arranged in such a way that
they do not allow such sentences. We disagree with this view and argue that
the language of relativity theory should be extended to include such
sentences. We make the conjecture that quantum field theory based on a
principle approach to relativity should be replaced by a constructive quantum
field theory to ground the dynamical model we propose. We do not know the
formalism of this constructive quantum field theory. However, we have made
predictions about several features of this theory based on our dynamical
model.

While developing the general theory of relativity, Einstein’s primary goal was
to generalize the theory of relativity in such a way as to ensure the equality
of all reference frames in the description of nature, regardless of their
motion states [56]. Of course, general relativity has a second important goal,
which is to harmonize the Newtonian theory of gravitation with the theory of
relativity. The equivalence principle provided a strong clue that these two
goals could be achieved together. In later years, however, the first goal of
general relativity received less attention and was less studied. In our
opinion, if we judge general relativity considering Einstein’s first goal,
there are some gaps in the path from the study of accelerated motion to
general relativity. Closing these gaps requires knowing the details of
acceleration processes and is only possible with a dynamical approach to
relativity. The kinematic approach to special relativity deals with inertial
frames and the transformations between them, but ignores the previous
acceleration processes of frames. However, the details of the acceleration
process are essential to understand what happens to the moving rods and
clocks. The kinematic approach leaves some questions unanswered. For example,
consider two inertial frames that are at rest with respect to each other. Then
one of the frames accelerates and come to rest in another inertial frame. Let
us denote the inertial frames before and after acceleration as $S_{0}$ and
$S_{1}$ respectively. Relative to $S_{0}$, the rods and clocks on $S_{1}$ have
changed with respect to their pre-acceleration state. The clocks have become
asynchronous and the rods have contracted. But what physical effects cause
clocks and rods to change? For example, the clocks can be modeled with
rotating wheels so that one period of the wheel represents a unit of time.
Then, where is the torque that changes the rotational phase of the wheels? Our
dynamical model provides answers to these questions by considering the details
of the acceleration process. Using our dynamical model, we have shown that the
equivalence principle and gravitational redshift is necessary to explain
special relativistic clock desynchronization. This result is interesting and
shows that special relativity by itself is an incomplete theory. Einstein knew
that special relativity was incomplete and that’s why he tried to generalize
his theory. Our dynamical model has contributed to filling some of the gaps in
the path from the study of accelerated frames in special relativity to general
relativity.

## Appendix A: Derivation of formulas (2.1) and (2.4) for infinitesimal proper
distance

Let’s consider the clocks A and B as the left and right ends of a rigid rod of
proper length $\delta\ell$, where $\delta\ell$ is infinitesimal. From the
equations (4.35) and (4.36) it can be shown that the velocity difference at
points A and B is infinitesimal:

$\displaystyle v_{A}-v_{B}=\delta
v=\frac{g^{2}\;\delta\ell\;t}{c^{2}\left(1+\frac{g^{2}t^{2}}{c^{2}}\right)^{3/2}}.$
(0.41)

Since the difference of $\gamma$ factors at A and B points is also
infinitesimal, using $\gamma_{A}$ or $\gamma_{B}$ in the calculation of length
contraction makes no difference. Using equations (4.35) and (4.36) we get

$\displaystyle\delta
x=x_{B}-x_{A}=\frac{\delta\ell}{\sqrt{1+\frac{g^{2}t^{2}}{c^{2}}}}=\frac{\delta\ell}{\gamma_{A}}$
(0.42)

where

$\displaystyle\gamma_{A}=\frac{1}{\sqrt{1-\frac{v_{A}^{2}}{c^{2}}}}=\sqrt{1+\frac{g^{2}t^{2}}{c^{2}}}.$
(0.43)

Thus, the special relativistic length contraction formula is verified.

Similarly, let’s calculate the time shift between clocks A and B, observed by
$S_{0}$ at time $t=t_{0A}=\frac{c}{g}\beta_{0}\gamma_{0}$. According to an
observer at $S_{0}$ , the total proper time clock A ticked from $t=0$ to
$t=t_{0A}$ is given by eqn.(4.32). On the other hand, the total proper time
clock B ticked from $t=0$ to $t=t_{0A}$ is given by

$\displaystyle\tau_{B}=\int_{0}^{t_{0A}}\sqrt{1-\frac{v_{B}^{2}}{c^{2}}}dt_{B}=\left(\frac{c}{g}+\frac{\delta\ell}{c}\right)\ln\left[\frac{c^{2}\beta_{0}\gamma_{0}+\sqrt{(c^{2}+g\delta\ell)^{2}+c^{4}\beta_{0}^{2}\gamma_{0}^{2}}}{c^{2}+g\delta\ell}\right].$
(0.44)

For the infinitesimal $\delta\ell$ the above expression reduces to

$\displaystyle\tau_{B}=\left(\frac{c}{g}+\frac{\delta\ell}{c}\right)\left\\{\frac{1}{2}\ln\left[\frac{1+\beta_{0}}{1-\beta_{0}}\right]-\frac{g\delta\ell\beta_{0}}{c^{2}}\right\\}$
(0.45)

which gives

$\displaystyle\Delta\tau=\tau_{B}-\tau_{A}=\frac{\delta\ell}{2c}\ln\left[\frac{1+\beta_{0}}{1-\beta_{0}}\right]-\frac{\delta\ell\beta_{0}}{c}.$
(0.46)

Thus eqn.(2.4) is obtained for infinitesimal $\delta\ell$.

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Figure 1: The clocks A and B separated by a proper length $\ell$ are initially
at rest and synchronized in an inertial frame $S_{0}$. Then, they accelerate
rigidly and come to rest in an other inertial frame $S_{1}$ with a velocity
$v_{0}$ relative to $S_{0}$. The accelerating frame of clocks is represented
by $S^{\prime}$. Figure 2: Schematic diagram showing the superposed
trajectories of the electron. Solid lines represent electrons and wavy lines
represent photons. Interaction vertices are indicated by black dots. Each
trajectory can be expressed as a sequence of photon interaction events in
space-time.  Figure 3: Scheme of Hardy’s experimental setup in the $S_{0}$
frame of reference. The BS symbol represents beam splitters and MR mirrors.
$C^{+},D^{+},C^{-},D^{-}$ are detectors. The inertial frames $S_{+}$ and
$S_{-}$ are moving with velocities $v_{+}$ and $v_{-}$ relative to $S_{0}$
respectively.  Figure 4: Panel (a) shows inertial
$S^{\prime}_{k}\;;k=1,2,..,N$ frames immediately after sequential jump steps.
$t^{\prime}$ represents the proper time of the accelerating $S^{\prime}$
frame. Panel (b) shows the rear and front ends of the rigid rod bouncing at
$\Delta v$ speed by interaction. The figures are presented according to the
observation of the observer $S_{0}$. Figure 5: Panel (a) shows the
acceleration of rockets relative to the inertial observer $S_{0}$. $S_{a}$ and
$S_{b}$ represent local frames attached to rockets A and B. $S_{a}$ and
$S_{b}$ experience proper accelerations $g$ and
$\frac{g}{1+\frac{g\ell}{c^{2}}}$, respectively. Panel (b) shows the same
acceleration process relative to $S_{a}$ and $S_{b}$ frames. An observer at
$S_{b}$ observes that a light wave of wavelength $\lambda$ emitted from A is
redshifted by $\lambda(1+\frac{g\ell}{c^{2}})$.


# Second Order Corrections to the Magnetic Moment of Electron at Finite
Temperature

Samina S. Masood∗ and Mahnaz Q. Haseeb∗∗
∗Department of Physics, University of Houston Clear Lake, Houston TX 77058,
<EMAIL_ADDRESS>
∗∗Department of Physics, COMSATS Institute of Information Technology,
Islamabad
<EMAIL_ADDRESS>
###### Abstract

Magnetic moment of electron at finite temperature is directly related to the
modified electron mass in the background heat bath. Magnetic moment of
electron gets modified at finite temperature also, when it couples with the
magnetic field, through its temperature dependent physical mass. We show that
the second order corrections to the magnetic moment of electron is a
complicated function of temperature. We calculate the self-mass induced
thermal contributions to the magnetic moment of electron, up to the two loop
level, for temperatures valid around the era of primordial nucleosynthesis. A
comparison of thermal behavior of the magnetic moment is also quantitatively
studied in detail, around the temperatures below and above the nucleosynthesis
temperature.

##  1 Introduction

Quantum Electrodynamics (QED) is well known as the simplest representative and
the most accurate gauge theory. Thermal medium effects are incorporated in QED
by taking into account the vacuum fluctuations of propagating particles along
with the hot particles in the background heat bath. For simplicity, all the
particles in the background are assumed to be in thermal equilibrium with the
heat bath. These particles are virtually created and annihilated continuously
due to the effects of heat bath at finite temperature. The interactions with
the background electrons, positrons and photons are included through
statistical distribution functions of fermions and bosons, known as Fermi-
Dirac distribution and Bose-Einstein distribution. These distribution
functions represent the possibility of exchange of virtual particles with the
real hot particles from the heat bath. Electromagnetic interactions of
particles get modified at finite temperature because of the many-body aspects
of the statistical background possessed by the hot medium. This replaces the
notion of one-particle systems adopted for particle interactions in the vacuum
with many particle aspects.

The techniques for handling the particle interactions in the background medium
have extensively evolved over the last few decades and are a part of standard
literature (see for example [1], [2] and references therein). It has been
explicitly demonstrated [3-10] that in the hot background, the density-of-
states factors have to be modified to include the real emission and absorption
of particles which are in thermal equilibrium with the heat bath. The validity
of QED renormalization in a background of particles is refined by comparing
the second order (in $\alpha$) corrections with the first order corrections
from the heat bath. It is explicitly seen, as expected, that one-loop
radiative corrections are significantly larger than the two-loop corrections.
Our scheme of calculations is based on the real part of the propagator and the
results are valid, at least below the decoupling temperature [11], i.e.,
around 2 MeV.

Renormalization techniques of vacuum theory are extended to include finite
temperature effects in a standard manner. Regular renormalization procedure
for QED in vacuum is used at finite temperature to study the background
effects on electron mass, charge and wave function renormalization constants.
The modifications in the electromagnetic properties are estimated in terms of
the renormalized values of QED parameters up to the two loop level [3-19].

Feynman rules at finite temperature remain the same as those in vacuum except
that the particle propagators are appropriately modified. We work in Minkowski
space where the Green’s functions depend on real Minkowski momenta $p^{\mu}$.
Therefore, the dynamical processes such as particles propagating in the heat
bath may be more conveniently dealt with. Moreover, in the real-time
formulation, the thermal corrections can be easily kept separate from the
vacuum corrections. Therefore, order by order cancellation of temperature
dependent singularities, and the convergence of perturbative expansion can be
straight away tracked down.

In this paper, thermal contributions to the anomalous magnetic moment of
electron are specifically studied. The magnetic moment of electron modifies
through the radiative corrections to the electron mass both at the one loop
and two loop levels. We analyze net effect from the first order and the second
order radiative contributions, including both the irreducible and disconnected
graphs up to the two loop level. Second order selfmass corrections due to
finite temperature (of the order $\alpha^{2}$) are used here to estimate
finite temperature effects on the magnetic moment of electron.

##  2 Self-Mass of an Electron at Finite Temperature

The renormalized mass of electron is represented by a physical mass given as

$m_{phys}=m+\delta m,$ (1)

where $m$ is the electron mass at zero temperature. Radiatively corrected
physical mass up to order $\alpha^{2}$ is

$m_{phys}\cong m+\delta m^{(1)}+\delta m^{(2)},$

with $\delta m^{(1)}$ and $\delta m^{(2)}$ as the shifts in the electron mass
at one and two loop level respectively. The physical mass is deduced by
locating pole of the propagator $\frac{i(\leavevmode\hbox{\hbox
to0.0pt{\thinspace/\hss}{$p$}}+m)}{p^{2}-m^{2}+i\varepsilon}$. For this
purpose, all finite terms in electron self-energy are combined together. The
physical mass of the electron at one loop was obtained by writing

$\Sigma(p)=A(p)E\gamma_{0}-B(p)\vec{p}.\vec{\gamma}-C(p),$ (2)

where $A(p)$, $B(p)$, and $C(p)$ are the relevant coefficients. Taking the
inverse of the propagator with momentum and mass term separated as

$S^{-1}(p)=(1-A)E\gamma_{0}-(1-B)\vec{p}.\vec{\gamma}-(m-C).$ (3)

The temperature-dependent radiative corrections to the electron mass up to the
first order in $\alpha$, are obtained from the temperature modified
propagator. These corrections are rewritten in the form of boson and fermion
loop integrals at the one loop level as

$\displaystyle E^{2}-|\mathbf{p}|^{2}$ $\displaystyle=$ $\displaystyle
m^{2}+\frac{\alpha}{2\pi^{2}}\left(I.p+J_{B}.p+m^{2}J_{A}\right)$ (4)
$\displaystyle\equiv$ $\displaystyle m_{phys}^{2},$

where

$I.p=\frac{4\pi^{3}T^{2}}{3},$ (5)

and

$J_{B}.p=8\pi\left[\frac{m}{\beta}a(m\beta)-\frac{m^{2}}{2}b(m\beta)-\frac{1}{\beta^{2}}c(m\beta)\right].$
(6)

Thus up to the first order in $\alpha,$ thermal corrections to the mass of
electron were obtained in ref. [3] to be

$m_{phys}^{2}=m^{2}\left[1-\frac{6\alpha}{\pi}b(m\beta)\right]+\frac{4\alpha}{\pi}mT\text{
}a(m\beta)+\frac{2}{3}\alpha\pi
T^{2}\left[1-\frac{6}{\pi^{2}}c(m\beta)\right].$ (7)

The first order correction at finite temperature is calculated as

$\displaystyle\frac{\delta m}{m}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2m^{2}}\left(m_{phys}^{2}-m^{2}\right)$ (8)
$\displaystyle\simeq$ $\displaystyle\frac{\alpha\pi
T^{2}}{3m^{2}}\left[1-\frac{6}{\pi^{2}}c(m\beta)\right]+\frac{2\alpha}{\pi}\frac{T}{m}a(m\beta)-\frac{3\alpha}{\pi}b(m\beta),$

with $\frac{\delta m}{m}$ as the relative shift in electron mass due to finite
temperature which was originally determined in ref. [3] with

$a(m\beta)=\ln(1+e^{-m\beta}),$ (9)
$b(m\beta)=\mathop{\displaystyle\sum}\limits_{n=1}^{\infty}(-1)^{n}\mathop{\mathrm{E}i}(-nm\beta),$
(10)
$c(m\beta)=\mathop{\displaystyle\sum}\limits_{n=1}^{\infty}(-1)^{n}\frac{e^{-nm\beta}}{n^{2}},$
(11)

At low temperature, the functions $a(m\beta)$, $b(m\beta)$, and $c(m\beta)$
fall off in powers of $e^{-m\beta}$ in comparison with
$\left(\frac{T}{m}\right)^{2}$ and can be neglected so that

$\frac{\delta m}{m}\overset{T\ll m}{\longrightarrow}\frac{\alpha\pi
T^{2}}{3m^{2}}.$ (12)

Moreover, in the high-temperature limit, $a(m\beta)$ and $b(m\beta)$ are
vanishingly small whereas $c(m\beta)\longrightarrow-\pi^{2}/12$, yielding

$\frac{\delta m}{m}\overset{T>m}{\longrightarrow}\frac{\alpha\pi
T^{2}}{2m^{2}}.$ (13)

Eq. (8) is valid for large temperatures relevant in QED including $T\sim$ $m.$
This range of temperature is particularly interesting from the point of view
of primordial nucleosynthesis. It has been found that some parameters in the
early universe such as the energy density and the helium abundance parameter
$Y$ become slowly varying functions of temperature [20-22] whereas they remain
constant in both extreme limits given by $T\ll m$ and $T\gg m$.

Using the same procedure as the one used for one loop calculations, the
relative shift in electron mass at the two loop level was obtained in ref.
[17]. This relative shift in electron mass introduces temperature dependence
in the magnetic moment of electron up to two loops. However, the two-loop
order result is very complicated and cannot be easily simplified. Therefore,
we will use the complete expression for the two loop calculations of electron
selfmass, near the nucleosynthesis temperature, given in refs. [12-14]. In the
following section, we compute the magnetic moment of electron from the self-
mass of electron, up to the two loop level, in thermal background.

##  3 Magnetic Moment of Electron in the Heat Bath

The anomalous magnetic moment of an electron is generated due to the coupling
of electron with the magnetic field through the radiative corrections. Some of
these results that are used here were given in ref. [17]. The electromagnetic
coupling is affected by the electron mass and the radiative corrections to the
electron mass. The coupling of electron mass with the external magnetic field
is regulated through mass of the particle itself. It is known from the
calculation of the radiative corrections that the self-mass corrections to the
electron are contributed by the distribution of hot bosons and fermions in the
background medium. This effect, in turn, changes the electromagnetic
properties of the medium itself. Therefore the magnetic moment gets changed
with the finite temperature effects. The magnetic moment of electron is
related to the relative shift in electron mass at finite temperature
$\frac{\delta m}{m}$ as:

$\mu_{a}=\frac{\alpha}{2\pi}-\frac{2}{3}\frac{\delta m}{m}.$ (14)

The leading order contributions to the magnetic moment up to the one loop
level is

$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{2}{3}\alpha\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-b(m\beta)\right]$
(15)

which can be shown to be:

$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{2}{9}\frac{\alpha\pi T^{2}}{m^{2}}$ (16)

for $T<m$ while it becomes:

$\mu_{a}=\frac{\alpha}{2\pi}-\ \frac{1}{3}\frac{\alpha\pi T^{2}}{m^{2}}$ (17)

for $T>m.$ First order in $\alpha$ contribution to $\mu_{a}$ around the
temperature range relevant for primordial nucleosynthesis (i.e., $T\sim m$),
soon after the big bang is given by expression in eq. (15). The two loop
contribution to the magnetic moment can simply be added to the magnetic moment
in terms of the relative shift in electron mass $\frac{\delta m^{(2)}}{m}$ as

$\mu_{a}=\frac{\alpha}{2\pi}-\frac{2}{3}\left(\frac{\delta
m^{(1)}}{m}+\frac{\delta m^{(2)}}{m}\right).$ (18)

Now using the expression for $\frac{\delta m}{m}$ at finite temperature in
ref. [14], we get thermal contributions to the magnetic moment up to the two-
loop level as

$\displaystyle\mu_{a}$ $\displaystyle=$ $\displaystyle\frac{\alpha}{2\pi}-\
\frac{2}{3}\alpha\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
(19) $\displaystyle-\frac{2\alpha^{2}}{3}\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
$\displaystyle\times\left[\frac{\pi
T^{2}}{3m^{2}}\left\\{1-\frac{6}{\pi}c(m\beta)\right\\}+\frac{2}{\pi}\frac{T}{m}a(m\beta)-\frac{3}{\pi}b(m\beta)\right]$
$\displaystyle-\frac{4}{3}\alpha^{2}\mathop{\displaystyle\sum}\limits_{r=1}^{\infty}[-\frac{m^{2}\beta^{2}}{\pi^{2}}c(m\beta)+T^{2}\\{\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\text{
}(-1)^{n+r+1}\frac{\pi\text{ }}{6mEv}\frac{e^{-\beta(rE+mn)}}{n}$
$\displaystyle-\frac{3}{8}(-1)^{r}\frac{e^{-r\beta
E}}{E^{2}v^{2}}[\frac{9E^{2}}{2m^{2}}+6\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}+4\mathop{\displaystyle\sum}\limits_{n,s=3}^{r+1}\frac{1}{ns}+(-1)^{s-r}\\{\frac{9E}{m}\left(3+4\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}\right)$
$\displaystyle+2\left(\frac{E^{2}v^{2}}{m^{2}}-3\right)\left(9+18\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}\frac{1}{s}+8\mathop{\displaystyle\sum}\limits_{n,s=3}^{r+1}\frac{1}{ns}\right)\\}]+\frac{4}{E^{2}v^{2}}\\}$
$\displaystyle-\frac{T}{m}\\{\frac{\pi m\text{
}}{6Ev}\mathop{\displaystyle\sum}\limits_{s=2}^{r+1}\mathop{\displaystyle\sum}\limits_{n=1}^{s+1}\frac{e^{-\beta(rE+mn)}}{n}\left[1\
-\left\\{(-1)^{r+n}-(-1)^{s+n}\right\\}\right]$
$\displaystyle+[\left\\{\mathop{\mathrm{E}i}(-m\beta)-\mathop{\mathrm{E}i}(-2m\beta)\right\\}\\{\frac{9E}{4}\left(\frac{E}{E^{2}v^{2}}-\frac{1}{m}\right)$
$\displaystyle+\left(\frac{5E}{m}-21+\frac{E^{2}}{2m^{2}}\right)\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\\}$
$\displaystyle+\left\\{\frac{9}{4v^{2}}-\mathop{\displaystyle\sum}\limits_{n=1}^{s+1}\left[\mathop{\displaystyle\sum}\limits_{s=3}^{r+1}1-E^{2}\left(\frac{1}{2m^{2}}+\frac{3}{E^{2}v^{2}}\right)+\frac{3E}{m}\right]\right\\}(-1)^{s}\mathop{\mathrm{E}i}\text{
}(-sm\beta)]$
$\displaystyle+\frac{e^{-rm\beta}}{m}\\{\left[\frac{9E}{2v^{2}}+2\left(\frac{3E}{v^{2}}+\frac{3E^{2}v^{2}}{m}-5E\right)\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\right]\mathop{\displaystyle\sum}\limits_{s=1}^{\infty}\sinh
sm\beta$
$\displaystyle-\frac{3m^{3}}{E^{2}v^{2}}\left(\frac{3}{4}-\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\right)\mathop{\displaystyle\sum}\limits_{s=1}^{\infty}\cosh
sm\beta\\}]\\}+\frac{1}{m^{2}}\\{\frac{9m}{4E^{2}v^{2}}\left(E^{3}+\frac{m^{3}}{2}\right)$
$\displaystyle+\left[\frac{3m}{E^{2}v^{2}}(E^{3}+m^{3})+5mE-3E^{2}v^{2}\right]\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\frac{1}{n}\\}\left\\{\mathop{\mathrm{E}i}(-m\beta)-2\mathop{\mathrm{E}i}(-2m\beta)\right\\}$
$\displaystyle-\frac{1}{m^{2}}\mathop{\displaystyle\sum}\limits_{n=3}^{r+1}\\{\mathop{\displaystyle\sum}\limits_{s=1}^{r+1}\frac{(-1)^{s}}{n}[\frac{m^{2}re^{-sm\beta}}{2}$
$\displaystyle+\left\\{sE\left(2m-\frac{E^{2}}{m}\right)+\frac{m^{2}(s-r)}{2}\right\\}\mathop{\mathrm{E}i}(-sm\beta)]$
$\displaystyle-\frac{\pi m^{3}\text{ }}{3Ev}\left[e^{-\beta rE}\text{
}(-1)^{n+r}(n+1)\
-\mathop{\displaystyle\sum}\limits_{s=2}^{r+1}(-1)^{n+s}\right]\mathop{\mathrm{E}i}(-nm\beta)\\}].$

It can be clearly seen from eq. (19) that the second order corrections are
suppressed by at least two orders of magnitudes as compared to the one loop
contributions. Dependence of the self-mass induced thermal contributions to
the anomalous magnetic moment of electron is very complicated at the two loop
level, as indicated by eq. (19). The exact estimate of this magnetic moment
for application to the primordial nucleosynthesis is very involved and
probably is not really so significant at two loop level. However, low
temperature $(T<m)$ and high temperature $(T>m)$ values of the magnetic moment
can be quantitatively analyzed to prove the validity of the renormalization
scheme. We give the plotting for magnetic moment $\mu_{a}$ vs $\frac{T}{m}$ in
low temperature and the high temperature regions in the next section. This
analysis indicates that the magnetic moment of electron changes its behavior
around nucleosynthesis.

##  4 Results and Discussion

The electron mass acquires a significant contribution from the heat bath even
for temperatures that are smaller than the electron mass. However, this
dependence becomes very complicated as soon as the background temperature
approaches the value of electron mass. One loop corrections to the electron
mass at finite temperature are presented in eq. (8). The low ($T<m$) and high
($T>m$) temperature values of self-mass of electron at the one loop level are
given in eqs. (12) and (13), respectively as limiting cases of eq. (8).
Incorporating the second order relation for the physical mass of electron [17]
into eq. (14) leads to eq. (19) which gives the general form of the anomalous
magnetic moment at finite temperature up to order $\alpha^{2}$. When an
electron couples with the magnetic field at finite temperature, a nonzero
contribution to the magnetic moment is picked up due to the coupling of
electron mass with the thermal background. Eq. (14) presents the acceptable
relation of the magnetic moment with that of the self-mass of electron.

Figure 1: Low temperature behavior of the magnetic moment of electron at the
two loop level.

Figure 2: High temperature behavior of the magnetic moment of electron at the
two loop level.

The behavior of the magnetic moment of electron near the nucleosynthesis
temperatures (eq.(19)) is very complicated and it can be fitted through a
single mathematical function under some special conditions only. However, we
can extract the quantitative behavior of magnetic moment of electron for low
and high temperatures in a comparatively simple form. Using previously studied
second order contributions to the electron mass at low temperature $(T<m)$,
leading order contributions to the magnetic moment of electron can be computed
as

$\mu_{a}\overset{T<m}{\longrightarrow}-\ \frac{2}{9}\frac{\alpha\pi
T^{2}}{m^{2}}-10\alpha^{2}\left(\frac{T^{2}}{m^{2}}\right),$ (20)

whereas, the leading order contributions at high temperature $(T>m)$ comes out
to be

$\mu_{a}\overset{T>m}{\longrightarrow}-\ \frac{1}{3}\frac{\alpha\pi
T^{2}}{m^{2}}-\frac{\alpha^{2}\pi^{2}}{6}\left(\frac{T^{2}}{m^{2}}\right)^{2}+\frac{\alpha^{2}}{6}\frac{m^{2}}{T^{2}}.$
(21)

Eqs. (20) and (21) are used for a quantitative study of magnetic moment of
electron at low temperature and high temperature, respectively. We plot the
temperature dependence of magnetic moment of electron versus $\frac{T}{m}.$ A
plot of eq. (20) is given in fig. 1, whereas, eq. (21) is plotted in fig. 2\.
Both of these graphs give a sort of quadratic behavior in the negative sense.

Eqs. (21) and (22), fig. 1 and fig. 2. indicate the difference between low
temperature and high temperature behavior. Major difference in the behavior
occurs due to the $m^{2}/T^{2}$ term at high temperature. Contribution of this
term reduces with increasing temperatures and becomes totally ignorable at
very high temperatures. This is obvious from fig. 1 and fig. 2 that the
magnetic moment of electron falls off rapidly with temperature after
nucelosynthesis as compared to that before nucleosynthesis. This rapid
decrease in magnetic moment after the nucleosynthesis is not the same as it is
compared in eqs. (16) and (17), implying that the one loop and two loop
behaviors are not exactly similar.

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